A Spectroscopy Toolkit:
Simulation and Inversion Methods
in Diatomic Molecule Spectroscopy
Robert J. Le Roy
Guelph-Waterloo Center for Graduate Work in Chemistry and Biochemistry,University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
c© Robert J. Le Roy
July 24, 2013
Chapter 1
Introduction
The patterns of energy levels and transition intensities observed in molecular spectra are key tools for
understanding molecular structure and properties, for fingerprinting and identifying unknown species, and
for monitoring the concentrations of particular chemical components or molecular states during chemical
reactions. These patterns are determined by two distinctly different types of molecular properties: the
nuclear exchange symmetry determined by the molecular structure and the character of the electronic
wavefunction, and the potential energy function characterizing the forces between the component atoms.
The symmetry properties determine the basic patterns of the frequencies and intensities associated with
rotational transitions in molecular spectra, and their analysis can be used to determine the equilibrium
structure of even quite complex molecules. The potential energy functions, on the other hand, govern the
number and spacings of the vibrational level energies, the patterns of vibrational transition intensities,
and the shapes of continuum spectra. As a result, analyses of experimental measurements of the latter
properties can be used to determine bond dissociation energies, to determine how the effective or average
structure changes with the degree of vibrational excitation, and to determine the potential energy function
itself.
Most spectroscopy textbooks offer quite thorough treatments of the spectroscopic manifestations of
the equilibrium structure and symmetry properties of molecules. However, the analogous treatments of
the relationships between the intermolecular potential and the distribution of vibrational levels, patterns
of vibrational band intensities, or the nature of bound→ continuum spectra, are often relatively cursory.
Moreover, analyses of rotational effects in spectra usually presume that the molecule has a well-defined fairly
rigid structure, something which is true only for molecules in very low vibrational states (and sometimes not
even then!), since vibrational motion can distort a molecule to shapes far from its equilibrium or average
configuration. As a result, the significance of information obtained from rotational effects in molecular
spectra is often also strongly contingent on a sound understanding of how those rotational properties are
affected by vibrational motion. Thus, analysis of rotational spectra also implicitly presume a knowledge
of the nature of the potential energy function and how it governs the dynamics of the molecule.
The classic 1950 book of Herzberg [1] covered all aspects of this subject in a comprehensive manner,
appropriate to its day, and it deservedly remains one of the best-selling books in of all science. However,
recent decades have brought new understanding of certain types of phenomena, such as the characteristic
near-dissociation behaviour of vibrational energy level spacings, rotational constants and other properties.
Moreover, since 1950 the development of computers and computational tools has led to a revolution in our
ability to readily simulate and analyse spectroscopic data, and to invert them to yield detailed information
regarding intermolecular potentials. However, most of these developments are only minimally reflected in
existing texts and monographs.
2
A primary objective of the present notes is to convey a rigorous and thorough understanding of the
relationship between intermolecular potential energy functions and the discrete and continuum spectra of
diatomic molecules. This is done using both quantum mechanical and semiclassical formalisms. While the
former is of course most accurate and rigorous, the latter often offers much clearer insight into how the
potential energy function affects molecular properties, as well as being the basis for virtually all quantitative
“inversion” procedures. Our attention is focussed on diatomic molecules because it is easiest to describe
them with high accuracy and rigor. At the same time, virtually all of the types of phenomena discussed
here also arise in normal polyatomic and Van der Waals molecule spectra, and since most of us do not
intuitively think in more than two dimensions, our explanations for them often come mainly from our
intuitive understanding of diatomic systems. Moreover, many of the quantitative methods and models
developed for describing various diatomic molecule phenomena are also useful for the more complicated
systems.
A complementary objective is the presentation and illustrative application of theoretical methods and
computer programs, both for doing “forward” calculations to simulate discrete or continuum spectra, and
for inverting experimental data to determine the underlying potential energy curves. While such methods
are widely applied in the literature, there is at present no thorough description of them in existing texts or
monographs, so another purpose of the present work is to serve as a sourcebook regarding such theoretical
and computational tools.
The spectroscopic manifestations of a number of the types of phenomena to be considered are schemat-
ically illustrated, together with the associated supporting potential curves, in Fig. 1.1. These phenomena
include (RJL ... discussion to be expanded):
1. bound state level patterns and how they are governed by the nature of the potential curves
2. bound–bound transition intensities and “Franck-Condon factors”
3. tunneling predissociation of quasibound levels
4. bound→ continuum transition intensities
5. curve-crossing and “Feshbach” predissociation
The practical utility of the material covered herein is underlined by the fact that the analysis of
spectroscopic data is the best single source of information regarding both intra- and intermolecular forces.
The wide range of application of spectroscopic methods also means that a knowledge of the methods and
tools discussed described herein will be useful to workers in fields ranging from astrophysics, to plasma
diagnostics, to atmospheric chemistry. If one knows the inter/intramolecular forces, one can in principle
predict all of chemistry. For reaction dynamics and condensed phase phenomena, this is a rather big “in
principle”, since the computations required for fully predicting such phenomena are unmanageably massive,
but this principle continues to motivate work in this field.
The background assumed here is a year of calculus, some familiarity with differential equations, and a
basic one-semester course in quantum mechanics. While additional background in molecular spectroscopy
would also be useful, it is not essential. Since the course also introduces a variety of practical computational
tools, modest familiarity with some computer which supports a FORTRAN compiler would also be helpful.
However, the programs described may be treated as “black boxes”, and all the user really need know is
how to use a basic editor (e.g., a UNIX system editor such as pico or vi, or a PC word processor “editor”
such as WORD or WordPerfect) for preparing the necessary input data files, and how to cause a compiled
(FORTRAN) program to run with a given input data file on the computer system of choice.
3
Unit and Notation Conventions
An effort is made throughout these notes to adhere to a consistent notation and set of units. In particular,
unless citation of literature results requires otherwise, energies are assumed to be in the spectroscopists’
usual unit of wavenumbers, E [cm−1] = E [J]/hc , where h is Planck’s constant and c the speed of light,
and distance is assumed to be in Angstroms, where 1 A= 10−10 m. However, we prefer not to clutter our
expressions by including explicit factors converting to SI units. Fortunately, the physical constants usually
appear in the form of the ubiquitous term �2/2m, where m is a mass, and this factor is written in this form
whenever it appears. Using current (1999) values of the physical constants [2], this term may be written
in our “spectroscopists’s units” as (�2/2m) [cm−1 A2] = 16.85762909(±0.0077 ppm)/m [u] ; the relevant
numerical factor is incorporated in the various computer programs described below, all of which use cm−1
and A as the default energy and length units for input and output quantities.
The other widely used units convention in molecular physics is the quantum chemist’s “atomic units”, or
au, and conversions to our spectroscopic units are often necessary. For the convenience of the user we record
here the fact that [2] the atomic unit of energy is “hartee’s” Eh, with 1Eh = 2R∞ = 219 474.631 370 98 (±7.6×10−6 ppm) [cm−1] , where R∞ is Rydberg’s constant, and the atomic unit of length is the “bohr” (usually
denoted a0), where 1 [a0] = 0.529 177 2083 (±0.0037 ppm) [A]. This means that if a theoretical value of
the coefficient Cm of an inverse-power potential energy term Cm/Rm (see Chapters 2 and 4) is reported
in atomic units or “au”, it may be converted to our spectroscopists’ units by multiplying by the fac-
tor 219 474.631 370 98 × (0.529 177 2083)m . Similarly, when the electron charge appears, the appropriate
conversion factor is e2/4πε0 = Eh a0 = 116 140.9727 (±0.0037 ppm) [cm−1 A]In addition, throughout the following, vector quantities are identified through the use of bold fonts (as
in rb), and unit vectors through addition of a “bar” over the vector name (as in rb = rb/|rb|). Similarly,
symbols representing operators will usually be identified by a “hat” (as in Htot); note too that operators
may be either vector or scalar quantities (e.g., J and J2). As usual, ∇q =(
∂∂xq
, ∂∂yq
, ∂∂zq
)is the vector
gradient operator acting on the three cartesian coordinates of a vector q , whose magnitude is q = |q| , and
∇2q =
∂2
∂xq2+
∂2
∂yq2+
∂2
∂zq2(1.1)
is the usual “del squared” second partial derivative operator acting those coordinates. To avoid clutter,
however, the “hat” notation is not shown for ∇q , ∇2q or any explicit differential operators, and it is
usually also omitted from multiplicative scalar operators (such as the potential energy).
Bibliography
G. Herzberg, Spectra of Diatomic Molecules” (Van Nostrand, Toronto, 1950); QC451.H46 (UW).
D.M. Hirst, Potential Energy Surfaces (Taylor and Francis, 1985); QD461.5.H57 (UW).
R.J. Le Roy, “Energy Levels of a Diatomic near Dissociation”, Ch3 (pp. 113-176) of Molecular Spec-
troscopy, Volume 1 edited by R.F. Barrow, D.A. Long and D.J. Millen (a Specialist Periodical Report
of the Chemical Society of London, 1973). QC454.M6M65X (UW).
H. Lefebvre-Brion and R.W. Field, Perturbations in the Spectra of Diatomic Molecules (Academic Press,
1986); QD96.M65L44 (UW).
G.C. Maitland, M. Rigby, E.B. Smith and W. Wakeham, Intermolecular Forces. Their Origin and De-
termination (Oxford U. Press, 1981); QD461.I47X (UW).
4
J.N. Murrell, S. Carter, S.C. Farantos, P. Huxley and A.J.C. Varandas, Molecular Potential Energy
Functions (Wiley, 1984); QD461.5.M65 (UW).
M. Rigby, E.B. Smith, W.A. Wakeham and G.C. Maitland, The Forces Between Molecules (Oxford Uni-
versity Press, 1986); QD461.F73 (UW).
J. Tellinghuisen, The Franck-Condon Principle in Bound-Free Transitions, Chapter 7 of Photodissociation
and Photoionization, Advances in Chemical Physics 60 (1985), K.P. Lawley editor. QD453.A27
(UW).
J. O. Hirschfelder (editor), Intermolecular Forces, or Advances in Chemical Physics 12 (1967), [esp.
Chapter 1 on “The Nature of Intermolecular Forces”, by J. O. Hirschfelder and W. J. Meath, and
Chapter 7 on “Methods for the Determination of Intermolecular Forces”, by E. A. Mason and L.
Monchick]; QD453.A27 (UW).
References
[1] G. Herzberg, Spectra of Diatomic Molecules (Van Nostrand, Toronto, 1950).
[2] P. J. Mohr and B. N. Taylor, Rev. Mod. Phys. 72, 351 (2000), cODATA Recommended Values of the
Fundamental Physical Constants: 1998.
5
Chapter 2
Origin of and Models for Interaction
Potentials
2.1 The Born-Oppenheimer Separation
Introductory quantum mechanics and spectroscopy courses tend to take the existence of potential energy
functions for granted, and the Schrodinger equation governing nuclear motion is often written down as if it
were known a priori. This is not the case, and in order to provide a proper context for our later discussions,
we must begin by carefully considering the question: “What is an intermolecular potential?”. This involves
a detailed review of the Born-Oppenheimer approximation, which was developed to allows a separable
treatment of the electronic and nuclear degrees of freedom of a molecule [1, 2]. This theoretical discussion
is also a necessary foundation for our later examination (in Chapter 7) of Born-Oppenheimer breakdown
effects, whose quantitative analysis is becoming an increasingly important topic in modern spectroscopy.
Because of their importance in our discussion of the characteristic near-dissociation behaviour of molecular
properties, this chapter also reviews the theory of long-range interatomic forces, before concluding with a
survey of commonly-used analytic models for potential energy functions.
While it yields a widely quoted result, it is often not fully realized that what is commonly known as
the Born-Oppenheimer separation of electronic and nuclear coordinates actually consists of two distinct
steps. The first is the separation of the motion of the overall center of mass from the relative motions of
the particles within the system, and the second the actual separation of the electronic and nuclear motion
degrees of freedom. We will see that this first step gives rise to many of the terms commonly associated
with “Born-Oppenheimer breakdown,” and that their precise form depends on the choice of the body-fixed
coordinates used for specifying the positions of the electrons relative to the molecular framework.
2.1.1 Separating off the Centre of Mass
We begin with the textbook quantum mechanical description of a diatomic molecule, using laboratory
frame coordinates for all of the particles. In the absence of external fields, the total Hamiltonian for
such a system consists of a kinetic energy operator for each particle plus a sum of the multiplicative scalar
operators associated with the various contributions to the total electrostatic potential energy of the system.
If the laboratory-fixed coordinates of nuclei “a” and “b” with masses ma and mb are labeled ra and rb ,
respectively, and the analogous coordinates of the N electrons of mass me are denoted ri , for i = 1, ...N ,
6
this total space-fixed Hamiltonian has the form:
Htot = −∑α=a,b
�2
2mα∇2
rα −�2
2me
N∑i=1
∇2ri + Ue (2.1)
where the total electrostatic potential
Ue = Ue (r, {rai}, {rbi}, {rij}) =1
4πε0
⎡⎣Za Zb e2
r−
N∑i=1
(Za e
2
rai+Zb e
2
rbi
)+
N−1∑i=1
N∑j=i+1
e2
rij
⎤⎦ (2.2)
is the usual sum of nucleus-nucleus repulsion, electron-nucleus attraction and electron-electron repulsion
terms depending only on the relative interparticle coordinates r ≡ |rab| = |rb− ra| , rαi ≡ |rαi | = |ri− rα|(for α = a or b ), and rij ≡ |rij | = |rj − ri| , and on the charges Za e and Zb e of the two nuclei, where
e is the electron charge.
Since Ue depends only on the relative distances between the various particles, the contributions to
the space-fixed Hamiltonian from the three coordinates for the overall centre of mass of the system can
be separated from those for the remaining 3N + 3 relative coordinates. However, there are a number of
plausible possible choices for the set of relative coordinates, with different choices giving rise to different
types of cross terms in the relative velocities or momenta of the various particles. The merits of some
possible choices have been discussed by Jepsen and Hirschfelder [3, 4], by Froman [5], and by Pack [6, 7, 8],
and will not be considered in detail here. However, it is important to note that this choice does have
a bearing on the definition of the effective or reduced mass which appears in the effective Schrodinger
equation for vibrational motion, and determines the nature of the cross terms which contribute to the
breakdown of the Born-Oppenheimer approximation.
Perhaps the simplest choice of relative coordinates is the centre of mass of the nuclei system (CMN)
illustrated in Fig. 2.1, in which the positions of all electrons are expressed relative to that of the centre of
mass of the nuclei rCMN :
rni ≡ ri − rCMN = ri − [ma ra +mb rb] /(ma +mb) (2.3)
while the coordinates for the two nuclei are replaced by their relative separation
r ≡ rab = rb − ra (2.4)
and the position of the overall centre of mass is
c ≡(ma ra +mb rb +
N∑i=1
me ri
)/M (2.5)
where M = ma+mb+Nme is the total mass of the system. The familiar chain rule of partial differentiation
then allows the partial derivatives with respect to the laboratory-fixed coordinates to be expressed in terms
of those with respect to the centre of mass and relative position coordinates:
∇a ≡ ∇ra =ma
M∇c −∇r − ma
ma +mb
N∑i=1
∇ni (2.6)
∇b ≡ ∇rb =mb
M∇c +∇r − mb
ma +mb
N∑i=1
∇ni (2.7)
7
Figure 2.1. Laboratory and relative coordinate systems for a diatomic molecule.
∇i ≡ ∇ri = ∇ni +me
M∇c (2.8)
where ∇ni ≡ ∇rni . On substituting these definitions into Eq. (2.1) and collecting related terms, the total
space-fixed Hamiltonian to be written as
Htot = − �2
2M∇2
c −�2
2μn∇2
r −�2
2me
N∑i=1
∇2ni + Ue + Hpol (2.9)
where
Hpol = HCMNpol = − �
2
2(ma +mb)
⎛⎝ N∑i=1
∇2ni + 2
N−1∑i=1
N∑j=i+1
∇ni·∇nj
⎞⎠ (2.10)
and μn = mamb/(ma+mb) is the usual reduced mass of the two nuclei. Note that the particular collection
of momentum operator cross terms appearing in Eq. (2.10) is a reflection of the fact that we have chosen
to express the electron coordinates relative to the centre of mass of the nuclei. These terms involve only
8
partial derivatives with respect to electron coordinates, and are called “mass polarization” terms. We shall
see below that use of other sets of relative coordinates gives rise to other forms for Hpol .
Since all of the dependence on c is contained in the first term on the right hand side of Eq. (2.9), the
simple separation-of-variables arguments applied in a first quantum course to the problem of a particle in a
two- or three-dimensional box show that the total wave function may be written as the separable product
Ψtot = ψtr(c)Ψint . Here, ψtr(c) = A exp(−ikc · c) is the free particle translational wavefunction solution
of the differential equation
− (�2/2M) ∇2c ψtr(c) = Etr ψtr(c) (2.11)
where Etr = (�2/2M)|kc|2 is the translational kinetic energy of the system as a whole, which is a constant
of motion.
After separating off the translational motion of the centre of mass, the rest of the system dynamics is
described by the internal motion Schrodinger equation
HintΨint ={−(�2/2μn)∇2
r + He({rni}; r) + Hpol
}Ψint = EintΨint (2.12)
which depends only on the relative coordinates r and {rni}, and we have chosen to group together the
terms which act most directly on the electron coordinates:
He({rni}; r) = HCMNe ({rni}; r) = − (�2/2me)
N∑i=1
∇2ni + Ue (2.13)
The superscript “CMN” appears here to remind us that the origin chosen for the electronic coordinates is
the centre of mass of the nuclei. The operator He({rni}; r) consists of a kinetic energy operator for each
electron, plus a potential energy operator which includes all terms depending on the electron coordinates,
so it may be thought of as a Hamiltonian for the motion of the electrons relative to the two nuclei. However,
the fact that Hpol also acts on the electron coordinates and that Ue depends on the internuclear distance r
means that this problem is not strictly separable. The complications arising from this give rise to interesting
physical phenomena. Before considering them further, however, let us briefly examine the implications of
other possible choices for the coordinates defining the relative motion of the particles in the system.
2.1.2 Other Choices for the Relative Coordinates
The general forms of Eqs. (2.9) and (2.12) remain the same for virtually all sensible choices of the coordi-
nates defining the relative motion of the electrons and nuclei. However, the associated definitions of Hpol
tend to be quite different. Consider, for example, the case in which the electron coordinates are expressed
relative to the midpoint between the two nuclei, rMN :
rmi = ri − rMN = ri − (ra + rb)/2 (2.14)
This choice would be most appropriate for nominally homonuclear molecules such as diatomic hydrogen
or bromine, where the bond mid-point is the molecular centre of charge for all isotopomers, independent
of the amount of vibrational stretching. As a result, it seems most appropriate to express the electron
coordinates relative to this fixed point on the internuclear axis.
If the molecule-fixed electron coordinates of Eq. (2.3) are replaced by those of Eq. (2.14), the resulting
versions of Eqs. (2.12) and (2.13) are unchanged except that the kinetic energy operators for the electrons
appearing in the latter, (−�2/2me)∇2mi , are expressed in the new relative coordinates. However, the
9
correction term operator Hpol now has the form:
HMNpol = − �
2
8μn
⎛⎝ N∑i=1
∇2mi + 2
N−1∑i=1
N∑j=i+1
∇mi · ∇mj
⎞⎠− �2
2
(mb −ma
mamb
)∇r ·
N∑i=1
∇mi (2.15)
While the first terms appearing here are the same type of mass polarization terms comprising HCMNpol , they
are now combined with a velocity-dependent term which takes account of the “wobbling” of the bond
mid-point relative to the nuclear centre of mass which occurs for isotopomers formed from different atomic
isotopes. It is the coupling of the electronic and nuclear motion due to this last term which gives rise to
the weak dipole moment of the HD molecule. Note, however, that this last term disappears and Eq. (2.15)
becomes identical to Eq. (2.10) if the two masses ma and mb are identical.
A number of other possible definitions of the relative electron coordinates, which give rise to other
versions of Hpol, have been used in the literature [3, 4, 6]. However, the differences between Eqs. (2.10) and
(2.15) suffice to illustrate the sorts of effects such choices can have. In any case, the following discussion will
be based on the CMN coordinates of Eqs. (2.3)-(2.5) and the internal motion Hamiltonian of Eqs. (2.10),
(2.12) and (2.13).
2.1.3 Separating off the Electronic Degrees of Freedom
Since He depends only parametrically on the (relative) nuclear coordinate r, at any specified value of r
one can in principle solve the electronic Schrodinger equation
HeΦs({rni}; r) = Ws(r)Φs({rni}; r) (2.16)
to determine the set of eigenfunctions {Φs({rni}; r)} and energies {Ws(r)} associated with the set of
electronic states {s} , for that particular internuclear separation. This is precisely analogous to solving
the electronic Schrodinger equation for an atom, where one knows that there are many different electronic
states with sometimes different and sometimes degenerate energies. The only difference for the molecule is
that this electronic Schrodinger equation will have different sets of solutions for different relative nuclear
coordinates. Note that since the electrons are considered to be moving relative to fixed nuclei, the electronic
eigenvalue depends only on the magnitude of r, and not its orientation.
From the normal properties of differential equations, we know that the set of functions {Φs({rni}; r)}must form a complete basis set over the electronic coordinates of the system at internuclear distance r.
Thus, the total eigenfunction of Hint may always be exactly expressed in terms of these infinite sets of
basis functions:
Ψint =∑t
Φt({rni}; r)Xt(r) (2.17)
Substituting Eq. (2.17) into Eq. (2.12), premultiplying by the complex conjugate of one of the electronic
wave functions (say, Φs({rni}; r) ), and integrating over the electronic coordinates, one obtains the inho-
mogeneous differential equation:
− �2
2μn∇2
r Xs(r) + [Ws(r) + ΔWs,s(r)]Xs(r)− E Xs(r) = −∑
t(s �=s)
ΔWs,t(r) Xt(r) (2.18)
where the functions ΔWs,t(r) consist of matrix elements of Hpol and of terms arising from the r–dependence
of the electronic wavefunctions:
ΔWs,t(r) = (−�2/2μn)(〈Φs|∇2
rΦt〉e + 2〈Φs|∇rΦt〉e · ∇r
)+ 〈Φs|Hpol|Φt〉e (2.19)
10
where the inner products 〈...〉e involves integration over all of the electronic coordinates.
There is one equation of the form of Eq. (2.18) for each electronic state of the system “ s ”, and solution
of the resulting set of coupled differential equations yields the exact eigenfunctions and eigenvalues of
Hint . Moreover, there exist a number of practical computational methods for solving such sets of coupled
equations. However, the “Born-Oppenheimer breakdown” correction terms contributing to ΔWs,t(r) are
difficult to determine accurately, either from the electronic wavefunctions or from empirical analysis of
experimental data. Fortunately, such terms are usually relatively small, and can often either be neglected
or be estimated by approximate methods.
What is commonly called the “Born-Oppenheimer approximation” is actually a set of approximations
which involve neglecting some or all of these correction terms, in order to obtain the effective radial
Schrodinger equation used in most practical applications. In particular, in what is called the “adiabatic
approximation”,1 the wave function expansion of Eq. (2.17) is approximated by a single term
Ψint ≈ Φs({rni}; r) Xs(r) (2.20)
In this case the coupled equations of Eq. (2.18) collapse to a set of uncoupled homogeneous differential
equations, one for each electronic state of the system:
− �2
2μn∇2
r Xs(r) + [Ws(r) + ΔWs,s(r)]Xs(r) = E Xs(r) (2.21)
Moreover, if the electronic wavefunction Φs is taken to be purely real valued, then 〈Φs|∇rΦs〉 = 0 and
ΔWs,s(r) is simply a scalar function [9, 10].
The left hand side of Eq. (2.21) has the form of a conventional quantum mechanical Hamiltonian opera-
tor acting on a function. It is the sum of a kinetic energy differential operator plus a scalar (multiplicative)
operator which we may associate with the effective potential energy of the system:
V (r) = Vs(r) ≡ Ws(r) + ΔWs,s(r) (2.22)
As a result, for a given electronic state, the locus of eigenvaluesWs(r) of the electronic Schrodinger equation
(2.16), plus the associated diagonal or “adiabatic” correction ΔWs,s(r), defines the effective potential energy
function governing the motion of the nuclei. In principle one may therefore determine the effective potential
function governing the radial or vibrational motion associated with a given electronic state by “simply”
solving the electronic Schrodinger equation of Eq. (2.16) on an appropriate mesh of r values, and using the
resulting wavefunctions Φs({rni}; r) to calculate the matrix elements contributing to ΔWs,s(r).
It is important to realize that the only approximation introduced to date is the assumption that
the wave function expansion of Eq. (2.17) may be replaced by the simple product function of Eq. (2.20).
Within this “adiabatic approximation”, the electronic energies Ws(r) and the electronic matrix elements
〈...〉e contributing to ΔWs,s(r) will be exactly the same for all isotopomers of a given species. However, the
nuclear mass scaling seen in Eqs. (2.10) and (2.15) means that the “diagonal correction terms” ΔWs,s(r)
will be scaled differently for different isotopomers. As a result, the effective electronic potential energy
functions for different isotopomers of a given molecular species will differ slightly from one another.
Making the further approximation of neglecting the diagonal correction term ΔWs,s(r) yields the
“clamped nuclei” approximation. In this case the potential energy function for a given electronic state
is simply the electronic energy Ws(r), which is exactly the same for all isotopomers of a given chemical
1 It is difficult to make a plausible connection between this definition and the use of the adjective “adiabatic”
in classical thermodynamics. However, the terminology is very deeply entrenched in both fields, so the user must
simply accept both definitions and allow the context of a discussion indicate which is relevant.
11
Figure 2.2. Clamped nuclei potentials for the lowest 18 electronic states of Li2 (from Ref. [11]).
species. Fig. 2.2 shows potential energy curves for the lowest 18 electronic states of Li2 calculated within this
approximation. Note that in common parlance the term “Born-Oppenheimer approximation” or “Born-
Oppenheimer potential” is sometimes applied to both of these results, so it is not always clear precisely
what potential is being used. Such ambiguities are avoided here through use of the terms “adiabatic” and
“clamped nuclei”, as appropriate.
For the simple case of a 1Σ electronic state, where the electrons have no net orbital or spin angular
momentum, both the electronic eigenvalues Ws(r) and the diagonal and off-diagonal matrix elements
ΔWs,t(r) depend only on the scalar distance r. Additional terms arising for other types of electronic states
give rise to effects such as lambda doubling and spin-orbit splitting, which are discussed in Chapter 7 [12].
However, limiting our discussion to the simpler case of 1Σ states facilitates a less cluttered examination of
the fundamental relationships between potential energy functions and molecular spectra, and this is the
view taken in the next few chapters. Note that since within either of these approximations the differential
equations for the different electronic states are completely independent of one another. Hence we usually
omit the subscript label s from the symbol for the effective potential energy function V (r), and simply
rely on the context to identify which electronic state is being considered.
2.1.4 Choice of Reduced Mass and The Effective Adiabatic Potential
Up to this point in our discussion, the reduced mass which has been used is the reduced mass of the two
nuclei, and when solving the radial Schrodinger equation using ab initio calculated potentials and adiabatic
potential correction functions, this is certainly appropriate [9, 13, 14]. However, at large internuclear
distances it seems intuitively physically more appropriate to consider the relative motion of whole atoms.
12
Moreover, while accurate masses are known for the nuclei of small atoms, those for heavy atoms are
generally not so well determined. As a result, almost all treatments of the dynamics of vibrating-rotating
molecules or of molecular collision phenomena, as well as empirical treatments relating data for different
isotopomers of a given species or applying formal inversion procedures, use the reduced mass of the two
atoms, μat =MaMb/(Ma +Mb) , where Ma and Mb are the total masses of the individual atoms.
Fortunately, this conventional choice of reduced mass also has been shown to have a sound theoretical
basis. A somewhat more complicated alternate formulation of the Born-Oppenheimer separation yields
an effective radial Schodinger equation with the same form as Eq. (2.21), but with μn replaced by μat(see Chapter 7) [15, 16, 17]. This approach gives rise to a slightly different theoretical definition of the
terms contributing to the effective adiabatic potential correction ΔWs,s(r), and it also introduces atomic-
mass-dependent “non-adiabatic” correction contributions to the kinetic energy term and the centrifugal
potential. However, with minor additional assumptions, the resulting relative nuclear motion differential
equation has exactly the same form as Eq. (2.21), the only apparent difference being the replacement of
the reduced mass of the two nuclei μn by the reduced mass of the two atoms μat . Note that for the sake of
notational simplicity, from this point onward the atomic reduced mass will simply be written as μ ≡ μat .
2.1.5 Separating off the Rotational Coordinates
The Hamiltonian operator appearing in Eq. (2.21) clearly depends on the orientation of the internuclear
separation vector r through the∇2r differential operator. This is precisely the same situation encountered in
the treatment of the hydrogen atom found in any standard introductory quantum mechanics text [18, 19].
As in the derivation for the hydrogen atom, we note that in spherical polar coordinates
∇2r =
∂2
∂r2+
2
r
∂
∂r− 1
�2 r2L2 (2.23)
where
L2 = −�2(∂2
∂θ2+ cot θ
∂
∂θ+
1
sin2 θ
∂2
∂φ2
)(2.24)
is the usual squared total angular momentum operator associated with the rotation of the axis r, and θ and
φ are the usual polar and azimuthal angles defining the orientation of r relative to the laboratory frame
axes [18, 19]. Since L2 commutes with the Hamiltonian operator of Eq. (2.21), the eigenfunctions of the
latter will also be eigenfunctions of L2. Basic angular momentum theory tells us that the eigenfunctions
of L2 are the familiar spherical harmonic functions YJM(θ, φ), and that its eigenvalues are J(J +1)�2 , for
all non-negative integer values of J :
L2 YJM (θ, φ) = J(J + 1) �2 YJM (θ, φ) (2.25)
As in the standard textbook discussion of the hydrogen atom, one may readily show that the eigen-
functions of Eq. (2.21) may be written as (where again the electronic state label subscript s is omitted):
X (r) = Xs(r) = r−1 ψ(r) YJM(θ, φ) (2.26)
On substituting this into Eq. (2.21), applying Eq. (2.25), and removing the common factor r−1 YJM(θ, φ)
from the result, one obtains the effective radial Schrodinger equation:
− �2
2μ
d2 ψ(r)
dr2+
(V (r) +
J(J + 1) �2
2μ r2
)ψ(r) = E ψ(r) (2.27)
13
Note that while X (r) is the full relative-motion wavefunction, which (for bound states) is normalized over
three-dimensional space,
〈X (r)|X (r)〉 ≡∫ ∞0
r2 dr
∫ π
0sin θ dθ
∫ 2π
0dφX (r)∗ X (r) = 1 (2.28)
the factor of r−1 appearing in its definition means that ψ(r) normalizes as a simple effective one-dimensional
function:
〈ψ(r)|ψ(r)〉 ≡∫ ∞0
ψ(r)∗ ψ(r) dr = 1 (2.29)
Equation (2.27) is the effective radial Schrodinger equation underlying much of the material presented
in the next few chapters. Because of the “Born-Oppenheimer breakdown” (B-O-B) effects referred to
above, the effective potential V (r) for a given molecular electronic state will actually be slightly different
for different isotopomers, and for very high precision work B-O-B corrections must be taken into account.
Moreover, additional effects must be considered for molecular states with a non-zero spin and/or orbital
electronic angular momentum. One simple extension, appropriate when the projection of the total elec-
tronic angular momentum on the axis of the molecule Ω� is non-zero, is simply to replace the factor
[J(J +1)] in Eq. (2.27) (and in all expressions based on it) by [J(J +1)−Ω2]. Some of these effects will be
considered in Chapter 7. For the present, however, all such embellishments will be ignored, and Eqs. (2.25),
(2.26) and (2.27) will be used to describe the rotational and radial or vibrational motion of a diatomic
molecule.
Equation (2.27) clearly has the form of the elementary Schrodinger equation for a particle moving in
one dimension subject to the effective potential
VJ(r) = V (r) + J(J + 1) �2/ 2μ r2 (2.30)
The first term is the total electronic potential of Eq. (2.22), while the second is a centrifugal potential
associated with the rotation of the molecular axis r. Thus, the effect of increasing the rate of molecular
rotation is to add an increasingly strong repulsive centrifugal term J(J + 1) �2/2μ r2 to the basic (rota-
tionless) potential energy curve for a given electronic state. We will see below (in Chapter 3) that it is
this change in the effective supporting potential which gives rise to rotational energy level spacings, and
that the repulsive centrifugal potential can eventually overcome even the strongest attractive electronic
potential, causing a molecule to dissociate. This is illustrated in Fig. 2.3 for the case of HgH.
2.2 Nature of the Electronic Potential Energy Function
2.2.1 General Features and Labeling Conventions
Figure 2.2 illustrates the fact that a given diatomic molecule has a different potential energy curve associ-
ated with each of its many possible electronic states. The absolute energy at the asymptote of each curve
is of course the energy of the atoms formed when that molecular state dissociates. The differences between
the different asymptotes are therefore simply the spacings between the energy levels of the component
atoms, which are well known [20]. This figure also clearly shows that a wide variety of potential energy
curve shapes may arise. While the ground state usually has a simple single-minimum potential, excited
state curves may have a “shelf” (see, e.g., the 3 1Σ+g or 4 1Σ+
g curves in Fig. 2.2), may have double min-
ima (see, e.g., the 2 1Σ+u curve in Fig. 2.2), or may be mainly or completely repulsive. Transitions among
bound states of potentials whose minimum energy lies below their asymptote give rise to discrete (line)
14
υ=0
υ=1
υ=2
υ=3υ=0
υ=1
υ=0
VJ=28(r)
VJ=0(r)
VJ=21(r)
2 3 4 50
1000
2000
3000
4000
5000
6000
7000
8000
r /Å
energy/cm
-1
HgH (X 1Σ+)
Figure 2.3. Centrifugally-distorted effective potentials for the ground electronic state of HgH as-
sociated with different values of the rotational quantum number J . Horizontal lines show the
vibrational energies for v = 0 − 4 and how they are shifted by the centrifugal potenetial of
Eq. (2.30).
spectra, while the continuum eigenstates at energies above the asymptotes of either repulsive potentials or
potentials with attractive wells give rise to continuum spectra.
The number of different potentials which can arise from a particular pair of atoms is determined by
their total electronic degeneracy. For example, a ground-state 2S Li atom is doubly degenerate, so the total
number of electronic species which can be formed from two such atoms is 2× 2 = 4. However, some of the
many possible combinations of atomic states are always grouped together in single molecular states. Thus,
when two ground state Li atoms interact they give rise to only two molecular states, the singly-degenerate
1 1Σ+g (X) ground state seen on the left hand side of Fig. 2.2 and the triply degenerate 1 3Σ+
u (a) state at
the bottom of the right hand side of this figure. Note, that the single degeneracy of the 1 1Σ+g (X) state
plus the triple degeneracy of the 1 3Σ+u (a) state adds up to 4, so that all possible combinations of the
associated atomic states are accounted for.
Just as different electronic states of theHatom are labeled by the symmetry of the electron orbital (s,
p, d, etc.), so the electronic states of a diatomic molecule are labeled by the symmetries of their electronic
wavefunctions. The fact that the spin and orbital motions can couple to each other and to the axis of the
molecule in a variety of ways means that the electronic symmetry labels for the molecule (e.g., 1Σ+g ,
3Π1u)
are somewhat more complex than those for a one-electron atom, but the principle is the same. Moreover,
just as an atom has a ladder of states with the same electronic symmetry, but ever increasing energy (e.g.,
1s, 2s, 3s, 4s, ... etc.), so a molecule has a ladder of molecular states of the same symmetry. While this
convention is not yet universal, it is becoming increasingly common to label such states with an integer
indicating their energy ordering; e.g., 1 1Σ+g , 2
1Σ+g , 3
1Σ+g , ... etc. Both this approach and a traditional
alphabetic way of labeling electronic states are illustrated in Fig. 2.2.
15
In that traditional method of state labeling, the ground state is always labeled X, and in order of
increasing energy, other states of the same spin multiplicity are labeled A, B, C, ... etc., while states
of different spin multiplicity are labeled a, b, c, ... etc., in order of increasing energy. However, those
alphabetic labels were affixed to states in the order in which they were experimentally discovered, and over
time some were found to be out of energy order. Moreover, additional states were discovered for which no
conventional alphabetic label was designated. Fortunately, the development and widespread application of
electronic structure computational methods in recent years has allowed a rigorous cataloging of electronic
states for many systems, and led to the increasingly popular unambiguous numerical labeling referred to
above. In any case, for the next few chapters these symmetry labels will often be used to identify the
various molecular states being discussed, but they will usually be treated simply as identity tags, and a
more detailed discussion of their significance will be postponed to Chapter 7.
The variety of potential energy curve shapes seen in Fig. 2.2 immediately suggests that it is unlikely
that any unique closed-form expression for such potentials would exist. In spite of this, a considerable
effort has been expended over the years in developing and testing a wide variety of analytic representations
of potential energy functions, a number of which are described in Section 2.2.3. For weakly-bound “Van
der Waals” systems where the attractive portion of the interaction is mainly due to classical electrostatic
or induction interactions and/or dispersion terms, this type of approach has sometimes been successful to
quite high precision. However, for chemically bound systems and most excited states, no general a priori
closed-form description of the system seems possible, and an accurate description requires either extensive
high quality electronic structure calculations to generate values of V (r) point-by-point, or flexible analytic
functions to be fitted either to such ab initio points or to various types of experimental data.
The only exception to this situation occur at long range or very short range, where the interaction
is, respectively, either weak or totally dominated by a single well-understood type of interaction, and
simple analytic expressions for the potential form may be obtained. The next two subsections summarize
our understanding of the nature of these long-range and very short-range intermolecular interactions. It
will be seen in Chapter 4 that our understanding of the strength and precise analytic form of the longest-
range attractive terms in an intermolecular potential have a remarkable impact on our ability to empirically
determine accurate molecular dissociation energies, and to make meaningful predictions about the number,
energies and other properties of levels lying above the highest ones observed.
2.2.2 Long-Range Behaviour – The Inverse-Power Expansion
It has long been known that if two atoms lie sufficiently far apart that their electron clouds overlap
negligibly, and one can ignore electronic coupling and fine-structure effects, their interaction energy may
be expanded as
V (r) = D−∑m
Cm/rm (2.31)
where D is the energy at the potential asymptote, the powers m are positive integers, and the nature of
the atomic species which a given molecular state yields on dissociation determines which powers contribute
to Eq. (2.31), and also sometimes determines their sign. Moreover, perturbation theory yields explicit
expressions for the Cm constants in terms of the properties of the isolated atoms and the symmetry of the
particular molecular state [21, 22, 9]. However, it is also well known that the expansion of Eq. (2.31) is an
“asymptotic series”, which means that although the first few terms often yield a fairly good estimate of
V (r), as m → ∞ the sum always diverges. In addition, at shorter distances where the higher (inverse)
power terms become relatively important, increasing overlap of the electron clouds on the two atoms both
requires the addition of electron exchange contributions to the interaction, and leads to breakdown of the
16
inverse-power form of most of these terms.
In view of the above, for practical spectroscopic applications it is usually most appropriate to focus
only on the first one or two non-zero inverse-power contributions to Eq. (2.31), since at distances where
higher-power terms become important the inverse-power form itself begins to break down and exchange
interactions, which one cannot represent in terms of independent particle properties, start to become
important. The present section therefore presents a compact summary of the rules determining what these
leading terms will be and how some of their coefficients may be calculated. (More complete discussions
may be found in Refs. [9, 21, 22, 23, 24] and other sources.) Since we are mainly interested in the leading
inverse-power terms, only first- and second-order perturbation energies are considered; the former can yield
terms with m ≥ 1 while the latter yield even-power terms for m ≥ 4 . Contributions from third- and
higher-order perturbation theory yield terms with powers at least 3 higher than that for the leading term,
and hence would tend to contribute substantially to V (r) only at smaller distances where Eq. (2.31) begins
to break down.
The first-order perturbation theory contributions to Eq. (2.31) are terms corresponding to the elec-
trostatic interactions between permanent electric moments (charge, dipole, quadrupole, ... etc.) on the
component species, and the associated inverse powers are the same as those occurring for the interaction
between such charge distributions in classical physics. The second-order perturbation theory interactions
involve only even values of m, and are of two types. The first is the “induction” interaction between a
permanent electric moment on one particle and the electron distribution on the other; terms of this type
also have explicit classical analogs. The second type of second-order term is the non-classical “dispersion”
interaction, which may be thought of as arising when the instantaneous electric moment due to a momen-
tary electron configuration on one species induces a moment on the other, and then interacts with it, with
this interaction being averaged over the full electronic configurations. However, while it is convenient to
classify the various types of long-range interactions according to how they arise, for present purposes it is
more convenient to itemize them according to the (inverse) power of the long-range term they give rise to.
An m = 1 term will contribute to Eq. (2.31) only when both atoms have a permanent charges, as in an
ion pair state of I+ I−. In this case the coulomb interaction coefficient is C1 = −ZaZb e2/4πε0 =
−116 140.97Za Zb [cm−1 A], where Za and Za are the (±) integer number of charges on atoms a and
b, respectively, and e is the electron charge.
An m = 2 term arises classically from the interaction between a permanent charge (e.g., an atomic ion)
and a permanent dipole moment. Although no atom truly possesses a permanent dipole moment,
an electronically excited one-electron atom such as excited H or He+, will sometimes behave as if it
does. This can occur when the atom is in an excited level for which eigenstates with orbital angular
momentum quantum numbers differing by one (e.g., 2s with 2p, or 3p with 3d) are degenerate. In
this case the presence of the interaction partner causes a mixing of degenerate states of different
symmetry to yield a hybrid atomic orbital which is effectively dipolar [9].2 The resulting species will
then interact as if it had a permanent dipole, and if its partner is an ion, it will contribute an m = 2
term to Eq. (2.31). This could occur, for example, in the interaction between Ar+ and H∗(n = 2),
where “n” is the usual hydrogenic-atom principle quantum number.
An m = 3 term would arise classically from the interaction between two permanent dipole moments.
The discussion of the preceeding paragraph indicates that this could occur in the interaction of two
2Consider, for example, the sum of a 2s with a 2p hydrogenic atomic orbital; the algebraic cancellation of ±wavefunction contributions would yield a hybrid orbital with more net electron density on one side of the nodal
surface than on the other.
17
electronically excited one-electron atoms, each of which is in a dipolar hybrid state, such as H∗(n=2)
or Li++(n = 2) with H∗(n = 3). However, a much broader range of cases involves the interaction
between a pair of atoms of the same species in different atomic states between which electric dipole
transitions are allowed [25, 26, 9]. In this case, the ‘resonance’ mixing of the wavefunctions for two
equivalent atoms whose total orbital angular momentum quantum numbers differ by one (i.e., S with
P , or P with D) effectively makes them act as if they both had permanent dipole moments, and an
R−3 interaction energy arises. In this case the sign of the C3 coefficient is determined by the symmetry
of the particular molecular state and its magnitude by the intensity of the associated dipole-allowed
atomic transition [9]. Numerical values of the C3 coefficients for a variety of homonuclear alkali atom
systems may be found in Ref. [27].
Another type of r−3 term can arise from the first-order interaction between an ion and a particle
with a permanent quadrupole moment (e.g., with a P–state atom). For this case C3 ∝ Za eQb ,
where Za e is the charge on the ion and Qb the permanent quadrupole moment on its interaction
partner [22, 28], and the proportionality constant depends on whether the molecular state formed
from these species has Σ or Π symmetry.
An m = 4 term in Eq. (2.31) could arise in first order from the interaction of an ion with a particle having
a permanent octupole moment (e.g., a D–state atom), or between an particle with (or acting as if
it had) a permanent dipole moment and a species having a permanent quadrupole moment (e.g.,
H∗(n=2) interacting with a P–state atom such as ground state Al, C or Br). In both of these cases
the associated C4 interaction coefficients would be proportional to the product of the two charge
moments with a factor defined by the symmetry of the particular molecular state [22].
A more common type of r−4 potential term arises as the second-order charge-induced dipole interac-
tion between an ion and the electron distribution of its interaction partner. As with other induction
contributions, this type of term arises from the classical electrostatic interaction of two charge dis-
tributions. For this case C4 =(Za
2 e2/4πε0)αbd/2 , where Za e is the charge on the ion (atom–a)
and αbd the dipole polarizability of particle–b. This type of term is quite important, as it is often the
leading (lowest-power) long-range potential term for molecular ions. For the common case in which
the polarizability is given in units A3 [29], values of this coefficient in conventional spectroscopists’
units are given by C4 = 58070.5Za2 αb
d [cm−1 A4]. Note that the numerical factor appearing here
is exactly one half of that appearing above in the expression for the m = 1 Coulomb interaction
coefficient C1 .
An m = 5 term can arise in first order from the classical electrostatic interaction of two permanent
quadrupole moments. Thus, except for particular molecular states for which the C5 coefficient
is accidentally zero for reasons of symmetry (e.g., for the ground electronic state of Cl2 or other
halogen diatomics [30]), it will always contribute to the long-range potential whenever neither of the
interacting atoms is in an S state. As with all first-order interactions, the associated C5 coefficient
is proportional to the product of the associated permanent moments with a factor depending on the
symmetry of the particular molecular state, or more particularly, as the product of an electronic state
symmetry factor times 〈re2〉a 〈re2〉b , where 〈re2〉α is the expectation value of square of the electron
radius in the unfilled valence shell of atom–α [31]. The multiplicative symmetry factors for a wide
range of different atomic partners and variety of different coupling cases have been summarized by
Chang [30], so this most common type of C5 coefficient can be fairly readily calculated for most cases
of interest.
Another type of r−5 interaction is the first-order resonance interaction energy between two atoms
18
of the same species in electronic states coupled by a quadrupole-type interaction (e.g., between
like atoms in S and D, or in P and F , states). However, since electric quadrupole spectroscopic
transitions are very much weaker than electric dipole ones, this type of interaction will tend to be
much weaker than the resonance dipole interaction which gives rise to r−3 terms, as discussed above.
Thus, even when it can arise, this type of r−5 interaction will usually be sufficiently weak that even
though it is in principle longer-range than the r−6 dispersion energy term (see below), at ‘normal’
distances its influence will be negligible.
Two other situations which can give rise to R−5 long-range potential terms are the interaction
of an atom acting as if it had permanent dipole moment (e.g., H∗(n≥ 2) ) with an atom having a
permanent octupole moment, or the first-order interaction of an ion with an atom having a permanent
hexadecapole moment (i.e., an F–state atom), but these are much less common.
“Dispersion energy” terms with m = 6, 8, 10, ... etc., arise in second-order perturbation theory and
contribute to all interactions between atomic systems (except for the trivial case when one particle
is a bare charged nucleus). Moreover, for the case of two uncharged S–state atoms there are no
first-order or induction contributions, and these dispersion terms are the leading (longest-range)
contributions to Eq. (2.31). Since they arise in second-order perturbation theory, for a pair of ground-
state atoms the associated potential coefficients are always positive (attractive) [21, 22]. If one or
both atoms are electronically excited, coefficients of either sign may arise, but for cases involving
low levels of atomic excitation they are almost always attractive. Moreover, for states with the same
type of orbital symmetry approaching the same asymptote, these values are usually quite similar to
(within a few % of) one another.
For ground-state atoms, realistic estimates of C6 coefficients may be generated simply from a knowl-
edge of atomic polarizabilities and ionization potentials [24, 32], while a variety of methods for gener-
ating highly accurate values of the dispersion coefficients for both ground and excited state systems
have now been developed. However, those more accurate methods are more sophisticated than the
level of the present discussion, and their application requires a high degree of expertise. Thus, while
extensive tables of highly accurate C6, C8 and C10 dispersion coefficients for many molecular sys-
tems may be found in the literature (see, e.g., Refs. [33, 34, 35, 36, 27, 37, 38]), calculations of high
accuracy are a matter for experts.
Fortunately, for some important applications a high degree of accuracy is not required, and even the
10–30% accuracy of some approximate methods will suffice. The two most widely-used such methods
are the venerable London formula [22, 24],
C6 =3 IPa IPb
2(IPa + IPb)αad α
dd (2.32)
where IPa is the ionization energy of atom–a (in cm−1) and αad its dipole polarizability (in A3), and
the Slater-Kirkwood formula[24, 32, 40]
C6 =(3/2) αa
d αbd
(αad/Na)1/2 + (αb
d/Nb)1/2Eh (a0)
3/2 =126 729.372 αa
d αbd
(αad/Na)1/2 + (αb
d/Nb)1/2(2.33)
where Na and Nb are the “numbers of equivalent electron oscillators” in atoms–a and b, respectively
[32], and (for polarizabilities in A3) the final version of this expression gives C6 in the familiar
spectroscopists’ units of cm−1 A6. These Na values are expected to be approximately the same for
all atomic species in a given isoelectronic series (e.g., for Ar, Cl−, K+ and Ca+2) [32], and good
19
Table 2.1 Test of London and Slater-Kirkwood formulae for C6 constants for ground state homonuclear
diatomic molecules.
% error in C6
C6/cm−1A6 Slater-Kirkwood
species theory Ref. London (Nvalencea ) Nfitted
a
He2 7.060 × 103 [39] −11.5 17.8 1.442
Ne2 33.13 × 103 [39] −38.4 34.6 4.416
Ar2 323.9 × 103 [39] −20.7 16.3 5.912
Kr2 641.0 × 103 [39] −18.5 9.5 6.674
Xe2 1439. × 103 [39] −16.6 1.3 7.798
H2 31.32 × 103 [33] 16.8 10.2 0.824
Li2 6.689 × 106 [27] 187.9 13.5 0.777
Na2 7.094 × 106 [27] 144.0 2.4 0.954
K2 18.38 × 106 [27] 169.1 −1.4 1.029
Rb2 21.33 × 106 [27] 165.0 −3.4 1.071
Cs2 30.51 × 106 [27] 174.2 −4.4 1.095
Be2 1.060 × 106 [39] 66.8 12.0 1.594
Mg2 3.055 × 106 [39] 70.1 1.2 1.952
Ca2 13.42 × 106 [39] 43.2 −16.5 2.871
Cl2 502.1 × 103 [35] −25.8 7.5 6.061
Br2 627.4 × 103 [35] 6.0 42.3 3.455
I2 2.11 × 106 [34] −14.2 −1.7 7.241
RMS error 91.9 16.2
first estimate of them is simply the (integer) number of electrons in the valence shell of that atom,
Na=Nvalencea .
For a number of simple systems for which accurate theoretical C6 values are known, the two middle
columns of Table 2.1 list the percentage errors in C6 values generated from the London and the basic
(i.e., with Na =Nvalencea ) Slater-Kirkwood formulae using polarizabilities and ionization potentials
from a standard source [29]. It is clear that the Slater-Kirkwood formula is on average more than a
factor of five better than the London formula, and that within the basic Na=Nvalencea approximation,
it yields predictions for these 17 species with average errors of less than 20%. Moreover, a simple
rearrangement of Eq. (2.33) would allow the accurate dispersion coefficients available for homonuclear
species Ca,a6 to define an effective of fitted number of equivalent electrons, Na=N
effa for each type
of atom (last column of Table 2.1). While use of the latter in Eq. (2.33) would of course yield the
accurate theoretical value for that homonuclear diatom, Ca,a6 , those fitted values should also yield
much more reliable Slater-Kirkwood estimates of Ca,b6 for associated heteronuclear species. This is
mathematically exactly equivalent to applying the combining rule
Ca,b6 = 2Ca,a
6 Cb,b6
/[(αb
d/αad)C
a,a6 + (αa
d/αbd)C
b,b6
](2.34)
to a set of accurate theoretical values Ca,a6 and Cb,b
6 for homonuclear species a–a and b–b. For the 88
heteronuclear dimers formed by hydrogen, the five alkali atoms, the five rare gas atom and Be, Mg
20
and Ca,3 Slater-Kirkwood C6 values calculated in this way agreed with accurate theoretical values
[39, 27, 41] with a root-mean-square (RMS) discrepancy of only 3.4%; in contrast, the RMS error
in those calculated using the basic Na = Nvalencea = 1 values was 9.5%. (Even the latter is quite
good!) In view of the fact that Na is expected to be essentially the same for isoelectronic species
in the same electronic state, these scaled effective N effa values clearly offer a very powerful way of
using basic atomic polarizabilities to generate realistic estimates of C6 constants for a wide range of
species for which accurate ab initio values are not yet available.
One other type of r−6 interaction which deserves mention here is the second-order charge-induced
quadrupole term which will arise whenever an ion interacts with an atomic charge distribution. For
a molecular ion a interacting with atomic species b,
C ind6 =
Za2 e2
4πε0αbq/2 =
Za2 αb
q
2Eh a0 = 58070.5 Za
2 αbq (2.35)
where αbq is the quadrupole polarizability of species–b, and for αb
q in units A5, the last version of
this expression gives C ind6 in units cm−1A6. This term is fairly important, since for many ionic
molecules the r−6 interaction is the second longest-range contribution to the sum in Eq. (2.31), and
for molecular cations the dispersion contribution to the r−6 interaction is relatively small, so this
induction term will dominate (e.g., see Table 7-2-1 of Ref. [32]). Note that the numerical factor
appearing here is exactly the same as that associated with the m=4 induction coefficient C4=Cind4 .
Limits to Validity of the Inverse-Power Expansion
As mentioned above, the simple inverse-power expansion of Eq. (2.31) is formally only valid at distances
sufficiently large that the overlap of the electronic wavefunctions of the two atoms can be neglected. As r
decreases this approximation always eventually breaks down. In particular, detailed studies of a number
of hydrogen and helium systems by Meath and co-workers [42, 43, 33] showed that the dispersion energy
terms which asymptotically take on m = 6, 8 and 10 inverse-power behaviour overestimate the strength
of the associated interactions at smaller distances, and that the onset of this breakdown occurs at larger
distances with increasing values of the power m.
The combination of the above behaviour with neglect of exchange interactions, plus the fact that
Eq. (2.31) is an asymptotic series, means that for any given system there is a characteristic distance
inside of which the inverse-power expansion representation for the potential should no longer be used. An
examination of results for a number of systems involving interactions of S–state atoms led to the suggestion
[44] that this limiting distance rLR should be defined as
rLR = 2[〈re2〉a1/2 + 〈re2〉b1/2
](2.36)
where 〈re2〉a is the expectation value of the square of the radius of the electrons in the valence shell of
atom a [45]. More recent work examining interactions of non–S–state atom yielded the more sophisticated
expression for this characteristic cut-off radius [46]:
rLR−m = 2√3[〈nlm|z2|nlm〉1/2a + 〈n′l′m′|z2|n′l′m′〉1/2b
](2.37)
3 The predictions for Na-Be, Na-Mg and Na-Ca were omitted from these averages, because the fact that the upper
and lower bounds which define them differ by a factor of two makes the theoretical values [39] too ill-defined to allow
meaningful comparisons.
21
where n, l and m are the usual principal, orbital and azimuthal atomic orbital quantum numbers of the
outermost (valence) electrons on the two atoms.
Although the expansion of Eq. (2.31) does become invalid at small distances, a desire to incorporate
this fundamentally correct inverse-power long-range behaviour into realistic models for intermolecular
interaction potentials instigated a number of efforts to quantitatively model this “damping” behaviour by
introducing theoretically-based correction functions into Eq. (2.31):
V (r) = D−∑m
Dm(r)Cm/rm (2.38)
where Dm(r) is a monotonic function which grows from zero at r = 0 , rises steeply at some distance
characteristic of the molecular species, and approaches unity as r →∞ . The models for Dm(r) functions
proposed in the literature were typically determined by examining the contributions of the m = 6 , 8 and
10 dispersion terms to the shallow 3Σ+u interaction of two ground-state hydrogen atoms, and generalized
to other species through an empirical rule based on some characteristic of the system. Unfortunately
most such proposed forms based their inter-system scaling on properties of potential minimum or repulsive
wall well for a pure Van der Waals interaction [47, 33, 48]. This made that scaling inappropriate for use
in describing damping in chemically-bound systems. A noteworthy exception is the form proposed by
Douketis et al.[49]
Dm(r) =[1− exp
(−3.968(ρab r)/m− 0.389(ρab r)2/√m)]m
(2.39)
where ρab = 2ρa ρb/(ρa + ρb) , and the atom-specific universal scaling parameter ρa is defined in terms
of a ratio of the ionization energy of atomic species–a to that of ground-state atomic hydrogen ρa =
(IPa/IPH)2/3 .
For the weakly-bound Van der Waals molecule Ar2 , the upper segment of Fig. 2.4 illustrates the nature
of the damping functions of Eq. (2.39); the lower segment of this figure shows the overall Ar2 potential
energy function [50] on the same radial scale. For this particular system, rLR = 4.1 A; thus, it is clear that
the criteria of Eqs. (2.36) and (2.37) are quite conservative, and refers only to the boundary of the region
where the full undamped inverse-power sum of Eq. (2.31) can be trusted. Note too that while this example
involves a simple Van der Waals molecule, the nature of the scaling parameter associated with allows is to
be used for interactions of any ground or excited-state atoms or ions.
2.2.3 Very Short-Range Behaviour – Approaching the United Atom
Very short internuclear distances is another limiting region for which explicit analytic expressions for the
intermolecular have been obtained. The interaction in this region is clearly dominated by the coulomb
repulsion between the two nuclei, +Zb Zb e2/4πε0 r , while the molecular electronic wavefunction is simply
a somewhat distorted version of the wavefunction for an atom with nuclear charge +(Za + Zb) and the
same total number of electrons. In this region the potential may be expanded in the form
V (r) = Za Zb e2/4πε0 r +Wua +W2 r
2 +W3 r3 + ... (2.40)
where Wua is the energy of the electronic state of the atom to which that particular molecular state
correlates, there is no linear term, and W2, W3, ... etc. may be calculated from the electron density of the
united atom [51, 52].4 For example, for the ground state of H2, Wua is the energy of a ground state He
atom, and for HCl it is that of an Ar atom.
4More particularly, if the united atom is in an S state, both W2 and W3 are equal to a numerical constant times
the value of the electron density at the nucleus [53].
22
0 2 4 6 8-100
-75
-50
-25
0
25
R /Α
V(r)
Ar2
°
/cm-1
0.0
0.2
0.4
0.6
0.8
1.0
Dm(r)
m=68
1012
14
Figure 2.4. Upper: damping functions for the ground state of Ar2 generated from Eq. (2.39).
Lower: overall potential for ground-state Ar2.[50]
While theoretically very interesting, detailed numerical studies have shown that region in which Eq. (2.40)
can be trusted is very narrow; Pathak and Thakkar show that it has an upper bound of ca. 0.26 A[54].
At these distances the interaction energies are extremely high, and it may even be argued that the very
concept of a potential energy function is compromised by the breakdown of the Born-Oppenheimer approx-
imation. Nonetheless, some empirical potential can have completely spurious behaviour as r→ 0 which
may interfere with their use in practical calculations unless ad hoc fix-up procedures are implemented.
It has also been found that imposing this correct limiting behaviour can yield potentials which give an
improved description of very high energy collision data [50]. However, such niceties are of little practical
concern in describing potential energy functions in the region where they affect normal spectroscopic data.
In spite of the above caveat, the idea that the electronic wave function of a collapsing diatomic molecule
becomes that of the associated united atom is quite useful in another context. In particular, the dependence
of the Ar–H2 interaction energy on the H2 bond length is a matter of considerable importance, since that
is the component of the interaction which causes vibrational inelasticity in Ar+H2 collisions. However,
experiment only yields information about this behaviour for H2 bond lengths near equilibrium. Thus, in
modeling the full three-dimensional H2–Ar potential energy surface it was found very helpful to build in
the constraint that as the H2 bond length approaches zero this potential surface smoothly became the
accurately-known one-dimensional He–Ar potential energy curve [55].
23
2.3 Models For the Electronic Potential Energy Function
2.3.1 Few-Parameter Analytic Model Potentials
A wide range of analytic functions for representing ordinary single-minimum potential energy curves have
been proposed over the years. Appendix I of Ref. [24] describes more than two dozen such potential forms,
and a number of painstaking comparisons of the capabilities of various several-parameter forms have been
reported [56, 57]. However, exact analytic expressions for vibrational level energies can be derived only
for a handful of these functions, and analogous exact expression for the rotational energies are available
for only one or two of them. The simplest of these forms are defined by two or three parameters which
typically characterize the well depth De , the position of the potential minimum Re, and the well shape.
The present section describes a few of them, because some are still widely used as simple models or as
the basis of flexible many-parameter potential functions, and/or because our knowledge of exact analytic
expressions for their eigenvalues makes them useful reference systems. For all but the Coulomb potential,
which is infinitely deep, these expressions are written with the energy zero at the potential minimum.
(i) A harmonic oscillator potential has the form
VHO(r) = 12 k (r − re)2 (2.41)
where k is the harmonic force constant and re the bond equilibrium length. This ubiquitous model rep-
resents virtually all small amplitude vibrational motion, and provides a basis for describing a host of
apparently unrelated problems in physics. Its level energies are given by the expression
E(v) = ωe (v +12) =
√2k (�2/2μ) (v + 1
2) (2.42)
and exact closed form expressions are known for its eigenfunctions and for the expectation values and
matrix elements of various operators [18, 19, 58].
(ii) The Coulomb potential which describes a hydrogenic (one-electron) atom and the long-range interaction
in any ion-pair system, is
V (r) = D− C1/r (2.43)
where C1 = Za Zb e2/4πε0. In this particular case, the rotational dependence of the vibrational eigenvalues5
E(v, J) = D− (2μ/�2)(C1/2)2/(v + J + 1)2 (2.44)
and of the eigenfunctions, expectation values and matrix elements are also exactly known [18, 19, 58].
While this potential is quite unrealistic for describing a molecular state near its equilibrium distance, the
pattern of behaviour for levels approaching dissociation provides an important prototype (see Chapter 5).
(iii) Square-well or step-function potentials are widely discussed in introductory quantum texts because
they can so readily be treated analytically, and they are tractable physical models for illustrating a wide
variety of phenomena. The simplest function of this type is the “particle-in-a-box” potential
Vsw(r) =
{0 for 0 ≤ r < L
∞ for r ≤ 0 and r ≥ L (2.45)
5For consistency with the molecular notation used herein, the conventional hydrogenic atom principle quantum
number is expanded as n=v+J+1, and the usual particle-in-a-box quantum number n is replaced by n=v + 1.
24
with eigenfunctions ψv(r) =√
2/L sin{(v + 1)π r/L} and eigenvalues5
E(v) = π2 (�2/2μ) (v + 1)2/L2 (2.46)
(iv) The Morse potential is a sum of attractive and repulsive exponential terms
VMorse(r) = De
[e−2β(r−re) − 2 e−β(r−re) + 1
]= De
[1− e−β(r−re)
]2(2.47)
whose vibrational eigenvalues have an exact quadratic dependence on v,
E(v) = ωe (v +12)− ωexe (v +
12)
2 (2.48)
where ωe = 2β√
De(�2/2μ) and ωexe = β2 (�2/2μ) . While it is very widely used, one drawback is
that its exponential form disagrees with the known inverse-power behaviour of long-range intermolecular
potentials.
(v) Lennard-Jones(m,n) or LJ(m,n) potentials are a family of potentials with inverse-power attractive and
repulsive terms
VLJ(r) = Cm/rm − Cn/r
n +De = De
[n
m− n(rer
)m − m
m− n(rer
)n+ 1
](2.49)
where m > n are both are positive integers, and Cn = [m/(m − n)]De (re)n . While there is no general
exact quantal expression for the vibrational energies of this potential, it has been very widely used for
modeling molecular interactions. Reasons for this include the simplicity of the form (especially for fixed
m = 2n ), the fact that it allows one to impose the correct long-range behaviour on the potential by setting
n equal to the power of the first non-zero term in Eq. (2.31), and the fact that for m = 2n radial integrals
involving the factor exp{−V (r)/kB T} can be evaluated analytically.ref. FRM However, the fact that it
attributes all of the attraction to a single inverse-power term effectively means that if D and re are correct,
the Cn constant must be too large and the outer branch of the attractive well will be too broad.
2.3.2 “HFD”-type Theory-Based Potential Forms
In the past quarter century, the contrast between the deficiencies of traditional few-parameter model
potentials and the increasing availability and accuracy of experimental scattering and spectroscopic data
for weakly-bound Van der Waals molecules has instigated the development of new model potential functions
which make optimum use of available theoretical information. All forms of this type are based on the
following assumptions:
• the attractive portion of the potential is almost solely due to the “physical” inverse-power multipolar
interactions of Section 2.2.2;
• the associated Cm potential coefficients for these interactions can be calculated theoretically, and
realistic models for the damping behaviour can be devised (see above);
• the overall potential may reasonably be represented as a sum of attractive multipolar terms plus a
repulsive electron overlap/exchange term .
If was found that ab initio points obtained from a simple Hartree-Fock or self-consistent field (SCF)
electronic structure calculation could usually be accurately represented by a simple repulsive exponential
25
function, which when added to a sum of damped attractive multipolar interaction terms, gave semi-
quantitative predictions of the overall potential energy functions for a wide range of systems. Since ac-
curate Hartree-Fock calculations are readily feasible for a wide range of systems for which high quality
“supermolecule” calculations would be prohibitively expensive, this approach has been widely adopted as
a means of generating realistic a priori estimates of potential energy curves and surfaces for Van der Waals
type systems [47, 59, 60, 61, 48, 62]. Although the repulsive interaction in some models is obtained from
a better-than-Hartree-Fock calculation [60, 63], this type of potential function still tends to be called a
“Hartree-Fock-dispersion” (HFD) form.
For the case of atom-atom interactions, this class of potentials is typically written in the form
VHFD(r) = Ae−βr −∑m
Dm(r)Cm/rm (2.50)
where the repulsive exponential term is determined from a fit to results of ab initio Hartree-Fock calcula-
tions, the Cm coefficients are taken from theory, and the Dm(r) damping functions are represented by a
model of the type discussed in Section 2.2.2. Variations of this form in which the exponent coefficient is
represented by a linear function of r, the repulsive term multiplied by a polynomial in r or some simple
power of r, and the damping functions either treated as being independent of m or factored into an m–
dependent function times a common overall damping function, have also been used. For atom-molecule or
molecule-molecule interactions the coefficients A, β and {Cm} are represented by relative-orientation and
(when appropriate) monomer bond-length dependent functions [64, 65, 66, 67].
2.3.3 Flexible General Empirical Potential Functions
Power Series Forms
A type of potential function which has seen wide use since the early 1930’s is the Dunham-type [68, 69]
power series expansion
VDun(r) = a0 ξ2[1 + a1 ξ + a2 ξ
2 + a3 ξ3 + ...
](2.51)
where ξ = (r − re)/re . This type of potential is essentially a Taylor series expansion about a harmonic-
oscillator leading term, and if enough terms are included, can give an arbitrarily accurate representation
of a potential over any finite range of r. Moreover, within the third-order JWKB approximation, Dunham
derived explicit relations expressing the coefficients of a double power series expansion for the vibration-
rotation energies in terms of the potential expansion parameters {am} (see Chapter 4). Unfortunately this
polynomial form has a finite radius of convergence, and at large r will always diverge to ±∞.
One response to the shortcomings of the Dunham form has been to replace the Dunham radial expansion
variable ξ by other forms which are better-behaved as r→∞ , such as (r − re)/r [70], [1− (re/r)p] [71],
2(r − re)/(r + re) [72] or by the generalized variable suggested by Surkus [73]:
y(r; p) = (rp − rep) / (rp + rep) (2.52)
By mapping Taylor series expansions in these variables onto the conventional expansion in ξ, explicit
Dunham-type expressions for the vibration-rotation energies can be obtained for these forms too. Moreover,
because of the nature of these expansion variables, these later forms all approach a finite limit with some
sort of inverse-power behaviour as r→∞. However, while it is technically possible to impose constraints
on such forms to remove all but the theoretically-predicted inverse-power terms (beginning at m=5 or 6 for
most neutral molecules) [74], it tends to be somewhat impractical, since the very high-order polynomials
obtained often have spurious non-physical behaviour in the region between the interval spanned by the
data and the large–r limit where the imposed inverse-power behaviour takes over.
26
Generalized Morse-Type Potential Functions
While power series forms of the type described above are still very much in active use, the past decade
has seen the growing exploitation of a new type of potential form based on a generalization of the Morse
potential of Eq. (2.47) in which the exponent parameter is expanded a power series in some r–dependent
expansion variable:
β = β(r) ≡ β0 + β1 y(r) + β2 y(r)2 + β3 y(r)
3 + ... (2.53)
The great advantage of this type of form is that the Morse potential structure gives the basic shape of
the potential, and only modest smooth variations in β(r) are required to give immense flexibility in the
describing details of the shape. In contrast, in simple power-series potential forms the polynomial expansion
is responsible both for imposing the basic structure and for accounting for details of the shape.
In the earliest applications of this type form for providing an accurate description of extensive high
resolution spectroscopic data, the variable y(r) was written as a simple power series in (r − re) (or in the
Dunham variable ξ) [75, 76]. While this necessarily gives rise to pathological behaviour at large r if the
highest-order coefficient βm(max) happens to be negative [76], it was remarkably successful in providing very
accurate potential function representations over very wide range of r in terms of a relatively small number
of empirical parameter. More recent work in which this expansion parameter was represented using the
better-behaved Ogilvie–Tipping [72] variable y(r; p = 1) has been equally successful [77, 78]. Since this
variable has the finite range [−1,+1], expansions based on it have a reduced tendency to show pathological
behaviour beyond the range of the experimental data [79].
“Morse-Lennard-Jones” Potential Functions
While they have been very successful in providing extremely accurate potential functions valid over a wide
range of r, generalized Morse-type potentials have the fundamental weakness that they cannot incorpo-
rate the correct theoretically-known limiting inverse-power behaviour of intermolecular potential energy
functions. This concern led to the development of an additional type of potential known as the “Morse-
Lennard-Jones” or MLJ function [80]:6
VMLJ(r) = De
[1− (re/r)
n e−β(y) y]2
(2.54)
where β(y) is an expansion of the form of Eq. (2.53). It is immediately clear that if n=0 this potential
collapses (to within a multiplicative constant) to the “modified Morse oscillator” (MMO) generalized Morse-
type potential [82], while if β(y) = 0 is becomes the simple Lennard-Jones(2n, n) potential of Eq. (2.49).
However, when both are non-zero it has a generic single-minimum type of form with the limiting long-range
behaviour
VMLJ(r) � De − Cn/rn (2.55)
where Cn = 2De(re)n e−β∞ with (using Eq. (2.53)) β∞ ≡ limr→∞ β(y) =
∑i βi . Thus, this form incor-
porates the correct theoretically-predicted limiting asymptotic behaviour, and the quantity β∞ may be
defined to yield a given theoretically-known Cn constant, while at the same time the flexible polynomial
expansion for β(y) can be fitted to yield virtually any desired behaviour (including double-minimum or
“shelf” behaviour) for the shape of the attractive potential well.
6 In the original work this function was called the “modified Lennard-Jones oscillator” potential because of a wish
to emphasize the inverse-power long-range behaviour [80, 81], but because of the symmetry noted here it seems better
to redefine the leading term in this acronym.
27
Early applications of this type of form used the simple expansion variable y(r; p = 1)=(r−re)/(r+re) ,and while quite successful in representing the potential over the range of the experimental data, β(y) tended
to have implausible extrema between that interval and the very long range region where the limiting
behaviour of Eq.(2.55) took over [80, 81, 83, 84, 85, 86]. That unsatisfactory behaviour was corrected for
by introducing a switching function to link the polynomial expansion for β(y) in the data region to the
limiting value β∞ determined by the known long-range potential constant Cn. More recently, however,
it has been found that simply using the Surkus expansion variable y(r; p) with p= 2 or 3 removes this
problem, and allows the MLJ function with a simply polynomial expansion for β(y) be used for all distances
[87].
Exercises
2.1 Using the familiar chain rule of partial differentiation, show explicitly how Eq. (2.6) is obtained from
Eqs. (2.3)–(2.5).
Hint: first consider the x cartesian coordinate of ∇a,∂∂xa
; when you have a result for that component,
the others may be written down by analogy.
2.2 Recalling that ∇2q = ∇q · ∇q and using the dot product rule of vector calculus, show explicitly how
Eq. (2.9) is readily obtained on substituting Eqs. (2.6)–(2.8) into Eq. (2.1).
2.3 For the case in which the electron coordinates are expressed relative to the midpoint of the nuclei (the
MN system, see §II.A.2), the complete set of coordinates introduced in place of those in the original
laboratory frame are the relative electron coordinates of Eq. (2.14) plus the centre of mass and relative
nuclear coordinates of Eqs. (2.4) and (2.5). Using these definitions and the chain rule of partial
differentiation, derive the analog of Eq. (2.6) in which the partial derivative operator ∇a = ∇ra is
expanded in terms of ∇c, ∇r and the set {∇mi} .
2.4 For a case in which the relative electron coordinates are all centred on atom–a, what is the form of
Hpol = Hapol ?
2.5 For the ground states of N+2 and C+
2 , identify all of the types of interactions which contribute to the
first three non-zero inverse-power terms in the long-range potential? For the case of N+2 , what is the
value of the leading (longest-range) Cm coefficient, in units cm−1 Am?
[Note: for the energies and symmetries of electronic states of atoms and atomic ions, see Ref. [20].]
2.6 Conventional expressions for the vibrational energies of a molecule count vibrational levels upwards
from the potential minimum where v=−12 . However, a very useful alternate viewpoint (see Chapter
4) is to count levels downward from the dissociation limit.
a) For a Morse oscillator, the vibrational level spacings get smaller with increasing energy (increas-
ing v). By taking the derivative of Eq. (2.48) with respect to v and setting it equal to zero,
determine an expression for (vD + 12) , where vD is the vibrational quantum number for which
the energy derivative equals zero.
b) Substitute this expression for v = vD into Eq. (2.48) and use the definitions of ωe and ωexefrom the notes to determine an expression for the vibrational energy associated with v = vD .
c) In Eq. (2.48, divide both sides by −ωexe, complete the square on the right hand side, and
determine a compact expression for EMorse(v) in terms of De, ωexe and (vD − v).
28
2.7 For the Van der Waals molecule Ar2 , early bulk property studies had determined estimates of
De = 85.6 cm−1 and re = 3.866 A[Mason & Rice, J. Chem. Phys. 22, 843 (1954)].
a) For a general Lennard-Jones(12,6) potential function:
1. determine an expression for the second derivative of the potential evaluated at its minimum.
2. By matching this potential to a harmonic oscillator with the same curvature at the mini-
mum, determine an (approximate) expression for ωe for this potential.
3. Using this potential as a model for the Ar2 interaction, determine a numerical estimate for
ωe (in cm−1) for this species.
4. What would your estimate of ωe be if you had used a Lennard-Jones(9,6) function for the
potential?
5. Calculate the C6 coefficient associated with this potential and compare it to values calcu-
lated using the London and Slater-Kirkwood formulae of Eqs. (2.32) and (2.33).
b) Early spectroscopic experiments gave an estimate of ωe = 31 cm−1 for Ar2 . Combining this
value with the above estimates of De and re , and assuming a Morse model for the potential
energy function, determine the values of the Morse exponent parameter β (in units A−1), andof the anharmonicity parameter ωexe (in cm−1).
c) Better spectroscopic experiments later showed that for Ar2 the values of ωe and ωexe were
actually 30.68 and 2.56 cm−1, respectively [Tanaka & Yoshino, J. Chem. Phys. 53, 2012 (1970)].
Using this information, determine an alternative estimate for De for this molecule.
Note: the discrepancies between the ωe values of parts a) and b), and the De value of part c) and the
scattering experiment value, illustrate the types of apparent inconsistencies found when using
simple models for intermolecular potentials.
2.8 For the ground state of NaAr+ (which dissociates to Na+ +Ar), identify and give numerical estimates
for all contributions to the two longest-range terms in the intermolecular potential.
[Note: dipole polarizabilities of neutral atoms may be found in Refs. [29, 39], those for some ions in
an Appendix III of Ref. [32], and quadrupole polarizabilities of some atoms in Ref. [39].]
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32
Chapter 3
Quantum Mechanical Treatment of
Discrete Level Properties and Spectra ...
or
Solving the Radial Schrodinger Equation
3.1 The Nature of Vibration-Rotation Energy Levels: Po-
tential Shape and Vibrational Level Spacings
We now turn our attention to the radial Schrodinger equation
− �2
2μ
d2 ψ(r)
dr2+ U(r)ψ(r) = E ψ(r) (3.1)
where the effective potential U(r) may or may not include the centrifugal term associated with molecular
rotation (see Eq. (2.30)). As discussed in §2.3, exact analytic solutions of Eq. (3.1) are known for a number
of simple potential energy function. Particularly familiar examples are the square well or particle-in-a-box
potential, the harmonic oscillator, the Coulomb or H-atom potential, and the Morse potential. Moreover,
standard numerical methods allow us to generate solutions of this equation which are accurate to virtually
any desired level of precision for virtually any potential energy function (see §3.3). If the potential is
defined by its values (obtained, perhaps, from electronic structure calculations) at some discrete mesh of
distances, numerical interpolation is required to generate a smooth continuous function, and this step will
in general introduce some “numerical noise” into the results, but accurate solutions may still be obtained.
One of the key signatures of a given potential energy function is the pattern of vibrational level energies.
This information is most often presented graphically as a plot of the energy spacings between adjacent
vibrational levels, ΔGv+1/2=G(v + 1)−G(v) , where G(v)=Evib(v) is the energy of vibrational level–v.
In such “Birge-Sponer plots”[1, 2], the ordinate values are plotted at half-integer values of v, since they are
a first-difference approximation to the derivative of the vibrational energy with respect to v at that point,
ΔGv+1/2≈dG(v)/dv|v+1/2 . For a model system approximately based on ground state Ar2 [De=100 cm−1,re=3.7234 A and μ=20u], the upper half of Fig. 3.1 shows several potential functions whose parameters
were chosen so that they would all have exactly the same spacing (of 25 cm−1) between the two lowest
levels. The lower half of this figure then compares Birge-Sponer plots for these potentials.1 As may be
1Since spectroscopic notation is used to label the energy levels as v = 0, 1, 2, ... etc., for the square well and
33
0 2 4 6 8 100
10
20
30
40
ΔGv+1/2
v
Harmonic Oscillator
Morse
LJ(13,6)
Coulomb
Square Well
vD(LJ)
vD(Morse)
Quartic Oscillator
3 4 5 6
-100
-50
0
R /A
energyLJ(13,6)
Morse
Harmonic Oscillator
Square Well
QuarticOscillator
°
Figure 3.1. Top: Model potentials having a well depth, equilibrium distance and fundamental
vibrational spacing approximately matching ground state Ar2. Bottom: Vibrational spacings
for the model potentials shown above.
34
expected, the different shapes of these potential functions give rise to different patterns for their vibrational
level spacings. However, the differences between these patterns may be readily understood in terms of the
properties of a square well potential.
A square-well potential whose well extends from r=0 to r=L has eigenfunctions ψv(r)=√
2/L sin[(v+
1)π r/L] , eigenvalues G(v)=(�2/2μ)π2 (v+1)2/L2 and vibrational spacings ΔGv+1/2=(�2/2μ)π2 (2v+
3)/L2 . It is clear that making the box wider (i.e., increasing L) will lower the energies and make the
vibrational spacings smaller. This is illustrated by the results in the first two segments of Fig. 3.2, which
compares the eigenvalues and eigenfunctions of two square-well potentials whose bond lengths differ by a
factor of 1.5. In this particular case, this broadening of the well reduces all vibrational energies by a factor
of 1/(1.5)2 and makes the v=2 and 5 levels of the broader well precisely isoenergetic with the v=1 and
3 levels of the narrower well, respectively (see Eq. (2.46)). The third segment of Fig. 3.2 shows that for the
original (narrower) box, exactly the same energy shift is accomplished by increasing the effective particle
mass by a factor of (1.5)2. This is a prototype illustration of isotope effects on vibrational level energies.
The more realistic smooth potentials shown in Figs. 3.1 clearly may all be thought of as being square-
well type potentials in which the box length increases smoothly with energy. Thus, the vibrational spacings
of all of those smooth potentials increase with v more slowly than those for a square well potential, and most
actually decease with increasing v. Because of its central position in such trends, we take the harmonic
oscillator potential, whose vibrational spacings are known to be independent of v, as a reference. For
potentials whose widths increase with energy more slowly than is the case for a harmonic oscillator (i.e.,
the quartic and square well potentials), the vibrational spacings increase with energy, and conversely, for
potentials whose widths grow more rapidly than a harmonic oscillator, the vibrational spacings decrease
with increasing energy. Thus, the patterns of the vibrational level spacings in a potential energy well are
clearly a direct reflection of the rate at which the width of the well increases with energy.2
The above discussion suggests that an experimentally determined pattern of vibrational level energies
can determine the potential energy function for a given system. This is only a half truth, since an infinite
number of potential wells with different inner-wall shapes could have exactly the same width vs. energy
behaviour, and hence exactly the same pattern of vibrational level energies. It turns out that this ambiguity
is resolved by simultaneously taking into account the magnitude and vibrational dependence of the pattern
of energies for the rotational sub-levels for each v. However, there is no (known) direct way of utilizing
this information in a quantum mechanical analysis. On the other hand, we will see in Chapter 4 that
semiclassical methods provide a very direct and easily applied solution to this problem in a form known as
the “RKR inversion procedure”.
A final point here concerns the form to use for a general expression for vibrational energies. We saw in
§2.3.1 that for a harmonic oscillator the energy can be written as E(v)=ωe(v +12 ) , and that for a Morse
oscillator it can be written as E(v)=ωe(v+12)−ωexe(v+
12)
2 . This would seem to suggest that in a more
general case the vibrational energy would be given by a polynomial expansion in powers of (v + 12). Such
forms had long been used in practice, with quadratic and higher-order terms being empirically associated
with deviation of the potential from harmonic oscillator behaviour. In 1932 Dunham showed [3, 4] that this
polynomial expansion form could be derived within the semiclassical or WKB approximation (see Chapter
4), and two decades later Kilpatrick and Kilpatrick [5, 6] showed that it may also be justified quantum
mechanically. In particular, they showed that for a potential expressed as a Taylor series expansion about
r=re (as in Eq. (2.51)), if terms of order higher than two are treated as a perturbation, quantummechanical
perturbation theory yields a level energy expression of the form
Coulomb potential cases the conventional quantum number n is replaced by n = v + 1 .2A more quantitative proof of this assertion will be found in the derivation of the RKR procedure in Chapter 4.
35
-120
-80
-40
0
υ=0
υ=2
υ=1
υ=3
υ=4
υ=6
υ=8υ=12
Kr2 (X1Σg
+)
3 5 7 9 11
-120
-80
-40
0
r /Α
υ=0
υ=2
υ=1
υ=3υ=4
υ=6υ=8
υ=12
°
energy/cm
-1
0 0 0L 1.5 L Lx→ x→x→
energy
μ =2.25μ =1 μ =1
υ=3
υ=2
υ=1
υ=0
υ⎯ 6
5
4
3
2
1
0
↑
Figure 3.2. Left: Eigenvalues and wavefunctions for a particle of mass µ in three square well
potentials. Right: Eigenvalues, wavefunctions (lower segment) and probability amplitude
(upper segment) for some levels of ground state Kr2.
36
E(v)/ωe = (v + 12) +
(Be
ωe
){a24
[6(v + 1
2)2 + 1.5
]− (a14
)2 [30(v + 1
2)2 + 3.5
]}(3.2)
+
(Be
ωe
)2{a44
[20(v + 1
2 )3 + 25(v + 1
2)]− (a2
4
)2 [68(v + 1
2)3 + 67(v + 1
2)]
− a14
a34
[280(v + 1
2)3 + 190(v + 1
2 )]+
(a14
2)2 a2
4
[1800(v + 1
2)3 + 918(v + 1
2 )]
−(a14
)4 [2820(v + 1
2)3 + 1155(v + 1
2)]}
+ O
{(Be
ωe
)3}
where ωe=√4 a0Be , Be=�
2/2μ re2 and the coefficients {ai} are as defined by Eq. (2.51). This equation
is clearly somewhat unwieldy to use for practical data analysis, to say nothing of the expressions obtained
when the eleven quartic polynomials of order O{(Be/ωe)3}, or the numerous quintic polynomials terms of
order O{(Be/ωe)4} are included. However, it does show that the use of polynomials in (v+ 1
2 ) to represent
the v–dependence of vibrational energies and other molecular properties has a sound theoretical basis. At
the same time, the intimidating level of algebraic complexity encountered, even without any generalization
to take account of rotational effects [6], clearly provided strong motivation for the widespread adoption of
the types of direct numerical methods described later in this chapter.
3.2 Bound Level Boundary Conditions and Normalization
The Right half of Fig. 3.2 shows the pattern of vibrational level energies and the nature of the wavefunctions
and of the probability amplitude for a number of the vibrational levels of Kr2, calculated from the accurate
HFD-type potential function of Ref. [7]. Note that the highest bound level of this potential is v = 15 ,
which is bound by less than 0.001 cm−1. It is immediately clear that the results seen in Figs. 3.2 and 3.3
share many common features. In particular:
• the wavefunction for the lowest level has onee loop and no internal nodes, and as one progresses up
the vibrational ladder, each wavefunction has one more loop than the preceding one;
• the average oscillation wavelength decreases as the vibrational energy increases;
• the wavefunctions have little (Fig. 3.3) or no (Fig. 3.2) amplitude in the classically forbidden region
where V (r)>E ;
• widening the “box” or potential well lowers the level energy and the level spacing;
• increasing the particle mass also lowers the energies and reduces the levels spacings.
All of these properties also hold for the eigenfunctions and eigenvalues of other standard model potentials
such as the square well, harmonic oscillator or Coulomb potential (see §2.3.1).3Another way of thinking of this behaviour is to consider the properties of the re-arranged version of
Eq. (3.1):
ψ′′(r) = − {(2μ/�2)[E − U(r)]}ψ(r) (3.3)
where the primes (′) denote differentiation with respect to r. This equation tells us that for any given
numerical value of the wavefunction, ψ(r), its rate of curvature depends on the magnitude of the quantity
3 One minor caveat is the apparent anomaly that for a Coulomb potential D−C1/r ,, increasing the mass actually
increases the level spacing between a given pair of levels, because in this case the energies of the lower levels decrease
more than do those for higher v values.
37
|E − U(r)|. If E < U(r) this curvature has the same algebraic sign as the wavefunction itself, and hence
ψ(r) will curve away from the axis (as in exponential-type functions); if E > U(r) it will have the
algebraic sign opposite to that of the wavefunction value, and hence the curvature will be towards the
axis (as in sine or cosine-type functions). In the latter case the dependence of the rate of curvature on
the magnitude of |E − U(r)| explains why the wavefunctions in Figs. 3.2 and 3.2 oscillate more rapidly
for higher vibrational levels. It also explains the distance dependence of the oscillation amplitude seen in
Kr2 segments of Fig. 3.2, which is particularly evident for the higher vibrational levels. In particular, as
r increases towards the outer turning point, the ever-decreasing magnitude of |E − U(r)| steadily reduces
the rate of wavefunction curvature (i.e., the rate at which the magnitude of the slope |ψ′(r)| decreases), sothe function grows to a higher amplitude extremum before it turns over.
The preceding discussion illustrates the point that the properties of step-wise flat or “square-step”
potentials can serve as a good model system for understanding normal (smooth potential) systems, and
justifies our use of a model of this type for illustrating aspects of wavefunction normalization and boundary
conditions. Let us therefore consider the square-step potential energy function of Fig. 3.3:
Vstep(r) = ∞ for r < a (3.4)
= −De for a ≤ r < a+ L
= 0 for L ≤ rFrom elementary quantum mechanics we know that at energies E < 0 the wavefunctions of allowed levels
have the general form:
ψ(r) = 0 for r < a (3.5)
= A2 sin(k2 r) +B2 cos(k2 r) for a ≤ r < a+ L (3.6)
= A3 ek3 r +B3 e
−k3 r for a ≤ r < a+ L (3.7)
where k2 =√(2μ/�2)(De + E) , k3 =
√(2μ/�2)(−E) , and Ai and Bi are constants; note that placing
the energy zero at the potential asymptote makes the value of E a negative quantity for bound states.
Application of the usual boundary condition that the wavefunction be continuous at r=a , that it be both
continuous and smooth (i.e., have a continuous first derivative) at r=a+L , and that it die off as r →∞yields the result that the allowed eigenvalues must satisfy the equation
(De + E)/E = tan−1(√
2μ/�2 (De + E)L2)
(3.8)
and the wavefunction of the third-lowest level which satisfies this condition is shown in Fig. 3.3. However,
the question of why the wavefunction is required to be both continuous and smooth at r=a+L , but only
to be continuous at r=a , is usually not discussed. Also, the requirement that ψ(r)=0 for r<a is usually
presented as a given, justified (if at all) by the physical argument that the particle cannot penetrate an
infinitely high potential barrier. We shall re-examine these conditions in order to present a more systematic
justification which will be useful in more general cases.
Let us start by considering the nature of the wavefunction in the region r > a+L . The general solution
in this region, as given by Eq. (3.7), consists of an exponentially increasing and an exponentially decreasing
term, and it is the requirement that acceptable solutions must be “normalizable” or “square integrable”,
i.e., that
〈ψ(r)|ψ(r)〉 ≡∫ ∞0
ψ∗(r)ψ(r) dr = 1 (3.9)
which introduces the condition that A3 = 0 , and hence that the solution for r > a + L must be a pure
decreasing exponential function.
38
a a+r→
-De
De
0.0
∞
V(r)
L
Figure 3.3. Step function model potential and sample eigenfunctions.
Let us now consider the nature of the wavefunction in the region r < a . If instead of being infinitely
high in this region, the potential had the large positive value +V1, the general form of the wavefunction
here would be similar to that of Eq. (3.7):
ψ(r) = A1 ek1 r +B1 e
−k1 r (3.10)
where k1 =√(2μ/�2)(V1 − E) , and again A1 and B1 are constants. In the limit when the potential
energy in this region V1 → ∞ , it is clear that for any positive value of r the second exponential term in
Eq. (3.10) will go to zero, while the first one will blow up and become singular. Thus, if the normalizability
requirement of Eq. (3.9) is to be satisfied, necessarily A1=0 , in which case one is left with the result that
ψ(r)=0 throughout this interval.
Another way of obtaining this result is in terms of the properties of Eq. (3.3). On the interval r<a , as
V1 →∞ the rate of curvature away from the axis will necessarily approach infinity unless the wavefunction
itself is identically zero. Since a wavefunction whose values grew away from the axis infinitely rapidly could
not be square integrable, it is merely the wavefunction normalization condition which requires than ψ(r)
be identically zero on this interval.
The preceding discussion explains the nature of the wavefunction on the intervals r < a and r > (a+L) ,
and the requirement that acceptable wavefunction must be continuous everywhere explains why the various
wavefunction segments must meet at the internal boundaries r=a and (a+ L). However, a question still
to be answered is why ψ(r) is required to be smooth (i.e., have a continuous first derivative) at r=(a+L) ,
but not at r=a . To resolve this question, let us consider the nature of the wavefunction solution on the
narrow interval [(a+ L)− δ , (a+ L) + δ] .
(. . . argument to be completed . . . )
39
The above discussion illustrates the fact that the “boundary conditions” whose invocation gives rise
to quantization of eigenvalues are not independent conditions, but simply arise from the requirement that
acceptable solutions of our differential equation Eq. (3.1) or (3.3) must by continuous and normalizable, a
pair of requirements often joined under the label “well behaved”.
In conclusion, therefore, it is clear that the “boundary conditions” which are often introduced in an ad
hoc manner in elementary quantum mechanics treatments are not independent conditions, but rather are
conditions which may be simply derived from the requirements that ψ(r) must be a solution of Eq. (3.1)
and that it must be “well-behaved”, which is a shorthand way of saying normalizable and continuous.
3.3 Continuum Level Boundary Conditions and Normal-
ization
40
3.4 Quasibound Levels: Their Nature and Determination
We have seen that above the potential asymptote, the radial Schrodinger equation has allowed solutions
at all possible energies. In regions where E > V (r) those solutions will be oscillatory sinusoidal functions
whose frequency of oscillation (rate of curvature) is proportional to the local value of√E − V (r). However,
if the potential energy function protrudes above the asymptote at a distance outside the potential minimum,
as illustrated by the heavy dotted V (r) function shown in the insert panel of Fig. 3.4, these solutions have
somewhat more complicated behaviour. In particular, in the ‘classically allowed’ regions where E > V (r)
they will be the conventional oscillatory functions which for ‘step-function’ potentials are exactly defined
as some linear combination of sine and cosine functions. In contrast, in the ‘classically forbidden’ region
underneath the barrier (where E < V (r) ), the solution will be some linear combination of decreasing and
increasing exponential functions
ψbarr(r) = B eκ r + C e−κ r , (3.11)
where for a step-function potential, κ =√
(2μ/�2)(V (r)− E) .
The outer panel of Fig. 3.4 shows wavefunction at a range of collision energies for a particle of mass
μ = 1 [u] moving on the step function potential shown in the figure insert. These functions are readily
determined by matching the values and slopes of the sinusoidal and exponential-type solutions at the
0 1 2 3 4
1208
1218
1228
1198
1178
1188 barrier
energy/cm
−1
r /Å
0 1 2 3 40
500
1000
1500
2000V(r)
Figure 3.4. Illustration of a broad “quasibound level” in a step function potential.
41
potential step positions, and then normalizing the resulting functions to have unit asymptotic amplitude.
It is immediately clear that aside from shifts in node position, the wavefunction solutions at distances past
the barrier are qualitatively similar at all energies. Moreover, the nodal structure of the wavefunctions
over the potential well changes very little with energy. However, the amplitude of the wavefunction in the
well region clearly shows a very dramatic dependence on energy.
The reason for this dramatic variation of the wavefunction’s ‘internal amplitude’ with collision energy
may be understood by considering the nature of those solutions in the barrier region. If the phase of the
inner-well wavefunction when it reaches the barrier is such that it smoothly joins with a pure decreasing
exponential function when it reaches the barrier (i.e., if B = 0 in Eq. (3.11)), then one obtains the maximum
possible amplitude reduction as the wavefunction propagates outward across the barrier. Although the
positions of the wavefunction nodes and extrema in the well region vary relatively slowly with energy,
the behaviour of the wavefunction in the barrier region is a very sensitive function of the precise phase
of the wavefunction at the matching point. In particular, even for very small values of parameter B,
the exponential growth of the first term in Eq. (3.11) with increasing r quickly brings it to dominate the
wavefunction. Since the normalization of continuum wavefunctions is defined by their oscillatory amplitude
at large distances, this means that they will have significant amplitude over the well only for very narrow
windows of energy.
As will be discussed in Chapter 6, the intensity of bound↔continuum spectroscopic transitions involving
potential energy functions with barriers usually depends mainly on wavefunction overlap in the inner-
well region. Thus, the behaviour discussed above will give rise sharp intensity maxima in the energy-
dependence of continuum spectra. When the amplitude ratios are sufficiently larger, the resulting peaks
in the continuum spectra become indistinguishable from lines associated with discrete transitions between
truly bound levels. As a result, the energies associated with these internal amplitude maxima have come
Figure 3.5. Illustration of bound and quasibound levels of HgH, and the centrifugally distorted potentials
supporting them (Figures taken from Herzberg [1]).
42
to be known as “quasibound levels” of the supporting potential. The case of HgH illustrated in Fig. 3.5 is
a classic example of this behaviour; in this case, all of the levels at energies above the horizontal dashed
line near 1000 cm−1 are quasibound.
The example presented in Fig. 3.4 is quite simplistic, in that the barrier width is a constant at all
energies. In contrast, in realistic physical situations such as those illustrated by Fig. 3.5, barrier widths are
usually much greater at low energies. Moreover, the fact that the magnitude of the exponent coefficient κ
of Eq. (3.11) is proportional to the effective mass√μ means that the variation of inner-well amplitude with
energy will be much more dramatic for species with large reduced mass. Nonetheless, even for diatomic
hydrogen, for which μ ≈ 0.5 [u], the amplitude growth associated with propagation of the wavefunction
inward from the asymptotic region is usually by many orders of magnitude.
43
3.4.1 How To Locate Quasibound Levels?
The most efficient methods for calculating the widths or predissociation lifetimes of quasibound levels are
semiclassical methods, and will be discussed in Chapter 5.
44
3.4.2 Another Viewpoint: Quasibound Levels and Phase Shifts
Figure 3.6. Phase shifts as functions of energy showing resonance structure associated with quasibound
levels. [1]).
45
3.5 Exact Numerical Calculations for an Arbitrary
Potential: Program LEVEL
For the present discussion, it is convenient to rewrite Eq. (3.1) in the form
d2 ψ(r)
dr2=
2μ
�2[U(r)− E]ψ(r) =
[U(r)− E]ψ(r) (3.12)
We wish to approximate the solution to this equation by a finite difference expression, replacing the
continuous functions U(r) and ψ(r) by their values Ui = U(ri) and ψi = ψ(ri) on a finite grid of equally
spaced radial distances
ri = rmin + (i− 1)h i = 1, 2, 3, . . . , N − 1, N (3.13)
Adopting the notation
ψ[n]i ≡ dn ψ
drn
∣∣∣∣r=ri
(3.14)
the function ψ(r) may be expanded as a Taylor series about any given point, say r = ri . Consider the
resulting expansions for the two mesh points adjacent to ri :
ψi+1 =∑∞
k=0hk
k! ψ[k]i = ψi + hψ
[1]i +
h2
2!ψ[2]i +
h3
3!ψ[3]i +
h4
4!ψ[4]i + . . . (3.15)
ψi−1 =∑∞
k=0(−h)kk! ψ
[k]i = ψi − hψ
[1]i +
h2
2!ψ[2]i − h3
3!ψ[3]i +
h4
4!ψ[4]i + . . . (3.16)
If these expressions are added together, all of the odd-order terms cancel out and we obtain
ψi+1 + ψi−1 = 2∑k=0
h2k
(2k)!ψ[2k]i = 2ψi + h2 ψ
[2]i +
h4
12ψ[4]i + . . . (3.17)
Neglecting terms of order higher than two and using Eq. (3.12) to define ψ[2]i then yields the wavefunction
propagation formula
ψi+1 = 2ψi − ψi−1 + h2[Ui − E
]ψi (3.18)
for which the error term is approximately (h4/12)ψ[4]i . This allows us to obtain a value for the wavefunction
at mesh point ri+1 from a knowledge of its values (ψi and ψi−1) at two adjacent grid points. If one has
suitable wavefunction starting values, ψ1 and ψ2, and values of the potential energy function Ui are known
at all grid points, propagation of this formula will yield a set of wavefunction values spanning the whole
interval of interest.
The propagation formula of Eq. (3.18) is very straightforward to derive, but the error term associated
with each step is proportional to (h4/12)ψ[4]i . An analogous, but much more accurate integration formula
expressed in terms of the auxiliary function values
Yi =
[1 − h2
12
(Ui − E
)]ψi (3.19)
is
Yi+1 = 2Yi − Yi−1 + h2(Ui − E
)ψi . (3.20)
This propagation formula, generally known as the Numerov procedure [8], has much better convergence
properties that does Eq. (3.18), since the leading error term is −(h6/240)ψ[6]i [9]. Moreover, the resulting
46
Yi values are readily converted to the associated wavefunction values ψi. Of course decreasing the step
size means that more steps will be required to cross a given interval, and it has been shown that the
cumulative error associated with solving the Schrodinger equation using this approach is in fact proportional
to h4.[10] Nonetheless, the Numerov propagation approach is probably the most widely used wavefunction
propagation scheme used for solving the radial Schrodinger equation, and it is the core procedure in program
Level (discussed below).
While Eq. (3.20) allows us to propagate ψi across any interval on which the potential energy function
is known, making practical use of it requires us to have realistic initial wavefunction values at two mesh
points. In practice this is usually not a very difficult problem. Since potential energy functions always
grow very rapidly at small internuclear distances, . . . . . .
Energy
i h di i l S h di i iFigure 3.7. Propagated wavefunctions at energies slightly below (Etrial < En), slightly above (Etrial >
En), and very close to (Etrial ≈ En) the exact v = 1 eigenvalue for this potential energy function.
47
3.0 3.5 4.0 4.5 5.0 5.5-100
-90
-80
-70
-60
-50
-40
R /A
energy/ cm-1
1st trial energy
2nd trial energy
4th trial energy
converged energy
V(R)
Figure 3.8. Wavefunctions at the first three trial energies, converging on the v=1 eigenvalue.
48
Figure 3.9. Mesh size dependence of the error in numerically calculated eigenvalues of a Morse model
potential for HCl in reduced units (1unit = 61.4557 cm−1), from Ref. [11].
49
3.6 Practical Bound-State Calculations: Program LEVEL
LEVEL is a general purpose computer program that can numerically solve the radial Schrodinger equation
to determine the eigenvalues and eigenfunctions of virtually any radial one-dimensional potential energy
function, and calculate expectation values or overlap matrix elements of any specified properties of interest.
It is written in Fortran, and the source code and a user manual may be freely downloaded through the
“Computer Programs” link at http://leroy.uwaterloo.ca.
People who have an account on the Waterloo theoretical chemistry compute server scienide2.uwaterloo.ca
may perform LEVEL calculation by preparing an appropriate input data file to specify the physical sys-
tem, the potential energy function(s), and the properties to be calculated, and storing it as a filename with
the suffix .5 (e.g., nameRb2.5). The command
rlev nameRb2
will then cause the program to run using that specified input data file. The main output will be written
to the file nameRb2.6, while optional (depending on the input instructions) supplementary output files are
written with filenames nameRb2.7, nameRb2.8, nameRb2.6, . . . etc.
The present section describes a number of features of program LEVEL and some practical consid-
erations associated with its use. First of all, we note that the core calculations are based on use of the
Numerov wavefunction propagator described in § 3.5. It it therefore clear that a user must specify values
the inner and outer bounds of the range of numerical integration, RMIN and RMAX, respectively, as well as
the radial mesh or stepsize RH = h. As discussed earlier, plausible initial values of the first two of these
parameters would be RMIN ≈ 12 re and RMAX ≈ 2re − 3 re , where re is the position of the potential energy
minimum. If the latter is too small or the former is either too large or too small, the program will print
warnings advising the user to make appropriate changes. In almost all practical cases, while making RMAX
‘excessively’ large leads to larger CPU times, it does not affect the accuracy or stability of the calculations.
It is convenient to characterize the mesh size to use for any particular physical problem by specifying
the number of radial mesh points between the most closely spaced wavefunction nodes of the highest energy
level being treated. As a result, we may define
RH =π
Nn
(2μ max{E−V (r)}
�2
)1/2 =π
Nn
(μ max{E−V (r)}
16.85762920
)1/2 (3.21)
in which max{E − V (r)} is the maximum (positive) value of the radial kinetic energy for the highest
vibrational level of interest, and the last version of thus expression assumes that the units of energy,
distance and mass are cm−1, A and amu, respectively. In general max{E − V (r)} ≤ De , where De is the
potential well depth, and it is often convenient to simply set max{E−V (r)} = De in Eq. (3.21). Experience
has shown that ‘stable’ eigenvalue calculations are generally achieved with Nn ≈ 15−20. However, a user’s
particular application may require much higher precision than is obtained with this choice of Nn, so the
user should always experiment by examining the results obtained with increasing values of Nn until the
desired precision is achieved. As discussed in § 3.5, the cumulative error of a Numerov-based eigenvalue
calculation scales as the fourth power of the radial integration stepsize, h4.
The potential energy function(s) to be used in any particular LEVEL calculation may be defined
either by an analytic function or by some array of points. In particular, the current version of the code
can generate eight different types of analytic potential energy functions, including most of those described
in §2.3. A user can also readily replace the normal program subroutine POTGEN with a subroutine of
their own to generate any other desired analytic function (see program manual for details [12]). In many
cases, however, our knowledge of the potential function for a given molecular state consists of an array of
50
potential function values at particular radial distances. This is the form of potential functions obtained
from electronic structure calculations or from the semiclassical “RKR” inversion procedure to be discussed
in §5.3.The Numerov wavefunction propagator requires a knowledge of the potential energy function on the
dense array of equally spaced distances defined by the input parameters RMIN, RMAX amd RH. For analytic
potential functions it is quite straightforward to generate this required array of potential function values.
However, the sets of potential function values obtained from electronic structure calculations or the RKR
inversion procedure are generally not located at equally spaced distances, and are reported on a much less
dense grid than is required for an accurate Numerov propagation calculation. Thus, for such ‘pointwise’
potential, the program must interpolate over the given set of potential energies to obtain the dense grid of
values required by the wavefunction propagation procedure. Moreover, those input points usually span a
narrower range of radial distances than is required for properly converged eigenvalue calculations, so it is
necessary to perform some realistic extrapolation in regions outside the domain of the given set of potential
function values.
Program LEVEL offers two main choices regarding how the numerical interpolation over a given set
of potential function values is to be performed. The first, invoked by setting input parameter NUSE = 0 , is
to define the required array of potential function values by performing a cubic spline interpolation through
the complete set of NTP input potential function values. Unfortunately the third derivative of a cubic spline
function is discontinuous at each of the NTP input turning-point distances, and their higher derivatives are
all zero everywhere, so this approach is not ideal. However, it may be the best choice when the precision
and smoothness of the input points is limited, as is often the case with ab initio potential function values.
The second type of interpolation procedure, invoked be setting NUSE > 0 , involves performing piecewise
polynomial interpolation. In this case, interpolation to determine a potential function value at a specified
distance r would first identify the NUSE/2 given turning points to the left and right of this specified distance,
and then define the desired potential function value V (r) in terms of an order-(NUSE−1) polynomial through
those NUSE selected points. In contrast with spline interpolation, with such piecewise polynomials there
is no absolute assurance that the first and second derivatives of the resulting function will be smooth at
each of the input point distances. However, use of NUSE = 6, 8 or 10 ensures the existence of high-order
derivatives, and if the input points are numerically very smooth (as is the case with ‘RKR’ turning points,
see § 5), this approach is usually the best.
One further consideration is the fact that the very steep nature of the short-range repulsive wall of
most potentials presents challenges to any numerical interpolation scheme, since the ordinate (potential
function) values change very rapidly in a region where the abscissa (radial distance) values are very close
together. A technique that has been found useful for addressing this problem is to actually perform the
interpolation over the modified array of function values (ri)2 V (ri) , and then divide the resulting function
value by r2 to obtain the desired V (r) value. Program LEVEL performs the interpolation this way if the
user sets input parameter IR2 > 0 .
One troubling feature associated with interpolation over pointwise potentials it that one cannot defini-
tively determine a unique ‘best method’. In general, numerical results obtained using different interpolation
methods will always differ from one another, at some level of precision. While it may be possible to rule out
some methods, in the end there will always be some level of what is called “interpolation noise” associated
with the final calculated results. An illustration of such problems is provided by Fig. 3.10, which presents
the results of a study of the effect of interpolation noise on the calculation of vibrational eigenvalues for
the ground state of H2 using the best available (in 1968) 87-point ab initio clamped nuclei potential energy
function [13]. The left-hand side of this figure shows the error in the interpolated value obtained if each
of the given function values is omitted, one at a time, and an interpolation over the remaining points used
51
Figure 3.10. ‘Interpolation noise’ in numerical eigenvalue calculations for ground state H2 [14]. Left Seg-
ment: error in a “known” potential function value obtained when on omitting it from the input array
and using piecewise polynomial and other interpolation schemes to predict it. Solid lines are for inter-
polation over the potential values themselves and dashed lines for interpolation over r2 V (r). Right
Segment: discrepancies in vibrational eigenvalues calculated using various interpolation schemes,
relative to results obtained from piecewise 8-point (7th order) interpolation over r2 V (r).
to estimate the potential function there. The main resulting conclusions are the fact that cubic-spline
interpolation is less reliable than the use of 6-point, 8-point or 10-point piecewise polynomials, and that
interpolating over r2 V (r) (dotted lines) rather than over the potential function itself (solid lines) yields
much more reliable interpolation in the short-range region.
The right-hand side of Fig. 3.10 shows the differences of the eigenvalues calculated by eight other
schemes from those obtained using 8-point piecewise polynomial interpolation over the quantities ri2 V (ri)
generated from the given potential function values. We see here that the results obtained for 6-point
interpolation over r2 V (r) and 8- and 10-point interpolation over either r2 V (r) or V (r) itself yield results
which agree among themselves to within approximately ±0.02 cm−1. It was therefore concluded that
the accuracy of vibration/rotation eigenvalues calculated from this potential could never be better than
52
±0.02 cm−1. These results illustrates the fact that independent of other considerations, unavoidable
‘interpolation noise’ places a limit to the precision obtainable in numerical calculations using pointwise
potential energy functions. It is therefore clear that anyone working with such potentials should perform
Table 3.1 Input data file structure for program LEVEL. For detailed parameter definitions see §4 of theprogram manual [12].
#1 READ(5,*,END=999) IAN1, IMN1, IAN2, IMN2, CHARGE, NUMPOT
#2a IF(IAN1.LE.0) READ(5,*) NAME1, MASS1
#2b IF(IAN2.LE.0) READ(5,*) NAME2, MASS2
#3 READ(5,*,END=999) TITL
#4 READ(5,*) RH, RMIN, RMAX, EPS
DO IPOT= 1, NUMPOT
#5 READ(5,*) NTP, LPPOT, OMEGA, VLIM
IF(NTP.GT.0) THEN
#6 READ(5,*) NUSE, IR2, ILR, NCN, CNN
#7 READ(5,*) RFACT, EFACT, VSHIFT
#8 READ(5,*) (XI(I), YI(I), I= 1,NTP)
ELSE
#9 READ(5,*) IPOTL, PPAR, QPAR, NSR, NLR, NCMM, IBOB
#10 READ(5,*) DSCM, REQ, Rref
#11 IF(IPOTL.GE.4) READ(5,*) (MMLR(I), CMM(I),I= 1, NCMM)
#12 IF(NVARB.GT.0) READ(5,*) (PARM(I), I=1,NVARB)
IF(IBOB.GT.0) THEN
#13 READ(5,*) MN1R, MN2R, PAD, QAD, NU1, NU2, PNA, NT1, NT2
#14 IF(NU1.GE.0) READ(5,*) U1INF, (U1(I), I=0,NU1)
#15 IF(NU2.GE.0) READ(5,*) U2INF, (U2(I), I=0,NU2)
#16 IF(NT1.GE.0) READ(5,*) T1INF, (T1(I), I=0,NT1)
#17 IF(NT2.GE.0) READ(5,*) T2INF, (T2(I), I=0,NT2)
ENDIF
ENDIF
ENDDO
#18 READ(5,*) NLEV1, AUTO1, LCDC, LXPCT, NJM, JDJR, IWR, LPRWF
NLEV= MAX(1,NLEV1)
#19a IF(AUTO1.GT.0) READ(5,*) (IV(I), IJ(I), I= 1,NLEV)
#19b IF(AUTO1.LE.0) READ(5,*) (IV(I), IJ(I), GV(I), I= 1,NLEV)
IF((LXPCT.NE.0).AND.(LXPCT.NE.-1)) THEN
#20 READ(5,*) MORDR, IRFN, DREF
#21 IF((IABS(IRFN).LE.9).AND.(MORDR.GE.0)) READ(5,*) (DM(J), J= 0,MORDR)
IF(IRFN.GE.10) THEN
#22 READ(5,*) NRFN, RFLIM
#23 READ(5,*) NUSEF, ILRF, NCNF, CNNF
#24 READ(5,*) RFACTF, MFACTF
#25 READ(5,*) (XIF(I), YIF(I), I= 1,IRFN)
ENDIF
ENDIF
IF(IABS(LXPCT).GE.3) THEN
#26 READ(5,*) NLEV2, AUTO2, J2DL, J2DU, J2DD
#27a IF(AUTO2.GT.0) READ(5,*) (IV2(I),I= 1,NLEV2)
#27b IF(AUTO2.GT.0) READ(5,*) (IV2(I),GV(I),I= 1,NLEV2)
ENDIF
53
tests analogous to those illustrated by Fig. 3.10 to ascertain the effect of interpolation noise on their results.
The table on the previous page summarizes the structure of the input instruction file for execution of
program LEVEL. Fortunately, only a fraction of the possible input parameters are usually needed in any
particular case. Details regarding the various Read statements and the meaning of the input parameters
are presented in the program manual [12]. However, additional comments about some of the parameters
input through Read’s #5 and 6 may be helpful.
Most of the analytic potential energy function forms which may be invoked as options in program
LEVEL express the potential energy function relative to the energy of its asymptote. To put calculated
results on some absolute energy scale, it is therefore necessary to specify a fixed absolute energy for that
asymptote. This is the quantity input through Read #5 as parameter VLIM (in units cm−1). If a user
wishes to set the zero of energy at the potential minimum, then they should set VLIM = De , where De is
the potential well depth; if they wish the absolute energy to lie at the potential asymptote they should set
VLIM = 0. A value of VLIM must also be specified for a pointwise potential, and parameter VSHIFT of Read#7 allows a user to shift energies of the actual given potential function values to make them consistent
with the chosen reference energy for that state, VLIM. For calculations involving two different potential
energy functions, it is important that the correct relative energies of their asymptotes be reflected in the
two input values of VLIM.
Finally, we note that the available set of known function values defining a given pointwise potential
may not span the entire radial range from RMIN to RMAX required for the desired calculations. When this
occurs, LEVEL will automatically extrapolate to fill in the range. In particular, if the domain of the
given potential function values does not extend all the way in to RMIN, LEVEL extrapolates inward with
an exponential function fitted to the three innermost turning points. Similarly, if that domain does not
extend all the way to RMAX, LEVEL extrapolate outward towards the specified asymptote VLIM using the
type of analytic function specified by the user through parameters ILR, NCN and CNN that are input through
Read #6.
3.7 Effect of Rotation on Vibrational Levels:
Centrifugal Distortion Constants
Figures 2.3 and 3.5 and the associated discussion shows that the the effect of molecular rotation on
vibrational energy levels occurs through the inclusion of the centrifugal potential [J(J + 1)]�2/(2μr2) in
the overall effective potential energy function VJ(r) of Eq. (2.30). This centrifugal potential raises the
energy of the inner portion of the potential well by more than it does the outer, which has the net effect of
shifting the average bond length for a given level to larger r, as well as increasing the level energy. However,
this viewpoint differs from the conventional picture of rotational energies as fine structure associated with
each vibrational level that is exemplified by the conventional “band constant” vibration-rotation level
energy expression
E(v, J) = Gv + [J(J + 1)]Bv − [J(J + 1)]2Dv + [J(J + 1)]3Hv + [J(J + 1)]4Lv + . . .
≡∑m=0
Km(v) [J(J + 1)]m (3.22)
Here all of Bv, Dv , Hv, Lv, . . . etc., may be called “rotational constants”, but it is more precise to
refer to Bv as the “inertial rotational constant” and the others as “centrifugal distortion constants”. The
combination of all the rotational constants for a given vibrational level with the associated pure vibrational
energy Gv comprise the “band constants” for that level.
54
It has long been customary to determine estimates of the values of the band constants associated with
the vibrational levels of a given potential energy function from empirical fits to experimental data. However,
in the 1970’s it became widely understood that together with the vibrational energies Gv, the values of the
inertial rotational and centrifugal distortion constants are all properties of the potential energy function
that may readily be calculated from solutions to the radial Schrodinger equation:
− �2
2μ
d2 ψ(r)
dr2+
[V (r) +
J(J + 1)�2
2μr2
]ψ(r) = E ψ(r) (3.23)
It had long been known that if the centrifugal potential was treated as a perturbation
H ′(r) = [J(J + 1)]�2/(2μr2) ≡ [J(J + 1)]H ′(r) (3.24)
the inertial rotation constant Bv was defined by the first-order perturbation energy [1]:
ΔE(1)vJ =
⟨ψ(0)j (r)
∣∣∣∣J(J + 1)�2
2μr2
∣∣∣∣ψ(0)v
⟩=
J(J + 1)�2
2μ
⟨ψ(0)j (r)
∣∣∣∣ 1r2∣∣∣∣ψ(0)
v
⟩
= J(J + 1)
[�2
2μ
⟨1
r2
⟩v
]≡ J(J + 1)Bv (3.25)
in which the zero’th order wavefunctions ψ(0)v (r) are the eigenfunctions of the Schrodinger equation (3.23)
for J = 0. Then in 1973 Albritton et al. [15] pointed out that the centrifugal distortion constants may
be defined in terms of the analogous higher-order perturbation energies. In particular, straightforward
application of the textbook expressions yielded by Rayleigh-Schrodinger perturbation theory yields the
expressions
Dv =
(�2
2μ
)2 ∑u �=v
∣∣∣⟨ψ(0)u (r)
∣∣ 1r2
∣∣ψ(0)v (r)
⟩∣∣∣2E
(0)u − E(0)
v
(3.26)
Hv =
(�2
2μ
)3 ∑u �=v
∑t�=v
⟨ψ(0)v
∣∣ 1r2
∣∣ψ(0)u
⟩⟨ψ(0)u
∣∣ 1r2
∣∣ψ(0)t
⟩⟨ψ(0)t
∣∣ 1r2
∣∣ψ(0)v
⟩(E
(0)u − E(0)
v )(E0)t − E(0)
v )(3.27)
− Bv
(�2
2μ
)2 ∑u �=v
∣∣∣⟨ψ(0)u
∣∣ 1r2
∣∣ψ(0)v
⟩∣∣∣2(E
(0)u − E(0)
v
)2Lv = −
(�2
2μ
)4 ∑u �=v
∑t�=v
∑s �=v
⟨ψ(0)v
∣∣ 1r2
∣∣ψ(0)u
⟩⟨ψ(0)u
∣∣ 1r2
∣∣ψ(0)t
⟩⟨ψ(0)t
∣∣ 1r2
∣∣ψ(0)s
⟩⟨ψ(0)s
∣∣ 1r2
∣∣ψ(0)v
⟩(E
(0)u − E(0)
v )(E0)t − E(0)
v )(E0)s − E(0)
v )(3.28)
+ 2Bv
(�2
2μ
)3 ∑u �=v
∑t�=v
⟨ψ(0)v
∣∣ 1r2
∣∣ψ(0)u
⟩⟨ψ(0)u
∣∣ 1r2
∣∣ψ(0)t
⟩⟨ψ(0)t
∣∣ 1r2
∣∣ψ(0)v
⟩(E
(0)u − E(0)
v
)2 (E
0)t − E(0)
v
)− Bv
2
(�2
2μ
)2 ∑u �=v
∣∣∣⟨ψ(0)u
∣∣ 1r2
∣∣ψ(0)v
⟩∣∣∣2(E
(0)u −E(0)
v
)3 + Dv
(�2
2μ
)2 ∑u �=v
∣∣∣⟨ψ(0)u
∣∣ 1r2
∣∣ψ(0)v
⟩∣∣∣2(E
(0)u − E(0)
v
)2As discussed earlier, for any given potential energy function V (r) it is a routine matter to solve the
radial Schrodinger equation to determine any or all of the eigenvalues E(0)v and eigenfunctions ψ
(0)v (r) of
55
Figure 3.11. Comparison of Dv and Hv values for the B 3Π+0u state of I2 calculated from Eq. (3.27) while
neglecting the continuum (dashed curves labelled P), with values calculated by “energy-derivative
fitting” [16].
the ‘unperturbed’ system defined by Eq. (3.23) with J = 0. As a result, it is in principle a straightforward
matter to calculate the matrix elements of 1/r2 appearing in Eqs. (3.26)–(3.28) and then evaluate the as-
sociated sums. However, all of these sum implicitly include the continuum of states above the potential
asymptote, as well as all of the bound states. While the evaluation of matrix elements involving continuum
state wavefunctions introduces no great difficulties (see Chapter 6), performing the integration over all
possible continuum energies may be quite tedious. Fortunately, neglect of contributions from the contin-
uum has have little effect on calculated distortion constants for the lower vibrational levels. However,
it introduces ever increasing errors for vibrational levels approaching the top of the potential well. This
problem is illustrated by Fig. 3.11 which compares Dv and Hv values calculated from Eqs. (3.26) and (3.27)
with those obtained by a more robust (though somewhat ad hoc) method. It is clear that neglect of the
‘downward pressure’ from continuum levels means that perturbation theory calculations that neglect the
continuum will be increasingly in error for levels lying in the upper portion of a potential well. More-
over, the increasing complexity of Eqs. (3.26)–(3.28) discourages one from extending this approach to yet
higher-order distortion constants.
Fortunately, a better quantum mechanical approach for the calculation of centrifugal distortion con-
stants has been developed which is computationally more efficient, and avoids the shortcomings associated
with the use of Eqs. (3.26)–(3.28). In Rayleigh-Schrodinger perturbation theory the wavefunction correc-
tions obtained in first-order ψ(1)v (r), second-order ψ
(2)v (r), third-order ψ
(3)v (r), . . . etc., are expressed as
expansions in a basis set consisting of the zero-th order wavefunctions ψ(0)v (r), which are assumed to form
a complete set:
ψ(k)v (r) =
∑v′
a(k)v′ ψ
(0)v′ (r) . (3.29)
It is the fact that the need for completeness requires inclusion of the continuum and the clutter of multiple
layers of summation which make the above approach unwieldy. However, in 1981 Hutson pointed out that
one may solve for the perturbed wavefunctions directly, without having to resort basis set expansions. In
56
particular, he showed that the first-, second- and third-order wavefunction corrections are defined as the
solutions of the linear inhomogeneous differential equations
− �2
2μ
d2 ψ(1)v (r)
dr2+[V (r)− E(0)
v
]ψ(1)v (r) =
(Bv − �
2
2μr2
)ψ(0)v (r) (3.30)
− �2
2μ
d2 ψ(2)v (r)
dr2+[V (r)− E(0)
v
]ψ(2)v (r) =
(Bv − �
2
2μr2
)ψ(1)v (r)−Dv ψ
(0)v (r) (3.31)
− �2
2μ
d2 ψ(3)v (r)
dr2+[V (r)− E(0)
v
]ψ(3)v (r) =
(Bv − �
2
2μr2
)ψ(2)v (r)−Dv ψ
(1)v (r) +Hv ψ
(0)v (r) (3.32)
In Eq. (3.30), the function on the right hand side of the equation is known, since the zero’th order solutions
for the unperturbed system of Eq. (3.23) are readily obtained. Similarly, in Eqs. (3.31) and (3.32) the
quantities on the right-hand side of the equation are known, since the each of the lower-order perturbation
functions ψ(1)v (r) and ψ
(2)v (r) is, in turn, defined by the preceding equation. It turns out that linear
inhomogeneous equations of this type may be readily solved by standard numerical methods, on of which
is a version of the Numerov propagator method used for solving the regular radial Schrodinger equation of
Eq. (3.23) [17].
Once one has obtained the wavefunction perturbations of various orders by solving Eqs. (3.30)–(3.32),
it is a straightforward matter to generate expressions for the centrifugal distortion constants [18]:
Dv = −⟨ψ(0)v (r)
∣∣∣H ′ −Bv
∣∣∣ψ(0)v (r)
⟩(3.33)
Hv =⟨ψ(1)v (r)
∣∣∣H ′ −Bv
∣∣∣ψ(1)v (r)
⟩+ 2Dv
⟨ψ(0)v (r)|ψ(1)
v (r)⟩
(3.34)
Lv =⟨ψ(1)v (r)
∣∣∣H ′ −Bv
∣∣∣ψ(2)v (r)
⟩+ Dv
(⟨ψ(1)v (r)|ψ(1)
v (r)⟩+⟨ψ(0)v (r)|ψ(2)
v (r)⟩)
(3.35)
− 2Hv
⟨ψ(0)v (r)|ψ(1)
v (r)⟩
Mv =⟨ψ(2)v (r)
∣∣∣H ′ −Bv
∣∣∣ψ(2)v (r)
⟩+ 2Dv
⟨ψ(1)v (r)|ψ(2)
v (r)⟩
(3.36)
− Hv
(2⟨ψ(0)v (r)|ψ(2)
v (r)⟩−⟨ψ(1)v (r)|ψ(1)
v (r)⟩)− 2Lv
⟨ψ(0)v (r)|ψ(1)
v (r)⟩
Moreover, the complexity of this approach is relatively modest (compared to that associated with Eqs. (3.26)–
(3.28), and it is a relatively straightforward matter to use the third-order wavefunctions to generate anal-
ogous expressions for Nv and Ov , or to generate the fourth-order wavefunction perturbation and use it to
generate even higher-order constants.
Why does this matter ?
57
3.8 Vibrational Spacing Patterns
and Molecular Dissociation Energies
υ=6
5
4
3
2
1
0
V(r)
D?
⇓
⇑
A
⎥⎥⎥⎥
?0 5 10 15 20 250
100
200
300
400
500
υ
ΔGυ+½
B
Figure 3.12. Illustrative application of a Birge-Sponer plot (B) for determining the molecular
dissociation energy.
58
Exercises
3.1 For each of the four potentials: (i) the harmonic oscillator, (ii) the quartic oscillator V (r) =
K(r − re)4 , (iii) the Coulomb potential, and (iv) the Morse potential, determine an analytic
expression showing how the width of the well (i.e., the distance between the outer and inner walls)
depends on energy, and illustrate your results in a plot. Comment on the relationship between your
results and those shown in Fig. 3.1
3.2 Could there exist a potential for which a Birge-Sponer plot with initial positive slope would have
positive curvature? Justify your conclusion.
3.3 Consider the step function potential: Vstep(r) = ∞ for r < 0
= 0 for 0 ≤ r < L
= V0 for L ≤ r < 2L
= ∞ for 2L ≤ r
a) At energies E<V0 on the subinterval L ≤ r < 2L ,
(i) What is the form of the general solution to the radial Schrodinger equation?
(ii) What is the form of this solution after application of the outer boundary condition?
b) For a case in which the v=0 level lay at an energy of V0/5 , roughly estimate the energies of
levels v=1− 3 , and sketch their wave functions on a diagram of the potential energy function.
c) At energies E>V0 , under what condition (if any) might the oscillatory wave function amplitude
at r<L be greater or equal to that at r>L ? If never, explain why not.
d) If the properties of the system (i.e., the values of μ, V0 and L ) were such that the v=2 level
had an energy of exactly V0: (i) determine the analytic form of its wavefunction on the interval
L≤r<2L , and (ii) sketch the wavefunction at that energy on a potential energy disgram.
References
[1] G. Herzberg, Spectra of Diatomic Molecules (Van Nostrand, New York, 1950).
[2] R. T. Birge and H. Sponer, Phys. Rev. 28, 259 (1926).
[3] J. L. Dunham, Phys. Rev. 41, 713 (1932).
[4] J. L. Dunham, Phys. Rev. 41, 721 (1932).
[5] J. E. Kilpatrick and M. F. Kilpatrick, J. Chem. Phys. 19, 930 (1951).
[6] J. E. Kilpatrick, J. Chem. Phys. 30, 801 (1959).
[7] R. A. Aziz and M. J. Slaman, Mol. Phys. 58, 679 (1986).
[8] B. Numerov, Pubs. Observati=oire Central Astrophys. Russ. 2, 188 (1933).
[9] J. W. Cooley, Math. Computations 15, 363 (1961).
[10] I. H. Sloan, J. Comp. Phys. 2, 414 (1968).
59
[11] J. Cashion, J. Chem. Phys. 39, 1872 (1963).
[12] R. J. Le Roy, Level 8.0: A Computer Program for Solving the Radial Schrodinger Equation for
Bound and Quasibound Levels, University of Waterloo Chemical Physics Research Report CP-663
(2007); see http://leroy.uwaterloo.ca/programs/.
[13] W. Kolos and L. Wolniewicz, J. Chem. Phys. 49, 404 (1968).
[14] R. J. Le Roy and R. B. Bernstein, J. Chem. Phys. 49, 4312 (1968).
[15] D. L. Albritton, W. J. Harrop, A. L. Schmeltekopf, and R. N. Zare, J. Mol. Spectrosc. 46, 25 (1973).
[16] J. D. Brown, G. Burns, and R. J. Le Roy, Can. J. Phys. 51, 1664 (1973).
[17] K. Smith, The Calculation of Atomic Collision Processes (Wiley-Interscience, New York, 1971), Chap-
ter 4.
[18] J. M. Hutson, J. Phys. B: At. Mol. Phys. 14, 851 (1981).
60
Chapter 4
Transition Energies, Intensity Patterns,
and Selection Rules
4.1 Classical Theory
4.1.1 The wave nature of light
Although we know that the energy and momentum of light are quantized, one of the enduring mysteries
of nature is that many of its properties are best described using a wave theory. The classical electromag-
netic theory of radiation explains light as a wave phenomenon consisting of a coordinated combination of
oscillating electric and magnetic waves propagating through space with speed c. These oscillating fields
perturb the molecules which see them, and cause some molecules to undergo transitions from one energy
level to another.
The electric field may be represented as:
�E(�r, t) = �E0 cos(�k · �r − ω t+ φ0) (4.1)
and the complementary magnetic field as
�H(�r, t) = �H0 cos(�k · �r − ω t+ φ0) (4.2)
in which
• �k is a vector pointing in the direction of propagation of the light
• �E0 is a vector perpendicular to �k whose direction defines the plane of polarization of the light,
and whose magnitude defines its intensity
• �H0 ∝ �k× �E0 defines the plane of polarization of the oscillating magnetic field as being orthogonal
to both the direction of propagation of the light and the oscillating electric field.
The magnetic field of light is what drives transitions in NMR and ESR spectroscopy, but the present
discussion considers only the rotational, vibrational and electronic transitions driven by the oscillat-
ing electric field.
• ω = 2π ν [radians/sec] is the angular frequency of the light, where the oscillation frequency of the
electric field is ν [sec−1] or ν [hz].
• k = |�k| = 2π/λ [radians/m] is the wave vector of the light, where λ is its wavelength.
• The speed of propagation of the light wave is therefore c = ω/k = ν λ
61
0 2 4 6 8 10
field
viewed at
fixed point
in space
time /10-15 s
+E0
-E0
0
E(x,t)
oscillationperiod 1/ν
0.0 0.5 1.0 1.5 2.0 2.5 3.0
field
viewed at
fixed point
in time
distance /μm
+E0
-E0
0
E(x,t)
wavelength λ
Figure 4.1: The electric field of light oscillates in space and in time.
Since the speed of light in a vacuum c0 = 2.997 824 58×108 [m/s] is a constant of nature, its frequency
ν , wavelength λ , wavevector k and angular frequency ω are all interrelated, and any one of them will
uniquely define all of the others. However, another property which is very commonly used to to characterize
light is its wavenumber :
ν ≡ 1/λ = ν/c = {no. of wavelengths per unit distance} = {the inverse of the wavelength}The wavenumber of a give type of light is clearly proportional to its frequency, and hence also to the
magnitude of its energy quanta. This has led to its use as an energy unit. In a strict SI units universe,
the wavenumber would have units m−1 , the number of wavelengths per meter. However, the virtually
universal usage in spectroscopy is to express ν in units cm−1 (the no. of wavelengths which fits into
one cm). As a result, molecular constants describing the vibrational spacings, rotational energy levels and
bond dissociation energies of molecules are most often expressed in units cm−1.Most measurements are actually made in air or other media in which the actual speed of light c = c0/η
is slower than c0 (and hence λ = λ0/η , and ν = η ν0 ), where η = η(ν) is the refractive index of the
particular medium. However, they are always reported in vacuum wavenumbers ν0 or vacuum wavelengths
λ0 , after being corrected by the appropriate value of η . To reduce notational clutter the symbols c , λ
and ν , without the subscript ‘ 0 ’, are henceforth used to represent the speed, wavelength and wavenumber
of light in vacuum.
4.1.2 Classical Description of Light-Induced Transitions
Consider a molecule at some point in space with light shining on it. The oscillating electric (or magnetic)
field of the light will perturb its energy levels and may cause transitions. We consider only transitions in
lowest order of the electric field strength, “electric dipole transitions”.
What transitions are driven by the oscillating electric field of light ?
• We know that different types of repetitive physical motion (rotation, vibration) have different char-
acteristic frequencies (periods) associated with them.
62
↑vertical
componentof
moleculardipole
time→
+
+
+
−+
+ +
− −
−− −
directionof
dipole
orientationof
molecule
Figure 4.2: Behaviour of the vertical component of the dipole field of a polar diatomic molecule
rotating clockwise in the plane of the paper.
• In classical physics, a pendulum (or swing) has a characteristic period (or frequency) of oscillation.
If one provides a small push at regular intervals, precisely in phase with the natural motion, the
oscillating system will absorb energy and the amplitude of motion will increase.
• If one attempts to ‘drive’ such an oscillating system with out-of-phase impulses, no net energy is
picked up from the impulse source.
Thus, within this viewpoint, the frequency of the light required to excite particular types of physical motion
must match the natural frequency of that phenomenon.
A. Molecular Rotation
Consider a polar molecule such as HCl rotating in space. In the absence of any external applied forces it
will rotate with a constant angular velocity, and the angular momentum will remain constant. However:
• The component of the molecular dipole pointing in any given direction (say, in vertical direction in
the plane of the page in Fig. 4.2) will clearly oscillate sinusoidally with time.
• If the electric field of the light oscillates in phase with this natural sinusoidal motion, the rotating
energy can absorb light from that oscillating electric field which effectively “pushes” the molecule
periodically (like pushing a swing or pendulum).
• Rotational transitions are typically driven by relatively low energy light correspond to wavenumbers
of ca. 0.1 − 50 cm−1, and appear in what is called the “microwave” region of the spectrum. If such
transitions occur, the molecule is said to be “microwave active” or “rotationally active”.
• However, molecules with no permanent dipole moment (e.g., N2, CH4, CO2) have no dipole for the
oscillating electric field of the light to “push” against. Thus, they will have no (first-order) interaction
with the electric field of the light, and hence no absorption of energy will occur.
Conclude: molecules with no permanent dipole moment have no pure rotational spectrum, and hence are
“rotationally inactive” or “microwave inactive”, while molecules which do have permanent dipole moments
do have allowed pure rotational spectra. This is an fundamental physical “selection rule” of rotational
spectroscopy.1
1Exceptions to this rule can arise when “centrifugal distortion” distorts molecules such as CH4 and causes them
to have a weak dipole moment.
63
↑vertical
componentof
moleculardipole
time→
equilibriumdipole moment
+
− − − − − − − −
++
++
++
+
strengthof
dipole
lengthof
molecule
Figure 4.3: Dipole moment of a vibrating polar diatomic molecule whose alignment is fixed in space.
B. Molecular Vibrations
(i) A molecule (such as HCl) that has a permanent dipole moment whose magnitude changes as it vibrates
also provides something for the electric field of light to “push” against (see Fig. 4.3). If the frequency
of the light is in phase with the natural period of vibrational motion, the molecule can absorb energy
from the light field and become vibrationally excited.
This type of motion typically has natural frequencies in the range 3000 − 100 000 GHz, and hence
appears in the “infrared” region of the spectrum at light ‘energies’ of 100−4000 cm−1. We therefore
speak of molecules such as HCl and NaI as being “infrared active”, while N2, H2 and I2 are “infrared
inactive”.
Note that what matters is not the magnitude of the dipole moment, but rather the fact that it
changes with time.
symmetric stretch
antisymmetric stretch
bending mode
(ii) A molecule with no permanent dipole moment can still have an
allowed infrared (or allowed vibrational) spectrum if an instantaneous
dipole moment arises which oscillates during the course of its
vibrational motion.
Consider CO2, which has the three types of vibrational motions shown
in the Figure on the right.
In the course of the symmetric stretching vibrational motion, the
symmetry of the molecules is maintained. Hence, there is no oscillating
dipole for the electric field of light to push against, so this vibrational
mode is not infrared active.
CO O
O
OO
O C
C
δ−δ−
δ−
δ− δ−
δ−
2δ+
2δ+
2δ+
stretched
equilibrium
compressed
However, both the bending vibration and the asymmetric stretch vibrational mode, the distortion
of the structure which occurs during the course of the motion means that the bond dipoles will not
always cancel one another, so both of these modes are vibrationally or infrared active.
64
Conclude: vibrational modes of polyatomic molecules in which a molecular dipole moment changes in the
course of the vibrational motion are infrared active (i.e., their vibrational transitions are “electric dipole
allowed), independent of whether or not the molecule has a permanent dipole moment.
C. Electronic Transitions
The Bohr/Rutherford model of an atom implies that it consists of a positively charged nucleus with the
negatively charged electrons orbiting around it, like planets orbit around the sun, and it is a straightforward
matter to extend the same picture to electrons in molecules. This constant circulation of charge means that
every atom or molecule will have a very high frequency oscillating dipole moment. As a result, within this
type of classical model, we conclude that every atom or molecule will have allowed electronic transitions.
Since this motion is very rapid, these transitions have relatively high frequency, and appear in the visibly,
ultraviolet regions of the spectrum where light has ‘energies’ of 104 − 106 cm−1 per photon.
4.1.3 How can we explain the relative magnitudes of the energies of
rotational, vibrational, and electronic transitions?
We saw in the discussion of Chapter 2 that the internal motion of a diatomic molecule could be exactly
described in terms of the motion of a pseudo-particle of mass μ moving in space. For vibration this is
motion along the radial coordinate r, while for rotation it is orbital motion about the centre of mass
of the system at a radius of re. The orbital motion of a particle of mass μ at a radius of re, may be
thought of as motion of a particle trapped in a box of length L = πre . In contrast, the box governing the
(radial) amplitude of the vibrational motion has a length a fraction of re (say ∼ 0.2 − 0.4 re). Since the
box length associated with rotational motion is an order of magnitude larger than that for vibration, this
particle-in-a-box model tells us that rotational level spacings will be of order 100 times smaller than those
for vibration. Moreover, since the values of μ for common molecules are typically in the lower portion of
the range 1−100 u, and re values are typically 1−3 A, the ‘particle-in-a-box’ energy level formula predicts
that vibrational level spacing will be of order 50 − 5000 [cm−1] and rotational level spacings in the range
0.05 − 50 [cm−1], which is what we call the microwave region of the spectrum.
Similar arguments explain the relative magnitude of the transition energies for electronic spectra. First
of all, the electron has a mass of me ≈ 5.486 × 10−4 u, while a diatomic molecule reduced mass will have
magnitudes of μ = 1− 100 u. Within a particle-in-a-box picture, for a fixed box size this difference would
make the energy level spacings for the electron roughly 104 − 105 times larger than those associated with
nuclear motion. However, the radius of the orbit for one of the outermost electrons in a molecule (the first
one to be excited) would be at least twice the equilibrium internuclear separation re , and this difference
increases the effective box length relative to that for rotation (which we saw was πre) by a factor of two
and hence reduces the level spacing by a factor of four. Thus, electronic excitation energies are expected
to be of order 2 500−25 000 times larger than rotational level spacings for the same molecule, which would
make them 25− 250 times larger than its vibrational level spacings. Thus, we see that with very simplistic
particle-in-a-box reasoning, we can rationalize the relative magnitudes of the photon energies required to
drive rotational, vibrational and electronic transitions.
65
4.2 Einstein’s Theory of Absorption and Emission
Our understanding of the relationship between absorption and emission intensities is based on a theory
which Einstein published in 1916, in which those processes are treated using language which we normally
associate with chemical kinetics [1]. The theory begins by considering a collection of identical systems
(molecules) which have two possible states “u” and “l” with energies Eu and El, respectively, where
Eu > El . This system is assumed to be in thermal equilibrium with bath of light with radiation density
(or intensity) ρ(ν, T ) that has units energy density per unit frequency, [J/m3 · s−1]. Since the system is in
thermal equilibrium at temperature T, then the populations Nu and Nl of levels in these two states, are
related by the usual Boltzmann expression:
Nu/Nl = e−(Eu−El)/kBT = e−h νul/kBT (4.3)
where as usual ΔE = Eu − El = h νul and kB is Boltzmann’s constant.
Under the perturbing influence of the light field, the molecules may undergo three possible processes:
absorption from state l to state u, stimulated emission from state u to state l, and spontaneous emission
from u to l. Since the first process requires the absorption of a quantum of radiation from the field, the
net rate will be proportional to the product of the number of initial-state molecules times the intensity of
the light at the frequency which can drive the transition:
dNu
dt
∣∣∣∣abs.
= Bu←lNl ρ(νul, T ) (4.4)
where Bu←l is a “rate constant” known as the Einstein absorption coefficient. Similarly, the molecules
initially in the excited state may be stimulated by the oscillating fields of the light to undergo a transition
to the lower state, and the rate at which the upper level is depopulated by this process is again governed
by a second-order rate lawdNu
dt
∣∣∣∣stim.
= −Bu→lNu ρ(νul, T ) (4.5)
where Bu→l is the Einstein stimulated emission coefficient. Finally, the molecules initially in the upper
state may spontaneously emit a photon, independent of the presence of the radiation field, and the rate at
which the upper state is depopulated by this mechanism is given by a simple first-order rate law
dNu
dt
∣∣∣∣spon.
= −Au→lNu (4.6)
where Au→l is the Einstein emission coefficient. We note that conservation of energy requires that the
absorption process be driven by light whose frequency exactly matches the energy spacing between the
two levels. However, it is often unstated that a key postulate of the Einstein theory is the assumption that
that stimulated emission process also requires the energy of the incident light photons to exactly match
that level spacing, although this is not required by energy conservation considerations [2].
Since the three above processes may occur at the same time, the total rate of change of the upper state
population is the sum of these rates
dNu
dt
∣∣∣∣tot.
=dNu
dt
∣∣∣∣abs.
+dNu
dt
∣∣∣∣stim.
+dNu
dt
∣∣∣∣spon.
(4.7)
However, for a system in thermal equilibrium this total rate of population change must be identically zero,
which means that
Bu←lNl ρ(νul, T )−Bu→lNu ρ(νul, T )−Au→lNu = 0 (4.8)
66
Rearranging this expression and making use of Eq. (4.3) one obtains:
Nu
Nl=
Bu←l ρ(νul, T )
Bu→l ρ(νul, T ) +Au→l= e−hνul/kBT (4.9)
and, rearranging Eq. (4.9) to solve for ρ(νul, T ) yields:
ρ(νul, T ) =Au→l
Bu←l e−hνul/kBT −Bu→l(4.10)
However, if the system of molecules is truly in thermal equilibrium, if must also be in equilibrium with the
radiation field, and the Planck radiation law taught us that for a radiation field in equilibrium with its
surroundings at a temperature T:
ρ(ν, T ) =8π h ν3
c31
ehν/kBT − 1(4.11)
It is immediately clear that these two expressions for the radiation density distribution at equilibrium can
only both be satisfied if
Bu←l = Bu→l ≡ Bul (4.12)
and
Au→l ≡ Aul =8π h νul
3
c3Bul (4.13)
These are the key relationships between absorption and emission rate coefficients yielded by the Einstein
theory.
4.3 Rotational and Vibrational Transition Intensities and
Selection Rules
4.3.1 Electronic/Nuclear and Angular/Radial Separability
The next step is to relate Aul and/or Bul to properties of the molecule by applying time-dependent
perturbation theory to a system consisting of a molecular charge distribution subject to an oscillating
electric field. This details of this derivation are not relevant to the present discussion, so we will only quote
the result [2, 3] that
Bul =8π3
(4πε0)3h3
∣∣∣∣∫τtot
Ψ∗int,u MΨint,l dτtot
∣∣∣∣2 =8π3
(4πε0)3h3|Mul|2 (4.14)
where Ψint is the total internal motion wavefunction of Eqs. (2.12) and (2.17)-(2.20), dτtot = dτel× dτnuc is
a generalized volume element for all electronic (“el”) and nuclear (“nuc”) coordinates, and the integration
volume τtot is the whole domain of all coordinates. The operator in this expression is the total instantaneous
dipole moment function of the system
M = M({rni}, r) =∑j
qj rj (4.15)
where the summation over j runs over all charged particles (electrons and nuclei) of the system. Similarly,
Eq. (4.13) allow us to write
Aul =64π4νul
3
(4πε0)3h c3|Mul|2 = 3.136 × 10−7 ν 3
ul |Mul|2 (4.16)
67
where Aul has units s−1, and the numerical factor in the last version of this equation assumes that νul has
units cm−1 and Mul has units Debye.2
We now invoke the “adiabatic” form of the Born-Oppenheimer separation of Eq. (2.20), which says that
for electronic state s, Ψint,s = Φs({rni}; r) Xs(r) . On substituting this product function into the definition
of Mul one obtains
Mul =
∫τtot
[Φu({rni}; r) Xu(r)]∗ M [Φl({rni}; r) Xl(r)] dτtot
=
∫τnuc
Xu(r)∗[∫
τel
Φu∗ M Φl dτel
]Xl(r) dτnuc
=
∫τnuc
Xu(r)∗ �M(r)Xl(r) dτnuc (4.17)
The quantity �M(r) is called the “transition moment function” for transitions between different electronic
states (Φu �= Φl ), and is simply the dipole moment of the molecule for rotational or vibrational-rotational
transitions within a single electronic state (Φu = Φl ). For simplicity, the name transition moment function
is sometimes used for both cases. In principle it may be calculated from of the electronic wavefunctions.
However, while such calculations are fairly routine for obtaining the diagonal (Φu = Φl ) dipole moment
function, especially for ground electronic states, even today it can still be a fairly severe challenge to
compute reliable transition moment functions for electronic transitions (see, e.g., Ref. [4]).
For the sake of simplicity, the present discussion is restricted to explicit consideration of transitions
in which the electronic states of the upper and lower levels (which may be the same state) are both Σ
states, which means that their electronic wavefunctions have axial symmetry about the internuclear axis.
Generalizations to other cases are in principle straightforward but algebraically much more complex, so
while the results for other cases will be quoted, only the derivation for this simpler case will be presented.
For Σ−Σ electronic transitions, the axial symmetry of the electronic wavefunctions means that matrix
elements of any vector operator must also have axial symmetry. For this case, the (vector) transition dipole
function must therefore point along the internuclear axis, which means that it may be written in the form
�M(r) = M(r)× [sin θ cosφ ex + sin θ sinφ ey + cos θ ez] (4.18)
where M(r) is a scalar function of a scalar variable, the ej are orthonormal unit vectors pointing along
the x, y and z laboratory coordinate axes, and θ and φ are the usual spherical coordinate angles defining
the orientation r of the bond axis. If we now introduce the angular-radial wavefunction factorization of
Eq. (2.26), Xs(r) = r−1 ψ(r) YJM(r) , the matrix element of Eq. (4.17) may be expanded as a product of
a radial overlap integral times a (vector) sum of angular overlap integrals:
Mul =
(∫ ∞0
ψu(r)∗M(r)ψl(r) dr
)×{ex
∫{θ,φ}
YJ ′M ′(r) sin θ cosφYJ ′′M ′′(r) d2r
+ ey
∫{θ,φ}
YJ ′M ′(r) sin θ sinφYJ ′′M ′′(r) d2r + ez
∫{θ,φ}
YJ ′M ′(r) cos θ YJ ′′M ′′(r) d2r
}= Mrad
ul ×{exM
xul + ey M
yul + ez M
zul
}(4.19)
The actual transition intensity then depends on the scalar quantity
|Mul|2 =(M rad
ul
)2×{|Mx
ul|2 +∣∣My
ul
∣∣2 + |Mzul|2}
(4.20)
2 1 Debye= 3.335 64× 10−30C ·m.
68
Let us now evaluate the components of Eq. (4.20) and see how rotational and vibrational selection rules
arise.
4.3.2 Rotational Intensities and Selection Rules
Let us first consider Mzul. The recurrence relations
(2J + 1) cos θ P|M |J (cos θ) = (J + |M |)P |M |J−1(cos θ) + (J − |M |+ 1)P
|M |J+1(cos θ) (4.21)
and normalization property
〈P |M |J |P |M |J 〉 = (J + 12 ) (J − |M |)!/(J + |M |)! (4.22)
of the Legendre polynomials [2] which comprise the θ–dependent part of the normalized total angular
wavefunction YJM (r) = NJM eiMφ PMJ (cos θ) yield the relationship
cos θ YJM (r) =
√(J + |M |)(J − |M |)(2J + 1)(2J − 1)
YJ−1,M(r) +
√(J + 1 + |M |)(J + 1− |M |)
(2J + 1)(2J + 3)YJ+1,M(r) (4.23)
Substituting this result into the expression for Mzul then yields
Mzul ≡
∫{θ,φ}
YJ ′M ′(r) cos θ YJ ′′M ′′(r) d2r
=
√(J ′′ + |M ′′|)(J ′′ − |M ′′|)
(2J ′′ + 1)(2J ′′ − 1)〈YJ ′,M ′ |YJ ′′−1,M ′′〉+
√(J ′′ + 1 + |M ′′|)(J ′′ + 1− |M ′′|)
(2J ′′ + 1)(2J ′′ + 3)〈YJ ′,M ′ |YJ ′′+1,M ′′〉
=
√(J> + |M |)(J> − |M |)(2J> + 1)(2J> − 1)
δJ ′,J ′′±1 δM ′,M ′′ (4.24)
where J>=max{J ′, J ′′} , M =M ′=M ′′ , and δi,j =1 for i= j and =0 whenever i �= j . In other words,
this gives us the selection rule that Mzul=0 unless ΔJ=±1 and ΔM=0 .
For the x and y components of the angular contribution to the transition moment, the recurrence
relations [2]
sin θ P|M |−1J (cos θ) =
1
2J + 1
{P|M |J+1(cos θ)− P |M |J−1(cos θ)
}(4.25)
sin θ P|M |+1J (cos θ) =
(J + |M |)(J + |M |+ 1)
2J + 1P|M |J−1(cos θ)−
(J − |M |)(J − |M |+ 1)
2J + 1P|M |J+1(cos θ)
and the Euler relations cos φ=(eiφ + e−iφ
)/2 and sinφ=
(eiφ − e−iφ) /2i yield expressions for Mx
ul and
Myul analogous to Eq. (4.24), which in turn yield the selection rules that these matrix elements are identically
zero unless ΔJ=±1 and ΔM=±1 .The final step of this derivation addresses the fact that Eq. (4.20) shows we must sum the squares of
the three angular intensity components. Moreover, since the M rotational sublevels associated with a given
J value are degenerate, to obtain the total intensity of a given J ′ ← J ′′ transition one must sum that
result over all possible values of M′ and M′′:
S{ΔJ}J ′′ =
∑M ′
∑M ′′
{∣∣MxJ ′,M ′;J ′′,M ′′
∣∣2 + ∣∣∣MyJ ′,M ′;J ′′,M ′′
∣∣∣2 + ∣∣MzJ ′,M ′;J ′′,M ′′
∣∣2} (4.26)
where the superscript “{ΔJ}” refers to a symbol denoting the value of ΔJ ; in particular, {ΔJ} = P , Q or
R for ΔJ =−1 , 0 or +1, respectively. The net effect of these summations yields the rotational intensity
69
Table 4.1: Honl-London rotational line intensity factors for P (J ′ = J ′′ − 1), Q (J ′ = J ′′) and R
(J ′= J ′′ + 1) transitions associated with singlet–singlet electronic transitions in which the change
in the electronic angular momentum projection on the internuclear axis is ΔΛ=Λ′ − Λ′′ [5, 6, 3].
ΔΛ = 0 SRJ = (J ′′+1+Λ′′)(J ′′+1−Λ′′)
J ′′+1= (J>+Λ>)(J>−Λ>)
J>
SQJ = (2J ′′+1)Λ′′2
J ′′(J ′′+1)= (2J>+1)Λ>
2
J>(J>+1)
SPJ = (J ′′+Λ′′)(J ′′−Λ′′)
J ′′ = (J>+Λ>)(J>−Λ>)J>
ΔΛ = +1 SRJ = (J ′′+2+Λ′′)(J ′′+1+Λ′′)
4(J ′′+1)= (J>+Λ>)(J>+Λ>−1)
4J>
SQJ = (J ′′+1+Λ′′)(J ′′−Λ′′)(2J ′′+1)
4J ′′(J ′′+1)= (J>+Λ>)(J>−Λ>+1)(2J>+1)
4J>(J>+1)
SPJ = (J ′′−1−Λ′′)(J ′′−Λ′′)
4J ′′ = (J>−Λ>)(J>−Λ>+1)4J>
ΔΛ = −1 SRJ = (J ′′+2−Λ′′)(J ′′+1−Λ′′)
4(J ′′+1)= (J>+1−Λ>)(J>−Λ>)
4J>
SQJ = (J ′′+1−Λ′′)(J ′′+Λ′′)(2J ′′+1)
4J ′′(J ′′+1)= (J>+Λ>)(J>−Λ>+1)(2J>+1)
4J>(J>+1)
SPJ = (J ′′−1+Λ′′)(J ′′+Λ′′)
4J ′′ = (J>−1+Λ>)(J>+Λ>)4J>
factor expressions in Table 4.1; the results derived above are those in the first and third rows for the case
of Λ′=Λ′′=0 . In this singlet–singlet case, SRJ = J ′′ + 1 , SP
J = J ′′ and of course SQJ = 0 , so the total
rotational intensity for absorption from a given J ′′ level is (2J ′′ + 1), the total rotational degeneracy of
that level.
Note that the expressions given here utilize the universal spectroscopic convention that the higher-
energy level in a transition is labeled with a single prime (′), the lower level by double primes (′′), and that
a quantum number difference ΔQ always refers to upper-level-value minus lower-level-value. An analogous
associated convention is that whenever two levels are linked in a symbol or notation, the quantum number
or label for the upper level precedes that for the lower one (see, e.e., Eqs. (4.8)). Note too that in the third
column of Table 4.1, Λ> =max(Λ′,Λ′′) . This notation is introduced to illustrate the symmetry between
the expressions for P–branch and R–branch intensity expressions, and between those for ΔΛ=−1 and
+1. Moreover, rotational transitions are always labeled by the lower-level quantum numbers, here J ′′ andΛ′′, and by a letter (here P , Q or R) identifying the value of ΔJ=J ′ − J ′′ for that transition.
For the more general cases in which the electronic angular momentum projection quantum number Λ
is non-zero for one or both of the upper and lower electronic states, the rotational intensity derivation is in
principle the same as that shown above. However, it is algebraically more complex because the electronic
angular momentum vector is initially specified in body-fixed rather than the space-fixed (or laboratory)
coordinates used above, and even more complicated expressions are obtained when non-singlet electronic
states are involved.
4.3.3 Vibrational Intensities and Selection Rules
We now return to Eq. (4.20) and consider the contribution of the radial overlap integral Mradul to the overall
transition intensity. It is of course clear that independent of any other considerations, the transition will
be forbidden (transition intensity will be identically zero) if the electronic transition moment function
70
or dipole moment function �M(r) = 0 . This will occur for vibration-rotation transitions within a given
electronic state if the molecule is homonuclear and there is no dipole moment, and it will be true for
electronic transitions for which the dipole radiation field does not couple the two states (e.g., between a5Δ and a 1Σ state). However, more detailed selection rules may also be discerned.
It was shown earlier that for Σ − Σ transitions the transition moment function �M(r) may be written
as the product of a scalar function M(r) times a vector sum whose terms depend on the orientation of
that moment relative to the axis of the molecule. This is true in general, although in the more general
case that transition dipole vector may not point along the axis of the molecule. This latter consideration
gives rise to different rotational selection rules and Honl-London rotational intensity factors, but does not
affect the fact that the radial matrix element Mradul is the overlap integral between the initial- and final-level
vibrational wavefunctions and the scalar transition strength operator M(r). Since any continuous function
may be approximated (at least locally) by a Taylor series, we can write
M(r) = d0 + d1 (r − re) + d2 (r − re)2 + ... =∑i=0
di (r − re)i (4.27)
where re is the equilibrium internuclear distance at the minimum of the potential energy function. Substi-
tuting this expansion into the definition of Mradul yields
Mradul =
∫ ∞0
ψu(r)∗M(r)ψl(r) dr (4.28)
=∑i
di
∫ ∞0
(r − re)i ψv′,J ′(r)∗ ψv′′,J ′′(r) dr
= d0
∫ ∞0
ψu(r)∗ ψl(r) dr + d1
∫ ∞0
(r − re)ψu(r)∗ ψl(r) dr + d2
∫ ∞0
(r − re)2 ψu(r)∗ ψl(r) dr + ...
Let us first consider vibration-rotation transitions among levels of a given electronic state. If we ignore
the very small shift and distortion due to the slight difference (ΔJ =±1) in the centrifugal contribution
to the effective radial potential, the wavefunctions ψv′,J ′(r) and ψv′′,J ′′(r) are eigenfunctions of exactly
the same differential equation. From the fundamental theorems of quantum mechanics, we know that
eigenfunctions of a given Hamiltonian form (or can be made into) an orthonormal set, which means that
(writing J=J ′=J ′′) ∫ ∞0
ψv′,J(r)∗ ψv′′,J(r) dr = 〈ψv′,J(r)|ψv′′,J(r)〉 = δv′,v′′ (4.29)
For pure rotational transitions, v′=v′′ , the orthogonality properties of the radial eigenfunctions tend
to mean that for i ≥ 1 , 〈ψv,J(r)|(r − re)i|ψv,J(r)〉 � 〈ψv,J (r)|ψv,J (r)〉 . This in turn means that for such
transitions it is mainly the magnitude of |d0|, i.e., the strength of the dipole moment at the equilibrium
bond length, which determines the absolute intensity of the transition.
For vibration-rotation transitions ( v′ �=v′′ ), wavefunction orthogonality makes 〈ψv′,J(r)|ψv′′,J(r)〉=0 ,
so the transition intensity is completely independent of the absolute strength (|d0|) of the dipole moment
function,3 and depends rather on the strength of the distance-dependent terms. This is a strong, but
qualitative selection rule for vibration-rotation spectra; to obtain quantum-number based restrictions, we
will have to turn to a specific model.
3 Careful readers will note that this is not precisely true, because the slight (ΔJ =±1 ) difference in centrifugal
potentials makes ψv′,J′(r) and ψv′′,J′′(r) not precisely orthogonal (Exercise: test this with numerical calculations!).
However, this effect is extremely small, and is totally masked by contributions from other terms.
71
We know that essentially all small-amplitude vibrational motion may be semi-quantitatively described
within a harmonic oscillator approximation. The recurrence properties of the Hermite polynomial contri-
bution to the harmonic oscillator wavefunction [2, 7],4
xHn(x) = Hn+1(x) + 2nHn−1(x) (4.30)
where x is a scaled distance coordinate proportional to (r − re). Combining this result with the orthonor-
mality of such wavefunctions means that if M(r) is strictly a linear function of (r − re), the only non-zero
contribution to Mradul occurs when Δv = ±1 , and that it comes entirely from the linear contribution to
M(r). In other words, the magnitude of the vibrational transition intensity depends only on the derivative
of the dipole moment with respect to r (d1), and is completely independent of its magnitude at re (d0).
This explains why the infrared (i.e., vibrational) transitions of CO, which has an extremely small dipole
moment, are of roughly the same intensity as those of HF, although the microwave or pure rotational
spectrum of CO is more than two orders of magnitude weaker. Ab initio electronic structure calculations
confirm that the radial derivatives of their dipole moment functions are roughly the same size,rj−Ref? while
the magnitude of the CO dipole at re is very small [8].5
A straightforward extension of the above argument immediately shows that if the associated coefficients
di are sufficiently large, the quadratic and cubic contributions to Eq. (4.27) would give rise to strong
Δv=±2 , and Δv=±1 and 3 transitions, respectively. However, while the dipole moment function for
a real system will never be exactly linear in (r − re), the quadratic and higher-order contributions to
Eq. (4.27) will generally be much smaller than the linear one, especially on a modest sized interval near re.
Moreover, in a non-harmonic system the eigenfunction recurrence relation of Eq. (4.30) will break down,
and |Δv| > 1 transitions are also allowed, even for the case of a purely linear dipole moment function.
However, such transitions are always much weaker than those for Δv = ±1 . Thus, as is observed in
practice, the dominant vibrational transitions will always be those between adjacent vibrational levels.
To a good first approximation, one can normally expect the dipole moment function to be approximately
linear across the width of the potential well. This gives rise to another intensity propensity or approximate
selection rule which is exact for any truly symmetric potential well (e.g., harmonic oscillator or particle-in-
a-box), and semi-quantitative for near-symmetric wells. It is based simply on the fact that in such wells,
wavefunctions corresponding to even values of v will be symmetric with regard to reflection through re,
while those with odd values of v will be antisymmetric. Since the constant term on the dipole function
makes no contribution to the intensities (see above), and the linear term will be antisymmetric, for any
transition with Δv even the overall integrand of the Mradul integral will be antisymmetric about r=re , and
hence the integral will equal zero. This gives the additional selection rule that for any (near-)symmetric
well, only Δv =±1 , ±3, ±5, ... etc. transitions are possible. For truly symmetric vibrational motions
such as the bending vibrations of CO2 or the umbrella inversion vibration of NH3, this Δv=odd selection
rule is exact, since the dipole moment function itself will also be exactly antisymmetric. However, for
asymmetric and strongly anharmonic potential energy functions such as a normal Morse or LJ(m,n) curve,
this Δv=odd propensity is completely lost and we are left with only the underlying strong preference for
Δv=1 .
In summary, therefore, the quasi-harmonic nature of vibrational motion near a potential minimum
means that Δv=±1 vibrational transitions will usually be strongly preferred, and that transitions with
|Δv| > 1 usually become rapidly very much weaker with increasing values of |Δv|. Nonetheless, real
4 As discussed in Chapter 2, the conventional harmonic oscillator quantum number n is equal to v+1.5 Indeed, for many years there was even a controversy regarding regarding whether that dipole pointed towards
the C or the O atom.
72
υ=0
υ=1
υ=2
υ=3
υ=4
υ=5
υ=6υ=7
fundamental
first overtone
second overtone
third overtone
hot bands}
DD
rer→
V (r)Morse
0
e
↑
zero point energy
Figure 4.4: Naming of various types of vibrational transitions of a diatomic molecule.
molecules deviate increasing from harmonic behaviour at higher energies, so such ‘overtone’ transitions
transitions are often observed. Figure 4.4 illustrated the naming conventions used for vibrational transi-
tions.
4.3.4 Selection Rules and Intensities in Electronic Spectra
For transitions between levels of different electronic states, the electronic/angular/radial separability of the
total internal motion wavefunction still applies. As a result, the transition dipole function is still separable
into a product of a scalar radial and a vector angular part. The latter gives rise to the rotational selection
rule intensities of Table 4.1, while the former depends on the the vibrational matrix element of Eq. (4.28).
In this case, however, the radial wavefunction ψu(r)=ψv′,J ′(r) and ψl(r)=ψv′′,J ′′(r) are eigenfunctions
of completely different potential energy function, so that the formal orthogonality properties of Eq. (4.29)
are completely irrelevant. Indeed, the only vibrational “selection rule” which can be said to apply is
that the transition intensity will zero unless the intervals in which the upper- and lower-state vibrational
wavefunctions have non-negligible amplitude overlap significantly. This point will be examined further in
Section 4.6.
4.4 Frequencies and Intensities in Pure Rotational Spectra
4.4.1 Frequency Patterns in Pure Rotational Spectra
Recall that the energies of the rotational sublevels associated with a given vibrational level may be described
by the band constant expression of Eq. (3.22)
E(v, J) = Gv + Bv[J(J + 1)] − Dv[J(J + 1)]2 + Hv[J(J + 1)]3 + Lv[J(J + 1)]4 + . . .
= Gv + Fv(J) (4.31)
73
Figure 4.5: Pure rotational absorption spectrum of CO gas. The small peaks are transitions asso-
ciated with the less abundant minor isotopologues.
in which Bv � |Dv| � |Hv| � |Lv| � |Mv | � . . .. Given the selection rule ΔJ = ±1, the energies of the
allowed transitions are
νrotJ = E(v, J + 1) − E(v, J) = Fv(J + 1) − Fv(J) ≡ ΔFv(J)
= 2Bv(J + 1) − 4Dv(J + 1)3 + . . . (4.32)
This expression shows that to a first approximation (neglecting centrifugal distortion), the energies of pure
rotational transitions increase linearly with J . Similarly, the spacing between adjacent line in the spectrum
may be written as
Δν rotJ ≡ Δ2Fv(J) = ΔF (J) − ΔF (J − 1)
= 2Bv − 4Dv(3J3 + 3J + 1) + . . . (4.33)
This in turn shows that to a first approximation, the lines in a rotational spectrum will be equally spaced;
this behaviour is clearly demonstrated in the segment of the rotational spectrum of CO shown in Fig. 4.5.
The existence of a set of approximately equally spaced lines is a characteristic signature of a pure
rotational spectrum, or of the rotational substructure in a vibrational or electronic spectrum. However,
three types of effects can complicate this simple picture.
Complication 1. Isotopologues
“Isotopologues” of a given molecular species are chemically identical molecules formed from different
combinations of the isotopes of the atoms of which it is comprised: e.g., 12C 16O, vs. 13C 16O, vs. 12C 17O,
vs. 12C 18O. To very high accuracy, the electronic potential V (r) is the same for all isotopologues of a given
molecular state, so the equilibrium bond lengths re should also be (almost) the same for all isotopologues.
However, the reduced mass μ appearing in the definitions of the rotational constants, Eqs. (3.25)–(3.28)
mean that the level energy expression will differ for different isotopologues, so they will have different
rotational constants Bv , and hence also different rotational line spacings. Sometimes these differences will
be fairly large, as for HCl and DCl, for which:
μ(H 35Cl) =(
11.0078250 + 1
34.968853
)−1= 0.979593 u
μ(D 35Cl) =(
12.014102 + 1
34968853
)−1= 1.904413 u
74
Figure 4.6: Pure rotational spectrum of hot HF molecules.
and sometimes they will be small, as for CO , for which μ(12C16O) = 6.8562086 u while μ(13C16O) =
7.1724125 u. However, their net effect is that all else being equal, rotational transition energies and line
spacings for different isotopologues of a given chemical species scale approximately as the inverse of the
associated values of the reduced mass μ.
Complication 2. Vibrational Stretching
The discussion of §3.7 showed that values of the rotational constants vary from one vibrational level to
the next. For the inertial rotational constant this behaviour is conventionally described by the expression
Bv = Be − αe(v + 1/2) + γe(v + 1/2)2 + δe(v + 1/2)3 + . . . (4.34)
The negative sign in front of the second term on Eq. (4.34) reflects the fact that the average molecular bond
length usually increases with the degree of vibrational excitation, which means that 〈1/r2〉v (and hence
Bv) generally decreases with increasing values of v. Thus, when excited vibrational levels are thermally
populated, the spectrum will contain multiple sets of approximately equally spaced lines superimposed on,
but slightly displaced from one another. This behaviour is clearly illustrated by the emission spectrum of
HF seen in Fig. 4.6.
Complication 3. Centrifugal Distortion
As was illustrated by Figs. 2.3 and 3.5, with increasing values of J the associated increases in the
centrifugal potential has the effect of shifting all vibrational levels to higher energy and increasing the
associated average bond lengths. The fact that the strength of the centrifugal potential scales as [J(J +1)]
explains why the conventional band constant expression for the rotational energy Eq. (4.31) is written as
a power series in this factor. As is illustrated by Eq. (4.33), the centrifugal distortion constants have the
effect of making rotational line spacings decrease with increasing J . However, the fact that Bv � Dv
means that unless J is relatively large, to a good first approximation, line spacings are approximately
constant in pure rotation spectra.
75
0 5 10 15 20 250
5
10
J
J (T)↓
emmax
J (T)↓
popmax J (T)
↓
absmax
fυ(J,T)∝ Iem
∝ Iabs
e⎯⎯⎯0.1−Fυ(J)/kBT (2J+1)
Figure 4.7: Boltzmann rotational population distribution for CO at T = 293K.
4.4.2 Intensity Patterns in Pure Rotational Spectra
The Honl-London factors listed in Table 4.1 were obtained by summing over contributions from all possible
(M ′,M ′′) sublevels. Thus, in order to separate off the effect of rotational M -degeneracy one can divide
each of these values by the factor (2J +1) to obtain the average degeneracy per initial-state M value. For
the simple case of absorption for Λ′′ = λ′ = 0 transitions this yields average P and R intensity factors of
SRJ = (J ′′ + 1)/(2J ′′ + 1) and S
PJ = J ′′/(2J ′′ + 1) for all transitions originating in a given (J ′′,M ′′) state.
It is immediately clear that except at very small J ′′ values, these ‘per M -state’ intensity factors become
essentially independent of J ′′.In view of the above, we might expect the intensity of any given spectroscopic transition to be propor-
tional to the molecular population in the initial state. From statistical thermodynamics we know that for
a system in thermal equilibrium, of all the molecules in vibrational level v at temperature T , the fraction
in rotational sublevel J is
fv(J, T ) = (2J + 1) e−Fv(J)/kBT/qv(T ) (4.35)
where
qv(T ) =∑J=0
(2J + 1) e−Fv(J)/kBT (4.36)
is the rotational partition function associated with that vibrational level. The nature of this population
distribution (points and solid curve) and of the two functions defining it (dot-dash line and dot-dot-dash
curve) are illustrated by Fig. 4.7. If we make the approximation of neglecting centrifugal distortion, so that
Fv(J) ≈ Bv[J(J + 1)], taking the derivative of Eq. (4.35) with respect to T and setting it equal to zero
shows that the value of J for the most highly populated rotational level will be
Jpopmax(T ) =
√kBT
2Bv− 1
2(4.37)
Thus, within the ‘rigid rotor’ approximation we might expect this expression to define the most intense
line in a rotational spectrum.
In practice, however, two other considerations mean that the values of J associated with the most
intense lines in pure rotational absorption and emission spectra are significantly larger than would be
76
predicted by the population maximum defined by Eq. (4.37). In particular, since many detectors measure
the energy of the emitted/absorbed light, the observed transition intensity effectively includes a factor of
νrotJ . Moreover, stimulated emission can occur at the same time as stimulated absorption, so the observed
intensity will depend on the difference between the population of the upper (J ′ = J ′′ + 1) and lower
(J ′′ = J) levels of the transition. As a result, dependence of the intensity of pure rotational transitions on
J and T is given by
Iabs(J, T ) ∝ J + 1
2J + 1νrotJ fv(J, T )
[1− e−[Fv(J ′)−Fv(J)]/kBT
]≈ 2Bv(J + 1)2 e−BvJ(J+1)/kBT
[1− e−[Fv(J ′)−Fv(J)]/kBT
]. (4.38)
The nature of this function is illustrated by the dotted curve in Fig. 4.7. Setting the derivative of Iabs(J, T )
with respect to J equal to zero, for the usual case in which the rotational level spacing ≈ 2Bv(J + 1) is
smaller than kBT we can write
Jabsmax(T ) =
(3kBT
2Bv
)1/2 [1 − 1
2(J + 1)− (J + 1/2)Bv
3kBT+ . . .
]1/2− 1
2
≈√3 Jpop
max(T ) . (4.39)
Following similar arguments, for the case of spontaneous emission with the intensity being measured
in units of energy per unit time, the presence of the ν 3ul factor in Eq. (4.13) means that the dependence of
the emission intensity on temperature and the upper-state quantum J ′ is given by
Iem(J, T ) ∝ J ′
2J ′ + 1(νrotJ )4 fv(J
′, T )
∝ 8(Bv)4 (J ′)5 e−BvJ ′(J ′+1)/kBT (4.40)
The nature of this function is illustrated by the dashed curve in Fig. 4.7. Requiring the derivative of
Iem(J, T ) with respect to J ′ to equal zero, and recalling that J ≡ J ′′ = J ′ − 1 leads to the result that
Jemmax(T ) =
(5kBT
2Bv
)1/2 [1 +
1
2(J + 1)
]−1/2− 1
≈√5 Jpop
max(T ) . (4.41)
Thus, we see that the positions of the intensity maxima in both pure rotational absorption and emission
spectra differ very substantially both from each other and from the J value associated with the maximum
in the thermal population distribution. This is very different from the situation for rotational structure in
vibrational and electronic spectra, where the values of Jmax are determined almost totally by the population
factor of Eq. (4.35).
4.5 Frequencies and Intensities in Vibration-Rotation
Spectra
Under most circumstances, rotational level spacings ≈ 2Bv(J + 1) are much smaller than kBT , so any
vibrational transition will consist of a “band” or group of transitions associated with different J sublevels
77
Table 4.2: Labels identifying rotational branches in vibration-rotation and electronic spectroscopy.
ΔJ ≡ J ′ − J ′′ − 3 − 2 − 1 0 1 2 3
label N O P Q R S T
of the initial and final vibrational level. The energies of such transitions in vibration-rotation (or infrared)
spectra may be written as
ν = E(v′, J ′) − E(v′′, J ′′) =[G(v′) + Fv′(J
′)]− [G(v′′) + Fv′′(J
′′)]
= ν0(v′, v′′) +
[Fv′(J
′)− Fv′′(J′′)], (4.42)
in which ν0(v′, v′′) ≡ [G(v′)−G(v′′)] is the “band origin”, which is the energy of a hypothetical (forbidden
by selection rules) pure vibrational transition between levels (v′′, J ′′ = 0) and (v′, J ′ = 0). The rotational
contributions to the overall transition energies appear as distinct “branches” characterized by the associated
changes in J . Ordinary electric dipole transitions normally allow only ΔJ = ±1 for transitions in electronic
states with zero total electronic angular momentum, and ΔJ = 0, ±1 for states with non-zero electronic
angular momentum. However, other selection rules apply for Raman transitions and for more complex
molecules, so the nomenclature for labeling rotational branches includes the wide variety of cases listed
in Table 4.2. Thus, P (22) labels a transition with J ′′ = 22 and J ′ = 21, while S(5) is the name for a
transition with J ′′ = 5 and J ′ = 7.
Since the leading centrifugal distortion constant for a given vibrational level (Dv) is typically 3–5 orders
of magnitude smaller than the corresponding Bv value, it is usually a good first approximation to ignore
centrifugal distortion terms. In this case, the transition energies in the two allowed rotational branches of a
vibrational transition in a molecular state with zero electronic orbital angular momentum may be written
as
P(1)P(2)P(3)P(4)
P(5)P(6)
P(7) R(0)
R(1)R(2) R(3) R(4)
R(5)R(6)
R(7)
∼ν / cm−1 →
P(8)
spectrum
∼ν0↓
bandgap
←⎯→
J ′=0234
5
6
7
J ′′=0234
5
6
7⎪
⎩
⎧
⎪
⎪⎪⎪⎪⎪⎨⎪⎪⎪
⎩
⎧
υ′
υ′′
↑energy
⎧
P branchΔ J = −1
R branchΔ J = +1
⎪
⎪⎪
⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪
⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪⎩ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎧⎪ ⎪
Figure 4.8: Schematic illustration of the energy level pattern and spectrum for an infrared band.
78
Figure 4.9: Room temperature infrared absorption spectra of D35Cl (stronger line of each pair) and of
D37Cl (relative abundance ∼ 32%).
νP (J′′) = ν0(v
′, v′′) +[Bv′(J
′′ − 1)(J ′′)−Bv′′ J′′(J ′′ + 1)
]= ν0(v
′, v′′) − [Bv′′ +Bv′ ] (J′′) − [Bv′′ −Bv′ ] (J
′′)2 (4.43)
νR(J′′) = ν0(v
′, v′′) +[Bv′(J
′′ + 1)(J ′′ + 2) − Bv′′ J′′(J ′′ + 1)
]= ν0(v
′, v′′) + [Bv′′ +Bv′ ] (J′′ + 1) − [Bv′′ −Bv′ ] (J
′′ + 1)2 . (4.44)
The vibrational transition propensities discussed in §4.4.3 told us that Δv = ±1 transitions are strongly
preferred, and we generally expect that Bv values for adjacent vibrational levels will be very similar, so the
coefficient of the quadratic terms in Eqs. (4.43) and (4.44) will be relatively small. As a result, to a first
approximation the P– and R–branches of a vibrational band will consist of equally spaced lines marching,
respectively, to the red (lower energy) and to the blue (higher energy) of the band origin. This behaviour
is schematically illustrated by Fig. 4.8.
Of course at higher values of J ′′ the quadratic terms in Eqs. (4.43) and (4.44) will become increasingly
important. For the P−branch this means that the line spacings will gradually increase with increasing J ′′,while for the R−branch the line spacings will become progressively smaller. This behaviour may be readily
discerned in the infrared absorption spectrum of DCl shown in Fig. 4.9. Note that as was the case for pure
rotational spectra, the existence of minor isotopologues will be reflected in sets of transition energies are
displaced slightly from the main peaks due to the isotopic band-origin shift and isotopic differences in the
values of the rotational constants.
For sufficiently large values of J ′′ the quadratic term in Eq. (4.44) will become comparable in magnitude
to the lineal leading term, and the R−branch transition energies will turn around and march off to the
red (to lower energies). When this occurs there will be a pile-up of lines at the turnaround point which
gives rise to an intensity buildup known as a “band head”. Examples of this behaviour are seen in
the infrared emission spectrum of NaCl shown in Fig. 4.10. This type of rotational branch turnaround is
observed relatively rarely in infrared spectra, since the small values of the difference [B′′v−B)′v] for adjacentvibrational levels means that the turnaround would only occur for very high−J levels whose transition
intensities are very low at ‘normal’ temperatures. This is the case for the DCl spectrum shown in Fig. 4.9.
79
Figure 4.10: NaCl emission spectra showing band heads for the fundamental and first hot bands.
The near-harmonic behaviour of most molecules near their potential minimum means that the spacings
between neighbouring vibrational levels usually change slowly with v. As a result, the width of the
rotational structure in a vibrational band is usually much larger than the shift of the band origin from one
v′′ value to the next. As a result, if the temperature is sufficiently high that multiple vibrational levels are
populated, all vibrational bands associated with a given Δv value will overlap with one another. This is
the case for the ΔV = −1 emission spectra of GeO shown in Fig, 11. This spectrum is doubly complicated
because there are five relatively abundant isotopes of Ge. Nevertheless, Bernath and co-workers [9] were
able to assign and analyze transitions for the fundamental band and for the first seven Δv = −1 hot bands
of this species: (v′, v′′) = (2, 1) − (8, 7).
Finally, we note that in contrast with the situation for pure rotational spectra, the intensity maxima
in the P– and R–branches of a vibration-rotation band are almost totally determined by the initial-state
population considerations, and hence is defined by Eq. (4.37). This occurs because the band origin term in
Eqs. (4.43) and (4.44) is much larger than the J ′′-dependent terms, so the fractional change in the transition
energies is very small from one J ′′ value to the next. Moreover, since the vibrational energy spacing is
Figure 4.11: High temperature (1800K) emission spectrum of GeO. Left: overview spectrum; Right:
dispersed segment of the fundamental band showing isotopomer assignments.
80
much larger than kBT , the analog of the second exponential term in Eq. (4.38) will be � 1.
4.6 Frequency and Intensity Patterns in Electronic Spectra
Figure 4.12 presents a schematic overview of the nature of electronic spectra and their vibrational and
rotational substructure. In general, the upper and lower electronic state each has its own potential energy
curve whose distinct radial position and shape gives rise to its own pattern of vibrational and rotational
level spacings. Pure rotational (or microwave) transitions occur between adjacent rotational sublevels
within a ‘stack’ associated with a single vibrational level of a single state, as seen on the right hand side
of the figure. Vibration-rotation (or infrared) transitions occur between rotational sublevels of different
vibrational levels of a single potential energy curve, as illustrated by the R(7) line of the first overtone (2, 0)
band shown in this figure. Electronic transitions are then transitions between vibration-rotation sublevels
in different electronic states, as illustrated by the P (10) line of the (3, 1) band of the electronic transition
labelled ‘B’ in this figure. Note that as in infrared spectra, the set of all rotational transitions associated
with a given upper (v′) and lower (v′′) level is called a “band” and is labelled by the two vibrational
quantum numbers, with the label for the level at higher energy being written first, as in (v′, v′′) or v′−v′′.The fact that the vibrational level spacings are usually quite different in the upper and lower electronic
states means that the various vibrational bands do not overlap extensively, and we usually do not encounter
the type of extreme congestion seen in Fig. 4.11.
Figure 4.12: Schematic illustration of the basis for vibrational and rotational structure in electronic
spectra.
81
Figure 4.13: R–branch turnaround in the (0,0) band of the A 1Σ+ −X 1Σ+ spectrum of CuD.
4.6.1 Rotational Structure in Electronic Spectra
Transition energies in electronic spectra are described in essentially the same manner as in vibration-
rotation spectroscopy, with the energies of P– and R–transitions being given by Eqs. (4.43) and (4.44).
In this case, however, the definition of the band origin includes the difference between the values of the
electronic energies Te associated with the minima of the two potential energy functions (see Fig. 4.12):
ν0(v′, v′′) = [T ′e +G′(v′)] − [T ′′e +G′′(v′′)] = ΔT ′e + G′(v′) − G′′(v′′) . (4.45)
Moreover, if one or both states is associated with non-zero electronic angular momentum, Q–branch (ΔJ =
0) transitions with energies
νQ(J′′) = ν0(v
′, v′′) − [B′′v′′ −B′v′
]J(J + 1) +
[D′′v′′ −D′v′
][J(J + 1)]2 + . . . (4.46)
are also allowed.
One significant difference from the type of band structure seen in vibrational spectroscopy is the fact
that in electronic spectroscopy the type of band turnaround seen in Fog. 4.10 is now the rule, rather than
the exception. In particular, since the upper and lower state inertial rotation constants B′v′ and B′′v′′ are
associated with different potential energy curves there is no expectation that the differences |B′′v′′ − B′v′ |which define the quadratic coefficients in Eqs. (4.43) and (4.44) will be small. As a result, the band head
associated with branch turnaround generally occurs at relatively low J ′′ values, and as a result, the band
head will lie quite close to the band origin. Moreover, in electronic spectra we encounter many bands for
which B′′v′′ < B′v′ , in which case it is the P–branch that turns around (to the blue).
A band in which it is the R–branch that turns around is called a “red-shaded band”, and if it is the
P -branch that turns around it is a “blue-shaded” band. For the former we can readily determine where the
turnaround occurs by setting the first derivative of Eq. (4.44) with respect to J equal to zero and solving
for J = JRh to obtain
(JRh + 1) =
(B′′v′′ +B′v′
)/2(B′′v′′ −B′v′
). (4.47)
82
Substituting this result into Eq. (4.44) then yields an expression for the position of the band head
νR(JRh ) = ν0(v
′, v′′) +
(B′′v′′ +B′v′
)24(B′′v′′ −B′v′
) , (4.48)
and it is a straightforward matter to obtain analogous expressions characterizing the P–branch turnaround
of blue-shaded bands. Figure 4.13 illustrated this turnaround behaviour for a band in the electronic
spectrum of CuD.
Finally, we note that as was the case for vibration-rotation spectra, the intensity maxima in the
rotational branches encountered in electronic spectroscopy depend only on the population distribution in
the initial levels. As was the case for vibration-rotation spectra, the fact that ν(v′, v′′) is very much larger
than the rotational contributions to the transition energies means that the intensity maxima for a system
in thermal equilibrium will be defined by Eq. (4.37).
4.6.2 Vibrational Structure in Electronic Spectra
In electronic spectra, the wavefunction orthogonality properties which gave rise to the very strong Δv=
±1 selection rule of vibrational IR and Raman spectroscopy no longer apply, since vibrational radial
wavefunctions of different potentials are not, in general, required or expected to be orthogonal to one
another (though near-orthogonality may happen by accident). Thus, between two electronic states, in
principle any v′ ↔ v′′ transition is allowed. In practice, however, for reasons discussed below, there
are often patterns of preferred vibrational quantum number change, and observed bands often appear as
“progressions” or “sequences”, where
• a vibrational progression refers to a series of (v′, v′′) bands with a common v′ or a common v′′
value, and
• a vibrational sequence refers to a series of (v′, v′′) bands with a common
Δv = v′ − v′′ value (see figure).
The results obtained for a given electronic band system are often summarized in what is called a
“Deslandres table”, an example of which is presented in Table 4.3 (see also Bernath’s Table 9.3 [3]). In
Δυ = −2sequence
υ ′′= 0progression
v ′= 5progression
υ′′=0246
υ ′=0
246
Figure 4.14: Illustrative definition of vibrational sequences and progressions in electronic spectra.
83
Table 4.3: Deslandres table for PN
v′′ = 0 1 2 3 4 5
v′= 0 39689.8 1323.3 38376.5 1307.8 — — — —
1087.4 1909.7 1086.8
1 40786.2 1319.0 39467.2 1311.7 38155.5 1284.2 36861.3 — —
1072.9 1069.0 1071.6
2 41859.1 1322.9 40536.2 — 37932.9 1280.4 36652.5 1265.3 35387.2
1061.2 1060.0 1059.2
3 — 41597.4 1309.1 40288.3 — 37712.5 1266.1 36446.4
1042.9 1042.9
4 — — 41331.2 — 38756.4 —
5 — — — 41066.1 — 38519.4
1015.9
6 — — — 42082.0 — —
7 — — — — 41798.3
8 — — — — — 41522.6
such tables, the main entries (shown in normal ‘roman’ font) are the “origins” of the various (v′, v′′)bands, ν0(v
′, v′′), the between-row entries (in italic font) are the spacings between adjacent vibrational
levels in the upper-state, and the between-column entries (also on italic font) are the analogous spacings
between adjacent vibrational levels in the lower electronic state. Ideally, all the between-row entries in a
given row would be identical, and the same would be true of all the between-column entries in a given
column. However, no experimental measurement is perfect. Moreover, rotational selection rules forbid
{J ′ = 0} ↔ {J ′′ = 0} transitions, so there can be no direct measurement of ν0(v′, v′′) . As a result, our
estimates of band origin energies are obtained from fits to sets of observed individual rotational-vibration-
electronic transitions, and hence will always have some statistical uncertainties associated with them.
For the range of v′ and v′′ levels considered, the Deslandres table for the C 3Πu − B 3Πg system of
N2 shown in Bernath’s Table 9.3 is completely filled in [3]. However, the much sparser set of results seen
in Table 4.3, in which the observed bands are distributed along what can be viewed as the two arms of a
diagonal parabola, is much more typical. The reason for this strange intensity patterns will be discussed
below.
Because of the order(s) of magnitude differences in the sizes of electronic, vibrational, and rotational
level spacings, the spectrum for a given electronic transition is usually spread out over a wide wavenumber
(or wavelength) range, as is illustrated by the low resolution A 1Σ+ − X 1Σ+ spectrum of SrS seen in
Fig. 4.15. As might be expected, bands associated with a particular vibrational sequence (i.e., for Δv = +2,
+1 or 0) are relatively close together, while those in a given vibrational progression are relatively far apart.
We also see that in this case the bands of the Δv=2 and 1 sequences are distinctly more intense that
those of the Δv=0 sequence; reasons for the latter will be discussed below.
84
Figure 4.15: Band structure in a low resolution spectrum for the A 1Σ+ − X 1Σ+ band system of SrS.
[Bernath’s Fig. 9.14 [3]].
Vibrational Propensity Rules in Electronic Spectra:
The Classical Franck-Condon Principle
The discussion of §4.3 showed that the vibrational or radial contribution to a transition probability is
defined by the square of the radial overlap integral (or ‘matrix element’) of Eq. (4.28). For transitions
between levels of different electronic states, the fact that the upper- and lower-state radial wavefunctions
are associated with different potential energy functions means that the wavefunction orthogonality consid-
erations discussed in §4.3.3 have no relevance, so there is no Δv propensity rule of the type that applies to
vibrational transitions between levels of a single potential energy curve. Thus, the only rigorous general
v′ − v′′ selection rule is that the transition intensity will zero unless the intervals in which the upper- and
lower-state vibrational wavefunctions have non-negligible amplitude overlap significantly.
While the above ‘selection rule’ is rigorous and general, it provides no guidance for understanding the
extremely wide range of intensities observed for the (v′, v′′) transitions it does not forbid. To obtain a
good qualitative understanding of the patterns of observed transition intensities, it is useful to turn to the
classical Franck-Condon principle which states that
“Nuclear positions and momentum are are conserved in an optical transition.”
Some justification for this statement is provided by consideration of the magnitudes of the quantities
involved. The time associated with an optical absorption or emission process is ∼ 10−15 s, while the
period of vibrational motion is roughly 10−8 − 10−7 s and rotational motion is 1–2 orders of magnitude
slower than that. Thus, during the absorption process the nuclei have no time to move, so the transition
occurs ‘vertically’, at a fixed radial distance on a potential energy diagram. Similarly, the Compton
relationship shows that photons of visible light have momenta of order 10−27 kg ·m/s, while the average
radial momentum for a vibrating diatomic molecule will be of order 10−21 kg ·m/s. Thus absorption or
emission of a photon cannot significantly affect the radial momentum of a vibrating molecule. This means
that for the system at some instantaneous radial configuration, a transition is only possible if the radial
momentum, and hence also the radial kinetic energy [E − V (r)], is the same immediately before and after
the transition. As is illustrated by Fig. 4.16, for a given (v′, v′′) transition there will usually exist only one
such “stationary point”.
The above argument determines the region of internuclear distance the molecule must find itself in for
85
Figure 4.16: Definition of the “stationary point” for a particular (v′, v′′) electronic transition.
a given a particular (v′, v′′) electronic transition to occur (the stationary point). However, it says nothing
about the relative probability or itensity of such a transition. Within a classical picture, at any instant
the radial speed vr of a vibrating molecule is related to the radial kinetic energy by the expression
KErad =1
2μ (vr)
2 . (4.49)
Hence, the time that the vibrating molecule spends with its internuclear distance in the interval between
r and r + dr is
dτ ≡ fv(r) dr =
(1
vr
)dr =
√μ
2[E − V (r)]dr . (4.50)
Figure 4.17 shows the nature of the distributions function fv(r) for an arbitrary vibrational level. It is
clear that fv(r) → ∞ at the inner and outer turning point of every level. This in turn tells us that
V(r)
G(υ)
fυ(r)
KEυ( )outerturningpoint
innerturningpoint r
∞∞
r / Å
↑energy
Figure 4.17: Classical prediction for the amount of time fv(r) δr that a vibrating molecule spends within
distance δr of radius r.
86
(within this classical picture) the molecule spends most of its time with its bond length close to one of
those turning points. As a result, while a transition can in principle occur at any radial configuration, they
will be most intense if the stationary point for that transition is near one of the classical turning points
where the vibrating molecule spends most of its time. This leads to the qualitative selection rule that
“vibrational transitions in electronic spectra will be most intense when the upper
and lower vibrational level have turning points that are nearly coincident.”
Figure 4.18 shows potential energy curves, level energies, and associated turning points for the B 3Π0+u=
X 1Σ+g system of Br2. The above argument tells us that emission from v′(B) = 5 should be most intense
into v′′ = 36 and v′′ = 6, and that emission from v′(B) = 10 should be most intense into v′′ ≈ 48 and
v′′ = 2. Similarly, it predicts that absorption from v′′ = 0 should be most intense for ibrational levels
lying near dissociation.
One refinement of this discussion is the fact that the radial probability distribution for the lowest
(v = 0) vibrational level of any state should be treated differently than those for higher vibrational levels.
In particular, as is seen in Fig. 3.2, the wavefunction, and hence the quantum mechanical radial probability
distribution for the v = 0 level is centred at the potential minimum, midway between the turning points
for that level. However, because this is the lowest vibrational level the radial momemtum at that point is
relatvely small. Thus, when applying the above principle to transitions involving the v = 0 level of one
potential or the other, we expect that the post intense transitions will be those in which that potential
Figure 4.18: Potential curves, vibrational levels and their turning points for the B −X system pf Br2.
87
minimum lies at (roughly) the same internuclear distance as an inner or outer turning point on the other
potential.
Franck-Condon Factors and Vibrational Intensity Patterns
* Oscillatory Cancellation
*F-C effects on rotational intensities; e.g. the v′(B) = 10← v′′(X) = 10 band of I2* quantal description & FCF’s (Nicholls) calculate slice along v’=9, v’=10, ...
* Condon parabola and the Classical F-C principle
Table 4.4: Deslandres table for PN (taken from Herzberg [6]).
v′′ = 0 1 2 3 4 5
v′= 0 39689.8 1323.3 38376.5 1307.8 — — — —
1087.4 1909.7 1086.8
1 40786.2 1319.0 39467.2 1311.7 38155.5 1284.2 36861.3 — —
1072.9 1069.0 1071.6
2 41859.1 1322.9 40536.2 — 37932.9 1280.4 36652.5 1265.3 35387.2
1061.2 1060.0 1059.2
3 — 41597.4 1309.1 40288.3 — 37712.5 1266.1 36446.4
1042.9 1042.9
4 — — 41331.2 — 38756.4 —
5 — — — 41066.1 — 38519.4
1015.9
6 — — — 42082.0 — —
7 — — — — 41798.3
8 — — — — — 41522.6
88
Figure 4.19: Franck-Condon factor surface for the B 3Π0+u −X 1Σ+g band system of I2.
89
40000 45000 50000 55000
D(0,0)
(1,0)(2,0)
(3,0) (4,0) (5,0) (6,0) (8,0)(10,0)
(12,0)
C(0,0)(1,0) (2,0) (3,0)
(4,0)(5,0)
(6,0) (8,0)(10,0) (12,0)
(0,0)I /20 B
(2,0)(1,0) (3,0) (4,0)
A(0,0)
(1,0)
(2,0)
(3,0)(5,0)
ν / cm−1
BA C D
Figure 4.20: Illustrative FCF patterns.
90
Franck-Condon Factor Contributions to Rotational Intensities in Electronic Spectra
Figure 4.21: Schematic illustration of the centrifugal displacement of vibrational wavefunctions.
91
Figure 4.22: Centrifugally shifting wavefunctions and overlap integral for lines in the (v′, v′′) = (5, 15)
band in the B 3Π0+u −X 1Σ+g spectrum of I2.
92
Exercises
4.1 Consider the levels of the particle-in-a-box potential of Eq. (2.45) for a system with the linear transition
moment function M(r) = d0 + d2(r − L2 ) .
a) Using the exact analytic wavefunctions for this system, derive an expression for the intensity
matrix elements Mradul (v
′, v”).
b) As a rough model for the molecule Ar2, setting μ = 20u and L = 2 A, calculate the relative
intensities |Mradul (v′, v”)|2 for v′′=0 and 5, for each of Δv=1− 5 .
4.2 For the Morse model potential for Ar2 of Exercise 2.7-b) (De = 85.6 cm−1, re = 3.866 A and ωe =
31 cm−1, assuming a dipole moment function which is precisely linear in (r − re), determine the
relative intensities of Δv=1− 6 transitions for v′′=0 and 2.
References
[1] A. Einstein, Verh. d. Deutsch. Phys. Ges. 18, 318 (1916); A. Einstein, Phys. Z. bf 18, 121 (1917).
[2] L. Pauling and E. B. Wilson, Introduction to Quantum Mechanics (McGraw-Hill, New York, 1935).
[3] P. F. Bernath, Spectra of Atoms and Molecules (Oxford University Press, Oxford, 1995).
[4] A. S. Alekseyev, H.-P. Liebermann, D. B. Kokh, and R. J. Buenker, J. Chem. Phys. 113, 6174 (2000).
[5] H. Honl and F. London, Z. Physik 33, 803 (1925).
[6] G. Herzberg, Spectra of Diatomic Molecules (Van Nostrand, New York, 1950).
[7] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Ninth
Printing.
[8] D. R. Lide, Editor, Handbook of Chemistry and Physics (CRC Press, Ann Arbor, MI, USA, 1993),
57’th Edition.
[9] E. G. Lee, J. Y. Seto, T. Hirao, P. F. Bernath, and R. J. Le Roy, J. Mol. Spectrosc. 194, 197 (1999).
93
Chapter 5
Semiclassical Treatment of Bound States
5.1 Semiclassical Solution of the Radial Schrodinger
Equation
The numerical methods discussed in the previous chapter are reliable, efficient, and exact quantum me-
chanical procedures for calculating properties of a diatomic molecule from a given potential energy function
or set of functions. Moreover, those methods are at the core of least-squares fit procedures for determining
analytic potential energy functions from experimental data. However, there is no known way of performing
an exact formal inversion of the radial Schrodinger equation to allow one to determine a potential energy
function from experimental vibration-rotation data. Moreover, while qualitative ideas regarding the effects
of isotope substitution or modifications of the potential may be obtained using quantum mechanical per-
turbation theory arguments, it is difficult to utilize such considerations in the quantitative interpretation
of experimental data.
Semiclassical or “phase integral” or “WKB” (for Weltzel, Kraemers and Brillouin) or “JWKB” (with
with Jeffries added) methods are “asymptotic” approximation methods for solving differential equations
that have been well known to mathematicians for more than a century. They were already well developed
in the early days of quantum mechanics, and while not exact, they often provide quite accurate results,
particularly for cases in which the effective (reduced) mass of the system is fairly large. Moreover, the
tractability of these methods made them useful long before the spread of digital computers made modern
numerical methods practical, and the explicit expressions they yield are an important source of physical
insight. In particular, semiclassical theory is the basis for much of our understanding of isotope effects on
molecular spectra and for some of the most widely used data inversion procedures in molecular physics
[1, 2, 3, 4].
5.1.1 The First-Order Wave Function
The starting point of this discussion is the familiar effective one-dimensional radial Schrodinger equation
of Eq. (3.1), slightly re-arranged to give the form
d2 ψ(r)
dr2+
2μ
�2[E − U(r)]ψ(r) = 0 (5.1)
where (again) the effective potential U(r) may include a centrifugal term, as in Eq. (3.23). The semiclassical
approach begins by defining the eigenfunction to have the form
ψ(r) = ei S(r)/� (5.2)
94
where as usual i ≡ (−1)1/2. Taking first and second derivatives of ψ(r), substituting them into Eq. (5.1),
and dropping the common factor ei S(r)/� then yields the following differential equation for S(r),
i�d2 Sdr2
−(dSdr
)2
+ 2μ [E − U(r)] = 0 (5.3)
which is a non-linear “Riccati” equation. This step introduces no approximations, since it merely replaces
a differential equation for the unknown function ψ(r) by one for the unknown function S(r), and a solution
for the latter is readily transformed into one for the former.
We now note that the constant � is quite small. This suggests that the first term in Eq. (5.3) is “small”
relative to the second one, ∣∣∣∣i� d2 Sdr2∣∣∣∣ � (
dSdr
)2
(5.4)
and hence that as a first approximation (partially corrected for below), the former may be neglected.
Omitting that term, defining
Q(r)2 =(2μ/�2
)[E − U(r)] (5.5)
and taking the square root on both sides of Eq. (5.3) yields the simple differential equation
dS(0)dr
= ± �Q(r) = ±√
2μ [E − U(r)] (5.6)
where S(0)(r) is our “zeroth order” estimate of the exact solution to Eq. (5.3). For a known potential energy
function U(r), the solution of Eq. (5.6) is simply
S(0)(r) = ± �
∫ r
Q(x) dx = ±∫ r √
2μ [E − U(x)] dx (5.7)
The next step is to correct for our neglect of the second derivative term in the original equation,
Eq. (5.3). To this end, we first note that use of the approximate solution S(0)(r) allows the (small) second-
derivative term to be approximated by
i�d2 S(r)dr2
≈ i�d2 S(0)(r)dr2
= ± i�2 dQ(r)
dr= ± i�2Q′(r) (5.8)
where the prime (′) denotes differentiation with respect to r. Substituting this result into Eq. (5.3) and
rearranging it then yields a differential equation for an improved “first-order” estimate of S(r),
dS(1)dr
= ± [�2Q(r)2 ± i�2Q′(r)]1/2 (5.9)
Again, the right side of this equation consists of a known function of the potential, so it may simply be
integrated to yield
S(1)(r) = ± �
∫ r [Q(x)2 ± iQ′(x)]1/2 dx (5.10)
= ± �
∫ r
Q(x)[1± iQ′(x)/Q(x)2
]1/2dx
Within the zeroth -order assumption that S(r) ≈ S(0)(r) , the approximation of Eq. (5.4) implies that∣∣Q′(r)/Q(r)2∣∣� 1 . This in turn implies that the second factor in the above integrand may be approximated
95
Figure 5.1. Comparison of exact quantal wave functions (solid curves) with first-order semiclassi-
cal wavefunctions (dotted curves) for the v = 0 and 2 levels of a harmonic oscillator potential
(from Fig. 2.1 of Ref. [4]).
by the leading terms of a binomial expansion, and the resulting expression evaluated as before:
S(1)(r) ≈ ±�∫ r
Q(x)
{1± i
2
Q′(x)Q(x)2
+ ...
}dx
= ±�∫ r
Q(x) dx+i�
2
∫ r Q′(x)Q(x)
dx
= ±�∫ r
Q(x) dx+i�
2lnQ(r)− i� lnC (5.11)
in which lnC is an integration constant. Finally, we substitute this approximation for the exact S(r) intoEq. (5.2) to obtain the first “first-order” semiclassical approximation for the wave function
ψ(1)(r) = exp
{± i∫ r
Q(x) dx
}× exp
{i2 ln
√Q(r)
}× exp (lnC)
=C√|Q(r)| exp
{± i∫ r
Q(x) dx
}(5.12)
The definition of Q(r) implies that it, and hence any integral with it as integrand, is a pure (math-
ematically) real quantity in “classically-allowed” regions where E > U(r) , and an “imaginary” quan-
tity in “classically-forbidden” regions where E < U(r) . From the familiar Euler relation of calculus
( eiφ = cosφ+ i sinφ ), it is therefore clear that the semiclassical wave function of Eq. (5.12) is an oscilla-
tory sinusoidal function in classically-allowed regions, and is a linear combination of exponentially growing
and exponentially dying functions in classically forbidden regions. Thus, this result is a natural generaliza-
tion of the familiar exact analytic results for step-function potentials, and on intervals where the potential
is completely flat, the semiclassical result coincides with those exact solutions.
96
We have noted above that the condition that must be satisfied if Eq. (5.12) is to be a good approximation
to the exact wavefunction is that
|Q′(r)/Q(r)2| =√
�2/2μ∣∣∣U ′(r)/[E − U(r)]3/2
∣∣∣ � 1 (5.13)
This condition will hold whenever the potential is relatively slowly varying (i.e., when |U ′(r)| is small) or
|E − U(r)| relatively large, but it will always break down near a “turning point”, which is defined as a
point where E = U(r) . This breakdown is seen in Fig. 5.1, which compares exact quantum mechanical
wave functions (solid curves) with the associated first-order semiclassical wave functions (dashed curves)
for levels of a harmonic oscillator-type potential. While the first-order wave function has the correct
qualitative behaviour at distances away from the classical turning points, the fact that the amplitude
factor denominator Q(r) goes to zero at those points makes the semiclassical wave function of Eq. (5.12)
completely unrealistic there. Finding a solution to this dilemma, a means of avoiding or properly treating
these regions of singular behaviour, was the key to the use of semiclassical methods.
5.1.2 Connection Formulae and The First-Order or Bohr-Sommerfeld
Quantization Condition
To provide a framework for the present discussion, let us consider the nature of the wavefunction near
the inner turning point of one of the higher vibrational levels considered in Fig. 5.1. As in an elementary
quantum treatment of a step-function potential, we first consider the general form of the wave function in
different regions, apply limiting boundary conditions where appropriate, and then address the problem of
connecting the two separate descriptions. In particular, as illustrated in Fig. 5.2, the classically forbidden
region r < r1(E) is called “Region II” and the classically allowed region r > r1(E) “Region I”. From
Eq. (5.12) we see that the general solution in Region II may be written as
ψII(r) =1√|Q(r)|
{A exp
(∫ r
|Q(x)|dx)+B exp
(−∫ r
|Q(x)|dx)}
(5.14)
where A and B are constants. Similarly, the general solution in Region I may be written
ψI(r) =1√Q(r)
{C exp
(i
∫ r
Q(x) dx
)+D exp
(− i∫ r
Q(x) dx
)}=
1√Q(r)
{E cos
(∫ r
Q(x) dx
)+ F sin
(∫ r
Q(x) dx
)}(5.15)
where again C, D, E and F are scalar constants. It is of course clear that both of these expressions break
down near the turning point where Q(r)→ 0 ; this finding is no surprise, in view of the condition Eq. (5.13)
for the validity of the semiclassical wavefunction. This problem will be addressed by finding a solution to
the Schrodinger equation that is accurate near a turning point, and connecting it to these semiclassical
solutions that are valid far from the turning point(s).
On any narrow interval, such as the immediate neighbourhood of a turning point r= r1 , it is clearly
reasonable to approximate the actual potential by a linear function,
U(r) ≈ E + U ′(r1)(r − r1) = E + S×(r − r1) (5.16)
At an “inner” turning point where the potential slope S is negative (as in Fig. 5.2), introducing the change
of variable
z =(|S| 2μ/�2)1/3 (r1 − r) (5.17)
97
Figure 5.2. Comparison of exact quantal wave functions (solid curves) with first-order semiclassi-
cal wavefunctions (dotted curves) for the v = 0 , 2 and 4 levels of a harmonic oscillator [From
Fig. 2.1 of Ref. [4]].
transforms Eq. (5.1) into one of the well-known differential equations of mathematical physics [5],
d2 ψ(z)
dz2= z ψ(z) (5.18)
The exact eigenfunctions of Eq. (5.18) are the Airy functions Ai(z) and Bi(z), both of which have sinusoidal
oscillatory behaviour at negative values of z, while at positive z Ai(z) dies off in an exponential-like manner
while Bi(z) undergoes exponential-type growth. More particularly, if we introduce our linear potential
model into Eq. (5.5) and define the modified variables
Qa =√
(2μ/�2)[E − U(r)] =(|S| 2μ/�2)1/3 z1/2 (5.19)
for the classically-allowed region where E ≥ U(r) or z ≥ 0 , and
Qf =√
(2μ/�2)[U(r)− E] =(|S| 2μ/�2)1/3 |z|1/2 (5.20)
for the forbidden region where E ≤ U(r) or z ≤ 0 , the the well-known asymptotic forms of the Airy
function may be written as:
for positive z � 0 , Ai(z) ∝ 1√Qf (r)
× exp
{−∫ r1
rQf (x) dx
}(5.21)
Bi(z) ∝ 1√Qf (r)
× exp
{+
∫ r1
rQf (x) dx
}(5.22)
and
for negative z � 0 , Ai(z) ∝ 1√Qa(r)
× cos
{∫ r
r1
Qa(x) dx − π
4
}(5.23)
98
Bi(z) ∝ 1√Qa(r)
× cos
{∫ r
r1
Qa(x) dx+π
4
}(5.24)
Thus, it is clear that in the classically-forbidden region at large positive z values, the Airy function of
the first kind takes on the form of the exponentially-dying-off semiclassical solution of Eq. (5.15) which
satisfies the square-integrability boundary condition. This means in turn that the wavefunction away from
the turning point in the classically-allowed region at z � 0 will be given by Eq. (5.23), which matches the
form of the square-integrable component of the semiclassical wavefunction of Eq. (5.14) in this region.
If we now generalize the discussion and consider the two-turning-point problem of Fig. 5.1, it is clear
that with a little manipulation of variables we can treat the wavefunction near the outer turning point
r = r2(E) in exactly the same manner. In this case, the normalizable semiclassical wavefunction at r � r2
ψIII(r) =B
|E − U(r)|1/4 × exp
{−∫ r
r2
Qf (x) dx
}(5.25)
maps onto Ai(z), which in turn connects it to the semiclassical wavefunction in the allowed region:
ψII(r) =A
[E − U(r)]1/4× cos
{∫ r2
rQa(x) dx− π
4
}. (5.26)
However, by the normal rules of calculus,
cos
{∫ r2
rQa(x) dx− π
4
}= cos
{∫ r2
r1
Qa(x) dx −∫ r
r1
Qa(x) dx − π
4
}(5.27)
= sin
{∫ r2
r1
Qa(x) dx−∫ r
r1
Qa(x) dx +π
4
}= sin
{∫ r2
r1
Qa(x) dx
}cos
{∫ r
r1
Qa(x) dx − π
4
}− cos
{∫ r2
R1
Qa(x) dx
}sin
{∫ r
r1
Qa(x) dx − π
4
}.
For this wavefunction solution, which was connected from the outer turning point, to exactly match the
wavefunction Eq. (5.23) obtained by connecting through the inner turning point, necessarily
sin
{∫ r2
r1
Qa(x) dx
}= 1 and cos
{∫ r2
r1
Qa(x) dx
}= 0 . (5.28)
This will only be true if the integral forming the argument is precisely a half-integer multiple of π:∫ r2
r1
Qa(x) dx =√
2μ/�2∫ r2
r1
√E − U(x) dx = (v + 1
2 )π , (5.29)
where v is a non-negative integer (= 0, 1, 2, 3, ... etc.). This important result is known variously as the
the semiclassical, or first-order JWKB, or phase-integral, or Bohr-Sommerfeld quantization condition; it is
the foundation for all the results presented in the rest of this chapter.
The normal working form of the first-order semiclassical quantization condition is
v + 12 =
1
π
√2μ
�2
∫ r2(E)
r1(E)[E − U(r)]1/2 dr . (5.30)
For any given potential energy function U(r), this integral across the classically-allowed interval between
turning points r1(E) and r2(E) may readily be computed at any chosen energy E. The nature of this
99
0.0 0.2 0.4 0.6 0.8 1.0
1
2
3
4
5
6
7
8
9
10
11
υ+½
E /De
Eυ=3
Eυ=4
Eυ=2Eυ=1Eυ=0
Eυ=6
Eυ=5
B
00.8 1.2 1.6 2.0r / r
V(r)
← E
√E−V(r)
r2(E)r1(E)
⎯⎯⎯
A
e
Figure 5.3. Illustration of use of the semiclassical quantization condition to determine vibrational
level energies.
integrand for energy E on a given model potential is shown as the shaded region on Fig. 5.3A. The allowed
eigenvalues, the energies for which the wavefunction satisfies the normalizability requirement, are the
energies for which the right-hand side of this equation has precisely half-integer values (12 ,32 ,
52 ,
72 , ...
etc.). The use of this criterion to determine the eigenvalues of a model potential is illustrated by Fig. 5.3B,
where the solid curve shows how the left-hand side of Eq. (5.30) varies with energy for a model Lennard-
Jones(10,6) potential.1 Horizontal dotted lines are drawn at half-integer values of the ordinate, and the
energies at which they intersect the smooth curve (solid points) indicate the allowed vibrational eigenvalues
of this system.
5.1.3 Higher-Order Semiclassical Treatments
As implied above, the ordinary first-order wavefunction and quantization condition are the basis for much
of the physical insight and practical procedures associated with semiclassical methods. However, because
they involve approximations, there is a limit to the absolute accuracy which may be attained in first order.
This concern directs us to examine higher-order semiclassical methods.
In the preceding discussion we saw that when the zero’th order solution S(0)(r) was used to estimate
the initially-omitted second-derivative term in Eq. (5.3), we obtained the improved solution S(1)(r), whichin turn yielded the first-order wavefunction ψ(1)(r) of Eqs. (5.14) and (5.15). Similarly, one can use S(1)(r)to define an improved estimate of that second-derivative term, and apply the same procedure to determine
the second-order solution S(2)(r), and hence the second-order semiclassical wavefunction ψ(2)(r). This
procedure does work, and may in principle be iterated to high order; however, it is algebraically rather
tedious.
A more systematic approach to the derivation of high-order semiclassical methods has been developed
1 The precise model is an LJ(10,6) potential with well depth De = 10 000 cm−1 and re = 1 A, and the reduced
mass used was that for 6,6Li2.
100
by Froman and Froman and co-workers [2]. Their approach begins by assuming that the solution to
Eq. (5.3) may be expanded as a power series in the quantity � :
S(r) = S0(r) + �S1(r) + �2 S2(r) + �
3 S3(r) + . . . . (5.31)
Note that when comparing this expansion with the solutions of Eqs. (5.7)-(5.11), we see that S0(r) =
S(0)(r) , while S(1)(r) is the sum of the first two terms in Eq. (5.31), S(2)(r) the sum of the first three
terms, . . . etc. Substituting the expansion of Eq. (5.31) into Eq. (5.3) yields the differential equation(i�d2 S0dr2
+ i�2d2 S1dr2
+ i�3d2 S2dr2
+ i�4d2 S3dr2
+ · · ·)
(5.32)
−(dS0dr
+ �dS1dr
+ �2 dS2dr
+ �3 dS3dr
+ · · ·)2
+ 2μ [E − U(r)] = 0 .
Eq. (5.32) will only be satisfied for all values of � (considered as an expansion variable) if the terms
contributing to each order in powers of � are individually zero. In particular, setting terms of order �0 to
zero implies
−(dS0dr
)2
+ 2μ [E − U(r)] = 0 , (5.33)
which is equivalent to Eq. (5.6), and hence yields the zero’th order solution S0(r) = S(0)(r) of Eq. (5.7).
Collecting terms of order �1 and setting their sum equal to zero yields the differential equation
−i d2 S0dr2
+ 2dS0dr
dS1dr
= 0 . (5.34)
Utilizing our expression for S0(r), rearranging Eq. (5.34), and integrating the resulting expression for
dS1/dr then yields
S1(r) = (i/2) lnQ (5.35)
One may readily verify that substitution of the sum S(r) = S0(r) + �S1(r) into Eq. (5.2) yields the
first-order wavefunction of Eq. (5.12).
Next, collecting terms of order �2 and setting their sum equal to zero yields the differential equation
− i d2 S1dr2
+
(dS1dr
)2
+ 2dS0dr
dS2dr
= 0 (5.36)
which in turn may be rearranged to yield
dS2(r)dr
=2 (Q′)2 −Q′′Q
4Q3=
√�2/2μ
4
U ′′(r)[E − U(r)]3/2
(5.37)
Similarly, for each successive higher orderm, collecting terms of order �m gives an expression for dSm(r)/dr
in terms of powers of Q(r) and its derivatives, or in powers of terms of [E −U(r)] and its derivatives. The
net result of this treatment is a general high-order semiclassical wavefunction of the form
ψsc(r) = exp
{± i∫ r
[y0(x) + y1(x) + y2(x) + · · ·] dx}
(5.38)
Through some complex manipulations Froman and Froman and collaborators showed that the odd-order
terms y2m+1 effectively determine the pre-exponential wavefunction amplitude factor, that they may be
101
expressed in term of combinations of the even-order terms y2m, and that a series re-summation yields the
general result that
ψsc(r) =A
[ y0 + y2 + y4 + . . . ]]1/2exp
{± i∫ r
[y0 + y2 + y4 + · · · ] dx}
=A√ysc(r)
exp
[± i∫ r
ysc(x) dx
](5.39)
where ysc(r) = y0(r) + y2(r) + y4(r) + . . . , with
y0(r) =
(2μ
�2
)1/2
[E − U(r)]1/2 (5.40)
y2(r) =1
48
(�2
2μ
)1/2U ′′(r)
[E − U(r)]3/2(5.41)
y4(r) =1
1536
(�2
2μ
)3/25U ′(r)U ′′′(r)− 7U ′′(r)2
[E − U(r)]7/2(5.42)
and analogous, albeit increasingly complicated expressions for higher-order terms. Note, that within the
systematics of this treatment, the possible solutions are based on inclusion of consistent orders of y2m(r) in
the exponent integral and amplitude factor, are identified as being “first-order” when only y0(r) appears,
“third-order” when the argument is [y0 + y2] , fifth-order when [y0 + y2 + y4] is used, ... etc. Thus, one
typically refers to the higher-order treatments in terms of odd-number orders in which the same set of
terms is used to define both the wavefunction phase and its amplitude.
This is a very elegant result in that the form of the resulting semiclassical wavefunction is the same in
any order, the only difference being the number of terms included in the expansion for ysc(r) As with the
simple first-order result of Eq. (5.12), ψsc(r) will be an oscillatory sinusoidal function of r in the classically-
allowed region where Q2(r) = (2μ/�2)[E − U(r)] > 0 , and a linear combination of exponentially growing
and dying terms when Q2(r) < 0 Moreover, application of the same type of connection formula arguments
used above yields the high-order semiclassical quantization condition
v + 12 =
1
2π
(2μ
�2
)1/2{∮[E − U(r)]1/2 dr +
1
48
(�2
2μ
)∮U ′′(r)
[E − U(r)]3/2dr (5.43)
+1
1536
(�2
2μ
)2 ∮5U ′(r)U ′′′(r)− 7U ′′(r)2
[E − U(r)]7/2dr + · · ·
}= Δ1(E) + Δ3(E) + Δ5(E) + · · ·
where as before, the allowed eigenvalues are the energies for which the sum of terms on the right-hand
side is precisely a half integer. In terms of Fig. 5.3, the solid curve shows the energy dependence of Δ1(E),
while the long-dash curve schematically shows the energy dependence of the sum Δ1(E) + Δ3(E) . The
small higher-order terms Δ3(E), Δ5(E), ... etc., will give rise to small shifts in the eigenvalues, defined as
the energies at which the resulting curve has precisely half-integer values. Such differences are in general
too small to see on a plot such as Fig. 5.3. However, Table 5.1 shows the results of a precise numerical test
of the accuracy of the first-, third- and fifth-order quantization condition for levels of a model potential of
depth De = 10000 cm−1 with parameters approximately resembling a chemically-bound hydride molecule
[6].2 This suggests that for strongly-bound hydride molecules, the eigenvalue errors associated with a
2 More precisely, this model system involved a reduced mass of 1 u and an LJ(12,6) potential with De =
10 000 cm−1 and re = 4.1058 0432 A [6].
102
first-order treatment are of order 1 cm−1, while those associated with third- and fifth-order quantization
are, respectively, four and eight orders of magnitude smaller. Indeed, semiclassical calculations of this type
have been performed for orders up to thirteen [7]! For species of larger reduced mass, however, Eq. (5.43)
shows that the importance of the higher-order terms is sharply reduced, and to “normal” levels of accuracy,
for systems of reduced mass μ � 20 u, first-order methods may often be treated as being virtually exact.
One intriguing apparent anomaly of the results in Table 5.1 is the fact that the errors in the first-order
semiclassical eigenvalues seem to be approaching zero for levels approaching the dissociation limit ( v=23
is the highest bound level supported by this potential), while the errors in the higher-order eigenvalues grow
become larger rapidly for levels approaching dissociation. Indeed, Table 5.2 shows that for semiclassical
orders higher than seven, the error in the v = 23 eigenvalue becomes larger with increasing order [7].
As discussed in Refs. [6] and [7], this intriguing behaviour may be understood using the near-dissociation
theory to be discussed in Section 5.4.
Table 5.1 Test of first-, third-, fifth- and seventh-order semiclassical eigenvalues for a model LJ(12,6)
potential with De = 10000 cm−1.
[Esemiclassical − Equantal] / cm−1
v E(v) 1st order 3rd order 5th order 7th order
0 589.5397 −0.8584 0.0000 5118 0.0000 0000 0.0000 0000
1 1699.9792 −0.8249 0.0000 5206 0.0000 0000 0.0000 0000
2 2723.5430 −0.7909 0.0000 5297 0.0000 0000 0.0000 0000
3 3663.0705 −0.7563 0.0000 5391 0.0000 0000 0.0000 0000
4 4521.4796 −0.7211 0.0000 5491 0.0000 0000 0.0000 0000
5 5301.7709 −0.6854 0.0000 5598 0.0000 0000 0.0000 0000
6 6007.0316 −0.6492 0.0000 5716 0.0000 0000 0.0000 0000
7 6640.4393 −0.6124 0.0000 5848 0.0000 0000 0.0000 0000
8 7205.2662 −0.5752 0.0000 5998 0.0000 0000 0.0000 0000
9 7704.8829 −0.5376 0.0000 6173 0.0000 0000 0.0000 0000
10 8142.7630 −0.4995 0.0000 6383 0.0000 0000 0.0000 0000
11 8522.4859 −0.4611 0.0000 6640 −0.0000 0001 0.0000 0000
12 8847.7411 −0.4225 0.0000 6960 −0.0000 0001 0.0000 0000
13 9122.3309 −0.3837 0.0000 7365 −0.0000 0002 0.0000 0000
14 9350.1727 −0.3447 0.0000 7889 −0.0000 0003 0.0000 0000
15 9535.3009 −0.3058 0.0000 8579 −0.0000 0004 0.0000 0000
16 9681.8669 −0.2670 0.0000 9507 −0.0000 0006 0.0000 0000
17 9794.1384 −0.2285 0.0001 0785 −0.0000 0010 0.0000 0000
18 9876.4963 −0.1904 0.0001 2610 −0.0000 0018 0.0000 0000
19 9933.4298 −0.1528 0.0001 5351 −0.0000 0034 0.0000 0000
20 9969.5286 −0.1160 0.0001 9805 −0.0000 0075 −0.0000 000121 9989.4725 −0.0800 0.0002 8101 −0.0000 0222 −0.0000 000722 9998.0166 −0.0449 0.0004 8423 −0.0000 1234 −0.0000 008923 9999.9730 −0.0102 0.0015 8813 −0.0007 0317 0.0001 9692
103
Table 5.2 Test of very high-order semiclassical eigenvalues for the highest levels of the LJ(12,6) potential
function of Table 5.1.
[Esemiclassical − Equantal] / cm−1
v 3rd order 5th order 7th order 9th order 11th order 13th order
20 0.0001 9805 −0.0000 0075 −0.0000 0001 0.0000 0001 0.0000 0000 0.0000 0000
21 0.0002 8101 −0.0000 0222 −0.0000 0007 0.0000 0002 0.0000 0000 0.0000 0000
22 0.0004 8423 −0.0000 1234 −0.0000 0089 0.0000 0056 −0.0000 0016 0.0000 0001
23 0.0015 8813 −0.0007 0317 0.0001 9692 0.0027 6925 −0.0076 0611 a
a In a 13’th order treatment, this level does not exist.
104
5.2 Dunham Theory for Diatomic MoleculesBy the early 1930’s a number of simple analytic potential energy functions had been devised for which
exact analytic expressions had been derived for the vibrational energies, and (rarely) sometimes also for the
energies of the rotational sublevels. However, such few-parameter potentials could not accurately represent
real systems, and could not quantitatively explain extensive high-quality spectroscopic data. A critically
important development for molecular spectroscopy was then provided by two seminal papers published
by J.L. Dunham in 1932. The first of these presented a general derivation of the semiclassical or WKB
quantization condition to what we here call fifth order [8], while the second applied that quantization
condition expression to a flexible general model for a potential energy function to obtain an explicit
expression for the vibration-rotation energies in terms of the vibration-rotation quantum numbers and the
parameters of the potential [9].
The potential energy form used by Dunham was a simple power series expansion about a harmonic
oscillator leading term:
VDun(r) = a0 x2[1 + a1 x+ a2 x
2 + a3 x3 + · · ·] = a0 x
2
(1 +
∑i=1
ai xi
), (5.44)
in which x ≡ (r − re)/re . Using this same reduced variable, the centrifugal potential may be written as
�2
2μ
J(J + 1)
r2=
�2
2μ re2J(J + 1)
(r/re)2= Be
J(J + 1)
(1 + x2)
= Be [J(J + 1)][1− 2x+ 3x2 − 4x3 + · · · ] . (5.45)
This allows the overall effective centrifugally-distorted potential VJ(r) of Eq. (2.27) or (3.23) may be ex-
pressed as a power series in x with coefficients defined in terms of [J(J + 1)] and the {ai} values. Substi-tuting this power series expansion into the integrands of the various terms in the quantization condition of
Eq. (5.43), making appropriate changes of variables, and applying techniques for integration in the complex
plane [9, 10], he found that the quantization condition could be written as a power series in the energy
v + 12 = A0 +A1E +A2E
2 +A3E3 · · · , (5.46)
in which the expansion coefficients {Ai} are explicit functions of the potential expansion parameters {ai}and of [J(J + 1)]. Performing reversion of series and collecting terms of the same order in powers of
[J(J + 1)] then yields the famous Dunham expression for the level energies of a rotating vibrator:
E(v, J) =∑m=0
∑l=0
Yl,m (v + 12 )
l [J(J + 1)]m , (5.47)
in which the {Yl,m} coefficients are explicit known functions of the potential expansion parameters {ai}and the reduced mass μ.
The conventional way of writing the expressions for the {Yl,m} coefficients involves use of two scaling
parameters:
Be = �2/2μ re
2 (5.48)
ωe = ωe(Harm.Osc.) ≡√
4 a0 (�2/2μ re2) =√
4 a0 Be (5.49)
These are, respectively, the inertial rotational constant for a hypothetical level lying precisely at the
potential minimum, and the vibrational frequency for a rigid rotor/harmonic oscillator model of the system;
within the first-order semiclassical approximation they would equal the leading coefficients Y0,1 and Y1,0,
105
respectively. They approximately describe the magnitude of rotational and vibrational level spacings of
the lowest level, and since typically(Be/ωe
)� 1 , their ratio serves as a parameter to indicate the relative
magnitudes of various terms in the following expression.
The original paper of Dunham [9] presents the following expressions for the fifteen Yl,m’s corresponding
to (l +m) ≤ 4 :
Y0,0 = 18 ωe
(Be/ωe
) [3 a2 − 7 a1
2/4 +O (Be/ωe
)2](5.50)
Y1,0 = ωe
[1 + 1
4
(Be/ωe
)2 (25 a4 − 95 a1 a3/2− 67 a2
2/4 + 459 a12 a2/8− 1155 a1
4/64)
+O (Be/ωe
)4](5.51)
Y2,0 = 12 ωe
(Be/ωe
) [3(a2 − 5a1
2/4) + 12
(Be/ωe
)2(245 a6 − 1365 a1 a5/2− 885 a2 a4/2
− 1085 a32/4 + 8535 a1
2 a4/8 + 1707 a23/8 + 7335 a1 a2 a3/4− 23865 a1
3 a3/16
−62013 a12 a22 + 239985 a14 a2/128 − 209055 a1
6/512)+O (Be/ωe
)4](5.52)
Y3,0 = 12 ωe
(Be/ωe
)2 [10 a4 − 35 a1 a3 − 17 a2
2/2 + 225 a12 a2/4− 705 a1
4/32 +O (Be/ωe
)2](5.53)
Y4,0 = 5ωe
(Be/ωe
)3 [7 a6/2− 63 a1 a5/4− 33 a2 a4/4− 63 a3
2/8 + 543 a12 a4/16
+ 75 a23/16 + 483 a1 a2 a3/8− 1953 a1
3 a3/32 − 4989 a12 a2
2/64
+23265 a14 a2/256 − 23151 a1
6/1024 +O (Be/ωe
)2](5.54)
Y0,1 = Be
[1 + 1
2
(Be/ωe
)2 (15 + 14 a1 − 9 a2 + 15 a3 − 23 a1 a2 + 21(a1
2 + a33)/2
)+O (Be/ωe
)4](5.55)
Y1,1 = Be
(Be/ωe
) [6(1 + a1) +
(Be/ωe
)2(175 + 285 a1 − 335 a2/2 + 190 a3 − 225 a4/2
+ 175 a5 + 2295 a12/8− 459 a1 a2 + 1425 a1 a3/4− 795 a1 a4/2 + 1005 a2
2/8
− 715 a2 a3/2 + 1155 a13/4 − 9639 a1
2 a2/16 + 5145 a12 a3/8 + 4677 a1 a2
2/8
−14259 a13 a2/16 + 31185(a14 + a1
5)/128)+O (Be/ωe
)4](5.56)
Y2,1 = 6Be
(Be/ωe
)2 [5 + 10 a1 − 3 a2 + 5 a3 − 13 a1 a2 + 15(a1
2 + a13)/2 +O (Be/ωe
)2](5.57)
Y3,1 = 20Be
(Be/ωe
)3 [7 + 21 a1 − 17 a2/2 + 14 a3 − 9 a4/2 + 7 a5 + 225 a1
2/8− 45 a1 a2
+ 105 a1 a3/4− 51 a1 a4/2 + 51 a22/8− 45 a2 a3/2 + 141 a1
3/4 − 945 a12 a2/16
+435 a12 a3/8 + 411 a1 a2
2/8− 1509 a13 a2/16 + 3807(a1
4 + a15)/128 +O (Be/ωe
)2](5.58)
Y0,2 = −4Be
(Be/ωe
)2 [1 + 1
2
(Be/ωe
)2(163 + 199 a1 − 119 a2 + 90 a3 − 45 a4 − 207 a1 a2
+205 a1 a3/2− 333 a12 a2/2 + 693 a1
2/4 + 46 a22 + 126(a1
3 + a14/2)
)+O (Be/ωe
)4](5.59)
Y1,2 = −12Be
(Be/ωe
)3 [19/2 + 9 a1 + 9 a1
2/2− 4 a2 +O(Be/ωe
)2](5.60)
Y2,2 = −24Be
(Be/ωe
)4 [65 + 125 a1 − 61 a2 + 30 a3 − 15 a4 + 495 a1
2/4− 117 a1 a2 + 26 a22
+95 a1 a3/2− 207 a12 a2/2 + 90(a1
3 + a14/2) +O (Be/ωe
)2](5.61)
Y0,3 = 16Be
(Be/ωe
)4 [3 + a1 +O
(Be/ωe
)2](5.62)
Y1,3 = 12Be
(Be/ωe
)5 [233 + 279 a1 + 189 a1
2 + 63 a13 − 88 a1 a2 − 120 a2 + 80 a3/3 +O
(Be/ωe
)2](5.63)
Y0,4 = 64Be
(Be/ωe
)6 [13 + 9 a1 − a2 + 9 a1
2/4 +O (Be/ωe
)2](5.64)
106
In subsequent years Sandeman [11] and others [10] have extended these expressions to include contributions
of higher-order in(Be/ωe
)2, and those due to potential expansion coefficients ai for i > 6 .
Note that conventional usage defines Be=Y0,1 and ωe=Y1,0 as the actual “true” experimental values
of these constants, so while the quantities Be and ωe comprise the dominant contributions to those terms,
they are not identical to them except in the limit when(Be/ωe
)2 → 0 . On the other hand, if one does
neglect terms of order O (Be/ωe
)2, one can show that
Y0,0 =Be − ωexe
4+ωe αe
12Be+
(ωe αe
12Be
)2
=Y0,1 + Y2,0
4− Y1,0 Y1,1
12Y0,1+
(Y1,0 Y1,112Y0,1
)2
(5.65)
and a number of other relationships among these constants may be generated.
Consideration of Eqs. (5.50)-(5.64) shows that the magnitudes of various coefficients, and also of terms
neglected by truncation, depend on powers of the ratio (Be/ωe). This quantity is usually quite small,
with values of 1.4 × 10−2 for ground-state H2, 2.7 × 10−3 for HF, 1.9 × 10−3 for Li2, 1.0 × 10−3 for Na2,
9.1 × 10−4 for O2, 8.5 × 10−4 for N2, 6.1 × 10−3 for K2, 4.0 × 10−4 for Rb2 and 1.7 × 10−4 for ground
state I2 [12]. Thus, it is clear that the higher-order corrections will be relatively unimportant for heavier
(large–μ) species. This
5.2.1 Successes of the Dunham Result
• Provided sound theoretical basis for the empirical level-energy expressions which had come into
common use based on intuitive generalization of exact expressions for idealized models.
Basis of virtually all practical spectroscopic analysis since 1932
• Can be directly used to determine potential expansion parameters {ai} from experimental Yl,m’s.
For an arbitrary analytic potential, expanding it as a Taylor series about re and using the derivatives
at re to define a set of Dunham {ai}, which may then be used to obtain expressions for (at least) the
first few Yl,m’s in terms of the parameters of that potential. E.g., apply this to a simple (constant
β(r) = β0 ) MLJ potential! For a Morse potential, can show that aj = (−a0)j 2j+2/(J + 2)! . Also
allows one to get rotational dependence for a harmonic oscillator.
107
• Shows that the potential {ai} values may be determined from only a knowledge of the {Yl,0} and
{Yl,1} coefficients, and hence that the CDC’s and their expansion coefficients {Yl,m} for m ≥ 2 are
em derived quantities, effectively determined by the G(v) and Bv expansions. However, this is not
algebraically transparent!
• Expressions have been derived for expectation values and matrix elements of powers of x between
different vibration-rotation levels as functions of the Dunham {ai} parameters and the associated
quantum numbers [13]. Also explicit expressions for “Herman-Wallis factors”:
F v′v′′ = 〈ψv′,J ′(r)|M(r)|ψv′′,J ′′(r)〉2/〈ψv′,0(r)|M(r)|ψv′′ ,0(r)〉2 .
• The leading contribution to each coefficient is the basis of the conventional simple first-order semi-
classical mass-scaling
Explicitly shows how the simple first-order mass scaling breaks down, and that the importance of
this breakdown decreases with the factor (Be/ωe)2
Isotope Effects in First-Order Semiclassical Treatment
Within the first-order quantization condition of Eq. (5.30), the allowed eigenvalues of the system are the
energies E for which the integral on the right hand side of this equation, multiplied by the appropriate
collection of constants, is precisely a half integer (12 ,32 ,
52 ,
72 , ... etc.). At the same time, as is shown by
Fig. 5.3, the integral itself is a smooth function of the energy, and the right hand side of this equation will
increase smoothly from a value of zero at the potential minimum to some limiting value (usually finite, see
§5.4). It is also clear that one cam remove the mass dependence from the right hand side of that equation
by writing
η(E) ≡ v + 12√μ
=1
π
√2
�2
∫ r2(E)
r1(E)[E − U(r)]1/2 dr (5.66)
where the function η(E) is exactly the same for all isotopologues associated with a given potential energy
function. This means that a given total energy E that corresponds to level v1 of isotopologue–1 will also
be associated with level
v2(μ2) =√μ2/μ1
(v1(μ1) +
12
) − 12 (5.67)
108
of isotopologue–2. Of course non-integer vibrational quantum numbers are not physically allowed within
quantum mechanics. However, within a semiclassical treatment E(v) is a smooth function of v, and we
can legitimately talk about the energies associated with fractional v. Thus, in the context of a plot such as
that seen in Fig. 5.3, the allowed eigenvalues for some minor isotopologue would be the energies associated
with fractional values of v2(μ2) +12 calculated from Eq. (5.67).
If we start from a set of results of th type presented in Fig. 5.3, it is clear that we can represent v, and
hence also η(E), as power series in E. The standard mathematical technique of reversion of series then
allows us to write
E(η) = U1,0, η + U2,0 η2 + U3,0 η
3 + . . . =∑�=1
U�,0 η� (5.68)
=U1,0
μ1/2(v + 1/2) +
U2,0
(μ1/2)2(v + 1/2)2 +
U3,0
(μ1/2)3(v + 1/2)3 + . . . =
∑�=1
U�,0
(μ1/2)�(v + 1/2)�
Similarly, we see that the rotational quantum number J always enters the theory in the form of the
factor [J(J + 1)]/μ. In particular, a potential energy function received precisely the same amount of
centrifugal distortion for isotopologue–1 with quantum number J1 as from isotopologue–2 with quantum
number value J2 defined by the relationship
J2(J1 + 1)/μ2 = J1(J1 + 1)/μ1 , (5.69)
or
J2 =
√μ2u1
J1(J1 + 1) + 14 −
1
2(5.70)
This implies, for example, that the J = 2 level of D2 has approximately the same amount of rotational
energy as the J1 = 1.3034 rotational level of H2.
Of course, levels associated with fractional quantum number values do not exist. However, the above
considerations implicitly tell us that within th first-order WKB approximation, a knowledge of the v–
and/or J–dependence of properties of one isotopologue implicitly provides information about that depen-
dence for all other isotopologues of that species.
5.2.2 Problems with the Dunham-type Approach
• Polynomial potential form bad at large r – always goes to ±∞ !
• The full Dunham expressions are quite complex – too unwieldy to use for high accuracy analyses
except very near the potential minimum
• Much effort focussed (dubiously) on determination of precise re values ...
• Correction terms correct for JWKB breakdown, but not Born-Oppenheimer breakdown.
Addressing Some of the Problems with the Dunham Approach
• Alternate Dunham-like potential forms based on alternate expansion variables: SPF, O-T, Surkus,
...
• If we give up attempting to write out expressions for Yl,m’s, especially for high-order terms, and trust
computer algebra, can get accurate and internally consistent fits with CDC’s implicitly constrained
to be consistent.
109
5.3 The “RKR” Inversion Procedure
5.3.1 Derivation
By 1929, even before Dunham’s work, the drawbacks of trying to work with model potentials for which
exact analytic quantum mechanical eigenvalue solutions existed were becoming evident, since the few such
functions which were known were not flexible enough to account fully for experimental data. As a result,
researchers started to investigate the use of semiclassical methods. A pioneer in this area was O. Oldenberg
[14] who began using semiclassical methods to test proposed potentials for various molecular states. In
particular, using a given model for the potential and an initial trial energy E for a given vibrational level, he
would plot the integrand of the semiclassical quantization condition integral of Eq. (5.30), measure its area
using a ‘planimeter’, and then repeat the procedure for different energies until the quantization condition
was satisfied. Comparing sets of vibrational energies determined in this way with experiment allowed him
to test various possible model potentials for a given molecular state. An alternate approach would be to
choose a given vibrational energy G(v) and vary the potential shape, repeatedly plotting and measuring
the area of the integrand in Eq. (5.30) until the quantization condition was satisfied. Since a harmonic
oscillator or Morse model should usually provide a good description of the ground state and first excited
vibrational level, moving up the well this procedure would essentially require one to make trial-and-error
determinations of the inner and outer turning points of higher levels one at a time.
It was soon noted that this process contained a fundamental ambiguity, in that the area of the integrand
may be changed by modifying the position of the turning point at either the inner or outer end of the
range [15]. This would allow an infinite number of different potentials to be obtained, all of which were
consistent with a given set of experimental vibrational energies. Thus, in addition to the quantization
condition, a second condition was required if one was to obtain a unique solution. R. Rydberg [15] chose
to follow a suggestion of Hulthen that the second condition be based on the value of the inertial rotational
constant Bv. we saw in Eq. (3.25) that the value of Bv is determines by the expectation value of 1/r2 for
vibrational level v. In a semiclassical treatment, the average value of any property f(r) is defined as
〈f(r)〉 =
∫ r2
r1
f(r)
[E − V (r)]1/2dr
/∫ r2
r1
1
[E − V (r)]1/2dr (5.71)
Thus, requiring that Bv values calculated from the ratio of integrals
Bv =�2
2μ
∫ r2
r1
1
r2[E − V (r)]1/2dr
/∫ r2
r1
1
[E − V (r)]1/2dr (5.72)
agree with experiment provided the necessary second constraint, allowing the construction of a unique
potential energy function. Figure 5.4 (taken from Rydberg’s 1931 paper) illustrates his proposed graphical
method for determining a unique potential energy function that would be consistent with the vibrational
energies Gv and Bv values for a given set of levels.
The first formal derivation of what is now known as the “RKR” method was due to O. Klein [16], and
it is outlined here. It starts from the first-order JWKB or Bohr-Sommerfeld quantization condition:
v + 12 =
1
π
√2μ
�2
∫ r2
r1
[E − V (r)]1/2 dr . (5.73)
For the purpose of this derivation it is notationally convenient to replace v by v′ and E by E′. We now
take the derivative of this expression with respect to energy E′, and divide the range of integration into
two parts to separate the repulsive and attractive regions:
dv′
dE′=
1
2π
√2μ
�2
{∫ re
r1
dr
[E′ − V (r)]1/2+
∫ r2
re
dr
[E′ − V (r)]1/2
}(5.74)
110
Figure 5.4. Figure from Rydberg’s 1931 paper [15] that schematically illustrates his graphical
inversion procedure.
For a well-behaved single-minimum potential, on each of the intervals [r1, re] and [re, r2] there is a unique
monotonic relationship between the distance variable r and the value of the potential energy function,
u = V (r) . We can therefore re-write Eq. (5.74) with u replacing r as the independent variable in the two
integrals:
dv′
dE′=
1
2π
√2μ
�2
{∫ 0
E′
1
[E′ − u]1/2dr1(u)
dudu +
∫ E′
0
1
[E′ − u]1/2dr2(u)
dudu
}(5.75)
=1
2π
√2μ
�2
∫ E′
0
(dr2(u)
du− dr1(u)
du
)du
[E′ − u]1/2
We now introduce a mathematical technique that is sometimes called an Abelian transformation of the
first kind; it involves pre-multiplying both sides of Eq. (5.75) by the factor dE′/[E−E′]1/2 and integrating
111
E′ from 0 to E:∫ E
0
(dv′/dE′) dE′
[E − E′]1/2 =
∫ v(E)
vmin
dv′
[E(v)− E(v′)]1/2(5.76)
=1
2π
√2μ
�2
∫ E
0dE′
{∫ E′
0
(dr2(u)
du− dr1(u)
du
)du
[(E − E′)(E′ − u)]1/2}
in which vmin = v(E=0) is the (non-integer) effective vibrational quantum number index associated with
the potential minimum. If we then change the order of the double integration, and utilize the standard
mathematical identity ∫ b
a
dx
[(b− x)(x− a)]1/2 = π (5.77)
we obtain∫ v(E)
vmin
dv′
[E(v) − E(v′)]1/2=
1
2π
√2μ
�2
∫ E
0du
{(dr2(u)
du− dr1(u)
du
) ∫ E
u
dE′
[(E − E′)(E′ − u)]1/2}
=1
2
√2μ
�2
{∫ E
0
dr2(u)
dudu −
∫ E
0
dr1(u)
dudu
}=
1
2
√2μ
�2
{∫ r2(E)
re
dr −∫ r1(E)
re
dr
}
=1
2
√2μ
�2[(r2(E) − re)− (r1(E)− re)]
=1
2
√2μ
�2[re(E(v)) − r1(E(v))] (5.78)
Rearranging this expression yields the first or “vibrational” RKR equation
r2(v) − r1(v) = 2
√�2
2μ
∫ v
vmin
dv′
[E(v)− E(v′)]1/2= 2f (5.79)
The derivation of the second or “rotational” RKR equation proceeds in the same way as that for
the vibrational RKR equation, except that we first have to perform some manipulations to obtain the
appropriate starting equation. The starting point is the recognition that for a rotating molecule J > 0
and the effective centrifugally-distorted potential appearing in the quantization condition of Eq. (5.73) is
VJ(r) = V (r) +�2
2μ
[J(J + 1)]
r2(5.80)
so the quantization condition may be re-written as
v(E, J) + 12 =
1
π
√2μ
�2
∫ r2
r1
[E − V (r)− �
2
2μ
[J(J + 1)]
r2
]1/2dr (5.81)
For a given value of J , Eq. (5.81) tells us that there exists a unique mapping between v and E, and the
chain rule of calculus tell us that in this case, for any function F(E, J),(∂F(E, J)
∂[J(J + 1)]
)E
=
(∂E
∂[J(J + 1)]
)v
(∂F
∂E
)J
(5.82)
112
Applying this chain rule relationship to Eq. (5.81) then yields(∂v
∂[J(J + 1)]
)E
=
(∂E
∂[J(J + 1)]
)v
(∂v
∂E
)J
= − 1
2π
√�2
2μ
∫ r2
r1
dr
r2[E − V (r)− �2
2μ[J(J+1)]
r2
]1/2 (5.83)
From the standard definition of the inertial rotational constant, we know that ∂E(v,J)∂[J(J+1)]
∣∣∣J=0≡ Bv , so
Eq. (5.83) becomes
Bv × dv
dE= − 1
2π
√�2
2μ
∫ r2
r1
dr
r2 [E − V (r)]1/2
(5.84)
in which the partial derivative has been replaced by an exact derivative, since when J is fixed (at J =0)
there in only one independent variable.
In the derivation of the rotational RKR equation, Eq. (5.84) provides a starting point which is the
precise analog of Eq. (5.73) in the derivation of the RKR “f integral” result of Eq. (5.79). Proceeding
precisely as before: (i) replace variable names E and v with E′ and v′, respectively, (ii) split the range of
integration into two parts at re, (iii) change the variable of integration from r to u=V (r) , (iv) multiply
by dE′/(E − E′)1/2 and integrate E′ from 0 to E, (v) change the order of integration and apply the
identity of Eq. (5.77), and (vi) rearrange the result appropriately, then yields the second or “rotational”
RKR equation:
1
r1(v)− 1
r2(v)= 2
√2μ
�2
∫ v
vmin
Bv′ dv′
[E(v) − E(v′)]1/2= 2g (5.85)
Combining Eqs. (5.79) and (5.85) then yields the final turning point expressions
r2(v) =(f2 + f/g
)1/2+ f (5.86)
r1(v) =(f2 + f/g
)1/2 − f (5.87)
Thus, for any case in which we have smooth functions which accurately describe the dependence on v of
the vibrational energy and inertial rotational constant Bv, Eqs. (5.79) and (5.79)-(5.87) may be used to
generate the potential energy function in a pointwise manner.
5.3.2 Practical Implementation of the RKR Method
In spite of their elegance and obvious potential utility, Klein’s equations saw little practical use for over
three decades. One reason for this would have been the practical difficulty of evaluating the Klein integrals
accurately prior to the advent of digital computers. The nature of this problem is illustrated by the plots
for the ground electronic state of Ca2 shown in Fig. 5.5. Panels A and B show the nature of the Gv′ and
Bv′ functions, while Panel C shows the integrands of Eqs. (5.79) and (5.85) for a representative vibrational
level, v=26. Although the areas under these curves are finite, the fact that the integrands go to infinity
at the upper bound makes an accurate evaluation of these integrals somewhat challenging.
In 1947 A.L.G. Rees pointed out that the two Klein integrals could be evaluated in closed form if
G(v) and Bv were represented by sets of quadratic polynomials in v for different segments of the range of
integration [17]. This contribution led to his name being attached to what became known ad the “Rydberg-
Klein-Rees” (or RKR) method, but the clutter of having to fit data piecewise to sets of quadratics meant
that it still saw little use. Finally, by the early 1960’s a number of groups had developed computer programs
for evaluating these integrals for any user-selected expressions for G(v) and Bv , and the ‘RKR’ method
quickly grew to become ubiquitously associated with diatomic molecule data analyses. However, truly
113
0.00
0.01
0.02
0.03
0.04
Bυ’
A
0 10 20 300
250
500
750
1000
υ’
Gυ’
↑υ
B
0 10 200.00
0.05
0.10
0.15
0.20
0.25
υ’
∞
⎯⎯⎯⎯⎯⎯140 × Bυ’ √Gυ −Gυ’
⎯⎯⎯⎯⎯⎯1√Gυ −Gυ’
↑υ
C
0 1 2 3 4 50.0
0.1
0.2
0.3
y
⎯⎯⎯⎯⎯⎯√ υ − υ’√Gυ −Gυ’
⎯⎯⎯
⎯⎯⎯√ υ − υ’
40×√Gυ −Gυ’
Bυ’ ⎯⎯⎯
D⎯⎯⎯
Figure 5.5 Panels A and B: spectroscopic properties of Ca2. Panel C: Integrands of the Klein integrals
of Eqs. (5.79) and (5.85) for level v=26 of Ca2; a numerical factor of 40 is introduced in order to
place the two integrands on the same vertical scale. Panel D: Integrands of the transformed Klein
integrals of Eqs. (5.88) and (5.89) for the case considered in Panel C. Units for energy are cm−1 in
all panels.
efficient techniques for evaluating the Klein integrals which take proper account of the singularities in the
integrand were not reported until 1972 [18, 19, 20].
One conceptually simple technique for evaluating the RKR integrals accurately is simply to introduce
a transformation that removes the singularities. For example, introduction of the auxiliary variable y =√v − v′ transforms Eqs. (5.79) and (5.85) into the forms
r2(v) − r1(v) = 4
√�2
2μ
∫ √v+1/2
0
{√v − v′
Gv −Gv′
}dy = 2 f (5.88)
1
r1(v)− 1
r2(v)= 4
√2μ
�2
∫ √v+1/2
0
{Bv′
√v − v′
Gv −Gv′
}dy = 2 g . (5.89)
As is illustrated by Panel D of Fig. 5.5, the integrands in these expressions are smooth and well behaved
and have no singularities(!), so a very modest amount of computational effort can yield turning points
converged to machine precision. A particularly convenient procedure is to apply a simple N–point Gauss-
Legendre quadrature procedure to the whole interval, and then bisect that interval and apply the same
procedure to both halves. At each such stage of subdivision the error will decrease by a factor of 1/2N−2
[5]; for N=12 this means an error reduction by three orders of magnitude at each stage of bisection.
It is important to remember that although the experimental data are only associated with integer
values of v, the vibrational energies Gv and rotational constants Bv in these integrals must be treated as
114
continuous functions of v. Moreover, as illustrated by Fig. 5.5B, the quantization integral of Eq. (5.73)
may be evaluated for any energy E (or Gv), independent of whether or not it corresponds to an integer
value of v. Thus, we are free to solve the RKR equations and evaluate turning points for any chosen
mesh of integer or non-integer v values. This is quite important, since solving the Schrodinger equation
numerically requires an interpolation procedure to provide a mesh of accurate potential function values at
distances that will not correspond to calculated turning points. If the evaluation procedure were restricted
to turning points at integer v, such interpolations would often have limited accuracy, in spite of the fact
that the calculated turning points would be smooth to machine precision.
Two other practical considerations intrude upon the use of RKR potentials. One is the perhaps
obvious, but sometimes overlooked point that calculated turning points cannot really be trusted beyond
the vibrational range of the experimental data used to determine the Gv and Bv functions. This restriction
is partially lifted if ‘near-dissociation expansions’ of the type described in § 5.4 are used to represent Gv
and Bv. However, use of the resulting potential to generate reliable solutions to the radial Schrodinger
equation would still require functions for extrapolating inward and outward to be attached smoothly at
the ends of the range of calculated turning points.
The second practical concern arises from the fact that shortcomings of the experimentally-derived
functions characterizing Gv and Bv will give rise to errors in calculated RKR turning points. Since the
repulsive inner wall of a potential function is very steep, especially at high energies, such errors often
manifest themselves as non-physical behaviour of the inner wall of the potential. For example, rather
than have a (negative) slope and positive curvature that vary slowly with energy, the inner wall might
pass through an inflection point and take on negative curvature, or it might turn outward with increasing
energy, with the slope becoming positive. In practice, the experimental Gv function is usually defined
with greater relative accuracy than is the Bv function. However, whatever the source of the problem, a
modest degree of inappropriate behaviour of either the Gv or Bv function can give rise to non-physical
behaviour of the inner wall of the potential, as the expected monotonic increase in slope with energy will
greatly amplify the effect of even very small errors in the f and/or g integrals. Thus, the behaviour of
the inner wall of any calculated RKR potential should always be examined, and if the slope deviates from
smooth behaviour with positive curvature, it should be smoothed or replaced with a physically sensible
extrapolating function.
Although small relative errors in the f or g integral can make the curvature or slope of the high-
energy inner wall change in an unacceptable non-physical manner, the rapid growth of the f integral
with increasing Gv means that the width of the potential [r2(v) − r1(v)] as a function of energy may
still be relatively well defined by Eq. (5.79) or (5.88), even when the directly calculated inner potential
wall is unreliable. In this case, combining this directly-calculated well-width function with a reasonable
extrapolated inner potential wall would yield a ‘best’ estimate of the upper portion of the potential (a
procedure first introduced by Verma [21]). Similarly, even in the complete absence of rotational data, a
combination of the well-width information yielded by the calculated f integrals with an inner wall defined
by a model such as a Morse potential can give a realistic overall potential function [22]. A ‘black box’
computer code (accompanied by a manual) for performing RKR calculations, which allows the use of a
variety of possible expressions for Gv and Bv and takes account of the practical concerns described above,
is available on the www [23].
Finally, it is also important to remember that the manipulations of Eq. (5.73) to obtain the RKR
equations (5.79) and (5.85) (or equivalently, (5.88) and (5.89)) are mathematically exact! In other words,
within the first-order semiclassical or WKB approximation [4], this method yields a unique potential energy
function that exactly reflects the input functions representing the v-dependence of the vibrational energy
Gv and inertial rotational constant Bv. A nagging weakness, however, is the fact that the quantization
115
Table 5.3 Root mean square errors in vibrational level spacings and rotational constants calculated from
RKR potentials for selected molecules.
molecule μ De vmaxG(vmax)
Deerr{ΔGv+1/2} % err{Bv}
[cm−1] [cm−1]
BeH 0.906 17590 9 0.895 0.527 0.031
N2 7.002 79845 20 0.529 0.052 0.0026
Ca2 19.981 1102 25 0.916 0.00079 0.0021
Rb2 42.456 3993 85 0.916 0.00017 0.0013
condition of Eq. (5.73) is not exact, so quantum mechanical properties of an RKR potential will not agree
precisely with the input Gv and Bv data used to generate that potential.
Table 5.3 illustrates this point for four species for which accurate and extensive Gv and Bv functions
are available from the literature. Those functions were used to generate RKR potentials, after which an
exact quantum procedure [24] was used to calculate the associated vibrational level spacings (ΔGv+1/2)
and inertial rotational constants (Bv). The two final columns of this table show the root mean square
differences between those calculated quantities and the values implied by the Gv and Bv functions used to
generate the original RKR potential. In each case the range considered was truncated at G(vmax) which
is the smaller of the upper end to the range of the experimental data used to determine the Gv and Bv
functions, or the point at which the onset of irregular behaviour of the inner-wall turning points (see above)
required smoothing and inward extrapolation to be applied.
These results show that errors in RKR potentials due to neglect of the higher-order terms in Eq. (5.43)
are largest for species with small reduced mass. For a hydride they are quite significant, but their im-
portance drops rapidly with increasing reduced mass, and for μ � 20u (Ca2 and Rb2) the vibrational
spacing discrepancies are smaller than typical experimental vibrational energy uncertainties. However,
such discrepancies add up, and even for these ‘heavy’ species the accumulated error in the vibrational
energy can be significant. Overall, although the situation is less satisfactory for light molecules, the first-
order semiclassical nature of the RKR procedure has only a modest negative effect on the quality of the
resulting potential, or of quantities calculated from it. At the same time, that fact that RKR potentials
are defined as sets of many-digit turning points, often need to have their inner wall smoothed, and always
need extrapolation functions attached at their inner and outer ends, are persistent inconveniences. These
problems are resolved, however, by use of the methodology described in §7.4.Although the higher-order quantization condition of Eq. (5.43) is not amenable to the exact inversion
procedures described above, it has been suggested that better-than-first-order results could be obtained
simply by replacing the lower bound on the integrals of Eqs. (5.79) and (5.85) by vmin=−12 − δvmin from
Eq. (5.96) [25]. Unfortunately, tests analogous to those of Table 5.3 show that although this procedure
does give somewhat better results near the potential minimum, the discrepancies at higher v are larger
than those obtained with the usual first-order method.
116
5.3.3 Program RKR1
Table 5.4 Input data file structure for program RKR1. For detailed parameter definitions see §4 of the
program manual [23].
#1 READ(5,*,END=99) IAN1, IMN1, IAN2, IMN2, CHARGE, NDEGv, NDEBv
#2a IF((IAN1.LE.0).OR.(IAN1.GT.109) READ(5,*) NAME1, MASS1
#2b IF((IAN2.LE.0).OR.(IAN2.GT.109) READ(5,*) NAME2, MASS2
#3 READ(5,*) TITLE
IF((NDEGv.EQ.0).OR.(NDEGv.EQ.2)) THEN
#4 READ(5,*) LMAXGv
#5 READ(5,*) (YL0(L),L= 1,LMAXGv)
ENDIF
#6 IF(NDEGv.GE.2) READ(5,*) VS, DVS, DLIM
IF(NDEGv.GE.1) THEN
#7 READ(5,*) NLR, ITYPE, IZP0, IZQ0, NP0, NQ0, VD, XCN0
#8 IF(NP0.GT.0) READ(5,*) (P0(I),I= 1,NP0)
#9 IF(NQ0.GT.0) READ(5,*) (Q0(I),I= 1,NQ0)
ENDIF
#10 IF(NDEBv.LT.0) READ(5,*) Req
IF((NDEBv.EQ.0).OR.(NDEBv.EQ.2)) THEN
#11 READ(5,*) LMAXBv
#12 IF(LMAXBv.GE.0) READ(5,*) (YL1(L),L= 0,LMAXBv)
ENDIF
IF(NDEBv.GE.1) THEN
#13 READ(5,*) ITYPB, IZP1, IZQ1, NP1, NQ1, XCN1
#14 IF(NP1.GT.0) READ(5,*) (P1(I),I= 1,NP1)
#15 IF(NQ1.GT.0) READ(5,*) (Q1(I),I= 1,NP1)
ENDIF
#16 READ(5,*) Kaiser, NSV, VEXT
DO J= 1,NSV
#17 READ(5,*) V1(I), DV(I), V2(I)
ENDDO
117
5.3.4 A Higher-Order RKR Method
The discussion of §5.1 pointed out that the first-ordinary semiclassical approximation that is the basis
of the standard RKR treatment is not exact. In practise this means that if an exact quantal calculation
(using, say, program LEVEL) is performed using an RKR potential, the resulting eigenvalues and Bv values
would differ from the values used to define the RKR potential. An illustration of this point is presented in
Table 5.5, which compares quantal vibrational energies and Bv values calculated from an RKR potential
for the ground X 1Σ+g state of Li2 with the “experimental” values defined by the Gv and Bv expressions
used to define that RKR potential. In the high-order quantization condition of Eq. (5.43) we see that the
prefactors in front of the third- and fifth-order terms have additional powers of μ in the denominator, so
we expect that this problem will be most serious for species with small reduced mass μ.
In §5.1.3 it was shown that use of higher-order quantization condition(s) could yield results with
virtually any desired level of accuracy. Unfortunately, to date no-one has been able to derive a direct
higher-order analog of the conventional first-order RKR equations of Eqs. (5.79) and (5.85). However, a
practical ‘working’ higher-order RKR method has been developed that is based on the iterative scheme
described below. This scheme starts from the third-order quantization condition defined by the first line
of Eq. (5.43). As in the Rydberg derivation we begin by taking the derivative of both sides of this equation
with respect to E:
∂ v
∂E=
1
4π
(2μ
�2
)1/2 ∮ 1
[E − U(r)]1/2dr − 1
64
(�2
2μ
)∮U ′′(r)
[E − U(r)]5/2dr (5.90)
We now shift the third-order term to the left-hand side of this equation and treat the first-order term in
the same manner as in the standard derivation, breaking the interval [r1, r2] into the two intervals [r1, re]
and [re, r2], and replacing r with u=V (r) as the independent variable:
∂ v
∂E+
1
64
(�2
2μ
)∮U ′′(r)
[E − U(r)]5/2dr =
1
2π
(2μ
�2
)1/2 ∫ E′
0
(dr2(u)
du− dr1(u)
du
)du
[E′ − u]1/2 (5.91)
As before, on pre-multiplying by dE′/(E − E′)1/2 and integrating E′ from zero to E the right-hand side
of this equation becomes 12
√�2/2μ [r2(v) − r1(v)], and we obtain a third-order analog of the first RKR
equation:
r2(v) − r1(v) = 2
√2μ
�2
∫ v
vmin
dv′
[E(v) − E(v′)]1/2
{1 +
√�2
2μ
ωv′
64π
∮V ′′(r)
[E − V (r)]5/2dr
}(5.92)
Table 5.5 Test of quantal vibrational energies and Bv values (both in cm−1) calculated from an RKR
potential for the X 1Σ+g state of Li2.
experiment quantal calculation
v Gv−G0 Bv err{Gv−G0} err{Bv}0 0.0 0.669 0483 0.0 0.000 0021
2 687.2085 0.654 7883 -0.0108 0.000 0032
4 1353.3617 0.640 1942 -0.0227 0.000 0036
6 1998.0408 0.625 2054 -0.0357 0.000 0037
8 2620.7309 0.609 7378 -0.0493 0.000 0039
10 3220.7965 0.593 5845 -0.0644 0.000 0055
12 3797.4505 0.576 9157 -0.0820 0.000 0106
14 4349.7377 0.559 2782 -0.1044 0.000 0209
16 4876.4994 0.540 5959 -0.1319 0.000 0343
118
in which ωv′ ≡ dE/dv′ is the derivative of the energy with respect to v′.If we introduce the notation
Im,nk,l (E) =
∮[∂mV (r)/∂rm]n
rl [E − V (r)]k+1/2
dr (5.93)
Eq. (5.92) may be written as
r2(v)− r1(v) = 2
√2μ
�2
∫ v
vmin
1
[E(v) − E(v′)]1/2
{1 +
√�2
2μ
ωv′
64πI2,12,0 (E
′)
}dv′ (5.94)
and an analogous manipulation of the partial derivative of the third-order quantization condition with
respect to [J(J + 1)] yields the companion result:
1
r1(v)− 1
r2(v)= 2
√�2
2μ
∫ v
vmin
1
[E(v) − E(v′)]1/2
{1 +
(2μ
�2
)3/2 ωv′
64πBv′I2,12,2 (E
′)
}dv′ (5.95)
It is clear that if we omit the terms in curly parentheses {. . .}, Eqs. (5.94) and (5.95) become the simple
first-order RKR equations of Eqs. ((5.79) and (5.85). Moreover, the correction terms involve the partial
derivatives of the energy with respect to v and [J(J + 1)] (ωv and Bv, respectively), quantities that are
known from experiment, and energy-dependent integrals I2,12,l (E) that may be calculated from a knowledge
of the potential energy function. This means that higher-order estimates of the turning points may be
generated by an iterative procedure in which:
(i) A conventional first-order RKR calculation is used to determine a preliminary estimate of the potential
energy function. Calculations using that potential may then readily generate values of the integrals
I2,12,0 (E) and I2,12,2 (E) at any specified energy.
(ii) A third-order RKR calculation using Eqs. (5.94) and (5.95) may then be performed in which the
requisite values of I2,12,0 (E′) and I2,12,2 (E
′) are evaluated at each (v′) quadrature point.
(iii) This procedure may then be iterated, with the requisite values of I2,12,0 (E′) and I2,12,2 (E
′) being evaluatedusing the improved potential energy function generated in the preceding cycle.
Application of the above procedure to the ground electronic state of Li2 yielded the results shown in Tables
5.6 and 5.7.
A widely used approximate version of a third-order RKR procedure focuses on the definition of the
lower bound of the first-order RKR integrals of Eqs. (5.79) and (5.85). The form of the Bohr-Sommerfeld
quantization condition of Eqs. (5.30) or (5.73) shows that in first order, the energy at the potential minimum
is associated with the quantum number value v = −1/2. However, Dunham’s original derivation for the
expansion parameters in the vibration-rotation level energy expression of Eq. (5.47) included the constant
term Y0,0 whose presence effectively means that the potential minimum corresponds to v = −1/2−Y0,0/Y1,0 .On applying the standard Dunham relations relating the potential function expansion coefficients Yl,m (or
conventional spectroscopic) parameters this yields:
v = vmin = − 12 − δvmin = − 1
2 −{Y0,1 + Y2,0
4Y1,0− Y1,1
12Y0,1+
Y1,0Y0,1
(Y1,1
12Y0,1
)2}
(5.96)
= − 12 −
{Be − ωexe
4ωe+
αe
12Be+
ωe
Be
(αe
12Be
)2}
.
119
Table 5.6 Changes in the inner and outer turning points calculated for ground-state Li2 on iterating the
higher-order RKR procedure of Eqs. (5.79) and (5.85), were ITER is the number of iterative cycles.
1010×Δr1 /A 1010×Δr2 /AITER = 1 2 3 4 5 1 2 3 4 5
v = 0 -2852 6 -1 0 0 -1998 -2 0 0 0
2 -3429 35 -1 0 0 -1453 2 0 0 0
4 -4080 70 2 -1 0 -1031 7 0 0 0
6 -5063 99 9 -3 2 -629 11 0 0 0
8 -6495 88 29 -10 4 -251 14 1 -1 0
10 -8387 22 76 -28 14 79 14 3 -1 0
12 -10596 -217 182 -71 53 331 8 7 -3 1
14 -12750 -775 409 -181 119 477 -8 15 -6 4
16 -14122 -1932 884 -449 132 491 -44 31 -14 8
18 -13459 -4169 1852 -987 142 361 -112 60 -29 11
20 -8727 -8313 3765 -1865 645 87 -234 113 -57 19
22 3258 -15739 7332 -3252 1866 -307 -444 210 -101 42
24 27337 -28669 13539 -5797 2318 -763 -796 375 -171 78
26 70712 -50512 23760 -9931 -962 -1175 -1369 646 -281 90
28 143879 -86230 39983 -15718 -10604 -1372 -2283 1070 -458 -13
This is the definition of vmin than must be used with the third-order RKR equations of Eqs. (5.94) and
(5.95)
While the standard first-order RKR equations of Eqs. (5.79) and (5.85) are normally applied using the
first-order definition vmin = −1/2 , Kaiser proposed that a better-than-first-order result would be obtained
if one applied the normal first-order method, but defined vmin using the third-order result of Eq. (5.96)
[25]. His word showed that this was indeed the case for the lowest vibrational levels of HCl. However, more
extensive tests of the type associated with Tables 5.6 and 5.7 show that while use of what is known as the
‘Kaiser correction’ improves the potential function near the minimum, it tends to yield worse results at
higher energies.
Table 5.7 Errors in the quantal eigenvalues for ground-state Li2 calculated from higher-order
RKR potentials after ITER cycles of the iterative procedure described on p. 119.
err{Gv −Gv=0}v Gv−G0 ITER = 0 4 8 12 16 20
0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
2 687.2085 -0.0108 -0.0002 0.0000 -0.0001 -0.0001 -0.0001
4 1353.3617 -0.0227 -0.0000 0.0000 0.0001 0.0001 0.0001
6 1998.0408 -0.0357 -0.0002 -0.0001 0.0000 0.0000 0.0000
8 2620.7309 -0.0493 -0.0007 -0.0005 -0.0003 -0.0003 -0.0002
10 3220.7965 -0.0644 -0.0020 -0.0013 -0.0009 -0.0007 -0.0007
12 3797.4505 -0.0820 -0.0052 -0.0029 -0.0019 -0.0014 -0.0011
14 4349.7377 -0.1044 -0.0118 -0.0060 -0.0033 -0.0020 -0.0014
16 4876.4994 -0.1319 -0.0221 -0.0091 -0.0030 -0.0001 -0.0012
120
5.4 Near-Dissociation Theory (NDT) and its Implications
5.4.1 NDT Expression for the Vibrational Energies
The preceding discussion shows that the RKR method can give a quite accurate potential energy functions
spanning the range of vibrational energies for which experimental data are available. However, it offers no
advice regarding how to address the question illustrated in Panel A of Fig. 6.4: that is, how to estimate
the distance from the highest observed vibrational level to the dissociation limit, and how to estimate the
number, energies, and other properties of levels lying above that highest observed vibrational level.
Panel B of Fig. 6.4 illustrates a graphical means for addressing this question which was introduced by
Birge and Sponer in 1926 [26] and remained the method of choice for most of the following half century. In a
Birge-Sponer plot the vibrational level spacings ΔGv+1/2 ≡ Gv+1 −Gv are plotted against the vibrational
quantum number, with the points placed at half-integer values of the abscissa. On this diagram, the
numerical ΔGv+1/2 value is equal to the area of the narrow vertical rectangle whose upper edge is centred
at that point. As a result, the sum of the areas of the six illustrated rectangles is equal to the sum of the
six ΔGv+1/2 values, which is, of course, the distance from level v=0 to level v=6 . It is immediately clear
that the area under a smooth curve through these points from v=0 to 6 is a very good approximation
to that energy difference. Birge and Sponer then pointed out that if this curve was extrapolated to cut
the v axis, the area under the curve in the extrapolation region would be a very good approximation to
the distance from the highest observed level to the dissociation limit. Moreover, the points at which the
extrapolated curve crossed half-integer v value gives predicted vibrational spacings for unobserved levels
extending all the way to the limit. If these predictions were correct, an RKR potential based on the
resulting extrapolated Gv values could be calculated for the whole well.
The only problem with Birge-Sponer plots is the uncertainty regarding how to perform the extrapola-
tion, a problem which remained an open question for 44 years. The dash-dot-dot line on Fig. 5.6B shows
υ=6
5
4
3
2
1
0
V(r)
D?
⇓
⇑
A
⎥⎥⎥⎥
?0 5 10 15 20 250
100
200
300
400
500
υ
ΔGυ+½
B
Figure 5.6 Panel A: Schematic illustration of the extrapolation problem of determining th dissociation
energy. Panel B: A Birge-Sponer plot in which the shaded area illustrates the uncertainty associated
with conventional vibrational extrapolation.
121
a linear extrapolation through the last two experimental points, while the dotted curves bounding the
shaded region are plausible alternative extrapolations, one with negative and one with positive curvature.
The ratio of the area of the shaded region to the overall area under the curve in the extrapolation region is
then an indication of the relative uncertainty in the distance from the last observed level to the dissociation
limit. Unfortunately, it is clear that this uncertainty could be as large as 50-100% !
A solution to this extrapolation problem was finally reported in 1970 [27]. It was based on the realization
that another type of potential for which an explicit analytic expression for the vibrational level energies
may be obtained from Eq. (5.30) is the attractive inverse-power function V (r)=D − Cn/rn whose form
matches the limiting long-range behaviour of all intermolecular interactions. As was true for the RKR
method, the derivation is remarkably straightforward.
Since the nature of distribution of vibrational levels near dissociation is being sought, the derivation
begins by taking the derivative of Eq. (5.30) with respect to the vibrational level energy to obtain an
expression for the density of states at energy Gv (for J=0 ):
dv
dGv=
1
2π
√2μ
�2
∫ r2(v)
r1(v)
dr
[Gv − V (r)]1/2. (5.97)
Consider now the nature of the integrand appearing in Eq. (5.97). For a model Lennard-Jones(12,6)
potential function
VLJ(r) =C12
r12− C6
r6+ De = De
[(rer
)6 − 1
]2(5.98)
that supports 24 vibrational levels, the lower panel of Fig. 6.5 shows a plot of that potential and indicates
the positions of the energies and turning points of selected levels. The upper panel then shows the nature of
the integrand in Eq. (5.97) for those four levels; note that while the integrand goes to infinity at both turning
points, the area under the curve is always finite. It is immediately clear that for the higher vibrational
levels, the area under the curve – and hence the value of the integral – is increasingly dominated by the
nature of the integrand (i.e., of the potential) in the long-range region near the outer turning point.
From the early days of quantum mechanics it has been known that at long range all atomic and
molecular interaction potentials become a sum of inverse-power terms
V (r) � D −∑m≥n
Cm/rm =⇒{very large r} D− Cn/r
n , (5.99)
in which the powers m and coefficients Cm are determined by the nature of the interacting atoms. (A
brief summary of the rules governing which terms appear in this sum for a given case is presented in the
Appendix.) This suggests that for levels whose outer turning points lie at sufficiently large r for the
leading (Cn/rn) term to dominate the interaction, it would be a reasonable approximation to replace V (r)
in Eq. (5.97) by the simple function V (r) ≈ D− Cn/rn to obtain
dv
dGv≈ 1
2π
√2μ
�2
∫ r2(v)
r1(v)
dr
[Gv − (D− Cn/rn)]1/2. (5.100)
By making the substitution y= r/r2(v) and noting that [Gv − V (r2(v))]=0 , and hence that [D −Gv]=
Cn/[r2(v)]n, Eq. (5.100) becomes
dv
dGv≈ 1
2π
√2μ
�2
(Cn)1/n
[D−Gv}(n+2)/2n
∫ 1
r1/r2
dy
(y−n − 1)1/2. (5.101)
The dotted curve in the Upper Panel of Fig. 5.7 shows what happens to the exact integrand of Eq. (5.97)
for level v = 20 if the actual potential is replaced by the single inverse-power term D − C6/r6. It is
122
0.0
10.0
20.0
30.0
{[Gυ −V(r)] /D }−1/2e
∞ ∞∞∞ ∞
−Gυ=8= 0.721
−G14= 0.935
−G18= 0.988
−G20= 0.997
1.0 1.5 2.0 2.5 3.00.0
0.2
0.4
0.6
0.8
1.0
V(r) /D
r / r
e
e
r2(20)r2(18)r2(14)
r2(8)
υ=20
υ=8
υ=14υ= 18
[D −C6 /r6 ] /De e
Figure 5.7 Lower Panel: A 23-level LJ(12,6) potential with selected level energies and turning points
labelled. Upper Panel: Integrand of Eq. (5.97) for selected levels with Gv≡Gv/De. (Adapted from
Fig. 2 of Ref. [3].)
immediately clear that both the effect of this substitution on the value of this integrand and the effect
of replacing the lower bound of the integral in Eq. (5.101) by zero will be very small, and will become
increasingly negligible for higher vibrational levels (here, v=21− 23). By making use of the mathematical
identity ∫ 1
0
dy
(y−n − 1)1/2=
π
n
Γ(12 +
1n
)Γ(1 + 1
n
) (5.102)
and inverting the resulting expression, the basic near-dissociation theory (NDT) result is obtained:
dGv
dv=
{2n√π
(Cn)1/n
√�2
2μ
Γ(1 + 1
n
)Γ(12 +
1n
)} [D−Gv](n+2)/2n ≡ Kn [D−Gv]
(n+2)/2n . (5.103)
It is usually more convenient to work with the integrated form of this equation; this is the central
result:
Gv = D − X0(n) (vD − v)2n/(n−2) , (5.104)
in which X0(n)=((n−2)2n Kn
)2n/(n−2). For n>2 the integration constant vD takes on physical significance
as the non-integer effective vibrational index associated with the dissociation limit – the intercept of the
correctly extrapolated Birge-Sponer plot for the given system – and its integer part vD is the index of
the highest vibrational level supported by the given potential. For n = 1 this expression becomes the
123
Bohr eigenvalue formula for the levels of a Coulomb potential, and vD(n = 1) = − (1 + δ) , where δ is
the Rydberg quantum defect. An attractive n = 2 long-range potential is not physically possible for
a diatomic molecule, but integration of Eq. (5.103) for that case gives essentially the same exponential
eigenvalue expression known from quantum mechanics. In particular, on setting n= 2 and rearranging
Eq. (5.103) we obtain ∫dGv
D−Gv= − ln{D−Gv} =
∫K2 dv = K2(v − vD) , (5.105)
or
D−Gv = (D−GvD) e−K2(v−vD) . (5.106)
where in this case the integration constant vD is an integer identifying an arbitrarily chosen reference level.
These results show that for molecular states with n > 2 there exist a finite number of bound levels, but
if n ≤ 2 an infinite number of discrete levels lie below dissociation.
In order to express these results in a more practical form, it is convenient to take the first derivative of
Eq. (5.104) to obtaindGv
dv=
[(2nn−2
)X0(n)
](vD − v)(n+2)/(n−2) . (5.107)
Because the vibrational level energies and level spacings are the actual physical observables, the fact that
[dGv′/dv′]v′=v+
12≈ ΔG
v+12
allows Eqs. (5.108–5.109) to be rearranged to yield the expressions
(ΔGv+1/2
)(n+2)/2n= [Kn]
2n/(n+2) (D−Gv+1/2) (5.108)
(D−Gv)(n−2)/2n = [X0(n)]
(n−2)/2n (vD − v) (5.109)(ΔGv+1/2
)(n−2)/(n+2)=
[(2nn−2
)X0(n)
](n−2)/(n+2)(vD − v − 1
2) . (5.110)
Near-dissociation theory therefore predicts that if the observable quantities on the left hand side of these
equations are plotted vs. the vibrational mid-point energy Gv+1/2≈ 12(Gv+1 +Gv) (for Eq.(5.108)) or the
vibrational quantum number v (for the other two equations), for levels lying close to dissociation those
plots should be precisely linear, with slopes defined by the constants Kn or X0(n) (i.e., by μ, n and Cn),
while the intercept determines either the energy at the dissociation limit D or the vibrational intercept
vD. Plots of this type, sometimes called ‘Le Roy–Bernstein plots’, are often used to illustrate applications
of near-dissociation theory.
The B 3Π+0u state of I2 dissociates to yield one atom in the 2P3/2 electronic state and one 2P3/2 atom.
Thus, the limiting long-range behaviour of the potential energy function for this species (see Eq. (5.99)) is
dominated by an inverse-power term with n=5 (see §2.2.2). In 1931 Weldon Brown reported measurements
of band heads for B-state levels extending up to v=72, and for many years a conventional Birge-Sponer
extrapolation of his results determined our best estimate of the I2 dissociation energy. As may be seen in
the upper-right panel of Fig. 5.8, this approach would suggest vD ≈ 77 .
Figure 5.8 compares a conventional Birge-Sponer plot (upper right panel) for this system with the
NDT-type plots suggested by Eqs. (5.108–5.110). It is clear that all three of the NDT plots show the
expected limiting linear behaviour, and that the two plots on the left-hand side extrapolate to the same
value of vD ≈ 87.7(±0.4). Moreover, while it cannot be discerned visually, the slopes of all three NDT plots
correspond to the same value of C5 = 3.1(±0.2)×105 cm−1 A5. Thus, the conventional linear Birge-Sponer
extrapolation was a factor of three in error regarding the number of bound levels lying above Brown’s last
observed level, and this analysis yielded the first ever experimental determination of the C5 coefficient for
this system [28]. The analysis of Fig. 5.8 was based on band-head data that extended only up to v=72,
124
Figure 5.8 Top Panels: Comparison of a Birge-Sponer plot with the NDT plot suggested by Eq. (5.110)
for the B 3Π+0u state of I2 . Lower panels: NDT plots suggested by Eqs. (5.109) and (5.108) for
B(3Π+0u)-state I2 .
125
0 2 4 6 80
10
20
30
0.0
1.0
2.0
3.0
4.0
5.0
ΔGυ+½
(ΔGυ+½ )½
υ
υDlinear
Birge-Sponerextrapolation
theoreticalslope [3X0(6)]
1/2
Ar2(X1Σg
+)
B-S extrapolationimplied by NDT
Figure 5.9 Illustrative application of NDT to data for Ar2. Left Axis: square points, dash-dot-dot line
and dotted curve; Right Axis: round points and solid line. (Adapted from Fig. 1 of Ref. [32].)
but subsequent high resolution work [29, 30] that included rotational data for levels up to v=82 confirmed
that indeed vD = 87 , and within the estimated uncertainties, the recommended dissociation energy [30]
agrees with the initial NDT value that was based on the band-head data [28].
5.4.2 NDT and the Determination of Molecular Dissociation Energies
One type of application of these results is summarized by Fig. 5.9. It illustrates an NDT treatment of
data for the ground electronic state of the very weakly bound Van der Waals molecule Ar2, which was
first observed in 1970 [31]. The square symbols represent the experimental vibrational level spacings and
the dash–dot–dot line is the conventional linear Birge-Sponer (B-S) extrapolation (left-hand ordinate axis)
reported by the experimentalists, while the shaded area defines their estimate of the distance from the
highest observed level (v= 4) to the dissociation limit. This approach clearly predicts that v= 5 is the
highest bound level of this molecule.
As with all molecular states formed from atoms in electronic S states, n=6 for the ground electronic
state of Ar2 (see §2.2.2). The round symbols in Fig. 5.9 then show exactly those same experimental data
plotted (against the right-hand axis) in the manner suggested by Eq. (5.110). Since the data for the lowest
bound levels are not expected to obey the NDT equation, a simple linear fit to these data could not be
trusted to provide a reliable extrapolation. However, an accurate value of the C6 coefficient for this species
was available from ab initio quantum mechanical calculations, so the expected limiting slope of this plot
could be predicted from the resulting known value of the X0(n) coefficient. The solid line on this plot shows
the NDT prediction of the extrapolation obtained when a line with this theoretical slope passes through
the experimental datum for v = 3 . The fact that the second-last point also lies on this line while those
for the two larger level spacings only gradually deviate from it attests to the validity of this extrapolation.
The value of vD=8.27 implied by this NDT extrapolation shows that this molecule actually has 50% more
bound levels than were implied by the linear B-S type extrapolation, and comparison of the shaded area
126
0 20 40 60 80
1.6
2.0
2.4
2.8
[D−Gυ ]1/6
υ−m
174Yb2(11Σu
+)
Figure 5.10 Illustrative application of NDT to data for a state of Yb2 for which n=3 . The unspecified
integer m indicates that the absolute vibrational assignment is not known.
with the area under the dotted curve in the extrapolation region shows that the estimate of the distance
from the highest observed level to dissociation yielded by the traditional B-S extrapolation was more than
a factor of two too small [32].
A second illustrative application of NDT is the use of Eq. (5.109) in the analysis of ‘photoassociation
spectroscopy’ (PAS) data, for which the measured observable is the binding energy [D − Gv]. The 1 1Σ+u
state of Yb2 dissociates to yield one 1S0 atom and one 1P1 atom, a case for which n=3 (see §2.2.2). Hence,Eq. (5.109) shows that for levels lying near dissociation, a plot of [D−Gv]
1/6 is expected to be linear with
a slope of [X0(3)]1/6 determined by the value of the C3 coefficient for this state, and the intercept by its
vD value. Figure 5.10 shows a plot of this type based on the recent results of Takahashi and co-workers
[33]. The precise linearity of the points on Fig. 6.7 over a range of almost 80 vibrational levels is a very
strong endorsement of the validity of Eqs. (5.104–5.110), and it illustrates the fact that NDT provides the
most reliable method known for experimentally determining values of long-range Cn potential function
coefficients.
5.4.3 NDT Expressions for Rotational Constants and Other Properties
Near-dissociation theory expressions analogous to Eq. (5.104) have been reported for a number of other
properties, such as expectation values of the kinetic energy or of powers of the internuclear distance,
and for values of the rotational constants Bv, Dv , Hv, . . . , etc. For the inertial rotational constant Bv
and the expectation values of powers of r and other properties, the derivation is precisely analogous to
that presented in §5.4.1 for the vibrational energy distribution. In particular, Eq. (5.72) shows that the
first-order semiclassical expression for Bv consists of a ratio of two phase integrals. The integral in the
denominator is the same one appearing in the expression for the density of states Eq. (5.97) that served
as the starting point for deriving the NDT expression for the vibrational energies. The integral in the
numerator has precisely the same form, except that its integrand includes a weight function of r−2.
127
Figure 5.11 Integrands of the integrals in Eq. (5.72) for three levels of the 24-level LJ(12,6) potential of
Fig. 5.7; the k = 0 curves are for the integral in the numerator and k = 2 for the integral in the
denominator.
Figure 5.11 presents plots of these two integrands for levels of the same 24-level model LJ(12,6) potential
considered in Fig. 5.7. While the behaviour is not so pronounced as for the case of the density of states
integral of Eq. (5.97), it is again clear that for levels near dissociation the values of the numerator (k=0)
integrals will be dominated by the nature of the integrand near the outer turning points. Thus an expression
defining the near-dissociation behaviour of this numerator integral may be derived in exactly the same
manner. Combining this result with our NDT expression for the density of states yields the result [34]:
Bv = X1(n) (vD − v)n+2n−2 = X1(n) (vD − v)
2nn−2−2 (5.111)
in which the constant X1(n) has exactly the same structure as the constant X0(n) appearing in Eq. (5.104):
Xm(n) = Xk(n)/[μn (Cn)
2](n+2)/(n−2)
(5.112)
and the Xk(n) are known numerical factors [34, 3].
The qualitative differences between the solid and dashed curves in Fig. 5.11 clearly shows that the inte-
grand in the neighbourhood of the outer turning point make a less dominant contribution to the numerator
integral in the definition of Bv that was the case for the density-of-states integral in the denominator. As
a result, we may expect the NDT expression for Bv to be less accurate than was the case for the NDT
expression for the vibrational energies. This is confirmed by the results for the B 3Π+0u state of 35,35Cl2
shown in Fig. 5.12. As for the analogous state of I2 considered in Fig. 5.8, the limiting long-range potential
for this state of Cl2 has n = 5 behaviour. The upper panel of Fig. 5.12 shows that the vibrational spacings
indeed display the limiting linear behaviour predicted by Eq. (5.110), with the slope of this plot determin-
ing the C5 coefficient for this state. However, while the Bv values do extrapolate to the same vD value, the
limiting slope is not consistent with that implied by the C5 value determined from the vibrational spacings
plot in the upper panel.
128
Figure 5.12 NDT plots for the vibrational level spacings and Bv values for the highest observed levels of
the B 3Π+0u state of 35,35Cl2.
In view of the form of the semiclassical expectation value for any property f(r) presented in Eq. (5.71),
it is clear that deriving NDT expressions for powers of r would proceed exactly as was the case for
Bv. However, for the centrifugal distortion constants the derivation is somewhat more complicated. In
particular, recalling the band-constant expression for the vibration-rotation energies
E(v, J) = Gv + Bv[J(J + 1)] − Dv[J(J + 1)]2 + Hv[J(J + 1)]3 + . . . (5.113)
we see that we can define an effective J-dependent inertial rotational constant as
Beffv (J) ≡ ∂E(v, J)
∂[J(J + 1)]= Bv − 2Dv [J(J + 1)] + 3Hv[J(J + 1)]2 + . . . (5.114)
and hence can write
−Deffv (J) ≡ 1
2
∂Beffv (J)
∂[J(J + 1)]=
1
2
∂2E(v, J)
∂[J(J + 1)]2(5.115)
If we now set J=0 and introduce the phase-integral definition of Bv from Eq. (5.72), we obtain
Dv = − 1
2
∂Beffv (J)
∂[J(J + 1)]
∣∣∣∣J=0
= − 1
2
(�2
2μ
)∂
∂[J(J + 1)]
⎧⎨⎩∫ r2r1
drr2[D−VJ(r)]
1/2∫ r2r1
dr[D−VJ(r)]
1/2
⎫⎬⎭ (5.116)
129
Treating the expressions obtained on taking derivatives of the integrals appearing in Eqs. (5.116) is
distinctly more complicated that for the integrals appearing in the definition of Bv, because the integrals
on the real line would have non-integrable singularities at the two turning points. Moreover, the resulting
expressions contain an additional factor of 1/r2 in the integrand which further de-weights the interval near
the outer turning point. Nonetheless, after the same approximations are applied we obtain an expression
for the limiting near-dissociation behaviour of the leading centrifugal distortion constant Dv that has the
same algebraic structure as our NDT expressions for Gv and Bv, namely:
−Dv = X2(n) (vD − v)2nn−2−4 . (5.117)
Analogous treatments of the second-, third-, and higher-order derivatives of Beffv (J) with respect to [J(J +
1)] yield analogous NDT expressions for higher-order centrifugal distortion constants,
Hv = X3(n) (vD − v)2nn−2−6 (5.118)
Lv = X4(n) (vD − v)2nn−2−8 , (5.119)
...
in all of which Xm(n) is defined as in Eq. (5.112) [3].
As was mentioned for the case of the Dv constant, the integrands of the semiclassical phase integrals
appearing in Eq. (5.116) gain an additional factor of 1/r2 upon each order of differentiation with respect to
[J(J+1)]. The associated relative deweighting of the integrand near the outer turning point means that the
region in which the limiting expressions of Eqs. (5.117)-(5.119) may be expected to be valid is pushed ever
closer to dissociation. Nonetheless, they do define the correct behaviour for the limit in which v → vD .
In particular, they show that for n = 5 and 6, all centrifugal distortion contributions to the level energies
→ −∞ as v → vD ; for n = 4 Dv approaches a constant in this limit while the higher-order distortion
constants → −∞ , while for n = 3 Dv → 0 as (vD − v)2, Hv approaches a constant, and higher-order
distortion constants → −∞ in this limit. The existence of this fundamental limiting behaviour was not
suspected before the development of NDT.
The limiting singular behaviour of centrifugal distortion constants raises a question regarding the valid-
ity of the conventional power series expression for the J-dependence of rotational level energies. However,
we know that as J increases the total number of vibrational levels supported by the effective potential
VJ(r) decreases systematically, as ever more levels are spilled out of the well. As a result, as v increases
and approached vD, the highest J values that can be reached before that level is spilled out of the well
decreases systematically. In particular, exact quantum mechanical expressions for LJ(2n− 2, n) potentials
and semiclassical results for any potential with an inverse-power D−Cn/rn long-range tail show that the
highest J value reached before a given v-level passes above the dissociation limit and becomes quasibound
is defined by
[J(J + 1)]vD = (n − 2)2 (vD − v)2 . (5.120)
The fact that this factor dies off as (vD−v)2 exactly cancels the additional power of 1/(vD−v)2 associated
with the limiting behaviour of centrifugal distortion constant of increasing order. Thus, although centrifugal
distortion constants approach −∞ for levels approaching dissociation, their maximum contribution to
vibration-rotation level energies remains finite.
130
5.4.4 Near-Dissociation Expansions
The two cases considered above both represent situations in which experimental data are available for levels
lying sufficiently close to dissociation that NDT may be expected to be valid there. However, for the much
more common situations in which this is not true, NDT still offers a valuable means for obtaining optimal
estimates of the distance from the highest observed levels to dissociation, and of the number and energies
of unobserved levels. In particular, ‘near-dissociation expansion’ expressions (NDEs), which combine the
limiting functional behaviour of Eq. (5.104) with empirical expansions which account for deviations from
that limiting behaviour, were introduced to address this problem. Most work with NDEs has involved the
use of rational polynomials in the variable (vD − v) :
Gv = D − X0(n) (vD − v)2n/(n−2) [L/M ]s . (5.121)
The power ‘ s ’ in Eq (5.121) is set at either s=1 (to yield ‘outer’ expansions) or s=2n/(n− 2) (to yield
‘inner’ expansions), while [L/M ] is given by
[L/M ] =1 +
∑Li=1 pt+i (vD − v)t+i
1 +∑M
j=1 qt+j (vD − v)t+j, (5.122)
with the value of t being determined by the theoretically known form of the leading correction to the
limiting behaviour of Eq. (5.104) [35].
The fundamental ansatz underlying the use of NDEs is that fitting experimental data to expressions that
incorporate the correct theoretically known limiting near-dissociation behaviour (such as Eq. (5.121)) will
yield more realistic estimates of the physically significant extrapolation parameters D and vD than could
otherwise be obtained. In effect, it replaces blind empirical extrapolation using Dunham-type polynomials,
with interpolation between experimental data for levels in the lower part of the potential well and the
exactly known functional behaviour near the dissociation limit. Moreover, such expressions often provide
more compact representations of the data than do conventional power series in (v + 12).
Figure 5.13 summarizes the results of performing NDE fits to experimental data for the A 2Π state of
MgAr+ [36]. Since this species is a molecular ion, the (inverse) power of the leading term in its long-range
potential is n=4 , and since at least one of its dissociation fragments is in an S state, the power of the
second term is m=6 (see Appendix). For this case, theory shows that the power t in Eq. (5.122) should be
t=2 [35]. Theory also tells us that for any molecular ion, the value of the C4 coefficient in atomic units is
α/2, with α being the polarizability of the neutral dissociation fragment, so that the value of the limiting
NDT coefficient X0(4) is readily obtained. Moreover, a good theoretical estimate of the C6 coefficient could
be generated for this state, so a realistic value of the leading-deviation coefficient p2 (for fixed q2=0) could
also obtained [36].
The dotted line in Fig. 5.13 shows the limiting slope [4X0(4)]1/3 defined by the known C4 coefficient,
while the dot-dash curve labelled “linear B-S” shows the extrapolation behaviour implied by a linear Birge-
Sponer plot. The cluster of seven dashed curves shows the results of NDE fits for different {L,M, s} in
which the C4 coefficient was held fixed at the theoretical value and two {pi, qj} parameters were allowed to
vary, with no C6-based constraint being applied to the p2 value. The cluster of nine solid curves then shows
the results of fits in which both X0(C4) and p2(C4, C6) were fixed at the theoretical values, and again two
{pi, qj} parameters were allowed to vary (as well as vD and D). The quality of fit for all of these cases was
essentially the same. It is clear that for a given type of model (i.e., only X0(4) fixed, vs. X0(4) and p2 fixed),
the NDE models corresponding to different choices of {L,M, s} are in reasonably good agreement with
one another. However, the difference between the extrapolation behaviour for these two classes of models
shows that when better theoretical constraints are applied, significantly better extrapolation behaviour is
131
0 10 20 30 40 50 600
1
2
3
4
5
6
υ
(ΔGυ+½ )1/3
theoreticalslope [4X0(4)]
1/3
free C4
linear B-Sextrapolation
fixed C4 & C6
fixed C4
Mg+-Ar (A 2Π)
↑υD
Figure 5.13 Illustrative application of NDE fitting to data for the A 2Π state of MgAr+.
attained. For reference, the dash-dot-dot curve labelled “free C4” shows that in the absence either of a
realistic value of the leading long-range Cn coefficient or of data for levels lying near dissociation, NDE fits
can give quite unrealistic extrapolations, and should not be trusted.
A final point raised by the above example is the question of model-dependence, which is an ever-
present, but usually ignored problem in scientific data analysis. While all of the nine models corresponding
to “fixed C4 & C6” give fits to the data of equivalent quality, they all extrapolate slightly differently, and
the associated values of the physically interesting parameters D and vD differ by substantially more than
the parameter uncertainty associated with any individual fit. In cases such as this there is no possibility
of selecting a unique ‘best’ model, since there is no physical basis for choosing one set of {L,M, s} valuesover another. The best that one can do is to consider as wide a range of models as possible, and then
average the resulting values of the physically interesting parameters and estimate their uncertainties based
on both the variance about their mean and the uncertainties in the individual values. A practical scheme
for accomplishing this which was introduced in Ref. [36] led to the value of vD=58.4(±1.2) indicated by
the pointer at the bottom of Fig. 5.13.
Upon completion of a study such as that illustrated by the results shown in Fig. 5.13, a representa-
tive ‘optimal’ NDE function for the vibrational energies could then be chosen and employed in an RKR
calculation to generate a potential spanning essentially the entire potential energy well. Analyses of this
type have been carried out for a number of molecular systems. In general, fits of vibrational energies and
rotational constants to NDEs tend to be somewhat more compact that conventional Dunham polynomials
– fewer parameters being required to yield a given quality of fit. However, the inter-parameter correlation
increases rapidly with the number of free {pi, qj} parameters, and it becomes increasingly difficult to obtain
sufficiently realistic preliminary estimates of those parameters for the non-linear fit to be stable.
Tellinghuisen and Ashmore addressed this fit stability problem by introducing ‘mixed representations’
132
for Gv and Bv, in which conventional Dunham polynomials are used at low–v and NDEs at high–v, with
a switching function merging the two domains [37, 38]. Such representations certainly work, and they
have been implemented in standard data analysis [39] and RKR programs [23]. However, the increased
‘clutter’ associated with these mixed representations makes them somewhat inconvenient to use, and (to
date) neither pure NDEs nor these mixed representations have been widely adopted. Indeed, in recent
years the whole approach of attempting to provide global descriptions of molecular vibrational-rotational
energies using expansions in terms of vibration-rotation quantum numbers is increasingly being supplanted
by the ‘direct-potential-fit’ approach described in Chapter 7.
133
5.4.5 Extending Near-Dissociation Theory
A wide range of applications have confirmed that the limiting functional behaviour of vibrational level
energies predicted by NDT is quantitatively correct, and can be used to determine optimal estimates of
the physically interesting properties D, vD, and the limiting Cn coefficient, as well as to make reliable
predictions of the number and energies of missing levels lying above the highest one observed. Nonethe-
less, although the analogous expressions for Bv values and centrifugal distortion constants appear to be
functionally correct, they are much less accurate than the corresponding expression for the vibrational
energies. Moreover, deviations from the vibrational NDT equations always occur as (vD − v) increases,
and we would like to explain that behavior.
It is also interesting to note that our limiting NDT equation appears to work “better than it should”.
In particular, consider the B 3Π+0u state of I2 which dissociates to yield one iodine atom in each of the 2P3/2
and 2P3/2 states. The discussion of § 2.2.2 tells us that the leading terms in the long-range potential for
this species will be
V (r) � D − C5/r5 − C6/r
6 − C8/r8 − . . . (5.123)
Figure 5.8 seem to show that the energies of levels v � 55 of this state accurately obey the limiting
NDT equation for vibrational energies. As a further test, Barrow and Yee performed fits of experimental
vibrational energies for v ≥ 64 to Eq. (5.108) using different choices for the power n. As seen in Fig. 5.14,
this confirmed that the data for this system do obey the vibrational NDT equation for n = 5 . However, we
know that the RKR inversion procedure of § 5.3 can be used to generate a set of potential energy function
points that is not based on any particular analytic model for the potential energy function. Fits to RKR
turning points for levels v = 64 − 77 to Eq. (5.123) with C5 held fixed at the value determined from the
vibrational NDT analysis then yielded estimates of C6 and C8 for this state. The resulting sets of Cn
coefficients then determine the relative magnitudes of the various terms contribution to Eq. (5.123). Figure
66 68 70 72 74 76
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
υ
deviation from a fit to Eq. (5.108)
n=4
n=6
n=5
I2(B3Π+
0u)
Figure 5.14 Test of NDT power law for the B 3Π+0u state of I2 for which theory predicts n = 5 .
[Based on data from Barrow and Yee [40]]
134
Figure 5.16 Relative magnitudes of inverse-power terms contribution to the long-range potential
for the B 3Π+0u state of I2. [Figure 6 of Barrow and Yee [29].]
5.15 shows that over most of the domain spanned by the outer turning points for levels v = 64 − 77 ,
the limiting C5/r5 term is actually only responsible for 50–70% of the interaction energy. It therefore
seems somewhat strange that the pattern of vibrational levels energies would be perfectly explained by an
expression that assumes that the long-range potential consists purely of the C5/r5 term.
The various questions posed above are at least partly explained if we extend NDT to take account of the
leading deviations from the limiting near-dissociation behaviour. From the nature of the integrand for the
uppermost levels considered in Fig. 5.7 (on p. 123), it seems unlikely that the leading corrections would be
associated with the neglected singularity at the inner turning point or setting the lower bound of integration
in Eq. (5.101) to r1/r2 = 0 . A more important source of error would probably be changes in the integrand
and outer turning point due to the presence of the additional (higher-power) contributions to Eq. (5.123).
This point is confirmed by Fig. 5.17, which shows how the integrand of the density-of-states integral of
Eq. (5.97) (upper panel) is affected when the full interaction potential (solid curves) is replaced by various
inverse-power-sum approximations to it. If we compare the dotted curve with the short-dash curve in the
upper panel, we see that the replacing D − C5/r5 by the two-term potential D − C5/r
5 − C6/r6 has
two competing effects. First of all, the value of the integrand at any particular distance becomes smaller,
which would tend to make the integral become smaller; secondly, the outer turning point shifts to a larger
distance, which would have the effect of making the integral become larger. The same two competing
tendencies are associated on the inclusion of higher-power terms on our approximation to the long-range
potential.
The limiting NDT results discussed above were derived using the assumption that for levels lying
135
Figure 5.17 Lower: comparison of the full RKR potential (solid curve) with various inverse-power-
sum approximations to the long-range potential for B-state I2. Upper: for the v = 60 level
of B-state I2, integrand of the density-of-states integral of Eq. (5.97) associated with those
different representations of the potential energy function. Vertical dashed lines in the upper
panel and vertical arrows in the lower panel indicate the turning point positions.
136
near dissociation, the properties of interest depend mainly on the nature of the potential energy function
near the outer turning point, and that the potential function there could be realistically approximated by
the leading (longest-range) inverse-power term Cn/rn. The content of Fig. 5.17 strongly suggest that the
leading corrections to those results will be due to the next higher-order term in the long-range potential.
We therefore begin our extended theory by assuming that the long-range potential is represented by two
inverse-power term
V (r) = D − Cn
rn− Cm
rm(5.124)
in which m > n. By the definition of the outer turning point, we can write
Gv = V (r2(v)) = D − Cn
[r2(v)]n− Cm
[r2(v)]m= D − Cn
[r2(v)]n{1 + α} (5.125)
which α is the ratio of the second to the first long-range term, evaluated at the outer turning point.
Rearrangement of Eq. (5.125) allows us to write
r2(v) =
(Cn
D−GV
)1/n
(1 + α)1/n . (5.126)
We may then re-write our definition for α in the form
α ≡ Cm/[r2(v)]m
Cn/[r2(v)]n=
Cm
(Cn)m/n
(D−Gv
1 + α
)(m−n)/n. (5.127)
If we now repeat the NDT derivation as before, the analog of Eq. (5.101) is
dv
dGv≈ 1
2π
√2μ
�2
(Cn)1/n
[D−Gv ]n+22n
{(1 + α)
n+22n
∫ 1
r1/r2
dy
[(y−n − 1) + α(y−m − 1)]1/2
}. (5.128)
In this expression, the factor (1 + α)(n+2)/2n in front of the integral reflects the shift of the outer turning
point from the one-term to the two-term potential, while the term associated with the factor of α in the
integrand reflect the change in the integrand. Within the assumption that α < 1 , we can expand the
integral in Eq. (5.128) as a Taylor series about α = 0 , yielding∫ 1
0
dy
(y−n − 1)1/2− α
2
∫ 1
0
(y−m − 1) dy
(y−n − 1)3/2+
3α2
8
∫ 1
0
(y−m − 1)2 dy
(y−n − 1)5/2+ . . . (5.129)
Applying a binomial expansion to the factor (1 + α)(n+2)/2n then yields the result that
dGv
dv= Kn[D−Gv ]
n+22n[1 + f1(n,m)α + f2(n,m)α2 + . . .
]−1(5.130)
and some further manipulation yields the result [35]:
Gv = D − X0(n)[vD − v]2nn−2
{1 + f1(n,m)
Cm
Cn
(X0(n)
Cn
)m−n2
(vD − v)2(m−n)
n−2 + . . .
}(5.131)
In this final result, the factor f1(n,m) is the difference between two terms, one associated with the shift
of the outer turning point on introducing the second long-range term, and the other due to the change in
the integrand. A remarkable result is the fact that when the two leading long-range powers differ by 1,
these two contributions precisely cancel one another; i.e., f1(n, n + 1) = 0 . This accidental cancellation
explains why the vibrational spacings for B(3Π0+u)-state I2, for which (n,m) = (5, 6). appear to obey the
137
simple limiting NDT law in regions where we know that higher-order long-range terms make significant
contribution to the potential energy function at the outer turning point (see Fig. 5.16). For other cases
such as (m,n) = (6, 8) or (4, 6) f1(n,m) is non-zero, but some cancellation of these two effects means that
again, the limiting NDT result for the vibrational energies will work better that one might have expected.
In the above derivation, the Taylor series expansion of Eq. (5.129) involves terms of the form∫ 1
0
(y−m− 1)k
(y−n − 1)k+1/2dy . (5.132)
While this integrand is always integrable at the upper bound where y → 1 , as it approached the lower
bound the integrand is proportional to 1/yk(m−n)−n/2, and hence for large enough values of k it will
yield a non-integrable singularity. Indeed, this problem occurs for the first (k = 1) correction term for the
physically important cases (n,m) = (1, 4) or (3, 6). However, a little more calculus shows that the problem
occurs because the leading correction depends on a fractional power of α and that for (n,m) = (3, 6) we
obtain [41]dGv
dv=
dGv
dv
∣∣∣∣NDT
/[1− 1.76737α5/6
](5.133)
while for (n,m) = (1, 4) we obtain [41]
dGv
dv=
dGv
dv
∣∣∣∣NDT
/[1− 0.603394α1/2
](5.134)
Unfortunately, while Eqs. (5.133) and (5.134) do correctly describe the onset of the deviation from
limiting NDT behaviour, the range of α over which they provide a semi-quantitative description of those
deviations is quite narrow (α � 0.05). Thus, further work is necessary if we are to obtain useful working
expressions to characterize the deviation from limiting near-dissociation behaviour for these challenging
cases.
138
5.5 Semiclassical Treatment of Quasibound Level Widths
It is well known that for an electronic state with a rotationless potential barrier or a centrifugal barrier
which protrudes above the asymptote at distances larger than the potential minimum, the molecule may
predissociate by tunneling through that barrier, which causes shifts of those “quasibound” levels and
broadening of associated lines. These tunneling predissociation line widths are experimental observables
which depend upon the potential energy function, and hence can be included with the transition frequencies
in the data set used to determine the potential energy function. However, to do so one requires efficient
and accurate methods for determining their energies and calculating their widths which can readily be
incorporated into the direct-fit analysis procedure.
Strictly speaking, the energies and widths of quasibound levels observed in “discrete” spectra can
only be accurately predicted from a detailed quantum mechanical bound→continuum simulation of the
intensity profile associated with each transition into that level. In practice, however, that approach would
be too cumbersome to incorporate into an automated least-squares data analysis procedure. Fortunately,
efficient and accurate approximate methods for locating quasibound levels and calculating their widths
were developed a number of years ago [42, 43, 44]. In particular, as discussed in § 3.3, it was found that
very accurate level energies could be obtained by combining the usual wavefunction boundary condition at
r → 0 with the requirement that at the outermost classical turning point r3(Ev,J) (lying on the outer wall
of the potential barrier at energy Ev,J ), the wavefunction must behave like an Airy function of the second
kind, Bi(r − r3(Ev,J )). This turns the problem of locating such metastable levels into the same standard
two-boundary-condition eigenvalue problem associated with truly bound levels, and allows application of
the standard very rapidly converging Numerov-Cooley predictor-corrector procedure.[45, 46] Tests showed
that quasibound level energies obtained in this way agree with those yielded by the best alternate methods
to within ca. 5% of the associated level width, discrepancies smaller than those usually associated with
experimental determination of such line positions. This method of located quasibound levels is implemented
in the widely-used eigenvalue subroutine SCHRQ that is the core subroutine in program LEVEL [24].
A simple physical description of the tunneling predissociation process considers the molecule in qua-
sibound level {v, J} at total energy E = Ev,J (where Ev,J > D) as being trapped behind the potential
barrier, and vibrating in the well bounded by classical turning points r1(Ev,J ) and r2(Ev,J ). The molecule
will collide with the barrier with a frequency of 1/tvib(Ev,J ) [s−1], where tvib(Ev,J) is the period of vi-
bration for that level, and on each collision there is a finite probability κ(Ev,J ) that quantum mechanical
tunneling will allow predissociation to occur. As a result, the tunneling predissociation lifetime will be
τtp = tvib(Ev,J )/κ(Ev,J ) , and the associated (FWHM) level width will be
Γ tpv,J = �/τtp = �κ(Ev,J )/tvib(Ev,J ) (5.135)
In a simple classical treatment
tclvib(Ev,J) = �
√2μ
�2
∫ r2(v,J)
r1(v,J)[Ev,J − V (r)]−1/2 dr (5.136)
and the conventional first-order semiclassical treatment of barrier tunneling yields[47, 48]
κsc(Ev,J ) = e−ε(Ev,J ) (5.137)
where
ε(Ev,J ) = 2
√2μ
�2
∫ r3(v,J)
r2(v,J)[V (r)− Ev,J ]
1/2 dr (5.138)
139
and ri(v, J) for i=1−3 are the three classical turning points, the distances where the potential energy equals
the total energy Ev,J of this level [42]. This formulation gives realistic estimates of level widths, particularly
for long-lived levels lying well below the potential barrier maximum. However, its simple (semi)classical
origin introduces unphysical artifacts, such as a singularity in tclvib(Ev,J ) and tunneling efficiency of 100%
when Ev,J lies precisely at the barrier maximum energy.
Fortunately, a uniform semiclassical treatment defined in terms of the same phase integrals appearing
in Eqs. (5.136) and (5.138) yields the much more accurate expressions [44]
κun(Ev,J ) = 4
([1 + e−ε(Ev,J )
]1/2 − 1[1 + e−ε(Ev,J )
]1/2+ 1
)(5.139)
and [49, 43]
tunvib(Ev,J ) = tclvib(Ev,J )− �
2π[ln {ε(Ev,J )/2π} − X]
∂ε(Ev,J )
∂Ev,J(5.140)
where
X ≡ 2πdArg Γ(12 − i ε(Ev,J )/2π
)dε(Ev,J )
(5.141)
= ψ(12 ) +∑k=0
1/(k + 1
2){[
2π(k + 12)/ε(Ev,J )
]2+ 1}
(5.142)
in which Γ(a+ iy) is a gamma function with complex argument, and the summation definition of X is ob-
tained using Eq. (6.1.27) of Ref. [5], where ψ(x) is the digamma function and ψ(12 ) = −1.96351 00260 21423 ...[5]. Note that for a level energy at the barrier maximum, the singularities in tclvib(Ev,J ) and the logarithm
term in Eq. (5.140) precisely cancel one another [49, 43, 44]. Note too that for the larger values of ε(Ev,J )
(e.g., value for which e−ε(Ev,J) � 10−5), more accurate numerical values of κun(Ev,J ) may be obtained by
evaluating the numerator Eq. (5.139) using the expansion[1 + e−ε(Ev,J)
]1/2 − 1 � 12e−ε(Ev,J )
(1− 1
4e−ε(Ev,J ) + 1
8e−2ε(Ev,J ) + ...
)(5.143)
In summary, therefore, the recommended method for calculating the tunneling predissociation level
width Γ tpv,J (or tunneling lifetime τtp) is to use Eq. (5.135), with κ(Ev,J ) and tvib(Ev,J ) calculated using
Eqs. (5.139) and (5.140), respectively. Detailed numerical tests reported in Refs. [42, 43, 44] show that for
levels lying below a potential barrier maximum, this approach yields results as accurate as can be obtained
without performing a detailed numerical bound→continuum simulation of the particular transition being
probed. However, as an estimate of the effects of uncertainty due to the (neglected) background phase
shift [44] and that associated with the precise determination of the energy Ev,J at which κun(Ev,J) and
tunvib(Ev,J ) are being evaluated, one should probably associate a computational uncertainty of a few percent
(say ca. 5%) with level widths calculated in this way.
The closed-form expression for level widths obtained above is particularly useful when optimizing
the parameters defining a model potential energy function using a least-squares fit to experimental data
which include measured predissociation level widths. It allows the partial derivatives with respect to
parameters of the potential required for the least-squares procedure to be computed directly, which is a
much more efficient and accurate approach than using numerical differentiation. Since the phase integrals
and related quantities discussed above are all implicitly functions of the parameters {pi} defining the
potential, ε(Ev,J ) = ε(Ev,J ; {pi}) , ... etc., straightforward use of the chain rule yields:
∂Γ tpv,J
∂pj
∣∣∣∣∣{pi;i �=j}
=�
tunvib(Ev,J )
{∂κun(Ev,J )
∂Ev,J
∣∣∣∣{pi}
∂Ev,J
∂pj
∣∣∣∣{pi;i �=j}
+∂κun(Ev,J )
∂pj
∣∣∣∣{pi;i �=j},Ev,J
}(5.144)
140
− �κun(Ev,J )[tunvib(Ev,J)
]2{∂tunvib(Ev,J)
∂Ev,J
∣∣∣∣{pi}
∂Ev,J
∂pj
∣∣∣∣{pi;i �=j}
+∂tunvib(Ev,J)
∂pj
∣∣∣∣{pi;i �=j},Ev,J
}where
∂κun(Ev,J )
∂Ev,J
∣∣∣∣{pi}
=dκun(Ev,J )
dε(Ev,J )
∂ε(Ev,J )
∂Ev,J
∣∣∣∣{pi}
(5.145)
∂κun(Ev,J )
∂pj
∣∣∣∣{pi;i �=j},Ev,J
=dκun(Ev,J )
dε(Ev,J )
∂ε(Ev,J )
∂pi
∣∣∣∣{pj ;j �=i},Ev,J
(5.146)
withdκun(Ev,J )
dε(Ev,J )= −e−ε(Ev,J)
/(1 + e−ε(Ev,J)
)1/2 [(1 + e−ε(Ev,J )
)1/2+ 1
]2(5.147)
and
∂ε(Ev,J )
∂Ev,J
∣∣∣∣{pi}
= −√
2μ
�2
∫ r3(v,J)
r2(v,J)[V (r)− Ev,J ]
−1/2 dr (5.148)
∂ε(Ev,J )
∂pj
∣∣∣∣{pi;i �=j},Ev,J
= +
√2μ
�2
∫ r3(v,J)
r2(v,J)
∂V (r)
∂pj[V (r)− Ev,J ]
−1/2 dr (5.149)
The phase integrals appearing in Eqs. (5.148) and (5.149) have the same type of integrable singularity
at the turning points seen in Eq. (5.136), which can readily be evaluated by standard numerical methods
(see e.g., §25.4.38 and 25.4.40 of Ref. [5]). However, the analogous partial derivatives of tclvib(Ev,J ) are not
quite so standard, since differentiation of the integrand yields non-integrable singularities at the turning
points. Fortunately, methods for treating high-order phase integrals of this type were developed some years
ago [50, 51]. The most efficient of these is the technique developed by Pajunen and Child [50, 52]; it begins
by noting that line integrals of the type appearing in Eqs. (5.136), (5.148) and (5.149) may be written as
contour integrals in the complex plane:
Ik({f}) =∫ rb
ra
dr f(r)/ |Ev,J − V (r)|k+12 = 1
2
∮Υ(ra,rb)
dr f(r)/ |Ev,J − V (r)|k+12 (5.150)
where Υ(ra, rb) is a contour in the complex plane surrounding the segment of the real line running between
the two turning points ra and rb. In this notation, the basic integrals introduced above may be written as:
tclvib(Ev,J ) = �
√2μ
�2Iwell0 ({1}) and ε(Ev,J ) = 2
√2μ
�2Ibarr−1 ({1}) (5.151)
Use of the recurrence relations [3] ∂Iwellk ({f})/ ∂Ev,J = −(k+ 12 ) I
wellk+1({f}) and ∂Iwellk ({1})/ ∂pj = +(k+
12) I
wellk+1
({∂V (r)∂pj− ∂Ev,J
∂pj
})(and for Ibarrk ({f}) , the same expressions with the opposite sign) then allows
us to write
∂ε(Ev,J )
∂Ev,J= −
√2μ
�2Ibarr0 ({1}) (5.152)
∂ε(Ev,J )
∂pj=
√2μ
�2Ibarr0
({∂V (r)
∂pj− ∂Ev,J
∂pj
})(5.153)
∂2ε(Ev,J )
∂Ev,J ∂pj= −1
2
√2μ
�2Ibarr1
({∂V (r)
∂pj− ∂Ev,J
∂pj
})(5.154)
∂tclvib(Ev,J )
∂Ev,J
∣∣∣∣{pi}
= −�
2
√2μ
�2Iwell1 ({1}) (5.155)
∂tclvib(Ev,J )
∂pj
∣∣∣∣{pi;i �=j},Ev,J
=�
2
√2μ
�2Iwell1 ({∂V/∂pj}) = − �
2
√2μ
�2
∂Iwell0 ({∂V/∂pj})∂Ev,J
∣∣∣∣{pi}
(5.156)
141
Using the compact integral notation, one may then write
Γ tpv,J = κ(Ev,J )
/√2μ
�2
{Iwell0 ({1}) +
1
2π
[ln
{ε(Ev,J )
2π
}− X
]Ibarr0 ({1})
}(5.157)
(5.158)
and hence the partial derivatives required for a least-squares optimization procedure would be
∂Γ tpv,J
∂pj
∣∣∣∣∣{pi;i �=j}
=�
tunvib(Ev,J )
dκun(Ev,J)
dε(Ev,J )
√2μ
�2Ibarr0
({∂V (r)
∂pj− ∂Ev,J
∂pj
})(5.159)
− �2 κun(Ev,J )[tunvib(Ev,J )
]2{√
2μ/�2
2Iwell1
({∂V (r)
∂pj− ∂Ev,J
∂pj
})
+
√2μ/�2
2πIbarr0 ({1})
[√2μ/�2
ε(Ev,J )Ibarr0
({∂V (r)
∂pj− ∂Ev,J
∂pj
})− ∂X
∂pj
]
−√
2μ/�2
4π
[ln
{ε(Ev,J )
2π
}− X
]Ibarr1
({∂V (r)
∂pj− ∂Ev,J
∂pj
})}where
∂X
∂pj=
8π2√
2μ/�2
[ε(Ev,J )]3 Ibarr0
({∂V (r)
∂pj− ∂Ev,J
∂pj
}) ∑k=0
k + 12{[
2π(k + 12)/ε(Ev,J )
]2+ 1}2 (5.160)
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144
Chapter 6
Photodissociation, Predissociation and
Bound→Continuum Emission
6.1 General Considerations
In our discussion of conventional discrete spectra, attention was focussed mainly on the line positions that
determine the relative level energies. In discrete spectra, almost all of the information about molecular
structure and potential energy functions comes from the line spacings, while the transition intensities play
the subsidiary role of providing information about the transition moment function. In bound→ continuum
spectra the situation is qualitatively quite different, as the pattern of the intensity vs. wavelength or
frequency carries all of the information about both the potential energy function and the transition moment
function. As illustrated by Fig. 6.1, bound→ continuum spectra may arise in either absorption or emission.
In either case, the lack of energy quantization for the final state means that transitions are associated with
Figure 6.1. Schematic illustration of bound↔continuum absorption and emission spectra.
145
a continuum of transition energies.
Historically, continuum spectra have received much less attention than discrete spectra. This is prob-
ably largely because thermal photodissociation spectra show relatively little structure, and it is relatively
difficult to obtain quantitative molecular information from them. Nonetheless, repulsive potential energy
functions abound in nature: all bound state potentials have repulsive walls, electronic states with purely (or
mainly) repulsive potential functions are very common, and all molecular collision events involve the repul-
sive region of the relevant potential energy function. Thus, simulation and inversion of bound→ continuum
phenomena is a much more important problem than is generally realized.
For absorption of light of frequency ν by molecules in vibration-rotation level (v, J) with energy E(v, J)
of electronic state “i”, which undergo transitions to continuum levels at energy Es = E(v, J) +hν of final
electronic state “s”, the absorption cross section (in units [cm2/molecule]) is [1, 2]
σν(v, J ;Es) =8π3 ν gs3h c
∑J ′
SJ ′J
2J + 1
∣∣〈ψEs,J ′ |Msi(r)|ψv,J 〉∣∣2 (6.1)
in which gs is the electronic degeneracy of final electronic state–s and Msi(r) is the transition moment
function. This quantity is related to the attenuation of a beam of light with initial intensity I0(ν) upon
passing through a cell of length L [cm] that contains a population of N [molec/cm3] of the absorbing
species:
I(ν) = I0(ν) e−σν(v,J ;Es)LN (6.2)
Similarly, for spontaneous emission from vibration-rotation level (v, J) with energy E(v, J) of the initial
electronic state into a continuum at energy Es = E(v, J) − hν on (lower) final electronic state s, the
Einstein coefficient for spontaneous emission (in units [s−1/cm−1]) is [2]
Aν(v, J, ;Es) =64π4 ν3 gs
3h
∑J ′
SJ ′J
2J + 1
∣∣〈ψEs,J ′ |Msi(r)|ψv,J 〉∣∣2 (6.3)
A related bound→ continuum phenomenon is the radiationless isoenergetic predissociation of a given
discrete vibration-rotation level (v, J) of some initial electronic state i into a continuum level of final
electronic state s. For this process the “golden rule” predissociation rate is (in units s−1):
k(v, J) =4π2 gsh
|〈ψEs,J |Msi(r)|ψv,J 〉|2 (6.4)
where in this case the final-state energy Es = E(v, J) . Although the nature of this process is quite different
than continuum absorption or emission, the fact that it is driven by the same type of bound↔ continuum
wavefunction overlap integral means that it may readily be treated by the same computational tools.
In all of Eqs. (6.1), (6.3) and (6.4), the density of states associated with the continuum wavefunctions
is assumed to have been incorporated into the asymptotic normalization of the continuum wavefunction:
ψEs,J ′(r) �(
2μ/�2
π2 [Es − V∞s ]
)1/4
sin(ksr + ηJ ′) (6.5)
in which ks =√{[2μ/�2][Es − V∞s ]} and V∞s ≡ limr→∞ Vs(r) . However, in practical calculations it is
usually more convenient to normalize the continuum wavefunction to have unit asymptotic amplitude, in
which case each of the above expression will acquire the density-of-states factor
ρ(Es) =
√2μ/�2
π2 [Es − V∞s ](6.6)
146
If we return to our general bound–continuum expressions, we see that if we omit the physical constants,
all of Eqs. (6.1), (6.3) and (6.4) become
{intensity/rate} ∝ νp∣∣∣∣∫ ∞
0ψEs,J ′(r)Msi(r)ψv,J (r) dr
∣∣∣∣2 (6.7)
in which p = 1 for absorption, p = 3 for emission, and p = 0 for predissociation. For the forward
problem of calculating transition intensities (or predissociation rates) for a system whose potential energy
and transition moment functions are known, prediction of the property of interest is a straightforward
matter.
• Solve the radial Schrodinger equation for the two electronic states to obtain ψEs,J ′(r) and ψv,J (r),
and evaluate the overlap integral on Eq. (6.7).
• Sum over all final-state J ′ values allowed by the rotational selection rules. Since all allowed J → J ′
transitions occur at the same energy (unlike the situation in discrete spectroscopy), they all contribute
to the same observable.
• Sum over all final electronic states ‘s’ accessible at the given final-state energy Es = Ev,J ±hν , with‘+’ for absorption, ‘−’ for emission and no hν factor for predissociation.
• For absorption, we must also sum over all populated initial-state (v, J) levels, since in general most
such levels can contribute to the net absorption at a given frequency.
The inverse problem of determining transition moment function(s) and repulsive final-state potential(s)
is much more challenging. In particular, there exist no exact quantum mechanical inversion schemes.
Semiclassical RKR-like inversions schemes have been devised for cases in which a structured absorption
or emission continuum associated with a single initial-state vibrational level (such as that illustrated in
Fig. 6.1) is clearly resolved. However this is a relatively uncommon case. Thus, in most cases it is necessary
to apply a procedure in which simulated data generated from assumed potential energy and transition
moment functions are compared with experiment, and parameters defining those functions are optimized
by a non-linear least-squares fit procedure. However, in order to understand and model such data, it is
important to have some intuitive understanding of what the observable pattern of intensities depend on.
6.2 Approximate Treatments of Bound→Continuum
Transitions
6.2.1 The “delta-function” and “reflection” approximations
Let us begin by considering the absorption process illustrated by Fig. 6.2. As we see there, the number of
intensity peaks in the absorption spectrum equals the number of extrema in the initial-state wavefunction.
As is shown in Fig. 6.1, the intensity peaks tend to pile up on one another at energies approaching the
asymptote of the final-state potential. However, the type of one-to-one mapping of wave-function extrema
to intensity maxima seen in Fig. 6.2 is the more general rule.
The existence of this apparent mapping behaviour is readily explained by the ‘delta function approxima-
tion’ for the final-state wavefunction. As is schematically illustrated by Fig. 6.3, a continuum wavefunction
will have its greatest amplitude in the innermost loop centred at the classical turning point rt(Es). Thus,
it would seem a reasonable zero’th order approximation to approximate that wavefunction by a ‘delta
147
Figure 6.2. Illustration of bound→ continuum absorption from initial-state vibrational level v = 5 emis-
sion spectra.
function’ δ(r − rt(Es)), which is a normalizable function that is equal to zero everywhere except where its
argument equals zero:∫ ∞0{δ(r − rt(Es))} dr = 1 and δ(r − rt(Es)) = 0 for r �= rt(Es) (6.8)
Figure 6.3. Schematic illustration the delta-function approximation for a continuum state wavefunction.
148
If we approximate the continuum wavefunction in Eq. (6.7) by a delta function, our expression collapses to
{intensity/rate} ∝ νp |Msi(rt(Es))ψv,J (rt(Es)) |2 ∝ |ψv,J(rt(Es)) |2 (6.9)
This clearly explains the one-to-one mapping of wavefunction extrema onto intensity maxima referred to
above.
This result gives rise to what is called the “reflection approximation” for determining a repulsive-
state potential from a structured absorption or emission continuum associated with a single initial-state
vibrational level. Application of this approach proceeds by the following steps. On an energy versus
distance diagram such as Fig. 6.2:
1. draw the initial-state wavefunction along the horizontal axis at an energy of zero;
2. plot the observed absorption/emission intensity versus energy along the vertical axis;
3. draw a horizontal line through each intensity extremum and a vertical line through each wavefunction
extremum;
4. the crossing point of each corresponding pair (first, second, third, . . . , etc.) of vertical and horizontal
lines defines a point on the repulsive potential energy curve.
This approach is called the ‘reflection approximation’ since it effectively assumes that the observed struc-
tured continuum is a reflection of the square of the initial-state wavefunction in a mirror that runs along
the final-state potential curve.
The reflection approximation is intuitively very appealing and does qualitatively explain the pattern of
bound→ continuum absorption or emission intensity originating in a particular vibrational level. However,
it is often not very accurate, especially for transitions into continuum energies lying near the final-state
potential asymptote. This point is illustrated by Fig. 6.4, which compares results obtained using three
different methods of calculating the spectrum for bound→ continuum emission from the v = 12 level of
the D 3Σ+ state of NaK into the a 3Σ+ state continuum using assumed-known potential energy functions.
Panel A shows the results of a calculation using exact quantum mechanical wavefunctions; Panel C shows
the results of a reflection approximation calculation, and Panel B shows results obtained using a ‘uniform
harmonic approximation’ method developed by M.S. Child [3] that is described below. It is clear that
Child’s approximate method almost exactly mimics the exact quantal results, the reflection approximation
is substantially in error.
6.2.2 Child’s “Uniform Harmonic Approximation”
The source of the shortcomings of the reflection approximation lies in its incomplete application of the
Franck-Condon principle. In particular, we should recall that the Franck-Condon principle states that
both the position and the momentum of the nuclei should be conserved in an electronic transition. This
means that a transition with energy E from vibrational level v′′ of a given initial electronic state with
potential energy curve V1(r) into the continuum of final electronic state with potential function V2(r)
only can occur when the internuclear distance has a value Rx(E) at which the momentum (and hence
also the kinetic energy [E − V (r)]) is the same in both states. This situation is illustrated by Fig. 6.2
where we see that the distance Rx(E) at which momentum is conserved is displaced slightly from the
turning point R2(E) at energy E on the final-state potential. Thus, in contrast with the picture associated
with the reflection approximation, for a transition from bound level Ev,J into a continuum level at energy
149
Figure 6.4. Comparison of exact quantal absorption intensity calculation, Case A, with predictions gener-
ated using the reflection approximation, Case C, and using Child’s uniform harmonic approximation,
Case B.
E, the main contribution to the overlap matrix element is associated with the momentum-conserving or
“stationary-phase” point where
E − Vcont(r) = Ev,J − Vbound(r) (6.10)
This stationary-phase point will clearly have different locations for each different initial- and final-state
energy.
Based on his appreciation of the importance of this stationary-phase point, Child and Hunt developed
a “uniform harmonic approximation” for bound→ continuum transition amplitudes [4, 5], which is based,
in part, on a mapping of the phase of the initial-state vibrational wavefunction onto that for a harmonic
oscillator wavefunction for the same v. The resulting expression is
Mv,E =
∫ ∞0
ψE(r)M(r)ψv(r) dr (6.11)
≈ M(rx(E))
[ω
uxΔFx
]1/2[2v + 1− ξ(E)2]
1/4 φv(ξ(E)) , (6.12)
in which M(rx(E)) is the value of the transition moment function at the stationary phase point, ω is the
classical bound-state oscillation frequency for level v, ux is the radial momentum and ΔFx the difference
between the slopes of the upper and lower-state potentials at the stationary phase point, and φv(ξ(E)) is
the value of the v’th harmonic oscillator wavefunction at a phase value ξ(E) associated with phase integrals
over intervals of the initial- and final-state potentials that meet at rx(E) [3]. It is clear from Fig. 6.4 that
Child’s approximate method almost exactly mimics the exact quantal results.
Starting from this result, Child developed an ‘RKR-like’ inversion procedure which is based on a
mapping of the extrema of the dimensionless harmonic oscillator variable ξ for the given vibrational level
onto the observed transition intensity maxima. This allowed him to convert the knowledge of the type of
150
Figure 6.5. Illustrative mapping of simulated photodissociation intensities originating in the v = 5 level
of Br2(X1Σ+
g ) onto the extrema of the harmonic oscillator wavefunction for v = 5.
ξ(E) mapping seen in Fig. 6.5 into a knowledge of the repulsive final-state potential energy function V2(r).
In particular, in terms of the notation of Fig. 6.2, this gave rise to the following expression for calculating
points on the (repulsive) final-state potential [3]:
r2(E) = 12 [b1 + rx(E)] +
{integrals involving ξ(E) and the known initial−state potential
}(6.13)
An illustrative application to a structured emission continuum that was initially attributed to the
v = 12 level of the D 1Π state of NaK is shown in Fig. 6.6. The dash–dot curve seen there was obtained
from a reflection approximation analysis and the solid curve from Child’s uniform harmonic approximation
inversion procedure. As shown by Panel (a) of Fig. 6.7, the positions and relative heights of the first 10
peaks in the resulting spectrum are in excellent agreement with experiment.
Unfortunately, while this application confirmed that the inversion procedure does work, there was a
problem with this result. In particular, independent experimental work had yielded discrete vibration-
rotation data for the lower (a 3Σ+) state of this transition. Those data were analyzed to obtain Dunham
Y�,m parameters which in turn were used to generate an RKR potential for the potential energy well of this
state. The inner wall of that potential is shown as a dotted curve in Fig. 6.6, and it is clearly incompatible
with the curve yielded by the inversion procedure. The dashed curve in the lower portion of Fig. 6.6 shows
a repulsive potential wall that was constrained both to smoothly join the RKR curve and to place the
lowest-energy emission peak in the right position. However, the upper panel of Fig. 6.7 shows that the
continuum spectrum implied by this (dashed plus dotted) repulsive curve was in very poor agreement with
experiment.
While the above situation was initially disturbing, there turned out to be a clear physical explanation
151
Figure 6.6. Model associated with a structured emission continuum that was originally assigned as being
due to the v = 12 level of the D 1Π state. The solid repulsive a 3Σ+ potential was obtained using
the Child inversion procedure and the dash–dot curve by using the reflection approximation. The
dotted curve is the inner wall of the bound portion of the a-state potential obtained from an RKR
analysis while the dashed curve is a ‘best’ Child-type analysis constrained to be consistent with the
RKR curve.
of this apparent discrepancy. In particular, later work showed that the observed structures emission
continuum was actually associated with emission from the v = 13 vibrational level of the d 3Π1 electronic
state which lies at almost the same energy as the v = 12 level of the D 1Π state; see Fig. 6.8. A later
extension of the uniform harmonic inversion procedure was developed which allowed one to determine an
initial-state attractive potential well from an experimental structured emission continuum and an assumed-
known repulsive final-state potential [6]. Application of this procedure led to the determination of both a
potential energy well for the previously unknown d 3Π1 state and the repulsive potential wall of the a 3Σ+
state. Thus, as is often the case in science, an apparent inconsistency or disagreement of a model with
experimental data became a source of new molecular information.
In spite of the success of the uniform harmonic inversion procedures of Child and co-workers, the
utility of this type of method is limited because the type of structured continuum input data it requires is
relatively difficult to obtain. Moreover, when such spectra extend to the final-state dissociation threshold
there can be ambiguity in ascertaining the correct vibrational assignment for the initial-state level, as was
the case for the NaK spectrum discussed above.
152
Figure 6.7. Comparison of experimental peak positions and relative intensities (vertical bars) with: Panel
A – simulations based on the solid curve in Fig. 6.6, and Panel (b) – simulations based on the dashed
and dotted curve in Fig. 6.6.
Figure 6.8. Suggested re-assignment of the upper state of Breford and Engelke’s [7] NaK structured
emission continuum.
153
6.3 Quantal Treatment of Absorption from a Thermal
Initial-State Population
6.3.1 Nature of the Problem
A more general approach to the treatment of bound→ continuum spectra is to model them by performing
an exact quantum mechanical simulation. This approach turns out to be particularly appropriate for con-
tinuum absorption from a thermal vibration-rotation population in some initial electronic state. Allowing
for the presence of several different (continuum) final electronic states, the overall absorption cross-section
for this case may be written as
σtotν (T ) =∑v
∑J
Fv,J(T )∑s
σν(v, J ;Es, J′) (6.14)
in which Fv,J (T ) is the fraction of the thermal initial-state population in vibration-rotation level (v, J).
While it is a relatively straightforward matter to calculate the cross sections σν(v, J ;Es, J′) for individ-
ual initial-state vibration-rotation levels, the various summations appearing in Eq. (6.14) could make the
calculation of thermal absorption cross sections relatively tedious.
The discussion of § 6.2 showed that the continuum spectrum associated with absorption or emission
from a particular vibrational level has a structure that mimics the shape of the square of the vibrational
wavefunction for that level. However, when the absorption occurs from a thermal population spread over
multiple vibrational levels, all of that discrete structure is lost, and the resulting spectrum becomes a single
broad featureless peak. Figure 6.9 presents illustrations of spectra of this type. It is clear that the lack
of structure in this type of spectra makes the type of inversion method discussed in § 6.2 inapplicable.
Nonetheless, these spectra contain a substantial amount of information about the repulsive final-state
potential(s) and transition moment function(s). In particular, as temperature increases, the increasing
Figure 6.9. Absorption coefficients of Br2 illustrating the broad featureless nature and the marked tem-
perature dependence of thermal absorption spectra: points – measurements of Passchier et al. [8];
curves – simulation of Ref. [1].
154
Figure 6.10. Potential energy function and (insert on right hand side) room temperature absorption
coefficients of Br2 showing the latter’s decomposition into contributions from two separate electronic
transitions. The horizontal lines are the dissociation limits of the three electronic states.
relative contributions of absorption from higher vibrational levels will cause a broadening and shifting of
the spectrum, while (in this case) the shoulder on the low-energy side of the main peak reflects the presence
of transitions into two different final electronic states.
For the Br2 system whose spectra are shown in Fig. 6.9, the potential energy functions of the initial and
final electronic states are shown in Fig. 6.10. For this system the absorption occurs into two final electronic
states, one of which (C 1Π1u) is purely repulsive while the other (the B 3Π0+ustate) has a substantial
attractive well as well as a repulsive wall. The presence of the bound well for the latter means that at
transition energies below about 20000 cm−1 the spectrum will consist of discrete transitions into vibration-
rotation levels of the B 3Π0+ustate, rather than a continuum.
A 1976 analysis of the Br2 absorption coefficients shown in Fig. 6.9 serves as a ‘textbook’ example of the
application of fully quantum mechanical simulation methods to this type of problem [1]. It included clear
illustrative tests of a number of practical approximations that are commonly associated with simulations of
thermal absorption spectra. In particular, as mentioned above, while calculation of a particular absorption
cross-section σν(v, J ;Es, J′) is a relatively straightforward matter, the need to perform sums over all J ′
values allow by rotational selection rules and over the v and J values for all populated initial-state levels can
make such calculations somewhat time-consuming, and reducing that computational effort is an important
practical matter.
155
Figure 6.11. Illustrative test of approximations applied in simulations of thermal bound→ continuum
absorption. A. Calculated Br2 C1Π1u −X 1Σ+
g absorption coefficients at T = 298 and 713K.
B. Difference between simulated P -branch and Q-branch spectra, and between R-branch and Q-
branch spectra at T = 298K (solid curves) and 713K (dashed curves). C. Errors in approximate
summations over the thermal rotational population. D. Errors associated with truncating the sum
of the initial-state vibrational levels.
156
6.3.2 Collapsing the Sum over J ′
An assumption made in most bound-continuum calculations is the “Q-branch approximation” of neglecting
the J ′–dependence of the bound-continuum matrix elements in Eqs. (6.1) and (6.3) by assuming that over
the range of (J ′ − J) allowed by the selection rules,
〈ψEs,J ′ |Msi(r)|ψv,j〉 ≈ 〈ψEs,J |Msi(r)|ψv,J 〉 (6.15)
Since the Honl-London rotational intensity factors SJ ′J have the normalization property that
gs∑J ′
SJ ′J = (2J + 1) (6.16)
this would remove the factor SJ ′J /(2J + 1) and collapse the sum over J ′ Eqs. (6.1) and (6.14). Panel
B of Fig. 6.11 presents a test of this approximation for the case of the C 1Π1u − X 1Σ+g portion of the
visible absorption continuum of Br2 [1]. The curves labelled R–branch and P -branch show the errors
in the spectra calculated allowing, respectively, only R or only P transitions. The reflection symmetry
of the R and P curves for a given temperature indicates that replacing this sum over J ′ = J ± 1 by
a Q-branch approximation n that J ′ = J will introduce very little error. Indeed, while not resolvable
on the scale of Fig. 6.11 B, the discrepancies between Q-branch calculations and the proper sum over J ′
have approximately the same shape as the P -branch discrepancy curves seen here, but only ca. 1/50 the
amplitude. Thus, collapsing the sum over J ′ to the Q-branch result that J ′ = J will usually introduce
negligible errors to absorption coefficient calculations.
6.3.3 Collapsing the Sum over J
The most computationally expensive summation in Eq. (6.14) is the sum over all the initial-state J sublevels
associated with each value of v. For molecules with a relatively large reduced mass (such as Br2), such
sums could involve hundreds of J values. By reflection-approximation arguments it is clear that for a given
value of v the shapes of their spectra will be very similar, and will differ mainly because of the centrifugal
displacement of the initial-state wavefunction with increasing J . However, this displacement can be quite
large. In spite of this, a very common approximation has been simply to collapse the sum over J to a
single term, using either J = 0 or some representative thermally averaged J value. For the case of Br2,
errors associated with these two approximations are shown as the curves labelled “J =0” and NL=1” in
Panel C of Fig. 6.11. The resulting errors of order 1% of th intensity maximum re clearly unacceptably
large. However, simply performing the direct sum over J for each v until essentially all of the population
was accounted for could involve hundreds of J values, which would be computationally very expensive.
An efficient quadrature procedure for collapsing the sum over initial-state J values was introduced
in Ref. [1]. From the discussion in § 4.4, we recall that for a given vibrational level v the fraction of the
molecular population in rotational sublevel J is
fv(J, T ) = (2J + 1) e−Fv(J)/kBT/qv(T )
and that the overall rotational population distribution has the behaviour shown by the curve in Fig. 6.12.
Within the rigid-rotor approximation for the rotational level energies
E(v, J) = Gv + Bv[J(J + 1)] , (6.17)
one can show that the rotational partition function is qv(T ) = kBT/Bv , and a little further manipulation
shows that if we break the overall population distribution shown in Fig. 6.12 into NJ equal segments, the
157
J
J Tpopmax
F ,J T
Figure 6.12. Partitioning of the thermal population distribution into NJ equal segments.
average value of J in the M ’th segment is [1]
JMNJ
(T ) = − 12 +
√kBT
Bv
{√πNJ
2[erf(a(M,NJ ))− erf(a(M − 1, NJ ))] (6.18)
− (NJ −M) a(M,NJ ) + (NJ −M + 1) a(M − 1, NJ )}
= − 12 +
√kBT
Bvf(NJ ,M)
in which a(M,NJ ) =√
ln{NJ/(NJ −M)} , erf(z) ≡ (2/√π)∫ z0 e−t2dt is the well known ‘error function’
[9], and f(NJ ,M) is a numerical factor determined solely by the integers NJ and M . This results shows
how a sum over all (often hundreds!) populated J value may be replaced by a sum over equally weighted
contributions associated with the NJ representative quantum number values JMNJ
(T ). For the v = 0 level
of the X 1Σ+g state of Br2, Table 6.1 shows the values of J
MNJ
(T ) for T = 298 and 713K, while Panel
C of Fig. 6.11 shows that the error associated with use of this quadrature procedure dies off rapidly with
Table 6.1 JM
NJ(T ) value for the v = 0 level of 79,81Br2(X
1Σ+g ), for which Bv = 0.08094 cm−1.
T = 298K T = 713K
NJ = 1 44 69
2 26, 63 40, 98
3 20, 42, 71 31, 65, 110
4 17, 34, 50, 76 27, 53, 77, 118
5 15, 30, 42, 55, 80 24, 46, 65, 86, 124
7 13, 24, 33, 42, 51, 63, 85 20, 38. 51, 54, 79, 97, 132
10 10, 20, 27, 33, 39, 45, 51, 59, 16, 31, 41, 51, 60, 69, 80, 82,
69, 91 108, 140
15 8, 16, 21, 26, 30, 34, 38, 42, 13, 25, 33, 40, 46, 52, 58, 65,
46, 50, 55, 61, 67, 77, 96 71, 78, 85. 94, 104, 119, 149
158
increasing values of NJ . The speed of this convergence confirms that neglect of centrifugal distortion terms
in the theory used to generate the JMNJ
(T ) values introduced no significant errors into the results.
6.3.4 Truncating the Sum over v
Unfortunately, the fact that absorption spectrum originating in different vibrational levels have different
peak structures means that there is no simple way of correcting for errors associated with truncation of
the sum over v in Eq. (6.14). In particular, wherever that sum is truncated, the leading contribution to the
associated error has a peak structure associated with the absorption from the first vibrational level omitted
from the sum. This point is illustrated by Panel D of Fig. 6.11, which shows plots of the errors associated
with simulated spectra generated while truncating the sums over v after either 90% (VSUM=0.90) or 98%
(VSUM=0.98) of the vibrational population was included. Thus, we see that if the first 90% of the vibrational
population is included, the sum includes only v = 0 and 1 at 298K and v = 0 − 3 at T = 713K, while
including 98% of the vibrational population leaves us with error of � 0.5% of the intensity maximum. In
practical work one would typically wish to continue this sum until � 99.9% of the vibrational population
was included.
6.3.5 Results of a Br2 Absorption Coefficient Analysis
The discussion of § 6.3.1–6.3.4 shows that it is a relatively straightforward matter to calculate continuum
absorption coefficients associated with a thermal population of initial-state molecules. Thus, if models
for the final-state potential energy function(s) and transition moment function(s) are written is explicit
functions of a set of defined parameter {pj}, the calculated absorption coefficients will also be functions
of those parameters, σtot = σtot(ν, T ; {pj}) , and one can use a non-linear least-squares fit procedure to
optimize the parameters defining those models. In 1976 Le Roy et al. [1] did this for the thermal continuum
absorption spectrum of Br2 for which the experimental absorption coefficients are shown in Fig. 6.9. This
work yielded the potential energy curves and transition moment functions shown as solid curves in Fig.6.13,
where the estimated uncertainties in those results are shown as shaded regions. The dashed curves on the
Figure 6.13. Potential energy and transition moment functions determined from a quantum mechanical
direct-potential-fit analysis of the Br2 thermal absorption coefficients shown in Fig. 6.9 [1].
159
transition moment function plot (right hand side) show what the fitted functions would have appeared
to be if the NJ = 1 or J = 0 approximations had been used for performing the summation over J in
Eq. (6.14). The dashed potential function curves and dotted transition moment functions values wet the
results of an early (1937!) analysis of absorption coefficients for this system reported by N.S. Bayliss [10].
160
6.4 Feschbach Predissociation
In § 3.4 of these notes we discussed tunneling predissociation, in which a molecule can fall apart because its
wavefunction can tunnel through a potential energy barrier. That behaviour is entirely associated with a
single potential energy curve. In contrast, ‘Feschbach predissociation’ occurs when wavefunction solutions
associated with two different zero’th order potential energy function are mixed together by some interaction
term. Figure 6.14 presents two schematic illustrations of this type of behaviour. As in all situations of this
type, the wavefunction of a bound level on some upper-state potential energy curve becomes coupled to a
continuum wavefunction associated with an isoenergetic state on some lower-state potential. As shown by
Eq. (6.4), the rate of this process is described by exactly the same type of matrix element that is associated
with photodissociation or bound→ continuum emission
A recent example of this type of phenomenon is the predissociation of vibrational levels of the F 1Σ+g
state of Li2 that lie above the asymptote of the E 1Σ+g state. The potential functions and F -state vibrational
levels of this system are shown in the lower panel of Fig, 6.15. Antonova et al. [?] observed hundreds of
F -state vibration-rotation levels lying above the E-state asymptote and measured the broadening of those
levels due to predissociation. In this particular case the coupling between the bound and continuum
wavefunctions is driven by an operator
P = − �2
2μ
2∑i=1
{dWi(r)
dr+ Wi(r)
d
dr
}(6.19)
in which Wi(r) =Hi
4Hi + (r − rci )2. Note that insertion of this operator in place of a simple radial
function Msi(r) in Eq. (6.4) means that the argument of the overlap integral includes the derivative of
the bound or continuum radial wavefunction. The form of Eq. (6.19) reflects the fact that the inter-
state coupling in this system is associated with two avoided crossings of the ‘adiabatic’ 1Σ+g potential
Figure 6.14. Schematic illustrations of two types pf Feschbach predissociation.
161
-0.80
-0.40
0.00
0.40
0.80
We(R)
dWe(R)/dR
coup
ling
oper
ator
com
pone
nts
3 6 9 12 15
28000
30000
32000
34000
36000
38000
R /A
ener
gy/c
m-1
F 1g+ state
E 1g+ state
v=10
v=20
v=30
v=40
v=0
v=80
v=60
Li(2p) +Li(2p)
Li(2s) +Li(3s)
Figure 6.15. Predissociation from the F 1Σ+g state of Li2.
energy curves, one occurring near 4 A and the other near 11.6 A. The upper panel of Fig. 6.15 shows the
radial parts of this coupling operator determined from a least-squares fit to the observed level-broadening
predissociation rates.
Finally, Fig. 6.16 presents plots of the accumulating integrands of the overlap integrals defining the
predissociation rate for levels v(F ) = 36 and 88. It is interesting to see that the accumulating net value
162
of this integral undergoes step function changes at distances near th avoided crossings where the radial
momenta in the initial and final states are most similar to one another.
163
2 4 6 8 10 12 14 16
0R
b(R ) (R ) dR
0R
b(R )P (R )dR
R /A°
v= 88J= 5
2 4 6 8 10
0R
b(R ) (R ) dR
0R
b(R )P (R )dR
R /A°
v= 36J= 5
dWe/dR term
We d c/dR term
Figure 6.16. Cumulative overlap integrals associated with predissociation from the v(F ) = 36 (lower
panel) and v(F ) = 88 levels of the F 1Σ+g state of Li2.
164
6.5 Continuum-Continuum or Collision-Induced
Absorption (CIA) Spectra
165
6.6 Practical Bound→Continuum Calculations:
Program BCONT
166
References
[1] R. J. Le Roy, R. G. Macdonald, and G. Burns, J. Chem. Phys. 65, 1485 (1976).
[2] J. Tellinghuisen, in Photodissociation and Photoionization, Vol. 60 of Adv. Chem. Phys., edited by
K. P. Lawley (John Wiley & Sons Ltd., New York, 1985), pp. 299–369.
[3] M. S. Child, H. Essen, and R. J. Le Roy, J. Chem. Phys. 78, 6732 (1983).
[4] M. S. Child, Mol. Phys. 35, 759 (1978).
[5] P. M. Hunt and M. S. Child, Chem. Phys. Lett. 58, 202 (1978).
[6] M. S. Child and D. J. Nesbitt, Chem. Phys. Lett. 149, 404 (1988).
[7] E. J. Breford and F. Engelke, Chem. Phys. Lett. 53, 282 (1978).
[8] J. D. C. A. A. Passchier and N. W. Gregory, J. Phys. Chem. 71, 937 (1967).
[9] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970),
ninth Printing.
[10] N. S. Bayliss, Proc. Roy. Soc. (London) A 158, 551 (1937).
167
