1

LECTURE NOTES ON THE SPECTROSCOPY OF DIATOMIC

MOLECULES

L. Veseth

Department of Physics, University of Oslo, Norway

Abstract: The present lecture notes have been prepared to serve as a supplement to parts of

the text in Herzbergs celebrated book on the Spectra of Diatomic Molecules. The objective has

been to make a somewhat more detailed and tutorial discussion of central topics like

anharmonicity, the vibrating rotator and centrifugal distortion. Other important issues are the

more subtle terms in the theory that break the Born-Oppenheimer approximation, leading to

-doublingΛ and effective spin-rotation interaction. Spin-orbit and spin-spin effects are

included, starting from general microscopic expressions for these interactions. The spin-

dependent terms are included in the total rotational Hamiltonian, and matrix elements are

given for the Hund’s coupling case (a) basis. A rather comprehensive discussion of the magnetic

hyperfine structure is also included, based on a microscopic form of the interaction between

the electronic motion and spin, and the nuclear magnetic moments. Finally, there is a section

devoted to the interaction with external electric- and magnetic fields. Compared with the

atomic case, molecules represent an extra challenge since we have to work with a space-fixed

frame of reference, as well as a rotating one fixed in the molecule. Thus, extra care has to be

taken working out matrix elements for interactions that include space-fixed quantities like

external fields, as well as internal molecule fixed properties.

2

1. The Molecular Hamiltonian

The first step in the quantum mechanical treatment of a physical system is to write down its

Hamiltonian operator. Even for quite complex systems this pose no big problems, in particular if

only kinetic- and electrostatic energies are included. For a diatomic molecule the Hamiltonian

takes the following compact form ( 1=ℏ ):

2

2 2

1 1

1 1( , )

2 2

N

i rel

i

H V R r HM m

αα α= =

= − ∇ − ∇ + +∑ ∑

. (1.1)

Here α refers to the two nuclei, and i to the N electrons. ( , )V R r

represents the potential

energy due to all electrostatic interactions, and depends on all the nuclear coordinates

represented by R

, and the complete set of electronic coordinates included in r

. relH refers to

relativistic interactions, which will be neglected until a later section. The Hamiltonian as given in

(1.1) do not yield a tractable Schrödinger equation. The first transformation of the Hamiltonian

will be to introduce the coordinates of the center of mass of the molecule denoted by CMX

,

and to separate it into a part describing the center of mass motion, i.e. CMH , and a part

refering to the internal motions, i.e. intH . The Schrödinger equation then appears as

int int int int( ) ( ) ( , ) ( ) ( ) ( , )CM CM CM CM CM CMH H X R r E E X R r+ Ψ Ψ = + Ψ Ψ

, (1.2)

where and R r

now refer to the nuclear and electronic set of coordinates relative to the

center of mass. The Hamiltonian governing the center of mass motion takes the simple form

21

2CM CMH

M= − ∇ ,

2

1 1

N

i

i

M M mαα = =

= +∑ ∑ . (1.3)

The translational energy CME will have a continuous variation, and it accounts for the Doppler

broadening of the spectral lines. No further consideration will be given to the center of mass

motion. It might be mentioned that the separation of the Hamiltonian into center of mass and

internal motion parts is exact if relativistic interactions are neglected.

The second step is to transform the internal Hamiltonian, where in analogy with the classical

two body problem, the nuclear motion may be handled as a one body problem. Thus, the

nuclear motion is described by the reduced mass µ and the internuclear separation, which is

now and in the following denoted by R , and the two angles θ and ϕ which specify the

orientation of the molecular axis. After some rather lengthy coordinate transformations the

internal Hamiltonian is obtained as[1]:

3

2 2

int

1 , 11 2

1 1 1( , )

2 2( ) 2

N N

i i j R

i i j

H V R rm M M

θϕµ= =

= − ∇ − ∇ ⋅∇ − ∇ ++∑ ∑

. (1.4)

The reduced mass µ is given by the well known expression 1 2 1 2/ ( )M M M Mµ = + . The cross

term including a double summation over electrons is known as the specific mass shift term, as it

depends on the nuclear masses. It represents an interesting and important isotopic shift effect

in the atomic case, but is of less interest in molecules as it will be swamped by other more

dominant nuclear mass terms. Finally, the operator 2

Rθϕ∇ describing the nuclear kinetic energy

takes the form (which is also known from the H-atom)

22 2

2 2 2 2 2

2 2

2 2

1 1 1( ) (sin )

sin sin

1 1 ( ) ,

R RR R R R R

R KR R R R

θϕ θθ θ θ θ ϕ

∂ ∂ ∂ ∂ ∂∇ = + +

∂ ∂ ∂ ∂ ∂

∂ ∂= −

∂ ∂

(1.5)

where K

denotes the (mechanical) angular momentum of the nuclei. The Hamiltonian of (1.4)

might appear to be separable into an electronic part and a nuclear part el nuclH H with

21

2nucl RH θϕµ

= − ∇ , (1.6)

and elH as the remaining part of (1.4). In that case the molecular wave function could be

factorized as ( , , ) ( , )nucl elR R rθ ϕΨ = Ψ Ψ

, where the internuclear separation R enters the

electronic wave function as a parameter due to the dependence of the potential ( , )V R r

on R .

The Schrödinger equation would then take the form

( ) ( , ) ( , , ) ( , ) ( , , )el nucl el nucl el nuclH H R r R E R r Rθ ϕ θ ϕ+ Ψ Ψ = Ψ Ψ

, (1.7)

and is further transformed into

[ ( ) ( , )] ( , , ) ( , , )el el nucl el nucl nuclE R H R E Rθ ϕ θ ϕ+ Ψ Ψ Ψ = Ψ . (1.8)

In the most simple version of the Born-Oppenheimer approximation we now assume that

( , )el nucl el nuclH HΨ Ψ = , i.e. we neglect the dependence of ( , )el R rΨ

on the internuclear

distance R in this integral. What Born and Oppenheimer showed was that this approximation

only leads to a loss of terms of the order of magnitude /m µ . However, the most important

terms omitted in the simple approximation may be retained, and include in Eq.(1.8) as a

correction to the electronic energy ( )elE R , and we obtain a somewhat better approximation

which is usually called the adiabatic Born-Oppenheimer approximation. The corresponding

4

expression for ( )elE R is then the “best possible” potential curve. The adiabatic approximation

represent the highest accuracy we can obtain with a molecular wave function factorized in an

electronic and a nuclear part. We shall later see how the nuclear Hamiltonian will lead to a

coupling between different electronic wave functions, so that a complete solution of the

molecular Schrödinger equation will require an in principle infinite linear combination of such

electronic- nuclear product functions.

In the following we will assume that ( )elE R represent the best adiabatic approximation, and we

have to solve an equation that takes the form

( ( )) ( , , )nucl el nuclH E R E R θ ϕ+ = Ψ . (1.9)

We notice that the electronic energy ( )elE R effectively works as a potential energy for the

motion of the nuclei. Hence, ( )elE R is named as the molecular potential curve.

We will seek the solution of Eq.(1.9) for the nuclear motion. Inserting for nuclH we have

2

2 2

2 2

1 1[ ( ) ( )] ( , , ) ( , , )

2 2el nucl nuclR K E R R E R

R R R Rθ ϕ θ ϕ

µ µ∂ ∂

− + + Ψ = Ψ∂ ∂

ℏ. (1.10)

As 2K

only depends on the angles θ and ϕ we realize that the nuclear wave function has an

exact factorization of the form 1

( , , ) ( ) ( , )nucl vib KR RR

θ ϕ θ ϕΨ = Ψ Ψ , where the rotational part

( , )K θ ϕΨ is an eigenfunction of 2K

with eigenvalues ( 1)K K + . We then obtain the following

equation for the vibrational wave function:

2

2 2

1 1[ ( 1) ( )] ( ) ( )

2 2el vib vib

dK K E R R E R

dR Rµ µ− + + + Ψ = Ψ . (1.11)

In the equation above we will normally not have a known analytic expression for ( )elE R , it will

rather be available in the form of a numerical table, and the term containing 21/ R also poses

problems. The following approximations will be useful:

2 2

22

2

2( )1 1[1 ] , and

( )1( ) ( ) ( )

2e

e

e e

elel el e e

R R

R R

R R R

d E RE R E R R R

dR=

−= − + ⋅⋅⋅

= + − + ⋅⋅⋅

(1.12)

5

where eR is the internuclear separation that corresponds to the minimum of the potential

curve ( )elE R . This series expansion around eR is often referred to as a Dunham expansion.

Retaining only the first term in the series for 21/ R , and the first two terms in the series for

( ),elE R we obtain a nice and simple approximation to Eq.(1.11), i.e. a harmonic oscillator

potential and a constant term. The total molecular energy E is then obtained as

( ) ( 1/ 2) ( 1)el e e eE E R v B K Kω= + + + + , (1.13)

with the following expressions for the vibrational- and rotational constants and e eBω

respectively

1/22

2 2

( )1 1 and

2e

ee e

eR R

d E RB

dR Rω

µ µ=

= =

.

Now, we recall that we have set 1=ℏ , i.e. assumed atomic units (a.u.), which is a common

practice in atomic and molecular physics. This means that and e eBω both have unit energy. The

most common energy units in molecular spectroscopy are, however, wavenumber ( 1cm− ) or

frequency (MHz). 5 1 91 . . 2.1944 10 6.579 10 .a u cm MHz−≈ ⋅ ≈ ⋅

The rotational quantum number takes the values K=0,1,2,∙∙∙, and similar for the vibrational

quantum number v. We notice that the rotational constant eB is determined by the reduced

mass µ and the internuclear separation eR , whereas the vibrational constant eω is determined

by the shape of the potential curve. The following relations generally apply for the constants of

Eq.(1.13): ( )e e e eE R Bω>> >> . Eq.(1.13) yields a reasonable approximation to the total

molecular energies of a molecule with no electronic spin, i.e. for singlet states. We shall later see

that the equations will be somewhat more involved with an electronic spin different from zero. If

more terms from the series expansions in Eq.(1.12) are included as perturbations to our simple

model, significant improvements may be obtained for the total energy in Eq.(1.13). We should,

however, keep in mind that such improvements are all within the Born-Oppenheimer

approximation.

2. Higher order effects

2a. The anharmonic oscillator

We shall now apply perturbation theory to obtain corrections to the simple expression for the

energy given in Eq.(1.13). The technique is to include more terms from the series expansion in

Eq.(1.12). Now, let us recall that our unperturbed model is a harmonic oscillator and a rigid

6

rotator ( eR R= ). The unperturbed (zero order) wave functions are ( ) ( , )vib KR θ ϕΨ Ψ , where

the last factor is an eigenfunction for 2K

( the rigid rotator part of the Hamiltonian), i.e.

2

2( , ) ( 1) ( , ).

2K e K

e

KB K K

Rθ ϕ θ ϕ

µΨ = + Ψ

(2.1)

For the harmonic oscillator approximation the vibrational wave function ( )vib RΨ is replaced by

the harmonic oscillator eigenfunction ( ).v xΨ It is generally given by the expression

2

1/4

/21( ) ( )

2 !

x

v vv

x H x ev

βββ

π− Ψ =

, with emβ ω= . (2.2)

Here ex R R= − , v is the vibrational quantum number, and vH denotes the Hermite polynomial

of order v.

We might now in principle include as many terms from the series of Eq.(1.12) as we would find

relevant. However, that would actually be a rather messy approach, yielding lots of quite

insignificant cross terms. Our experience tells us that the omitted terms in the expansion for

( )elE R will strongly dominate the neglected terms from the expansion for 21/ R . Hence, we

first concentrate only on the next two terms in the expansion for ( )elE R that are not included

in Eq.(1.12). These terms are expressed as

3 4

3 4 3 4

3 4

( ) ( )1 1( ) ( )

6 24e e

el ele e

R R R R

d E R d E RR R R R gx jx

dR dR= =

− + − = −

, (2.3)

where we have replaced the third and fourth derivatives with the anharmonicity coefficients g

and j [2], which represent the deviation from the harmonic potential curve. Due to symmetry

reasons there is no first order contribution from the cubic anharmonicity term. To second order

the effect is:

3 2

(2) | ( , ) |u vv

u v v u

gxE

E E≠

Ψ Ψ=

−∑ , (2.4)

where and v uE E denote the harmonic oscillator energies of Eq.(1.13). The summation in the

expression above might be worked out in an explicit way by repeated use of the following

recursion formula for the Hermite polynomials:

1 1

1( ) ( ) ( )

2v v vxH x H x vH x+ −= + . (2.5)

7

The fourth power term yields a firs order contribution ( )(1) 4, .v v v

E jx= Ψ − Ψ

The final result takes the form:

(1) (2) 2( 1/ 2)v v e eE E x vω+ = − + , (2.6)

where e exω denotes the anharmonicity constant as given in Herzbergs book ([2], page 93).

2b. The Morse potential

The perturbative approach discussed in section 2a is expected to yield satisfactory results only

close to the minimum of the potential curve, where its shape is fairly harmonic. Instead of the

slowly converging series expansion for ( )elE R there are approximations that cover the whole

range of the potential curve up to the dissociation limit quite well. The most common of such

models is probably the Morse potential:

2

( )( ) 1 ea R R

Morse eE R D e

− − = − , (2.7)

where e

D is the dissociation energy, and a is a constant. Inserted in Eq.(1.11) for ( )el

E R the

Morse potential actually yields an exact solution (provided 21/ R is approximated by 21/ eR ),

and the vibrational energies are obtained as

1/2 22

,( 1 / 2) ( 1/ 2)

4

ev Morse

D aE a v v

c cπ µ π µ

= + − +

. (2.8)

The corresponding vibrational eigenfunctions are associated Laguerre polynomials, somewhat

related to the radial wave functions for the hydrogen atom. Observation of the vibrational

energies then enables a determinations of the two constants e

D and a, and furthermore a test

of the accuracy of the Morse potential.

2c. The vibrating and non-rigid rotator

We now continue the perturbation expansion, and include the first two neglected terms in the

series expansion for 21/ R in Eq.(1.12). These two terms represent a perturbation 'H , which is

given by

2

2

2

( )' 2 3e e

e e

e e

R R R RH B B K

R R

− −= − +

. (2.9)

Now, we have to clarify our unperturbed model. This is actually the anharmonic vibrator

discussed in section 2a, and the unperturbed vibrational wave functions are to second order

given by

8

( )3,

' ( ) ( ) ( )u v

v v u

u v v u

gxx x x

E E≠

Ψ ΨΨ = Ψ + Ψ

−∑ . (2.10)

In the equation above an insignificant contribution from the fourth power term 4jx− in Eq.(2.3)

is omitted. To first order there is a contribution to the rotational energy from 'H which takes

the form

( ) ( )( )

( )3

(1),

' , ' ' , ' 2 , 'u v

K v K v K v K v K u K v K

u v v u

gxE H H H

E E≠

Ψ Ψ= Ψ Ψ Ψ Ψ = Ψ Ψ Ψ Ψ + Ψ Ψ Ψ Ψ

−∑ . (2.11)

In the first term on the right hand side of Eq.(2.11) there will be a contribution only from the

second power term in Eq.(2.9), and the result is

( ) ( )2

2

2

3 6, ' ( 1) , ( 1/ 2) ( 1)e e

v K v K v v

e e

B BH K K x v K K

R ωΨ Ψ Ψ Ψ = + Ψ Ψ = + + . (2.12)

In the summation on the right hand side of Eq.(2.11) there is a dominant contribution from the

first power term in 'H from Eq.(2.9). A bit of calculation now yields the final result:

( ) ( )3 3 2

(1)

3

24 61/ 2 ( 1) 1/ 2 ( 1)e e e

K e

e e

B R g BE v K K v K Kα

ω ω

= + + + = − + +

. (2.13)

The constant e

α will generally be positive (negative g), and so (1)

KE represents a reduction of

the rotational energy. Generally we will also have | |e e

Bα << , and the term ( 1 / 2)evα− + may be

interpreted as a correction to the rotational constant e

B . Physically, this correction stems from

the fact that the mean internuclear separation increases with increasing vibration of the

molecule, and thereby a reduction of the rotational constant. This model is the “vibrating

rotator”.

We will now consider the second order correction to the rotational energy, which is given by:

( ) 2 2

(2)| ' , ' ' | | ( , ' ) |u K v K u K v K

K

u v u vv u v u

H HE

E E E E≠ ≠

Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ= ≈

− −∑ ∑ . (2.14)

Here it will be sufficient to include only the first term in the expression for 'H in Eq.(2.9),

hence ( ) ( )2, ' , ( 1)e

u K v K u v

e

BH x K K

RΨ Ψ Ψ Ψ = − Ψ Ψ + ,

and according to Eq.(2.5) there will only be contributions to the sum in Eq.(2.14) from u=v-1

and u=v+1. After some calculations the following result is obtained:

9

3 3

(2) 2 2 2 2 2 2

2 2

8 41( 1) ( 1) ( 1)

2 2

e eK e

e e e e

B Bv vE K K K K D K K

ω ω ω ω +

= − + + = − + = − +

. (2.15)

The new constant eD in Eq.(2.15) above is called the centrifugal distortion constant. This

correction physically arises from the fact that the molecule is stretched as a consequence of the

rotation. The rotational constant is reduced, and a lowering of the rotational energy follows, as

expressed by Eq.(2.15). From Eq.(2.15) we notice that the constant eD is completely

determined by two other well-defined constants:

3

2

4 ee

e

BD

ω= . (2.16)

The simple expression in Eq.(2.16) is known as Kratzers relation, and it offers a unique

opportunity to compare theory and experiment.

If we now collect all the derived energy terms in one formula, we have the following expression

for the combined electronic, vibrational and rotational energy:

[ ]2 2 2( ) ( 1/ 2) ( 1/ 2) ( 1/ 2) ( 1) ( 1) .el e e e e e e eE E R v x v B v K K D K Kω ω α= + + − + + − + + − + (2.17)

To exemplify the relative magnitudes of the various terms in Eq.(2.17), we include the

experimental values of all the constants for a typical example, i.e. the 1

gX +Σ electronic ground

state in the N2 molecule:

62358.027, 14.135, 1.9980, 0.0177, 5.74 10e e e e e ex B Dω ω α −= = = = = ⋅ , all values in cm-1

.

The contribution from the centrifugal distortion term ( eD -term) is then for K=20 (a modest

value) equal to 1.01cm-1

, whereas the experimental uncertainty is typically of the order of

magnitude 0.01cm-1

. The theoretical value of eD is from Eq.(2.16) equal to 5.738∙10-6

cm-1

.

2d. Isotope shifts

The molecular constants inherent in Eq.(2.17) all depend on the reduced mass µ, except for the

electronic energy ( )el eE R .Thus, they will have different values for different isotopic molecules.

If µ and µi denote the reduced masses of a molecule and one of its isotopic substitutes

respectively, we can make a list showing the isotope shifts in the various constants in terms of

the ratio 2 / iρ µ µ= :

10

2

2

3

3/2

4

2

1 ( ) /

1 ( ) /

1 ( ) /

1 ( ) /

1 ( ) / .

e e e

e e e e e e

e e e

e e e

e e e

i

x x i x

B B i B

i

D D i D

ω ω ω ρµ

ω ω ω ρµ

ρµ

α α α ρµ

ρµ

=

=

=

=

=

∼

∼

∼

∼

∼

The isotope shift will be particularly large if one of the two atoms in the molecule is hydrogen,

which in the isotopic substitution is replaced by deuterium. In this case 2 / 1/ 2iρ µ µ= ≈ . As an

example we list the observed isotope shifts for AuH and AuD:

( ) / 0.7093, ( ) / 0.5022, ( ) / 0.5029, ( ) / 0.3565,

( ) / 0.254.

e e e e e e e e e e

e e

i x i x B i B i

D i D

ω ω ω ω α α= = = =

=

For comparison the ratios are ρ=0.7089, ρ2=0.5025, ρ

3=0.3562 and ρ

4=0.2525. Thus, we see

that the isotope shifts predicted by the formulae in the list above, are in excellent agreement

with experiment.

3. Terms that break the BO-approximation. The Λ-doubling

We now return to the Schrödinger equation [Eq.(1.7)] which applies to a diatomic molecule

within the Born-Oppenheimer (BO) approximation:

22 2

2 2

( ) ( , ) ( , , ) ( , ) ( , , ),

1( ) .

2 2

el nucl el nucl el nucl

nucl

H H R r R E R r R

H R KR R R R

θ ϕ θ ϕ

µ µ

+ Ψ Ψ = Ψ Ψ

∂ ∂= − +

∂ ∂

ℏ (3.1)

Here we notice that the nuclear Hamiltonian nuclH acts on- and thereby also couples different

electronic wave functions due to the R-dependence of elΨ . This means that an exact solution

would require an in principle infinite linear combination of electronic wave functions

corresponding to different electronic states. This topic is handled in general textbooks on

molecular physics/quantum chemistry. Here we will handle the terms in the nuclear

Hamiltonian that break the BO-approximation as a perturbation to the BO-states. The first part

in the expression for nuclH in Eq.(3.1) applied to ( , )el R rΨ

will lead to corrections that are

11

independent of the rotation, i.e. just a small correction to the vibrational energy, and thereby a

redefinition of the vibrational constants. Hence, this term will be neglected in the following

discussion. The last term in the expression for nuclH is more interesting. We recall that the

nuclear angular momentum K

only represents the rotation of the bare nuclei. However, it will

none the less affect the electronic wave function as the electronic coordinates r

are defined

with reference to a frame fixed with regard to the nuclei, and thereby also rotating with the

nuclei. The total quantized orbital angular momentum N

of the molecule is now given by

N K L= +

, (3.2)

where L

is the total angular momentum ascribed to the orbital motions of all the electrons.

Thus, we have for the rotational Hamiltonian rotH :

2 2 2 2

2

12

2rotH K BK BN BL BN L

Rµ= = = + − ⋅

. (3.3)

Unlike the case for atoms, 2L

will not be quantized for a molecule, as there is no spherical

symmetry. The last two terms in Eq.(3.3) may now be handled as perturbations. However, the

term 2BL

is less interesting , as it does not contain the rotational angular momentum. Finally,

we concentrate on the last term

2

1' 2H BN L N L

Rµ= − ⋅ = − ⋅

. (3.4)

Before we start the perturbation calculation it will be useful to have a quick look at the

properties of the electronic wave functions. These may be classified according to the

eigenvalues Λ of zL , i.e. the component of the electronic orbital angular momentum along

the molecular axis. Denoting the various wave functions by the corresponding kets, we have for

the total molecular state |Ψ > within the BO-approximation:

| | | | |n v N nv NΨ >= Λ > > >= Λ > , (3.5)

where n denotes quantum numbers that are needed in addition to Λ to specify the electronic

state. We might rather intuitively agree that the states | and |nv N nv NΛ > −Λ > are

degenerate. Hence, it is important to obtain the linear combinations of these two states that

are correct zero-order states before we start the perturbation calculation. The total inversion

operator *E (parity) which inverts all electron and nuclear coordinates helps us to achieve this.

Actually, we have (cf. section 6)

* | ( 1) |nN sE nv N nv N

+Λ >= − −Λ > , (3.6)

12

where ns =0 for 0Λ ≠ . For 0 ( -states)Λ = Σ 0 for states, and 1 for states.n ns s+ −= Σ = Σ Finally,

we then have correct zero-order states in terms of the eigenstates of the parity operator:

( )1| | ( 1) |

2

snv N nv NΨ >= Λ > + − −Λ > , (3.7)

where s can take the values 0 or 1, and the parity is ( 1) nN s s+ +− (cf.Eq.(3.6)). The parity operator

*E is a symmetry operator, characterized by the commutation [ ]*, 0E H = , where H denotes

the complete molecular Hamiltonian. Hence, * and E H have simultaneous eigenstates, and

the two eigenstates of *E are the correct unperturbed (zero-order) states. In Eq.(3.7) Λ>0 is

assumed. For Λ=0 there is obviously no need to make linear combinations.

The next step now is to determine the matrix elements of the perturbation 'H of Eq.(3.4) in

terms of the basis states of Eqs.(3.5) and (3.7). We observe that 2', 0H N =

, which means

that there are no non-diagonal terms in N . The matrix elements needed generally take the

form | ' | 'H< Ψ Ψ > , with ' , | and | 'H Ψ > Ψ > respectively given by Eqs.(3.4) and (3.7). In a

more worked out form we have

'

'

1| ' | ' ( | ' | ' ' ' ( 1) | ' | ' ' '

2

( 1) | ' | ' ' ' ( 1) | ' | ' ' ' ).

s

s s s

H nv N H n v N nv N H n v N

nv N H n v N nv N H n v N+

< Ψ Ψ >= < Λ Λ > + − < Λ −Λ >

+ − < −Λ Λ > + − < −Λ −Λ > (3.8)

For diagonal elements Eq.(3.8) takes the simple form

| ' | | ' | ( 1) | ' | .sH nv N H nv N nv N H nv N< Ψ Ψ >=< Λ Λ > + − < Λ −Λ >

We recall that for 0Λ = there is no parity dependent term (cf. Eq.(3.6)), and for 1Λ ≥ we have

| ' | 0.nv N H nv N< Λ −Λ >= Hence, we conclude that there will be no parity dependent first

order contribution to the energy, i.e. no lifting of the degeneracy.

To second order the correction generally takes the form

2

(2) | | ' | ' |

'

HE

E E

< Ψ Ψ >=

−∑ , (3.9)

where the summation extends over all primed state | 'Ψ > with corresponding energies 'E .

Most interesting now are states with Λ≠0, since they are degenerate. For states with 1Λ > it

turns out that there are no parity dependent contributions even to second order, and the

degeneracy will only be lifted if the perturbation expansion is carried beyond the second order.

Hence, we will now consider Π states only ( )1Λ = , and note that there are parity dependent

13

terms different from zero only for ' 0Λ = . For ' 0Λ = the primed state corresponding to

Eq.(3.7) takes the form | ' | ' ' ' 0n v NΨ >= Λ = > , and the second order correction is expressed as

(2)

', '

2

' '

1(| | ' | ' ' ' 0

2

+( 1) | ' | ' ' ' 0 | ) / ( ).

n v

s

nv n v

E nv N H n v N

nv N H n v N E E

Π = < Λ Λ = >

− < −Λ Λ = > −

∑ (3.10)

To work out the matrix elements for 'H in the expression above, requires somewhat subtle

results from the theory of angular momenta [3]. The problem is that N

refers to a space fixed

coordinate system, whereas L

refers to a molecule fixed one. The result is here stated without

proof:

1/2| 2 | ' ' 1 | | ' ' 1 [( )( 1 )] .nv N BN L n v N nv BL n v N N+< Λ − ⋅ Λ − >=< Λ Λ− > +Λ + −Λ

(3.11)

Finally, we then obtain

( )

( )

2(2)

', ' ' '

| | | ' ' ' 0 |1 1 ( 1)

1 = 1 1 ( 1),

2

s

n v nv n v

s

nv BL n vE N N

E E

qN N

+Π

< Λ Λ = > = + − + −

+ − +

∑ (3.12)

when we follow common practice and let the new constant q be determined by twice the sum

in Eq.(3.12) above [2]. From Eq.(3.12) we see that one of the degenerate levels for the Π state

will get a second order correction of the form ( 1)qN N + , whereas there is no correction for the

other level. This effect is a consequence of the conservation of parity. For the ' 0Λ = state there

is a single non-degenerate level with a parity given by Eq.(3.6) for a given N -value, which then

interacts with the 1Λ = level with the same parity (cf. figure).

This resulting effect may be interpreted as a second order correction to the rotational energy

of one of the two 1Λ = levels. This lifting of the degeneracy of the two levels of opposite parity

is generally called Λ-doubling.

N +

N

' 0Λ =

+

-

1Λ =

14

There will also be a corresponding second order correction to the non-degenerate Λ=0 levels. In

that case the result is

2

(2)

', ' ' '

| | | ' ' ' 1 |2 ( 1) * ( 1).n v nv n v

nv BL n vE N N q N N

E E

+Σ

< Λ Λ = >= + = +

−∑ (3.13)

Also in this case we notice that the second order correction effectively adds a contribution to

the rotational energy. In a classical way the terms containing q and *q represent the

contribution from the electrons to the moment of inertia of the molecule. Basically, there is no

simple relation between q and *q . However, in some cases there is a dominant interaction

between one electronic and vibrational level | 1 and another level | ' ' ' 0nv n vΛ = > Λ = > , and

the result is *q q≈ − . In such favorable cases there is an opportunity to estimate the

unobservable constant *q from the observed Λ-doubling.

In other favorable cases, especially for hydrides, the orbital electronic angular momentum 2L

will have an approximate quantized value ( 1)L L + . In addition, there may be just one

electronic and vibrational state | ' ' 'n v v= Λ > that contributes to the sum that defines q in

Eq.(3.12). Altogether the following simple approximation is then obtained

2

'

2 ( 1)e

nv n v

B L Lq

E E

+≈

− . (3.14)

This simple approximation and the assumptions behind it is known as the hypothesis of pure

precession, as the classical analogy would be a L

vector that makes a precession motion

around the molecular axis with | |zL = Λ .

As an example of an observed Λ-doubling we will consider the hydride AlH, and its isotopic

substitution AlD. In this case there is an excited 1A Π state that predominantly interacts with

the 1X Σ ground state. For the 1A Π state in AlH the observed value of q is (8.0±0.2)∙10-3

cm-1

.

The Λ-doubling is then according to Eq.(3.12) 0.016cm-1

, 0.88cm-1

, and 3.36cm-1

respectively for

1, 10 and 20.N N N= = = Thus, for high N -values there is a Λ-doubling that is substantial

compared with an experimental accuracy of around 0.1cm-1

. The pure precession hypothesis

yields q =6.3∙10-3

cm-1

for L =1, i.e. 79% of the observed value. For AlD the observed value of q

is (2.3±0.1)∙10-3

cm-1

. The isotope ratio is (AlD) / (AlH) 0.29q q = . Since q according to

Eq.(3.12) is proportional to 2

eB , the isotope ratio should be equal to 4 0.27ρ = .

The classical analogy to the interaction 'H of Eq.(3.4) which is behind the Λ-doubling, is the

Coriolis force, which governs the motion of an object (e.g. an electron) in a rotating frame of

reference (e.g. the rotating molecule).

15

4. Spin-dependent effects

It is well-known that the spin (eigen spin) of the electrons is introduced via the Dirac equation.

For one electron atoms (H-atom) the Dirac equation was able to correctly predict the observed

spin-orbit coupling, as well as the magnetic moment of the electron. The one electron Dirac

equation is not readily generalized to many-electron systems. To the authors best knowledge it

is not possible to write down a relativistic many-electron equation in a closed form. The

standard procedure is to start from an approximate relativistic expression known as the Breit

equation, which is then reduced to a non-relativistic form. In this non-relativistic form the spin

( )is

of the individual electrons will make their explicit appearance. For an in depth treatment of

this rather complex topic the reader is referred to e.g. the textbook by Moss[4].

The most interesting and relevant of these non-relativistic spin-dependent terms may be

divided into two categories. Firstly there are terms that relate to a coupling between the orbital

motion ( )iℓ and the spin ( )js

of the individual electrons. These terms yield together what is

denoted the spin-orbit interaction, and a Hamiltonian that takes the microscopic form

,

( , )SO i j i j

i j

H a r r s= ⋅∑ ℓ . (4.1)

Here, ( , )i ja r r

is a rather complex function of the positions and i jr r

for electrons i and j

respectively. We shall not here be concerned with the explicit expression for ( , )i ja r r

. The

Hamiltonian of Eq.(4.1) is seen to include interactions between the orbital angular momentum

of one electron, and the spin of another, or so called spin-other-orbit interaction. However, the

dominant contribution will stem from same electron spin-orbit interaction.

The second category of spin-dependent terms are related to the spin-spin interaction, with a

microscopic Hamiltonian as follows:

22

2 2 5

3( )( ) ( )i ij j ij i j ij

SS

j i ij

s r s r s s reH

m c r>

⋅ ⋅ − ⋅= − ∑

. (4.2)

The spin-spin Hamiltonian above certainly resembles the classical expression for the interaction

between magnetic dipoles.

The next step is to work out the matrix elements of the spin-dependent interactions

and SO SSH H , which are handled as perturbations to the spin-free Hamiltonian of the preceding

sections. Then we need to decide on a molecular basis set that also includes spin quantum

numbers. Since the unperturbed Hamiltonian H does not include spin-effects, we will have

[ ]2, , 0zH S H S = =

, where S

refers to the total spin of all the electrons, and zS denote its

16

component along the molecular axis. This means that 2S

and zS have quantized values, and

the corresponding quantum numbers are and S Σ , according to standard notations. The spin

quantum numbers may be added to the basis states of Eq.(3.5). However, we now have a total

molecular angular momentum J

which is obtained by adding S

to N

of Eq.(3.2), i.e.

J N S= +

. (4.3)

Furthermore, we have 2, 0H J =

, which means that 2J

has quantized values with

corresponding quantum number J . From the theory of coupling of angular momenta we also

know that J may take the values | |, ,J N S N S= − ⋅⋅⋅ + . More subtle considerations show that

there are two sets of simultaneously quantized operators [2,3]. One set consists of the

quantities 2 2 2, , and zL N S J

, and the corresponding basis set which is denoted Hund’s coupling

case (b) takes the form (cf. Eq.(3.2))

| |b nv NSJΨ >= Λ > . (4.4)

The other set comprises the operators 2 2, , and z zL S S J

, which is referred to as Hund’s coupling

case (a), and the corresponding basis set is of the form

| |a nv SJΨ >= ΛΣ > . (4.5)

Now, matrix elements for the spin-dependent interactions in Eqs.(4.1) and (4.2) which contain

operators defined relative to a molecule fixed coordinate system, are most easily worked out in

the case (a) basis. We will first handle the spin-orbit Hamiltonian SOH in Eq.(4.1). The scalar

product i js⋅ ℓ is rewritten in terms of spherical components as

1

, ,

1

( 1)i j i js sµµ µ

µ−

=−

⋅ = −∑ ℓ ℓ .

We recall that any internal interaction Hamiltonian like SOH will commute with 2J

, i.e.

2, 0SO

H J =

. This means that 2J

will be conserved, and that there are only matrix elements

diagonal in J . Furthermore, we shall restrict ourselves to elements diagonal in S . By use of the

Wigner-Eckart theorem [3] it is rather straight forward to work out the matrix elements of SOH ,

and the result takes the form

| | ' ' ' ' ( , , ', ', ) | | ' ' ' 'SOnv SJ H n v SJ A n v n v S nv SJ L S n v SJ< ΛΣ Λ Σ >= < ΛΣ ⋅ Λ Σ >

, (4.6)

17

where ( , , ', ', )A n v n v S is a constant that depends on the electronic- and vibrational states and

S , but not on J . It is important to emphasize that Eq.(4.6) is only valid for elements diagonal

in S . The Hamiltonian SOH will also couple states with different S -values. However, to work

out such elements we have to resort to the microscopic form of SOH in Eq.(4.1). We see that

Eq.(4.6) might actually be obtained from an effective spin-orbit Hamiltonian of the form

, ( , , ', ', )SO effH A n v n v S L S= ⋅

. (4.7)

In the atomic case with quantized values of 2L

, an effective operator like that in Eq.(4.7) is only

valid for elements diagonal in L .

The most relevant application of Eq.(4.6) will be for elements diagonal in n and v, i.e. to

calculate the first order contribution from SOH for a given electronic and vibrational state.

Hence,

1

1

| | ' ' ( , , , , ) ( 1) | | ' 'SOnv SJ H nv SJ A n v n v S nv SJ L S nv SJµµ µ

µ−

=−

< ΛΣ Λ Σ >= − < ΛΣ Λ Σ >∑ , (4.8)

where the subscript µ indicates spherical components with reference to a molecule-fixed

coordinate system. Furthermore, non-zero elements in Eq.(4.8) requires ' 'Λ + Σ = Λ + Σ , and if

'Λ = Λ is assumed (same electronic state), then also 'Σ = Σ . In that case the element in

Eq.(4.8) is simply

| | ( , , , , )SOnv SJ H nv SJ A n v n v S< ΛΣ ΛΣ >= ΛΣ . (4.9)

Hence, it follows that there is no first order spin-orbit effect for Σ states (Λ =0).

Turning now to the spin-spin Hamiltonian of Eq.(4.2) it is quickly observed that this interaction

is rather intractable. It was shown long ago [5] that for Σ states it effectively takes the form

( )2 2

,

1

3SS effH S n Sλ = ⋅ −

, (4.10)

where n

is a unit vector along the internuclear axis. This expression is valid for computing

diagonal elements, i.e. for

2 1| | ( 1)

3SSnv SJ H nv SJ S Sλ < ΛΣ ΛΣ >= Σ − +

. (4.11)

For non-sigma states SSH may add a (small) contribution to the spin-orbit effect for doublet

states ( 1/ 2)S = , whereas for S ≥1 it might yield a separately observable effect in terms of an

asymmetry in the spin-orbit splitting (cf. Eq.(4.9)).

18

5. The molecular Hamiltonian including spin

The total molecular angular momentum J

including spin is now given by (cf. Eq.(4.3))

J K L S= + +

, (5.1)

and the rotational energy Hamiltonian is according to Eq.(3.3)

2 2 2 2 2

2

1( ) ( ) 2

2rot a a aH K B J L S B J J BJ BJ J BJ

Rµ= = − − = − = − ⋅ +

, (5.2)

with aJ L S= +

. Now, we have 2 2, , 0rot rotH J H S = =

, so non-zero matrix elements of

rotH are diagonal in and J S . Hence, we consider

2 2| | ' ' ' ' | | ' ' | | ' 'nv SJ BK n v SJ nv B n v SJ K SJ< ΛΣ Λ Σ >=< >< ΛΣ Λ Σ >

. (5.3)

Furthermore, we will in the following assume 'Λ = Λ , i.e. only consider matrix elements within

the spin manifold of a given electronic state. Thus,

2

, '| | ' ( 1)SJ J SJ J J δΣ Σ< ΛΣ ΛΣ >= +

(5.4)

2| 2 | 2( )aSJ J J SJ< ΛΣ − ⋅ ΛΣ >= − Λ+Σ

(5.5)

[ ] [ ]1/2 1/2| 2 | 1 ( 1) ( 1) ( 1) ( )( 1)

aSJ J J SJ S S J J< ΛΣ − ⋅ ΛΣ ± >= − + −Σ Σ ± + − Λ + Σ Λ + Σ ±

. (5.6)

Finally, for the last term in Eq.(5.2) 2 2 2( 2 )aBJ B L S L S= + + ⋅

,

2 2 and BL BS

will merely make

uninteresting contributions to the total electronic energy, and these terms are omitted. Thus,

we only retain the diagonal element

, '| 2 | ' 2SJ L S SJ δΣ Σ< ΛΣ ⋅ ΛΣ >= ΛΣ

. (5.7)

In Eq.(5.5) there is a term 22− Λ on the right hand side which may be omitted, as it merely

adds a constant contribution to the electronic energy. Returning to Eq.(5.3) we then have for

elements diagonal in n , v and Λ :

2 2| | | | ( 1) 2 2nv SJ BK nv SJ nv B nv J J < ΛΣ ΛΣ >=< > + − ΛΣ − Σ

,

[ ]

[ ]

1/22

1/2

| | 1 | | ( 1) ( 1)

( 1) ( )( 1) .

nv SJ BK nv SJ nv B nv S S

J J

< ΛΣ ΛΣ± >= − < > + −Σ Σ ±

× + − Λ +Σ Λ +Σ ±

(5.8)

19

To the element diagonal in Σ in Eq.(5.8) we shall also add the spin-orbit contribution from

Eq.(4.9), i.e.

| | ( , , , , )SOnv SJ H nv SJ A n v n v S< ΛΣ ΛΣ >= ΛΣ . (5.9)

The element off-diagonal in Σ (cf.Eq.(5.6)) is similar to the matrix element in Eq.(3.11), N

replaced by J and by Λ Λ +Σ . See also [3] and Hougen [6].

6. Symmetries of the molecular states

For a general diatomic molecule there are, as mentioned in section 3, two important symmetry

operations. The first one is inversion of the coordinates of all electrons and nuclei in a space-

fixed coordinate system (actually applies to all physical systems as long as the weak nuclear

force is ignored). The corresponding symmetry operation is in molecular physics and chemistry

usually denoted by *E (otherwise in physics by P). For a Hund’s case (a) basis state it follows

that [6,7]:

* | ( 1) |nJ S sE nv SJ nv SJ

− +ΛΣ >= − −Λ −Σ > , (6.1)

where ns is related to the other symmetry operation, namely reflection in a plane containing

the molecular axis. This reflection operation is denoted vσ , and we have [6]

| ( 1) |ns

v nv SJ nv SJσ ΛΣ >= − −Λ −Σ > . (6.2)

Here 0ns = for states with 0Λ ≠ , whereas 1 for states, and 0 for states.n ns s− += Σ = Σ

Symmetry operators means that [ ] [ ], * , 0vH E H σ= = , thus there are simultaneous

eigenstates for and *H E . Hund’s case(a) states are according to Eq.(6.1) easily organized into

eigenstates | aΦ > for *E through the linear combinations:

( )1| | ( 1) |

2

s

a nv SJ nv SJΦ >= ΛΣ > + − −Λ −Σ > , (6.3)

with eigenvalues (parity) given by

( )* | 1 |nJ S s s

a aE

− + +Φ >= − Φ > . (6.4)

Here s=0 and s=1 represent the two different parities for given values of , and nS J s . We also

notice that in Eq.(6.3) 0, and 0 for 0Λ ≥ Σ ≥ Λ = . Now, | | 'a aH< Φ Φ > will be zero if

20

| and | 'a aΦ > Φ > have different parities. Let | 'aΦ > be given by Eq.(6.3) with Σ replaced by

'Σ . Then we have

| | ' | | ' ( 1) | | 's

a aH nv SJ H nv SJ nv SJ H nv SJ< Φ Φ >=< ΛΣ ΛΣ > + − < ΛΣ −Λ −Σ > . (6.5)

First we notice that for 1Λ ≥ there is no parity dependent term. This means that to first order

in a Hund’s case (a) basis the energy levels are doubly degenerate. On the other hand for Σ

states there may be elements off-diagonal in the Σ quantum number. This means that levels

with the same and S J quantum numbers for a given electronic state may have different

energies, even to first order, according to their parities.

The doubly degenerate energy levels ( 0)Λ ≠ corresponding to the spin manifold of a given

electronic state are then to first order obtained from Eqs.(6.5), (5.8) and (5.9) by diagonalizing a

symmetric matrix of dimension 2 1S + . For Σ states the matter tends to be a little more

complex. For doublets ( 1/ 2, 1/ 2)S = Σ = Eq.(6.3) yields two states of opposite parity which do

not interact. There are two different matrix elements from Eq.(6.5) ( ' 1 / 2)Σ = Σ = , which in

turn lead to two levels of different parity and energy for the same and S J values. For triplets

(S=1) there are according to Eq.(6.3) two states of opposite parity for 1Σ = . The basis state

| 0 0nv SJΛ = Σ = > interacts with one of those 1Σ = basis states of the same parity (cf.

Eq.(6.4)), and the result will be three levels with different energies for the same and S J values.

The matrices that have to be diagonalized to yield the energies are in each case obtained from

Eqs.(5.8) and (5.9).

The effective spin-spin Hamiltonian of Eq.(4.10) with matrix element given in Eq.(4.11) also has

to be taken into account for Σ states with 1S ≥ . For doublets ( 1/ 2, | | 1 / 2)S = Σ = there is

merely an uninteresting contribution to the electronic energy.

The eigenstates obtained by diagonalizing the matrices, will be so called intermediate states

that take the form

| |S

S

nv SJ C nv SJΣΣ=−

Λ >= ΛΣ >∑ , (6.6)

where the coefficients CΣ depend on the rotational constant and the spin-orbit and spin-spin

coupling constant, besides the relevant quantum numbers.

An alternative to the Hund’s coupling case (a) basis is Hund’s coupling case (b) (cf. Eq.(4.4)). The

case (b) basis states may be expanded in terms of the case (a) basis as [3]

( )1/2| 2 1 ( 1) |S

J

S

N S Jnv NSJ N nv SJ−Λ−Σ

Σ=−

Λ >= + − ΛΣ > −Λ −Σ Λ +Σ

∑ . (6.7)

For Σ states the intermediate states obtained by diagonalizing the Hamiltonian matrix will be

Hund’s case (b) states (if the spin-spin interaction is neglected).

21

The double degeneracy of the states with 0Λ ≠ may be lifted if the perturbation expansion is

carried beyond the first order. Returning to the matrix element of Eq.(5.3) we notice that the

terms 2 2BL S BJ L⋅ − ⋅

from 2K

supplemented with the spin-orbit interaction of Eq.(4.6)

represent a perturbation 'H that breaks the BO approximation and couples different

electronic states off-diagonal in Λ , i.e.

' ( 2 ) 2H A B L S BJ L= + ⋅ − ⋅

. (6.8)

For Π states ( )1Λ = second order contributions from 'H yield the Λ -doubling for non-zero

spin (cf. section 3), similar to the zero-spin case (cf. Eq.(3.12)). For states with 1Λ > the

degeneracy in the levels of opposite parity is lifted only if the perturbation expansion is carried

beyond the second order. For Σ states where case (b) coupling applies, the second order

contribution from 'H will lead to a splitting of states with the same N -quantum number, but

different values of J . This splitting is often denoted ρ -doubling, and might effectively be

considered as arising from a spin-rotation interaction of the form N Sγ ⋅

, where γ is the spin-

rotation coupling constant. In practice the second order corrections that lead to the

doubling and doublingρΛ − − are included in the matrix that describes the spin manifold of a

given electronic state through a so called Van Vleck transformation. This is, however, a

somewhat technical matter that falls outside the scope of this presentation.

7. Diatomic magnetic hyperfine structure

The magnetic hyperfine structure arises from an interaction between the electrons and the

magnetic moments of the nuclei. Since the electron spin magnetic moment is involved, the

basic point of departure for a derivation of the magnetic hyperfine interaction is certainly the

Dirac equation for an electron in an external electromagnetic field. In this case the magnetic

field stems from the magnetic moment of the nucleus. We shall not here reproduce the rather

complex algebra that leads to the non-relativistic form of the interaction. In the Pauli

approximation it takes the form

( )M i i i i

i i

e eH A p s A

mc mc= ⋅ + ⋅ ∇×∑ ∑

, (7.1)

where iA

represents the magnetic field (vector potential) at the position ir

of electron i .

Furthermore, a nuclear magnetic moment µ

yields a vector potential given by

3( ) /i i iA r rµ= ×

, (7.2)

22

with I Ng Iµ µ=

, where Ig is the nuclear g-factor, and Nµ denoted the nuclear magneton.

Now, inserting Eq.(7.2) into (7.1) with special attention to the singularity at the origin, we get

the magnetic hyperfine Hamiltonian [8]

3

5 3

2 ( ) /

3( )( ) ( ) 2

8 2 ( ),

3

hf I B N i i i

i

i i i iI B N

i i i

I B N i i

i

H g I r p r

I r s r I sg

r r

g s I r

µ µ

µ µ

πµ µ δ

= ⋅ ×

⋅ ⋅ ⋅+ −

+ ⋅

∑

∑

∑

(7.3)

where Bµ denotes the Bohr magneton. The first part of the equation above is easily recognized

as the interaction between the nuclear spin magnetic moment and the magnetic field

generated by the electrons orbital motion. The second part is similar to the spin-spin

interaction of Eq.(4.2). The last part is the Fermi-contact interaction. It has to be taken into

account for states where there is a finite probability for the electron to be at the nucleus (e.g. s-

states for atoms). For a molecule with several nuclear spins there will be one term like Eq.(7.3)

for each nuclear spin, where ir

denotes the position of the electron relative to the relevant

nucleus. The first- and last part of Eq.(7.3) as well as the last term in the second part are readily

handled by the Wigner-Eckart theorem to yield effective hyperfine operators. The first term in

the second part, however, needs special attention. This term has to be expanded in terms of

spherical tensors centered at the relevant nucleus to make a more tractable expression.

Matrix elements of hfH that are diagonal in the electronic spin quantum number S will be the

most, or only relevant ones. For this purpose the hyperfine Hamiltonian can be replaced by the

following effective operator (cf. [8-9]):

( )

( )

,

2 2

1( )

2

1 .

2

hf eff z z

i i

H aI L b c I S b I S I S

d e I S e I Sφ φ

+ − − +

−− − + +

= ⋅ + + + +

+ +

(7.4)

The vector components in the expression above are all relative to a molecule fixed coordinate

system, with the z-axis along the molecular axis. In the last part of Eqs.(7.4) the exponentials 2ie φ± indicate second rank tensors that couple states with for 1±Λ Λ = , i.e. this part yields a

parity dependent hyperfine effect for Π states according to Eq.(6.3). The hyperfine constants

a , b , c , and d are in practice adjustable parameters obtained from experiments. Explicit

expressions are given in e.g. [8].

23

The effective hyperfine Hamiltonian of Eq.(7.4) also applies for a homonuclear molecule, but

now I

is the total nuclear spin, i.e. 1 2I I I= +

. In the expressions for the hyperfine constants

there will now be an (equal) contribution from both nuclei. For a heteronuclear molecule with

two nuclear spins there will be a contribution as in Eq.(7.4) from each nuclear spin, also with a

double set of hyperfine constants.

For a diatomic molecule with two nuclear spins different from zero, there is in addition a

nuclear spin-spin Hamiltonian like the electronic spin-spin Hamiltonian in Eq.(4.2). This effect

will, however, normally be very small compared to the magnetic effect discussed above. Nuclei

with spin 1/ 2I > will also have an electric quadrupole moment, which yields a (small) electric

hyperfine interaction. The nuclear spin-spin interaction and the electric hyperfine effect will,

however, not be discussed further here.

For a diatomic molecule with only one nuclear spin we have simultaneous quantization of

2 and zI I

, with quantum numbers and II Ω respectively. Adding these quantum numbers to

the Hund’s case (a) basis yields the new complete basis as | Inv SJIΛΣ Ω > . Then we have the

following matrix elements for ,hf effH diagonal in , , , , and n v S J IΛ :

,| | [ ( ) ]I hf eff I Inv SJI H nv SJI a b c< ΛΣ Ω ΛΣ Ω >= Λ + + Σ Ω , (7.5)

[ ]1/2,

1| | 1 1 ( )( 1)( 1)( ) .

2I hf eff I I Inv SJI H nv SJI b S S I I< ΛΣ Ω ΛΣ − Ω + >= + Σ −Σ + +Ω + −Ω

For Π states there will in addition be an element proportional to the constant d that couples

the two states of opposite parity. Σ states also need special attention as there is a parity

dependent element of the form (cf. Eq.(6.3)):

[ ]

,

1/2

1, ' 1, '

0 | | 0 ' '

1= ( )( 1)( 1)( ) .2 I I

I hf eff I

I I

nv SJI H nv

b S S I I δ δΣ− −Σ Ω + −Ω

< Λ= Σ Ω Λ= −Σ −Ω >

+Σ −Σ+ +Ω + −Ω (7.6)

The matrix elements for the hyperfine Hamiltonian of Eqs.(7.5) and (7.6) are then added to

those of the rotational, spin-orbit and spin-spin Hamiltonians from section 5. Thus, we obtain a

complete matrix within the spin manifold of a given electronic state that also includes the

hyperfine effects. The matrix elements from section 5 are all obviously diagonal in IΩ , and also

independent of and II Ω . Diagonalizing this matrix yields eigenstates with , and S JΛ as good

quantum numbers, and in addition a quantum number F corresponding to the total molecular

angular momentum F J I= +

. Hence, a rotational level with quantum number J will be split

24

into 2 1I + hyperfine substates, each labeled with a F quantum number in the range

, , ( )F J I J I J I= − ⋅⋅⋅ + ≥ .

8. Interaction with an external field

Equation (7.1) also applies to an atom or molecule exposed to a homogeneous magnetic field

B

. In this case

1( ) ,

2i i iA B r B A= × = ∇×

, (8.1)

and after a little rearrangement of terms we get the Zeeman Hamiltonian

2

Z

e eH B L B S

mc mc= ⋅ + ⋅

. (8.2)

If the system is exposed to an external homogeneous electric field E , the interaction

Hamiltonian (Stark effect) takes the form

SH D= − ⋅E . (8.3)

Here, the electric dipole moment D

is given by i i

i

D q r=∑

, and the sum extends over all

electrons as well as nuclei. For an electric neutral system the dipole moment D

is independent

of the origin.

We notice that for both and Z SH H there are terms of the form E I⋅

, where E

denotes a

vector given in an external space fixed coordinate system ( or )BE , and I

is an internal vector

( , or )L S D

, defined relative to a molecule fixed coordinate system. As in the preceding

sections we want to work out matrix elements for and Z SH H in a Hund’s case (a) basis. As

before we consider matrix elements within the spin manifold of a given electronic state, i.e.

elements diagonal in , , and n v SΛ . However, we now notice that 2, 0Z

H J ≠

and

2, 0S

H J ≠

, which means that J will not be quantized for a molecule subject to an external

field. Thus, we want to work out matrix elements of the general type:

| | ' ' 'J Jnv SJM E I nv SJ M< ΛΣ ⋅ ΛΣ >

, (8.4)

where we in the basis have included the quantum number JM corresponding to the space

fixed component of ZJ J

. Assuming that the external E

vector points along the space fixed Z

axis, we just need to consider the matrix element | | ' ' 'J Z Jnv SJM EI nv SJ M< ΛΣ ΛΣ > , where

E denotes the constant external field.

25

Now, the Hund’s case (a) basis states refer to a molecule fixed coordinate system, whereas ZI

denotes a space fixed component. Thus, the space fixed component has to be expanded in

terms of the molecule fixed ones. Then we finally obtain for our matrix element (cf. [3]):

1

1

' ' 1/2

| | ' ' ' | | '

1 '1 '( 1) [(2 1)(2 ' 1)] ,

0 ''J

J Z J

M

J J

nv SJM I nv SJ M nv I nv

J JJ JJ J

M M

µµ

µ

µ

=−

−Ω −

< ΛΣ ΛΣ >= < ΛΣ ΛΣ >

× − + + −Ω − −Ω

∑ (8.5)

where Iµ denotes the molecule fixed spherical component of the internal vector I

. Here

, ' 'Ω = Λ +Σ Ω = Λ + Σ . From the properties of the 3j-symbols in the expansion above we

notice that a non-zero result requires ' , ' or 1J JM M J J J= = ± , and furthermore that

' µΩ = Ω− , which means that ' µΣ = Σ − . This last condition implies that since for

or Z Z Z ZI L I D= = only 'Σ = Σ applies, we only have 0µ = , i.e. just one term in the sum in

Eq.(8.5). In the case Z ZI S= there may be off-diagonal elements in Σ , but also now just one

term in the sum ( ')µ = Σ −Σ .

It might also be of interest to express the matrix elements of external field interactions of

Eqs.(8.1) and (8.2) in terms of the parity eigenstates | aΦ > of Eq.(6.3). From Eq.(8.5) it is clear

that all matrix elements will be diagonal in the quantum number JM . Thus, we let | 'aΦ >

represent the parity eigenstate with and replaced by ' and 'J JΣ Σ respectively. For molecular

states with 0Λ > we then obtain according to Eqs.(6.3) and (6.1):

' ' 21| | ' 1 ( 1) ( 1) | | ' '

2

s s J J S

a Z a J Z JI nv SJM I nv SJ M+ + − < Φ Φ >= − − < ΛΣ ΛΣ > ∓ . (8.6)

Here, the upper sign applies to , and the lower to and Z Z Z ZI D L S= . Off-diagonal elements in

Σ are obviously non-zero only for Z ZI S= (space fixed components!).

For statesΣ − the result is somewhat more complex. For 0 and ' 0Σ > Σ > we obtain

' ' 2

' ' ' 2

1| | ' 1 ( 1) ( 1) 0 | | 0 ' '

2

1+ ( 1) 1 ( 1) ( 1) 0 | | 0 ' '2

s s J J S

a Z a J Z J

s s s J J S

J Z J

I nv SJM I nv SJ M

nv SJM I nv SJ M

+ + −

+ + −

< Φ Φ >= − − < Λ = Σ Λ = Σ >

− − − < Λ = Σ Λ = −Σ >

∓

∓

. (8.7)

Here, the upper sign once more applies for ZD , and the lower for and Z ZL S . The last part (with

'−Σ ) of Eq.(8.7) applies only for ZS , as this element is off-diagonal in Σ .

26

Finally, we have for the case 0 and 0, but ' 0Λ = Σ = Σ > :

' 21| | ' 1 ( 1) ( 1) 0 0 | | 0 ' '

2

s J J S

a Z a J Z JI nv SJM I nv SJ M+ − < Φ Φ >= + − − < Λ = Σ = Λ = Σ > , (8.8)

with a similar expression for 0, 0 and ' 0.Λ = Σ > Σ = The matrix element in Eq.(8.8) is off-

diagonal in Σ , and certainly applies only for ZS . Now, Eqs.(8.6)-(8.8) yield useful information

on how the parities and 's s have to be chosen for different rotational quantum numbers

and 'J J to obtain non-zero matrix elements. We notice that the rules are different for

and the axial vectors and Z Z ZD L S .

For homonuclear molecules there is the extra g-u symmetry, related to the inversion of the

electronic coordinates through the midpoint between the nuclei. From the electronic matrix

element | | 'nv I nvµ< ΛΣ ΛΣ > of Eq.(8.5) we furthermore notice that this diagonal element will

be zero for I Dµ µ= due to the g-u symmetry. Hence, there is no first order Stark effect for a

homonuclear diatomic molecule, corresponding to the fact that there is no permanent electric

dipole moment. The Stark effect has to be explained in terms of higher order effects due to

interaction with other electronic states.

In the final step the matrix elements of Eqs.(8.6)-(8.8) have to be expressed in terms of the

molecule fixed components by use of Eq.(8.5). Thereafter they are added to the elements of the

zero-field Hamiltonians of section 5 and 7 to yield the complete matrix for the total electronic

and nuclear spin manifold of a given electronic state. The eigenstates obtained by diagonalizing

the Hamiltonian matrix with external fields and hyperfine interaction will in principle only retain

FM as a good quantum number. However, for weak fields as well as ( )J F F J I= +

will serve

as adequate labels for the energy levels and eigenstates.

References:

1. H. Margenau and G. M. Murphy: “The Mathematics of Physics and Chemistry” (Van

Nostrand, Second edition, p.411).

2. G. Herzberg: “Spectra of Diatomic Molecules”.

3. L. Veseth: “Lectures on Group Theory: The rotation group R3 and the special unitary

group SU2, with applications to molecular rotation”.

4. R. E. Moss: “Advanced Molecular Quantum Mechanics” (Chapman and Hall, 1973).

5. H. A. Kramers: Zeitschrift fur Physik, 53, 422, 429 (1929).

6. J. T. Hougen: “The Calculation of Rotational Energies and Rotational Line intensities in

Diatomic Molecules” (Nat. Bur. Stand. Monograph 115, 1970).

7. M. Larsson: Physica Sripta 23, 835 (1981).

8. P. Kristiansen and L. Veseth: J. Chem. Phys. 84, 2711 (1986).

9. G. C. Dousmanis: Phys. Rev. 97, 967 (1955).

27