25
CHAPTER – 1
THEORIES AND TECHNIQUES
1.1. INTRODUCTION
Spectroscopy is the branch of science which deals with the interaction of
electromagnetic radiation and matter. The most important phenomenon of such
interaction is that energy is absorbed or emitted by the matter in discrete amounts
called quanta. The ways in which the measurements of radiation frequency (emitted
or absorbed) are made experimentally and the energy levels deduced from it comprise
the practice of spectroscopy. Currently, the spectroscopy is one of the most powerful
tools available for the study of atomic and molecular structures of wide range of
samples. Moreover, the spectroscopic methods of the different regions of the
electromagnetic spectrum are the powerful techniques available for the understanding
of molecular structure, nature of bonding between atoms, conformation analysis,
symmetry of molecular groups or ions in the crystals and so forth. In short,
spectroscopy has become an indispensable tool to present day physicists owing to the
availability of very sophisticated instruments with data processing arrangements.
Molecular spectroscopy deals with the interaction of electromagnetic radiation
with molecule. A molecule possesses various forms of energy (translational,
vibrational, rotational and electronic energies) due to its different kinds of motion and
intermolecular interactions. However, these are quantized and the interactions
between them are weak. Electromagnetic radiations can be allowed to interact with
the molecular energy levels and investigation of these interactions can provide various
information regarding their rotation, charge localization, molecular structure,
symmetry, vibration, etc. It is an established fact that the interaction of
electromagnetic energy with the vibrational energy levels of a molecule provides
amazing information on the molecular dynamics [1].
Vibrational spectroscopy is a valuable tool for the elucidation of molecular
structure. The study of vibrational spectroscopy has resulted in a large volume of
data on the vibrations of polyatomic molecules. It gives a dynamic picture of the
molecule. The position, bandwidth and intensity of the absorption bands may be
26
correlated with the electronic and molecular structure of the system. It also provides
important information about the intra molecular forces acting between the atoms in a
molecule, the inter-molecular forces in the condensed phase and the nature of the
chemical bond. It may be pointed here that, the goal of high resolution molecular
spectroscopy is the determination of molecular geometry and the potential energy
function. Vibration spectroscopy has also contributed significantly to the growth of
other areas such as polymer chemistry, catalysis, fast reaction dynamics, charge-
transfer complexes, etc [2].
The vibrational spectroscopy includes different types of spectra. Transitions
between electronic energy levels give the spectrum in the visible or ultraviolet region
and are referred to as electronic spectra. Vibrational spectra are due to transitions
between vibrational levels within the same electronic level which fall in the infrared
region. Transition between rotational levels within the same vibrational level give the
spectra in the far infrared or microwave region and is referred to as rotational spectra.
In addition, there are Nuclear Magnetic Resonance (NMR) and Electron Spin
Resonance (ESR) spectroscopy.
Among these various spectra, Infrared and Raman spectroscopy can provide
macroscopic as well as microscopic information on the whole dynamics of the
molecule, structures and their various functional groups. These are comparatively
sensitive technique, requiring small sample sizes to obtain spectra with a helpful
signal-to-noise ratio. Molecular vibrations, which occur due to the molecular dipole
moment, are visible in the infrared spectrum, while due to polarizability, appear in the
Raman spectrum. Hence it is noteworthy that these two methods should be used for a
complete vibrational analysis of a molecule [2].
In addition, one of the great vibrational techniques used for the complete study
of molecules is Fourier Transform Infrared spectroscopy (FT-IR) due to its fast, non-
destructive, and requirement of small sample quantities. On the other hand, using of
traditional Raman spectroscopy was limited on account of not only the laser induced
fluorescence and risk of sample destruction by light energy but also record of spectra
with good signal-to-noise ratio was often time-consuming. But, these problems were
rectified with the development of Fourier Transform Raman spectroscopy
(FT-Raman).
27
Thus, in the modern era of sciences, the Infrared and Raman spectroscopy,
which constitute the core of the vibrational spectroscopy, have found wide
applications in the precise identification and analysis of the compounds [3]. Both
methods have been applied to a wide variety of problems, which yield not only the
desired knowledge such as inter-nuclear distances, vibrational frequencies and force
constants, etc. but also certain vital thermo dynamical quantities such as entropy, free
energy, specific heat capacity, enthalpy etc.
The Ab initio HF and DFT methods for the analysis of molecular vibrations
have been of great service in the study of molecular dynamics. The computational
analysis is the universally accepted approach for interpretation of the spectral data of
the molecules. These theories have proved to be highly successful in describing
structural and electronic properties in a vast class of materials, ranging from atoms
and molecules to simple crystals to complex extended systems. These work falls
mainly into three classes: those dealing with the theory, those concerned with the
technical aspects of the numerical implementations, and vast majority of presenting
results.
Ab initio calculations of vibrational spectra are nowadays feasible for small
and medium size molecules, largely due to the development of analytical methods for
computing energy derivatives [4-7]. Moreover, large molecules are normally treated
at the harmonic level using Hartree-Fock calculations with basis sets of moderate size.
The scaled theoretical force fields may assist in the assignment of vibrational spectra
and allow predictions for unknown molecules, and the combination of theoretical and
experimental information may lead to more reliable force fields for known molecules.
1.2. MOTIVATION
The study of molecular dynamics belongs to a high priority open field of
research in the recent times. It is a technique to investigate equilibrium and transport
properties of many–body systems. The investigation of internal motion, dynamics of
molecules are emerging as a new field of research towards understanding of physics
of complex system and the development of sparkling technologies. The power of
quantum mechanical calculations and the increasing availability of fast personal
computers reduce the complexicity of theoretical methods and launch novel
possibilities for comprehensive and reliable vibrational analysis.
28
The vibrational spectroscopy, which includes FTIR and FT-Raman, is a
persuasive tool for the study of molecular structural properties and their functions at
the molecular level. Infrared spectroscopy is a time-honored research tool, which has
enjoyed a renaissance in recent years due to the introduction of Fourier transform
techniques. Because of high speed, sensitivity, mechanical simplicity, internal
calibration enhances FTIR technique most praiseworthy in the field of molecular
dynamics. The qualitative interpretation of IR spectra picturises the vibrational modes
of the molecule and has enfolded within much information on chemical structure.
The prologue of FT-Raman spectroscopy has brought a new impetus to Raman
spectroscopy, which provides exquisite structural insights into the molecular
structures. It has allowed the materials that were previously ―impossible‖ because of
laser-induced fluorescence and provides ready access in the extensive data handling
facilities that are available with a commercial FT-IR spectrometer. Using modern dual
instrument, in which switching between FT-Raman and FT-IR modes has been easily
taken with the help of computers; both spectra can be obtained simultaneously. With
the aid of microcomputers, the instruments are capable of performing the spectra with
high speed and high sensitivity with most accuracy. These instruments are exemplary
for accumulation, manipulation, presentation, and control of the data.
The Ab initio HF and DFT theories have proved to be highly successful in
describing structural and electronic properties in a vast class of materials, ranging
from atoms and molecules to simple crystals to complex extended systems. These
work falls mainly into three classes: those dealing with the theory, those concerned
with the technical aspects of the numerical implementations, and vast majority of
presenting results.
The Gaussian program [8-9] developed for HF and DFT calculations endow
with vast number of data such as the harmonic vibrational frequencies, optimized
molecular parameters (bond length, bond angle, dihedral angle), dipole moment,
rotational constants, Non linear optical parameters (first and second order
Polarizability, anisotropy) and thermodynamical parameters (zero point energy,
entropy, enthalpy and specific heat capacity.
29
The discussed concepts, advancements in techniques, scaled quantum
mechanical approach and the computerized programmes are the driving forces and
motivation behind this present work.
1.3. OBJECTIVES AND SCOPE OF THE WORK
There is high demand for research on materials. The medical and
pharmaceutical industries need lot of materials which possess high biological and
pharmacological action. The polymer industries need lot of new polymers which can
satisfy the growing demand in various fields connected with polymers. Auto
industries which include ships, submarines, airplanes, rockets, satellites, cars etc, need
a lot and lot variety of materials which suit various tear and wear conditions and
comforts. Hence, the research on materials at molecular level which is very important
for the thorough analysis of the properties of the materials is quite imperative.
In addition to these experimental facilities IR and Raman which are very
effective in analyzing the characteristics and properties of the materials, a good
theoretical method, based on quantum mechanical principles, has also been available,
which has enormous potentials to calculate all parameters, such as thermal,
mechanical, electrical, magnetic, thermodynamic etc., at present as ready built
package program Gaussian 03 W. These programs can be easily run on any Pentium
IV processor personal computers. In addition to the parameters mentioned above the
software also helps to optimize the molecular structure, predict energies, reduced
mass, IR intensity, Raman activity, depolarisation ratio, force constants, bond
strengths, wave numbers and HOMO-LUMO analysis.
Hence, the present research is undertaken to study certain industrially,
biologically and pharmaceutically very important compounds such as the derivatives
of benzene, pyridine, and naphthalene after a thorough study of literature, with the
tool of Fourier transform Infrared, Fourier transform Raman spectroscopy and the
Gaussian 03 program, under its standard quantum mechanical background.
1.4. INFRARED SPECTROSCOPY – INTRODUCTION
Infrared spectroscopy is widely used for the identification of organic
compounds because of the fact that their spectra are generally complex and provide
numerous maxima and minima that can be used for comparison purposes. Infrared
spectroscopy is generally concerned with the absorption of radiation incident upon a
30
sample. IR technique when coupled with intensity measurements may be used for
qualitative and quantitative analysis. Currently, this technique has become more
popular as compared to other physical techniques (X-ray diffraction, electron spin
resonance, etc.,) in the elucidation of the structure of unknown compounds.
The atoms in the molecule execute different types of vibrational motions. The
vibrational energy so produced should be quantized and corresponds to the infrared
region of the electromaganetic spectrum. When infrared radiation of the same
frequency is allowed to fall on the molecule, the system absorbs energy, causing
excitation of the molecule to higher vibrational levels. IR spectroscopy is used both to
gather information about the structure of a compound and as an analytical tool to
assess the purity of a compound.
For a molecule to absorb IR, the vibrations or rotations within a molecule must
cause a net change in the dipole moment of the molecule. This change in the dipole
moment arises due to the intraction of the alternating electrical field of the radiation
with fluctuations in the dipole moment of the molecule. If the frequency of the
radiation matches the vibrational frequency of the molecule then the radiation will be
absorbed, causing a change in the amplitude of molecular vibration.
1.4.1. Molecular Vibrations
A molecule is not a rigid assembly of atoms; it can be viewed as a system of
balls and springs of varying strengths, corresponding to the atoms and chemical bonds
of the molecule. So, the positions of atoms in molecules are not fixed; they are
subjected to number of different vibrations. These vibrations fall into the two main
catagories of stretching and bending. Stretching, in which the distance between two
atoms decreases or increases, but the atoms remain in the same bond axis, and
bending (or deformation), in which the position of the atom changes relative to the
original bond axis. Each of the vibrational motions of a molecule occurs with a certain
frequency, which is characteristic of the molecule and of the particular bond. The
stretching comprises of symmetric and asymmetric stretching whereas the bending
involves in-plane (such as rocking, scissoring vibrations) and out-of plane bending
(such as twisting, wagging) vibrations. The energy involved in a particular vibration
is characterized by the amplitude of the vibration, so that higher the vibrational
energy, larger will be the amplitude of the motion [7]. In addition to above said
31
vibrations, there are coupling vibrations if the vibrating bonds are joined to a single,
central atom.
According to the results of quantum mechanics, only certain vibrational
energies are allowed to the molecule, and thus only certain amplitudes are allowed.
Associated with each of the vibrational motions of the molecule, there is a series of
energy levels (or vibrational energy states). The molecule may be made to go from
lower energy level to a higher energy level by absorption of a quantum of
electromagnetic radiation, such that
𝐸𝑓𝑖𝑛𝑎𝑙 − 𝐸𝑖𝑛𝑖𝑡𝑖𝑎𝑙 = 𝑛 …… 1.0
During such transition, the molecule gains vibrational energy, and this is
manifested in an increase in the amplitude of the vibration. The frequency of light
required to cause a transition for a particular vibration is equal to the frequency of that
vibration. So, by measuring the vibrational frequency it is enough to measure the
frequencies of light which are absorbed by the molecule. As it happens, light of this
wavelength lies in the infrared region of the spectrum. This IR spectrum of the
corresponding molecule appears as a series of broad absorption bands of variable
intensity which provides structural information. Each absorption band in the spectrum
corresponds to a vibrational transition within the molecule, and gives a measure of the
frequency at which the vibration occurs.
1.4.2. Vibrational degrees of freedom
A molecule has as many degrees of freedom as the total degrees of freedom of
its individual atoms. Each atom has 3 degrees of freedom in the Cartesian coordinates
(x, y, z), necessary to describe its position with respect to a fixed point in the
molecule. A polyatomic molecule containing n atoms can be described by a set of
Cartesian coordinates has 3n degrees of freedom. In a nonlinear molecule, 3 of these
degrees are rotational and 3 are translational and the remaining 3n – 6 correspond to
fundamental vibrations; in a linear molecule, 2 degrees are rotational and 3 are
translational and the remaining 3n-5 corresponds to vibrational motions. Among the
3n – 6 vibrational modes, (n – 1) modes are bond stretching vibrations and the other
2n – 5 [(2n – 4) for a linear molecule] modes are angle-bending vibrations [10]. The
number of vibrational degrees of freedom gives the number of fundamental
32
vibrational frequencies of the molecule, or in other words, the number of normal
modes of vibrations [1].
1.4.3. Infrared Vibration Spectra
The infrared spectrum is formed by the absorption of electromagnetic
radiation at frequencies that correlate to the vibration of specific sets of chemical
bonds within a molecule. It is important to reflect on the distribution of energy
possessed by a molecule at any given moment, defined as the sum of the contributing
energy terms [23].
𝐸𝑇𝑜𝑡𝑎𝑙 = 𝐸𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑖𝑐 + 𝐸𝑣𝑖𝑏𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 + 𝐸𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 = + 𝐸𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑖𝑜𝑛𝑎𝑙 …… 1.1
The translational energy relates to the displacement of molecules in space as a
function of the normal thermal motions of matter. Rotational energy, which gives rise
to its own form of spectroscopy, is observed as the tumbling motion of a molecule,
which is the result of the absorption of energy within the microwave region. The
vibrational energy component is a higher energy term and corresponds to the
absorption of energy by a molecule as the component atoms vibrate about the mean
center of their chemical bonds. The electronic component is linked to the energy
transitions of electrons as they are distributed throughout the molecule, either
localized within specific bonds, or delocalized over structures, such as an aromatic
ring.
During the vibrational motion of a molecule the charge distribution undergoes
a periodic change, which changes the dipole moment periodically. For a particular
vibrational mode, in order to directly absorb infrared electromagnetic radiation, the
vibrational motion associated with that mode must produce a change in the dipole
moment of the molecule. Normal vibrations that are connected with a change of
dipole moment and therefore, appear in the infrared spectrum are called infrared
active modes, while vibrations for which the change of charge distribution is such that
no change of dipole moment arises and which, therefore, do not appear in the infrared
spectrum are called infrared inactive modes [3].
Let μx, μy and μz are the three components of the dipole moment μ of the
molecule in the direction of the axes, of a Cartesian coordinate system fixed in the
molecule, in a displaced position of the nuclei. If μx0, μy
0 and μz
0 are the components
33
of the dipole moment μ0 in the equilibrium position, then, for sufficiently small
displacements, we can expand μx as
μx = μx0
+
k
k
k
x
k
k
x
k
k
x zz
yy
xx
000
…… 1.2
Where the xk, yk and zk are the displacements coordinates of nucleus k. Similar
relations hold for μy and μz. If we use normal coordinates q1, q2, q3 …. qk we have
μx = μx0 +
2
0
2
2
02
1k
k
x
k
k
x qq
…… 1.3
where kkkk tqq 2cos0 …… 1.4
The values of μy and μz are also calculated in the similar way. Thus, the expression for
molecular dipole moment is
μ = μ0
+
2
0
2
2
02
1k
k
k
k
…… 1.5
According to equation (1.4) and (1.5), the dipole moment μ of the molecule
will change with the frequency νk of a normal vibration k if and only if at least one of
the derivatives
000
,,
k
z
k
y
k
x
qqq
is different from zero. The intensity of
this infrared fundamental band is proportional to the square of the vector representing
the change of the dipole moment for the corresponding modes of vibration near the
equilibrium position; that is,
2
0
2
0
2
0
k
z
k
y
k
x
qqqI
…… 1.6
The above discussion is based on the assumption that the vibration of the
molecule is simple harmonic. If the anharmonicity is taken into account, the
vibrational motion contain also the frequencies 2νk, 3νk , ... , and furthermore νk + νi ,
νk – νi , 2νk + νi , …. Therefore, in the infrared spectrum in addition to the
fundamentals, overtones and combination vibrations may also occur, if they are
connected with a change of dipole moment. However, they will, be much weaker than
34
the fundamentals, since the anharmonicites in general are slight, except for very large
amplitudes of the nuclei [1].
1.4.4. Infrared Selection Rule
Selection rule for the infrared activity can be obtained by expressing equation
(1.5) using the transition moment integral, as
k
jki
ok
jioij dqq
d
…… 1.7
neglecting the higher order terms. As μo is a constant and also due to the orthogonal of
the wave functions the first integral is zero except for i = j. Since we are considering a
transition from state i to j, the first term vanishes. Consequently, the permanent dipole
moment of the molecule has no effect on the vibrational transitions.
The second term gives the transition probability for the infrared absorption. In
this term, there is a factor 𝜕𝜇
𝜕𝑞𝑘
0which gives a change in dipole moment μ around the
equilibrium position during a vibration. For at least one of the components qk of μ, the
dipole moment must be non-zero. Accordingly, 𝜕𝜇
𝜕𝑞0≠ 0, then a normal mode qk will
be active in the infrared absorption spectrum [2].
1.5. RAMAN SPECTROSCOPY - INTRODUCTION
1.5.1. Raman Effect
In Raman effect, when monochromatic light is scattered by molecules, a small
fraction of the scattered light is observed to have different frequency from that of the
irradiating light. Raman effect has been important as a method for the explanation of
molecular structure, for locating various functional groups or chemical bonds in
molecules, and for the quantitative analysis of complex mixtures. Although Raman
spectra are related to infrared absorption spectra, a Raman spectrum arises in a quite
different manner and thus provides complementary information. Vibrations that are
active in Raman may be inactive in the infrared and vice versa. A unique feature of
Raman scattering is that each line has a characteristic polarization and polarization
data provide additional information related to molecular structure. When
electromagnetic radiation of energy content h irradiates a molecule the energy may
be transmitted, absorbed or scattered. The scattering mechanisms can be classified on
35
the basis of the difference between the energies of the incident and scattered photons.
If the energy of the incident photon is equal to that of the scattered one, the process is
called Rayleigh scattering. If the energy of the incident photon is different to that of
the scattered one, the process is called Raman scattering. When the substance is
illuminated by monochromatic light, the spectra of the scattered light consists of a
strong line (the exciting line) of the same frequency as the incident light together with
weaker lines on either side shifted from the strong line by frequencies ranging from a
few cm-1
to about 3500 cm-1
. The lines of frequency less than that of exciting line are
called Stokes lines, the others anti-Stokes lines. [11].
1.5.2. Classical Theory
The classical theory of Raman effect, while not wholly adequate, is worth
consideration since it leads to an understanding of the concept – the polarizability of a
molecule. For a molecular vibration to be Raman active there must be a change in the
polarizability of the molecule. A hypothetical atom with an spherically symmetrical
electron cloud has no permanent dipole moment. When such an atom is irradiated by
an electromagnetic radiation the electric field of the radiation displaces the electron
and positively charged nucleus in opposite directions. This separation is called
polarization and the atom now has an induced dipole moment. If E represents the
electric field of the radiation and μ' represents the induced dipole moment oriented
parallel to the direction of E, which can be described by its components
μ'x = αxx Ex + αxy Ey + αxz Ez
μ'y = αyx Ex + αyy Ey + αyz Ez …… 1.8
μ'z = αzx Ex + αzy Ey + αzz Ez
All αij are components of a tensor α, which projects one vector, the electric
fields vector E to produce another vector μ', the induced dipole moment. This can be
written in matrix notation as
…… 1.9
or μ' = α E …… 1.10
z
y
x
zzzyzx
yzyyyx
xzxyxx
z
y
x
E
E
E
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Since the electric field associated with the incident radiation of frequency νo
alternates the induced dipole moment also alternates at the same frequency νo. Thus
the molecule emits electromagnetic radiation of the frequency νo. Rayleigh scattering
is due to this process [3].
A molecule is not static; it continuously vibrates (also rotate) when radiations
fall on it. This vibratory motion of the molecule is capable of modulating the
polarizability of the molecule. Consequently, the induced dipole moment and
therefore also the amplitude of the emitted field is modulated by the frequency of the
vibration. The modulation of the polarizability of a molecule, vibrating with the
frequency νk is
tq
qkk
k
k
2cos0
0
0 …… 1.11
Here 0
kq represent the normal coordinates. As a consequence of irradiation,
molecular polarizability get modulated with the frequency of electric field νo of
incident light and gives the induced dipole moment as
ttEqq
tE kook
k
ook
2cos2cos2cos 0
0
0
…… 1.12
This is equivalent to
ttEqq
tE kokook
k
ook
2cos2cos2cos 0
0
0…… 1.13
According to this equation, the induced dipole moment acquires three
components νo, (νo – νk) and (νo + νk). When light is emitted by these induced dipoles,
the emitted light also posses the above frequencies. The first term in equation (1.12)
describes Rayleigh scattering, the second term concerns Stokes, and the third anti-
Stokes Raman scattering [12]. The intensity of the emitted radiation is proportional to
the square of the absolute value of the second derivative of the induced dipole
moment:
2
kS …… 1.14
Just as in case the infrared spectrum, if the anharmonicity is taken intro
account, in addition to the fundamentals, overtone and combination vibrations may
37
also appear as Raman shifts if they are connected with a change of polarizability.
When the anharmonicity is small, the intensity of Raman lines corresponding to
overtone and combination vibrations will be very small compared to those
corresponding to fundamentals.
1.5.3. Quantum Theory
Use of Raman effect is to obtain vibrational and rotational frequencies of
molecule. Quantum mechanics gives a qualitative description of the phenomenon of
Raman effect. The interaction of a photon of the incident light beam with the
molecule in its ground electronic and vibrational state (=O) may momentarily raise
the molecule to a time dependent quasi excited electronic state (or a virtual state)
whose height is above the initial energy level. Virtual states are those in which the
molecule has a very short mean lifetime and hence the uncertainity in energy is large
according to the Heisenberg uncertainity principle.
In the case of stokes line, the molecule at ground electronic state is excited to
the virtual electronic state, then radiates light in all directions except along the
direction of the incident light. On return to the ground electronic state, quantum
vibrational energy may remain with the scattering species and there will be a decrease
in the frequency of the scattered radiation.
Anti-Stokes lines arise when the molecule is already in an excited vibrational
state and is raised to quasi-excited state and reverts to ground electronic state on
scattering of photon. Scattered photon is the sum of the energy of incident photon and
the energy between the vibrational levels = 1 and = 0 and there will be an increase
in the frequency of the scattered radiation. In another instance, a molecule in the
ground state on interacting with a photon and attaining the virtual state may leave the
unstable electronic state and return to the ground electronic state. In this case of the
Rayleigh scattering, scattered photon has the same energy as the incident radiation.
The description of Raman Effect will be complete by the of use quantum
theory [11]. Suppose a molecule is initially in state i with the energy Ei and after
interaction with monochromatic radiation of angular frequency νo goes to state f
with energy Ef. From the conservation of energy
Ei + h νo = Ef + h (νo ± νfi ) …… 1.15
38
Where hνfi = Ef – Ei
and h(νo ± νfi) = h νs, the energy of the scattered photon..
If Ef > Ei, νs = νo - νfi (stokes) …… 1.16
If Ef < Ei νs = νo + νfi (anti-stokes) …… 1.17
The stoke lines are more intense than the anti-stokes lines, a consequence of
the different populations of molecule in the ground and first excited vibrational state,
as described by the Boltzmann distribution. Application of the Boltzmann distribution
law shows that the ratio of the intensities for the Stokes and the corresponding anti-
Stokes Raman lines is given by
kT
h
I
I fi
fio
fio
S
AS
exp
4
…… 1.18
where νfi is the magnitude of the Raman shift and νo is the frequency of the incident
beam. Thus the intensity of anti-stokes lines decreases greatly at higher vibrational
frequencies [12].
1.5.4. Raman Selection Rules
Another important form of vibrational spectroscopy is Raman spectroscopy,
which is complementary to infrared spectroscopy.The selection rules for Raman
spectroscopy are different to those for infrared spectroscopy, and in this case a net
change in bond polarizability must be observed for a transition to be Raman active.
The induced transition moment associated with a transition from the initial
state final state j and I are given by
djiij …… 1.19
Edjijiij …… 1.20
where ij is the induced dipole moment, ψi and ψj are the wave functions of the lower
and final excited vibrational states respectively [13]. Using equations (1.11) and
(1.20) we obtain
39
dqq
EdE jki
k k
jiij
0 …… 1.21
where qk is given by the equation (1.4). The first term of the above equation (1.21)
disappears for all values, except for ψi = ψj due to mutual orthogonal of the wave
functions. This first term accounts for Rayleigh scattering. The components of αo are
nonzero for all molecules and hence it follows that the Rayleigh scattering is never
forbidden. The second term shows that the transition probability depends on the
transition moment integrals involving the components of the molecular polarizability
matrix element αij which transform with the normal coordinate qk.
1.5.5. Mutual Exclusion Principle
To understand the presence and absence of lines in IR and Raman spectra, the
understanding of the Mutual exclusion principle is necessary. It gives the relation
between the symmetry of the molecular structure and their infrared and Raman
activity. The rule is: For molecules with a centre of symmetry, transitions that are
allowed in the infrared are forbidden in the Raman spectrum; conversely, transitions
that are allowed in the Raman spectrum are forbidden in the infrared. This rule
implies that if there is no centre of symmetry then some (but not necessarily all)
vibrations may be both Raman and infrared active. It should be realized that the above
rule does not imply that all transitions that are forbidden in the Raman scattering
active in the infrared.
The rule can also be explained as follows: in the infrared, only transitions
between states of opposite symmetry with respect to the centre of symmetry i are
allowed ( g ↔ u) whereas in the Raman spectrum, only transitions between states of
same symmetry with respect to i can takes place (g ↔ g, u ↔ u). This rule arises
mainly due to the fact that all the components of magnetic dipole moment μ, change
sign for a reflection at the centre of symmetry, but the components of polarizability α ,
which behave as the product of two components of induced dipole moment μ' remain
unchanged [1]. From the mutual exclusion rule, it is concluded that the observance of
Raman and infrared spectra showing no common lines implies that the molecule has
centre of symmetry. But, it has to be done cautiously because some times a vibration
may be Raman active but too weak to be observed. However, if some vibrations are
40
observed to give coincident infrared and Raman bands it is certain that the molecule
has no centre of symmetry.
1.6. INFRARED SPECTROSCOPY - INSTRUMENTATION
Infrared spectrometers generally fall under two categories namely: dispersive
and interferometer instruments. Conventional spectrometers are dispersive
instruments using prism or grating as dispersive elements. In order to overcome
shortcomings of these devices spectrometers based on interferometer technique were
developed. This type of instruments records the interferogram of the signal and later
with the help of Fourier transformation methods, it is converted into conventional
spectrum and these instruments are called FTIR spectrometers.
1.6.1 Conventional IR Spectrometers
The schematic diagram of IR spectrometer is as shown in the fig. 1.1. It
essentially consists of a source which generates light of desired wave numbers, a
monochromator (either a salt prism or a grating with finely spaced etched lines)
separates the source radiation into its constituent wavelengths and a slit selects the
collection of wavelengths that shine through the sample at any given time. In double
beam operation, a beam splitter separates the incident beam in two; half goes through
the sample, and half to a reference. The sample absorbs light according to its chemical
properties. A detector collects the radiation that comes out of the sample, and in
double-beam operation, compares its energy with that of the reference. The detector
generates an electrical signal proportional to the collected radiation and sends to an
analog recorder. A link between the monochromater and the recorder allows us to
record energy as a function of frequency or wavelength, depending on how the
recorder is calibrated [14].
Although very accurate instruments can be designed on these principles, there
are several important limitations. First, the monochromator/slit limits the amount of
signal that can be obtained at a particular resolution. To improve resolution, the slit
must be narrow but it decreases intensity. Second, there is no easy way to run multiple
scans to build up signal-to-noise ratios. Finally, the instrument must be repetitively
calibrated, because the analog connection between the monochromator position and
the recording device is subject to misalignment and wear [12].
41
Fig 1.1. The schematic diagram of IR spectrometer
1.7. FTIR SPECTROSCOPY – THEORY AND INTRUMENTATION
The Fourier theorem states that any complex wave can be viewed as a
superposition of series of sine and cosine waves. The Fourier Transform uses the
above concept to convert an interferogram into constituent vibrations. The Fourier
transformations connects two physical descriptions using the integral
deFtf
ti
)(2
1)( …… 1.22
This relates an f(t) - a function of time with F(ω) - a function of frequency. We
can also express the relation as,
…… 1.23
The F(ω) function gives the frequencies at which the signal is non-zero and
the f(t) function gives the corresponding time of the signal. Both of these functions
are suitable descriptions of a waveform or physical system. If one is known the other
function can be obtained from it. An interferogram, as mentioned earlier, is a function
dtetFF ti )()(
42
of time and hence it can be transformed into a function of frequency using the above
equation. The intensity detected by the detector of a FTIR instrument can be
mathematically expressed as
dtSxI ]2cos1)[()(0
…… 1.24
The above equation can also written as
dxxxIS
0
]2cos1)[(2)( …… 1.25
where x is the path difference. Thus using I(x) and S( )as the Fourier transform pair,
a Fourier transformation can be performed using the above equation on the
interferogram and a spectrum as a plot of percentage transmission against the wave
numbers can be obtained [12]. Since, all modern FT instruments are computer-
interfaced, a small computer program will do the transformation in a matter of
seconds (or less) and the output of the detector is digitized. The Discrete Fourier-
Transform (DFT) is an algorithm for doing the transform with discrete data, which
was used previously. The DFT is an order N2 calculation, meaning that the number of
multiplications is equal to the square of the number of data points. This algorithm has
been supplanted by Fast Fourier-Transform (FFT) algorithms, which reduce
redundancies and take much less computer time. The order of this calculation is
Nlog(N).
A Fourier transform infrared spectrometer consists of a source, an
interferometer (an addition which makes the instrument unique) and a detector. The
interferometer arrangement of FTIR is shown in the Fig. 1.2.
1.7.1. Interferometers
The interferometer is the heart of any FTIR instrument. It is this part which
analyses the infrared radiation and hence enables us to generate a spectrum. The
classical Michelson Interferometer involves a beam splitter – which sends the light in
two directions at right angles. One beam goes to a stationary mirror then returns back
to the beam splitter. The other goes to a moving mirror. The motion of the mirror
produces path difference with respect to the stationary mirror. When the two beams
43
meet again at the beam splitter, they recombine, but the difference in path lengths
creates constructive and destructive interference producing an interferogram.
Fig. 1.2. The interferometer arrangement of FTIR
The recombined beam passes through the sample. The sample absorbs all the
different wavelengths characteristic of its spectrum, and this subtracts specific
wavelengths from the interferogram. The detector now reports variation in energy
versus time for all wavelengths simultaneously. A laser beam (usually a He-Ne Laser
having a wavelength near 632.8nm) is superimposed to provide a reference for the
instrument operation i.e for the measurement of path difference.
1.7.2. Sources
Most of the FT instruments use a heated ceramic source. The composition of
the ceramic and the method of heating vary from instrument to instrument but the idea
is always the same - a ceramic is heated to suitable temperature to obtain IR radiations
of required range. Modern FTIR instruments use a conducting ceramic or a wire
44
heater coated with ceramic as source. The advantage with the heated ceramics is that
at high temperature they emit IR radiation of all wavelengths with reasonable
intensity. Care should be taken to maintain the temperature of the heater at constant
value and this is achieved by monitoring the source output, and using part of the
output signal as feedback to control the electrical power [15].
In recent years, tunable dye lasers are emerging as very precise sources with
resolution of 10-6
cm-1
. The wave number range of tunable dye lasers is however
restricted and hence their applicability is limited. The infrared spectroscopic analysis
uses the carbondioxide laser and Semiconductor lasers.
1.7.3. Detectors
Various detectors are encountered in FTIRs and the majority of which are
photo resistors i.e. they have a very high resistance in the dark and this falls as light
falls on them. The most sensitive are the Ge and InGaAs semi-conductor devices and
others are Golay cell, thermocouple, and pyroelectric detectors. Some of detectors
give adequate performance (an acceptable useful S:N ratio) at room temperature but
others must be cooled. Cooling detectors invariably reduces the amount of noise they
develop, but unfortunately cooling shifts the absorption edge towards shorter
wavelength. In an FTIR, one normally finds a TGS or similar detector for ordinary
use. If better results are required, one then resorts to the use of a cryogenically cooled
mercury cadmium telluride semi-conductor detector [15]. Thus, these are invariably
used in infrared microscopy and very often in diffuse reflection experiments.
1.7.4. Advantages of FTIR instruments
All of the source energy gets to the sample, improving the inherent signal-to-noise
ratio. This is usually called as energy advantage.
Resolution is limited by the design of the interferometer. The longer the path
of the moving mirror, the higher the resolution. Even the least expensive FT
instrument provides better resolution than all the best CW instruments were
capable of.
During spectral acquisition all the frequencies were recorded simultaneously
during the whole period of the detection and it is called multiplexing. This
method gives rise to multiplex advantage or Fellgett advantage [16].
45
The radiation from the source reaching the detector in an interferometer is not
limited by the entrance and exit slits as in a dispersive spectrometer thus the
brightness of the detected signal increases enormously which is called as
Jacquinot’s advantage or throughput advantage [17].
The digitization and computer interface allows multiple scans to be collected,
also dramatically improving the signal-to-noise ratio.
Most of the computer programs today allow further mathematical refinement
of the data: you can subtract a reference spectrum, correct the baseline, edit
spurious peaks or otherwise correct for sample limitations.
1.8. RAMAN SPECTROSCOPY - INSTRUMENTATION
The problem facing the development of Raman spectroscopic instrumentation
is the inherent weakness of the inelastic scattering. Therefore to produce a detectable
Raman signal a high-powered light source is required. In addition, Raman requires a
spectrometer with a very high degree of discrimination against the Rayleigh elastic
scattered light. Finally, since very few Raman photons are generated, the detection
system must be very sensitive to detect the Raman signal over the dark noise
background [12].
It was not until early sixties that the Modern renaissance took place with the
development of commercial continuous wave visible lasers. In recent years, micro
electronics revolution has further improved the technique with the developments of
stepper motor drives, photon counting devices, digital data acquisition techniques and
computer data processing and provided the Chemists and physicists a technique which
is more useful and versatile than infrared spectroscopy [18]. Some of the advantages
of Raman over infrared technique are: Raman spectroscopy is a scattering process, so
samples of any size or shape can be examined; Very small amounts of materials can
be examined; glass and closed containers can be used for sampling; fiber optics can be
used for remote sampling; aqueous solution can be analyzed. Recent development in
Fourier transform instrumentation now made these advantages available to
researchers.
1.8.1. Sources
Before the inventions of lasers, radiations emitted by the mercury arc,
especially at 435.8 and 404.7 nm, have been used for exciting Raman spectra [3].
46
Today, most types of lasers, like continuous wave (CW) and pulsed, gas, solid state,
semiconductor lasers, etc., with emission lines from the UV to the near-IR region, are
used as radiation sources for the excitation of Raman spectra. Especially argon ion
lasers with lines at 488 and 515 nm are presently employed. Near-IR Raman spectra
are excited mainly with a neodymium doped yttrium-aluminum garnet laser
(Nd:YAG), emitting at 1064 nm [12]. The main advantages of a laser light source for
Raman spectroscopy are: (i) directionality which makes focusing simple, (ii)
coherency which enhances the usable power, (iii) intensity which yields a high power
density and (v) monochromatic which eliminates multiple Raman lines.
1.8.2. Conventional Raman Spectrometers
Raman spectroscopy until recently relayed on conventional single-channel
spectrometers. They are designed to generate Raman signal and differentiate it from
the unwanted stronger Raleigh-scattered light from the weaker Raman signal and
count the Raman photons. These conventional Raman spectrometers are the simplest
and are readily available for routine work when interfering fluorescence is not a major
problem. It is essentially consists of a powerful laser irradiating in the visible region,
an illuminating chamber for the sample, a high performance light dispersion system to
resolve the more intense, elastically scattered light from the weak, in-elastically
scattered Raman signal, a light detection and amplification system capable of
detecting weak light levels and a recorder [3]. The Schematic diagram of Raman
spectrometer in as shown in fig 1.3
Fig. 1.3. The Schematic diagram of Raman spectrometer
47
1.9. FOURIER TRANSFORM RAMAN SPECTROMETER
FT-Raman spectrometers are designed to eliminate the fluorescence problem
encountered in conventional Raman spectroscopy [19]. The FT-Raman instrument has
the following components: (1) A near IR Laser excitation source, generally an Nd:
YAG laser working at 1.06 μm. (2) An interferometer equipped with an appropriate
beam splitter, made of glass, and a detector for the near –IR region. The detector is
usually InGaAs or Ge semiconductor detector. (3) A sample chamber with scattering
optics that match the input port of the Fourier transform instrument. (4) An optical
filter rejection of the Raleigh –scattered light. A schematic representation of such FT-
Raman instrument is shown in fig 1.4.
Utilizing an excitation frequency well below the threshold for any
fluorescence process eliminates fluorescence. To focus and align the invisible
Nd:YAG laser beam a visible He:Ne laser beam is co-aligned with the Nd:YAG
beam. Another method of optical alignment can be realized by using fiber optics [15].
With fiber-optic components, optical alignment is virtually eliminated which allows
rapid switching from one sample to another.
Fig. 1.4. Schematic representation of FT-Raman
One of the advantages of FT instrument is that it can collect all the scattered
radiation over the entire range of frequencies simultaneously during the whole period
of the detection and it is called multiplexing. This becomes a disadvantage, as the
intense Rayleigh line is the primary source of noise. Multiplexing redistributes the
48
noise associated with Rayleigh line across the entire spectrum by the FT-process and
this is called as multiplex disadvantage [20]. Interferometer can be combined with
Rayleigh line filters (notch filters) in order prevent the consequences of the multiplex
disadvantage. The Rayleigh line filters minimizes the amount of Rayleigh scattered
light entering the interferometer [21] and is essential for FT-Raman spectroscopy.
1.10. DUAL INSTRUMENTS
Due to the rapid developments in the instrumentation techniques, nowadays
both infrared and Raman spectrometers are incorporated into a single instrument
assembly and available commercially as a single package. The main advantages of
such instruments are: switching over from one technique to other is simple; they are
compact, and comparatively cheaper. Bruker's FTIR spectrometers IFS 66/S fitted
with FRA 106/S, Thermo Nicolet‘s Nexus/Magna FTIR & FTR systems, ABB
Bomen MB157 series these are some of the commercially available dual instrument
packages. Fig 1.5 shows the schematic diagram of Bruker's FTIR spectrometers IFS
66/S fitted with Raman module FRA 106/S, which is used in the present work.
Fig. 1.5. Schematic diagram of Bruker's IFS 66V with FRA 106/S
49
1.11. SAMPLE HANDLING TECHNIQUES
1.11.1. Infrared Spectroscopy
Recording of IR spectra of sold sample are more difficult because the particles
reflect and scatter the incident radiation and therefore transmittance is always low.
Three different techniques are employed commonly in recording the spectra.
1.11.1.1. Mull Technique
In this method, a slurry or mull of the substance is prepared by grinding it into
a fine powder and dispersing it in the mulling agent such as nujol,
hexachlorobutadiene and perfluorokerosene. It is smeared between two cell windows
which are then held together. The mull spreads out a s very thin film and the window
setup is placed in the path of the IR beam. The main disadvantage of this technique is
the interference due to the absorption bands of the mulling agent. For best results, the
size of the sample particles must be less than that of the wavelength of the radiation
used.
1.11.1.2. Pellet technique
The first in this method is to grind the sample very finely with potassium
bromide. The mixture is then pressed into transparent pellets with the help of suitable
dies. This is then placed in the IR beam in a suitable holder. This method has the
following advantages.
(i) Absence of interfering bands
(ii) Lower scattering losses
(iii)Higher resolution of spectra
(iv) Possibility of storage for future studies
(v) Ease in examination
(vi) Better control of concentration and homogeneity of sample
The main disadvantage is that anomalous spectra may result from physical and
chemical changes induced during grinding. A common change of this type is the
absorption of water from atmosphere.
50
1.11.1.3 Solid film
Spectra of solids may also be recorded by depositing a thin film of a solid on a
suitable window material. A concentrated layer of the solution of the substance is
allowed to evaporate slowly on the window material forming a thin uniform film.
Thickness can be adjusted by changing the concentration of the solution. Interference
caused by multiple reflections between parallel surfaces affect the accuracy of
measurements.
1.11.2. Raman Spectroscopy
Raman spectrometer but the important thing to be taken care of while
preparing sample is that they should be dust free can study gases, liquids and solids.
Glass is almost transparent in the Raman frequency region and thus samples in
different phases can be measured in glass or silica containers or capillaries [3].
1.11.2.1. Liquids
Liquids may be examined neat or in solution and normally liquids of about 0.3
ml enclosed in glass or silica containers or capillaries may be required for obtaining
good spectrum. Even though water cannot be used as solvent in IR studies, in the
Raman studies water is one of a good solvent. Thus spectra of aqueous solutions can
be easily studied and also spectra of water-soluble biological material can be easily
recorded [12].
1.11.2.2. Solids
Solid as poly crystalline material or as a single crystal can be studied with the
help of Raman technique. Solvents or alkyl halides or mull are not required for
recording the spectra. Solids in the form of fine powder enclosed in a glass or silica
fiber can be used. When the measurement is made as a single crystal, depending on
the orientation of the crystal axis and polarization of the incident radiation the spectra
may vary. Raman spectra can also be recorded for adsorbed species. Samples can also
be studied using the Raman technique at various pressures and temperatures [3].
To record the polarized spectra of single crystal, it is properly cut, polished,
mounted on a goniometry head and the scattered radiation is collected in the 90o
geometry. To investigate and represent the spectra for different orientations, the
following notations i(kl)j, called Porto‘s notation is normally used. The symbols I
51
and j represent the direction of propagation of the incident and scattered radiations
whereas the symbols k and l represent the direction of the electric vector of the
incident and scattered beams respectively. The symbols within the bracket also define
the components of the scattering tensor being measured [22].
1.12. APPLICATION OF GROUP THEORY TO VIBRATIONAL SPECTROSCOPY
Symmetry is a visual concept as reflected by the geometrical shapes of
molecules such as ammonia, benzene, etc. The link between molecular symmetry and
quantum mechanics is provided by the group theory. In vibrational spectroscopy,
group theory can be effectively used for: (1) determining the symmetry types of
normal mode vibrations of the molecule, (2) predicting the infrared and Raman
activity of a normal mode of vibration of a particular symmetry types and (3)
simplification of method of obtaining the relation between force constants and
vibrational frequencies [10]. The group theory was used, first time by Wigner (1930),
for the study of molecular vibrations [25].
The molecular symmetry is systematized quantitatively by introducing the
concept of ‗symmetry operation‘. A symmetry operation transforms the molecular
framework into an equivalent configuration or identical configuration. A symmetry
element is a geometrical entity such as point, an axis or a plane about which one or
more symmetry operations are carried out. Five kinds of fundamental symmetry
operations are utilized in specifying molecular symmetry. (i) Proper axis of symmetry
(Cn) – it is rotation once or several times by an angle θ = (2π/n) about the axis, (ii)
Plane of symmetry ( σ) – one or more reflections in the plane, (iii) Improper axis of
symmetry (Sn) rotation about an axis followed by reflection in a plane perpendicular
to the rotation axis, (iv) centre of symmetry (I) – inversion of all atoms through the
centre of symmetry, (v) identity element (E) – rotation of the molecule through 0° or
360° which leaves the molecule unchanged [10].
All the symmetry operations present in a molecule form a group and such
groups are called point groups. In a point group all the elements of symmetry present
in the molecule intersect at a common point and this point remains fixed under all the
symmetry operations of the molecule Although theoretically large numbers of such
groups are possible most molecules falls under dozen point groups. Some of the
common molecular point groups are classified as follows
52
Groups with no Cn axis: C1,Cs,Ci
Groups with a single Cn- asis : Cn, Cnh, Cnv and Sn
Groups with Cn axis and nC2 axes : Dn, Dnh, Dnd
Groups with more than one Cn axis : T, Th, Td, O1, Oh
Any symmetry operation about a symmetry element in a molecule involves the
transformation of a set of coordinates x, y & z of an atom into a set of new
coordinates x', y' & z‘. The two sets of coordinates can be related with the help set of
equations or a matrix. The matrix is referred as the transformation matrix and a
specific transformation matrix can represent each symmetry operation. Such matrices
for the various symmetry operations of a point group form a representation.
Representations can be divided into two types: Reducible representation and
irreducible representation. If the resulting matrices can be blocked into smaller
matrices, then the representation T is called a reducible representation. If it is not
possible to find a similarity transformation matrix, which will reduce the matrices of
representation T, then the presentation is said to be irreducible [10]. Every point
group consists of a certain number of irreducible representations. The characters of
matrices in the different irreducible representations of a point group can be listed in a
table known as character table. Character table plays a vital role in solving problems
such as molecular vibrations. The character table for a point group can be constructed
with the knowledge of properties of irreducible representations.
The construction of character table requires practice, expertise and knowledge
of theorems in group theory such as orthogonal theorems, theorems of representation
theory, etc. Without elaborating the procedure, the character table for Cs point groups
[13] Cs (the molecules chosen for the present thesis falls under these groups) is given
below:
In the tables A and B represents representation which is symmetric and anti-
Symmetric with respect to the main axis of rotation, ', '' –represents symmetric and
anti-symmetric with respect to a plane of symmetry. The last two columns of
character table list the infrared and Raman activity of the particular species.
Polarizability components are listed for Raman activity and translational and
rotational components are listed for infrared activity.
53
Based on the character table the selection rules for infrared and Raman activity
can be obtained with the help of group theory and quantum mechanics.
Table 1.1
Character Table for Cs point Group
Cs E h Activity
Infrared Raman
A’ 1 1 Tz, Ty, Rz
Tz, Rx, Ry
αxx, αyy, αzz, αxy
αyz, αxz A’’ 1 -1
1.13. INFRARED-ACTIVE VIBRATIONS
The point group of a molecule will have a definite number of symmetry
operations. These operations are of two types: Proper Rotations –a rotation through
an angle ± υ about some axis of symmetry and Improper Rotations – rotation
followed by a reflection in a plane perpendicular to the axis of rotation. A quantity
called character is necessary for the determination of the selection rules and number
of fundamentals of each vibration type. For a given vibration type there is a separate
character for each class of symmetry operations. These characters can be found from
the character table.
For a mode of vibration to be infrared active, the vibrational motion of that
mode must give rise to change in dipole moment. The character of the dipole moment
for the operation R, M(R) is
M R = 1 + 2 cos ϕ for proper rotation
M R = −1 + 2 cosϕ for improper rotation
where is the angle of rotation during the operation R.
The number of vibrations Ni(M) contributing to a change in dipole moment is
given by
𝑁𝑖 𝑀 = 1
𝑁𝑔Σ𝑛𝑒𝜒𝑀(𝑅)𝜒𝑖(𝑅) …… 1.26
where Ng is the number of elements in a point group, M the number of elements in
each class, i is the character of the vibration type, Ni the number of times the
54
character i of the vibration appear in M. The value of Ni if equal to zero than that
vibration type is infrared inactive otherwise active [13].
1.14. RAMAN ACTIVITY
For a vibration to be Raman active, the vibrational motion of that mode must
give rise to a change in polarizability. The character of the polarizability α(R) for the
operation R is given by
χα R = 2 + 2 cosϕ + 2 cosϕ for proper rotation
α R = 2 − 2 cosϕ + 2cos2ϕ for improper rotation
where is the angle of rotation during the operation R.
The number of vibrations Ni(M) contributing to a change in dipole moment is
given by
𝑁𝑖 𝛼 = 1
𝑁𝑔Σ𝑛𝑒𝜒𝛼(𝑅)𝜒𝑖(𝑅) …… 1.27
1.15. QUANTUM MECHANICAL HARMONIC FORCE FIELDS
Harmonic force fields of polyatomic molecules play an important role in
several branches of molecular spectroscopy. The prediction and interpretation of
vibratinoal frequencies, infrared and Raman intensities, vibration structure in UV and
photoelectron, effects on molecular geometries, dipole moment, rotational constants
and polarizability makes it more effective in quantum mechanical calculations.
If the number of parameters is increased, the fitting procedure often converges
to an unphysical solution. This can only be counteracted by increasing the number of
independent experimental observables. Moreover, unless a completely general
harmonic force field can be used, which is possible only for the smallest molecules,
there is always an arbitrariness in the choice of the terms retained [25].
The primary aim of vibrational analysis is to theoretically calculate the
vibrational frequencies of a molecule from the force constants. But the vibrational
frequencies are easily observable from IR and Raman spectra. With some difficulty it
is possible to compute the force constants from the observed frequencies. To achieve
this, the observed vibrational frequencies should be first correctly assigned to the
symmetry species of a molecular point group. Force constants can also be calculated
55
from molecular data like Coriolis coupling constants, centrifugal distortion constants
and mean amplitude of vibration.
For many molecules it is not possible to evaluate all the force constants from
the experimental data and it becomes necessary to reduce the force constants. This is
accomplished by making assumptions about the nature of the potential energy
function. Some of the important restrictive force fields are:
1.15.1. Central Force Field (CFF)
This force field function accounts only for the forces in molecules among the
atoms along the lines joining them. According to this filed, the potential energy is the
quadratic function of change in distance between the nuclei. It fails to account for
angle forces, bending vibrations and out-of-plane vibrations [26]. The harmonic
frequencies from this theory differs more from the experimental frequencies.
1.15.2. Valence Force Field (VFF)
The Valence Force Field (V.F.F.) is the simplest model in which only the
diagonal force constants in a valence co-ordinate are assumed to differ from zero. So,
this force field considers only the forces associated with valence bonds. In this
approximation forces between the non-bonded atoms are not considered. Here the
number of force constants that have to obtain is usually less than the number of
observed frequencies [27].
1.15.3. Simple General Valence Force Field (SGVFF)
This force field is superior to central force filed which the extension of
valence force field and it takes into account the interaction force constants. This
method employs stretching and bending force constants and also the interaction force
constants between them [2]. But this force excludes the forces between non-bonded
atoms. The number of interaction of force constants becomes increasingly larger as
the molecule is large. Some of this difficulty can be overcome by neglecting
interaction terms of frequencies of the same symmetry, which are widely separated
from one another. This method also neglects the force between the non-bonded atoms.
This is one of the effective potential functions employed for normal coordinate
calculations. The simple general valence force field (SGVFF) has been shown to be
very effective in normal coordinate analysis of hydrocarbons, peptides, amides and
pyrimidine bases [28]. The potential energy function in this model is expressed as
56
𝑉 =1
2 𝑓𝑟𝑖
𝑟𝑖 2 +𝑖 𝑓𝛼𝑖
𝛼𝑖 2
𝑖 …… 1.28
where r and a are the changes in bond lengths and bond angles respectively, fr. and fα
are the respective stretching and bending force constants.
1.15.4. General Valence Force Field (GVFF)
In this force field, the interaction constants such as stretch, bend-bend, stretch
bend were introduced in the simple valence force field potential functions in order to
reproduce accurate vibrational frequencies. In this model the potential energy
function, which includes all interaction terms in addition to the valence forces is given
by the expression
𝑉 =1
2 𝑓𝑟𝑖
𝑟𝑖 2 +𝑖 𝑓𝛼𝑖
𝛼𝑖 2
𝑖 + 𝑓𝑟𝑖𝑟𝑗 𝑟𝑖𝑟𝑗 𝑖 + 𝑓𝛼𝑖𝛼𝑗
𝛼𝑖𝛼𝑗 𝑖≠𝑗 +
𝑓𝑟𝛼 𝑟𝛼 𝑖 ...… 1.29
where r and α are the changes in bond lengths and bond angles, respectively. The
force constants fr , fα , frr, fαα, frα refers to principal stretching, bending force constants,
stretch-stretch, bend-bend and stretch-bend interactions respectively. Since the force
constant is directly transferred from one molecule to other, it is very convenient force
filed for usage.
1.15.5. Urey-Bradley Force Field (UBFF)
Basically it is a GVFF superimposed with some repulsive force constants
between non-bonded atoms. It is the combination of central force field and valence
force field. This force field includes stretching (K) , angle bending (H), torsion (Y)
and repulsive force constants (F). In UB model [29], in addition to conventional bond
stretching and angle bending force constants, all interaction between bond stretching
coordinates, bond stretching and angle bending coordinates are defined interms of
interactions between non bonded atoms . Pure UBFF is unsatisfactory for polyatomic
molecules except tetrahedral (Td) molecules and it has been found necessary to
introduce additional valence type interactions.
𝑉 =1
2 𝐾𝑖𝑗 𝑟𝑖𝑗
2+
1
2 𝐻𝑖𝑗𝑘 𝛼𝑖𝑗𝑘
2+
1
2 𝑌 𝑡𝑖𝑗𝑘 +
1
2 𝐹𝑖𝑗 𝑅𝑖𝑗
2
…… 1.30
57
where r, α, t and R are the changes in bond lengths, bond angles, angle of internal
rotation and distance between non-bonded atom pairs, respectively. Shimanouchi [27]
explained the validity of this force field. In this model, the Valence Force Field was
applied between non-bonded nuclei. The advantages of this force field are:
Potential energy is explained with some parameters
Transfereable force constants within the related molecules
Determination of force constants for complex molecules are also possible
Due to the inadequacy in the exactness of determination frequencies, this force
field was modified and referred as Modified Urey Bradley Force Field.
1.15.6. Oribital Valence Force Field (OVFF)
Another model, which has been less widely applied, is the Orbital Valency
Force Field (O.V.F.F.) developed by Linnett et al. [30], in which the force constants
controlling displacements in the angle bending co-ordinates are expressed in terms of
parameters describing changes in the overlap of the atomic orbitals that make up the
bonds. This force field is a modified version of valance force field in which the same
interaction constants were used for out-of plane bending vibrations as in the in-plane
bending vibrations. In this field, it is assumed that the bond forming orbitals of an
atom X are at definite angles to each other and a most stable bond is formed when one
of these orbitals overlaps the bond forming orbitals of another atom Y to the
maximum extent possible. If now Y is displaced perpendicular to the bond, a force
will be set up tending to restore it to the most stable position. The potential energy
function is expressed as,
𝑉 =1
2𝐾 𝑟𝑖
2 + 𝐾𝛼′ 𝛽𝑖
2 + 𝐴 𝑅𝑗𝑘 2
𝑗𝑘
− 𝐵 𝑅𝑗𝑘 + 𝐵′ 𝑟𝑖
𝑗𝑘
…… 1.31
where r , R and i are the changes in bond lengths, the distance between non-bonded
atom pairs and the angular displacement respectively. The symbol K, Kα’, B and A
stands for the stretching, bending, non-bonding and repulsion force constants
respectively.
58
1.15.7. Hybrid Orbital Force Field (HOFF)
In this type, the prediction of stretch-bend interaction force constants was
explained on the basis that the angle deformation leads to a change in hybridization in
the bonding orbitals of the central atom. Interaction force constants between bond-
stretching and angle-bending co-ordinates in polyatomic molecules changes the
hybridization due to orbital-following of the bending co-ordinate and also changes
bond length. According to Mills [31], a method is described for using this model
quantitatively to reduce the number of independent force constants in the potential
function of a polyatomic molecule, by relating stretch-bend interaction constants to
the corresponding diagonal stretching constants. The model is applied to the
tetrahedral four co-ordinated carbon atom and to the trigonal planar three coordinated
carbon atom. The relation between stretching force constants is given by the
expression
𝐹𝑖𝑗 =−𝛿𝑅𝑖
𝛿𝜆 𝑖 𝛿𝜆 𝑖
𝛿𝛼 𝑖 𝐹𝑖𝑗 …… 1.32
where, Ri and αi refers to internal stretching and bending coordinates respectively. λ i
is the hybridization parameter associated with Ri.
1.16. MOLECULAR ORBITAL THEORY
In many body electronic structure calculations, the nuclei of the
molecules or clusters are treated as fixed, in abeyance with the Born-
Oppenheimer approximation, and that generate a static potential V in which
electrons are moving. A stationary electronic state is described by a wave
function ψ(r1, r2 ….rn), satisfying the many electron time- independent
Schrödinger Equation
𝐻 Ψ = 𝑇 + 𝑉 + 𝑈 𝜓 = − ħ2
2𝑚
𝑁𝑖 ∇𝑖
2 + 𝑈𝑁𝑖>𝑗 𝑟 𝑖 , 𝑟 𝑗 Ψ = 𝐸Ψ …… 1.33
where H is the Hamiltonian, E is the total energy, T is the kinetic energy, V is the
potential energy from the external field due to positively charged nuclei and U is the
electron- electron interaction energy. The operators T and U are called Universal
operators as they are the same for any N- electron system, while V is system
dependent.
59
In theory, solving the Schrödinger equation yields wave functions ψ that
describe the system fully. Unfortunately the Schrödinger equation cannot be solved
analytically for many body systems, so approximations are to be made. Typical
solution algorithms therefore is to fix the atomic positions and then solve the
Schrödinger equation for a specific value of the atomic coordinates ri. This is repeated
many times to obtain the energy of the system as a function of the atomic coordinates.
The complicated many particle equations are not separable into simpler single
particle equations because of the interaction term U. However the single-particle
approximation simplifies calculations notably, as it does not account for electron-
electron interaction. This method is known as Hartree-Fock method.
1.16.1. Hartree Fock Method (HF)
Hartree Fock method is the basis of molecular orbital (MO) theory, which
explains the motion of electron by a single particle function which does not depend
explicitly on the instantaneous motions of the other electrons. Hartree-Fock theory is
the Self Consistent Field (SCF) method often provides better approximations to the
electronic Schrödinger equations for multi electron atom or molecule as described in
Born-Oppenheimer approximation. In HF method, the approximations were
conducted by iterations which give rise to SCF method.
Ab initio calculations of vibrational spectra are nowadays feasible for small,
medium and large size molecules due to the development of analytical methods for
computing energy derivatives. Larger molecules are normally treated at the harmonic
level using Hartree-Fock calculations with basis sets of moderate size. The direct
calculation of vibrational frequencies by ab initio computations can be of considerable
help in the interpretation of experimental vibrational spectra. In larger molecules it is
virtually impossible to reliably assign vibrational fundamentals without input from
theory. Theory can also suggest frequencies that can be used as ―fingerprints‖ for the
presence of particular conformers, isomers, or compounds. The computed normal
modes can be used to estimate IR and Raman intensities from dipole and
polarizability derivatives, as well as vibrational averaging effects on molecular
geometries and properties. Comparison of calculated and experimental vibrational
spectra has become one of the principal means of identifying unusual molecules [25].
60
In HF approximation, the full molecular wave function is a function of
electrons and coordinates of each nucleus. Here, the relativistic effect is completely
neglected and the finite basis set is assumed to approximately complete. The solution
is assumed to be a linear combination of a finite number of orthogonal basis
functions. Each energy eigen function is explained by a single Slater determinant. The
electron correlation is completely neglected for the electron of opposite spin but the
electrons with parallel spin were taken into consideration. The Hartree Fock operator
is expressed as follows :
𝐹 Φ𝑗 1 = 𝐻 𝑐𝑜𝑟𝑒
1 + 2 𝑗 𝑖 1 − 𝐾 𝑗 (1)
𝑛
2
𝑗 =1 …… 1.34
Where
𝐹 Φ𝑗 1 …… 1.35
is the one-electron Fock operator generated by the orbitals ϕj ,
𝐻 𝑐𝑜𝑟𝑒
1 = −1
2∇1
2 − 𝑍𝑎
𝑟1𝑎𝑎 …… 1.36
is the one-electron core Hamiltonian,
𝑗 𝑖 1 …… 1.37
is the Coulomb operator, defining the electron-electron repulsion energy due
to the orbital of the jth electron,
𝐾 𝑗 1 …… 1.38
is the exchange operator, defining the electron exchange energy.
Finding the Hartree–Fock one-electron wave functions is now equivalent to
solving the eigenfunction equation:
𝐹 Φ𝑗 1 = ∈𝑖 ∅𝑖 1 , …… 1.39
where ∅𝑖(1) are a set of one electron wave funcatins called the Hartree-Fock
molecular orbitals.
Moreover, this method uses Slater type of orbitals (STO) which has the form
𝑺𝑻𝑶 = 𝝃𝟑
𝝅𝟎.𝟓 𝒆−𝝃 …… 1.40
61
where ξ is orbital exponent which reflects the spatial extent of the orbital. The HF
method is also called, especially in the older literature, the self-consistent field
method (SCF) as the solutions to the resulting non-linear equations behave as if each
particle is subjected to the mean field created by all other particles. The equations are
almost universally solved by means of an iterative, fixed-point type algorithm.
It is important to remember that STO leads to very tedious calculations. Thus
S.F.Boys developed an alternative type of orbital called Gaussian type orbital
(GTO) for calculations, which are of the form
𝑮𝑻𝑶 = 𝟐𝝌
𝝅𝟎.𝟕𝟓 𝒆 −𝝌𝒓𝟐 …… 1.41
The difference between STO and GTO lies in the spatial coordinate r. The
GTO has square of r so that the product of one Gaussian gives another Gaussian.
Ultimately it is found that more the combination of Gaussians, more the accuracy of
the equations.
This type Restricted Hartree Fock method is applicable for molecules having
closed shell system with all orbitals are doubly occupied. But in the case of Open
shell systems, where some of the electrons are not paired, can be dealt with either
restricted open-shell Hartree–Fock (ROHF) or Unrestricted Hartree–Fock (UHF)
method.
The general form of Hartree Fock equation is
−1
2∇2 + 𝑉𝑒𝑥𝑡 𝑟 +
𝜌 𝑟′
𝑟 − 𝑟′ 𝑑𝑟′ ∅𝑖 𝑟 + 𝑉𝑥 𝑟, 𝑟′ ∅𝑖 𝑟′ 𝑑𝑟′ =∈𝑖 ∅𝑖 𝑟
…… 1.42
where the non-local exchange potential Vx is such that
𝑉𝑥 𝑟, 𝑟′ ∅𝑖 𝑟′ 𝑑𝑟 ′ = − ∅𝑗 𝑟
𝑁𝑖 ∅𝑗
∗ 𝑟′
𝑟−𝑟′ ∅𝑖 𝑟′ 𝑑𝑟 ′ …… 1.43
Additional modifications have been implemented with the previous Hartree-
Fock approximations to account for electron correlation. There are three methods,
among several, that account for electron correlation: Configuration-Interaction,
Moller-Plesset perturbation, and Density Functional Theory.
62
1.16.2. Configuration Interaction (CI) Method
In Hartree – Fock, it is assumed that a wave function can be written as a
product of one – electron wavefunctions. In Configuration interaction (CI) method
(2), some multi – electron wave functions are added back into the basic Hartree –
Fock method to account for their coordinated motion. This is done systematically by
constructing multi – electron wave functions as a sum of the Hartree – Fock
wavefunctions for different electronic states. A slight improvement of this method is
CISD where one or two electron is assumed to be in excited states. S and D stand for
single and double.
1.16.3. Couple Cluster (CC)
This method is conceptually similar but differs mathematically in the
construction of multi electron wave functions. Unlike in CI, CC uses an exponential
ansatz to guarantee size extensively of the solution. The calculations may also include
single, double and triple excitations, leading to CCSD and CCSDT methods.
1.16.4. Moller – Plesset Perturbation Theory (MPPT)
The alternative to multi – electron wavefunctions is to choose an
approximation for the correlation energy. In this theory, an additional term is added to
the total energy and is treated as a perturbation. The perturbed wave function and
perturbed energy are expressed as a power series to the nth order (MPn) such as MP2,
MP4 etc, although it is not always convergent with increasing the power n.
1.16.5. Density Functional Theory (DFT)
For the past 30 years density functional theory has been the dominant method
for the quantum mechanical simulation of periodic systems. In recent years it has also
been adopted by quantum chemists and is now very widely used for the simulation of
energy surfaces in molecules [32].
DFT is increasingly used in the ab initio calculation of molecular properties.
DFT codes are increasing versatility, efficiency and availability. DFT calculations
exhibit an attractive ratio of accuracy of computational cost. In particular, DFT is
increasingly used in calculating harmonic force fields and vibratinoal frequencies.
The introduction of analytical gradient techniques [33] permitted numerically accurate
calculations. Very recently, the introduction of analytical second-derivative
63
techniques [34] has substantially improved the efficiency of calculations. DFT
frequencies are much more accurate than SCF frequencies and are comparable in
accuracy to MP2 frequencies. DFT can obviously be expected to become
increasingly the method of choice.
The accuracy of DFT calculations depends on the density functional adopted.
At this time, many density functional are available varying substantially in
sophistication and accuracy. Broadly, they can be grouped into three classes (1) local
(2) nonlocal and (3) hybrid. Local functional for the exchange and correlation
functional were the first to be used. Nonlocal (gradient) corrections were then added.
Most recently, a number of functional have been introduced in which some
percentage of exact (HF) exchang
e is admixed. In the most sophisticated of these hybrid functional, based on the
adiabatic connection method of Becke [35], the values of three weighting factors are
determined by optimizing the fit of predicted properties to experiment [36].
The main aim of quantum mechanical calculations is to solve the Schrödinger
equation and determine the 3N dimensional wavefunction in order to compute the
ground state energy. The ground state energy is completely determined by the
diagonal elements of the first order density matrix which is termed as the charge
density. There are two theorerms which lead to the fundamental statement of Density
Functional Theory (DFT). They are
Hohenburg-Kohn Theorems
Kohn and Sham
1.16.5.1. Hohenburg – Kohn theorem
According to Hohenburg-Kohn [37] first theorem, the electron density
determines the external potential (to within an additive constant). If this statement is
true then the electron density uniquely determines the Hamiltonian operator.
According to second theorem (variation principle), For any positive definite
trial density, t, such that
ρt r dr = N then E ρt ≥ Eo …… 1.44
where E[t] is the energy is the functional of density and Eo is the ground state
energy.
64
These two theorems lead to the fundamental statement of density functional
theory.
From the form of the Schrödinger equation it can be seen that the energy
functional contains three terms – the kinetic energy, the interaction with the external
potential and the electron-electron interaction. But, the kinetic and electron-electron
functionals are unknown. It has the general form
E ρ = T ρ + Vext ρ + vee (ρ) …… 1.45
1.16.5.2. Kohn-Sham Equation
Kohn and Sham [38] proposed an approximation for the kinetic and electron-
electron functional. According this Kohn – Sham the above equation is rearranged as
E ρ = T ρ + Vext ρ + Vee ρ + Exc (ρ) …… 1.46
where Exc is simply the sum of the error made in using a non-interacting
kinetic energy and the error made in treating the electron-electron interaction
classically. Writing the functional explicitly in terms of the density built from non-
interacting orbitals and applying the variational theorem, they proposed the following
equation
−1
2∇2 + Vext r +
ρ r′
r−r′ dr′ + Vxc (r) ∅i r =∈i ∅i r …… 1.47
where Vxc is a multiplicative potential which the functional derivative of the exchange
correlation energy with respect to the density
Vxc r = δExc ρ
δρ …… 1.48
This non-linear equations describes the behavior of non-interacting electors in
an effective local potential. These K-S equations have the same structure as the
Hartree –Fock equatins with non local exchange potential replaced by the local
exchange correlation potential Vxc. As stated above Exc contains an element of the
kinetic energy and is not the sum of the exchange and correlation energies as
explained in Hartree-Fock and correlated wavefunction theories.
Moreover, the Kohn-Sham approach achieves an exact correspondence of the
density and ground state energy of a system consisting of non-interacting Fermions
and the ―real‖ many body system described by the Schrödinger equation. The
65
correspondence of the charge density and energy of the many-body and the non-
interacting system is only exact if the exact functional is known. In this sense Kohn-
Sham density functional theory is an empirical methodology.
1.16.5.3. Local Density Approximation (LDA)
The generation of approximations for Exc has lead to a large and still rapidly
expanding field of research. There are now many different flavours of functional
available which are more or less appropriate for any particular study. The major
problem with DFT is that the exact functionals for exchange and correlation are not
known except for the free electron gas. However, approximations exist which permit
the calculation of certain physical quantities quite accurately. In physics the most
widely used approximation is the local-density approximation (LDA), where the
functional depends only on the density at the coordinate where the functional is
evaluated. The local exchange correlation energy per electron might be approximated
as a simple function of the local charge density (say, Exc()). It has the general form,
Exc ρ ≈ ρ(r)Exc ρ r dr …… 1.49
where, Exc() is the exchange and correlation energy density of the uniform electron
gas density () is the Local Density Approximation. With LDA Exc() is a function of
only the local value of the density. It can be separated into exchange and correlation
contributions
∈xc ρ = ϵx ρ + ϵc ρ …… 1.50
The remarkable performance of the LDA is a consequence of its reasonable
description of the spherically averaged exchange correlation hole coupled with the
tendency for errors in the exchange energy density to be cancelled by errors in the
correlation energy density.
1.16.5.4. Generalized Gradient Approximation (GGA)
The local density approximation can be considered to be the zeroth order
approximation to the semi-classical expansion of the density matrix in terms of the
density and its derivatives [39]. A natural progression beyond the LDA is thus to the
gradient expansion approximation (GEA) in which first order gradient terms in the
expansion are included. This results in an approximation for the exchange hole [39]
which has a number of unphysical properties
66
In the generalised gradient approximation (GGA) a functional form is adopted
which ensures the normalisation condition and that the exchange hole is negative
definite [40,41]. This leads to an energy functional that depends on both the density
and its gradient but retains the analytic properties of the exchange correlation hole
inherent in the LDA.
The general form of GGA is
Exc ρ ≈ ρ(r)Exc ρ, ∇ρ dr …… 1.51
A number of functionals within the GGA family have been developed such as
BLYP, PBE, HCTH etc. Using the latter (GGA) very good results for molecular
geometries and ground-state energies have been achieved.
1.16.5.5. Meta GGA Functionals
Recently functionals that depend explicitly on the semi-local information in
the Laplacian of the spin density or of the local kinetic energy density have been
developed [42,43,44]. Such functionals are generally referred to as meta-GGA
functionals. Potentially more accurate than the GGA functionals are the meta-GGA
functionals. These functionals include a further term in the expansion, depending on
the density, the gradient of the density and the Laplacian (second derivative) of the
density.
The form of the functional is
Exc ρ ≈ ρ(r)Exc ρ, ∇ρ , ∇2ρ, τ dr …… 1.52
where, τ is the kinetic energy density. It is given by the equation as follows.
τ =1
2 ∇φi
2i …… 1.53
1.16.5.6. Hybrid Exchange Functionals
Difficulties in expressing the exchange part of the energy can be relieved by
including a component of the exact exchange energy calculated from Hartree-Fock
theory. Functionals of this type are known as hybrid functionals.
Exc ≈ a EFock + b ExcGGA …… 1.54
where a and b are the coefficients, determined by reference to a system for which the
exact result is known. Becke adopted this approach [45] in the definition of a new
67
functional with coefficients determined by a fit to the observed atomisation energies,
ionisation potentials, proton affinities and total atomic energies for a number of small
molecules [45]. The resultant (three parameter) energy functional is,
Exc = ExcLDA + 0.2 Ex
Fock − ExLDA + 0.72∆Ex
B88 + 0.81∆EcPW 91 …… 1.55
Here, ∆ExB88 and ∆Ec
PW 91widely used GGA corrections [46,47] to the LDA exchange
and correlation energies respectively.
Hybrid functionals of this type are now very widely used in quantum
mechanical applications with the B3LYP functional in which the parameterisation is
as given above but with a different GGA treatment of correlation [45]) being the most
notable. Computed binding energies, geometries and frequencies are systematically
more reliable than the best GGA functionals.
The most popular B3LYP [48] scheme is a hybrid functional method in which
the mixture of exact HF exchange and approximate DFT exchange is commonly
employed to increase performance. Several different mixing ratios have been
advocated. Becke Half-and-Half LYP uses a 1:1 ratio of HF and DFT exchange
energies. ie
Exc =1
2Ex HF +
1
2Ex Becke88 + Ec LYP …… 1.56
The most commonly employed hybrid method is the Becke 3- parameter
scheme (B3). The scheme is represented as
Exc = 0.2Ex HF + 0.8Ex LSDA + 0.72Ex Becke88 +
0.81Ex LYP + 0.19Ec LYP …… 1.57
Becke derived the parameters by fitting to a set of thermo chemical data, the
G1 molecule set. When B3 is paired with a correlation function other than LYP, the
LYP coefficients are retained. Becke also developed a one parameter fit (B). Another
option is to use modern valence bond methods. The MPW1PW91 method was
developed by Barone and Adamo. It employs a modified PW91 exchange functional
with original PW91 correlation functional and employs a HF and DFT exchange ratio
of 0.25: 0.75.
68
1.17. BASIS SETS
A basis set [49] is a set of functions used to create the molecular orbitals,
which are expanded as a linear combination of similar functions with the weights or
coefficients to be determined.
One characteristic of a molecule that explains a great deal about the properties
is their molecular orbitals. The following diagram must be considered in order to
calculate the molecular orbitals.
One of the three major decisions is which of the basis set to use. There are two
general categories of basis sets:
Minimal basis sets :A basis set that describes only the most basic aspects of
the orbitals.
Extended basis sets : A basis set with a much more detailed description.
Basis sets were first developed by J.C. Slater. Slater fit linear least-squares to
data that could be easily calculated. The general expression for a basis function is
given as:
69
Basis Function = N * e(-alpha * r)
where:
N = normalization constant
alpha = orbital exponent
r = radius in angstroms
All basis set equations in the form STO-NG (where N represents the number
of GTOs combined to approximate the STO) are considered to be "minimal" basis
sets. The "extended" basis sets, then, are the ones that consider the higher orbitals of
the molecule and account for size and shape of molecular charge distributions.
There are several types of extended basis sets:
Double-Zeta, Triple-Zeta, Quadruple-Zeta
Split-Valence
Polarized Sets
Diffuse Sets
1.17.1. Double-Zeta, Triple-Zeta, Quadruple-Zeta
Previously with the minimal basis sets, all orbitals are approximated to be of the
same shape. However, this is not true. The double-zeta basis set is important because
it allows treating each orbital separately when Hartree-Fock calculation is conducted.
This gives us a more accurate representation of each orbital. In order to do this, each
atomic orbital is expressed as the sum of two Slater-type orbitals (STOs). The two
equations are the same except for the value of 𝜁(zeta). The zeta value accounts for
how diffuse (large) the orbital is. The two STOs are then added in some proportion.
The constant‘d‘ determines how much each STO will count towards the final orbital.
Thus, the size of the atomic orbital can range anywhere between the value of either of
the two STOs. For example, let's look at the following example of a 2s orbital:
𝛷2𝑠 𝑟 = 𝛷2𝑠𝑆𝑇𝑂 𝑟, 𝜁1 + 𝑑𝛷2𝑠
𝑆𝑇𝑂 𝑟, 𝜁2 …… 1.58
In this case, each STO represents a different sized orbital because the zetas are
different. The ‗d‘ accounts for the percentage of the second STO to add in. The linear
combination then gives the atomic orbital. Since each of the two equations is the
same, the symmetry remains constant.
70
The triple and quadruple-zeta basis sets work the same way, except the use of
three and four Slater equations instead of two. The typical trade-off applies here as
well, better accuracy for .more time or work.
1.17.2. Split-Valence
Often it takes too much effort to calculate a double-zeta for every orbital.
Instead, it can be simplified by calculating a double-zeta only for the valence orbital.
Since the inner-shell electrons aren't as vital to the calculation, they are described with
a single Slater Orbital. This method is called a split-valence basis set. A few
examples of common split-valence basis sets are 3-21G, 4-31G, and 6-31G.
An example is given below. It will be of help to understand the subject. Here,
a 3-21G basis set is used to calculate for the carbon atom. This means 3 summing
Gaussians for the inner shell orbital, two Gaussians for the first STO of the valence
orbital and 1 Gaussian for the second STO. Here is the output file from the Gaussian
Basis Set Order Form for carbon given a 3-21G basis set.
3 - 21G
The number of Gaussian functions summed in the
second STO
The number of Gaussian
functions that comprise the first STO of the double zeta
The number of Gaussian functions summed to
describe the inner shell
orbital
71
There is another common method of displaying data. Notice the numbers are
labeled so it is easy to match this data with the corresponding data in the output file.
Once a basis set output file is retrieved, these numbers can be used to calculate
equations. For a carbon, three equations will be needed: 1s orbital, 2s orbital, and 2p
orbital.
This equation combines the 3 GTO orbitals that define the 1s orbital.
Φ1s r = d1siΦ1sGF3
i=1 r, α1si
= 0.6176Φ1sGF r, 172.256 + 0.3587 Φ1s
GF r, 25.910
+ 0.7007Φ1sGF (r, 5.533) …… 1.59
This equation combines the 2 GTO orbitals that make up the first STO of the
double-zeta, plus the 1 GTO that represents the second STO for the 2s orbital.
Φ2s r = d2siΦ2sGF3
i=1 r, α2si + d′2sΦ2s
GF r, α′2s
= −0.395Φ2sGF r, 3.664 + 1.215 Φ2s
GF r, 0.771
+ 1.000Φ2sGF (r, 0.195) …… 1.60
This equation combines the 2 GTO orbitals that make up the first STO of the
double-zeta, plus 1 GTO that represents the second STO for the 2p orbital.
Φ2s r = d2pi Φ2pGF
3
i=1 r, α2pi + d′2pΦ2p
GF r, α′2p
= −0.395Φ2pGF r, 3.664 + 1.215 Φ2p
GF r, 0.771
+ 1.000Φ1sGF (r, 0.195) …… 1.61
Now, using these three equations, the LCAO can be calculated for the carbon
atom.
72
1.17.3. Polarized Sets
To approximate Hartree-Fock orbitals for first-row atoms (Li-Ne), one does
not need d and higher angular momentum functions. However, in molecular
environments orbitals become distorted from their atomic shapes. This is polarization.
To describe effects of polarization one needs to add polarization functions (functions
of higher angular momentum than any occupied atomic orbital). Moreover, Polarized
basis sets are considered good general purpose basis sets for semi-quantitative
calculations on ground state neutral molecules without non-bonding effects.
In the previous basis sets atomic orbitals are treated as existing only as 's', 'p',
'd', 'f' etc. Although those basis sets are good approximations, a better approximation
is to acknowledge and account for the fact that sometimes orbitals share qualities of 's'
and 'p' orbitals or 'p' and 'd', etc. and not necessarily have characteristics of only one
or the other. As atoms are brought close together, their charge distribution causes a
polarization effect (the positive charge is drawn to one side while the negative charge
is drawn to the other) which distorts the shape of the atomic orbitals. In this case, 's'
orbitals begin to have a little of the 'p' flavor and 'p' orbitals begin to have a little of
the 'd' flavor. One asterisk (*) at the end of a basis set denotes that polarization has
been taken into account in the 'p' orbitals. Notice in the graphics below the difference
between the representation of the 'p' orbital for the 6-31G and the 6-31G* basis sets.
The polarized basis set represents the orbital as more than just 'p', by adding a little 'd'.
Original 'p' orbital Modified 'p' orbital
Two asterisks (**) means that polarization has taken into account the‘s‘
orbitals in addition to the 'p' orbitals. Below is another illustration of the difference of
the two methods.
73
Original 's' orbital Modified 's' orbital
A split-valence double-zeta basis set with a single polarization function much
better than non-polarized basis sets. This is the smallest basis one should use for
qualitatively correct calculations. 6-31G* does not include polarization function on
the hydrogen, but 6-31G** does. New convention is to denote 6-31G* as 6-31G(d)
and 6-31G** as 6-31G(d,p).
1.17.3. Diffuse Sets
In chemistry the valence electrons are the main concern which interacts with
other molecules. However, many of the basis sets that are talked about previously
concentrate on the main energy located in the inner shell electrons. This is the main
area under the wave function curve. However, when an atom is in an anion or in an
excited state, the loosely bond electrons, which are responsible for the energy in the
tail of the wave function, become much more important. To compensate for this area,
computational scientists use diffuse functions. One needs to include diffuse functions
(functions with small exponents, hence large radial extent) to predict properties of
anions accurately. Diffuse functions are also necessary to describe certain non-
bonding interactions. These basis sets utilize very small exponents to clarify the
properties of the tail. Differences between diffuse basis sets are mostly due to the
differences in their core and not in their diffuse component.
Diffuse basis sets are represented by the '+' signs. One '+' means the 'p' orbitals
are accounted, while '++' signals mean both 'p' and 's' orbitals, (much like the asterisks
in the polarization basis sets).
6-31+G** - same as 6-31G** plus one set of s and p diffuse functions on first
row atoms.
6-31++G** - same as 6-31+G** plus a diffuse function on hydrogen
aug-cc-pVXZ - same as cc-pVXZ plus one diffuse function per each angular
momentum present in cc-pVXZ.
= 6-31G + = 6-31G**
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1.18. GAUSSIAN – SOFTWARE OVERVIEW
Gaussian [50] is a computational software program initially released in 1970
by John Pople and research group at Carnegie-Mellon University. It has been
continuously updated since then. The recent version is Gaussian 09. The name
originates from the word Gaussian function or orbital, a choice made to improve the
computing capacity of then existing software which used a function or orbital called
Slatter type. Gaussian functions are widely used in statistics where they describe the
normal distributions. In this Quantum computation, it represents the wave function of
the ground state of a harmonic oscillator. The linear combinations of such Gaussian
functions for a molecular orbital is called as Gaussian orbital. The Gaussian software
program is used by Physicists, Chemists, Chemical engineers, Biochemists and others
for research in established and emerging areas of molecular physics or chemistry.
Starting from the basic laws of quantum mechanics, Gaussian predicts the energies,
molecular structures, vibrational frequencies and other molecular properties derived
from these basic quantities. It can be used to study molecules and reactions under a
wide range of conditions, including both stable and short lived intermediate
compounds.
Computational techniques consist of three areas: Ab-initio methods, semi-
empirical methods, and molecular mechanics. Molecular mechanics utilizes classical
physics to solve large systems of molecules and is considered the least accurate due to
the fact that no electron behavior is factored in. Semi-empirical methods are more
accurate because of utilization of quantum physics to account for some of the electron
behavior, but its scope is still limited since it relies on extensive approximations and
empirical parameters. Ab-initio methods are based purely on quantum physics and use
no approximations from classical physics to describe the electronic structure of the
molecule very accurately. The drawback of using ab initio methods is that the
computations are extremely taxing and so is limited to much smaller systems such as
individual molecules. However, ab-initio methods give a lot of information on the
electronic structure without having to actually synthesize the molecule
experimentally. The fundamental idea behind ab initio calculations is to solve
Schrodinger‘s equation with a set of mathematical functions called a ―basis set‖.
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1.18.1. Capabilities of the software
The technical capabilities of Gaussian Software are listed below.
The energy calculations were carried out by different methods such as
Molecular mechanics calculations, Semi-empirical calculations, Self
Consistent Field calculations, Correlation energy calculations using Moller-
plesset perturbation theory, Correlation energy calculation using CI, CID
CISD techniques, Density functional theory
Analytic computation of force constants, polarizabilities, hyper polarizabilities
and dipole moment derivatives analytically for RHF, DFT, MP2 etc.
Harmonic vibrational analysis and thermochemistry analysis using arbitrary
isotopes, temperature and pressure
Analysis of normal modes in internal coordinates
Determination of IR and Raman Intensities for vibrational transitions, pre-
resonance Raman intensities
To determine Harmonic vibration-rotation coupling constants
To calculate Anharmonic vibration and vibration-rotation coupling constants
Mulliken population analysis, APT analysis, electrostatic potentials and
electro-static derived charges
Static and frequency dependent polarizabilities and hyperpolarizabilites for HF
and DFT
NMR shielding tensors and molecular susceptibilities using the SCF,DFT and
MP2 methods
Calculation of spin-spin coupling constants at HF and DFT level
Vibrational circular dichroism intensities
To calculate nuclear quadrupole constants, rotational constants, quartic
centrifugal distortion terms
HOMO – LUMO analysis
1.19. GAUSSVIEW – SOFTWARE OVERVIEW
GaussView [51] is a graphical user interface designed to prepare input for
submission to Gaussian and to examine graphically the output that Gaussian produces.
GaussView is not integrated with the computational module of Gaussian, but rather is
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a front-end/back-end processor to aid in the use of Gaussian. GaussView provides
three main benefits to Gaussian users [52].
First, through its advanced visualization facility, GaussView allows to rapidly
sketch in even very large molecules, then rotate, translate and zoom in on these
molecules through simple mouse operations. It can also import standard molecule file
formats such as PDB files.
Secondly, GaussView makes it easy to set up many types of Gaussian
calculations. It makes preparing complex input easy for both routine job types and
advanced methods like ONIOM, STQN transition structure optimizations, CASSCF
calculations, periodic boundary conditions (PBC) calculations, and many more.
Lastly, to run default and named calculation templates known as schemesto
speed up the job setup process.
Finally, GaussView is used to examine the results of Gaussian calculations
using a variety of graphical techniques. Gaussian results that can be viewed
graphically include the following:
Optimized molecular structures.
Molecular orbitals.
Electron density surfaces from any computed density.
Electrostatic potential surfaces.
Surfaces for magnetic properties.
Surfaces may also be viewed as contours.
Atomic charges and dipole moments.
Animation of the normal modes corresponding to vibrational frequencies.
IR, Raman, NMR, VCD and other spectra.
Molecular stereochemistry information.
Animation of geometry optimizations, IRC reaction path following, potential
energy surface scans, and ADMP and BOMD trajectories. Two variable scans
can also be displayed as 3D plots.
Plots of the total energy and other data from the same job types as in the
previous item.
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1.20. DESCRIPTION OF THE PRESENT RESEARCH WORK
The outline of the complete work presented in the thesis involves the
following algorithm:
To cope up with the current trends in this field, the research begins with the
molecule having 14 atoms and extends upto 27 atoms. Totally eight molecules
have been chosen for the study.
The FT-IR spectra of the compounds were recorded in the range of 4000 to 10
cm-1
. The spectral resolution was ± 2 cm-1
. The FT-Raman spectra were also
recorded in the same range, in the same instrument with FRA 106 Raman
module equipped with Nd: YAG laser source operating at 1.064 μm line
widths, with 200 mW powers. The spectra were recorded with the scanning
speed of 30 cm-1
min-1
with spectral width 2cm-1
. The frequencies of all sharp
bands were accurate to ± 1 cm-1
.
With the aid of above discussed experimental techniques, these molecules
were subjected to new trends of theoretical methods based on quantum
mechanical computations such as Ab initio HF, MP2 and DFT for the spectral
analyses.
The quantum mechanical computations were carried out using Gaussian 03
programs with appropriate basis sets. The Gauss view software was used to
input the necessary data.
The molecular optimization, optimized geometrical parameters such as bond
length, bond angle and dihedral angles were calculated.
Potential energy surface scan and structural confirmation based on energy
were also carried out.
The results obtained from the computation were sorted out, the different
parameters were tabulated, and the theoretical frequencies were scaled up with
appropriate scaling factors in comparison with the experimental frequencies.
The fundamental frequencies has been assigned to different modes of
vibrations based on the expected range, literature values, TED, possibility of
vibrations in the compound and group theory principles.
The computed normal modes were used to estimate IR and Raman intensities
from dipole and polarizability derivatives, as well as vibrational averaging
effects on molecular geometries and properties.
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Thermodynamical parameters (entropy, enthalpy specific capacity etc),
rotational constants, atomic charges and Non linear optical parameters
(polarizabilty, anisotropy, first and second order hyper polarizabilites) were
calculated for the molecules.
UV-VIS analysis, Frontier Molecular Orbitals (HOMO-LUMO) analysis and
molecular electrostatic potential interpretation were carried out.
In order to compare the visual view of the parameters, the graphs are plotted
using appropriate software for theoretical frequencies, theoretical intensities
and other structural parameters.
The deviations between the experimental and theoretical values are noted and
the deviations are analysed with the help of literatures on structurally related
molecules.