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

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

qq

qq

…… 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

36

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‖.

75

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.

78

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.