Chapter If: Methodology

Chapter 11: Metbodology

The numerical results obtained in the present thesis are based on the

computational experiments conducted on personal computers and Sillcon Graphics

workstations using various software program packages. Although the methodologies are

used as they are implemented in the programs, a clear understanding of the principles

behind them is obviously important to meaningfully apply them to chemical problems.

Such a knowledge is vital to judge the suitability of a given computational procedure and

also the strengths and limitations of it in a given context. The present chapter makes an

attempt to give a brief description of the various theoretical methodologies employed."'

All the calculations (except molecular mechanics calculation on sumanene, in chapter V)

are based on quantum mechanical principles using semiempirical, ab initio and density

functional theory (DFT). The work embodied in the thesis involves addressing a number

of questions which encompasses patterning the observed data, refining, understanding of

a few concepts, and also designing new chemical entities and pathways. A quick outline

of the quantum mechanical principles, which provide the platform to build the

computational techniques, is given first. This is followed by a description on obtaining

and characterization of stationary points, optimization procedures, frequency calculations

and calculation of thennochemical corrections. It is well known that the only chemical

problem that has exact analyhcal solutions is the time independent form of hydrogen

atom in the absence of external effects.' Therefore, approximations manifest themselves

in all the computational methodologies in some way or the other. All molecular orbital

based methods work within the Born-Oppenheimer approximation, which allows the

separation of the nuclear and electronic parts of the molecular ~amiltonian.~ Decoupling

of the nuclear and electronic parts is an essential first step in the electronic theory and this

is necessary to analytically solve the Schrodinger equation even for the simplest

molecular ion, HA This approximation does not introduce any errors where the

electronic state functions do not critically depend on the nuclear motions. However as is

inevitable for any approximation under some conditions, Born-Oppenheimer

approximation breaks down in regions of the potential energy surface where the ordering

of the electronic state functions become critically dependent on the minute displacement

Chapter 11: Methodology 2 1

of the nuclear positions. All the chemical problems considered in the present thesis are

restricted primarily to the ground state properties of the systems, and in all these systems

clear separation of excited and ground states enables that Born-Oppenheimer

approximation is valid.

Various computational methods based on molecular orbital (MO) theory have

evolved over the years, which differ in the way the approximations are implemented. The

computational methods employed in the present thesis are mainly based on Hartree-Fock

(HF), second order Moller-Plesset perturbation (MP2), coupled cluster (CCSD(T)),

density functional theory and semiempirical (AM1, MNDO and PM3) calculations. A

range of basis sets is used in conjunction with these theoretical methodologies at various

stages. Brief accounts of the methods employed here and ~ o ~ g u r a t i o n interaction (CI),

multiconfigurational self-consistent field (MCSCF) and multireference configuration

interaction (MRCI) methods are provided in the present chapter.

After the Born-Oppenheimer approximation, the main obstacle to solve the many

electron problem is the non-separability of variables due to the electron-electron

repulsion term. Thus, constructing the many electron wavefunction from one and two-

electron wavefunctions necessitated further approximations. The two principles in

electronic structure methods are the valence bond (VB) and molecular orbital (MO)

theories, which uses the two- and one-electron wavefunctions respectively. Although, VB

theory is closer to chemical thinking, the mathematical equations are highly complicated

and also the VB wavefunctions do not form basis for irreducible representations. On the

other hand, the MO wavefunctions are basis for irreducible representation: enabling us to

use the elegant group theoretical principles. Thus, MO approaches are more efficient than

the VB approaches.

METHODOLOGIES

Hartree-Fock Method (HF):

n e Hartree-Fock approximation is a crucial first step to tackle the many electron

wavebctions in the molecular orbital theory. The first step in the HF approximation is

that the wavefunction of a 'N' electron system is represented by a single Slate[

determinant of N dimension, YsD, with one-electron wavefunctions/molecular orbitals

Chapter 11: Methodology 22

(4) as the components (eq. 2.1).1° The use of Slater determinant as the wavefunction is to

take into account the antisymmetry requirement of the fermionic systems."

14!(1) 420) .,. 4,dl)I

The orthonormal one-electron wavefunctions (4,) in the Slater determinant are expressed

as a linear combination of the basis functions (eq. 2.2).

where, c, are the expansion coefficients of the basis functions X, and ' M is the total

number of basis functions

The energy of the system is given by the following equation,

where, Y is the normalized wavefunction, H is the molecular Hamiltonian.

The variational theorem is used to determine the coefficients to obtain optimum orbitals

so as to minimize the energy, E. Mathematically,

aE - = 0 ; for all p and i ...( 2.4) &a

The essence of the HartreeFock approximation is that the many electron problem

is reduced effectively to a one-electron problem by considering an effective one-electron

Hamiltonian (Fock operator). This is achieved by introducing the mean field

approximation, which proposes an average potential (v, (i) ) experienced by an electron,

i, in the mean field of the other electrons.

The Fock operator, is given by,

'0' is the number of atoms in the molecule.

Chapter IS: Methodology 23

The operators corresponding to the electron repulsion terms are divided into two types,

j,(i) and %,(i) are called the Coulomb and exchange operators respectively (eq. 2 . 5 ~

and 2.6d).

The one-electron wave function (4,) is then obtained by solving,

A 4 , = & , # , ; i = l 9 2 ,.,., N , . .(2.6)

E, are the molecular orbital energies.

The vanational condition leads to a set of algebraic equations for c, called the Roothaan-

Hall equations (eq. 2.7).12

or rewritten as FC = SCE . . .(2.7b)

where, = 1, 2, . . .M), M is the number of basis functions. p, v, A and a are indices of

the basis functions x,, X , ,yA and X, respectively. In this chapter, we denote the basis

functions with these indices.

In the above equations, F is the Fock matrix with F,,as the matrix elements (eq. 2.8b). E

is a diagonal matrix with the molecular orbital energies (E,) as the diagonal elements. S is

the overlap matrix and S,,. are the overlap matrix elements.

Here, HrVm is the matrix containing the energy of each electron exclusively in the field of

the nuclei given by,

1 " Z Hr = ( p I - - ~ 2 - z L I v ) A-I r ! ~

where, ZA is the atomic number of atom A.

ChDptv 11: Methodology 24

and the matrix Pi, contains the one-electrun densities, which is given by

The Roothaan-Hall equations (2.7b) are non-linear since the Fock operator itself depends

on the one-electron wavefunctions; hence the equations have to be solved iteratively. An

initial guess of the spin orbitals is generated to evaluate the mean potential in the Fock

matrix, which in turn generates an improved wavefunction through an iterative process to

reach self-consistency hown as the self-consistent field (SCF) procedure.

Restricted openshell HF (ROHF) and unrestricted HF (UHF): The foregoing discussion

of Hartree-Fock formalism can be extended to the open shell systems in two different

ways, namely the restricted open shell HF (ROHF) and unrestricted HF (UHF)

procedures.'3 The ROHF procedure restricts that the a and Pobitals are identical. In the

UHF procedure, the a and P orbitals are individually optimized. Although UHF method

gives lower energies compared to ROHF, the wavefunction is not a true eigenfunction of

the spin operator, d2. In other words, the wavefunctions do not represent pure spin

states; for example, a UHF wavefunction obtained for a doublet may be contaminated by

those of the higher spin multiplicity, which is referred to as the spin contamination.

ROHF procedure yields pure spin states, but gives a higher energy for the system.

However, ROHF cannot account for spin polarization.

Electron Correlation

Hartree-Fock method does not take into account the instantaneous electrostatic

repulsion among the electrons since the electron-electron repulsion is treated by the mean

field approximation. The difference between the exact nonrelativtstic energy (LC,) and

the HF energy (EHF) at the Hartree-Fock limit is the correlation energy (E,,,,,).'~

Et, = E,.,! - EHF . . .(2.9)

Two types of electron correlation exist; Dynamic electron correlation' this arises

due to the improper treatment of the electron repulsion between the electrons in motion.

Nondynamic or stalic electron cowelation: It is related to the fact that in certain

circumstances a single Slater determinant is not a good reference wavefunction to the true

ground state wavefunction and one should go for multi-determinantal approach as a

Chapter 11: Methodology 25

starting point. In cases, where the ground and excited states are clearly separated, only

dynamic correlation becomes relevant.

Due to the non-inclusion of dynamic electron correlation, the electrons tend to

accumulate around the nuclei of the system strongly thereby overestimating the

stabilization between the electron and nuclei. This leads to the underestimation of the

bond lengths in the molecules, and overestimation of the harmonic force constants and

harmonic frequencies. The HF method is found to overestimate the harmonic frequencies

uniformly by about 10-11% and thus a scaling factor 0.89 is recommended.." Various

theoret~cal methods, which include electron correlation, are discussed in the following

sections. While the post-SCF methods are good for treating electron correlation, they are

computationally expensive. Density functional theory seems to be ideally suited to treat

electron correlation, as it is computationally economic.

Electron correlation is taken into account by an may of theoretical methodologies

such as configuration interaction (CI), multi configurational self consistent field

(MCSCF), Moller-Plesset perturbation theory (MPn) and coupled cluster theory (CC).

The cmently popular pure and hybrid density functional theory (DFT) approaches also

include electron correlation, but with a less computational effort compared to the other

post-HF methods.

Density Functional Theory (DFT):

The density functional theory approaches are based on the principle that for a

given set of nuclear coordinates, the energy and all properties of the ground state are

solely determined by the electron den~i ty . '~ ' '~ In other words, the ground state energy of a

system is a functional only of the electron density as shown by Hohenberg and Kohn in

the 1 9 6 0 ~ . ' ~ The advantage of the DFT procedure over the traditional SCF procedure is

that, in the SCF procedure, the wavefunction is a function of 3N variables; whereas in

DFT, the electron density is a function of only three variables independent of the number

of electrons in the system. Hohenberg-Kohn theorem states that a functional exists, which

connects the electron density and the energy, the form of which is not known exactly.

merefore, the challenge in DFT is to design accurate functional^ relating the energy and

electron density.

Chapter 11: Methodology 26

In the early 19Ih century, attempts were made to use the electron density for

calculating properties of atoms and molecules. Two such methods are the Thomas-Fe-

(TF)~' and Thomas-Fermi-Dirac (TFD) m ~ d e l s . ~ ~ . ~ ' In these two models, the kinetic

energy functional used is that of a noninteracting electron, and the potential due to

electron-electron and nuclei-electron interactions are treated classically.

Kohn-Sham formalism elegantly demonstrates that the electrons can be treated

noninteracting in the kinetic energy functional and the exchange-correlation functional

may be modified accordingly so as to include the kinetic energy difference between the

reallinteracting and the noninteracting systems.22 The concept of one-electron orbitals

proposed by Kohn and Sham provided the breakthrough in the field of D F T . ~ ~ This KS-

DFT procedure is very similar to the HF methodology in the sense that the KS-orbitals

are obtained by an iterative process, analogous to the one-electron wavefunctions in HF,

to achieve self consistency. This is associated with the minimization of energy with

respect to the electron density.

The energy of the system is separated into kinetic energy of the non-interacting

system (T,), the energy due to the nuclei (E,,,), the electrostatic electron-electron

repulsion (J) and the last term containing the correlation, exchange and the part of kinetic

energy not covered by T, (Ex,) (eq. 2.10); all are functionals of the electron density @).

~,bl=~sbI+~"~bl+~bl+~~,bl ...( 2.10)

Much of the research in DFT following Kohn-Sham formalism centered on the

development of exchange and correlation functionals. If a suitable functional for the

exchange-correlation part is available, a set of orthonormal KS-obitals can be generated

to minimize the total energy. Since the coulomb part (m]) and the exchange-comelation

functional (E,b]) in turn depend on the total electron density, an iterative procss is

employed to achieve self consistency similar to the SCF procedure. usually the

exchange-comelation functional is divided into two parts, namely the exchange and the

functionals (eq. 2.11). Several functionals were proposed and are discussed

below.

E,,[p] = E,[p] + E,[p] = lp(r)&,[p(r)~dr + ~P(~)E , [P ( r )~dr . . .(2.11)

In the above equation, EX and E, are the energy Per particle.

Chapter 11: Methodolqy 27

Local dens@ methods: Local density approximation (LDA) assumes that the electron

density can be treated as a uniform gas locally (eq. 2.12a). Hence, the correlation

functional fitted to the uniform electron gas is employed. The exchange functional within

the LDA is given by,

lil

here, C" = 4 77

This approach fails when the a and P spin densities are not equal; Local spin density

approximation (LSDA) is used for the open shell systems, where the a and P electron

densities are treated separately. The LSDA functional is given by,

&y" [ p ] = -2"'c,[pb" + p;'] ...( 2.13)

One of the correlation functionals designed based on the LSDA approximation is the one

proposed by Vosko, Wilk and Nusair referred to as the VWN f~nctional.~' The L(S)DA

based methods tend to underestimate the exchange energy and overestimate the

correlation energy; however, they are comparable to the HF method in terms of accuracy.

Gradient corrected methods: Real progress in the application of DFT to molecular

systems began with the introduction of non-local (gradient) corrections. Before the

advent of the gradient corrected methods, DFT was used mainly by the solid state

physicists and not by the computational chemists. When the exchange and the correlation

functionals are made dependent on both electron densities and their gradients, the

procedwe seems to model the inhomogeneous (interacting) nature of the electrons. These

methods are termed as the gradient corrected methods, non-local methods or the

generalized gradient approximation (GGA). Inclusion of the electron density gradients

takes into account the interacting nature of the electrons, which is not the case in the

L(S)DA based approaches. Perdew and Wang reported two such exchange functionals.

PW86 (eq. 2.14) andPW91 (eq. 2 .161.~~

c r a b =C:Dl(l+ax2 +bx4 +a6))"" ...( 2.14)

Choptv 11: Methodology 28

where, al.3, b are constants and x is defined in eq. 2.15

Becke proposed a functional (B or B88) whch aptly reproduces the asymptotic behavior

for the energy density (eq. 2.17)." One of the popular functionals for the correlation part

is the one proposed by Lee. Yang and Pam, called the LYP correlation functiona~.'~

x is similar to that defined in the eq. 2.15 and P i s an empirical parameter, determined by

fitting to the experimental atomic data.

All the functionals that we have discussed till now are based on pure density

functional theory. In the next section, we will discuss the hybrid methods which make use

of the exchange part from the HF formalism for the exchange functional.

Hybrid mefhods: All the pure DFT methods apply approximate functionals for the whole

energy. In general, the exchange contributions are significantly large compared to the

contribution from the electron correlation effects. Hence, precisely defining the exchange

functional is important. Instead of designing approximate functionals for both the

exchange and correlation parts, the exact exchange energy can bc derived from the HF

formalism and on the other hand, the correlation part alone may be expressed in terms of

approximate functionals. Such methods are called the hybrid methods. Half-and-half

(H+H)*' and the more popular Becke's three parameter functional (Becke3 or ~ 3 ) ' ' given

by Becke are some of the functionals in hybrid methods.

The H+H functional is given by,

E? 1s the exchange functional derived from the HF methodology.

The B3 functional contains both exchange and correlation functional terms; the exchange

functional consists a combination of exchange term from HF, LSDA and a gradient

correction term whereas the correlation functional contains a LSDA form and a gradient

correction term (eq. 2.19).

Chapter If: Methodology 29

In this equation, the three empirical parameters, a, b and care calculated by fitting to the

experimental data and hence called the Becke3 parameter functional.

Semiempirical Methods:

Most of the modern semiempirical approaches in general rue formulated within

the same conceptual framework as the ab rnlllo HF method, hut the computat~onally

electron repulsion integrals are either neglected or approximated to increase the

computat~onal e f f i c i enc~ .~~ .~ ' In order to compensate for the errors caused by these

approximations, empirical parameters are introduced in the remaining integrals and the

parameters are optimized to reproduce the reference data known as the parameterization

procedure.

Zero differential overlap (ZDO) is the central assumption in most of the modem

semiempirical methodologies (eq. 2.20a).

( P V I LO) = 6 , , 6 2 , ( ~ v l AD) ...( 2.20a)

According to ZDO all two electron integrals vanish unless j ~ v and L=n, i.e., all the

three- and four-center electron repulsion integrals are neglected. Several approximations

have been proposed based on ZDO namely and much work was due to Pople and Dewar

and their coworkers. Three levels of approximation are available: CNDO (complete

neglect of differential overlap):3 INDO (intermediate neglect of differential overlap),'4

and NDDO (neglect of diatomic differential overlap)." In the following sections, we

discuss the various approximations like CNDO, INDO and NDDO, and finally a

comparison of the three NDDO based methods namely, MNDO?' AMI" and P M ~ " IS

given.

Complete neglect of differential overlap (CNDO): The CNDO approximation was

proposed by Pople, Santry and Segal in 1965 and the approximations involved in this

procedure are listed below:

( I ) The overlap matrix in the Roothan-Hall equations is replaced by a unit matrix, where

all overlap integrals (S,d with p + v are neglected.

Chapter II: Methodology 31

(pv I LO), where . x,, and X, are centered on one atom XL and X, in another atom are

retained. Introducing this approximation leads to inclusion of a large number of integrals

compared to those from the CNDO and INDO approximations.

MNDO, AM1 and PM3:

All the three methods namely, rnomfied neglect of diatomic overlap (MNDO),

Austin model 1 (AMI) and parametric model 3 (PM3 or MNDO-PM3) are based on the

NDDO approximation. The major improvements that were made from MNDO to AM1

and AM1 to PM3 are discussed in this section. The three main differences between the

MNDO and the AM1 methods are: (a) a set of gaussian functions is added to the core-

repulsion in the AM1 method. This was included to take into account the overestimation

of the core-core repulsion term in the MNDO method, which leads to the overestimation

of the activation energies. The inability of the MNDO method to reproduce the hydrogen

bonding is rectified by the improvement of the core-core repulsion tern. (b) the

parameterization procedure is different in the two methods. In the parameterization of

MNDO, the overlap terms (P, and P,), without considering whether they involve s or p

type orbitals and Slater orbital exponents (6, and 6,) for the second row elements are

assumed to be equal. Whereas in the AM1 procedure. the exponents are varied durlng the

parameterization procedure. The inclusion of the Gaussian functions to the core-core

repulsion and removing the equating the overlap terms and orbital exponents increased

the number of to be optimized from 7 (MNDO) to 13-16 (AMI) per atom. (c) The AM1

parameterization is done by including more number of molecules compared to the

MNDO, where only 34 molecules were included.

The parameterization of the MNDO and AM1 methods is done manually, and

hence large number of molecules could not be included In the training set. Stewart and

co-workers made the parameterizatlon automatic In PM3, which allowed use hundreds of

atoms for the optimization of the parameters. Another improvement of PM3 over the

AM1 method is that the Gaussian functions ~ncluded in the core-repulsion term is

allowed to vary freely during the parameterizatlon. The number of gaussians used in the

core-core repulsion pan in PM3 is only 2 compared to the inclusion of 2-4 gaussians in

AM1.

Choptv 11: Methodology 32

Error trends observed in the results from semiempirical calculations are not

uniform, unlike in ab rn~tio procedures. Paramelenzation 1s not done for all the elements.

The accuracy of a given semiempirical method may be very different for different class

of compounds. Hence, the performance of the semiempirical methods to model a given

class of compounds should bc evaluated based on experimental data or results from ab

rnrlro calculations.

Although the success of DFT in solv~ng wide ranging problems is truly

spectacular, when a particular exchange or correlation functional fails to reproduce [he

results, there is no systematic way to Improve the results. Also, the procedure doesn't

guarantee for definitive improvenlent in results as the quality of the basis set is improved.

The undesirable feature In DFT procedure is that when a method fails In a given case.

there 1s no systematic way to improve it; a s~mllar scenario exists in semiempirical

methods also. Traditional ah inilro procedures have a mechanism to systemat~cally

improve the rel~ability of the computed numbers, although they are computationally not

economic. The CI and MP methodologies are two main routes to include the electron

correlation by taking the HF wavefunction as a reference. However, when a s~ngle

determinantal schemes are inadequate, e.g. a biradicaloid, one has to resort to CASSCF

methodology. These methods are described later in the chapter.

Configuration Interaction (CI)

The C1 method considers the expansion of the ground statc wavefunct~on by

including all the excited state Slater determinant^.^^.^^ This is achieved by expressing the

wavefunction as a l~near comhinat~on of the Slater dcterminants corresponding to the

ground and all possible excited states. The molecular orbitals derived Srom the )IF

method are used to generate the excited state determinants. The coefficients of each

Slater determinant arc optimized with respect to energy and the molecular orbital

coefficients are kept fixcd. These determinants may be classified as HF ground state

wavefunction, singly, doubly, triply, . . . N-tuply excited state determinants with reference

to the ground state determinant. Given a trial function for 'f', , the exact wavefunction

can be computed using the variational method.

vo =coy,,F + 2 1 r;,yr:, + C C caw: + C C c ; w a +... ...( 2.21) ,, r ,,>b ,..I ,,>h,, r > , > ,

Chapter II: Methodology 33

W : . 1,; . W.: . . . are the singly, doubly. tnply.. . excited wavefunctions respectively.

In the full CI, the number of determinants in the wavefunction exponentially

increases with the size of the system and hence is computationally demanding even for

small molecules. To overcome this problem, one has to truncate the CI expansion for the

wavefunction assuming that the determinants correspond~ng to the higher excited states

do not contribute to the wavefunction significantly; th~s procedure is referred to as

truncated CI. The procedure in which, only single and double excitations along with the

ground state determinant are included in total wavefunction, is referred to as CI with

singles and doubles substitutions (CISD). The contrlbut~on liom the s~ngle excitation to

the correlation energy is very small compared to that from double excitat~on for the

closed shell ground state systems. This IS due to the fact that ground stated determinant

does not mix with the singly excited Slater determinants (Br~llouin's theorem) due to

symmetry considerations. CID is another such truncated CI procedure, where only the

doubly excited state determinants are Included in the C1 expansion.

Size consistency and tnrnca~ed CI: A theoretical procedure is size consistent if the energy

of a many particle system is proportional to the numher of particles (P) in the hmit P + m. In other words, the energy of a dimer made of non-interacting monomers should be

the sum of the individual energies of the monomers. A full CI calculation is size

consistent; however, any truncated CI is not." Many schemes for the correction to the

energies obtained by CID and CISD were reported to solve the size cor~sistency problem

of which Davidson's correction has been widely used."4 4'

Moller-Plesset Perturbation Theory (MP):

In this methodology, the HF wavefunctlon IS taken as the zeroth-order solution

and electron conelation is considered exphcitly by treating the difference between the

exact Hamiltonian and the HF Harniltonian as a perturbation.4h In other words, this

method heats electron correlation as a perturbatlnn to the HF problem.4'.4"i'he

perturbation expansion works well only when the differences in energy between exact

and HF wavefunctions is not very high. In other words, in Systems where electron

correlation contributes substantially then penurbative series converges very slowly

resulting in an oscillatory behavior.

Chapter If: Methodology 34

The true Hamiltonian (H) is split into two namely the unpert- Hamiltonian

and a perturbation term (eq. 2.22). The unperturbed or the zeroth order Hamiltoninn, Hh

is the Fock operator obtained from the HF method.

H = H , + / w ' ...( 2.22)

A is a variable parameter, which describes the extent of the perturbation and the

~erturbation term, H' in MP theory IS given by,

1 where GHf = - 1 (J,, - K,, )

2 , , In the above equation, the first term cannot be calculated unless we adopt the mean field

approximation as in the HF method. If we were to use this approximation, the expectation

value of the perturbation term becomes zero. 1'0 overcome this difficulty, IWO new

operators for H, and H' are obtained by subtracting G"' term (eq. 2.23a) from the

original H , and H ' operators (2.23). The average values of the resulting Hamiltonians

have a finite value when calculated based on the mean field approximation.

In this formalism, the energy, E and the wave function. Y are expanded In the form of

Taylor series,

E = E'"' +X"' +A2E'" +TE"' +,.,RE'"' +... . ..(2.24)

Y = Y f o ' + ~ ~ " ' + P Y ' 2 ' + R ~ " ' +...+ RY'"' +... ... (2.25)

here, d"' and Y("' are the nth order correction to the HF energy and wavefunction

respectively. The zeroth order wavefunction is the HF determinant and the zeroth order

energy would be the sum of the HF molecular orb~tal energies. The first order correction

to the energy is the average value of the perturbation operator operated over the zeroth

order wavefunction, when summed up to the occupied molecular orbital energies gives

the HF energy. Thus, inclusion of electron correlation starts only from the second order.

MP methods are computationally more attractive than CI methods. Also. very

agreement between the results obtained from MP6 and full CI calculations has been

ob~erved.~' In contrast to the truncated C1 methods, MPn procedures are size extensive.

MP theory leads to a hierarchy of well-defined methods, which provide increasing

Chapter I1 Methodology 35

accuracy w~ th mcreasrng order of n Sometimes, the permrbat~on serles does not show a

monotoruc convergence towards a llrmtlng value, Instead, as we Increase the order of the

perturbat~on, osclllahons In a glven property are observed '" Coupled Cluster Method (CC):

Over the past 30 years, coupled cluster (CC) theory has developed as perhaps the

most rel~able computat~onal method for the predrct~on of molecular propenles of systems

whose wavefunct~ons are not multtdeterm~nantal In nature 54 Contrary to the CI

procedure, In whch the exc~ted state determinants are exphc~tly cons~dered, the CC

method uses the exponentla1 funct~on to generate the excltcd states (eq 2 26a)

In thls method, the wavefunct~on 1s glven as.

Yrr = eTY6 (2 26a)

" 1 where, er = - T" , (2 26b)

nd nt

Tls called the cluster operator, whlch IS glven as, T = T, + T: + T, + + T, , and Yo IS a

slngle Slater deternunant, usually the HF wavefunct~on The operator TI generates all the

slngly exc~ted state detemnants, TZ generates all the doubly cxc~ted state deternunants,

so on so forth The phys~cal forms of the components In the clustcr operator are.

and so on The coeffic~cnts, I : , 1;" are the coefficients and often called the cluster

ampl~tudes From the above equations, the expnnent~al of the cluster operator. T can be

rewritten as,

(2 29)

The first term In eq 2 29 glves the reference ground state wavefunct~on, the

second term, T,, generates all the slngly exc~ted states, the terms In the first parenthes~s

generate the doubly exc~ted states, the terms In the second parenthes~s generate all the

tnply exc~ted states and so on The rnarn d~fference between the CI and CC approaches IS

Chapter II Methodology 36

the manner In which the excltat~ons are treated For example, quadruply excited state can

be generated only by /V,"&) m CI (eq 2 21), whereas In the CC methodology the

quadruply excited states arise due to T4 and also vla the products of the lower cluster

operators, namely the TI, T2 and Tj T h ~ s essentrally means that coupled cluster theory

provldes a better descrlpt~on of electron correlation effects compared to the CI method

These addltlonal terms In the CC methods make this procedure st= extens~ve wh~ch IS

lacklng m truncated CI methods The energy of the system through the coupled cluster

procedure IS glven as,

E,, =(Yo 1 e 'He7 /Yo) (2 30)

lncludlng all TN In the T operator generates all posslble exclted state determinants

whlch 1s In pnnc~ple equivalent to the full CI procedure As we have seen before In the

full CI procedure, this 1s not pract~cal even for molecules w~th few non-hydrogen atoms

theretore, truncated CC methods are enecessary '"' lnclud~ng the tr~ply exc~ted state detemlnants IS computatiooally expenslve

However, cons~dering the importance of the tr~ply exc~ted state detemnants, several

approxlmate schemes were proposed to account for the tnple excltatlons, whlch can be

class~fied as ~terat~ve and nonlterative schemes

Iferafzve schemes In the lteratlve schemes, the T I effects are calculated dunng the

Iterative solut~on of the coupled cluster equatlons w ~ t h certaln approxlmatlons The

computatiooally expenslve terms and those terms, whlch do not make s~gn~ficant

contr~bution to the energy are neglected The importance of the varlous contr~but~ons

based on whlch they are neglected or Included 1s est~mated uslng perturbat~on theory

Depending on the lerms excludediapproxlmated In the tr~ples equatlons, many models are

avdllable (CCSDT-1, CCSDT-2, CCSDT-3 and CC3) " These ~terat~ve schemes showed

good performance, however, the computat~onal effort IS st111 unaffordable s~nce the tnples

contr~but~on IS calculated ~teratlvely

Nonrierafrve schemes In the nonlteratlve schemes, the T3 effects are estlrnated

nonlteratlvely and then added to the CCSD energy Two such popular ~rocedures are

CCSD+T(CCSD)'~ and CCSD(T)" methods In the CCSD+T(CCSD) approach, the

triples conhlbut~on 1s calculated from the MP4 method In calculating the triples

contrlhutlon from the fourth order energy from MP theory, the CCSD ampllmdes are

Chapter II Methodology 37

used m place of the pemcrbat~on coefficients to account for the overestlmatlon of the

tnples Tlus reduces the error caused by the overeshmatlon of the energy due to the

tnples excltahon ohtamed using the MP4 method The CCSD(T) approach Includes a

correction term due to the coupllng of the singles and tnples from fifth order penurbatlon

theory, m addt~on to all the terms ~ncluded vla CCSD+T(CCSD) methodology T h ~ s

extra term Included here IS actually posltlve, whlch Increases the energy compared to that

of the CCSD+T(CCSD) method This compensates for the overestlmatlon of the electron

correlat~on by the pure fourth order perturbatwe term The tnple excitation ampl~tudes

are calculated only once in the nonlterattve schemes, however these methods are shown

to be as good as the more expenswe ~terat~ve methods Therefore, CCSD(T) method.

wh~ch uses nomterative tnples has been a rel~able and economical optlon among CC

procedures

Quadratic Configuration Interaction (QCISD) Quadrat~c CI IS a procedure

formulated by Pople and coworkers to make the CISD method s~ze consistent '"n the

CISD method, the h e a r equahons conslstmg of the configurat~on expansion coeffic~ents

are solved iterahvely In QCISD, these equations are mod~fied by ~nclud~ng some

quadrahc terms, whch 1s equivalent to ~nclud~ng Slater determinants corresponding to

h~gher exctted states, wh~ch makes QCISD size extenslve It was shown that ~t 1s

equ~valent to the CCSD procedure, where not so Important integrals are neglected

S~rmlar to the CCSD(T) approach, tnples contribut~on to QCISD may be ~ncluded uslng a

nonlteratlve scheme, whlch is the QCISD(T) method

Multi-Configurational Self Consistent Field Method (MCSCF):

MCSCF method 1s slmilar to the C1 procedure " In MCSCF methodology, the

wavefunction 1s vanationally optlmlzed w~th respect to s~multaneous var~atlons of both

the orbitals and the configurat~on coeffic~ents It is to be noted that only the configuration

coeffic~ents are the variables in the C1 methods, whercas the MOs are kept fixed But,

vanahanal principle suggests that the orh~tal coeffic~ents In the molecular orb~tals should

be vaned m order to mnlmze the energy, whlch 1s cons~dered In the MCSCF

methodology Ths method w~ t h more configurat~ons Included In the wavefunctlon 1s a

better approach to evaluate the correlat~on effects compared to the CI procedure

However, ~nclus~on of all poss~ble configurat~ons in the wavefunchon and optrm~zlng the

Choptcr II Methodology 38

MOs and configwatton CQeff~lentS IS far from realtshc even for very small molecules

Since the MOs are opmlzed In th~s procedure, tnclus~on of only a small number of

configuratton state functtons seem to recover the electron correlatton One of the wldely

used MCSCF method 1s the complete acttve space self conststent field (CASSCF)

methodology @ The acttve space lncludes a selected wtndow compnstng a few hrgh ly~ng

filled molecular orbltals and a few low lyrng vtrtual molecular orbttals and the rematnlng

MOs are rn the lnactlve space The cmc~al task In uslng the CASSCF method IS to select

the approprtate orbltals to be rncluded rn the actlve space CASSCF(m.n) refers to the

tncluston of 'm' orbttals tn the acttve space wtth 'n' electrons and all configurattons

correspondlng to the dtstrtbut~on of 'n' electrons In the 'm' orbltals are tncluded In the

wavefuntton MCSCF methodology IS wtdely used for exctted state calculattons, however

often suffer from SCF convergence problems CASSCF procedure tends to overesttmate

the dlradtcalold character, which can be mlnlm~zed by tncludmg all valence electrons In

the actrvc space

CASPT2

The electron correlatton defictt from the CASSCF calculatton due to the tncluston

of only ltmtted number of contiguratron state funct~ons are taken care by the perturbatton

scheme 6' The first step IS to perform a CASSCF calculatton wtth the glven basts set and

the next 1s to employ second order Moller-Plesset pembatton theory, MP2 Thus. ~t

recovers both nondynanuc and dynannc electron correlattons Thts procedure has been

applted to a large number of chemtcal systems espectally In the study of electrontc

spectroscopy O2

Multi-Reference Configuration Interaction (MRCI)

In the normal Ci procedure, the electrons are exctted from the ground state Slater

determtnant by takrng the HF wavefunctton as the reference The procedure In whlch

many configurat~ons are chosen as the reference, based on which the other cxctted state

determtnants are generated, IS called the multtreference CI (MRCI) method '' For

example, a MCSCF wavefunct~on may be chosen as the reference and all determtnants

corresponding to the excited state confimattons from each of the reference deternunants

are In the CI expanston The MRCI procedure generates accurate

w t e r fI Methodology 39

wavefunchons, however It 1s computahonally very demanding even when -cat& up to

only smgles and doubles substtt~~t~ons

The other approach to deal w~th the larger molecules IS to adopt a hybnd

approach These hybnd methods dlv~de the entlre system lnto two or three dtfferent

reglons and apply different levels of approxlmatlon to each of the layers * 67 Solvent ENects

Most of the chemlcal and all the b~ologlcal processes lnvolve mtermolecular

tnteracttons and solvent effects But, computations In general are done on lndlv~dual

molecules m gas phase, slnce fust prtnctples calculatton on an assembly of molecules

w~ th mtercalated solvent molecules 1s computat~onally expensive Thls 1s based on the

assumptton that the effect due to the tntermolecular tnteractlons and solvent molecules on

a chenucal entlty IS nearly the same throughout the course of the process Var~ous

procedures for treating the solvent effects are available In general these methods can be

class~fied Into two types, (a) continuum models and (b) expllcrt solvent models

Continuum models In these models, the substrate or the molecule under study IS

immersed m a &electnc m d u m and the solvent molecules are not cons~dered

expltcltly Self-conststent reactton field (SCRF) 1s one such method, where the solvent

effect IS treated as a perturbatton The molecule 1s treated quantum mechanically and a

potentla1 IS Included In the Hamlltonlan as a perturbat~on, whlch accounts for the solvent

effect Vanous other continuum solvent models such as COSMO" and polarizable

conttnuum models70 are avatlable and are tmplemented tn var~ous quantum mechanical

software packages

Expliat solvent models In the expllclt solvent models, the tnteract~on of the substrate

w~th the solvent molecules 1s calculated expllc~tly Models of thls type are used In

systems where the tnteractton between the substrate and the solvent 1s cruc~al Molecular

dynamics and Monte Carlo stmulat~ons constder thc solvent expllc~tly

BASIS SETS

Vanous bas~s sets have been proposed over the years to descrlbe the molecular

orb~tals by a Imear combtnatton of these funchons These bas~s functions are centered on

each of the nucle~ present tn the molecule Earher the Slater T p e Orbltals (STOs) were

used to represent basts funchons, as they are s~mrlar to the hydrogenlc orb~tals '*

Chapter II Methodology 40

Calculahng the huo-electron Integrals lnvolvlng the Slater type o&ltals are

cowutatlonall~ expensive Altemat~vely, Gauss~an type orbitals (GTos) are used to

represent the basis f~nchons " A comb~nat~on of a set of GTOs wtth different exponents

and coeficlents can nnmic the ST0 type basis funct~ons The computat~onal effort w~th

even SIX GTO representing one ST0 1s much less than that uslng the or~nglnal Stater type

funct~on The maln drawback o l the Gausslan functlon 1s that it has a zero slope at the

nucleus, whle the S T 0 has a cusp, hence GTO do not represent the proper behavior near

nucle~ Var~ous types of bass sets are avallable depend~ng on the number of basls

functlons and the way the bas~s funct~ons are defined Some of the bas~s sets that were

employed In the present thew are bnefed tn thls sectlon

Pople Type Basis Sets

Mrntmal basts sers In the rmnlmal basls sets, each orb~tal In the atoms of the molecule IS

represented by only one bas~s funct~on Several mlnlmum bas~s sets are avallable,

however, the most commonly used mln~mal basis sets are the STO-nG basls sets

developed by Pople and coworkers In thls case, each of the basis funct~on 1s a llnear

combmat~on of 'n' number of pnmltlve GTOs STO-nG bass sets w~th n = 2-6 are

ava~lable," of whlch STO-3G 1s the most w~dely used one STO-nG bass sets have some

defects rnalnly because, there 1s not enough flex~bli~ty and ~mportantly, the exponents

used for s and p orbltals are the same

Double-{and &ended bas~s sers Double-6 bass set IS the one which assigns two basls

functlons for each of the orbltal, for a glven atom If these orb~tals are represented by

more than two bas16 functlons, then the bass sets are referred to as extended bas~s sets

Slmllarly In tnple-< basis sets, all the orb~tals are represented by threc bas~s functlons and

so on Thus, the rmmmal bas~s set may be v~ewed as a s~ngle< bas~s set The extended

bas~s sets allow h~gher flexiblllty for the orbital exponents, whlch correlate to the sue of

the orb~tals and hence a better descr~pt~on

Spltr-valence basts sets Spllt valence basls sets are the ones whlch asslgo one bass

funchon for the core orb~tals and the valencc part 1s modeled by two ( s~ l l t valence

double-6 basis set) or three bas~s functlons (spl~t valence double-< basis set) " (a) k-nlG

basis sels ( 3 - 2 1 ~ , 6 - 3 1 ~ ) Here the core basls functlons are modeled by 'k' number of

prlmhve GTOS The valence part 1s modeled by two basis functions where the inner Pan

Chapter 11: Methodology 41

and the outer part consists of 'n' and '1' numbers of primitive GTOs respectively. @) k-

nlm basis sels (6-31 1G): These basis sets are used to represent the valence orbitals by

three basis functions which are conbaction of 'n', 'I' and 'm' number of primitive GTOs

and the core orbitals as in k-nlG basis set..

Polarization functions: The presence of other nuclei and due to their interactions, the

electron density will be polarized in a molecule. This effect is included by adding basis

functions of higher quantum n~mber. '~ Thus, the polarization functions allow flexibility

in the shape of the orbitals. For example, a pure s orbital would be distorted and tends to

have definite p character. Hence, s orbital can be polarized by including a p type basis

function and p orbital can be polarized by including a d type function and so on. These

basis sets, which incorporate Functions of higher angular quanhlm number than those are

needed by the atom, are polarization basis sets. The polarization functions are denoted by

an asterix sign after the 'G' in the above defined basis sets (e.g. 3-21G*, 6-31G8, 6-

3 lG**. . .). A single asterix sign denotes the addition of d-type polarization functions to

the heavy atoms and the addition of second asterix imply the inclusion of p-type

functions to the hydrogen atoms.

Dt&e firnctions: In some cases, particularly in anions, the electron density is spread

farther in space. So, we require basis functions, which account for the differences of the

orbitals, is achieved by including basis function with smaller exponents.77 These extra

functions are called the diffuse functions, which are denoted by a '+' sign before 'G' in

the above defmed basis sets (e.g. 6-31tG*, 6-31++G* ...). The inclusion of these

functions is important in highly charged systems, molecules with lone pairs, etc. '+' sign

denotes addition of a set of s- and p- diffuse functions to the non-hydrogen atoms,

whereas '++' denotes the addition of diffuse functions to all the atoms including the

hydrogens (a diffuse s-function is added to hydrogens).

One basic limitation of Pople's basis sets is that they are less satisfactory when

applied in conjunction with methods including electron correlation. The Dunning's basis

sets78 and the segmented basis sets like, Atomic Natural Orbital (ANO) type basis sets7'

have been proposed to overcome these difficulties.

Chapter 11: Methodolcgy 43

E ~ ' = EMP4 + dEt + aZdi+ A E ~ ' + mHLC + (0.8929 ZPE) ...( 2.31)

The zero point energy (ZPE) is obtained at the HFl6-31G* level.

AEt = E M P ~ I ~ - ~ I I + ~ . * - EMP~I~-31 IG*. . . .(2.32) ~ ~ 2 d f = EMP~I~-31 l~(2dl.p) - E~ktb-31 IG.. . . .(2.33)

AE"' = E~clso(~ya-311c.. - EURI(I-~I I<;- . . .(2.34)

The MP4 calculations involving larger basis sets and QCISD(T) calculations are single

point calculations on MP216-31G* optimized geometries.

A E ~ ~ ~ = 4.19% - 5.9511~ ...( 2.35)

n, and ng are the number of a and p valence electron respectively.

GI theory fails to quantitatively predict the dissociation energies of ionic

molecules, singlet-triplet energy differences of carbenes, SiH2, NH3 and the atomization

energies of hypervalent molecules. To take into account of these deficiencies, Gaussian-2

(G2) theory was developed. G2 theory involves the same steps as in GI procedure and

includes more correction terms to the computed energy using the GI theory (eq. 2.36 -

2.38).

E G ~ = EGI + A1 + A2 + 1.14npnl, . . .(2.36)

AI = {E(MP~I~-31 IG(2dLp) E(MP~/~-~IIG..)I - {E(MP~I~.~II+c..- E(M~216-311~..)

- (E(M~216-311~(2df,~) - E(MPM-31 IG..)} . . .(2.37)

A2 = E(MPZI~-31 1 ~ ( 3 d f , ~ ~ ) - E(MPZ~-31 1 ~ ( 2 d r , ~ ) ...( 2.38)

n,,,, 1s the number of paired electrons.

Complete Basis Set Methods:

The complete basis set based methods such as CBS-4, CBS-q, CBS-Q and CBS-

APNO extrapolate the results obtained using the calculations done with smaller basis sets

to the complete basis set, with infinite number of basis functions.82 In this model, the

correlation energy is explicitly extrapolated, unlike in GlIG2 methods, where the

correction is an additive scheme based on results obtained using high level calculations.

The second order MP correlation energy is calculated using a limited natural orbital

expansion to reach the asymptotic limit.

Chapter 11 Mcthcdolcgy 44

In the present study, the energehcs of the skeletally subshtuted benzenes and the valence

Isomers of heterobenzenes are computed at the CCSD(T)/6-31G* level The Increment of

energy obtamed using the MP2 level whle chanpng the bas~s set from 6-31G* to 6-

311+G** bas~s set IS added to the CCSD(T) energy to obta~n the best estlmate of the

energy, from whch the relatlve energles are calculated Thus, the relatlve energles are

expected to be s~mllar to those obtalned at the CCSD(T)I6-31 1+GW level

Optimization Techniques

All the quantum chmcal molecules, whlch we have discussed above deal w~th

the electromc structure of a molecule for a glven nuclear coordlnates One has to compute

the electromc energy by altering the nuclear coordnates to attain a nunlmum, whlch 1s

done by the ophmlzatlon techn~ques 8184 Ophmlzat~on IS a process, whlch refers to

fmdlng statlonary polnts on a potentla1 energy surface, I e polnts where the first

derlvatlves of energy wlth respect to all the nuclear coordlnates are zero The statlonary

polnt obta~ned may be a mnlmum, where all the second derlvatlves are posltlve, a

transltlon state, where only one of the second denvatlves 1s negatlve, or hlgher order

saddle point, whch possesses more than one negatlve second denvatlve The stralght

forward way for ophrmzatlon n to vary one of the vanables at a tlme until the funct~on

reaches a nunimum, and then cont~nue the same procedure by changlng the other

vanables

For example, ~f T IS a funct~on of x,, x2, x,, the task IS to find all the values when If 1s

the mlnlmum

In other words, 3 = 0 for all x,, In add~t~on, all the second derlvates should be posltlve ax,

for a mlnlmum on the potential energy surface

Normally this 1s not practical, as the number of Independent vanables lnvolved In an '0'

atom system IS 30-6 (30-5 Incase of h e a r molecules) Slnce the coordlnates are

coupled, many ~teratlons over the all the vanables w~ll be necessary and become

cumbersome slnce more number of vanables are lnvolved Several optlmlzatlon

Chapter II Methodology 45

procedures to overcome these d~f icul t~w are aviulable and can be class~fied Into three

types, (a) non-denvate methods, (b) first order denvatwe methods and (c) second order

derlvatlve methods 86

Non-derivative methods

Grrd search, 1s an example of an non-denvat~ve mlnlmlzatlon algorithm In th~s

method a cublc gr~d 1s placed upon the surface and the value of the functlon at each node

1s calculated The gnd polnt w~th lowest energy 1s chosen as the mlnlmum Clearly. the

quallty of a grld search depends on the dens~ty of the gnd mesh, hlgher the density of the

gnd, larger 1s the computat~onal expense and accuracy Intt~ally, a gr~d wrth less denslty

can be used and once a lower energy 1s chosen, the m~nimlzatlon can be refined by

lncreaslng the density of the gnd Smce, we are already ~n the vlclnlty of the best polnt,

the convergence of the method further Improves

First order derivative methods

These procedures use the first denvatlve of the mult~&mens~onal energy surface

to dlrect the search towards the nearest local mlnlmurn In other words, the lnformatlon

about the slope but not about the curvature (whlch IS glven by the second denvatlve) 1s

used d u n g the optlmlzatlon procedure Steepest descent and conjugated gradlent

methods are examples of these methods In general, these methods Iterate over the

following equatlon In order to perform the m~nlmlzatlon

R, = R , , + I , S, (2 39)

Where R, 1s the new pos~tlon at step k, R,, 1s the posltlon at the prevlous step k-1, 1s the

slze of the step to be taken at step k and S, 1s the d~rect~on of that step

Steepest-Descent Method In each step of the steepest-descent method, the gradent, the

first derivative of the energy, gk IS calculated and a displacement 1s added to all the

coordinates In a dlrectlon oppos~te to the gradlent (I e , m the d~rectlon of the force) In

terms of the general scheme outllned above th~s means,

S, =-g , (2 40)

Choptw 11: Methodology 46

The step size. lk, is increased if the new conformation has a lower energy and decreased

otherwise.

Conjugated Gradient Method: Conjugated Gradients is another first order derivative

method for optimization. Unlike steepest descent algorithm, which uses only the current

gradient, this procedure uses information of the gradients from the previous steps also.

The first step is similar to the steepest descent algorithm in the sense that, a certain

displacement is added to all the coordinates from the information derived from the first

derivative.

i.e., Sk = - gk ; only fork = 1

For all steps k > 1 the direction of the step is a weighted average of the current gradient

and the previous step direction, i.e.,

Sk =-gk + h S k 4 .. .(2.41)

b, is the ratio of the magnitudes of the current and previous gradients. This information

compensates the lack of information about the curvature of the surface, i.e. the second

derivatives. Many algorithms based on conjugate gradient technique, such as Fletcher-

Reeves, Polak-Ribiere and Hestenes-Stiefel methods are available. These methods differ

in the way in which the weight factor, bk is defined. The convergence obtained by these

procedures is better compared to the steepest descent method.

Second order derivative methods:

The second order derivative methods use both the first and second derivatives of

the energy with respect to the nuclear coordinates during the minimiiation process. This

means that for a molecule of 0 atoms it requires not only the vector of 30 first

derivatives to be calculated but also the Hessian mamx of (30)' second derivatives.

Newton-Raphson method is a second order derivative method.

Newton-Raphson (NR): The basic idea in Newton Raphson minimization for a one-

dimensional case can be represented as follows,

Chapter 11: Methodology 47

where X,,, is the next position, X, is the current position and F ' ( X , ) and F'(X,) are

the first and second derivatives at X, . Near a minimum, all the Hessian eigenvalues are

postive and the step direction is taken to be opposite to the gradient direction to reach the

minima. If one of the Hessian eigenvalues is negative, the step in this direction will be

along the gradient component, and thus increase the function value. In this case, the

optimization may end up at a stationary point with one negative Hessian eigenvalue.

Although this minimization procedure is very accurate and converges very well it is

computationally expensive to apply to large systems. The need to calculate the Hess~an

matrix and calculating the inverse of this matrix at every iteration makes this algorithm

computationally 'expensive'.

In principle, optimization should yield stationary points, where all the first derivatives of

energy with respect to the nuclear coordinates should vanish. Practically, this is not

possible especially if the number of degrees of freedom is large. Hence, a threshold value

for the gradients is fixed for its root mean square value so that the optimization process

can be terminated below the threshold value.

Frequency Calculations:

Frequency calculations are performed by constructing the of the Hessian mahix

and subsequent diagonalization. Hessian matrix is a 3 0 dimensional matrix, for a 0-atom

system, containing the force constants, i.e., second derivatives of energy with respect to

the displacement of atoms in the Cartesian coordinates. The first step after obtaining the

Hessian matrix is that the matrix elements, the force constants are converted to mass

weighted coordinates. Diagonalizing the mass weighted coordinates gives 3 0 eigenvalues

and 3 0 eigenvectors. These 3 0 eigenvectors are the normal modes of the vibrational

frequencies, and rotational and translational modes. The 3 0 eigenvalues correspond to

the fundamental frequencies of the given system for the translational, rotational and

vibrational motions. For a stationary point, the freqeuencies corresponding to the

rotational and translational modes should be zero. The next step is to identify the modes

corresponding to the rotational and translational motions and separate them from the

vibrational modes. This is done by calculating the moments of inertia and diagonalizing

Cbptcr 11: Methodology 48

the resulting tensor, from which the vectors corresponding to the rotational and

translational motions can be identified. After these are separated out, the Hessian matrix

in the mass weighted c o o d i t e s is transformed to the intemal coordinates. The

vibrational part corresponding to the internal coordinates is diagonalized, and the

eigenvalues (42 d) and the eigenvectors are obtained which are the normal modes of the

vibrational frequencies in the internal coordinates. From the eigenvalues, the vibrational

frequencies corresponding to the 30-6 or 30-5 normal modes are calculated.

Calculation of Thermochemical Properties:

The thennochemical properties of a system can be calculated using the standard

procedures once the total partition function (Q) is k n ~ w n . ~ ' ~ ~ ' In this section, we derive

the total partition function The total partition function is given by,

Q = 4,4,4,4, . ..(2.43)

q,, q,, q , and q, are the electronic, vibrational. rotational and translational partition

functions respectively.

Electronic partition function (9,):

The electronic partition function is normally written as,

where E, is the energy of the th level and w, the degeneracy.

We assume that the difference between the ground state energy and first excited state is

greater than ksT and hence the excited states do not make any significant contribution to

the total partition function. This approximation fails especially in cases where the excited

states are closer in energy to the ground state.

So q, reduces to, q, = o,, which is the ground state degeneracy of the system

Vibrationalpartition function (9,):

All the 30-6 (30-5 for linear molecules) vibrational modes contribute to the vibrational

partition function. In calculating the q,, only the modes with real frequencies are

considered and those modes corresponding to the imaginary frequencies are neglected.

The partition function for a given vibrational normal mode, K is,

Choptu I1 Methodology 49

Here, OVK IS the vlbrat~onal temperature glven by. O>,, = %, VK IS the harmonic k,

frequency correspondlng to the normal mode, K The total v~brahonal partltron funct~on

1s grven by the product of all the partltlon funct~ons correspondlng to normal modes of all

the real frequenc~es

The contnbut~on from the molecular vlbrat~ons to the Internal energy 1s glven by,

In t h~s equation, the first term 1s the zero pant vlbratlonal energy

Rotational partltronfunctron (q,)

The rotat~onal partition funct~on (4,) is d~fferent for atoms, Imear molecules and nonlinear

molecules q, for atoms IS equal to 1 and the contnbut~on of q, to the entropy and internal

energy a zero For h e a r molecules, q, 1s glven by,

, I IS the moment of mema where @, = -- 8azIk,

For a nonllnear molecule, the rotat~onal partltron funct~on IS,

Translatronalpartlrron functzon (9,) The translational partltlon funct~on IS,

where V 1s glven by ksT P

In the calculat~ons of the thermochem~cal properties, they are treated as non-

lnteract~ng partlcledldeal gas Usmg the partltlon funct~ons from the electroruc,

Chapter II Methodology 50

v~brabonal, rotahonal and translational wntnbubons, the t h e m h m c a l pmpert~es such

as entropy (S) internal energy (El, enthalpy (H), G~bb's free energy (G) and heat capaclty

(C,) can be calculated from the follow~ng equations (q 2 50-2 54)

a lnQ S = k .7 (m)+kJn~ (2 50)

E = k 8 ~ 2 [ s ) (2 51) Y

H=E+k .p (%) T (2 52)

G = H - T S (2 53)

(2 54)

The present chapter glves an overvlew of the computat~onal methodolog~es of

relevance today, wlth a spec~al reference to those employed tn the them It beglns w~ th a

conclse ~ntroduct~on to the HF theory and introduces the popular cholce of computat~onal

methodolog~es In recent times, namely DFT Although, the maln advantage of DFT 1s

~ncludng the electron correlat~on effects, the l~m~tat~ons of thls approach are exposed In

cases where the funct~onal fall to reproduce the results Thls IS followed by a dlscuss~on

on the post-SCF theones, wluch are certarn ways of improving the rel~abll~ty, albeit

computat~onally more expenslve A bnef descr~pt~on of the computat~onal methodolog~es

that are ~mplemented m the thesls is glven A d~scussion on the types of bas~s sets 1s

glven next Fmally, the opt~rmzation techn~ques, frequency calculat~ons and calculat~on

of thermodynamc properties are presented

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