1

ABSTRACT

This chapter is devoted to the general discussion related to the present

thesis, the systems considered, the literature survey and the methods of calculation

are explained.

2

PART (A)

MOLECULAR POLARIZABILITY

(1A.1) INTRODUCTION :

Semi-empiricial approaches are of considerable interest for studying

molecular structure and properties of compounds. For the macroscopic

estimates of the properties of molecules, the concept of characteristic structure

is shown to be extremely helpful. Molecular polarizabilty is considered to be

one of the important properties in view of its several applications. Hasse1

evaluated the polarizabilities for the lithium ion and helium atom by using the

Lenard-Johnes potential model involving several ground state as well as

perturbed state wave functions in terms of perturbing potential. Hirschfelder2

computed the polarizability of molecular hydrogen and diatomic molecular

hydrogen ion applying the variational method of Hylleraas3 and Hasse1 and

using the eigen functions proposed by Rosen4 and Wang5. It was found that

the Kirkwood method6 is applicable to the diatomic hydrogen ion and hence

polarizabilities were obtained for many internuclear separations by using the

eigen functions developed by Guillemin and Zener7. While comparing the

values calculated by Hirshfelder2 with those of Mrowka8 and Steenshold9,

Van Vleck10 and Atanosoff11 have pointed out that the method due to Hylleraas3

is not strictly applicable to approximate eigen functions and may give values

fluctuating either high or low to the real values. Buchingham12, based on

3

Kirkwood's variational method6 , made an effort to evaluate it in the case of

heavier atoms and obtained complex self consistent field wave functions. Bell

and Long13 have calculated the polarizabilities for H2+ and H2 molecule from

six differential wave functions used. Abatt and Bolton14 on the other hand

used the polarizability of N2 molecule as a criterion for determining the

molecular wave functions of a system by SCF method. Kolker and Karplus15

used ab initio method to calculate the polarizability tensor for a series, of

diatomic molecules of first row elements.

The use of δ-function potential model was initially made by Ruedenberg

and Parr16 and Ruedenberg and Scherr17. Later, Frost18, applied the δ-function

potential model to calculate energies of various systems with the introduction

of a branching condition. This was followed by Lippincott19 with a semi

empirical δ-function potential model. Lippincott and Dayhoff20 used the semi

empirical δ-function technique in predicting vibrational frequencies,

anharmonicities, bond dissociation energies and equilibrium internuclear

separations of diatomic molecules.

Lippincott and Stutman2l applied this semi empirical model to evaluate

polarizabilities of different components (bond parallel and bond perpendicular)

in order to compute the mean molecular polarizabilites for diatomic and

polyatomic molecules.

4

Balodi and Semwal22 applied this semi empirical to determine bond

and molecular polarizabilities of Ferrocene and its derivatives and interpreted

the results in terms of observations of Waite and Papadopoulous and

others23-24. We have also applied this method for the determination of bond

and molecular polarizabilities of buckminsterfullerene and a number of

aromatic compounds and compared the calculated values with those obtained

by Semwal and Balodi25 using Miller’s method. Semwal and Uniyal26-28 have

reported the molecular polarizability of C60, C70, C84 buckminsterfullerenes.

The model assumes a potential which is infinite at the nucleus and zero

everywhere else. The integral of the potential over all space, is finite and equal

to a parameter which is called ‘ the δ-function strength’ or ‘reduced

electronegativity’ and is analogous to the effective ‘nuclear charge’ used by

Slater.

At the nucleus, a δ-function wave function is generated representing

the probability amplitude of the electron for this isolated nucleus. These δ-

function atomic orbitals are then linearly combined to form molecular orbitals

with the restriction that only two atoms may interact at a time provided if bond

exists between them. The major advantage of a δ-function model lies in its

one dimensional nature. The model assumes every bond in a molecule to be

separate entity. Since the potential has a non-zero value at only two points

5

along any given bond, the problem of calculation are trivial.

Thus careful literature survey reveals that the methods of calculating

polarizabilities can be grouped under the heads (i) Variational method

(ii) Molecular orbital Method and (iii) Semiempirical δ-potential function

method.

A molecular system may be well realised as a cluster of positive and

negative charge distributions. The force of interaction between the positive

and negative charge distributions consists of (i) attraction forces and

(ii) repulsion forces. If r be the intermolecular separation and F(r) the force of

interaction, the inter molecular potential function Φ (r) may be defined as

Φ (r) = ∫r∞ F (r) dr (1A.1)

where F(r) is the force of attraction for r>>> and the force of repulsion

for r<<<, since the nature of F(r) is dependent on the intermolecular distance,

it has been classified29 into three categories as

(i) Short range forces - applicable when molecules come close enough for

the charge distributions to overlap, hence repulsive in nature.

(ii) Intermediate forces - applicable when there is only small overlap.

(iii) Long range forces - applicable when there is no overlap, i.e. the molecules

are far apart and hence attractive in nature (dispersion forces).

6

The dispersion forces in particular can be expressed in terms of

polarizabilities of molecules. Much importance has been attached to the

evaluation of intermolecular forces to understand wide variety of phenomena

in molecular systems. In the subsequent section, some of the intermolecular

potential functions and methods for evaluating average molecular polarizability

have been discussed. Numerous spectroscopic phenomena,. like hydrogen

bonding, spectroscopy of compressed gas, have also been explained on the

basis of intermolecular forces30-31.

7

(1A.2) PRINCIPAL OF COMPUTATIONAL PHYSICS AND

QUANTUM MECHANICAL MEHTODS :

Thus, computational physics emerged in the last one decade or so due

to the growing use of computers in physical sciences12-13. In this branch of

physics starting from the study of the nature of bonds and structures to the

estimation of physico-chemical properties of molecules, including the study

of behaviour of chemical reactions is done theoretically using computational

physics methods, modeling and stimulation schemes and combinatorial physics

approaches. In fact, there are three broad classifications of methods of

computational physics.

1. Quantum Mechanics: A quantum mechanical model of electronic

structure of a molecule, which involves solving the Schrodinger equation.

Quantum mechanics can be used to predict electronic properties of molecules,

such as dipole moments, polarizabilities, spectral transitions and

spectroscopy16.

2. Semi empirical Quantum Mechanical Methods: Semi empirical

molecular orbital methods have been developed which ignore or approximate

some of the integrals used in the Schrodinger wave equation. To compensate

for neglecting the integrals, the semi empirical methods introduce parameters

8

based on molecular data. The most commonly orbital package (MOPAC)

computer programs. For example, in the computer-aided physics18 for windows

software package includes several computational physics tools including a

package of MOPAC. The MOPAC offers two of the most important semi

empirical methods (PM3 and AMI) for prediction of heats of formation, ground

state geometries and ionization potentials. Their applications to real molecular

systems have to be tested. The hydrocarbons offer very good platform to test

such theories and packages.

Other semi empirical molecular orbital method for the study of electronic

polarization and related physico-chemical properties includes the use of Dirac

delta function model of bonding. Recently developed technique like density

functional theory could also find use in predicting the properties of

hydrocarbons. Rules of molecular connectivity, and graph theoretical

approaches also find use in predicting molecular properties.

3 Molecular Mechanics: Molecular mechanics (MM) is valuable tool for

predicting geometries and heats of formation of molecules for which force

fields are available. In a molecular mechanics method, the "static energy"

calculated for the minimum energy conformation is not the energy at the bottom

of the potential well but is a good approximation for the thermally averaged

energy. This is because force field parameters obtained by fitting to

9

experimental data are thermally averaged values. Simulation and modeling,

of course, deals with the chemical engineering aspects of a reaction using

principles of mathematics and conditions of reactions, its importance is vital

in applied physics.

10

(1A.3) GENERAL THEORIES OF BOND MOMENTS AND

MOLECULAR POLARISABILITIES:

In an insulator the electrons are firmly attached to individual atoms,

molecules or ions. When such an insulator is placed in an electric field, the

Electrons are pulled slightly in one direction while the nuclei are pulled in the

opposite direction. The relative displacement of charge in matter is referred

as ‘polarization’. The differences between multipole moments of the

undistorted charge distribution are known as the induced multipole moments.

For small external fields, due to interaction with an electromagnetic radiation,

E = E0 cos (2πυt) where E0 is the amplitude and ‘t’ the time an induced dipole

moment is created. The dipole moment (p) which is induced is propotional to

the electric field, the proportionality constant ‘α’ being termed the

‘polarizability’ of the molecule. It may be represented in the form

P = α E = E0 cos2 πνt (A1.2)

The polarization of matter due to the influence of an applied electric

field is due to two sources (i) the polarizability of individual molecules and

(ii) applied electric field.

When the carbon and hydrogen bond of molecules oscillates a large charge

transfer phenomenon occurs that makes the infrared intensity complicated to

understand. Happily, several schools have contributed to our knowledge of

the nature of CH bond dipoles. In general, the dipole of a bond AB is defined

as follows:

11

If a system placed in a uniform electric field experiences a twist , measured by

turning moment , which is proportional to the field strength , then the

proportionality factor for the maximum twist in a given field, which is

characteristic of the system, is called its electric dipole moment. Such an effect

can arise if the system has poles or centers of action of its positives and negative

charges (+e and e) at fixed position separated by a vector distance, for then

when is at right angle to the field, F (see Fig. 1.1) the turning moment, is Fe.

The electric dipole moment is e and commonly written. Now the atoms in a

bond AB are allowed to perturb each other, an electron from A shifts its mean

position a distance x Å towards B, while one from B similarly shifts (1-x) Å

towards A . The net moment created will be e(l-2x) , which is zero only when

x = ½.

The order of the magnitude of electric dipole moment which may arise

in molecules may be visualized by considering the electronic charge

(4.8 x 10-10 esu), so that if an electron moves I Å i.e., a distance of the order of

magnitude of the bond distance, a moment of 4.8 x 10-10 esu is created . This

is usually called 4.8 Debye unit or 4.8 x 10-18 D (3.335640 x 10-30 c-m). Now,

when dealing with the intensities of a Vibrational spectrum of hydrocarbons,

this basic quantity, i.e., the CH bond dipole, is going to be strongly affected

and, in particular, there oscillations. Methods for infrared intensity calculation

have been published5-15 and were the subject of review by R.P. Semwal, N.K.

Sanyal19, and others20-25.

12

It is worth mentioning that the infrared spectroscopy of hydrocarbons

and modern concept on bonding based on so called molecular orbital theory

coupled with the conceptual frame of spectral intensity prediction are readily

available for the solution of the problem of CH bond moments26-30 . We have

good reason for believing that the true CH bond moment is only about 0.4 D.

The immediate questions concern the theoretical and experimental basic for

believing the magnitudes of CH bond moment, Since 1958, when this value

was presented , the CH bond dipole has been the subject of extensive studies

by infrared spectroscopists and theoreticians , and the recent past divergent

opinions have been expressed concerning the magnitude and sense of CH

bond moments in hydrocarbons.

FIG. 1.1 The dipole in a uniform field and development of a dipole bybond formation.

As a result of this, many contradictory ideas dealing with the clarification

of a common progency, i.e., its sense and magnitude, have been presented.

+++++

—————

I

13

Since the parameters associated with a CH bond dipole represent basic

quantities in setting values of bond moments in substituted aliphatic and

aromatic molecules they are of paramount importance and, therefore , efforts

are perpetually being made to clarify and settle the situation. The development

of the problem of CH bond moments had been reviewed by many experts

including a most authoritative report by Orville-Thomas and coworkers who

have tried to reconcile the situation on the basis of infrared intensity analysis,

structural parameters, and most modern concepts of dipoles and their

interactions as encountered in the framework of molecular orbital theories.

The experimental determination of effective charges on atoms in molecules

has achieved widespread attention. The main impetus in this field, besides the

applications of NQR, ESCA, and the Stark effect, has been from infrared

intensity spectroscopy. The reviewer will confine his major coverage of the

discussion to infrared spectroscopy. Because the infrared absorption intensity

depends upon the dipole moments derivatives where Qi is the normal

coordinate of the i-th vibrational mode of the molecule, in principle it should

be possible to extract information about the polarities of bonds as well as their

magnitudes. Onces under stood, the bond moment parameters may be fruitfully

utilized in the prediction on infrared spectral intensities whose basic usefulness

is more or less established.

Result can be placed in a simple and more explicit form. The diagonal

form of αmol has two distinct components α|| and α⊥ parallel and perpendicular,

dµdx( )

14

respectively, to the bond axis. Silberstein's equations for this case are the

following which may also be derived from the above.

α = (αA + αB+4 αAαa/r²) / (1 - 4αA αa/r²) (1A.3)

α = (αA + αB+2 αA αa/r²) / (1 - 4αA αa / r²) (1A.4)

These relations illustrate certain essential features of the atom dipole interaction

model :

(i) the molecule becomes anisotropic even though the atoms are isotropic;

(ii) the predicted polarizability of a macula parallel to its long axis is generally

greater than that perpendicular to the long axis ; (iii) deviations from additivity

of polarizabilities become large as the atom polarizabilities approach r²

(1A.3.1) MILLERS AND SAVCHIK METHOD

Miller and Savchik5 proposed a new empirical approach to the calculation of

the average molecular polarizability by taking it as a square of the sum of

atomic hybrid components. In this work, we have applied the method

successfully to a number of aromatic hydrocarbons and the three fullerenes.

C60, C70, and C84. The functional form of the method is represented by the

equation.

α(abc) = [ r A ]² (Å)³ (1A.5)

15

(1A.3.2) CHARACTERISTIC C-C AND C-H BOND POLARIZATION

The polarization characteristic of (C-C) and (C-H) bonds are discussed

considering variation if bond length, bond multiplicity and state of hybridization

in case of hydrocarbon molecules. Choosing electron reshuffling in the bond

region as criterion, polarizability components are computed using Lippincott-

Stutman method. Computed values are found in good agreement with the

literature values of cyclo paraffins, isomers of benzene and some hetero

compounds.

Semi empirical approaches are of considerable interest for studying

molecular structure and properties of hydrocarbons. Bond lengths in

hydrocarbons have been the subjects of theoretical and experimental studies.

While delaing with αm values of hydrocarbons by the δfunction potential

technique, the parallel component of' polarizability (αp) are found to vary

proportional to 4th power of the (C-C) and (C-H) bond lengths. Because of

this, and particularly when bond characteristic are aggregated for total

polarizability value of molecule, it becomes important to examine fluctuation

in (αp) with respect to internuclear separation (R).

The chemical concepts of the state of hybridization, C(Sp³) - C (Sp³)

etc. ionicity of (C-X) bond and multiplicity (bond order n = 1,2,3, ....) lend

support to the concept of Bartell14. Hence while dealing with the polarizability

16

components to a first approximation, Lippincott-Stutman19 introduced the inter

nuclear separations term (R) in such a way that all the concepts of chemical

binding are automaticaly assimilated within the framework of the theory.

17

(1A.4) THE DELTA FUNCTION MODEL OF CHEMICAL BINDING

AS APPLIED TO POLARIZABILITY COMPUTATIONS

The delta (δ) function method utilises the principle of additivity of bond

polarizabilities and δ-function branching condition of Frost18. Initially

Lippincott and Dayhoff20 used semi empirical δ-function technique in predicting

vibrational frequencies, anharmonicities, bond dissociation energies and

equilibrium internuclear separations of diatomic molecules. Lippincott and

Stutman21 proposed a semi empirical model based on δ-function to evaluate

polarizabilities of different components (bond parallel and bond perpendicular)

and mean molecular polarizabilities of polyatomic molecules. The model

assumes a potential which is infinite at the nucleus and zero everywhere else.

The integral of the potential over all space is finite and equal to a parameter

which is called' the δ-function strength' or 'reduced electronegativity' and is

analogous to the effective nuclear charge used by Slater. At the nucleus a δ-

function wave function is generated representing the probability amplitude of

the electron for this isolated nucleus. The δ-function atomic orbitals are then

linearly combined to form molecular orbitals with the restriction that only two

bonded atoms may interact at a time. The major advantage of this model lies

in its one dimensional nature. Every bond in a molecule is regarded as separate

entity.

The form of assumed potential function acting on the nuclei of two

18

atoms may be written as

V = - [A1g δ(x - a/2) + A2g δ(x + a/2)] (1A.6)

Where A1, and A2 are the reduced electronegativities or δ function strengths of

the nucleus 1 and 2 respectively, x, the coordinate of motion, a, the δ-function

spacing, g, the unit δ-function strength (the case of hydrogen atom) and δ(x)

the δ-function, such that δ-function, such that δ(x) = 0 when x ≠ 0 and

δ(x) = ∞ when x = 0, and ∫0

∞ δ(x)dx = 1 for any argument of x. This implies

that the potential is zero everywhere except at the δ-function positions, i.e

x = ± a/2. The δ-function strength is related to separated atom energies, Ei, by

the relation A = ( -2Ei)½ which in turn is obtainable from the first ionization

potential or from the solution of the corresponding Schrodinger equation solved

by δ-function model. The latter yields.

Ψi = N [ exp (-Ci | Xi + a/2 |) ± exp (-Ci | Xi - a/2 |)]

N = (2/Ci) [{ 1 ± exp (-Cia) ± aCi exp (-Cia)}]½

Ci = (-2Ei)½ (1A.7)

Using Frost's δ-function branching condition18 the expression of Ci, for

homonuclear molecule is

Ci = Ag [1 ± exp (-Cia)] (1A.8)

Where plus and minus signs represent repulsive and attractive state respectively.

19

The above equations also yiels a limiting value of Ci as

Lim Ci = Ag = (-2Ei)½ a= ∞ (1A.9)

a → ∞

Lippincott and Dayhoff20 generated a 'super' one electron situation from the

corresponding n-electron situation to overcome the problem of obtaining Ci

values for individual electron. Hence the resultant, C, commonly written as

CR may be obtained as

CR= A (nN)½ for homonuclear cases)

CR12 = (CR1 CR2)½ = (n1, n2 N1 N2)

¼ (A1 A2)½

for heteronuclear cases) (1A.10)

Where A1, and A2 and δ-function strengths for atoms 1 and 2 in a bond, n1 and

n2 the principle quantum numbers for atoms 1 and 2, and N1, and N2 the number

of electrons contributing to the binding or two times the column number in the

periodical table in which the ith atom falls. The δ-function strengths for singly

bonded atoms differ from those with multiple bonded atoms due to the

difference in electronic distribution in the two cases. Lippincott and Dayhoff2O

have also found an empirical relation for, A, in terms of principal quantum

numbers.

20

(1A.5) ATOMIC POLARIZABILITY

According to the variational treatment of Hylleraas3 and Hasse1, the component

of polarizability is written as

αxx = (4n A/a0) [(x1 - <x>)² - (n-1) < (x1 - <x>) (x2 - <x>)>]² (1A.11)

where x is coordinate of any one of the n equivalence class of electrons

which falls in the first equivalence class, <x> is the average coordinate of any

one of these electrons and a0 is the radius of the first Bohr orbit for atomic

hydrogen. Under the assumptions that the δ-function potentials are

symmetrically placed and all the bonding electrons are equivalent the above

equation may be written as

αxx = (4nA/a0) (<xi2>)² (1A.12)

Using the fact that the polarizability of an atom is isotropic, i.e. <x²> =

<y²> = <z²> = (1/3) <r²>, and for a δ-function located at nucleus one may

have Ψ =Ne-A r (where N is the normalization factor = A3/2 n-½), one gets

<r²>= ∫0π

∫0 2π

∫0∞ Ψ r² Ψ∗ .r² sinθ dθ dφ dr = 3/A2 (1A.13)

so that <x²> = 1/A². Finally, the polarizability of the atom along X-axis

is given by

αxx = 4

a0³ A³ (1A.14)

21

(1A.5.1) BOND POLARIZABILITY

The mean polarizability, αM, for a diatomic molecule with an axis of

symmetry can be written as made up of bond parallel component, αII, and

bond perpendicular component, α⊥ of polarizability, i.e.

αM = (1/3) [αII + 2α⊥ ]. (1A. 15)

The component arises due to the contributions from (a) bond region

electrons, and (b) non bonded region electrons. The contribution of bond region

electrons is obtained by using a linear combination of atomi δ-function wave

functions representing the pair of nuclei involved in the bonding and is given

by

αIIb = 4n A12 (1/a0) ( <x²>)² (1A.16)

where n is the bond order, A12 the root mean square δ-function strength

of the two nuclei, and

<x²>=R²/4+1/2C2R (1A.17)

R being the equilibrium internucler distance. In case of heteronuclear

bond, the difference in electronegativities of atoms introduces charge density

which must be accounted for by applying a polarity correction to αIIb given by

σ = exp [(-1/4) (X1 - X2) ]² (1A.18)

22

Where X1, and X2 are the electronegativities of atoms 1 and 2 on

Pauling's scale32 Thus the corrected value of the parallel component of bond

polarizability is given

αIIp = ( αIIb) σ (1A.19)

The contribution of non-bonded electrons, i.e. the electrons in valence

shell not taking part in the bonding, is calculated by using Lewis-

Langmuir33-34a Octet rule with a modification by Linnett34b in terms of double

quartet of eletrons. The general expression may be written as

ΣαIIn = Σ fj αj (1A.20)

Where fj is the fraction of the electrons not involved in bonding for the

jth atom with atomic polarizability αj.

The a⊥ component i.e. the perpendicular component of bond

polarizability as obtained by Lippincott and Stutman21takes into account the

residual atomic polarizability degrees of freedom, which for an isolated atom

is three. In case of diatomic molecule, this component is simply the sum of the

two atomic polarizabilities. But if one of the atoms is less electronegative than

the other, the average magnitude of components perpendicular to the bond

will be function of the charge separation and atomic contributions will be

considered according to the Square of the respective electronegativities.

23

Extending the principle to polyatomic molecules, the analytical expression

for all the perpendicular components is given by :

ΣXj² αj

Σ2α⊥ = ndf (1A.21)ΣXj²

where ndf is a symmetry and geometry dependent quantity and is

denoted by a factor (3N-2b), gives the idea of residual atomic polarizability

degrees of freedom. The determination of ndf is based upon some basic

assumptions. Every bond which is formed between two atoms removes two

polarizability degrees of freedom with the restriction that - (i) if two bonds

are formed from the same atom and exist in a linear configuration, three atomic

polarizability degrees of freedom are lost, and (ii) if three bonds are formed

from the same atom and exist in a plane, then five atomic polarizability degrees

of freedom are lost.

24

(1A.6) CONTRIBUTION DUE TO NONBOND-REGION ELECTRONS

TO PARALLEL COMPONENT

In the valence shell of each atom the remaining electrons also contribute

to the bond parallel component, or αII. But there must be a residual polarizability

along the internuclear axis, and this additional component is expected to be

independent of bond length. A general expression to obtain this contribution

can be written as

Σ α11n = Σ fj αj (1A.22)

where fj is the fraction of the electrons of the jth atom which are not in

the bond region and αj is the atomic polarizability of the jth atom.

(1A.6.1) PERPENDICULAR COMPONENT

In the Lippincott and Stutman method the perpendicular component is

obtained by considering the residual atomic polarizability degrees of freedom

which for an isolated atom is three. Thus α⊥ = 2αA for a nonpolar diatomic

molecule and α⊥ = αA + αB for a polar AB molecule.

Let us consider two atoms A and B, A is less electronegative than B, the

average magnitude of components perpendicular to the bond will be a function

of the charge separation and atomic contributions will be considered according

to the square of the respective electro negativites. Hence, the analytical

expression in this case will be

25

2 (χ²A αA + χ²B αB)α⊥ = (1A.23)

χ²A + χ²B

The above equation is modified in the case of polyatomic molecules,

which is written as follows :

Σ2α⊥ = ndf Σxj² αj

Σxj² (1A.24)

where ndf is calculated by 3N-2nb when N is the total number of atoms

and nb is the total number of bonds in that atom.

(1A.6.2) MEAN POLARIZABILITY (αM)

The mean polarizability or average molecular polarizability is the sum

of the parallel component, the perpendicular components of bond

polarizabilities and the contribution of nonbonding electrons. So the αM is

expressed as :

αM = 1 ΣαII pi σ + Σfiαj + ndf

Σxj²αj (1A.25)

3 Σxj²

where

αIIp = 4n Ao R² + 1 ² e -(χ

A-χ

B)

a0 4 2CR²12 (1A.26)

In the above equation A12 is the δ-function strength for a bond, a0 is the

[ ]

[ ]

26

Bohr radius, the covalent bond character as defined by Pauling and n is the

bond order. The atomic polarizabilites α (in 10-25 cm³) is taken from the work

of Lippincott et al.

(1A.6.3) AVERAGE MOLECULAR POLARIZABILITY :

The average molecular polarizability, αM is independent of bond angles

and is obtained from contributions of (i) the parallel components due to bonding

as well as non-bonding electrons, and (ii) the perpendicular components of

bond polarizability. Thus αM is expressed as

αM = 1 Σi αIIb σ + Σj fj αj + ndf

Σj Xj² αj (1A.27)

3 j αj²

Where the summation refers to i bonds and j atoms.

= 8 x 1.1964242

= 9.5713936

A12 = √A1 A2

= √.846 x 1.00

= .9197826

<x>² = R²

+ a²0

4 2CR²12

= (1.11)²

+ .279841x10-16

4 9.5713936

][

27

= .331737 x 10-16

αIIb = 4n A12 [<x>²]²

a0

= 4 x 1 x .9197826 (.331737 x 10-16)²

.529 x 10-8

αIIb = 7.65 x 10-25

For C-H bond

σ = e-¼ (x1 - x2)²

= e-¼ (2.5-2.1)²

= e-0.04

= -0.4 x .4342942

= .07371768

= 1.982628

σ = .9607916

αIIp = αIIb σ

= 7.653814 x .9607916 x 10-25

= 7.353720747x10-25

to the case of Methane, their are four C-H bonds,

= 4 x7.353720747 x 10-25

= 29.41488 x 10-25

ΣαIIp = 29.41488 x 10-25

Σfj αj = 0

28

Σ2α⊥ = ndf Σxj² αj

Σxj²

= 7 (2.5)² x 9.78 + (2.1)² x 5.92 x 4

(2.5)² x 1 + (2.1)² x 4

= 7 x 6.929836

= 48.50885

αM = 1/3 [ΣαIIp + Σfj αj + Σ2α⊥ ]

= 1/3 [29.41488 x 10-25 + 0 + 48.50885]

= 1/3 [77.92374]

αM = 25.97458x10-25

][

29

PART (B)

ATOMIC HYBRIDIZATION AND MOLECULAR POLARIZABILITY

(1B.1) INTRODUCTION

The most successful empirical approaches to the calculation of molecular

polarizabilites include the environment through atomic hybridization about

the central atom or with the atoms forming a bond or group. For this purpose

the concept of atomic hybrid component (ahc) and atomic hybrid polarizability

(ahp) was introduced. The simplicity of the ahc and ahp atomic hybrid method

is understood by nothing that the molecular polarizability cannot be written as

a sum of atomic polarizabilites, when atoms are defined only by the atomic

number.

It is well known that the atomic polarizability cannot be added simply

to get molecular polarizability. For this, each atom must be assigned a

polarizability depending on the atoms to which they are bonded. Eisenlohr35

and Vogel36 have set up an extensive system of atomic refractions which have

been supplemented by Batsanov37 in which, for example, nitrogen in primary,

secondary and tertiary amines is assigned different values. Silberstein38 pointed

out that the ever growing hierarchy of rules indicating how to treat the

exceptions to the law of additivity, although helpful to the physicist, is the

dearest confession of non additivity, unless the atomic environment is

considered in great detail. The methods that consider atoms in atomic hybrid

states eliminate the need to develop an extensive set of rules that consider the

specific atoms involved in bonding.

30

(1B.2) ATOMIC HYBRID COMPONENT

To circumvent this problem a new empirical approach to the calculation

of the average molecular polarizability as a square of a sum of atomic hybrid

components is presented by Miller &Savchick39-40. It requires fewer parameters

than other methods. and has the functional form

α(ahc) = 4/N [ΣA τA]² (Å)³ (1B.28)

Where N is the number of electrons in the molecule, and ahc refers to

the atomic hybrid components τA of α for each atom in a particular hybrid

configuration. The summation proceeds over all atoms A in the molecule. In

the present formulation τA does not depend on atoms which are bonded to A,

but it does depend on the type of bonding through the hybridization of atomic

orbitals on A.

Semwal, Uniyal and Balodi26-28 have applied this method successfully

to a number of aromatic hydrocarbons and three fullerenes C60, C70, C84. We

have applied this method to determine the polarizability of some organometallic

compounds which have recently attracted the interest of those who search for

novel and efficient non-linear optical materials41-45.

The concept of bond polarizability was introduced here and Steiger

and Smyth46(a,b) obtained bond polarizabilities directly from the atomic

31

refractions47. In contrast, Denbig48 developed a system in which a unique bond

polarizability is obtained directly for each kind of chemical bond from molar

refractions R, for example R(C-H) is 1/4R (CH4) and for the alkanes

(n-1) R (C-C) + (2n + 2) R (C-H) = R (Cn H2n+2)

from which R(C-C) follows.

The empirical formula eq. (1B.28) is presented as an additional method

for the calculation of average molecular polarizabilities. It is easy to apply.

Fewer parameters are needed in this approach than in other methods because

only hybrid components are considered and not the specific atoms to which a

given atom is bonded. For each atom, A, one valuer τA is used for each type of

hybridization. The number of and type of bonds as well as lone pairs are

automatically incorportated into a parameter τA by the hybrid state of each

atom.

32

(IB.3) RATIONALIZATION OF THE EMPIRICAL APPROACH

The functional form of the empirical formula proposed in this

investigation can be rationalized, with the variational-perturbation approach

proposed by Hylleraas3 and Hasse1 and approximated by Hirschfelder, Curtiss,

and Bird29 for the calculation of the x component of molecular polarizability.

αXX = 4N/a0 [(x1-x)² - (N-1) (x1- x) (x2-x)]² (1B.29)

Where N is the number of electrons, x is the average position of an

electron in the x direction, (x1-x)² is the mean square deviation of an electron

from its average position, and (x1-x) (x2-x) is the average correlation between

two electrons in the x direction. The average value of the operator q, is

q = ∫Ψ∗qΨ dτ (1B.30)

it is calculated with the zeroth order wave function Ψ obtained from

perturbation theory. If the term in brackets in eq.(lB.29) is rewritten to include

a summation over all electrons i, j = 1, 2, 3, ...................,V, then

αxx = 4/Na0 [Σij (xi-xi) (xj-xj)]² (IB.31)

= 4/Na0 [ Lxx ]² (IB.32)

33

The term in brackets in eqs 1B.31 and 1B.32 reduces to

Lxx = Σij xi xj - (Σi xi)² (1B.33)

For closed shell systems the zeroth order wave function may be approximated

by an antisymmetrized product of molecular orbitals Ψµ, µ= 1,2,3, ................

and spin function η and η with z components equal to ± ½

Ψ = (N!)-½ A {Ψ1 (1) η (1) Ψ1 (2) η (2)............} (1B.34)

Substituting eq. (IB.34) in eq. (IB.33) yields

Lxx = - 2 ΣµΣν∫Ψ∗

x Ψν dτ ∫Ψ∗µ x Ψµ dτ + 2Σµ ∫Ψ∗

µ χ² Ψµ dτ (1B.35)

where µ and ν refer to the occupied molecular orbitals. The molecular orbitals

are expanded as a linear combination of atomic orbitals χAt

Ψµ = ΣA Σt CAtµ χAt (1B.36)

where t may refer to S, px, py and pz atomic orbitals or to hybrid atomic orbitals

on atoms A= 1,2,3----substituting eq IB.9 into eq. IB.8 and using the zero

differenntial overlap approximation yields.

Lxx = ΣA √a0 τAxx (1B.37)

34

Where √a0 τAxx =-2 ΣBΣtΣuΣvΣwΣµΣν C*Atµ CAuv C* Bvν CBwµ x ∫χ∗Atx χAu dτ

∫χ* Bv X χBw dτ + 2 ΣtΣuΣµ C*Atµ CAuµ x ∫χ*At X² χAu dτ (1B.38)

provides the link between the proposed empirical, formula, eq. 1B.28 and a

molecular orbital model.'Atomic orbitals t, u, v and w are centered on atoms

A and B in At, Au, Bv and Bw

It is convenient to assume that Ψµ, µ = 1,2,3.. are the localized molecular

orbitals studied extensively by England et al49a,b-50. Their orbitals, confined to

the region of a chemical bond, are written as a sum of hybrid atomic or π

orbitals. For acylic hydrocarbons two centre localized molecular orbitals are

found, whereas for condensed hyrbocarbons two to four center localized

molecular orbitals connected through a set of adjacent atoms are obtained.

Now, the contributions to τAxx consist of two types of terms. The first eq

(I B.38) connects atomic orbitals on a pair of atoms A and B and the second

involved only atomic orbitals on each atoms A. The first term may be viewed

as a bond contribution because the coefficients C*Atµ CBwµ and CAuν C*Bvν

connected through common localized molecular orbitals Ψµ, and Ψν are largest

if A and B are adjacent atoms. If these bond contributions, which arise for

each hybrid atomic orbital in Ψµ on atom. A connecting adjacent atoms, are

partitioned between the two atoms A and B, then one has a rationale for the

35

assignment of a unique value of τAux for each hybrid atomic state. If eq.

(1B. 38) is written as

√a0τAxx = -½ ΣBΣtΣuΣvΣw PAtBw P*AuBv ∫χ∗ AtX χAu dτ

x ∫χ∗Bv x χBw dτ + ΣtqAt ∫χ∗ AtX² χAt dτ (1B.39)

where

PAtBw = 2 Σµ C* Atµ C BWµ (1B.40)

is the bond order for doubly occupied orbitals qAt = PAt Av* which arises

because symmetry requires that t = u in the second term of eq (1B.38), is the

electronic charge, then the bond and atomic terms are more obvious. Bond

orders are small unless A and B are adjacent atoms.

This approximation ought to work well for covalently bonded systems

in which the electron pairs are nearly equally shared by two atoms. For very

polar systems, some modification of τAχχ will be required. However, in this

investigation, eq (1B.28) reproduces experimental values of average molecular

polarizabilities of organic compounds quite well.

To complete the rationalization of eq (1B.28) the component of

polarizability αXX obtained with eq (1B.29), and similarly αyy and αzz are

assumed to be calculated with the xyz coordinate system oriented along the

36

principal axis of the polarization epsoid. If the contribution of τAalong each

principal axis is assumed to be given by

τAkk.= √3 cosγkk τA (K = x,y, or = z) (1B.41)

where

Σkcos² γkk = 1 (1B.42)

then the average molecular polarizability

α = 1/3 Σkαkk (1B.43)

is obtained from

αkk = 4/N [ΣkτAkk ] ² (1B.44)

by combining eq (1B.32) with eq. (1B.37) and (1B.41)for K = x and

similarly for

K = y and z. The component τkkA are projections of √3τA onto the principal

axis K = x, y and z, or from the components of the polarizability

√αkk = cos γkk √3α (1B.45)

is the projection of 3α onto the principal axis.

In the case of condensed hydrocarbons, the localized molecular orbitals

are delocalized over more than two atoms. The contribution to each atomic

hybrid component is still obtained by partitioning the effect of terms involving

A and B over each atom. The assumption is made that there are two values τA

for the trtrtr π hyrbid states of carbon τc = 1.428 for one with a C-H bond and

37

τc = 1.800 for one in branched configuration as in graphite. This is the only

exception made in the formulation of the average molecular polarizability with

parameters τA for atoms, A, in their conventional states of hybridization and it

is required to achieve agreement between α(ahc) and α(exp).

An alternate approach to that proposed in this investigation, in which,

A represents atomic hybrid contributions, would be assumption over bond

contributions with a formula like eq 1B.28. In eq 1B.38 or 1B.39, terms

involving orbitals on atoms would be partitioned onto pairs of atoms connected

by a bond or several atoms encompassed by the localized molecular orbital in

the case of condensed hydrocarbons, and τ would be accordingly redefined.

The terms connecting pairs of atoms A and B are already cast into a form

which could be viewed as bond contribution. This approach was not explored.

It would require a set of parameters τA, for every pair of atoms with each

particular bonding type, and consequently there would be greater number of

parameters than required in the present empirical approach.

The Van der Waals radii can be correlated to the atomic polarizability

αA = 4/NA (τA)² (1B.46)

calculated with eq (1B.28) for each atomic hybrid contribution τA. Combining

eq (1B.46) with the Slater Kirkwood approximation50.

38

4 rA² ² (1B.47)αA =

a0 3

yields

√r²A = 3 (a0αA)¼ (1B.48) 2√

][

39

PART (C)

HUCKEL MOLECULAR ORBITAL METHOD

(1C.1) INTRODUCTION :

Several semi-empirical theories have emerged which are capable of

predicting the properties of molecules through mathematical equations of wave

mechanism. The most important of these theories belong to a class known as

the molecular orbital theory which is based on a set of approximation. The

simplest of these molecular methods is one developed by Huckel1 and known

as Huckel molecular orbital (HMO) method. Though this method is very

primitive and the crudest of all the molecular orbital methods, it has been

extensively and successfully used in several descriptions52-64. It has also been

argued that for a comparative study on π-electron systems, this method is as

good as any refined method65-70 and results obtained by it are if not exact are

very close to observed values.

The major advantage of the HMO method is that it is conceptually simple

and mathematically precise in the sense that it rests quite simply on Huckel’s

original assumption51 that to a first approximation only the π-electrons need

be considered in calculating the properties of such molecules as olefins,

polyenes, aromatic hydrocabons, and their derivatives including the compounds

containing cyclopentadienyl rings like chromocenes. In addition to it in this

method the diagonalization of secular determinants is relatively easier than

40

any other process involving iterative procedure, because the secular equations

has to be solved once.

Many attempts have been made71 to allow for the effects of σ-electrons

in some way or the other but all of these are not so successful, because the

inclusion of σ-electrons in MO treatment poses many problems. First, there is

the sheer magnitude of the calculations, given that the total number of valence

electrons in a molecule is very much greater than that of π-electrons in a

conjugated system of comparable size; benzene, for example contains only

six π-electrons, out of a total of thirty valence electrons, secondly, the MOs no

longer have the symmetry of those in π-system being three dimensional.

Thirdly, there is the problem of invariance in choice of coordinate axis and

finally there is the problem of predicting molecular geometry71.

In addition to these factors out of valence electrons containing σ and π

both, the σ-electrons are strongly localized in individual bonds and are

represented by bond orbitals of σ-symmetry. However, the π-electrons are

highly delocalized over carbon skeleton and chemically reactive. That is why

the π-electrons are treated in an explicit fashion and σ-electrons are regarded

as providing some sort of potential field for the π-electrons. Here every system

is divided into σ and π parts, later on we will treat all β as equal and all σ as

constant.

In the Huckel derivation, the individual MOs of a molecule are

41

eigenfunctions of the corresponding one electron Hartree-Fock operator H;

HΨµ = ∈µΨµ (1C.49)

where H is Hamiltonian operator, ∈µ is the energy of the µth state and

Ψµ is wave function of the µth molecular orbital. Instead of solving we

approximate the molecular orbitals by using the variation method with an

LCAO trial eigenfunction;

Ψµ = Σi aµi φi (1C.50)

where aµi is coefficient of the µth state for ith atom and φi is atomic

orbital for the ith atom. Assuming neglect of differential overlap, the

corresponding variation equations are

|Hij - ∈δij| = 0 (1C.51)

Σj aµj (Hij - ∈µδij) = 0, i = 1, 2, 3, --------N (1C.52)

where Hij = ∫φiHφjdτ (1C.53)

Instead of calculating the intergrals Hij, we treat them as parameters.

The diagonal element Hij is called a Coulomb integral and is supposed to have

a value αi characteristic of the AO φi and the atom of which is an AO, and

independent of the rest of the molecule, while the off-diagonal element Hij,

written as βij and called a resonance integral, is likewise assumed to have a

42

value characteristic only of the AOs φi, φj. It is usually assumed that

βij = 0 Unless φi, φj are AOs of two atoms (1C.54)

that are linked by covalent bond

on the basis of various intuitive arguments, it is also assumed that if φi

is an AO of an atom p, then

αi = Ionization potential of an electron occupying

the AO φi of an isolated atom p [ (≡W(φi)] (1C.55)

The ground state of a closed-shell molecule is supposed to correspond

to a situation where the 2N electrons are placed in pairs in the N MOs of

lowest energy; the total binding energy ET of the molecule is then equated to

the total orbital energy of the electrons, i.e.

ET = 2 Σµocc ∈µ (1C.56)

The energy Ep of an atom p is likewise equated to the total orbital energy

of its electrons; i.e.

Ep = Σip ni α (φi) (1C.57)

where the sum is over AOs φi of atom p, and where ni is the number of

electrons occupying the AOφi. If one is interested in the comparison of ground

43

state properties of a series of structurally related molecules, the MO theory is

the most suitable approach, however, a more convenient route is provided by

the Valence Bond (VB) method, which owes its origin to the pioneering work

of Heitler and London72 on the hydrogen molecule. The VB method was

extended in a more general fashion by Pauling73 and Slater74. A very

comprehensive and systematic presentation of the basic ideas and applications

of the VB method was made by Pauling75 in 1960.

In contrast to the MO theory which is based on the concept of electronic

configurations, VB method is directly linked with the common chemical

concepts like bonds, lone pairs, Lewis structures, resonance etc. In the MO

theory one works with the Slater determinants constructed from an orthonormal

set of MOs, while in the VB theory the building units for such determinants

are the AOs, which are not mutualy orthogonal. These determinants form a set

of non-orthogonal many electron functions corresponding to different pairing

schemes of electron spins. Starting from a given set of AOs the MO and VB

theories will lead to identical results provided all possible electronic

configurations are taken into account in the former and all possible pairing

schemes are considered in the later. Such a complete treatment being not feasible

for large molecules, one is forced to work with approximate wavefunctions,

such as the single-determinant wavefunction in the MO theory. There is no

such simple prescription for the approximate wave function in the VB theory.

44

Thus it is the mathematical simplicity for which the MO theory has become so

attractive to the theoretical physicists. However, the MOs are many-centre

functions and they do not represent as such the valence structures with which

physicist are so familiar with. Even a staunch protagonist of the MO theory

cannot undermine the utility of structural formulas in physics .

Wheland76 has shown that the inclusion of overlap between nearest

neighbour atoms in the HMO approximation does not lead to serious

mathematical complications provided all the overlap integrals are considered

equal. Maslen and Coulson77 have extended the Coulson and Longuet-Higgins

perturbation theory to Wheland’s modification. The secular determinant in

the Wheland modification now has the matrix elements (for alternant

hydrocarbons) α−∈ and β−∈∆ for diagonal elements and nearest neighbour

off-diagonal elements, respectively. One can define a quantity x by

x = α−∈ (1C.58)

β−∈∆

so that the Wheland secular determinant has the same form as in the

HMO approximation with the overlap omitted. The MO energies are thus given

by

45

∈j = α-xjβ

= α - xj (β−∆α)

(1C.59)

1-xj∆ 1-xj∆

If one defines the quantities yi and β’ by

yi = xj

β’ = β-∆α(1C.60)

1- xj∆

the MO energies have the form

∈j = α- yjβ’ (1C.61)

this is the same form as in the simple HMO approximation except that

the resonance parameter β is replaced with a new empirical parameter β’ to

obtain the MO energy in the Wheland modification one simply uses the roots

of the simple HMO approximation to compute the yj’s. The overlap integrals

can be calculated theoretically using Slater-type 2pπ AOs of carbon78. For

linear molecules one usually uses ∆ = 0.27 and for cyclic molecules ∆ = 0.25.

In general, one finds that the inclusion of overlap tends to crowd bonding

levels closer together and to spread antibonding levels farther apart. Also the

difference between the highest occupied MO and the lowest unoccupied MO

is increased (assuming β and β’ have the same value). This has the effect of

giving somewhat better agreement with spectroscopic data. Nevertheless, the

failure to elucidate the electronic interactions means that singlet and triplet

46

states are not differentiated, just as in the original HMO approximation.

Inclusion of overlap removes the pairing properties of the MO energies and

the MO coefficients. The new coefficients are given in terms of the simple

HMO coefficients by

brj = crj (1C.62)

(1-∆xj)½

Chirgwin and Coulson79. have shown that charge densities and bond

orders must be redefined in the Wheland modification. The proper definitions

are

qr = Σj n j (b²rj + Σrbrj br’j ∆ rr’) (1C.63)

prs = Σj nj [brj bsj + ½ (Σs’ ∆ss’ brjbs’j + Σr’ ∆rr’ bsj br’j)] (1C.64)

where r’ represents atoms adjacent to r (with the exception of s in the

case of prs) and s’ represents atoms adjacent to s (with the exception of r). It is

found that the above formulas lead to precisely the same charge densities and

bond orders as for the HMO approximation without overlap. Nevertheless,

the two methods do not generally lead to the same results. Ruedenberg80 has

carried out an extensive study of the Wheland treatment (which he calls the

tight-binding approximation) to show that this is a bonafide first approximation

to an accurate quantum-mechanical treatment, whereas the HMO

approximation in which nearest-neighbour overlap is not neglected. This

47

conclusion does not appear to be necessarily valid if the basis set used in the

simple HMO approximation can be justifiably, assumed to consist of

orthogonalized AOs. Amos and Crispin81 have also redefined polarizabilities

as per above modification.

Hoffman82 has investigated an interesting extension of the HMO

approximation in which the total electronic wave function is separated, not

into a sigma-elecron portion and a pi-electron portion, but rather into an inner-

shell nonbonding portion and a valence-shell portion. For a hydrocarbon of th

general formula CnHm and a total of 6n+m electrons, the total wave function

is written as

Ψ (1,2, ------, 6n+m) = a ΨN (1,2, -------, 2N) x ΨB (2n + 1, 2n + 2, --------,

6n + m) (1C.65)

where ΨN and ΨB are separately antisymmetrized and well behaved.

The function ΨN describes 2n electrons localized in ls AOs of the carbon atoms,

and ΨB describes the remaining 4n+m electrons, i.e., the valence electrons,

the function ΨB now plays the same role in the theory as Ψπ does in the σ−π

separation approximation. Each MO used to construct ΨB is given by the LCAO

form

Ψi = Σp φp cpi (1C.66)

48

The basis set used by Hoffmann consisted of STOs; m ls AOs of

hydrogen, n 2s AOs of carbon, and 3n 2p AOs of carbon, this is of course, the

minimal basis set. The effective Hamiltonian of the system then is

KB = Σµhµcore + Σµ<ν

1 (1C.67)

rµν

which sums over the electrons in the valence shell. The inner shell

nonbonding electrons now play the same role as sigma electrons in the HMO

method. As in the HMO method, one can define a suitable Hartree-Fock

operator and solve the corresponding Roothan equations. As in the HMO

approximation, the matrix elements of the Hartree-Fock operator are guessed

without elucidating the core hamiltonian, but now all the Fpq and ∆pq elements

are given explicit nonzero values, i.e. one does not invoke the nearest-neighbour

approximation for either Fpq or ∆pq. Hoffman choose the diagonal elements of

the Hartree-Fock operator equal to the negatives of the valence state Ionization

potentials associated with the relevant basis AO. For calculations on molecules

containing sp³ carbon atoms, Hoffmann used the values given by

-11.4 eV for 2pc AOs

Fpp = -21.4 eV for 2Sc AOs (1C.68)

-13.6 eV for lSH AOs

Hinze and Jaffee83 have also calculated valence-state energies which

should be useful in this context. Hoffman has appoximated the off diagonal

49

matrix elements by the relationship

Fpq = 0.5K (Epp + Fqq) ∆pq (1C.69)

As justified by Hoffman, the constant K must be chosen so as to satisfy

l<K<α (where α can be regarded as an average value of Fpp) in order to avoid

certain trivial or absurd results, Hoffman chose K=1.75. The overlap integrals

are calculated analytically. Since these depend upon the molecular atomic

coordinates, one can carry out the calculations as a function of the molecular

geometry. Since the treatment includes all the sigma electrons, which play a

major role in the geometry of the molecule, one might expect that the

approximation would predict molecular geometries quite well.

The total electronic energy can be written in the same form as for the

HMO approximation, namely,

EB = Σi ni ∈i + G (1C.70)

where G is the interelectronic term. The molecular energy is given by

E(R) = EB + Σk < 1

ZK Z1 (1C.71)

rkl

= Σini ∈i + Σk<l

ZkZl + G

(1C.72)

rkl

where R represents the collection of internuclear coordinates. Slater84

50

has observed that the sum of the nuclear-repulsion energies and the

interelectronic term is small and is roughly constant as the nuclear coordinates

change. This implies that the sum of the orbital energies behaves approximately

as the molecular energy does. Thus it should be possible to predict qualitatively

acceptable molecular geometries without elucidating the inter-electronic terms.

Failure to elucidate this term, however, will lead to serious quantitative and

qualitative difficulties in attempts to carry out spectroscopic correlations, just

as in the pi-electron HMO approximation.

Hoffman’s method will have to be investigated more thoroughly before

its advantages and disadvantages are fully revealed. The approach is certainly

a useful extension of the simple HMO approximation. It is applicable to

saturated as well as unsaturated hydrocarbons. In fact, the method should be

applicable to virtually any type of molecule. The calculations of this kind only

recently been attempted in a serious manner, and the best of these still fall

short of the success achieved in the pi-calculations. Nevertheless the results

already obtained have been encouraging enough to virtually ensure ultimate

success, and progress in this field is likely to be rapid in near future.

An MO method similar to that of Hoffman has been discussed by Pople

and Santry85 and applied to saturated hydrocarbons. Hartman86 has discussed

a modification of the HMO approximation which is not so far-reaching as that

of Hoffman. It is quite similar to ordinary approximation except that the

51

pi MOs are approximated by an extended basis set, that is, 2p and 3p AOs.

The latest extension of the HMO theory is by Cyvin87-91. Using two-

dimensional approach to HMO theory he has extended the use of the approach

to sp-hybridized carbon atoms in sigma-system in addition to the sp² hybridized

ones. Hence the extended theory covers systems with acetylenic C≡C bonds

and cumulated C=C bonds, in addition to conjugated C=C bonds and aromatics.

In the case of heteromolecules all the atoms and bonds are not same

and the difficulty is in choosing the values for coulomb and resonance

parameter. This is a difficult problem of pi-electron theory and the prospects

of solution appear none too bright41. We know that the coulombic parameter

of the xth atom is given by

αx = α + hxβ (1C.73)

where hx is electronegativity parameter and α, β are the reference

integrals in terms of which any coulombic integral can be expressed. One

might then hope that hx could be related to the ionization potentials of the

reference atom of the heteroatom42. Specifically one might hope to express the

electronegativity parameter as

hx = Ix - I

β (1C.74)

52

Where I is the ionization potential of the reference atom and Ix is the

ionization potential of the xth atom. The problem in using is that one must

choose valence state ionization potentials (not always easy to estimate) and

must assume an explicit value for β. Mulliken92 and Laforgue93 have suggested

a more effective and easy to use relationship

hx = χx - χc (1C.75)

where χx and χc is the electronegativity of the xth and carbon atom

respectively. The obvious disadvantage of using is that it implies that hx is the

same for a given atom in any molecule containing the same reference atom.

For example the value of hN would be used in pyridine as in pyrrole. One

should evaluate hx in terms of atom surrounding it in a given molecule, but we

have used this relationship in chapter 5. What appears to be a very sensible

way to evalutate the electronegativity parameter is through the method of

Streitwieser and Nair94 given by equation

hx(n) = (1-qx

(n-1)) ω (1C.76)

where qx(n-1) is charge density of the xth atom resulting from a calculation

based on a previously assumed value of hr. The new value of hx is then used to

recalculate qx and thereby to lead to yet another value of hx. The iterative

procedure is carried out until consecutive values of hx agree within previously

53

chosen limits. The quantity ω is itself an empirical parameter and is usually

chosen to have the value 1.4. This method has been reformulated by Doggett95,

by avoiding some of the convergence difficulties commonly encountered in

the original method. The resonance parameter is represented by the relationship

βcx = kcxβ (1C.77)

where kcx is called the bond parameter. We have calculated this parameter

by the following Lennard-Jones96 relationship

kcx = βcc

= E(c=x) - E(c-x) (1C.78)

βcx E(c = c) - E (c-c)

where the E’s are empirical bond energies for the bonds indicated and

obtained from the literature such as those given by Pauling75.

Another widely used relationship for kcx is

∆ cx (1C.79) kcx =

∆ cc

which was proposed by Wheland76. This later relationship assumes a

direct proportionality between the resonance integral and the corresponding

overlap integral. Ruedenberg has suggested that a more correct relationship

is80

54

kcx = ∆cx (1 - ∆cx) (1C.80)

∆cc (1-∆cc)

The difficulty with either (1C.79) or (1C.80) is that one assumes

∆cx = δrs in the HMO approximation. If the basis AOs are carbon 2pπ AOs then

∆cx ≠ 0, and the above relationships are satisfactory. However, it is then

inconsistent to neglect overlap for nearest neighbours. On the other hand, if

the basis AOs are orthogonalized, so that overlap formally disappears, the

above relationships lead to trivial reuslts.

One could also choose resonance parameters by an iterative technique

having some resemblence to the Streitwieser’s ω-technique for coulombic

parameters. Following Longuet-Higgins and Salem97, one could write

kcx = e-a(dcx -- d) (1C.81)

where dcx is the length of the cx bond, d is the length of a standard C-C

bond (the one in whcih β is based), and a is an empirical constant (usually

a-1 = 0.3106A). An interative scheme for obtaining self-consistent values of

the bond parameters could be set up by relating dcx to the bond order pcx.

Longuet-Higgins and Salem used the empirical relationship

dcx = 1.50 - 0.15pcx (1C.82)

which was based on alternate hydrocarbon calculations.

55

The simplifying assumption is usally made that all the atoms in a

heteromolecule are equivalent except for the heteroatom (s) and that all bonds

not involving heteroatoms are also equivalent. The atoms and bonds not

involving the heteroatom (s) then define the reference atom and the reference

bond. This assumption has the advantage of greatly reducing the number of

parameters which must be chosen empirically. It is rather pointless to argue

the justifiability of such an assumption on theoretical grounds, since the HMO

approximation is so overwhelmingly empirical. The simpler one can keep an

empirical scheme and still have it work the better.

Recently Gupta98-99 has tried to rationalize the parameters for heteroatoms

in HMO theory by the help of the concept of molecular connectivity given by

Kier and Hall61. He has determined various physical properties like ionization

potential, electron affinity, bond order and bond length of some heteroaromatic

systems by the help of classical values of parameters obtained empirically and

by using the molecular connectivity indices, on comparing both the results

with the available experimental values he concluded that there is better

parameterization by use of molecular connectivity than by the empirical

method. In chapter 5, we have also done the same kind of study for one of the

parameter i.e. h by taking different kind of systems and more in number, we

found that h(δ) and k lead better agreement with the observed values than

those by use of empirical h and k. Even after this investigation no final

56

conclusion can be drawn at this stage about which method gives better values

of both the parametes, till the extensive study on the various kind of systems is

done, the study to be carried out in future.

The vast literature of the HMO approximation is replete with examples

of different ways of choosing the electronegativity and bond parameters for

heteromolecules. There are those who argue for the use of auxiliary

electronegativity and bond parameteres i.e. the heteroatom is regarded as

polarizing other atoms and bonds in proportion to their proximities to the

heteroatom. For example, in pyridine one could use the additional parameter

h2=h6 to describe an inductive effect of the nitrogen atom on its carbon atom

neighbours. Similarly one could use the parameter k23 = k56 to describe a similar

effect on the bonds next to the C-N bonds. Unfortunately, the more empirical

parameters one is forced to choose, the harder it becomes to attach even a

remote physical significance to the choices, and the less meaningful the results

become. It has been found that the SCF solutions of Roothan’s equations for

heteromolecules are capable of shedding some light on the HMO parameter

problem. Lykos100 has discussed in greater detail the choice of parameters for

the generalized HMO method and Ohno101 has carried out a critical study of

this method and has showed that the usual methods of evaluating the one-

centre coulombic integrals and atomic core energies are quite good. Ohno has

also investigated semiempirical (as opposed to empirical) methods for

estimating core resonance integrals. Such a method would be valuable for

choosing core resonance integrals involving heteroatoms.

57

PART (D)

THE MOLECULAR CONNECTIVITY

(1D.1) INTRODUCTION

Several attempts have been made at a level below the quantum

mechanical to provide quantitative structure descriptions. Some of these are

classified as topological methods. Most of these approaches attempt to assign

numbers to parts of molecule and to sum these numbers in accordance with

molecular topology to yield indices. The values assigned to molecular fragments

may be calibrated against a physical property or even a biological activity,

these include methods due to Wiener102, Smolenski103, Franklin104, and Klages105.

Free and Wilson106 gave a method in which a set of dummy variables is used in

an attempt to capture the pattern of substituents. The most common of these in

terms of its wide spread application is the use of a physical property as a

surrogate for the structure107. One may call these approaches empirical, they

are lacking in a fundamental respect as structural descriptors. What is needed

is a universal, non-empirical weighting and counting of molecular fragments

to give numerical descriptors which reflect the structure, as evidenced by the

ability to relate physical or biological properties.

Such a method has been suggested in embryonic form by Randic108 and

developed mainly by Kier, Hall and Murray109-113. Kier and Hall has treated

the method in depth in a book and called this method molecular connectivity.

58

In the molecular connectivity method we seek direct relationships between

molecular structure and activity. The molecular connectivity indices are unit

less quantitative descriptors of molecular structure characteristics which are

basically geometric and topological in character. Hence no physicochemical

method or mechanism is presupposed in this approach.

The molecular connectivity method may be considered as a de novo

approach. The information content of the connectivity descriptors is the

structure of the molecule, that is the set of connections between the atoms

expressed in topological terms. Hence the significance to be attached to the

connectivity terms is similar to that given to any fundamental structure

description. The significance is based on topology, not on forces, properties

or other physicochemical type quantities.

(1D.2) MOLECULAR CONNECTIVITY IN STRUCTURAL

INFORMATION

The place of molecular connectivity in the hierarchy of structural

information can be illustrated as in Fig 1.2. At the most primitive level A the

information is limited to the elements present in the molecule. Hence we can

draw only certain limited generalizations derived from physical and chemical

experience with other molecules of the same chemical composition. We may

discuss the molecule in terms of hydrocarbon chemistry, in this case.

59

More information is conveyed at the level B description. The atom

combining ratio as well as the atom types in the molecule are presented in the

empirical formula. Such information indicates that the molecule is an alkane.

Other generalizations about properties, beyond that related level A, can now

be stated.

At level C the information contained in the molecular formula is greatly

increased. However a specific molecule is not identified because of the isomeric

possibilities. Nonetheless, we can now state approximate values of properties

such as solubility, density, molecular refraction as well as discuss chemical

reactivity.

At information level D there is presented for a first time the manner in

which the atoms are arranged or connected within the molecule. Thus, level D

reveals the molecular topology of two structural isomers, structural information

not apparent at level C. The chemist’s physical and chemical experience can

now be used to derive much information concerning the properties of these

molecules. It is at this level that much chemical intuition used by scientists is

focussed.

All of the information about the molecule is contained, in principle, in

the quantum mechanical description level E. This information takes the form

of numerical values and mathematical relations and includes electron

60

distribution probabilities and energetics.

However, at level D, the level of common chemical intuition, a very

large and useful amount of physical and chemical information can be obtained.

At this level it is clear that some properties are directly related to the number

of atoms in a series of molecules, whereas others follow a different pattern. It

is observed that some properties depend upon the branching pattern in a regular

fashion. Other topological features such as cyclization and heteroatom position

have well known influences on properties. Topological methods are based

upon these relationships. Molecular connectivity is a method for developing

correlation based on a formal developments of molecular topology using

elementary aspects of graph theory.

E Ψi = C1i φ1 + C2i φ2 + -----

Ψn = C1n φ1 + C2n φ2 + ----

D CH2 CH3

H3C CH2

CH3

CH

CH3 CH3

→

→

61

C C4H10

B CnH2n+2

A C, H

Fig 1.2 hierarchy of structural descriptions from the most primitive level

A, the elemental constituents of a molecule, to the most complete level E, the

quantum mechanical level D represents the topology of the molecule and is

the basis for numerical indices of structure.

62

(1D.3) GRAPH THEORETICAL AND QUANTUM MECHANICAL

APPROACHES

The distinction between a graph theoretical approach to the molecular

structure and the quantum chemical approach is in the nature of the parameters.

Both schemes aim at the same goal, which is a search for corrections between

the theoretical numbers and selected molecular properties, but the both are not

necessarily concerned with the same properties, and that the emphasis in graph

theory is on conceptual development while the quantum mechanical

calculations tend to be pragmatic, not fundamental. There is complementarily

between graph theoretical and quantum mechanical approaches to molecular

structure. Since molecular structural description is the quantification of

information pertaining to the atomic composition of the molecule such

information includes the number and kinds of atoms together with their

formally-bonded and spatial relationships, where as the most complete

quantitative representation of molecular structure is obtained by the use of

quantum mechanics. Several approaches based in quantum mechanism can be

applied to the graph theory and vice versa62-64, 114-116. In chapter 5 we determined

some physicochemical properties of conjugated aromatic hydrocarbons by

both the methods.

63

PART (E)

ORGANOMETALLIC COMPOUNDS

(1E.1) INTRODUCTION

The compounds in which organic groups are linked directly to the metal

through at least one carbon atom either directly (σ bond) or through a special

(π) type of bonding are called organometallic, in specific case when such

complexation is made by a arene ring; they are called π-arene metal complexes

when metals are transition metals. These compounds are π-arene transition

metal complex such as ferrocene, cobaltocene, chromocene, etc. The suffix

metallic in the terms organometallic is rather loosely interpreted to include

non-metallic elements such as boron, silicon, phosphorous, arsenic, selenium

and tellurium.

As in other areas of chemistry, there are some obvious exceptions to

this simple definition of organo-metallic compounds, for example, inspite of

the species like CaC2 Hg (CNO)2 and Fe(CN)6-4 having metal carbon bond,

these are not considered organo-metallics for traditional or/and some other

obvious reasons.

(1E.2)HISTORICAL ACCOUNT

‘Cacodyl’ (tetra methly diarsine, As2 (CH3)4 was probably the first

organometallic dertivative isolated as early as 1760 by a Prussian military

apothecary. Cadet de Glaussicourt.

64

In 1842 Robert Bunsen discovered an ‘organic-element’ (now called

radical) in the form of methyl group, which forms a compound with arsenic.

Edward Frankland (a British student of Bunsen) attempted to isolate

ethyl-radicals in 1849. This attempt of Frankland led to the synthesis of the

first organozinc compound diethylzinc.

Frankland was soon able to lay the foundations of organometallic

chemistry by extending his studies to other elements like mercury, cadmium,

tin-lead and silicon. These studies added much not only to the initial

development of organo-metallic chemistry, but incidentally a study of their

composition and empirical formula led to the first reliable assignments of atomic

weights of these elements.

The most notable use of organo-metallic (mainly organo aluminium)

compounds as polymerization catalyst for polyethylene and other hydrocarbon

polymers was crowned by the award of Nobel Prize to K. Ziegler and G. Natta

in 1963.

In 1900 Grignard reagents (organo magnessium halides) are discovered.

Lithium alkyls were discovered in 1917 by W. Schlenk and Holtz, by

now organolithium reagents have almost out classed Grignard reagents both

on synthetic and catalytic (industrial) fronts.

A signal recognization of the highly useful role of organometallics in

synthetic organic chemistry has been in the award Nobel prize to Prof. H.C.

Brown in 1977 for the discovery of now well known ‘hydroboration reactions’.

65

(1E.3) GENERAL IDEA OF ORGANOMETALLIC COMPOUNDS

Broadly organometallic compounds can be classified in two ways :

(1) Based on hapticity (from the Greekword haptien : meaning to fasten);

this can be indicated by the number of carbon atoms of the organic moiety,

which are within the bonding distance of the metal such as monohapto

(h1 or η1), dihapto (h² or η²) penta hapto (h5 or η5) organometallics etc.

(2) Based on the nature (extent of polarity) of the metal-carbon bond, on the

basis of which the structure and reactivity of specific organometallic

compound are often explained.

(1E.4) POLARITY OF THE BOND

In all the organo-metallic compound, the bond Mδ+-- Cδ- unlike the truly

covalent c-c bond) is polarized generally in the direction shown in the figure

1.3 and 1.4.

Calculating the percentage ionic character of M-C bond by the simple

Pauling formula : % Ionic character = 100 [1-exp {1/4 (χc-χm)}], where χc and

χm represent the electro negativities of carbon and metal respectively, the Si-c,

Al-C, Mg-C and Na-C bonds should have roughly 12, 22, 34 and 45% ionic

character respectively. For contrast, it may be pertinent to recall in this

connection that the C-C1 bond which has only 6% ionic character on Paulings

formulation, is considered to be quite polar in organic chemistry and many

66

reaction of chloro compounds are explained on that basis.

Similary a synchronous four centre (Fig. 1.3) or six centre (Fig. 1.4)

transition state of the type :

(1E.5) CHROMOCENE

(C5H5)2 Cr chromocene is obtained by reacting with anhydrous

chromium (III) Chloride with sodium cyclopentadienide in tetra hydrofuran

solution gives dark-red air-sensitive volatile biscyclopentadienyl-chromium

(chromocene), melting point 173°C. This chromium complex (C5H5)2Cr has a

sixteen-electron configuration, which is two less than the rare-gas configuration,

and two holes are unpaired, resulting in two unpaired electrons in accord with

the observed para magnetism.

(1E.6) FERROCENE

(C5H5)2 Fe is an orange, crystalline, diamagnetic solid, melting point

173º-174° C. Ferrocene is very volatile and can be purified by steam distillation

or sublimation. It has a high thermal stability decomposing only around 500.

Ferrocene was the first π-cyclopentadienyl derivative to be prepared. It

was discovered by kealy and Pauson as an unexpected product of the reaction

Mδ Rδ

yδ- xδ+

Mδ Rδ

Zδ C- C-

Cδ+

>

>

>

>

Fig. 1.3 Fig. 1.4

67

between cyclopentadienyl magnesium bromide and ferric chloride in dethyl

has been suggested for carbon-metal bond cleavage in their interaction with

polar substrates, X+-Y-. With the above mechanism, the influence of bond

polarity on reactivity is evident.

In this portion four organometallic compound viz cobaltocene,

chromocene, Ferrocene & Nickelocene are studied, their average moleuclar

polarizabilties are calcuated by the δ-function potential model. The structure

and general idea of five organometallic compound are as follows :

(1E.7) COBALTOCENE

Cobaltocene (C5H5)2 Co is formed by reacting with cobalt (II) chloride

with sodium cyclopentadienide in tetrahydro furan solution gives violet black

biscyclopentadienyl cobalt (cobaltocene), metling point is 173º-174º C,

cobaltocene has a nineteen - electron configuration (one in excess of the next

rare gas) and hence exhibits the expected paramagnetism for a complex with

one unpaired electron.

Ether solution in an unsuccessful attempt to prepare the still unknown

dihydrofulvalene C5H5-C5H5 by a coupling reaction. At about the same time,

ferrocene was independently prepared by Miller, Tebboth, and Tremaine in

low yield by passing cyclopentadiene vapour over a special iron-molybdenum

catalyst at elevated temperatures.

Wilkinson, Rosenblum, whiting and woodward postulated the

68

pentagonal antiprismatic structure for ferrocene, which was subsequently

confirmed by X-Rays crystallography.

(1E.8)NICKELOCENE

(C5H5)2 Ni, dark-green in colour, volatile crystalline solid. Nickelocene

has a twenty electron configuration, which is two greater than that of the next

rare gas.

69

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