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NUCLEAR SHIELDING CALCULATIONS

FOR

SOME FIRST-ROW, SECOND-ROW AND TRANSITION METAL ELEMENTS

A thesis submitted to the University of Surrey

for the degree of Doctor of Philosophy in the

Faculty of Biological and Chemical Sciences

by

BUNDIT NA-LAMPHUN

Spectroscopy section July 1982

Department of Chemistry

Faculty of Biological and Chemical Sciences

University of Surrey

Guildford, Surrey

ENGLAND.

ProQuest Number: 10804072

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Summary

The main purpose of this work is to calculate the isotropic

shielding of some nuclei, other than protons, of first- and second-

row atoms in the periodic table and some transition metal elements

in a wide variety of molecular environments, with a view to gaining

an understanding of the various electronic factors which determine

the observed nuclear shielding.

Chapter One introduces some general concepts. Chapter Two

presents a general servey of various Semi-empirical molecular orbital

methods. Various MO calculations of nuclear shielding are briefly

reviewed in Chapter Three with particular emphasis on Pople's GIAO-

MO approach. This chapter also contains a description of the theory

of medium effects on nuclear shieldings.

Chapter Four is concerned with approximate MO theories and

their application to the GIAO-MO method for Sum-Over-State (SOS)

results of some shieldings of first-row elements. The medium effect

models are used to explore the role of medium effects on nuclear

shielding. This exploration is supplemented by performing some

calculations on hydrogen-bonded models.

Chapter Five records the SOS results for the nuclear shielding

of some second-row elements in conjunction with the GIAO-MO method.

The solvaton model is also used in an attempt to improve on our

understanding of the relation between magnetic shielding and various

features of molecular electronic structure.

Chapter Six presents some shielding data for transition metal

elements obtained by means of Pople* s GIAO-MO method in conjunction

with the INDO/5R parameterization scheme for SOS results of some

inorganic molecules. A reasonable correlation between the calculated

and experimental chemical shifts, compared by mean of a least-squares

fit is obtained for some of the molecules considered.

ACKNOWLEDGEMENTS

I wish to express my deepest gratitude to my supervisor,

Dr G A Webb, for his continual advice and encouragement throughout

this work.

I am also grateful to my colleagues, Dr T Khin, Dr T Blair,

Dr D J Reynolds and Dr S Duangthai for useful discussions.

MY PARENTS AND WIFE

CONTENTS

Page

SUMMARY 2

ACKNOWLEDGEMENTS 4

CONTENTS 6

CHAPTER ONE GENERAL INTRODUCTION 9

1.1 General introduction 10

1.2 Basis functions

1.3 Basis set 12

1.4 Orbital exponent 14

1.5 Some computational details 14

CHAPTER TWO SEMI-EMPIRICAL MOLECULAR ORBITAL THEORIES 17

2.1 Introduction 18

2.2 The Hartree-Fock method 18

2.3 The analytical Hartree-Fock method

(The LCAO-SCF-MO method) 21

2.4 Semi-empirical LCAO-SCF-MO method 22

2.5 Zero Differential Overlap (ZDO) approximation 23

2.6 Complete Neglect of Differential Overlap (CNDO)

approximation 25

2.7 Intermediate Neglect of Differential Overlap

(INDO) approximation 31

2.8 CNDO/S parameterization 32

2.9 INDO/S parameterization 33

2.10 IND0/5R parameterization 36

CHAPTER THREE DEVELOPMENT OF SOME THEORIES OF NUCLEAR MAGNETIC

SHIELDING 42

Page3.1 Basic considerations of nuclear magnetic

shielding 43

3.2 Independent electron GIAO-MO method 52

3.3 Theory of solvent effects on nuclear shielding 59

CHAPTER FOUR ' SOME CALCULATIONS OF SHIELDING FOR FIRST-ROW NUCLEI. 62


4.2 Carbon shieldings 63

4.3 Nitrogen shieldings 79

4.4 Fluorine shieldings 88

4.5 Conclusions 94

CHAPTER FIVE SOME CALCULATIONS OF SHIELDING FOR SECOND-ROW NUCLEI. 95


5.2 Silicon shieldings 97

5.2.1 Introduction 97

5.2.2 Results and discussions 98

5.3 Phosphorus shieldings 123



5.3.3 The variation of shielding with molecular

conformation 136

5.3.4 Solvent effects 143

5.4 Conclusions 150

CHAPTER SIX CALCULATIONS OF SHIELDING FOR SOME TRANSITION METALS. 151


6.3 Vanadium shieldings 153


6.2.2 Molecular conformations used in the shielding

calculations 154

Page6.2.3 Results and discussions 155

6.3 Cobalt shieldings 162



calculations 163


6.4 Platinum shieldings 175



calculations 176


6.5 Conclusions 183

General conclusions and suggestions for further developments 184

Appendices 187

References 211

CHAPTER ONE

GENERAL INTRODUCTION

1.1 General introduction.

NMR spectroscopy has now undoubtedly become a major technique

for determining molecular structure. Various developments in the

technique and the increasing range of applications of the method to

structural analysis make its use widespread.

One of the common NMR parameters, known as the nuclear shielding,

provides a detailed insight into the chemical bonding and molecular

electronic structure. A great deal of attention has been paid to the

experimental measurement of, and theoretical interpretation of, the

nuclear shielding of various species^

Since the pioneering work of Ramsey^* the calculations of

nuclear shielding using semi-empirical theories at various levels of

approximations remain an active area of study. Indeed, the calculated

nuclear shielding, different shieldings for two nuclei of the same

isotope is chemical shift, could be of major practical value in the

identification of unknown molecules if the calculations reproduce the

experimental data reliably and quantitatively. Theoretical estimates

are usually based upon an isolated molecule as a model, whereas many

experimental chemical shifts are reported for liquid samples in which

solvent effects may be present. Therefore it is not reasonable to

expect of any theoretical treatment of magnetic shielding that it

exactly reproduces experimental values.

This work deals with the calculation of nuclear shielding for

some nuclei of the first- and second-row elements of the periodic

table and some transition metal elements, for which a large body of(1-5)experimental data is now available , in polyatomic molecules

with a view to understanding the various electronic factors governing

the observed shielding constants in the molecules of interest. The

calculated results are compared with the available experimental data

and with other appropriate theoretical treatments. Emphasis is placed

on Semi-empirical theories, which are practical at the present for

calculations of the magnetic shielding of larger molecules. Several

currents theories of magnetic shielding are briefly reviewed for

purposes of comparison. SI units are used throughout this work.

1.2 Basis functions.

There are many ways in which the M O ’s themselves may be

expressed. One of the most commonly used, and certainly one with a

great deal of intuitive appeal, is the LCAO (Linear Combination of(8)Atomic Orbitals) method in which each MO is taken to be a linear

combination of some starting set of atomic orbitals (AO's),

We require that the molecular orbitals, , form an orthonormal set,

which under the LCAO approximation demands that

where S is the overlap integral of atomic orbitals,d).. andUY) M v

(1.1)

/S (1.3)

C . is the LCAO coefficient of <}> in MO 4/. and 6 . . is the Kronecker datpi Tp l ij

By adjusting the number of basis functions, cf) , which appear in the

LCAO expansion one may improve the accuracy of the molecular orbitals

until the minimum energy is reached.

For the LCAO method it is necessay to have a convenient

analytical form for the atomic orbitals that appear in the basis set.(9)The analytical functions for the hydrogen atom are well known and

appear as the product of a radial part and an angular part ^ ^ ( 0 ,

The radial part is expressed as Laquerre polynomials of the radial—rrdistance, r, multipled by a decaying exponential e ^ where £ is

the orbital exponent. In choosing analytical forms for the atomic

functions of many-electron atoms, it is possible to use the hydrogenic

functional form with an adjusted orbital exponent to reflect the

electrostatic screening of the nucleus by electrons of the inner shell(9)as determined by the variational principle. Slater proposed a

much simpler analytical form for R ^ ( r ) and simple rules for evaluating

the orbital exponent C • The resulting functions are now widely known

as Slater Type Orbitals (STO) which are used throughout this work.

1.3 Basis set.

For many years there have been frequent discussions about the

extent of the involvement of 3d orbitals in the bonding of molecules

containing second-row a t o m s . G e n e r a l l y this involvement seems

greatest for atoms in higher valence states, but even for the low

valence state 3d orbitals can be needed in computational schemes

either as polarization functions or to take up deficiencies associated

with other aspects of the calculations. Much has been written on the(12-13)function of 3d orbitals m the chemistry of second-row elements

Although SCF ground state properties, namely total energy, orbital

energies, or atomic orbital populations, seems to be little influenced

by including 3d A O ' s ^ ^

The situation for excited states might be different because

singly excited configurations involving excitations to 3d AO's lieC 28)close in energy to low lying excited valence states. It is

(14)shown that 3d functions play an important role m an accurate

description of compounds containing second-row elements. A minimal

basis set, supplemented with a set of 3d functions on second-row atoms,

is seen to reproduce experimental geometries. They lead to the conclusion

that the 3d orbitals should be considered essential for the adequate

representation of the electronic structure. The idea that 3d orbitals

contract in an electron withdrawing environment and thus become more(12 15)effectively involved in bonding has been advanced. ’ Exploratory

calculations^^, have demonstrated the importance of 3d atomic

orbitals for second-row atoms in minimal basis set calculations.

Ab initio calculation with optimized scaling factors have shown that

such factors for 3d orbitals on second-row atoms vary widely with the

atoms to which the second-row atoms are bonded. As mentioned(21)elsewhere, the inclusion of 3d orbitals in a basis set causes

problems for Semi-empirical theories. This inclusion has also been

the main factor in hindering the progress of reliable Ab initio methods

for calculating the shielding constants of second-row nuclei. These

calculations become more expensive in computing time because of the

extended basis sets for the inner and valence electrons, including

3d orbitals.

1.4 Orbital exponent.

(22-23)Exploratory calculations have demonstrated the importance

of 3d atomic orbitals for second-row atoms in minimal basis set calcul

ations. While most of these papers deal with the ground state properties(24“29)of molecules. Several attempts have been made to parameterize CNDO and

(30-32) - •INDO methods to reproduce spectroscopic data for molecules containing

second-row atoms. However, none of the methods presented have incorporated

adjustments of the size of the 3d orbitals to the electronegativity of the(33)environment; this problem was first discussed by Kuehnlenz . Jaffe'

/ Q / et al extended the CNDO/S method to adjust the size of the 3d orbitals

to second-row elements by using a self-consistent optimization of a 3d

orbital exponents to account for the environment of the second-row

elements. They found that the variation of the 3d orbital exponents of

sulphur atoms is quite significant from one molecule to another which(35)is m agreement with Craig et al who suggested that the second-row

(36)3d orbitals are very sensitive to their chemical environment. Mitchell

. . . (37)has carried out electrostatic calculations similar to those of Craig

on X^PO, where X is F, Cl, C and H. Considerable increases in the

3d orbital exponent were found, these increases being in the order F > C( 38 )> C 1 > H . Keeton and Santry found that the 3d orbitals of Phosphorus

are rather insensitive to their molecular environment, and can therefore

by easily included in minimal basis set calculations with fixed exponents

which are used throughout the calculations reported in this work.

1.5 Some computational details.

The nuclear magnetic shielding calculations reported in the

present work were calculated by self-consistent perturbation methods.

The perturbation treatments as well as the determination of the

unperturbed SCF wavefunctions are carried out at the CNDO/S, INDO/S and

IND0/5R levels using Slater Type Atomic Orbitals (STO's).

For magnetic shielding calculations for first-row nuclei, the(3 9)method of computation was developed by Pople using Gauge Invariant

Atomic Orbitals (GIAO's); this method, utilized in previous s t u d i e s ^ ^ ’^ ^

has given results in satisfactory agreement with experimental data. In

the present calculations, the unperturbed wavefunction is obtained from

Semi-empirical calculations and all the integrals which appear in the

Self-consistent perturbation equations are obtained from standard

parameters based upon the respective LCAO-SCF-MO programs obtainable

from the "Quantum Chemistry Program Exchange (QCPE)" at the chemistry

department, University of Indiana, Bloominton, USA. An appropriate

program was written to calculate the medium effects on the molecular

shieldings.

Both CNDO/S and INDO/S programs were modified in this laboratory

in order to perform calculations of the shielding of second-row nuclei.

The SCF computations have been performed by the standard programs but

modified versions of the original QCPE programs were used for the CNDO/S

and INDO/S methods. For magnetic shieldings of second-row nuclei, the(39)method of computation was based upon the original Pople method for

first-row nuclei, using GIAO's.

Calculations on the transition metal elements were performed by/ / O *\

using the modified version of the original IND0/5R program for the

SCF computations. The modified version for the nuclear shielding

calculations of the transition metal elements was also prepared in(39)this laboratory, based upon the original Pople method for first-

and second-row nuclei, using GIAO's.

The calculations were performed on the "Prime" system computer

of the University of Surrey and on the CDC 7600 computers of the

universities of London and Manchester. The molecular conformations used

in the shielding calculations were obtained from standard bond lengths(4 3) . (44)and angles , from standard configuration data , by geometry

(4 5)optimization using a GEOMIN computer program , or by analogy with

similar types of compounds. Often combinations of these procedures were

needed in order to calculate the conformation, and with some types of

molecules more than one conformation was used and the shielding and

chemical shift results were compared with the experimental chemical

shift values.

CHAPTER TWO

SEMI-EMPIRICAL MOLECULAR ORBITAL THEORIES

2.1 Introduction.

Molecular orbital (MO) theory is the most widely applied

method for describing the electronic structure of molecules. It

provides an exact description of molecular electronic structure

for one-electron systems and also gives a good approximation for

many-electron molecules. Since most applications of MO theory do

not necessarily require an accurate knowledge of all of the M O ’s

for the system, a number of simplifications and approximations

are introduced in the theory and Semi-empirical methods have been

developed. As discussed elsewhere^ the calculations of

nuclear shielding have been done using a set of wavefunctions

derived by means of a Semi-empirical SCF-MO method.

2.2 The Hartree-Fock method.

Most of the available Semi-empirical all valence-electron(48-49)methods have been based upon the analytical Hartree-Fock formalism.

This method represents the best possible single determinant wave-

function that can be obtained and thus serves as a convenient

starting point for higher approximations. The Hartree-Fock formalism

often referred to as the LCAO-SCF-MO method in which the " many-

electron " wavefunction ip is written as an antisymmetrized product

of one-electron molecular orbitals Op.’s) formed usually from a

linear combination of atomic orbitals (cj) ’ s) according to

mip. = I C .(j) (2.1)yi £ yi y

Although the complete solution of the Hartree-Fock problem

requires an infinite basis set in the LCAO expansion in equation

(2 .1 ), good approximations can be achieved with a limited number

of atomic orbitals. The coefficients, C . , are determined by ayivariational procedure, i.e., chosen as to minimize the expression

= (2.2)

where E represents the expectation value of the electronic energyA

associated with the n electron Hamiltonian H of a given molecule.

Then the general approach is based on the variational principle and

involves a systematic determination of stationary values of the energy

of the system by adjusting an approximate many-electron wavefunction

through variation of all of its contributing one-electron molecular

orbitals, • *^n the determ*nant: until the energy E,

achieves its minimum value. Such orbitals are referred to as SCF or

Hartree-Fock molecular orbitals.

AThe electronic Hamiltonian for n electrons, H , in a molecule

is defined in the Born - Oppenheimer approximation as

h = yH(core)nci + i— y y I <2 -3)k 4tt£ k>p rkpo

(core)The quantity H (k) is the one-electron Hamiltonian for the kelectron moving in the field of the bare nuclei. This operator is

linear and Hermitian and has the form

H (core)p^ = (2>4)

2m 4,reo B rkB

where Z„ and r. „ represent the charge of nucleus B and the distance d Kr>

between the electron k and the nucleus B, respectively, and where 2 Z,h n 2 eVi a n d ---2m k a_r

B are respectively, the kinetic energy and4lreo kB thpotential energy operators for the k electron. In the equation

(2.3), r, is the distance between electrons k and p , and 9 kpe 1•j— represents the mutual repulsion operator between them.

o rkpSubstituting equation (2.3) into equation (2.2) the general

expression for the electronic energy is obtained as

n n nE = 2 TH.. + Y y (2J.• - K..) (2-5)11 L L ^ lj lj''

i=l i jwhich includes integrals over M O ’s, so equation (2.5) is derived on

the basis that the MO's form an orthonormal set.

In equation (2.5) represents the energy of an electron in

a M O ’s, , in the field of bare nuclei,

(core)| «|/|(k) H ( k J i ^ O O dTk (2.6)

The Coulomb integrals J . . and the exchange integrals K . . are definedij ijas

and

^ ( k ) ^ j (p ) _ i|ji (k)^(p) dxk dxikp

(2.7)

K y ■ I (k)dxkdxp (2.8)kp

After applying a unitary transformation to the M O ’s, 4^ , the

corresponding differential equations for the best forms of the MO* s

have the form

H (COrfib + I( 2Jj - Kj) lb. = e . i p . Y i iri (2.9)

or F 4^ = > i=l,2,..n (2.10)

These are known as the Hartree-Fock equations. In the equation

(2.10), F is the Fock Hamiltonian operator and is the energyt hof the i MO; the Coulomb operator, , and the exchange operator,

K . , are defined as J

= 'I'JGO - ^ ( p ) dt r, pkp

’I'iOO (2.11)

and

K ^ C k ) = ipt(p) - ^ ( p ) dip i|jj(k) (2 .1 2 )kp

2.3 The analytical Hartree-Fock method or the LCAO-SCF-MO method.

The LCAO approximation to these Hartree-Fock orbitals leads (8)to Roothaan* s equations which requires, for each molecular

orbital 4 , that the coefficients C ^ satisfy the following set

of simultaneous equation

7 ( F - e.S ) C . = 0 , for all u and all i (2.13)L K UV 1 U\K VI rV

where

F = <J> Fp(k) H(k) (k) dxk (2.17)

(yv|Aa) = <J>y (k)<f>v (k) - <|>x (p )<|>ct(p ) dTkdrp (2.18)rkP

and

P. , are the elements of the bond-order charge density matrix Aodefined as

Atfocc= 2 Y c. .c .L Ai oi (2.19)

The main obstacle to the rigorous solution of the Roothaan equations

for a medium-sized molecule lies in the formidable number of multi

centred integrals (yv|Ao) which arise even with the use of a

minimum basis set, and the difficulty involved in their evaluation.

2.4 Semi-empirical LCAO-SCF-MO methods.

As stated in the previous section, the most difficult part of

LCAO-SCF-MO calculations is the evaluation of a large number of multicentred

integrals of the types (yv|Aa) which arise even with the use of

a minimal basis set. Many of these integrals have very small values,

particularly those involving the overlap distribution <f) (1)<J> (1)p Vwhere y^V . Thus Roothaan's equations have been simplified by the

With the ZDO approximation, however, it was shown that in

order to retain the invariance of the wavefunction to the orthogonal

transformation among orbitals centred on the same atom, only certain

approximate schemes are permissible.

2.5 Zero Differential Overlap (ZDO) approximation.

electron repulsion integrals which are considered to be uniformly

small, greatly simplifying approximate SCF-MO schemes. Thus applying

the ZDO approximation

Zero Differential Overlap (ZDO) approximation.

ZDO approximation of Parr (50) allows a systematic neglect of

(yv| Acr) = (yy|AA) 6yv5Xo (2.20)

Also, the corresponding overlap integrals are given by

S,yv <f>u O)v O ) dTk = 6yv (2.21)

Although the core integrals

(2.22)

involve an overlap distribution, they are not neglected, but are

treated semi-empirically.

Applying the ZDO approximation to all atomic orbital pairs

greatly simplifies the closed-shell Roothaan's equation (2.13) to

give

7 F C . = e.C •L yv vi 1 yi

where the Fock matrix elements become

(2.23)

Fuu = H ° re)“ I Pyy(w|,j,j) +2 Pu (uy|u) (2‘24)

and Fuv = HS ° re)- 7 pUvtw lw i « (y } (2,25)yv yv i yv

Thus only the one- and two-electron integrals remain. This approach

is consistent in that the neglect of the overlap integral S in ther

normalization involving the associated charge distribution $,.4r*parallels the neglect of electron repulsion involving a similar

(47)distribution.

The ZDO may have an effect on the invariance restriction of

the molecular orbitals and their approximation by LCAO's. The

restriction arises because the wavefunction and calculated molecular

properties should be invariant to unitary transformation of the basis

functions. The transformation will involve the rotation of the

coordinate axis of the system. For a full calculation of LCAO-MO's,

the invariance is maintained, but this is general will not be the

case where approximations, such as the ZDO approximation are made to

the full set of Roothaan's equations. Certainly, rotational invariance

is highly desirable, especially for systems of low symmetry where the

choice of the coordinate axis would otherwise affect the calculated

energy.

2.6 The Complete Neglect of Differential Overlap (CNDO) approximation.

('52-53')The Complete Neglect of Differential Overlap (CNDO) method

is perhaps the most elementary all valence-electron theory retaining

the main features of electron repulsion. In the CNDO method, a mini

mum STO basis set is assumed. Only the valence AO's are explicitly

considered and all the inner shells are treated as part of the

non-polarizable core.

The CNDO method employs the following approximations:

1. The ZDO approximation is applied to the overlap matrix, ,

so that S is replaced everywhere with the unit matrix

Spv j 4 ^ 0 0 4 ^ 0 0 dT = 6yv (2.26)

Thus the atomic orbital basis set i-s treated as an orthonormal

set.

2. The ZDO approximation is also applied to all two-electron

integrals,

(yv|Xcr) - J 4^ 0 0 4 ^ 0 ) - <l> (p)<l>a (p) dxkdxr kp

■ V x a

= V V x o (2.27)

Approximations (1) and (2) by themselves destroy the rotational

invariance so a further approximation is needed.

3. The electron-interaction integrals y are assumed to'yvdepend only on the principal quantum numbers and orbital exponent

(22)of the orbitals considered . This is different than Pople1s original

suggestion that y depends only on the atoms containing orbitals p and 0y\>

3.1 In the case of the spd basis set in which the

s-,p- and d-functions have the same orbital exponent, the

two-electron integrals depend only on the nature of the

atoms A and B to which the orbitals belong.

(yy|XX) = , yon A, vonB .28)

is then an average of the electrostatic repulsions between

any electron on atom A and any electron on atom B. These

average Coulombic repulsion integrals are approximated by

the theoretical integrals calculated for the repulsion of

spherical symmetric charge distributions, S^, using STO1s

of orbital exponent £ and principal quantum number n.

Tab = J J nSl(sA>k) 1 n'SB(?B>p dTkdTP (2-29)rkP

For these basis sets, the invariance of the CNDO method of

the use of basis function is retained.

3.2 For an s-,p- and d-basis set in which the d-orbitals

have a different orbital exponent and principal quantum number

than the s- and p-orbitals, it is far more realistic to adopt

three average Coulomb integrals to describe all possible types

of electronic interactions.

YabCs,3) = | |nS^C?s ,k) - n ' s 2 ( ^ , p ) dTk dxprkp

Y A B ^ . d ) =

YAB^d ,d )

ns|(Cs,k) - n's|(?d,p) dxkdx,rkp

n ^ (C d ,k ) I n 's j jc e j .p ) dxkdxp J

rkp

-(2.30)

Using approximations (1) through (3), the unrestricted

equations for the Fock matrix elements are given by

.aW

H +W

P - PaAA yy YAA + E PBBYAB’ yon A (2,31) B(VA)

V v = Hyv yvYAB (2.32)

where is the total charge density on atom IFDD

BBB T (Pa + P3 )

“ yy w J(2.33)

The diagonal core matrix element, H , includes the interaction

of an electron in atomic orbital d> with the cores of the other

atoms. These can be conveniently separated into one- and two-centre

terms to give

H JAyy Cp| - \ v2 - — |y)

yA- I Cp| — |y)

B(*A) B

Uyy ■ I (p| A Id) (2.34)B(*A) B

where U is a one-centre term and Z is the core charge onyy a

atom A (in units of +e). U is essentially an atomic quantityyy( the energy of orbital <b in the bare field of the core of its

yown atom ) and could be evaluated from approximate atomic orbitals,

but it is chosen Semi-empirically. The potential arising from the

interaction of the electron in orbital cf> , where y belongs to

atom A, with the cores of the other atoms is usually written as

ZB— = VB (2.35)YB

For the off-diagonal core matrix elements, H , it is convenientyv

to distinguish cases where ^ and (|>âre on the same or different

•atoms. If both belong to atom A, H may be writtenyv

Huv = ”u v - I Cul — |v) (2.36)UV yV B(#A) yB

If <{> and 6 , are functions of the s, p, d , .... types then U ,y v yv

the one-electron matrix element will vanish by symmetry. The

remaining terms represent the interaction of the distribution d> d>Yy v

with the cores of the other atoms. Since CNDO neglects differential

overlap in the two-electron interaction integrals, it is consistent

to neglect those contributions which give rise to the next

approximation.

4. Monatomic differential overlap (j> <J>^(y=V, y,V belong to A)

is neglected in the interaction integrals involving the cores of other

atoms, that is

(y| Vg |v ) = 0 , where y=v, (2 .37 )

y,V belong to A

Further, the invariance condition requires that the diagonal elements

(y | Vg|v)be the same for all orbitals of the same principal quantum

number on atom A, which is generally written (y| V R |y) = V whereB AB

yon A and V is the interaction of any valence electron on atom A

with the core of atom B. In the original method, CNDO/1,

is calculated using the atom A valence s-orbital,

VAB = = ZB l SA ^ ¥ 1 B dh (2-38)

where r1T) is the distance of electron 1 from the B nucleus. CNDO/1 I d

calculations on diatomic molecules predict equilibrium distances

much too small and dissociation energies correspondingly too large.

This is found to be primarily due to a " penetration " effect in

which electrons in an orbital on one atom penetrate the shell of

another leading to net attraction.

(22)The CNDO/2 method corrected this deficiency by neglecting

the penetration integrals, “ ÂB* t le electron-core

potential integrals, V._. are not evaluated separately but are relatedA15to the electron-repulsion integrals by

VAB ' & l | I**) = V A B (2'39)

There is no real theoretical justification for neglecting the

penetration integrals, but it appears to compensate errors of the

opposite sign introduced by the neglect of overlap integrals, Sy \)and the neglect of inner-shell o r b i t a l s . W i t h these refinement,

the model can now be used to estimate equilibrium bond lengths quite

well. The elements of the core Hamiltonian are then given by

and yv

J ZnY AT3 (2.40)yy yy B ABB(M)

H = 0 , where y*v and y,v on A (2.41)

To complete the simplification of the calculation, one needs

the off-diagonal core matrix elements H where and <f>v are on

different atoms.

5. The three-centre two-electron integrals, ]> (y | V |v) ,cr±Afor y on A are neglected, leaving that part of the J core

Hamiltonian matrix commonly referred to as the resonance integral, Byv

V = Hyv = - 7 V 2 - VA - VB |v) , u on A, (2.42)V on B

In CNDO, the resonance integrals are estimated using the formula

Byv Hyv eAB Syv

2 » A + Syv (2.43)

where S is the overlap integral and (3° is an empirical parameter y vdependent on the nature of the atom and the principal quantum number

of the orbital.

Using these approximations, the CNDO unrestricted Fock matrix

elements are given by

F = U + ( P AA - Pa ) y AA + V ( P - 7 ) v (2.44)yy yy AA y y ; yAA L BB B TABB W )

- p“ Y^ b .where U, v on A (2.45)

and

,where y on Av on B

(2.46)

2.7 The Intermediate Neglect of Differential Overlap (INDO) approximation.

The Intermediate Neglect of Differential Overlap (INDO) method

refinement of the CNDO method, and uses the ideas of ZDO to a lesser

extent than does CNDO. In the CNDO method, two-electron exchange

integrals are neglected which leads to the inability of the method to

account for the separation of different spin states arising from the

same configuration, since this effect is closely associated with

electron interaction integrals of the exchange type.

To take some account of these exchange integrals, INDO retains

differential overlap provided the orbitals are on the same atom. Thus

one-centre integrals (yv|Aa) , y,v,A and a on A are no longer equated automatically to zero, but may of course still vanish by symmetry. The

inclusion of these exchange integrals in INDO leads to a substantial

improvement over CNDO in problems where the electron spin distribution

is important.

was developed by Pople, Beveridge and Dobosh (56) as an important

The general expressions in the INDO method for the unrestricted

Fock matrix elements can be shown to be

AF “ , = H,„, + I fP^ t o | A a ) - Pâ (yX|va)] + £ PRRYAR (2.47)yy yy “ Aa

AcrBB'AB

B(*A)

.aHyv = Hyv

A - 1

XapAa( H Xa) - P ^ a (yA|va) , y=/v, y,v on A (2.48)

Fyv = I ^ A + *B ) Syv ‘ PyvYAB> y°n A, von B

yv - P06 Yyv tAB (2.49)

2.8 CNDO/S parameterization

Del Bene and J a f f e ^ ^ have developed a method of calculation

by determining the transition energies from a CNDO calculation and(58 )then refining the results by means of Configuration Interaction,

to obtain the energies and wavefunctions of various excited states.

This CNDO/S procedure has the aim of predicting reliable singlet-

singlet transition energies. The differences between the CNDO/S

and CNDO/2 methods are the evaluation.of .the one-centre, integrals, Y ^ ,

electron-repulsion integrals, v AT,, and the resonance integrals, 8AB yv

In the CNDO/S scheme, the one-centre integrals, Y Aa » are

evaluated using the Pariser approximation, (59)

(yy|vv) = y m = IA - Aa , y and v on A (2.50)

where 1^ and A^ are the relevant valence orbital ionization potential

and electron affinity, respectively. In the CNDO/S method, the

uniformly charged sphere model of Pariser and P a r r ^ ^ is used to

evaluate YA g* The resonance integrals, which represent a

measure of the bonding energy between the orbitals <j> and ^ , is

taken to be proportional to the total overlap between atomic orbitals

d) and d> . Del Bene and J a f fe^*^ divided the total overlap SYy YV * yvinto two parts of tt-tt and d - d overlaps, denoted as S77 and S °yv yvrespectively. Furthermore, they assumed that the effective tt-tt

overlap would be screened differently to the d - d overlap. Accordingly,

the resonance integrals, g , are given by

5 , = 7 ( 6? + Bn )( s° + KS71 ) (2.51)yv 2 v A B J v yv yv J

where the bonding parameters (3° and (3° depend only on the natureA D

of atoms A and B respectively and they are adjusted so as to

reproduce the singlet-singlet transition energies of a given reference

molecule and K is taken to be 0.585. Furthermore, the parameter |3°

is readjusted so as to reproduce the singlet-singlet transition

energies of a given reference molecule. Other features of the original

CNDO/2 method are retained. The CNDO/S method has been satisfactorily

employed to account for the observed singlet-singlet transitions in

substituted conjugated h y d r o c a r b o n ^ ^ and heterocycles. ^(62)Improved results are obtained when the Nishimoto-Mataga approxi-

//* O mation is used for the two-centre Coulomb repulsion integrals, Y^g*

2.9 INDO/S parameterization

Krogh-Jesperson and R a t n e r ^ ^ introduced the INDO/S procedure

which can describe spin properties, which CNDO/S, since it includes

no exchange integral terms, can not. In principle the difference

between INDO/S and INDO is the introduction of the K parameter, as

in CNDO/S, to distinguish between the screening of effective tt-tt

overlap (K = 0.585), and the screening of effective d-d overlap

(K , = 1.000). The value of K can be rationalized in terms of local oscreening, and when K = 0.585 the Tr-resonance integral between

neighbouring carbon atoms is reduced from -4.17 eV to -2.44 eV for

the 2p TT-orbitals in b e n z e n e ^ w h i c h is very close to the value of

-2.39 eV used in the Pariser-Parr-Pople^ ” model of planar unsaturated

hydrocarbons. This less negative value will tend to raise the energy

of the occupied TT-orbitals, and thus to counteract the incorrect

tendency for the occupied tt-MO's to plunge below the cj-MO's, which

is observed in calculations based on the CNDO and INDO approximations.

The INDO/S Fock operator is the same type as the INDO Fock

operator of Pople and Beveridge^*^. The two-electron one-centre

exchange integrals are as for the INDO procedure. The two-electron

coulomb repulsion integrals, y .., were evaluated as on the case of theriri

CNDO/S procedure. The two-centre coulomb repulsion integrals, are

obtained from the Nishimoto-Mataga approximation as in CNDO/S. The

overlap integrals are calculated over a ST0 basis, as in the other ND0

schemes described in this Chapter. The monatomic core integrals, U ,FFare found semi-empirically in a similar manner to INDO.

The effect of including 3d orbitals in the basis set used for

calculations on molecules containing second-row atoms has been examined(22)by Santry and Segal . They recommend the use of an spd type of basis,

which a 3d radial function characterised by = 0.75 They also

recommend that the adjustable constants [3° in equation (2 .4 3 ) be made

subshell dependent, such as

where is the local core integral associated with the subshell M1M .

Within the framework of the CNDO/2 procedure for the off-dia

gonal Fock matrix elements, equation(2.43), S is calculated with

respect to a standard diatomic coordinate system ( the molecule is

rotated such that both atoms A and B bearing ({> and cj) , respectivelyylie on the Z-axis ), which means that (b and d> referred to they Yvmolecular coordinate system are expanded ( with expansion coefficients

a • and a • ) into the new set {^.} referred to the diatomic coordinate y i lsystem, i.e.,

S = <d) I d) > yv yy 1 Yv

= 7 Y a - a - <iD -1 iD -L L yi vj yi|Y3i J

= y y a .a . s . - (2.53)L L yi vj 13i j

As Hinze and J a f f e ^ ^ found, the electronegativities for

cj-orbitals are considerably larger than for TT-orbitals. No specific

account appears to be taken of this in the CNDO/S and INDO/S paramete-

rizations used here. The major effect is the introduction of the K

parameter, which results in a very substantial improvement in the. (40-41)shielding calculations for first-row nuclei. By multiplying

any of the . integrals in equation (2.53) by an arbitary constant,t

we arrive at modified integrals without destroying their

rotational invariance

s ’ = H a -a • S.. L .yv L L yi V3 13 13i j

= H a . a . S . ' . £ fj yi vj 13 (2.54)

1On replacing S by S we obtain the modified off-diagonal energy matrix yv yvelements,F (equation(2.43)) which are still rotationally invariant.

As is well known, the original CNDO/2 method (spd basis set), due

to exaggerate 3d participation, predicts the wrong molecular orbital

s e q u e n c e ^ ^ * ^ ^ . As in the original CNDO/INDO methods, Schulte and ( 28 )Schweig used RL . = 1.000 except when i and j refer to tt-AO* s, in

this case they empirically derived ^ .0.585. They predict the correct

sequence for phosphabenzene and Thiophen molecules by including the

aforementioned 3d AO correction by choosing K-^ = 0.300.

2.10 INDQ/5R parameterization

/ / n NBlair introduced the CIND0 program which has been concieved

as an all-valence M0 program for element up to Rn. Lack of specific

interest in the Lanthanide series and the programing difficulties

posed by f orbitals have led to the restrictive assumption of s-,

p- and d-valence subshell for every atom with atomic number greater

than or equal eleven.

In this program, in order to retain any degree of rigour, it

is clearly necessary under certain circumstances that different

radial functions be associated with different subshells of a given

valence shell. On spherically averaging the coulomb integrals,

equation (2.28) no longer applies, for this basis, if A or B is a

second-row atom because a more diffuse radial function is required

for the 3d orbitals.

For the case when A and B are first- and second-row atoms,

respectively, we have instead

Y^v > s (A) s(B) if <j>v is an s- or p-orbital "

s(A) d(B) if <(> is a d-orbitalY (2.55)

when A and B are both second-row atoms, there are clearly four

alternatives, two of which are identical if A = B.

The general expressions for the INDO method for the unrestricted

Fock matrix elements are given by

F“ , “ H„„ + I f f h n M X a ) - P£a (yX|ya)) + £ f Pw Yyv (2.56).a = n 1- / / J:,yy yy L L I Aa X a B(*A)v

A AF“v = I I Pxa ^ v lXa " PXa(y X lva) ’ for ^y and on A (2 -57>

X o

Fyv = ~ pyvCwlvv) » for <j> and <j>v on different (2.58)y^v orbitals

8 ftand similar expressions for the F and Fyy yv

The diagonal elements, H , of the core Hamiltonian arey yestimated using equation (2.34). The neglect of penetration, which

characterises the CNDO/2 method, is observed, so that

3 BV = 7 N y yy L v 1 yv (2.59)

v

where is the occupancy of the valence AO in the neutral ground

state of the isolated atom B. Since, with spherical averaging, the

value of v . is the same for all orbitals associated with a given ' pVsubshell "I" of B, we have

V = 7 N, y (2.60)yy L 1 'yv v '1

where is the occupation number of valence subshell 1 in the

ground state configuration of B.

For the case when A is a first-row atom or hydrogen, the values of

U appearing in equation (2.40) are calculated according to the originalppformulae, using the same values of — (I +A ) as tabulated in2 v y y;

reference (47). By contrast, lack of electron affinity data has

led to the following expression for atoms having an atomic number

equal to or greater than eleven

AU = - I - 7 N- y + Y . (2.61)yy y L v 'yv 'yX

v

when the CNDO option is chosen, y ^ is the simple, (spherically

averaged) one-centre repulsion integral. For the INDO option, y .pvis the average interaction energy (including exchange terms) between

an electron in subshell (n ,1 ) and an electron in subshell (n .1 ).y v* v'(9)From the formulation of Slater, for fn ,1 ) = (n .1 ) = (n.l).y y v v v >

we have

Yyv = F°(n,l,n,l) - (4U-1)'1 ? Ck (l,0,l,0)Fk (n,l,n,l) (2.62)k=2

while for (n ,1 ) # (n .1 J, we have v y * K v* v J 9

Y . = F°(n ,1 ,n ,1 ) -'yv v y* y* v* v 4(21y+l)(21v+l)

06 *1y CK(1 ,0,1 ,0) G (n ,1 ,n ,1 ) (2.63)L V > v > } \ y> y> v > ^k=0

where F^ and G k are the Slater-Condon parameters.

From equation (2.16), the elements of the matrix represen

tation of the Hartree-Fock Hamiltonian operator, F, are

Fpv = Hyv + PA0 - 7 (PXMXa

= Hyv + ^ PAA(yvlXX) - j I py a ( w | w )X=a y=X

v+a- 7 I pXv(uX|w) - 1 1 Pyv(uu|w)

y*X y=Xv=a v=a

Comparing equations (2.64) and (2.58), we have

epv = Huv + I px x M x x ) - \ I PyCT(w|va)X=g y=X

v*cr

' \ l PAvCyXl vv)

=a

(2

( 2

A simple and unbiased estimate of the bond-order matrix P

is obtained if we assume the given species to consist of non-inter

acting neutral atoms, in which case

PXa SXa NXX (2

then

yv H , +yv I NA(yv|XX) - j I N^(yy|vy) ~ \ l Nv (yv|vv)X=a y=X y£X=v

y#a=v v=a

H + , _yv L XX=oI (yv|XX) - y N (yy|vy) - j Nv(yv|vv) (2

.64)

.65)

. 66)

.67)

The Mulliken appro x i m a t i o n ^ ^ for the two-electron integrals

appearing in equation (2.67) is

M a x ) = \ sy v (YljX + YvX) (2.68)

and Mulliken's scheme for approximating multicentre integrals implies

that

then

If H S (H + H )m i n i m m -'yv 2 yvk yy "vv (2.69)

yv = 7 W \ v + + I Nr 7 s„vY,* + WX=0

X 2 yvk,yX TvX'

2 ^ y ’l Sy v ^ y y + Yyv^ ” 2 ^ v *2 Sy v ^ y v + Y )'vv^

- i s r h + h ) + yn, (y a +y y)2 yv yy vv L X u yX rvXJ

■ 7 N (y +y ) - -TT N (y +y ) 2 y v,yy ' \ iv J 2 v u yv yvvJ

From equations (2.34) and (2.59)

(2.70)

H = UW

then $yv

yy

7 S2 yv

B- I n yL v ’yv

v(U + U ) +. yy vv

AI n iY,

B■A'yA + I NX V

X X1 1 1 " 7 N y ■ N y - r y (N + N ) 2 y yy 2 v'vv 2 Tyvu y V

(2.71)

(2.72)

From this equation, the assumption of an isolated atom electron

distribution of the bond-order matrix elements is given by

equation (2.66). In that a given bond, as characterised by the

atoms it joins, may carry a characteristic charge separation into

any number of chemical situations, it may be possible to find a

single factor K able to correct for this charge separation in

any number of chemical situations. may also be used to correct

for the non-orthogonality of the AO's of atoms A and B at the

characteristic bond distance. Then

I c A B6yv 2 KAB Syv LUyy + Uvv + I Nx V + I NaA>A

X A

1 1 1o- N y - -s- N y “ -oY (N + N )2 y'yy 2 v'vv 2 Tyv^ y v J

(2.73)

CHAPTER THREE

DEVELOPMENT OF SOME THEORIES

NUCLEAR MAGNETIC SHIELDING

3.1 Basic considerations of nuclear magnetic shielding.

In order to discuss the variation of the chemical shift of a

given element we require a suitable theory for evaluating nuclear

shielding. L a m b ^ ^ considered that the shielding d arises from a

circulation of electrons around a nucleus in a magnetic field and

that this motion may be described as an effective rotation of the

whole electronic cloud about the direction of the applied field.

This motion results in a secondary magnetic field which opposes the

applied field. The components opposing the applied field are

integrated over all space to give a total induced field

F 2 iB . = -t— < - > B (3.1)ind 4tt 3m r. ol

where B is the applied field and < — > is the expectation value of1 ° ri th— , r. being the distance between the nucleus and the i electronr. llof charge +e and mass m resulting in the secondary field. The Lamb

expression for shielding is thus

u 2 1t i = (3.2)4tt 3m r.i .

This theory is strictly only applicable to atoms since it depends

on the spherical symmetry of the electric field of the nuclear

electric potential.

The energy E(BQ ,pN), associated with the electronic

Hamiltonian describing a closed shell molecule in the total magnetic

field due to a uniform external magnetic field, Bq , and the dipole

field arising from nuclear magnetic moment, pN , can be found by

solving the Schrodinger equation, for nucleus N .

K B ^ ) = E(B0,yN i K B ^ p J (3 .3 )

where is the wavefunction describing the molecule in the

presence of B_ and fi , . For small values of B0 and Tj- , E(B„,if )N N Nand ^ ( B ^ y ^ can be expressed as Taylor series about their zero

field values

’K 3 »vn) = 0)Bo + I E(0>1Vn L a a L a Naa a

+ H B^ 62,0)Bg - H B o E ^ L N3a$ a$

An alternative expression for the energy is given by(3 .5 )

Eflyjj = E (0) - I y Bo - 7 p Bon 'a a L ^Na aa a

N3a $ (3 .6 )

Here y - is a component of the permanent magnetic moment of the

molecule. The third term represents the direct inter

action between the external magnetic field and the

nuclear magnetic moment. The forth term describes the

diamagnetic polarization of the molecule; the total

magnetic moment (in direction a) associated with the

electronic currents induced by the external magnetic

field is Tv > where y « is a component of the|Aa$ $ Aa$molecular diamagnetic susceptibility tensor, y . The

secondary magnetic field (in direction a) at nucleus N

due to these electronic currents is where

is a component of the magnetic shielding tensor, .

Thus the total magnetic field experinced by nucleus N which

determines its NMR frequency is given by = Bo(l-G^) . It can

be clear from comparing equations (3.5) and (3.6) that the calculation

of requires the determination of .

The quantum mechanical nature of the magnetic resonance

parameters may be determined by comparing the spin Hamiltonian with

the energy expectation value of the complete Hamiltonian operator

and matching terms bi-linear in the field and spin terms. The

complete Hamiltonian operator at fixed nuclear positions has the

form

(P. + eA'(r.))2- 2Tz rT1 j y } n N

Here £ is a sum over all electrons and £ is a sum over all nuclei.j N

The last three terms represent the electron-nucleus, electron-electron,

and nucleus-nucleus contributions, respectively, to the potential energy

The vector potential describing the total magnetic field at

the position of electron j, A(r^), is given by

A(rp = B<,x ?. - I (pN x r . ) r73 (3.8)

where r^ is the distance vector from direction j to some arbitrary

origin

r\ , "r._ and R.IT_ are the vectors C?. - R*,), Or. - R-, ) andJN Jl ND j N j 1(Rn - respectively,

and Z , is the charge of nucleus n *N

Substituting equation (3.8) into equation (3.7), replacing

the quantum mechanical momentum by "iVj and working in the

Coulomb gauge^"^ (div A = 0) leads to

H(itj = + I BoHC1>°) + I p H(0’n “ a a L KNa aot a

£ | ' W + 1 l B&Ha3, yN3 +--- (3-9)a p a p

To simplify the notation, from this point on we consider a single

nuclear magnetic moment, y , only. Extension to several nuclearNmagnetic moments is straight forward. In equation (3.9), Bo and

B% are components of the external magnetic field a n d ^ ^ ’^ ^

(0,1) _a - i l L . rT3j jNa jn

Hae’X) = 7 I ( % *j„ 6ae ■ rjarjN6 5 rjN

^(3.10)

where L. = ( r. x V. )J 3 3 a

and L • , = ( r• x V- )jnoc 3N J

Here 6 D is the Rronecker delta and a, 8 are used to indicate expcartesian coordinates X, Y and Z.

Using equations (3.4) and (3.9) to compute

(3.11)

and comparing with equations (3.5) and (3.6) leads to quantum

mechanical expressions for the cartesian components of the nuclear

magnetic resonance parameters as

a8 a 1

where only the leading contributions are explicitly retained.

Alternative expressions for these parameters can be formaly

obtained as second derivatives of the energy by

a32F-(^.Pn )

(3.13)

B=y =0° NEquation (3.12) leads to the calculation of nuclear magnetic

shielding using Sum-Over-States (SOS) perturbation theory and

equation (3.13) leads to the calculation of nuclear magnetic shielding

using Finite Perturbation (FPT) theory^

Nuclear shielding is a tensor property. In the absence of any

kind of symmetry it requires nine independent components to fully describe(76)the shielding at a given nuclear site Quantum mechanics provides

an expression for the components of the shielding tensor. This was first (6 7)obtained by Ramsey * . In their most general form the Ramsey equations

for the shielding tensor component, ôr a chosen nucleus of a molecule

in its ground electronic state, can be written

tfa3 = V 1} + W 2) + *ae(1) + 4tt 2m k a8 ka k£-

i £ <0 |K 3' " V w |o>

e24ir 2m2 J [<Q| K ~ \ a l n><nl £ V 0>

-1

y 2o

(3.15)

+ <0|EPk6|nxn|Zrv“3L^|0>](E„ - EJ -1

L (3.16)

k k a 1 n

The two parts of 0^ are referred to as the diamagnetic shielding term,

c/L, and the paramagnetic shielding term, C^p. In equations (3.15) andaP aP*(3.16) the symbols pQ , e and m denote respectively the permeability of

free space, the electronic charge and mass, r^ is the distance of the

k ^*1 electron from the nucleus under consideration, L, and P. are thek korbital angular and linear-momentum operators for the k *'*1 electron.

1 0 > refers to the unperturbed electronic ground state of the molecule

and |n> to the excited states with energies of Eq and respectively.

The summations in equation (3.16) are taken over all of the excited

states including the continuum, 6^ is the Kronecker delta and i-s

the alternating tensor (= 1 if (3y6 is an even permutation of xyz, = -1

for an odd permutation and = 0 if any two of the labels j3y6 are identical)Cl nThe magnitudes of and depend on the location of the origin of the

vector potential of the external magnetic field since the shielding itself

can not be dependent on this location.

Equations (3.15) and (3.16) can readily be applied to atoms.

However, when applied to the molecules many difficulties are encountered

The first difficulty is that, in general, little is known about the

molecular eigenfunctions of either the high energy discrete states or

the continuum of a molecule. The second difficulty follows from the

first. In an atom it seems logical to choose the origin of the vector

potential as the atomic nucleus, but for a molecule this choice is not

easily made. It is not obvious whether to choose the origin as one of

the nuclei, as the molecular centre of mass, as the electronic centre

of charge, or as some other point. For medium sized molecules, Ramsey's

approach is disadvantageous in that the diamagnetic and paramagnetic

terms become large and of opposite sign. Consequently the calculation

of the resulting nuclear shielding is likely to be considerably in error.

The theory of nuclear magnetic shielding can be more conveniently

developed within the framework of the LCA0-SCF-M0 theory, where the

electronic ground state wavefunction \|/q of a closed-shell molecule

with 2n electrons is expressed as a normalized single Slater determinant

of doubly occupied Molecular Orbitals (MO's) \Jt as

1iPQ = (2n ! )~ 2 ^ ( 2 ) ^ ( 2 ) ......i(Jn (2)ifJn (2) (3.17)

4’1 (2n)i|;1 (2n) .. .^n (2n)i|;n (2n)

which is usually written as âr

indicalts that ij is associated with a |3-spinfunction. A further

approximation is made by expanding the MO's as a linear combination

It may be clearly seen that more accurate MO* s can be obtained

from the large basis sets of cj) functions. This, however, increases

the complexity of the calculations and frequently limits the

applications to those on small molecules. The MO's are therefore

usually simplest to apply and interpret if the basis set is minimal,

consisting of the least number of atomic orbitals required to

describe the molecular ground state. In equation (3.17) the anti

symmetrized product (AP) of the M O ’s is one possible configuration

or assignment of electrons to molecular spin orbitals and represents

the ground state configuration. In excited state configurations, the

electrons are assigned to higher molecular spin orbitals to form other

AP's. In the case of degenerate configurations, a set of several AP's

is often required to form the proper excited state wavefunctions.

Then the various integrals in the Ramsey equations (3.15) and

of atomic orbitals (cj) s)

(3.18)

(3.16) become integrals involving determinantal wavefunctions.

Using " Slater-Condon " r u l e s ^ ^ these many-electron expressions

may be reduced to one electron matrix elements, in which only

one-electron operators are involved. For a one-electron operator

like £ L ., the only excited states, ^ , that can give non-zero

matrix elements in equation (3.16) are those described by single( 78 )excited spin singlet configurations in which an electron is

promoted from an occupied iJk to an unoccupied ^ . These singlet

excited states are described by the function

^ C I ^2^1 ’ * * *î^k* * 1 4*2^2* * * *î^k* * * (3.19)

The matrix elements for a one-electron operator, are thus

/s occ’J’o I I Ljc I ’I’o > = 2 I K I h I V 1) >k i

►(3.20)

within this MO framework, may be expressed as

, u 2 occ 0 oA _ K o e r . . i / 2 0 n -3u o 7— — T I (r 6 c - r r D)r (3.21)ap 4tt m k î1 v aP a p |yi1and

U 2 OCC UnOCC *. 1 i I<£ _ _ £ > « 2 . 2 ( iEk c°^ - y 1ap 4rr 2 f* v 1 o Jm i k

- < <|,k I r ~ \ u . > )(3.22)

™ • 1 • • • ,1^ ( 0) 1 (0).The electronic singlet transition energies ( - E q ) are

•xprossed a:(«)

l„k(0) U ( 0)( E. '0) = e, - e • + 2 K., - J' k l ',vik ik (3.23)

where and are the SCF orbital energies and K^k and J^k are

the molecular exchange and Coulomb integrals, respectively. These

are defined by

K.ik ^(1)^(2) dlldT2 (3 .2 4 )

and

ik(3 .2 5 )

3.2 Independent electron GIA0-M0 method.

The difficulties associated with the gauge-dependent calculations

of nuclear shielding can be overcome by using an approach in which each

MO is composed of a linear combination of gauge-dependent atomic orbitals(39 82—84)as demonstrated by Pople ' It is unfortunate that this method

has been referred as a gauge-independent atomic orbital (GIAO) approach

in the literature. The dependence of the atomic orbitals on the gauge

provides nuclear shielding data which are gauge-independent. The MO's

, are given by

I C -X (3 .2 6 )

where ( - i(f) AuCr).? (3 .2 7 )

where A (r) is the vector potential associated with the electron in P

orbital p. By treating terms in the one-electron Hamiltonian

involving A as a perturbation, changes in the individual molecular

orbital energies are calculated, and hence nuclear shielding data

which are gauge-independent.

The local, non-local and interatomic contributions to nuclear( 81")shielding arise in the manner proposed by Saika and Slichter

Cj q = c7d0(loc.) + (jdQ(non-loc.) + Cjd Q( inter.) oc3 ap ap ap

+ CjPpCloc.) + CJ^( non-loc.) + inter.) (3.28)

The various diamagnetic and paramagnetic terms in equation (3.28)

are not directly comparable with expressions bearing these names in

R amsey's t heory.

The local terms arise from electronic currents localized on

the atom containing the nucleus of interest. Similary, the non-local

terms are contributions from the currents on neighbouring atoms.

The interatomic contribution terms are due to shielding currents not

localized on any of the atoms in the molecule, e.g., ring currents.

These latter two terms usually only produce a shielding contribution

for a few ppm at most, which is important for protons due to their

small range of chemical shifts. Other nuclei have chemical shift

ranges of several hundred ppm and thus interatomic contributions are

negligible by comparison.

(39 83)Pople * developed a MO theory of molecular diamagnetism

within the independent-electron framework which results in all explicit

two-centre two-electron interaction terms becoming zero, and all

two-centre overlap integrals being neglected. The gauge-independent

expressions for the local and non-local terms of the diamagnetic and

paramagnetic contributions are, for the shielding of nucleus A, given

by

and

d R(loc• )ap I PU1J < * u I r "3 (r2 6 ag - r a V l 4>u > ( 3 ' 29)8tt m uu ' y y

c1 p(non-loc.) = — --- Y YM(Â)Y

rM

- 5 ^ 2“m (V a g - 3RMyW (3.30)

tfPg(l°°.)2 2

y e h occ unocc

2TrmlEk(0) _ 1E (0) j o

-3^

- 1

y c. C C. C , <cf> I r L Icj>, ><4>11Lq14) > (3 .3 1 ) yvXa ^ kv k y

aa 3 (n o n - lo c .)y e2h2-2- t I I4tt m M(M) y

occ unoccf k ( 0 ) (0)E . - EJ D

- 1

AI Cjp Scv Cj a ScA La I VyXva J K

<<t>X I Le I V RM$ (RM 6ag " 3IW W(3.32)

where C's are the unperturbed LCAO coefficients of the atomic orbitals

y, V , X, c1 in the occupied and unoccupied MO's j and k, respectively.

All of the angular momentum integrals in equations (3.31) and (3.32)

are one-centre in character and are given in units of ft/i. The

evaluation of these integrals is facilitated by using the following

f t A Aexpressions or L and in spherical polar coordinates (in units

of fi/ i )

- ( SincJ) |q- + CotG Coscf) )

AL

AL

= + ( COS<J> Jg - Cote Sin<}> )

a_3cJ>

' (3.33)

The integrals are non-vanishing only between pairs of atomic orbitals

having the same angular momentum quantum number L. The matrix elements

required in expressions (3.31) and (3.32) are given in Appendix F for

p and d atomic orbitals.

theFor molecules containing atoms inv second-row and transition

metal elements, the expressions (3.29) to (3.32) are simplified when

s, p and d atomic orbitals are considered. The rotationally averaged

values of the diamagnetic and paramagnetic terms are then given by

<j a (1 o c .)

CjP( loc.)

(loc.) + d d (loc.) + (jd (loc.) xx yy zz

u e ro12-rrm W

cfp (locXX .) + c P (loc.) + (JP (loc.) yy zz

u e

6tt m: V o c c u n o c c (0 ) ! ( 0 ) -XT ~ I I C Ej ' Eo )Ti 1 J

ALA B

y y y c. c, c,,c. < y ,L £ (• jy kv kX -ja 1 „3yv B Xo J

(3.34)

| I v >< A I La I a >

FQe2h2 o c c u n o c c k (0 ) x (0 ) . - i-— — 1 1 ( Ej - E0 }6rr m j k

AA L B AI c* C, < y I -4 I v >y y C. ,C. < X I L I a > L t iy kv ' 3 1 £ e kX ja 1 a 1yv r B Xa J

2 2p e h occ unocc _ . ,I I < } E * W ~ E^ )

6-rrm j k J

(ca.p ca p “ ca.p c ^ (cb-p cb,p ~ °B -p C K p y ' )j y k z j z ARPy B j y k z j z k

+ ^CA.P CA P ” CA.P CA P ^ CB.P °B P " CB.P CB P ^j z k x j x k z B j z k x j x k z

+ ^CA.P CA P ” CA.P Ci P ^ CB-P CB,.p ~ °B -P CB.p ^ <r >Pj x k y j y k x B j x k y j y k x

+ j/3 (CA d C 2 “ CA.d 2°A d ^ CB .d CB d 2 “ CB.d 2C B d j y z k z j z k y z B j y z k z j z k y z

+ ^CA.d CA d “ CA.d CA d ^ ° B . d CB d " °B .d C B d ^j xy k xz j xz k xy B j xy k xz j xz k xy

(CA.d CA d 2 2 " CA.d 2 2CA d ^ CB .d CB d 2 2 ‘ CB.d 2 2C\ d ^j yz k x -y j x -y k yz B j yz k x -y j x -y k yz

+ ^ (CA.d 2CA d " CA.d CA d 2 ^ (CB.d 2C B d " °B .d C B d 2^j z k x z j x z k z B j z k x z j x z k z

(CA.d CA, d 2 2 " CA.d 2 2CA, d )jUc_ , CR , 0 _ - CR 9C )j xz k x -y j x -y k xz B B.d B d 2 2 B -d 2 2 B dJ J J J j xz k x -y j x -y k xz

+ (CA.d CA d " CA.d CA d ^ (CB.d CB d " °B .d 'B d ^j yz k xy j xy k yz B j yz k xy. j xy k yz

+ (CA.d °A d _ °A.d CA d ^ CB.d CB d “ °B .d C B d ^j xz k yz j yz k xz B j xz k yz j yz k xz

(CA.d 2 2°A d “ CA.d CA d 2 2 ^ (CB.d 2 2CB d " °B .d C B d 2 2)} <rj x -y k xy j xy k x -y B j x -y k xy j xy k x -y

(3.35)

where CA is the unperturbed LCAO coefficients of the np orbitaljPx Von atom A in molecular orbital j etc. The summation, ^ , m

Bequation (3.35) includes A, it is obvious that the summation will be

zero unless both atoms A and B possess p and/or d valence electrons.

The matrix elements in equation (3.34) are evaluated as

% \ r_1 i y =npA

n 2a.(3.36)

-3 -3The terms < r > and <r >, are the mean inverse cubes of the P ddistances of the valence p and d electrons from the nucleus. For

-3first-row nuclei, <r > ^ is usually evaluated by means of the. . . . (84) relationship

< r " 3> np,_ 2E,na L o J

(3.37)

In equations (3.36) and (3.37), Z is the effective nuclearnÂ

charge for the atomic orbital <j>, with principal quantum number n on

atom A and a is the Bohr radius. The value of Z may be obtained° nHA(9) .from Slater’s rules . According to these rules, s and p orbitals

of the atoms are expected to have the same effective nuclear charge

which is given by

z « = + 6Z «eff,npA eff,npA eff,npA (3.38)

where Z is the effective nuclear charge for atomic orbitaleff,npAp of atom A in molecule,

is the effective nuclear charge for isolated neutraleff,npAatom,

is the change in electronic population in subshell

on going from atomic to molecular situation which

is given by

and 6Zeff,npA

where is the population of i ^ subshell of atomic orbitals in a

given molecule ( = P ),ppis the population of ith subshell of an isolated atom

(= Valence electron),

in equation (3.39), a^ are constants obtained from Slater's rules,

and (Q^-N^) is the subshell population on going from the atomic to

the molecular situation.

As discussed elsewhere^ 89)^ semi_empir icai methods suffer

from an ambiguity in the choice of the 3d orbital exponent for second--3row nuclei and hence the value of <r > . The most of the calculations

P-3reported m the literature, <r is estimated by using STO's. These

give fair results for very light nuclei, but for most nuclei the values—3 f85)of <.r > obtained from the STO's are too small

P

From atomic theory, one knows that, for different atoms with-3the same valence subshell configurations, one expects < r > for theP

same subshell to increase with atomic number Z. However, the behaviour -3of < r > for 3p and 3d orbitals versus atomic number Z, cannot be

arrived at readily by using the same kind of reasoning. Self-consistent(86 87) ~3field calculations * on atoms show t h a t < r >0 is about two3p

_3 (88) orders of magnitude greater than <r It has been shown that-3< r >0 , increases on the removal of an electron from a neutral atom.3d

a similar effect has been n o t i c e d ^ ^ in molecules where the atom is

bonded to a highly electronegative atom.

The electronic singlet transition energies ( ^ E ^ ° ^ - Êq °^) in

equations (3.31), (3.32) and (3.35) are expressed as in equation (3.23).

P in equations (3.29) and (3.34) are the elements of the charge density PP

bond-order matrix defined by

3.3 Theory of solvent effects on nuclear shielding.

Observed nuclear shielding data are usually obtained from NMR

measurements on liquids. In this case the observed nuclear shielding

O' , is a sum of the shielding for the isolated molecule O’. , andobs isoa contribution due to the presence of the solvent, O' . The- solvsolvent contribution to the shielding may arise from five additive

effects, i.e.,

= 0 + 0 + 0 + t f + t f (3.41)solv b w a E c

Here tf arises from the bulk magnetic susceptibility of the solvents,

0 from solvent-solute Van der Waals interactions, w0 from the diamagnetic anisotropy of the solute molecules,

3.

0E from the electric field induced in a polarizable solvent

when the solute molecule has a permanent dipole moment,

and d from the formation of solute-solvent complexes through weak

chemical interactions.

The main difficulty in calculating solvent effects is that we

do not have a complete picture of the nature of the orientation of

solvent molecules around a solute molecule. In the present study we

restrict ourselves to the model introduced by K l o p m a n ^ ^ and latter(91)implemented by Germer within the framework of a semi-empirical

SCF approach. In the Klopman m o d e l ^ ^ the interaction between

solute and solvent molecules has been considered by means of an

imaginary particle called a "solvaton", S. This imaginary particle

presents the oriented solute distribution around each atom in the(90 91)solute molecule. In the "solvaton" theory it is assumed that 3

1. Upon additional of a solute at finite dilution to an aprotic

solvent of dielectric constant, £ , a number of charges ("solvatons")

are induced in the solvent.

2. Associated with each atomic centre of the solute molecule is a

"solvaton” whose charge is equal in magnitude but opposite in sign

to that of the atom to which it is attached.

3. There are no interactions between the "solvatons" themselves and

they can have any fractional or integral charge required.

A. The strength of the interaction between the "solvatons" and solute

molecules depends on the polar nature of the solvent and is a function

of the dielectric constant of the solvent.

On the basis of this model, the solvent interaction terms are

incorporated into the Hamiltonian of the system and this modified

Hamiltonian can be used in the Hartree-Fock SCF-MO formalism to

determine a wavefunction which reflects the solute-solvent interactions.

Therefore, the Hamiltonian, H, of a molecule with M electrons and N

nuclei consists of two parts, namely, the inherent term, anc*

A

A

Athe solvent interaction term, H and is given by

A A (3.42)H H.inh + H sol

where HA

and HAsol

where is the permittivity of free space

Z is the nuclear charge,n

Q g is the induced "solvaton" charge, and

r . and r , are the "solvaton"-electron and "solvaton"-nucleus si skdistances, respectively.

In order to evaluate r s^> two more assumptions have been made.

First, for AO's associated with the same atomic centre as a "solvaton",

r . is the Van der Waals radius of the particular atom type. Second, siif the AO's and "solvatons" are associated with different atomic

centres, the "solvaton" is assumed to be on the atomic centre associated

with the "solvaton" and r g^ is evaluated accordingly.

One drawback of the "solvaton" theory is ignorance of the possible

steric inhibition of the solvent which may occur for atoms in the bulk

of molecule. In addition this theory is unable to account for hydrogen

bonding effects of protic solvents.

CHAPTER FOUR

SOME CALCULATIONS OF SHIELDING FOR FIRST-ROW NUCLEI

4.1 Introduction.

There is a widespread interest^ in the theoretical

interpretation of nuclear shieldings at both ab initio and semi-

empirical MO levels, especially for first-row and second-row elements.(92)Although some nuclear shielding calculations by Ditchfield have

achieved a high level of success for some first-row elements using an

ab initio SCF method with FPT, this method is unfortunately limited to

those small molecules such as NH^ and HCN which can be treated with

extended basis s e t s ^ ^ .

In this regard, the computational simplicity of semi-empirical

methods seems attractive. The independent electron theory of molecular(39 83)diamagnetism developed by Pople ’ for closed shell systems provides

the most satisfactory model to date on which to apply semi-empirical MO

calculations by using gauge invariant atomic orbitals (GIAO-MO). Webb (93-104)et al have reported the results of such calculations on some

first-row nuclei in a variety of molecules by means of the INDO, CNDO/S

and INDO/S parameterization schemes.

In the present work, some theoretical results of nuclear shieldings

for some first-row nuclei in a variety of different electronic environments(39 83)are obtained by means of Pople's GIAO-MO method ' in conjunction with

the CNDO/S and INDO/S parameterization schemes. The results obtained are

analysed in an attempt to improve our understanding of the relation between

nuclear shielding and various features of molecular electronic structure.

4.2 Carbon shieldings.

The results of calculated carbon shieldings and chemical shifts,

with respect to benzene, are reported in Table 4.2.1 and are compared

• (105-115)with the experimental data where available

From Table 4.2.1, the calculated differences in the average

carbon shieldings arise almost entirely from changes in the local

paramagnetic contribution, cj (Ioc). The variation in the local diamag

netic, Cjd(loc), is within 0.5 ppm. The values of local and non-local

terms of diamagnetic and paramagnetic contributions are not listed

separately in Table 4.2.1. An over all average value of 295.5 ppm for

the local diamagnetic contribution is obtained from both CNDO/S and

INDO/S calculations. This is similar to the local diamagnetic contribu

tion calculated for some molecules, using similar methods of calculation

in which the changes in the local diamagnetic contribution are not of• ..• • j n (93-104)major significance for the first-row nuclei as demonstrated elsewhere

In Table 4.2.1 are the CNDO/S results for some carbon shieldings

and chemical shifts of compounds number 1 to 15, relative to benzene.

These are compared with the experimental results as shown in Figure 4.2.1

with a correlation coefficient of 0.80, standard deviation 21.25 ppm and

slope 1.08. For INDO/S calculations, the correlation coefficient being

0.82, standard deviation 20.23 ppm and slope 1.23. The over all agreement

from the INDO/S calculations is slightly better than the CNDO/S, except

the slope. The slope of the results from the INDO/S calculations is

slightly larger than the CNDO/S ones. This means the INDO/S chemical

shift calculations are numerically smaller than the experimental data in

all cases considered whereas the CNDO/S results are closer numerically

to the observed carbon chemical shifts. This is probably due to the orbital

energies being too widely spaced in the INDO/S calculations than the CNDO/S(41)ones as demonstrated elsewhere

A better agreement is obtained between the INDO/S results for the

carbon shieldings for compounds number 1 to 28 and the experimental data

TABLE

4.2.1

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s - \cooQ55CO<o

/'■NCO-o ’o55to

/-\CO/S

CO CO CO coO e co co CO oO G Q Q G!5 X 55 55 55 55M w CJ O M M

M_/ v_«0 o to «o b> O

HPU

X4-JC0)J40«H4->COb03COT3C3Oa6oco

D

xu

-CO00asojCJ

O

00o

p-l CM VO 00 o in VO on i—i o CM co o oon in <r in 00 o vO m 00 <r o o« m • • • • • • • • • •

VO in i—i <r in CO i— i VO LO <r o <r<r o m 00 <r r. VO <r o oi—i I i—i I r— i i— i I i—i 1 1 i—i 1 i—i i i—i i

co

00 ON vO 00 o VO i—t co o <r m o CM oCO ON i—1 CM CM ON in m CM VO CM 00 ON <r

• • • • • • • • • • •r-( On 00 <r <r o ON I-"- CO <r o ON CO <rin <r m <r in <r h-i— i 1 i— i I I r— i I i— i I I i— i I i— i 1 i

vOo

ino

co

coc p

CMcp

o

TOo£ 1l-JCDe

<?00oin

incoin<r

coi—i i—imH

cooQ53CP\D

in<ronI

VOONCO1

<rCT\I

s~\COoQ53o

inoon< r

oo

p'.ON

<ri—i

/-\COoQ !3 I—It)

o in in o 00i—i vO CO r—l CM ON o VO VO• • • • • • • • •in r- o ON ON in o 00 00o m <r o in <rI i—ii i—i I i—i I i—i i i—i I i—i

VO o 00 ON vO 00 o VO r mON 00 co ON CO <r O VO o• • * • • • • • mo in CO 1—1 CO in co coVO o <r <r VO o <r <ri I—1 1 r—l I i—i l 1—1 1 i—i I r—l

ON o m VO vO 00 o CM co ONo VO CM 00 ON o r—l 00 <r ON• • • • • • • • •*0 oo CM o 00 m i—1 o ONVO m 00 <r VO m •oI 1 i—i 1 H I 1 i—i I i—i

/~N S'N r~\ ON r~\ 0~NCO co CO CO CO CO CO CO"■— ■— ON ■—ô H o o o o H o o oQ P p Q Q « « PM o Q Q5Z X 53 !3 53 23 X 3 531—1 W CP CP 1—1 I— I w CP O Mv_/ V- v_o N_ v^ v—s<o cO "D «o ■O «o (O "O <o t)

on

<rI

r-'i—ionvOI

i-1 i—I<rI

s~\COo'O53<0

oVOVOI—Ir—lI

oinvOOr— lI

oCMOI

KBV_/<o

XIpCDPPJ • Hi->CO£>PCOTOiPPOa§o

CO__

■ O

'CMPC- cpac(NOCMCODPncjn

: 0

'c mECCManCslOCO

ancoO

o53

oo i—i o CN o <r in I-1 CO o <r in CO 00 o0\ LO 00 j'-' <r o 1—1 CNJ o CT\ m in CO CNJ# • • • • • • • • • • *vO in CO i—j co ON co CNJ cr\ in CO CNJtn 00I <r CO in 00 r-~ CO in 00 <r COi—i i—i I 1 i—i 1 rH I 1 i—i I r—l I 1

o00 cr> o\ ■—i o 0\ o o CM o o\ o cr> o00 <r CNJ <r <r CM cr\ CO <r <r <r r—i in vO 00

* • • • • • • • • • • • *o O'! CO <r CM o i—1 CO crv 00 CO CM 00in <f r>. <r <r CO t-- <r <r r. |Ni—i I i—i 1 1 i—i 1 i—i 1 I i—i 1 i—i 1 1

cOO

mo

o

coo

CMo

i—IO

T3 O ■G4-><Ue

1—1 CM m r. O o 1—1CM 00 <r in 00 i—i• • • • • • •H o\ o\ m CO co i—iin <r r—l co COr-4 I i—i i i—1 1 r-4 1

CO o CM O CM coo CO r. CO CO CM• • • • • •CO in co o\ 00 00<r 1—1 I 'd- i—i co1 o 1—1 1

1

00 in o i—i oin i—i CO CO <r o• • • • • • •i—i o 00 <r CO CMin oo <r CO 00 r—lr-4 I i—i 1 I 1

/N N /-“C /~N \CO CO CO CO CO CO_ S~S -_ "—COp oPi oPI oQ £ oQ oQIS is 125 25 X IS ISCO o I—I M w cj oc c_y c_y CX) O 3 <o o 'D <o

O CM o o r— l OC r-4 O0 0 CO CO CM r—1 CO <r# • • • • • • •

co CM CO 00 0 0 <r COco CO I— I <r <r r— lr— i 1 I—I1 r—l I i— i 1 r-4

1

o\ i—1 o CO 00 o Oa\ i— l CM r . co CO 0 0 o\m • • • • • • •

i— 4 oo I—1 O N 0 0 CM 00 CM0 0 I CM1 I 0 0 1 co1

0 0 o o 0 0 <r CO or>. P\ CO o CO o\ o CO• • • ■ • • • •in i—1 i—i CO <r <r CM00 i—1 1 i—i 0 0 i—ii 0 0 r— l1 1—1

s~\ / N /N ss /N / \CO CO CO CO CO coN ^ N"o o H o o o o Ho Q FM Q Q Q Q PM5S X X IS 13 P5 XM w W CJ O M M wc_y V c c ^ C_ c_/o <o "O <o (O O

S~\x4->g0)4-1•H4-1WP3cotPG0Oa6oo

O CODPcoOac(NOPC’-OCO•Hov— y

X:6

<0 ✓'Nffl mcoO35cnOffiT-O

GGG4-1

O13 00

TABLE

4.2.1

(con

tinu

ed)

ooo

o

MOo

ino

CN CO in n- COCO ON in <r• • • • •o 00 ON in <rLO CO CO o1—1 1 i—1 i i—i

cj

cno

CNO

cj

oo:4-><u6

00CNCNcn

00CNCN CO i—I

<rn-oini—i

coo Q 53 O c—'

ON00oCOI

ON000CO1

incoOnI

/~\cooQ53Oc_cO

ONinONCM

ONinONCN

coin<r

/scooQ23I—I v_■D

CO 00 O in COCM 00 CO 00 I—Ii | • • • • : • •ON 00 <r ON ON<r CO CO O <ri—i I 1—1 i l—l1 i—i

i—i m CO <r ON i—i 1—1 COin CM 00 o CM o CO• • • • • • • •in CO r» in CN CN CO COin o CO CO co 00 On coi r—l1 i—i i 1—1 1 1 H

i—i in i—i CN 1—1 CO in 00r. in o CO m CO in• • • • • • • •in CO in CO o CO in ONin o co CD co in ON CMi i—ii i—i 1 i—i i i i—l

co n- in CO CO 00 CN COr. o CO O I—I ON• • • • • • • •■—i co CN o CO CN CO CNin 00 <r r inI I i—i 1 i—i 1 I i—i

r ~\ \ /-~s /—\ <—NCO CO CO CO co CO/ N — /—N ,o H o o o o H oQ PM o Q Q Q P-i o!3 X 3 53 3 3 H 523M W CJ CJ i—i i—l w CJc_ N_/ N-> N_S v_y v—s<o <o TO o TO <o <o TD

<r

I

<rCNinCOI

ON

00ini

<roi—i 00 I

/'-NCOoQ53O

i—I0000CO

00CMco

00CO

co

inONin<ri—i

cooQ53

COON<rCOi

inooooinl

oininI

oCNI

ncoOQ53l—l c.«o

CO<r<roi—i i

oONooI

COCNn-00I

coco<rI

s~\HP-i

s~\Xv_>4->0CD34-)•H4->COOl3COTO03Oa6oo

Y71✓"soc_CNJ/—NoCO

DC- O

\

PM✓-Noc_CMoCOEC —o

IoC_yCMoCODC

- CO

o53

TABLE

A.2.1

(c

onti

nued

)o

o

uoo

o

COo

CMo

o

0OA4-)a)6

uo VO o CM CM VO r>.O vO CO CM i—1• • • • • • •o 00 CO UO CM o 00UO r^ CO vO i—l uor-4 I r-4 I i—I

1I—1 I

UO vO o CM CM CO <To VO CO <r CM o VO• • • • • • •o 00 CO uo CM o 00uo co VO i—1 uo r.rH 1 1—1 1 i—l

1I—l 1

O rH o CM VO I—1 CM<r O <r UO VO CM 00• • • • • • •<r CO i—i VO CM oCO vO co uo 00 CO VOrH 1 i—i 1 I 1—1 1

00 CO VO VOCO CO CO <t VO vO CM• • • • • • •<r CM i—i VO <r coCO vO CO uo 00 CO VOi—i 1 i—i 1 1 i—i 1

CO o CO uo i—4 00 COoo U0 CM CO CM uo I—1• • • • • • •CM i—1 CO <r CM i—4UO 00 <r UO 00|H 1 i— i 1 1 I—1 1

/~N /~\ f~\ ✓-S r~\ /"NCO CO CO CO CO CO_^ — ^ "■-o o o o H o oo Q Q Q P-i Q Q53 53 53 53 X 53 3o o M i—I w CP o

v _ / V— / v _ y~o <o "o <o tO t) to

CMONCOi—I

couoCOCO

CMco

VO<rCMCOi—I

UOr—lVO<ri—i

COoo531-1■D

CO o CM oON uo o i—1 VO1 1 ■ • • • •cn 00 cn UO CO<r CO VO 1—1i—i I i—i 1 1—1 1

cn 1—1 <r UO CM <r COCO 00 o VO CO <r ONm • • • • • •UO <r i—i ON o VO vOvO i—i uo <r VO o1 i—i i i—4 I rH 1 i—41

U0 i—i 00 ON rH CO <rvO 00 ON U0 <r uo CM• • • • • • •UO <r ON 00 cn uo U0vO i—4 <r r"-. co VO o1 i—1 1 i— i 1 i—i 1 1—1 1CM i—1 o r-1 ON i—1 VOCM O CO UO r- uo• • • • • • •00 H i—i o CM 00 COUO cn CO VO CO uo 001 1 i—i 1 rH I 1

00 i—i CM CO CO uo COuo o 1. CO O r-4 o• • • • • • •00 iH ON 00 CO cn couo cn CM uo CO uo 001 I r-4 1 1—1 l i

VO VO r- <r vO uoCM cn <r o 00 cn <r• • • • • • •CM CO CM i—i vO CM <rr- UO 00 <r r- r~1 I rH 1 i—i l I

/O /"O r~\ / \CO CO CO CO CO/-N ■»_ ✓—\o H o O o o HQ P Q Q Q Q P-j3 X 53 3 3 53 Ki— i W CP O l—l I—1 wv. vto O "O t O "D to <o

XV-4->PQJP4-J • H4-JC OOPCO0PPO(X6oo

If No

CMOCODC-o

\tov_/CMoCOE

*-CP

o53 CO <r

TABLE

A.2.1

(c

onti

nued

)o

o

o

LOo

o

COc_>

o

o

TOoo:4-)CUs

o 1—1 CNJ <r CN00 <r i—l CN vO• • • • • 1 1o cn o CN VOin r. <r VO oi—i 1 i—i 1 1—I 1

o i—i CNJ <f CN00 <r r—l C\J VO• . • • • 1 1o cn o CN voin <r vo Oi—i I i—i 1 i—I1

o i—i CNJ <T CN00 <r 1—1 CN VO• • • • • 1 1o cn o CN VOin n* <r VO oi—i 1 ■—i 1 1—1 1

CO <r CNJ <r CNVO CNJ 1—1 CN VO• • • • • 1 1O cn o CN VOin <r VO Oi—i I r—l 1 i—l1cn o r—l CO cn 00 oVO CO O I—l i—i o 00. • • • • • •00 CNJ 00 o CN r~4CNJ in co in 001—1 i 1—1 I I

I—l CNI 1—1 CO cn 00 o1 CO o r—l i—i o 00• • . • • • •00 n- CNJ 00 o CN i—iCNI in CO in 00 r'~-r-1 i r—l i 1

VO cn i—i o cn cn<r o in n«. VO o r-'-. * • • • • •CO CNJ co <r co inin 00 <r VOi—i 1 i—i I I

/'"N /'N N /'N /—NCO CO CO CO CO CO-^ "— /-VO o o o H o oQ Q Q Q Pm o QPS PS PS PS PS PSO O 1—l i—l W w l—lV_y V- V_y v_ID <0 "O O <o "D O

oCT\ini

ocninI

oid

/"NHPM

coOOvVO

coocnVO

coininvo

cnoisID

in oVO i—i| i • • •<r o VO<r cni—i I l

in <r i—i t. o00 <r cn cn cnm • • • •

<r 00 cn cn o

inoo

incooo

■ncoo'QPSi— l<o

<r<rooi

ino

cO

vo

CNJCOVOr-

incovoVO

/ NCOoQ PS I—I■O

<r<rCNJI

coin

cooQ PS i—l<o

O<rooi

oVOo\

/~NHPm

X!j->CQ)34-)• H4-)COO!3COTO33Oago

N.

L

t/ s O v_x CM ^NoCODO *- O

oN cQ TD • H6i—i

oN cd XJ • He

QJ I—l O N nJ T3 • Hs

!>»A4-1QJsIPS

oPS

TABLE

4.2.1

(con

tinu

ed)

o

o

VOo

ino

o

coo

o

o

TOo434-)<uS

ov 1—1 o ca OV or—l CO in OV O 001 | | 1 1 1 • • • I I 1 • • •VO CM o co O in<r pa OV <r IA. r—lr-i 1 1 i—i 1 i—l1

CO in o CO in CM i—i o CM VO o CM VO oo\ OV OV in co in 00 o CO CM VO 1—1 O 00 vo• • • • • • • • • • • • • • •CO OV in co o VO CM i—i o vo l-A CO o voVO VO r—l VO r—l I—1 vo VO r—l i—1

co in o co in CM 00 O o i—I l-A o CM VO oOV OV OV in co in O 00 t—l co in p-- O 00 VO• • • • • • • • • • • • • • •co ov in co o VO OV aT i—1 o CO in co o VOVO VO I—1 VO VO i—i i—i VO t—l ■—1

o 00 o CO in fA <t <r o o 00 o <r VO oo 00 OV OV OV m 1—1 i co CM VO <r o i—1 ov• • • • • • • • • • • • • • •I—1 CM CM CM o ov CO o CO co O co <r o coCM1 VO I—l 1—1 1 CM1 CM1 |A. 1 CM1

/•"A /-A /-A r ~\ ?~\ /—\ r s <AC/5 CO CO CO CO CO co CO CO CO/'A - /'"A A* /"A a^ A /''A ^ A /—No o’ H o o H o o H o o H o o HQ Q PM p Q Pm Q Q Pm Q Q Pm Q Q Pm3 53 x: 53 3 ft 3 3 3 3 3 3 Kl—l HH w M H W i—l M PM i—I M W M I—l PMa^ A_y v_* v A_ A> v— * A> A_ A_ v-/ v_ v_ At) O ■Q <o tO t) <o <o "D CO tO 'O O tO

✓"AX4-)CCDP4-)•H■PWPCOT3CPoPsoo

oNccJf-4

Pm

P O I—i

OISJn)UPM

CL) i—IoN0)U£■*•>Pm

£4-)Q)SI3

0 NctiUX>Pmi-134-)CDs1CO

0NCj£P-I(—I£4-J(Ue• HT31inco

o53 OCvl CMCM

00CM

TABLE

4.2.1

(con

tinu

ed)

i i i i i i i i i

o<r <r oCM CO CM• • •CO r-MCO i—l

i i i i l l i i i

o<r CO oo r— 1• • •CO O'\ 0000 1 I—1

I I I I I I I I I

inoCO in o<r in o\• • •CM 00 CM00 1 1

oo CM oCO r- CO• • •co ON 0000 1 1

00oCM <r o on On <r ON ON o CM CO o 00 in oO I—1 co CO i—1 i—i CO <r o m co in CO m <r# • • • • • • • • • • *o 00I CM i—l CO ON <r i—i CO o co co o CM00 i—i CO CM I r» i—1

CMc_>On ON o o CM o ON ON o in r". o CO in or-x o ON CM co m CO <r CM CO o in cO r

a a a • • • • • • • • •00 in ini <r o m CM i—i o <r o CM <r o ocO 1 i i 1 r. 1

oON ■—i o CO CM o ON ON O in CO o O CO o<r CO CM CO o CO <r ON CO in CO ON ON r-

a a a a • • • • • • • • •in i—i <r CO I—l i—1 CM CO co o CM o i—i1 CO CM r- I—1 CO 1 rM n~ i—i

T3OX iCUB

coo Q 53 I—I'w't)

s ~ \COoQ13 I—ltO

/'"NHPM

O

cooQ 53 I—I c_y tD

/'"NCOoQ53M<o

f \

CO CO CO CO CO CO-_ / ~ \ -.o o H o o H o oQ Q PM Q Q P4 Q o13 !3 X 53 13 X 53 531—1 I—1 H M M H M I—1c_\D <o <o \D <o O t> <o CO

Xv—'udd)M>•H4->CO-OPCOT3d53Oasoo

cui—ioNdT3d

d)d•HX 3*HMXPM

B53•Hd• HTO•HMXPM

CU T3 • HX0153CUd•H•HdXPM

doM

'd♦H0153CUd•r-JT3 • HuXPM

O <rCMinCM CDCM CM

00CM

Chem

ical

sh

ifts

, 6,

are

expr

esse

d with

resp

ect

to C

HA,

shifts

to high

freq

uenc

y are

po

siti

ve.

6bo -oO O

lh•H

o oI—I-co

- to

_o(uidd) S3JTL[S iBOTUiaqo pBînoiBO O o

QD

L o

CL).fl4-JCO4-1CO£•HctiCtOcdCO 4J <4-1 • H.flCO

o3o• Hea).dodo,Q3oI—Itt5isCUe

• H

CUP<XQ)

mooi—iPM

CN

a)Hpno•HPm

o.d•Pcu6coo«53o

cu.dP

PT5CU4-)njrMPoI—I03OCOcu 2 i-1aJ>

r09

O o - oo<5>

i—I

rH

CM

<r

•H

o o(rndd) sajxqs x130™ 91!0 psînoiBO

“ CM

oo

valu

es

calc

ulat

ed

by the

INDO/S

meth

od•

with a correlation coefficient of 0.92, standard deviation 18.56 ppm

and slope 1.24. The better agreement of the correlation coefficient

and standard deviation but slightly larger value of the slope may be

partly due to the types of the molecule considered, compounds number

19 to 28. In general, we find that the calculated trend in chemical

shifts are in agreement with the experimental data when compared with

the structure of the molecules considered. However, the calculated

shielding data appear to be dependent on the molecular conformation

and the accurate conformations are difficult to o b t a i n e ^ ^ ’^ ^ .

By comparing the carbon shieldings for compounds number 1 to 8

and 16 to 28, which contain only first-row nuclei, the calculated shielding

differences for the carbon atoms bonded to nitrogen and/or oxygen atoms

or Tr-bonded to another carbon atom are in good agreement in magnitude

with experimental data for both CNDO/S and INDO/S calculations. One

possible explanation for this is that the CNDO/S and INDO/S parameteri

zation schemes are parameterized to reproduce the transition energies

of Tr-electron s y s t e m s ^ ^ ’^ ^ and that it does not give as good a

reproduction of the correlations between the calculated and experimental

results in Ci-bonded system. This explanation could be used for the

carbon shieldings of compounds number 1 to 8 and 16 to 28. For compounds

number 9 to 15, which contain a second-row element, phosphorus, the spd

basis set is used in the calculations, the parameters employed are taken

from the original CND0/2 s e t ^ ^ \ The carbon shieldings of compounds

number 9 to 15 are lower in magnitude than those for compounds number

1 to 8 and 16 to 28. As a consequence of the presence of a nitrogen

lone pair and empty 3d valence ground state orbitals of the phosphorus

atom, which probably cause the electrons in the valence shell of nitrogen

to move into the empty 3d orbitals of phosphorus, i.e. back donation,

so that the chemical shift of carbon atoms in those compounds are lower

in magnitude than the experimental data. A general consideration of the

results given in Table 4.2.1 and Figures 4.2.1 and 4.2.2 for the CNDO/S

and INDO/S calculations, respectively, reveals that our calculations, by

means of the CNDO/S and INDO/S parameterization schemes, have reproduced

a correlation of calculated and experimental carbon chemical shifts in

the compounds considered.

4.3 Nitrogen shieldings.

In the present study, nitrogen shieldings have been calculated

for some molecules and ions using INDO/S parameters. The results of

nitrogen chemical shifts, with respect to nitromethane, are compared

with the experimental d a t a ^ ,‘ '^,‘ ^ and other theoretical works^"*"^

where available.

An investigation of hydrogen-bonding effects is also performed

in the present study. For this reason, the data of monomers, dimers

and polymers are collected in this section.

Table 4.3.1 shows the results of some calculations of the nitrogen

shielding of some imidazole, pyrazole, Indazole and pyrimidine compounds

and their derivatives. These compounds and their derivatives show two

types of nitrogen shielding those for pyridine-type and pyrrole-type

nitrogens. The pyridine-type shielding is smaller than that for the

pyrrole-type nitrogen. The difference in the shielding of the two types

of nitrogen atom is about 100 ppm and depends on the molecules and.. (3,116,117)medium

In order to consider specific solute-solvent interactions, the

calculations are performed on the hydrogen-bonded dimers and polymers

at a minimum energy INDO geometry obtained by means of the GEOMIN, (119)procedure

TABLE 4.3.1 The results of INDO/S calculations of nitrogen shieldings, O',

and chemical shifts, 6 , compared with experimental d a t a ^ ,‘*’ ,^ ‘

No. Compound Atom N(l) N(2) N(l)’ N(2)

1 Imidazole tf(cal) 61.09 0.566(cal) -172.42 -111.896(cal)(av) -142.166(expt) -166.40

0.56 61.09-111.89 -172.42

-142.16 -166.40

2 Imidazole+HÔ

X \ . —:»v \] /

3 Imidazolium Ion

Ni+Nf'

4 N-Methyl Imidazole

CH. N1 n 2\l_/

5 N-Methyl Imidazoleh 2o

CH- -s .--a',yV N2:"

.o^

(j(cal) 6(cal) 6(cal)(av) 6(expt)

(j(cal)6(cal)6(expt)

Cj(cal) 6(cal) 6(expt)

(j(cal) 6(cal) 6(expt)

66.07-177.40

3.45-114.78

-146.09-171.00

47.58-158.91-200.30

- Ni\ /

N-Methyl Imidazolium dCcal)Ion 6(cal)

6(cal)(av)

63.50-174.83-215.10

61.63-172.96-211.50

55.12-166.45

6.40-117.73-119.30

17.12-128.45-128.50

63.43-174.76

CHo. ✓ X\_J

6(expt)

7 Dimethyl Imidazolium (j(cal)6(cal) 6(expt)c h.

IonA < ^ c h 3

n .+ n 2\ L j

-170.61-201.60

54.07-165.40-204.50

8 4-Methyl Imidazole

N 2 N f '

CH

d ( cal) 67.77 6.246(cal) -179.10 -117.576(cal)(av) -148.346(expt -167.00

3.45 66.07-114.78 -177.40

-146.09 -171.00

47.58 -158.91 -200.30

6.40-117.73-119.30

17.12-128.45-128.50

63.43-174.76

63.50-174.83-215.10

61.63-172.96-211.50

55.12-166.45

-170.61-201.60

54.07-165.40-204.50

4.55 66.54■115.88 -177.87

-146.88 -161.50

9 4-Methyl Imidazole+ h 2o

H- / XN, n 2- "

CH

10 Pyrazole

n , - n 2

cj(cal) 69.53 9.196(cal) -180.86 -120.526(cal)(av) -150.696(expt) -172.80

(j(cal) 56.50 0.406(cal) -167.89 -111.736(cal)(av) -139.816(expt) -128.50

6.64 70.75-117.97 -182.08

-150.03 -164.10

0.40 56.50-111.73 -167.89

-139.81 -128.50

TABLE 4.3.1 (Continued)

No. Compound Atom N(l) N(2) N(l)’ N(2)’

11 Pyrazole+Ô

Ni-n 2

OH

12 Pyrazolium Ion

N|—N2/ \h

13 N-Methyl Pyrazole

oN!-N2c h /

14 N-Methyl Pyrazole+ h 2o

c " 2 r r+l-OH15 N-Methyl Pyrazolium

^ i ° nNi—In2

c h 3/

16 3-Methyl Pyrazole

N2-N1

d’(cal) 6(cal) 6(cal)(av) 6(expt)

cT(cal)6(cal)6(expt)



tf(cal)6(cal)6(expt)

tf(cal)6(cal)6(cal)(av)6(expt)

63.28-174.61

15.33-126.66

-150.64-132.80

58.64-169.97-178.80

54.98-166.31-174.60

59.72-171.05-176.00

66.57•177.90-171.10

6.49•117.82

2.82-114.15-70.30

18.81-130.14

- 88.20

57.60-168.93-183.80

65.09-176.42

-147.12-133.80

15.33 63.28-126.66 -174.61

-150.64 -132.80

58.64-169.97-178.80

2.82-114.15-70.30

18.81-130.14

- 88.20

57.60-168.93-183.80

61.72-173.05

54.98-166.31-174.60

59.72-171.05-176.00

66.57-177.90-171.10

9.27-120.60

-146.83-128.10

17 3-Methyl Pyrazole

CH3" 0 + H2°N.1Hi

OH18 3,5-Dimethyl

Pyrazolec h . CH,

N!— n2

19 3,5-Dimethyl Pyrazole + H_0 ch, ^ ^ \ ^ Ch3

hN1-N2

20 Indazole OH

N2

tf(cal) 20.42 70.086(cal) -131.75 -181.456(cal)(av) -156.886(expt) -142.10

cj(cal) 66.69 13.206(cal) -178.02 -124.536(cal)(av) -151.286(expt) -133.60

(j(cal) 69.74 23.986(cal) -181.07 -125.316(cal)(av) -158.196(expt) -144.00

c»(cal)6(cal)6(expt)

80.80-192.13-194.00

65.88 21.74-177.21 -133.07

-155.14 -139.50

13.20 66.69-124.53 -178.02

-151.28 -133.60

23.98 69.74-135.31 -181.07

-158.19 -144.00

28.25-139.58-58.90

TABLE 4.3.1 (Continued)

No. Compound Atom N(l) N(2) N(l)' N(2)’

21 Indazole 4-H20

Nf-VH—o h

23

22 Pyrimidine (1,3-Diazine)

'NPyrimidine

OH

HIOH

Cj(cal) 6(cal) 6(expt)

cT(cal)6(cal)(av)6(expt)

O’(cal)6(cal)6(expt)

83.71-195.04- 201.00

5.84-117.17-85.40

15.66-126.98-92.00

36.35-147.68-84.60

5.84-117.17-85.40

15.66-126.98-92.00

Chemical shifts,6 , are expressed with respect to c h 3n o 2 , shifts to high

frequency are positive.

200

«oO

OSoaONQ

SO

oo o

aa

CO4-3CH•H&COI—InJo• HscuJ GO

• dcu4-3 nJ i— i 2 ai—irta

cu4-3nji—IP V i—IaJococupI—ICCS>cu.a4-3

CO4->COC3•HctfClOctico4-3CH•H.dCO

, do

P3<UbQOd4-3 • Hg

aJ4-3f3cuS• HCUa><:cucho4-3oI—IPu

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(UupbD•HPu

• doJ G4-3Q )

eCOoQ55CU

. d

r*>£2

(uidd) s q j i q s x ^ p u a q o x'eauauii.iadxa

The results of shielding calculations are presented for some

nitrogen nuclei in Table 4.3.1. An analysis of the various terms

contributing to the shielding reveals that the change in the nitrogen

shielding arises almost entirely from the variation in the local

paramagnetic term, cj (1o c .), in all cases considered as in the carbon

shielding calculations. This is similar to the studying in different

molecules using CNDO/S^°^ and I N D O / S m e t h o d s of calculations.

Consequently, the contributions from the individual components of the

total shieldings are not listed separately in Table 4.3.1. The results

of chemical shifts in Table 4.3.1 are plotted in Figure 4.3.1. From a

closer consideration of Table 4.3.1 and Figure 4.3.1 we find that the

calculated trend in the chemical shifts are in agreement with the

experimental data, with a correlation coefficient of 0.80, standard

deviation 26.10 ppm and slope 1.72.

Pullman et a l ^ ^ ^ have reported the values of the magnetic

shieldings of the different nuclei of imidazole (compound number 1)

calculated with a minimal basis set using Gaussian function set and

ab initio self-consistent perturbation method for the isolated (compound

number 1) and hydrated (compound number 2) molecules. Their r e s u l t s ^ ”

show that hydrogen-bonds produce large variations of the shielding of

the nitrogen nuclei which are directly involved in intermolecular bonding.( 118}Hydrogen-bonding of HÔ in their calculations causes the nitrogen

shieldings to increase by about 15 ppm. This increase is too high when

compared to the experimental d a t a ^ ^ ^ which is about 4.5 ppm. The results

from the calculations in this laboratory by using the INDO/S parameterization

scheme and the GEOMIN procedure show that hydrogen bonding causes the

nitrogen shielding to increase in value about 4 ppm, which is in better

agreement with the experimental data (4.5 ppm)^116 than Pullman et al's

results (15 p p m ) ^ ^ \

a ^ (3,40,95,104,120) w ^ Âs discussed elsewhere , the' theory of the

shielding gives a good explanation when different types of nitrogen

functional groups are considered. However there has so far been no

satisfactory explanation of the shielding changes for the nitrogen atoms

in series of amine; H 2NCH^, H N ^ H ^ ^ and ^CHg)^. Theoretical

treatment give shielding trends for such series of molecules

which are in the opposite direction to the chemical shifts found

experimentally. The nitrogen shielding calculations^^* at the INDO,

CNDO/S and INDO/S levels are not in agreement with the amine experimental

data.

Little attention has been given to the influence of hydrogen

bonding on the shieldings of the atoms which donate electrons to the

hydrogen bond. The nitrogen resonances in some series of amines have(1-3) . . .been reported for pure liquids and solutions, it is found that the

signals occur at significantly lower applied field than the respective

resonance of the gaseous molecules. We anticipate that our study might

enhance our understanding of the interactions which influence the shielding

of the atom which commonly serve as electron-donors such as nitrogen atom

in hydrogen bonding.

In order to consider solute-solvent interactions, the calculations

are also performed on hydrogen-bonded dimers of the amine series at a(119)minimum energy INDO geometry obtained by means of the GEOMIN procedure

The results of calculations of shieldings as a function of dielectric

constant of the medium obtained by using the "solvaton" model are presented

for some amine nitrogen nuclei in Table 4.3.2. An analysis of the various

terms contributing to the shieldings reveals that shielding changes arise

from the variation of the local paramagnetic contribution in all cases

considered. A close consideration of the data given in Table 4.3.2 reveals

CO CO CO4 CO 00co1 COI co1

4-J P1

O H •4 •v *4PM CO CO coG X cr\ Ch

o w • • •• i-1 i—i CO 1—14-) CO f"» 00g CO co1 CO1V—4P1 l .

14-1 i—1 <r CO I—1 r—1 <r ON45 o o m ON o <r# • • • • • • •co o on i—1 CM CO ON i—i 0045 CO <r VO <r in <r VO <r

i—1 CM i—i CMi ■—i CMI i—i•o' 1

co t—1 <r co VO ■—i <r <rp o U0 <r in 00 CM4-1 • • • • • • •• H o on o CM CO ON o 00P <r <r vO <r in <r VO <rCO i—i CMi •—i CMi i—i CM| i—iI—1

145o•d 00 (—1 co vo 00 i—1a o CO r- co vo CO r ocu • • • • • • • •p o ON o CM CO ON o 00o CO <r VO <r in <r VO <r

rH CM i—i CM i—i CM i—iT5 1 i 1Grt•v CM UO co VO CM m CO

to o i—I <r i—i <r i—l <r• • • • • • • •o ON o CM CO ON oCO • p CM VO <r in <r VO <r40 s ~ \ G r-1 CM i—i CM i—i CM i—iG oo 45 1 i 1•H I P•o r—1 COi—I GCU 45 O CM m Ch CM CM in i—i•H P O O CO VO <r CO co VOP 4) • • • • • • • •CO 43 O O CO ON i—i CM 00 ON vo

•H rH in <r in <r in <rG r—1 p H CM i—i CM i—i CM r—icu 45 P i 1 i40 P oO G CUp cu i—IP S CU VO on co VO VO ON in• H •H • H o CM m CO l—l CM in inG P fP • • • • • • •

cu vO r'- CO o CM 00 mcw cu <r in <r in <r inO X t—i CM •—i CM r—i CM i—i

cu i 1 iCOG PO p♦H •H vO Ch CO vo VO Onp IS O ON CM <r Ch Ch CM45 • • • • ■ • • • •1—1 43 CM i—1 CO 00 I—I CO 00p CU <r in co <f <r m coo p r-1 CM i—i CM i—i CM i—iI—1 oJ i 1 145 cuO aCO oV O co CO 1—1 o co r

O <r in Ch <r VOo p • • • • • • •Q G i—1 <r m CM co <r in VO53 45 CO <5- CO <r co <r CMI—1 P i-l CM r—1 CM i—i CM i—l

CO 1 1 14-1 GO OOCOp o to •O to •O to •o to1—1 • Hp pCO p CMcu o EGp cu 531—1 CO CO CO o<D . cu 43 EC EG EG CMP • H G 53 O 53 EGH 43 PO + + + +CUCM g CO CM CO co• o EG EG EG EGCO u 13 53 53 53.• CO<r ECocui—iP •45 o i— I CM CO <rH 53

00 oo o CM vor'- vO inCO1 CO1 COi coI COi•so o<r <r• ■00 00

co1 COICM ON CM ON CM ON CM00 <r 00 m 00 o <f CO• • • • • • • • •ON oo ON in VO CO CO <rin in <r in <r in CM COCM1 1—1 CMi i—i CM1 i—i CMi i—1 CM1

<E <r <T r- om CM in co VO ON CM CM VO• • • • • • • • •ON 00 ON in vo CM <r CO <rin <r in <r m <r m CM coCMi i—i CMI i—i CMi i—i CMi i—1 CM1

o o o CO <r o co<r o <r CM in 00 i—i CM in• • • • • • • • •ON 00 ON m VO CM <r COin <r in <r in <r in CM COCM1 i—i CMi i—i CMi i—i CMI i—1 CM1

vo CO VO co vo <r <ro O ON CM VO ON ON CM• • • • • • • • •ON ON <r VO CM co CM <rin <T in <r in <r in CM COCMi i—1 CMi r—1 CMi i—i CMI i—l CM1

<r i—1 <r i—l <r CO VO oo o i—l <r o CO in ON• ■ • • • • • • • •00 VO 00 <r in CM CO CM coin <r in <r in <r in CM coCMI r—1 CM1 i—i CMi i—i CMI i—1 CM1

00 m 00 CM in CM m <r r-'00 in 00 O CO CM in ON CM• • a • • • a • •VO in VO CO <r i—1 CM I—l COin <r in <r in <r in CM coCMi i—i CMI i—i CMi i—i CMi i—1 CM1

o o <r o 00 I—100 <r 00 in 00 o <r l~ I—l• • • • a • a * •ON 00 ON 00 00 00 o<r CO <r CO <r CO <r I—l COCM1 1—1 CM1 i—l CM1 i—i CM1 I—l CM1

O o i—l <r <r I—l <ro VO o CO VO 00 i—i ON CM• • • • • • • • •00 VO 00 ON o o CM CO inco CM CO CM <r CO <f I—l CMCM1 i—1 CM1 i—1 CM1 i—i CM1 I—l CM1

<o to <o to o to •O to •o

o o o oCM CM CM CMEC EG EC EC+ + + +CO CM EP 53EC EC ^13 13 CM /-nCO C"h COeg co egO EC CJCJ> v pV—/

in vo oo

that our calculations, by means of the INDO/S parameterization scheme,

in conjunction with a hydrogen-bonded dimer and the "solvaton" model

have reproduced some correlations of the nitrogen shielding trends of

the amine series considered. In agreement with the experimental data,

the nitrogen shieldings in the neat liquids (compound number 3) are

found to have larger value than in HÔ (compound number 4). The nitrogen

shielding for the pure liquid of H^NCH^ (compound number 2) is smaller

than for the pure liquid of NH^ (compound number 1), indicating that the

introduction of a methyl group on the nitrogen atom induces a decreasing

nitrogen shielding in the amine series which probably modifi$sthe hydrogen-

bonding of the lone pair of nitrogen. The results in Table 4.3.2 imply

that hydrogen bonding plays an important role in the shieldings, especially

where the atom contains electron donor pairs for hydrogen bonding such as

in the case of nitrogen atom. However, the generally observed trend in

the nitrogen shielding of amines in HÔ (compounds number 5 to 8) is not

well reproduced. This is probably due to the medium considered, apart

from hydrogen bonding effect. By using the "solvaton” model, we have

obtained a general agreement between our calculations and the experimental

data when the dielectric constant of the medium is greater than 1.0

(isolated molecule). From the resux^s, we can see that the nitrogen

shielding of the amine series in HÔ increases as a consequence of

dielectric constant of the medium increases.

4.4 Fluorine shieldings.

The results of calculated fluorine shieldings and chemical shifts,

ect to Si](121-123)

with respect to SiF^, are compared with the experimental data where

available

The results of fluorine shieldings and chemical shifts are presented

in Tables 4.4.1 and 4.4.2 as obtained from CNDO/S and INDO/S calculations,

respectively. Tables 4.4.1 and 4.4.2 reveal that the maximum variation

of the local diamagnetic contribution to fluorine shieldings, cr^(loc),

is 2 ppm, which is less than 1% of the experimental fluorine shielding

differences for the molecules considered. The overall average value of

the local diamagnetic contribution of 473.7 and 473.3 ppm are obtained

by the CNDO/S and INDO/S calculations, respectively. These values are

in agreement with the results of calculations for fluorine nuclei in

different molecules obtained by similar m e t h o d s ^ ^ * ^ ^ . The agreement

between these values of the magnitude of the local diamagnetic contribution

to total shielding indicates that they are insensitive to the choice of

the wavefunctions and molecules considered. The correlations of the

fluorine chemical shifts by the CNDO/S and INDO/S results and the

experimental data are shown in Figures 4.4.1 and 4.4.2, respectively.

From Figures 4.4.1 and 4.4.2, there are two distinct sets of data,

set A (molecules number 1 to 6, 8 and 11) and set B (molecules number 7,

9, 10 and 12 to 16), with different correlation lines. The set A in

which the calculated chemical shift trend coincides with that of the

experimental data. These molecules have Cl and CH^ substitute on SiF^.

For the set B, the calculated chemical shift trend is opposite to that

of the experimental data contain H and as the substituents.

From Tables 4.4.1 and 4.4.2, we can see that, by computational

AP•H

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data, the effect of introducing a second or third substituent on silicon

reduces the shielding values uniformly as the first substitution. In

the chlorofluorosilanes, shieldings are almost uniformly smaller in

magnitude in the sequence SiF^, SiF^Cl, SiF^Cl^ and SiFCl^ and also the

calculated chemical shifts increase, which is the same trend as the (121-123)experimental data . For hydride and alkyl substituents, even-

though the magnitudes of the calculated shieldings for the second and

third substituents are less than for the first, as in chlorofluorosilane,

the trends of calculated chemical shifts are opposite to the experimental(121-123) (123)data . It has been suggested that the non-uniform and

reverse trends in the experimental chemical shifts in alkyl groups are

due to the electrons in valence shell orbitals from the alkyl group

carbon atom moving into the empty 3d orbitals of the silicon atom, which

is a P -d effect.TT TT

A.5 Conclusions.

The CNDO/S and INDO/S parameterization schemes used in the

present calculations for some first-row nuclei are those of Del Bene

and Jaffe', and Krogh-Jesperson and Ratner, respectively, in conjunction

with the Nishimoto-Mataga approximation which is used to calculate the

two-centre Coulomb repulsion integrals. From these parameterization

schemes, the valence states of the various atoms are chosen such that

they are one electron TT-donors in molecules which involve rr-bonding.

These are not expected to give a good reproduction of d -bonded systems.

Since the theoretical estimates are usually based upon an

"isolated" molecule as a model, it seems unreasonable to expect from

any theoretical treatment of magnetic shielding exact reproduction of

experimental data which are usually reported for liquid samples and are

susceptible to medium effects. In the present work, hydrogen-bonding

and the "solvaton" models have been employed to study medium effects.

It is demonstrated that the local paramagnetic contribution to the

shielding varies with the variation of dielectric constant of the medium.

For the amine series, we could reproduce the experimental trend by using

hydrogen-bonding model but neither the "isolated" molecule nor the

"solvaton" models.

It should be of interest to study the effect of the variation of

bond lengths and angles of molecules considered, which would be a further

step towards the accurate determination of shielding and chemical shift

trends in order to have the more accurate structure of molecules considered.

CHAPTER FIVE

SOME CALCULATIONS OF SHIELDING FOR SECOND-ROW NUCLEI


The general theory of shielding is found in Chapter 3. This

has been used within the framework of semi-empirical parameterization

schemes for some first-row elements in Chapter 4 and for some second-

row in this chapter. The usual way of treating shielding calculations

for second-row nuclei in this chapter is the same as for first-row nuclei.

(85)Jameson and Gutowsky have calculated the required matrix

elements for describing phosphorus shielding, and this has been used

by Gutowsky and L a r m a n ^ 23 as well as Letcher, Van Wazer and L a r m a n ^ 2^ 128)

for the numerical treatment of phosphorus chemical shift data. They were(129 130)followed by Radeglia et al ’ for calculations of silicon shielding

data. According to the treatment employed by these authors, the varying

shielding values of the second-row nuclei from one compound to another are

attributable to changes in the occupation of the 3p and/or 3d orbitals and

to variations in the average excitation energy. We found that there are

various transitions which contribute to the paramagnetic contribution of

total shielding. So it is not reasonable to use an average excitation

energy in interpreting the phenomenon of shielding calculations.

In the present study, second-row nuclear shieldings have been

calculated for a variety of molecules containing silicon and phosphorus

nuclei by means of Pople* s GIA0-M0-S0S model in conjunction with the

CNDO/S and INDO/S parameterization schemes which we employed. In this

case we used the parameters for the second-row nuclei from the original

CNDO/2 parameterization scheme^"^ as shown in Appendix B. The results

are compared with the experimental data and other theoretical work where

available(125"136).

5.2 Silicon shieldings.

5.2.1 Introduction.

Silicon has only one naturally occuring isotope with non-zero1spin, namely silicon-29, which has spin I = and a natural abundance

of A.70%, low NMR sensitivity (7.8x10 3 with respect to ^h )(153,154)^

The marked differences between the chemistry of carbon and silicon

have been pointed out in several investigations^3^ -^®) Particular

examples include the instability of Si-H bonds and the absence of Si-Si

double bonds. Since the 3d orbitals for second-row nuclei are unoccupied

in the ground state, inclusion of 3d orbitals in a description of silicon

molecular species of the most common examples of the use so-called higher

functions, or polarization functions, in the description of chemical, .. (12,147-150) , . . _ _ . . _ (149)bonding are considered m the present calculations. Coulson

has stressed the fact that the 3d functions should be very diffuse for the

silicon atom even in states in which they are occupied. He then argued

that the relative importance of the 3d functions should increase as

electron withdrawing (electronegative) species, such as oxygen or halides

are bonded to a given silicon atom. This is because the substituents tend

to remove electron density from the 3s and 3p orbitals of the silicon atom,

rending the silicon atom partially positive and causing the 3d functions(13)to contract. This sort of argument was first proposed by Craig et al

The most frequently discussed role for silicon 3d functions in

valence theory lies in bonding to more highly electronegative species

(F, Cl, 0 and S) with occupied p^ o r b i t a l s ^ ^ \ The argument in its

simplest form, is that the Si-X transfer in the <j system results in a

partially positive Si, which in turn, results in tighter 3d orbitals,

which therefore overlap more effectively with the p^ orbitals on X,

resulting in a rather strong interaction. Within an LCAO-MO

description, this picture is a sensible one to study the phenomenon4T ^ (150)of p -d interaction

TT TT

In one of the earliest discussions of silicon shieldings,(132 151)Lauterbur et al * graphically demonstrated the " sagging pattern"

exhibited by compounds of the type (CH^)nSiX^_n as n is varied from

0 to A and X is an electronegative group. The comparison of carbon

and silicon in the series (CH-) M(0R). , where M is either carbon or3 n 4-n(132-151)silicon atom and R is methyl group, studied by Lauterbur also

shows significant differences in substituent effects for the two nuclei,

carbon and silicon. Since Hunter and R e e v e s ^ " ^ published their

collection of chemical shift data, several attempts have been reported (129 130)by Radeglia et al ' , using semi-empirical MO calculations in

conjunction with Pople's GIA0-M0 model to explain the shieldings of

silicon nuclei.

5.2.2 Results and discussions.

The results of some silicon shieldings obtained by CNDO/S and

INDO/S calculations are reported in Tables 5.2.1 and 5.2.2, respectively.

The diamagnetic contribution to the non-local shielding data found to be

negligible and the contribution to the local diamagnetic shielding is

865 ppm on average for the silicon atoms in the molecules considered

in these calculations. This is similar to the diamagnetic contribution

calculated for different molecules, using the CND0/2 method of calculation

The diamagnetic and paramagnetic contributions for local and non-local -3terms and <r > for 3p and 3d orbitals are quoted separately in Tables

5.2.1 and 5.2.2 for the CNDO/S and INDO/S calculations, respectively.

For the paramagnetic term, the contributions from the 3p and 3d atomic

TABLE

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orbitals are also shown separately in Tables 5.2.1 and 5.2.2.

In the present work, the values of cj (1o c .) and total shielding

interpretation of the shielding calculations were fixed before hand.

The calculated chemical shifts are plotted against the experimental data

in Figures 5.2.1 and 5.2.2 for Tables 5.2.1 and 5.2.2, respectively.

From these figures, in general, the agreement of the magnitude and trend

of the calculated chemical shifts and the experimental results for the

series of substituted molecules is poor. From these figures, we can see

that there are two distinct sets with different correlation lines. The

calculated trend of the first set, for molecules number 1 to 27 is in

the same as that of the experimental data. But for the other molecules,

numbers 28 to 40, the trend is in the opposite direction. It is the

former set which is experimentally found to exhibit the "hanging chain"

or "sagging pattern" in the series of X nSiY^_n molecules, where n varies

from 0 to 4.

From Tables 5.2.1 and 5.2.2, the results obtained using the

CNDO/S and INDO/S parameters in the present calculations show that we

can reproduce the "hanging chain" or "sagging pattern". This is shown

in Figures 5.2.3 and 5.2.4 for the CNDO/S and INDO/S calculations,

respectively, where X is H and Y is F and Cl.

A closer consideration of the contribution from transition energies

of various symmetries to the paramagnetic contribution is demonstrated

in Table 5.2.3 for the species of molecules which show the "hanging chain"

trends from the CNDO/S calculations. Charge distributions for each orbital

and the average weighted value of the energies of all of the transitions

obtained for each molecule by weighting the transition energy in proportion

to the size of the corresponding contribution to the local paramagnetic

term, Cj (1o c .), are also reported in Table 5.2.3. Plots of the net charge

265

T3Cl) r - l4_> cd co fd O 1-1 i— I *H 4—1P g -H O Q) H P m cd o o

Q)nptuO•H

•aCl) r-l u cd co ( S o di—I -i—l M—iP £ <HCJ CD rC H £ CO Cti CJ CJ

CM

CDJ-l3CuO•H

on silicon atom and the average weighted value of transition energies

against the number of n in the series of molecules X SiY, ,where Xn 4-nis H and Y are F and Cl,are shown in Figures 5.2.5 and 5.2.6, respectively.

Figure 5.2.7 is the plot of the average weighted value of transition

energies against net charge on the silicon atom in the series of molecules

X SiY, . n 4-n

From Tables 5.2.1 and 5.2.2, we can see that the use of equation_3

(3.38) to calculate <r > for 3p and 3d orbitals from Slater atomic

exponents gives values an order of magnitude smaller than those obtained

by Whiffen et a]_(155,156)^ us^ng Hartree-Fock SCF atomic wavefunctions,

D e s c l a u x ^ ' ^ , using Relativistic Dirac-Fock equations, and experimentally (1 58 by Smith and Barnes from spin-orbit splitting data. The small values

-3of <r > for 3p and 3d orbitals result from the Slater functions which(85)has been noted elsewhere . If we include the factor due to the small

-3 .values of <r > for 3p and 3d orbitals m our calculations we can see

that the magnitude of the variation of the shieldings can be improved

but the overall agreement of the calculated chemical shifts with experi

mental data is still not good. This is probably, at least in part, due

to the estimated transition energies in the present calculations.

From Tables 5.2.1 and 5.2.2, the overall agreement of the shielding

differences obtained by the calculations and those observed experimentally,

molecules number 1 to 40 is not good. On closer consideration, we find

that we could have a good correlation between calculated and experimental

chemical shift values for each series of molecules in the first set, e.g.

molecules number 1 to 3, 5 to 9 and 24 to 27 etc. The reproduction of the

"hanging chain" pattern of chemical shifts of the molecules considered in

our calculations (molecules number 5 to 9) and the trend between the

calculated chemical shifts and experimental data represents a satisfactory

Table 5.2.3 The results of some CNDO/S calculations of charge densities, transition energies and contributions to the paramagnetic components of silicon atoms.

No. molecule s p d atom q q q totq transition transition

energy (^(loc.)

1 SiH4 Si 1.1158 2.4486 0.0541 3.6187 3 — 6 8.492 -18.669H 1.0953 1.0953 3 —»8 8.563 -56.667 X

H X average weighted value of 4 —* 7 8.462 -62.785\Si-H

h'

r N transition energies = 10.3033 e.v. net charge on Si = +0.3822

22

-*-7 —*8

8.5938.280

-36.510-38.973 y

3 —►6 8.492 -53.1733 —*• 7 9.115 -11.472

2 — 7 8.593 -27.2202 -*8 8.280 -44.090 z

4 —*6 8.381 -72.990

2 S iH F Si 1.2223 2.1534 0.0841 3.3596 5 -+10 11.436 -22.228F 1.9720 5.2492 7.2194 6 -» 8 7.025 -39.192 XH 1.1403 1.1403 7 — 8 7.025 -84.972

H\Si-F

H <«'H’

X average weighted value of 7 -*11 9.026 -18.970

Uy

transition energies = 16.454 net charge on Si = +0.6404

e.v. 56 6 7

-» 9 —*■ 8 — 11 — 8

11.4367.0259.026 7.025

-22.228-84.925-18.971-39.190

y

6 -10 8.350 -84.980 z7 — 9 8.350 -84.980

3 SiH2F2 Si 1.0857 1.8579 0.1369 3.0806 8 -13 12.402 -13.820F 1.9662 5.3918 7.2980 9 -11 6.967 -39.341 X

H 1.1617 1.1617 10 -11 6.828 -117.660

H\Si—H

Xuyaverage weighted value of 9 -13 10.343 -38.608transition energies = 19.165 e.v. 10 — 12 7.508 -136.217 y

F<-F' net charge on Si = +0.9194 10 — 14 8.992 -14.803

8 — 12 11.151 -18.8609 -11 6.967 -72.290 z

10 — 11 6.828 -62.500

'4 SiHF3 Si 1.0461 1.6042 0.1759 2.8263 12 — 16 12.378 -18.204F 1.9664 5.3692 7.3356 13 — 14 7.276 -10.912 X

H 1.1669 1.1669 13 -15 7.202 -131.236F\ X

L zy

average weighted value of 11 -+16 12.012 -18.758^ Si— H

F < 'F'


e.v. 1313

->14->15

7.2767.202

-129.885-11.026

y

6 -14 11.593 11.5507 — 15 10.080 12.450

11 — 14 5.058 -43.940 z11 — 15 8.042 -24.81012 — 14 10.670 -18.29012 — 15 6.727 -32.030

Table 5.2.3 (Continued)

No. molecule s p d atom q q q totq

y.'transition transition

energy CiP(loc.)

5 SiF6 Si 0.9960 1.6037 0.2092 2.6088 16 ->18 10.666 -30.263F 1.9660 5.3825 7.3685 15 ■+17 10.766 -22.277 X

F X average weighted value of 16 -*17 12.677 -13.329

\ i - FF^'

f'Y~^Z transition energies = 20.309

net charge on Si = +1.3912e.v. 16

1516

-*19-*17-*17

10.76010.76612.677

-30.036-15.697-18.915

y

6 -*18 10.192 11.3207 -*19 10.389 11.030

15 -*19 6.610 -59.350 z

16 -18 7.331 -53.010

6 SiH3Cl Si 1.2661 2.1905 0.0552 3.5096 3 -10 9.656 -26.676Cl 1.9570 5.1869 0.0267 7.1686 5 — 8 8.960 -25.269H 1.1073 1.1073 6 -11 6.527 -29.666 X

H\ X average weighted value of 7 — 11 6.527 -73.396\Si— ci

Hr jH'

Uy


e.v. 3667

- 9— 8 -11 -11

9.6578.9976.5276.527

-26.677-25.269-73.393-29.666

y

6 -10 9.383 -30.6895 - 9 9.383 -30.6906 -10 7.363 -61.826 z7 — 9 7.363 -61.826

7 SiH2Cl2 Si 1.2880 2.0369 0.0601 3.3850 5 -11 8.926 -15.505Cl 1.9566 5.2261 0.0236 7.2037 5 -12 8.875 -13.655H 1.1039 1.1039 6 -13 8.551 -16.799 X

H \ X average weighted value of 8 -13 7.562 -11.817\Si— H U z transition energies = 16.667 e .v. 10 -11 6.115 -23.337

Cl r j ynet charge on Si = +0.6150 10 -12 5.989 -72.623

Cl6689

-12-11-11-12

9.5697.9277.1657.003

-18.735-21.012-32.639-36.099

y

6 -13 10.196 -22.1295 -11 8.926 -27.729

10 -11 6.115 -62.503 z10 -12 6.985 -66.63010 -16 8.232 -12.720

3 SiHCl3 Si 1.1268 2.1753 0.0616 3.3636 6 -11 8.787 -10.617Cl 1.9519 5.2068 0.0271 7.1837 7 -16 7.703 -22.856 X

H 1.0852 1.0852 11 -16 6.660 -81.377

Cl\ X

uy

average weighted value of 5 -16 8.798 -11.070\?i-H

CI^YCl


e.v. 711

-15-15

7.7676.757

-21.305-78.670

y

12 -15 6.673 -56.273 213 — 16 6.569 -55.670


No. molecule atom q S q P q ^ totq transition transition

energy c^doc.)

9 SiCl. Si 1.1960 1.9669 0.0686 6 3.2296 16 ->17 5.856 -62.322Cl 1.9536 5.2162 0.0269 7.1927 15 ->19 5.891 -50.721

x average weighted value ofSi— ci I— >z transition energies = 16.030

Cl ^C| net charge on Si = +0.7706

16 ->18 5.969 -66.389e.v. 16 — 19 6.160 -6 7.560

y

15 -.18 6.227 -61.96316 - 1 7 6.035 -66.029

Transitions contributing less than 10 ppm to the paramagnetic term are not included.

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calculation in nuclear shielding by using Pople's GIAO-MO-SOS method,

in conjunction with the CNDO/S and INDO/S wavefunctions without any

extra adjustable parameters as used by W o l f ^ ^ ^ and Wolf and R a d e g l i a ^ ^ ^

In general, the qualitative agreement of the magnitude and trend of the

INDO/S calculated shieldings for the series of substituted molecules

show slightly poorer agreement with the experimental data than do the

CNDO/S ones. This seems to indicate that for the series of silicon

containing molecules considered, the poor correlation obtained from the

INDO/S calculations is due to the parameters used rather than the

neglecting of the one-centre exchange integrals. This is not the same

as in the first-row nuclear shielding calculations in which the correlation

between the calculated chemical shifts and experimental data is better

in the INDO/S than the CNDO/S calculations^^’ .

However, the results in Tables 5.2.1 and 5.2.2 for our calculations

do not reproduce all of the ’’hanging chain" pattern. This is, probably,

due to the unsuitable parameters for second-row elements for CNDO/S and

INDO/S parameterizations, employed from the original CNDO/2 program^~^,

the difference among CNDO/2, CNDO/S and INDO/S were discussed in Chapter

two.

tt , - i . (129,130,159,160) ^ .Wolf and Radeglia have performed their calculations

based on Pople’s model by using the AEE and SOS approximations to reproduce

the "hanging chain" pattern by using CNDO/2 wavefunctions with some extra

adjustable parameters. They could reproduce the pattern for a limited

number of compounds only. Thus it is not clear as to whether their

calculations by using CNDO/2 wavefunctions without such extra adjustable

parameters is able to reproduce this pattern.

From Tables 5.2.1 and 5.2.2, only the local paramagnetic terms for

3p and 3d orbitals vary from molecule to molecule depending upon the type

-3of substituent. From these tables, we can see that the values of <r >

for 3p and 3d atomic orbitals, calculated from Slater atomic orbitals,

vary from molecule to molecule, depending upon the type of substituent.

This is more reasonable than using a fixed value for all of the series-3of molecules considered. However the calculated magnitude of <r >

is 2 times smaller than those of W h i f f o n ^ ' ^ * , Desclaux^'*^ and

Smith and B a r n e ^ ' ^ \

A closer consideration of Tables 5.2.1 and 5.2.2 reveals that

the contributions from the 3d orbitals of silicon are significant,

depending upon the type of substituent. From these tables, we have to-3keep in mind first, that the values of <r are smaller than those

_3of <r by 5 times (as calculated from equation F.8 , Appendix F)

then the paramagnetic contribution from 3d orbitals is small when

compared to the paramagnetic contribution from 3p orbitals. So the

paramagnetic contributions of 3p and 3d orbitals depend upon the choice-3 . . .of values of <r > , apart from the parameters used m the parameterization

/0 5) _3schemes. SCF calculations on atoms show that <r > 0 is about two-3p

-3 -3orders of magnitude greater than <r > 3^* It has been shown that <r

increases on removal of an electron from a neutral atom. A similar effect (13)has been noticed m a molecule where a given atom is bonded to another

highly electronegative atom. A discussion of the contraction of 3d orbitals(149)is also given by Coulson . These conclusions are in agreement with

-3our results which show that the values of < r depend on the type of

substituent present.

For molecules in the X SiY. series, where X is H and Y is F, ifn 4-nwe consider the paramagnetic contribution for 3d orbitals from the equation

3(j (3d-loc.) = <r >OJ x 3d-contribution3d

we find that the 3d-contribution, as shown in bracket, for the silicon

atom increases from SiH.(17.513) to SiH F(22.342) to SiH0F_(29.635) to4 3 2 2SiHF3(33.090) to SiF^(33.061) whereas the 3p-contribution, also shown

in bracket, decreases from SiH.(164.7013) to SiH F(161.0234) to SiH0F_4 3 2 2(141.8505) to SiHF^(108.5309) to SiF^(51.4391), which shows the maximum

value of the 3d-contribution in this series is about 36% of the corres--3ponding 3p-contribution and so even a small value of < r > may lead

to a large contribution to the total shielding. So the paramagnetic

contribution from 3d-orbitals is not neglected in the present report.(85)This is in agreement with Jameson and Gutowsky

Table 5.2.3 considers charge distributions in the series of

molecules in the series X SiY. where X is H and Y is F and Cl, wen 4-nfind that in the series where X is H and Y is F, charge density in the

3d orbitals of the silicon atom increases while in the 3s and 3p orbitals

it decreases as the number of substituents, F increases. At the same

time, the charge densities of the fluorine and hydrogen atoms are nearly

constant. This is similar to the series of molecules where Y is Cl. In

this series of molecules, 3d charge densities of the silicon atoms are

nearly constant which is similar to that of the chlorine and hydrogen

atoms. We can see from Table 5.2.3 for each nucleus, that there are

several transitions which show contributions to the same order to the

local paramagnetic term, ^(loc.), so it would be incorrect to say, in

general, that a particular transition governs the nuclear shielding.

In Table 5.2.4, are presented the average weighted value of

transition energies, charge densities and total shieldings for the

silicon atoms in the series X SiY. where X is H and Y is F and Cl,n 4-nwith the variation of K , the numerical constant for rr-bonding overlaprrintegrals, equation (2.51), is also given. As discussed in Chapter two,

Table 5.2.4 The results of some CNDO/S calculations of average weighted value of transition energies,

charge densities and total shieldings of silicon with the variation of parameter.

No. molecule KTt 0.500 0.600 0.700 0.800 0.900 1.000

Transitionenergy 10.3033 10.3033 10.3033 10.3033 10.3033 10.3033

sq 1.1158 1.1158 1.1158 1.1158 1.1158 1.1158

1 SiH,QPdq

2.4486

0.0541

2.4486

0.0541

2.4486

0.0541

2.4486

0.0541

2.4486

0.0541

2.4486

0.0541totq 3.6187 3.6187 3.6187 3.6187 3.6187 3.6187.tot0 707.37 707.37 707.37 707.37 707.37 707.37

Transitionenergy 16.4503 16.4544 16.4610 16.4 704 16.4827 16.5093

sq 1.1228 1.1223 1.1216 1.1209 1.1200 1.1198

2 SiH3FqPdq

2.1564

0.0829

2.1529

0.0841

2.1488

0.0855

2.1444

0.0870

2.1397

0.0884

2.1363

0.0918totq 3.3619 3.3592 3.3560 3.3523 3.3482 3.3480.tot0 693.84 694.81 696.00 697.95 698.16 699.68


sq 1.0872 1.0854 1.0832 1.0808 1.0782 1.0758

3 SiH2F2qpdq

1.8638

0.1364

1.8568

0.1375

1.8494

0.1406

1.8416

0.1439

1.8336

0.1469

1.8262

0.1493totq 3.0855 3.0797 3.0732 3.0662 3.0588 3.0728rot0 698.22 701.07 704.13 707.34 710.67 714.69


sq 1.0484 1.0457 1.0425 1.0390 1.0352 1.0348

4 SiHF3qPdq

1.6100

0.1732

1.6031

0.1765

1.5968

0.1799

1.5917

0.1831

1.5878

0.1859

1.6145

0.1872totq 2.8314 2.8254 2.8193 2.8136 2.8090 2.8366.toto 730.95 728.13 738.59 749.12 755.79 763.82


sq 0.9991 0.9954 0.9912 0.9864 0.9813 0.9757

5 SiF4qPdq

1.4004

0.2064

1.4036

0.2097

1.4054

0.2124

1.4104

0.2148

1.4190

0.2161

1.4314

0.2169totq 2.6098 2.6087 2.6090 2.6114 2.6163 2.6240.tot0 784.45 797.01 807.30 815.97 823.73 831.26


No. molecule KTT 0.500 0.600 0.700 0.800 0.900 1.000


sq 1.2643 1.2641 1.2637 1.2632 1.2626 1.2619

6 SiH3ClqPA

2.1937 2.1899 2.1856 2.1809 2.1761 2.1708aq 0.0541 0.0552 0.0564 0.0578 0.0590 0.0604

totq 3.5121 3.5091 3.5057 3.5019 3.4978 3.4933cJt o t 699.59 700.06 701.45 701.93 702.80 703.13

T ransitionenergy 14.4425 14.4484 14.4564 14.4666 14.4791 14.4941

sq 1.2886 1.2879 1.2869 1.2857 1.2843 1.2827

7 SiH„Cl_qpA

2.0423 2.0359 2.0293 2.0224 2.0153 2.0083Z Z Qq 0.0583 0.0603 0.0624 0.0645 0.0667 0.0690

totq 3.3893 3.3842 3.3786 3.3726 3.3664 3.3600.toto 708.50 712.49 716.32 720.33 724.46 728.68


sq 1.1277 1.1266 1.1252 1.1233 1.1211 1.1184

8 SiHCl.qpA

2.1814 2.1742 2.1665 2.1586 2.1510 2.1438Uq 0.0598 0.0619 0.0639 0.0662 0.0683 0.0704

totq 3.3690 3.3626 3.3556 3.3481 3.3405 3.3328.tot0 726.98 733.29 739.70 746.15 752.66 759.72


sq 1.1971 1.1957 1.1939 1.1916 1.1887 1.1852

9 SiC1*qPA

1.9745 1.9632 1.9519 1.9410 1.9315 1.9242Qq 0.0663 0.0689 0.0718 0.0744 0.0770 0.0795

totq 3.2379 3.2279 3.2174 3.2070 3.1973 3.1889.toto 732.27 745.63 759.60 774.01 787.75 800.98

within the framework of the CNDO/S and INDO/S parameterization schemes,

the Tf-Tf overlap is screened differently to the d - d overlap. The extent

of this difference depends upon the value of parameter. So all of

the CNDO/S and INDO/S calculations have been performed using a value

of K = 0.585. In order to investigate the K parameter dependence of

silicon shieldings, we have performed some calculations using values of

K from 0.500 to 1.000, where 1.000 is the screened value of d - d bonding.TT

Comparison of the results given in Table 5.2.4 with those in

Table 5.2.3 reveals that the average weighted value of transition energies,

charge densities in 3d orbitals and total shielding increase as the value

of increases. Therefore the shielding calculations are numerically

larger when increases. A closer consideration of the increasing of

charge densities in 3d orbitals in Table 5.2.4 we found that the variation

in magnitude of charge densities in 3d orbitals for silicon atom is not

significant when the value of varies from 0.500 to 1.000.

5.3 Phosphorus shieldings.

5.3.1 Introduction.

Phosphorus-31 is known as one of the common NMR nuclei. It has-2100% natural abundance, low NMR sensitivity (6.25x10 with respect to

1 (173) 31 -7 -1 -1H) and a positive magnetogyric ratio (y P = 10.829x10 rad.T S )

There have been many attempts to establish empirical relationships

between phosphorus chemical shifts and such molecular quantities as bond

lengths and angles^ or electron d e n s i t i e s ^ ^ ^ . Yet there appear to

have been few attempts at theoretical interpretations. Gutowsky and (125)Larman treated a few phosphorus derivative using only 3s and 3p

orbitals and estimated the value of the average excitation energy for

each compound. However, Letcher and Van W azer^"^^ treated most of the

classes of phosphorus compounds for which there are appreciable phosphorus

chemical shift data, allowed 3d orbitals for bonding, and assumed that

the ratio of the average excitation energy to the expectation value of -3r was a constant for each coordination number, i.e. number of phosphorus

(126 i3i)substituents. Letcher and Van Wazer have proposed to deal with

3s and 3p contributions as being by far the most important in relation

to certain parameters, such as substituent electronegativities and bond

angles of phosphorus. As the latter are known in only a few compounds

the use of their assumption can only be qualitatives as is the case for

phosphorus anions^1^ ^ . Of these, the condition of only 3s and 3p orbitals

is better when dealing with trivalent phosphorus compounds, in these cases,

however, the role of the lone pair electrons is very i m p o r t a n t ^ . For

phosphorus involving tetra- or penta-coordination, the importance of the

3d electrons must not be underestimated^^' 166). consideration of 3d( 85)contributions on a theoretical basis was rather complex and was limited

i . _ . . (165,166) _ . .(167,168) .to qualitative considerations . Rajzman et al and

Pouzard et a l ^ " ^ ^ have performed calculations using Pople's GIAO method

together with eigenfunctions and eigenvalues obtained from a variation

of the CNDO/S method. They used <r ^(168) tê orbital

exponents^^^^ as adjustable parameters to fit calculated values for

phosphorus shieldings to the experimental ones. Ab initio calculations

have been performed for some small phosphorus containing molecules(170,171)

Zanasi^"^^ also performed the theoretical determination of magnetic

properties of small molecules by using a CoupledHartree-Fock method.


In the present study, phosphorus shieldings and their chemical

shifts, with respect to PH^, have been calculated for a variety of

molecules with the CNDO/S and INDO/S parameterization schemes. The

results are compared with experimental data and other theoretical works

where available.

Tables 5.3il and 5.3.2 present the results of some CNDO/S and

INDO/S calculations together with the experimental data. It is found

that the local diamagnetic contribution is constant to within 0.5% for

the phosphorus environments considered and that the non-local diamagnetic

contribution is negligible. An overall average of 952 ppm is obtained

for the local diamagnetic contribution from both CNDO/S and INDO/S

calculations, respectively. This is similar to the diamagnetic contri

bution calculated for different molecules, using the CNDO/2 m e t h o d ^ ~ ^ \

The diamagnetic and paramagnetic contributions for local and non-local-3terms and < r > for 3p and 3d orbitals are quoted in Tables 5.3.1 and

5.3.2 for the CNDO/S and INDO/S calculations, separately. For the

paramagnetic term, the contributions from the 3p and 3d atomic orbitals

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CO

r— lu o P N o o u p O CJ

CO PC U O /~\ CM U P CMP O CM PC u o /*—N r~\

c o o CM PC o o O O OOPC u p O CM PC PC PCCO CM u p a o o o'— ' PC PC u p u p u p U PPM PM fC P m P m P m

<r U O U D 0 0 CO 10

o o o o o oo o 00 o o 00• • • • • •

CO oo <r r u UOCM uo CO uo oo CO<r <r <r 00 oo oo

uo r-'- <r o uO oU0 CM uo o CM 00• ■ • • • •o <r o uO 00 U0<r UO uo <r o

i—iCMI—1

uo 00 uo o <t oCO 00 CO UO CM CM• • • • • •<r 00 <r CO CM oCM oo rH 1—1 UO <rUO UO uo uO UO uo

00 uo O CO r. CO<r uO UO o uo <r• m • • • •oi

o1

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<r oo 00 CMCM UO oo oo 00 rHoo1

ool

ool

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oo1

<ri

CO o CM o UO uoCO CO 1—1 CM

• • • • • •uo1

uo1 CM1

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001

uo1

CO rH ru . o rH CO00 CO 00 00 O 00CM CM CM CM 00 CM• • • • • •o O o o o o

uo o UO CO 1— 1 COU0 r- CO <r• • • • • •1—1 r'. <r I—1 o UOCM uo 00 oo 00 oCO oo oo CO ooI I I I I I

UO UO r. CM CO<r UO oo O o <r<r <r <r UO <r• • • • • •rH r—l rH i—i 1—1 rH

UO <r U0 uO rHO rH O O o O

• • • • • •O1

O1

O1

o1

o1

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<r UO CM CO <r rHUO CO UO uo• • • • • •

CM CM CM CO i—i CMU0 UO UO uo uo UOCO CO CO CO CO CO

S \ CMCM LOr-l PC LO C-NO CM CC COo CM U-Kco up o o£C co cm u p v pCJ rH rH r-H CO CMu p O O O fn PmP-i P-i PM P-i PM PM

r—l CM CO < f UO UOi—I t—I r—l c—I rH i I

to0g• H■P3OOv_

00inwPQ<H

r~N o o o o O O O o o O O O O O O o op ON o o O VO r'- ON o O 00 O On O O o OaX<1> r-l m •i—i H <r 00 CO in VO CM <f r-l ON VO <r U0

VO co vO ON o VO o vO VO UO VO 00 CO 00 o VO oO OO CM 00 00 <r 00 <r CO CO 00 CM CM CM CM 0M CM CM

CM (—1 in 00 i—i 00 o UO vO 00 O ON O UO VO rrl <ri—1 ON 00 co <r 00 ON ON CM ON co i—l vO co o VO 00ctJ v • a • • • • • a • • • • •a to oo oo o uo CO 00 1-1 ON CM vO U0 VO CM ONV—'o

OOin <r00 r-lCM ONi—1 OOr—l <r 1—1 1 VO VO VO i—li—1 1 I I uo1 <rr-l1i—l1 <fr-l1

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<r CM m <r 00 OO ON I—I <r <r CM 00 uo ON <r <r 00o oo in 00 t-M CM <r ON 00 i o ON 00 i-i UO vOor-l CO CM ON ON CM r-s 00 VO vO 1—1 uo vO ON VO i—1 CM

CM co <r CM CM VO <r in UO o o CM oo COOk 00 r-l <r <r <r co CM co 00 CO r-l CM CM CM i—i CM l—l■O 1 r-l1 t I i i 1 i l 1 1 1 1 1 I 1 1

ar-l CM <t 1—I i—i CM UO CO UO vO 00 oo CO CM <r 00 uo1 o rH ON CM UO o i—i o co CM in I—l ON 00 00 oo

X}00 ooI rH xAi in i—1 o <t CO CM CM co vO <f in UOv-/%

i i I CM1 r-lI i—iI l—l1 l—l1 CM1 CMi CM1 r-l1 CMI i—iI col’Oco r-l CO co On r» ON r- r-1 00 CM O uo 00 <r ON CM vO✓\ ON r- VO i"- o 00 ON 00 ON r-l o o ON H ON r—l

ooI CM CM CM CM CM 00 CM CM CM CM CO CO oo CM 00 CM COa u s a • a a a a a • • • • • °

i >o o o o o o o o o o o o o o o O O

o ON VO <r 00 1—1 00 in VO 00 <r m CM l-l VO co(—1 VO CM CM o 00 co CM 00 oo r-l co1 « « « a a a a a a a • • • • • •Oh On r-l i—1 r- U0 <r CO <r 00 CM o <r o <r I'-'00 tH CO <r r-. i—i o uo co <r <r <r 00 00 r—l l—l in Onv_/ 00 r-l <r <r <r 00 CM co CO co r—l r—l t—i CM l—l CM 1Pit> 1 i—1 1 i i i 1 1 I 1 i 1 1 i 1 1 1

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a a a a a a a a a a • • • •£ i—i l—l i—i I—1 l—l l—l 1-1 l—l l—l r-l r—l 1—1 1—1 i—l r-l i—i i—1

P‘{ CM i—i CM 00 UO l—l o 00 CM CM VO CM CO CM0 o O o O o o 1—1 o 1—1 O CM CO <r CM OO <r

T3\J ol O o O1 o1 o1 01 oi o1 O1 oi O1 oi O1 O1 o1 oi

o oo 00 in in 00 00 ON i—i r-l <r 00 i—1 CM CM VO vOo in oo oo r"> ON VO ON in ON <r LO r-l 00 CM in <r ONr_l a • m a a • • a • a • • •**__/ CM m oo co o CM CM CM CM o 1— 1 o 0M ON CM ON

to in in in in UO uo uo UO UO in uo uo in in <r U0"D

TO00o

ON

CM/'"VooPnCO

ON

oo/-\00pLH

ON

00/-\CMs~\OO53CJw

ON

PnCMs-\CMS~\00a:o

ON

CM 1—1 cj n CMr~\CO53o

ON

CMPnOCJ

ON

CMr-lCJ-\OCJ

ON

PnCM<r-\oo

ON

r-lCJCMS~\OCJ

ON

i—i CJ CM /—\COCJ

ON

coscCJ

ON

r—NCOffiCJCM

ON

00

ON

00oo53

ON ON

00/'“ Nco53

ON

Oh CJ !3 53 53 53 53 53 53 53 CM t—( r-l CJ 00 CJ 006 (jlh V_/ v_v vp V vp Pn CJ CJ w Pn V-/ PnOCJ

PM PM PM PM P-I PM PM PM PM P-I PMO PMO PMO PMO PMO PMCO PMCOo 00 ON O 1-1 CM co <r UO VO 00 ON O r—l CM 0053 i—i 1—1 r-l CM CM CM CM CM CM CM CM CM CM 00 00 co 00

The

chem

ical

shifts

are

repo

rted

with

resp

ect

to PH_,

shif

ts

to high

freq

uenc

y are

po

siti

ve

50

0

coo

to

« o

OSSo

Soso X c g °<N0 _ goMV Pk ^

<00,O*a>o®0 n O

.8

-8

8.cu

co4-14-1•HAco

i—iccSO•HeCU.CO•dCU4-3rtr-l2Or-lctia

So

s o S o

4-JCOCJ•HctibDa5co

CdoCU6Oacodno.aa,COo.aa.cuSoCO

LwoCO 4-3 CM • HJCCO

nJo•Hecu.coCOd8o.COOP<r-lctiCJCU6•HucuCUXcucuX

4-3

cmo4-3Or-lP-i

Sc?

(uidd) s ^jtlis xB0Tma^ 0 TBduauiTJtBdxa

CMCOLOcuCi3to•HpH

the

valu

es

calc

ulat

ed

by INDO/S

meth

od.

are shown separately in Tables 5.3.1 and 5.3.2. The calculated chemical

shifts are plotted against the experimental data in Figures 5.3.1 and

5.3.2 for Tables 5.3.1 and 5.3.2, respectively. From these figures,

the agreement between the calculated chemical shifts and the experimental

results for the series of molecules chosen is in general poor.

From Tables 5.3.1 and 5.3.2, we can see that there are two distinct

sets of results in the present calculations. The first set contains p * ^

(molecules number 1 to 26), which has smaller shielding than the reference,

PH^, and another set which contains P^ (molecules number 27 to 33) with

a larger shielding than the reference, PH^, while there is no such

distinction in the experimental results. This is probably due to theIII Vchoice of the parameterization for the P and P groups.

The results of the CNDO/S and INDO/S calculations in Tables 5.3.1

and 5.3.2 show that the variation of the magnitudes of the calculated

shieldings are too small. And because of this, the values of chemical

shifts also too small when compared with the experimental data. This-3is probably due to the transition energies and <r > , which play the

important role in the paramagnetic contribution as shown in Tables 5.3.3

and 5.3.4. We can see from Tables 5.3.1 and 5.3.2, the use of equation-3(3.38) to calculate <r > from Slater atomic exponents gives smaller

values than those obtained by Whiffen et using Hartree-Fock(158)SCF wavefunctions, Desclaux , using Relativistic Dirac-Fock equations,

(159)and those obtained experimentally by Smith and Barnes from spin-orbit

splitting data by about 2 times, which is the same as in the silicon-3calculations. The small values of <r > for 3p and 3d orbitals result

from Slater functions which has been noted e l s e w h e r e ^ " . If we include_ 3

the factor of the difference between the value of <r > from the calculated

and experimental results in our calculations we can see that the magnitude

Table 5.3.3 The results of some CNDO/S calculations of charge densities, transition energies and

contributions to the paramagnetic components of phosphorus atoms.

No. molecule s p d atom q q q totq transition transition

energy cJP(loc.)

i ph3 P 1.8721 2.9050 0.0403 H 1.0609

4.87141.0609

34

- 5 -* 7

6.1377.468

-209.342-99.334

X

Laverage weighted value of transition energies = 10.1286 e.v.

24

- 5- 6

6.1737.467

-209.346-99.333

y

2 -* 6 9.420 -10.0822 - 7 7.633 -122.9203 — 6 7.633 -122.9143 — 7 9.420 -10.082

2 PH2(CH3) P 1.8056 2.9781 0.0483 4.8320 5 -10 9.693 -20.332C 1.2256 2.9532 4.1788 6 - 8 6.431 -25.729Hp 1.0691 1.0691 6 - 9 7.075 -55.320H 0.9494 0.9494 7 -10 5.382 -210.336

X

. . . rP:..:; 1— *2< ' y

average weighted value of transition energies = 15.878 net charge on P - +0.1680

e.v.45 7 7

00 CO

00 os

t t

t t

9.6838.2417.7345.502

10.787-54.918-93.368

-223.125

y

5 -10 9.693 -26.7046 - 8 6.431 -134.8016 - 9 7.075 -85.899 z6 -11 8.962 -11.8507 -10 5.382 -23.912

3 PH(CH3)2 P 1.7398 3.0541 0.0493 4.8433 9 -12 3.909 -376.699C 1.2447 2.9518 4.1966 9 -19 11.500 -10.081 X

Hp 1.0691 1.0691 10 -*11 4.835 -83.628Hc 0.9462 0.9462 8 -11 8.632 -40.837

cwfv y th3

average weighted value of transition energies = 17.952 e.v.

99

-11-13

6.035 4 .694

-53.824-331.022

y

net charge on P = +0.1567 8 -12 7.258 -88.77010 -11 4.835 -27.370 z

10 -13 5.509 -167.811

* p(c h 3)3 P 1.7111 3.0617 0.0486 4.8214 11 -15 4.692 -114.286C 1.2540 2.9568 4.2109 11 -16 5.354 t 104.856 XHc 0.9520 0.9520 13 -14 4.736 -138.749

X

CH^ ... Ip.-.;: •— *2 ych3

average weighted value of 11 -15 4.692 -119.621transition energies = 19.077 net ctiarge on P = +0.1786

e.v. 1112

-16-14

5.3545.327

-100.162-123.369

y

12 -15 2.601 -111.94212 -16 3.906 -136.75313 -15 2.878 -185.54013 -16 5.386 -54.035

Transitions contributing less than 10 ppm to the paramagnetic term are not included.

of the variation of nuclear shieldings and chemical shifts can be

improved but the overall agreement of chemical shifts from our calculations

and the experimental data is still not good.

Rajzman and S i m o n ^ ^ ^ used Pople's GIAO-MO-SOS method in

conjunction with CNDO/S wavefunctions to calculate phosphorus nuclear

shieldings for some molecules in Tables 5.3.1 and 5.3.2 (molecules

number 1 to 4), they find that it is necessary to introduce 3d orbitals

for phosphorus. T h e y ^ ^ ^ used <r as an adjustable parameter to

fit the theoretical to the experimental data. Latter, Rajzman and

S i m o n ^ ^ ^ and Bernard-Moulin and P o u z a r d ^ ^ ^ performed further calculations

of phosphorus shieldings, also using Pople's GIAO-MO-SOS method. The

eigenfunctions for these latter calculations were obtained from CNDO/S

method with the orbital exponents of phosphorus taken as a function of

the electronegativity of the substituent. These exponents also contain

an empirical factor which can be varied from molecule to molecule, so as

to improve the correlation between the calculated and experimental values

of the phosphorus shieldings. Thus it is not clear as to whether their

calculations by using CNDO/S wavefunctions without those extra adjustable

parameters is able to reproduce experimental data.

A closer consideration of the contribution from transitions of

various symmetry to the paramagnetic term is demonstrated in Table 5.3.3

for a series of molecules (molecules number 1 to 4) from the CNDO/S

calculations. Charge densities for each orbital and the average weighted

value of transition energies are also reported in Table 5.3.3. From this

table, for each molecule, we can see that there are several transitions

which show contributions to the same order to the local paramagnetic term,

^ ( l o c .), so it would be incorrect to say, in general, that a particular

transition governs the nuclear shielding. Charge density in the 3d orbitals

Table 5.3.6 The results of some CNDO/S

charge densities and total

calculations of average weighted value of transition energies,

shieldings of phosphorus with the variation of K parameter.tr

No. molecule KTT 0.500 0.600 0.700 0.800 0.900 1.000


sq 1.8721 1.8721 1.8721 1.8721 1.8721 1.8721

1 PH3QP 2.9050 2.9050 2.9050 2.9050 2.9050 2.9050d

q 0.0603 0.0603 0.0603 0.0603 0.0603 0.0603tot

q 6.8176 6.8176 6.8176 6.8176 6.8176 6 .8176rot0 650.96 650.96 650.96 650.96 650.96 650.96

Transition

PH,(CH )

PH(CH )3 2

P(C"3)3

energy 15.8863 15.8765 15.8695 15.8633 15.8580 15.8536s

q 1.8055 1.8057 1.8059 1.8061 1.8063 1.8065

qP 2.9801 2.9776 2.9751 2.9725 2.9700 2.9673d

q 0.0679 0.0683 0.0688 0.0692 0.0697 0.0501tot

q 6.8336 6.8317 6.8297 6.8278 6.8258 6.8238

<jcoc 627.78 628.36 629.03 629.79 630.62 631.56


sq 1.7388 1.7600 1.7612 1.7626 1.7638 1.7651

qp 3.0586 3.0536 3.0683 3.0629 3.0375 3.0321d

q 0.0687 0.0696 0.0501 0.0508 0.0512 0.0520tot

q 6.8660 6.8628 6.8396 6.8361 6.8327 6.8292.tOto 556.85 557.66 558.67 560.00 562.00 566.60


sq 1.7086 1.7116 1.7167 1.7179 1.7211 1.7263

qp 3.0692 3.0606 3.0513 3.0623 3.0335 3.0265d

q 0.0683 0.068 7 0.0693 0.0698 0.0502 0.0509tot

q 6.8259 6.8206 6.8156 6.8101 6.8062 6.7996.tot0 602.95 692.58 593.91 610.21 616.26 615.76

of the phosphorus atom nearly constant while in the 3s orbital decreases

and 3p orbitals increase as the number of substituents, CH^ group, increases

in PH^ series.

From Tables 5.3.1 and 5.3.2, we find that the contribution from

the 3d atomic orbitals on the phosphorus atom depends upon the type of-3substituents, and is not as large as for silicon. The value of <r > ^

-3is less than < r > by 5 times, and the 3d contribution is small when

compared to the 3p contribution.

From Table 5.3.4, the average weighted value of transition energies,

charge densities and total shieldings for phosphorus atoms in molecules

number 1 to 4 are shown with the variation of K , the numerical constantTT

for tt-bonding overlap integrals. As discuss in silicon calculations,

the Tt-Tf bonding overlap is screened differently to the d - d bonding overlap.

All of the present calculations have been performed by using a value of

= 0.585 as in the silicon calculations. In order to investigate the

parameter dependence of phosphorus shieldings as a function of Tr-bonding,

we have performed some calculations using = 0.500 to 1.000, where 1.000

is equal to the screened value of d - d bonding. Comparison of the results

given in Table 5.3.4 to those in Table 5.3.3 reveals that the average

weighted value of transition energies decreases whereas charge densities

and total shieldings increase as the parameter increases.

5.3.3 The variation of shielding with molecular conformation.

In general, calculated shielding data for first-row elements

appear to be dependent on molecular conformation. One application of

a reliable theory of shielding calculation would be the determination

of molecular conformations. Molecules in solutions may not have the

same conformation as in the solid state. Also the conformation of a

molecule in solution may depend not only upon the solvent used but on

the concentration and temperature as well. Rotation about single bonds

and tortional angles may also occur, and the phenomenon of restricted

rotation, has been known for some time. Shielding values for second-

row elements may then best be calculated by determining a weighted

average of the shielding values obtained by using several different

conformations. The temperature at which the measurements are made may

also be important due to the possibility of small barriers to rotation.

The present work was under taken to calculate the shieldings of

the phosphorus atoms in and at different values of the

twist tortional angle between the two lone pairs on the phosphorus atoms

in and between the phosphorus atom and the CH^ group in PH^CC^H^).

Comparison of the results obtained using the spd basis set shows

that angular dependence of the shielding is mainly due to the paramagnetic

contribution. The diamagnetic term remains constant during the rotation

process in Tables 5.3.5 and 5.3.7.

According to the results reported here, the nuclear shielding is

highly dependent upon the rotation angle as shown in Figures 5.3.3 and

5.3.5. However, it may be pointed out that in rotation around the P-P

and P-C bonds, the bond angles and bond distances are kept at fixed values.

Such a constraint of the geometrical parameters, which are not allowed to

relax may have an influence on the nuclear shielding, t f ( total) vs f(^)

to an amount which has not been calculated up to now.

The calculations were carried out by using the INDO/S parameteri

zation scheme. The results are shown in Tables 5.3.5 and 5.3.7, plots

between the tortional angle and calculated shielding (j(total) for the

phosphorus atoms in and are shown in Figures 5.3.3 and

5.3.5, respectively.

The total energies given in Table 5.3.6 were calculated by varying

the tortional angle in 15° steps for ^y using the INDO-SCF method.

The tortional angles corresponding to the energy minimum are shown in

Figure 5.3.4.

From the above results, it is clear that the calculated shielding

depends upon the conformations considered, this study may help in under

standing the conformation of molecules considered.

TABLE

5.3*5

Phos

phor

us

shie

ldin

gs

calc

ulat

ed

in ppm

using

INDO/S

meth

od

for

Diph

osph

ine,

P0

H ,

as fu

ncti

on

of di

hedr

al

in <r VO LO VO in um oo On o oo m 00 <r cm i-i On00 • • <r • CM m • • •r H CM o • CO • • cn o oo

m i—i 00 o m On i inOn CM

1I CM

1vo

in <r VO UO r- CM i—1 <r <rin on o CO cn 00 <r on i—i invO • • <r • CM in • • •I-l CM o uo • • i—i o r—l

in i—i 00 o m On i VOOn CM

1i CM

1vO

vO <r uo r-- o in VO Ono CTv o oo <r 00 on i—1 00LO • • <r • CM <r • ■ •

CM o • • CM o Onm H 00 o in on 1 inOn CM

11 CM

1VO

VO <r in vo vo cn mLO On o oo O oo On <r i—i cnoo • • <r • CM cn • • •p-l CM o • r^ • • CM o o

in i—i cn o m o i inO'! CM

1i cn

iVO

VO in o vO r—l cno o\ o oo i—i 00 m <r CM cnCM • • <r • CM cn • • •r~ l CM o • o • • in o

LO r—l o o in o 1 <rOn CO

1i cn

ivo

in vo r-. CO CM o oin on o oo O 00 in on CM mo * • • CM cn • • •i—i CM o • 00 • • cn o on

in I—l on o m o 1 <ro\ CM

Ii cn

ivo

<r U0 00 r. in OnOn o co o 00 <r in r - l CMo • • <r • CM <r • • •

o\ CM o • in • • o o CMin t—i cn o m o 1 inon CM

Ii cn

iVO

00 <r <r on r> CM VO cncr\ o 00 CM 00 vo VO i—1 CM

in <r • CM cn • • •CM o • H • • VO o VOin i—i On o m on 1 inOn CM

1i CM

1vo

00 <r <r in 00 in o cnCh o 00 in 00 r^ 00 CM oo • • <r • CM cn • • *

vO CM o 00 • • cn o Onin r—l 00 o in on 1 UOon CM

1i CM

Ivo

On <r <r CO i—i On <r t—lOn o CO r-> 00 00 on i—i On

m • • <r • CM cn • • •<r CM o r. • • CM o On

in i—i 00 o m on i inCTv CM

1i CM

1vo

CTv <r CO cn r. cn on o ocn o 00 o 00 i <r CM <ro • • <r • CM cn • • •

CO CM o • cn • • <r o 00m i—i 00 o m on 1 mcn CM

li CM

IVO

cn <r CO CM i—i VO i—ian o oo <r 00 vo r - l r H

in • • <r • CM cn • * •i—i CM o • 00 • • 00 o

UO i—i cn o m on 1 incn CM

1i CM

1vO

vo CO CO O'- 00 CM OO cncn o co o 00 in m r—l cn• <r • CM cn • • •o CM o CO • • cn o Onm t H on o in o 1 <rcn CM

1i cn

iVO

cuI—l s~\tup CJ CJc o oni c-n a 1—1 TO 1—1 cn 1—1o /—\ CO 1 cn 1 CJ /-n nJi— l o r—l /\ a /\ X3 O i—1 4->cj r—l CJ CO 00 cn cn i— l CJ o

v_c 1 i V_y<

TO TO u a u a a a v_/CU t) "O \ y 'D \ y t> ~D "o U)Si• Ho

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TABL

E 5.

3.7

Pho

sph

orus

sh

ield

ings

ca

lcul

ated

in

pp

m us

ing

INDO

/S

meth

od

for

PH^C

^Ht-

, as

fu

ncti

on

of

dihe

dral

a

ngle

,

r-l <r c n <r c n 0 0 VO c no <r o i— i 0 0 0 0 n - i n o00 « <r • C N • • • •1— 1 c n o • <r • <r c n o c n

i n 1 i— i <r o I <r 1 oC h c n

ic ni

V O

C N < r C N <r C N cn < r i n c nL O <r o i— l < r 0 0 V O C N o ov O • • <r • C N • • • •i— I c n o » C N • <r o VO

i n i i— i <r o i <r 1 oG\ c n

ic ni

v o

<r c n r-l V O C N i n C N 1— 1 0 0o <r o r— l o 0 0 i n o v om • • < r • C N • * • •t— i c n o o • < r <r o 0 0

m i i— i < r o i < r 1 oO N c n

ic ni

v o

VO c n i— i i— i C N <r 0 0 <r c nm <r o r— l c n 0 0 < r V O oc n • < r • C N • • • *r-i c n o • O V • <r c n o c n

i n i i— i C N o i < f o0 ^ cn

1c ni

v O

m c n r— l C N c n c n c n C No <r o i— 1 v o 0 0 < r o o < rC\l « <r • C N • • • •r— i c n o • • <r C N o r— l

m 1 r— l c n o i < r i— 1cn c n

ic ni

V O

c n c n C N C N 0 0 c n r— l c nm < r o r— l c n 0 0 < r 0 0 i— l v oo • < r • C N • • • •i— i c n o • c n • <r r-v O i n

i n 1 i— i c n O i c n i— i< n c n

ic ni

v o

o C N c n m c n o m i— l < r< r O i— i < r 0 0 V O r— l i— l c n

o • < r • C N • • • *a \ c n o • v o • <r i-l o C N

i n 1 r-i C N o i c n C Ncr> c n

Ic ni

V O

0 0 r-l i n C N c n o c n 0 0 C Nc n O r— l C N 0 0 0 0 o V O

i n • • C N • • • •c n O • c n • <r o i ni n 1 r— l C N o i C N C No \ c n

ic nI

V O

v O o i n < r c n 0 0 c n C N i ncn o r— l m 0 0 <r O c n

o • <r • C N • • • •vo c n o • V O • < r i— t o i— i

in r-i C N o i c n C Nc n c n

ic ni

V O

<T o vO 0 0 c n i n v o <r i nc n o i— 1 C N 0 0 0 0 i— 1 o r-l

i n • <r • C N • • • •<j- <r o • c n • < r <r o < n

i n i— i C N o i c n 1 r— lc n c n

ic ni

vO

cn I— 1 n * vo c n c n n - cn c ncn o i— i c n 0 0 0 0 o <r

o • <r • C N • • • •c n c n o cn • <r <r o 00

i n i— i C N o i c n i l— lCT\ c n

Ic ni

VO

c n C N C N c n r-i <r C N cnc n O r— l 0 0 0 0 c n i— 1 <r

i n . • <r • C N • • • •rH cn o • cn • < r < r o 00

m r-l C N o i c n 1 HCT\ cn

ic ni

vO

c n C N C N c n r-l m cn VOc n o i— 1 i n 0 0 c n <r i— i• • <r • C N • • • •

o c n o O V • <r <r o 00m i— i C N o i c n i i— icn c n

ic ni

v o

CL)t— I r v00 o Uc o O S~\aJ r -N Cu r-l i— I 1— 1

o r~\ cn 1 c n 1 a r~\ ccJpH o r-l / \ (X TO o i— l 4->nJ I— l c cn cn c n c n r— l £ O

v_/ v_-/ i _ ' i V^r v_y 4-)TO TJ 'O Ci a, K c u aQJ "D ID V / i d \ / t) D \D \Dx:•r—lQ

630

5.3.A Solvent effects.

As a second-row element, phosphorus is subject to structural and

electronic influences similar to those experienced by silicon. However,

phosphorus posses some features that distinguish it from silicon, these

features result largely from the presence of the unshared electron pair

and are useful in structure elucidation. Thus, as nitrogen, hydrogen-

bonding of phosphorus probably changes the resonance positions, and the

magnitude and direction of this change can be characteristic of the

specific type of phosphorus compounds. In a similar manner, phosphorus

shieldings are frequently more sensitive than silicon to solvent

c h a n g e s ^ ^ 180)^ This fact may be exploited in studies on solvent effects

The polar effects of solvents on nuclear shielding may be accounted( 9 0 911for by means of the "solvaton" model ’ . I n the present study, the

"solvaton" model is applied for the first time to phosphorus shieldings

in the hope of providing a closer insight into the effects of solute-

solvent interactions.

In the present study, we have chosen phosphine and its derivatives

as a simple model to investigate solvent effects. Tables 5.3.7 to 5.3.12

represent the phosphine shieldings, calculated as a function of dielectric

constant of the medium. The "solvaton" model is used in conjunction with

the INDO/S parameterization scheme. The data given in Tables 5.3.7 to

5.3.12 reveal that both the diamagnetic and paramagnetic contributions to

the shielding vary with the dielectric constant of the medium. However,

changes in the diamagnetic term are not as pronounced as those in the

paramagnetic term. We have found that, from Tables 5.3.7 to 5.3.11, the

phosphorus shieldings decrease when the dielectric constant of the medium

increases from 1 to 80. The total electron density, q(total), the value -3of <r > for the 3p and 3d orbitals as a function of dielectric constant

of the medium are also presented in Tables 5.3.7 to 5.3.11. We can see

that, all of these factors vary as a consequence of dielectric constant

of the medium variation. An increase of dielectric constant of the

medium tends to decrease the electron density on the phosphorus atom.-3 -3This occurs with a simultaneous increment in <r and < r because3p 3d

of a net contraction of the 3p and 3d orbitals. On the other hand, the

average weighted value of transition energies decrease as the dielectric

constant of the medium increases. Thus the paramagnetic contribution

increases and total shielding decreases as the dielectric constant of the

medium increases.

In contrast, the results presented in Table 5.3.12 for OFF^> reveal

a shielding increases of 0.80 ppm when the dielectric constant of the

medium increases from 1 to 80 which show a very small sensitivity of

dielectric constant of the medium variation. Decreases in the paramagne

tic contribution to the phosphorus shielding of OPF^ are found to be due

to the variation of two factors. Firstly, due to the reduction of the- 3 - 3radii of the 3p and 3d orbitals, the <r > and <r factors tend to

increase as the dielectric constant of the medium increases. Secondly,

the transition energies increase with an increase in the dielectric constant

of the medium. Thus the paramagnetic contribution in OFF^ decreases and

the total shielding increases as the dielectric constant of the medium

increases.

Although these calculations have been able to show that the variation

of shieldings depends upon the solvent used by using the "solvaton" model,

it is to be noted that it would be unrealistic to base quantitative conclu

sion on the results obtained. However, they could be useful in providing

qualitative interpretation guidelines.

TABLE

5.3.8

Depe

nden

ce

of ca

lcul

ated

ph

osph

orus

sh

ield

ings

and

ch

emic

al

shifts

(ppm)

of PH

upon

diel

ectr

ic

o

OCO

oo<r

o•oco

o•p op CMcdPCOPOCJ OCJ O•H i—1PPOCUHCU o•H •Q co

o

CM

ot—l

TOaPOa£oo

CT\ o CM CM CO co CO o <r in CO(Oc o OO 00 ON <r CM o CO CM ON• <r • CM • • • • • r-lr-l o • MO • CO CO o r—l CO COm 1—1 o o I 1—1 <r i—l •on CO1 COi CO <r

G\ o CM in CO co ON o CO 00 00on o 00 o ON <r <r o CM <r o• • CM • • • . • • CMt—l o • CO • CO CM o CM CM COm t-l o o 1 r-l <r r—l •ON COi CO1 CO <r

o o t—l 00 CO co <T o i—i CO COo o 00 CO ON <r t—1 o ON i—1 r-l• • <r • CM • • • • • CMCM o in • CO CM o CO CM COin r-l o o I r-l <r t—l •ON COi CO1 co <r

1—1 o t-l t—i CO <r CM o CO i—1 CMo o 00 <r ON <r o co• <r • CM • • • • • CMCM o in • CO r-l o 00 r-l COin r-l o o 1 i—1 co r-l •ON coi CO1 CO <r

<r o ON 00 CO in i—1 o 1—1 O CMo o 00 ON <r co o in CO 00• • <r • CM • • • • • CMCM o co • CO o o ON O COm t—l o o i t—l CO r-l •ON CO1 CO1 CO <r

00 o !"• I—l m VO CO o <r ino o ON <r 00 o 00 00 <r• <r • CM • • • • • coCM o • CM • CO 00 o ON 00 COin t—i o o i o CO •ON co1 coI CO <r

<r o m in CO t—i o I''. o ONr—l o CO ON <r 00 o <r 00 CM• • <r • CM • • • • • <rCM o • CM • CO 00 o o 00 COin t—i O o 1 o <r •ON CO1 coI CO

o o in co in 00 o CM ON inCO o vO in ON <r ON o <r• • <r • CM • • • • • COCM o r—l • CO r o co r. cOin ■—i o o 1 o <r •ON CO1 co1 CO <r

t—i o CM o <r t—i o in o <rvO o in m ON <r o o CO o CO• • <r • CM • • • • • r-lCM o co • CO o o CM oin i—i ON o i o m •ON CM1 CO1 CO 'd

-r-\ /~No oo o /-*\/'"N ft 1—1 X I—l s~\ r-l i—iCJ /"\ co 1 CO 1 CJ Cd cdO !—1 /\ ft /\ TO o t—l p Pr—l p CO co CO CO t—1 p O <o Oc_ i C_> 1 c_/ C_ P PTO TO P ft p R ft RtJ ■D \ / tJ \/ t> id tD ■Q cr*

coP h

CUPa4a)PlhP<DXJm•Hxo4-J

COPM-i•HxCOor-l

a)r-lPcjcur-lO£TPcupcdr-lOCO•HCUXpoppocuftCOcuPXp

XcupPoftcupcupcdCOp4-i•HXcor-icd O • H£cuxocuxH

CU>•l—lp• r l

COofttuOP•r”lcuxi

TABLE

5.3.9

Depe

nden

ce

of ca

lcul

ated

ph

osph

orus

sh

ield

ings

and

ch

emic

al

shifts

(ppm)

of PH

0(CH_)

upon

Vj

4-1GGGCOgOOo•rHgGoCU i—I QJ • HXJ

O'v O cn cn H ov CN O cr\ O cno in O in i—i CTv cn m o CTV 00 r—l• • <r • CN • • • • • i—1o CN o 00 « m cn o 00 r'-.oo in i—i 1—1 O i CN CN •Ov cni cni VO O’

a\ o cn o\ r-l o 00 o CTV o r—lo in o in o CTv o o o o CN• • <r . CN • • • • * i—1o CN o 00 • in cn o o\ r><r m i—i l—l O i CN CN •cs\ cni cni vo O’

a\ o cn VO r—l o vO o m O’o m o in o CTV o O’ o r-l VO CN• <r • CN • • • • • i—io CN o 00 • in cn o Ov f->.CO m t—i 1—1 O i CN CN •Ov cni cni VO O’

o o cn o i—l o o o 00 r—lo VO o in o a\ O’ O’ o CN in cn<r • CN • • • • • i—iG o CN o 00 • in cn o O'v£ C\l in i—i r-i o i CN CN •G O'v cn cn VO O’G i iCOGo CN o CN cn o o\ CN o 00 i—i O'vo o VO o in VO a\ cn O o VO i—i VO• • <r • CN • • • • • r-lo o CN o • in cn o O'v•H H in i—i i—i o i CN CN •G O'v cn cn VO O’G I iOQJr—l o o o VO o i—i vo o i—l 00 CNQJ o VO o in CTv Ov O’ cn o CN in i—1•H « • O • CN • • • • • CNQ CO CN o VO • in CN o O VOin i—i i—l o i CN cn •G\ cni cn1 VO O’

00 o Ov 00 o CN o o VO cn ino VO o O’ O’ OV O’ OV o 00 o\ VO• • o • CN • • • • • CNCN o VO • in 1—1 o o inin 1—1 i—l o i CN cn •CTv cni cni vo O’

00 o o in Ov VO i—i o O’ in O’o o o in 00 O’ o o 00 O'v CN• • O’ • CN • • • • • O’CN CN o O’ • m o o CN cnm 1—1 r-l o i CN cn •OV cn1 cni vo O’

00 o o OV cn CN o O'v o CNo OV o cn VO 00 in CN o o O’• • o • CN • • • • • ri-H CN o o • in vo o VO oin i—l l—l o i i—1 cn •ov cni cni VO o

s~sV CJ /—\O o i—i /*—\ s~\ss Oh i—l 'G i—i G i—i i—1CJ s~\ cn 1 cn i G G Go 1—1 /\ Oh /\ XJ O 1—1 G Gi—1 G cn cn cn cn r—l G O <o Ov—✓ 1 __' i v—' V-/ G G•G TO G Oh G Oh Oht) "O \/ V t) tJ tJ G1

/ NC' cn03

c OOh v-/E CNC 00O P-I

> >oGQJ3GQJG4-1GQJ43tuO•H43o4->CO4-J4-1•H43CO

IIVd•»

QJ i—I3OQJi—lOe

T3CU

4-JaJi—IOCO• HQJ43GO4-J

4-JOQJD<COa>G

434-J• H &X 5QJ

4-JGOOhQ)G

QJ15CO

4-J4-4• H43CO

I—IGO• HsQJ43CJ

QJ43H be

ing

posi

tive

.

TABLE

5.3.10

Depe

nden

ce

of ca

lcul

ated

ph

osph

orus

sh

ield

ings

and

ch

emic

al

shifts

(ppm)

of PH

(CH_

)0 up

on

o

4-)$2cd4-JWcooo•H

4-4oQJ i—l 0J • H' O

LO o o m vo CTv <r in in <ro o o co r—l 00 O CM o VO CTv in• <r • CM • • • • • 00o co o CTv • <r co o CTv VO00 in l—l in o i VO i 00 •O'v coi CO1 m <T

in o o o VO o \ CTv in <r 00 CTvo o o CO 1—1 00 o i—1 o 00 in• <r • CM • • • • • 00o CO o CTv • <T CO o CTv vom r—l in o 1 VO i 00 «

o \ COi co1 in <r

vO o o r VO O m o CM CMo o o co o 00 i—l i—i o 00 00 VO<r • CM • • • • • 00o CO o CTv • <r co o CTv VOco LO 1—1 in o i VO i 00 •o\ COi coI in <r

vO o o CM VO o i—i in CM o CTvo o o CO O 00 i—i i—i o CTv VO• • <r . CM • • • • • 004_) o CO o CTv • <r co o CTv VO t'-

{2 CM in ■—i in o i VO i 00 •cd o \ co CO in4-> i 1COS3o o CTv 00 VO o 00 in in oa o o o CM m 00 r—l VO o CM co CTv• <t • CM • • • • • 00a o CO o 00 • <r CM o o VO• H rH in |—1 in o i VO i CTv •d o \ co CO in <r4-J i 1oQJiH o \ o 00 CTv vo r—l o in in r» 00QJ o o o CM O 00 i—l CM o 00 1—1•H • <f • CM • • • • • CTvp vo co o 00 • <r CM o o m rin 1—1 in o i VO i CTv •Ov COi CO1 in <r

rH o i—i in i—i CM in <r 00 coo i—* o CM in 00 i—i VO o co CM in• <r . CM • • • • • CTvCO o • <r 1—1 o i—i in r.m i—i in o i VO i CTv •CTv coi CO1 in <r

CO o <r r- in <r i—l in CTv CO 00o r-l o CM 00 00 i—i o o o in in<r • CM • • • • • oCM CO o in • <r o o co CO 00in ■—i in o i vo i CTv •CTv COi CO1 in <r

i—1 o 00 VO <r o r—l m CM oo co o r-l r—l 00 CM o VO o VO• <r « CM • • • • • CMfH CO o CM • <r VO o VO o 00m i—i in o i in i CTv •CTv COi COi . m <r

/~N no ao on cu l—l ■a r-l r—N I—1 l—la S " \ CO 1 CO 1 O cd cdo rH /\ cu /\ ' O O H 4-> +ji—i (2 CO CO CO CO r—l C2 o <o ov-/ 1 i V_ 4-J 4-1Td TO cu cu v_>■Q t) "0 \/ "D CT

T3 CM(2 /vCOo 32CU oeO 32O P-i

O$2Q)2a 1CU{-Imucu42GO•H42O4-4

CO■U4-i•H4 2COoi—i

QJi—l3O0)i—lo6•aaj4Jcdl—loco•HQJ4Jo4->

4->OQJCUCOa)u4 24-J•H&T3QJ4-JMOCUajMQJuCO4-J4-1•r-l42cocdu•HeQJ42OQJ42H

oj>• H 4-J• H CO OGO(2• HOJ43

TABLE

5.3.

11 De

pend

ence

of

calc

ulat

ed

phos

phor

us

shie

ldin

gs

and

chem

ical

shifts

(ppm)

of PF

upon

pddpcodooo•Hppo<U

x)

o ON <r VO i—i <r 00 <r CMo o o CM m o o VO in vO VO i—1• • in a cn a a a a ao I—I o a CM a CO o o ON in <r00 in i 1—1 vO o 1 1 i—i a

CT\ cni

cni

m <r

i—i r~ ON 00 VO 00 vo <r CM o CMo o o CM CM o o cn in o cn CM• in a cn a a a a ao t—1 o a CM a 00 o o o in <rin 1 1— 1 VO o 1 1 00 i—i a

ON cni

cni

m <r

i— i ON i— i VO 00 00 <r vo VO ONo o o CM CM O o CM in r—i i—l CM0 m a cn a a a a ao i—i o a CM a 00 o o o m <r00 m 1 1—1 VO o 1 1 00 i—i a

ON cnI

cni

in <r

CM f"- 00 o vo ON 00 <r CM o cno O o CM 00 o O 00 in <r ON <rin a cn a • a a a

o i—1 o a l—l a 00 ON o o <rd CN in 1 1—1 VO o 1 VO 1 00 i—i a

d ON cn cn in <rP i iCOdo in 00 VO ON VO <r <r 00 <ro o o o CM 1—1 o o CM m i—i i—i 00• a in a cn a a a a ao o i—i o a r—l • 00 ON o l—l <r <rl“4 m 1 1— 1 VO o 1 VO 1 00 i—i a

p ON cn cn in <rp I iod| !■ j ON r. in ON in o ON <r o CM ON<D o o o CM ON o 1—1 O m cn O cn•H a a in a cn a a a a a 00

VO 1—1 o a ON a 00 00 o CM cn <rt-H m 1 1—1 in o 1 VO 1 00 i—i a

ON cni

cnI

in

cn cn in in cn 00 <r cn ON 00o i—i o CM 00 o i—i ON m in oa in a cn a a a a a ONi—i o a 00 a 00 VO o cn l—l <rin 1 1— 1 in o 1 VO 1 00 l—l a

ON cni

cnI

in <f

VO o cn r. VO in in VOo CM o i—l ON o r—l o in <r 00 I—la a in a cn a a a a a l—l

CM 1— 1 o a <r a 00 cn o inin 1 r—l m o 1 VO r oo •ON cn

icni

in <r

in 00 cn l—l o VO VO CM o CMo m o o CM o cn in in cn o cna a m a cn a a a a a m

r H 1—1 o a • 00 in o in o inm 1 l—l <r o 1 in 1 ON a

ON cni

cni

in <r/ \

o oo o Na i—i XJ r—l /— \ 1—1 r—lo s~\ cn i cn 1 o d do 1— 1 / \ a / \ ■d o H P P1— 1 d cn cn cn cn 1—1 d O <o O

i i v_^ P PX* xi p a P pf V ^XD "D \ / \/ h > h> t) Xd zr

C*- cnc a

aEC

o

oda;do *dPchPd•dCuO•H&opcopm•HCO

orH

3OdOgxfdpcdi—ioCO•Hd.CPoppodaCOdp

.ap•HtsX}dpPoadPdpdcoPm•H,dCOI—Ido•pgdX iod&H iei

ng po

siti

ve.

TABLE

5.3.12

Depe

nden

ce

of ca

lcul

ated

ph

osph

orus

sh

ield

ings

and

ch

emic

al

shifts

(ppm)

of OPF»

upon

4-1anJ4-1cocooo•Hu4-1 CJ QJ i—I 0) •H Q

CTv rH in CO m 00 rH t— 1 VO O VOo 00 CO <r co CN in CO 00 oVO • CO • • • • • coo o o co CO t—i CN o o00 <rCTv i i—1 i—i tH 1

o CNI COtH1

i rH00 1 <r

CN i—i in CO m CO vO rH in CTv ino CTv CO <r CO co r- in co COvo • co • • • • • coo r o o • co CO i—i CN O o

<r <rCTv I rH tHi—1 1

o CN1

COtH1

I tH00

1 <r

CO pH <r co in t—i CO 1 uoo CTv CO r- <r CO CO r-' in co uoVO • CO • • • • • coo o o • co CO i—i CN o o00 <r

CTv 1 tH i—i tH 1

o CN1

COrH1

i i— 100

i <T

vO pH CO in in rH r i—i CN VO <ro CTv CO r <r CO <r 00 in CO r-s (TvVO • CO • • • • • COo O o co CO t—i CN o OCM <rCTv 1 tH |H

tHIo CN

1COrH1

i rH00

i <r

ui•s

4-1P3CtJ4-1WcooO•HM 4-1 O QJ i—I QJ •H d

<r t—i CTv CTv <r co CN tH CO rHo o co VO <r co uo O UO CN VO t—1

VO • co • • • • • UOo 00 o o • co <r i—1 CN o OH <r i 1—1 rH o CN CO 1 rH 1 •

OV rH1

1 i—iI 00 <r

VO i— 1 CO CTv CO rH 00 tH VO o r.o 1—1 co vo <r co l"- tH U0 tH VO VO

VO m CO • • • • • VOVO 00 o o • co <r t— 1 C N o o

i I—1 1— 1 o C N CO 1 tH 1 •CTv tH 1 rH 00 <r

dP33Oa£oo

o i— i <T tH o tH VO o CNo co co in in CO 00 <r U0 o UO VO

• VO • CO • • • • • 0000 o o • CO <r tH CN o o<rCTv

1 i—i rHi—1 1

o CN1 COrHI

1 tH00 1 <r

co i— 1 VO tH O rH in oo CO CO UO CN in o in 00 CN o

VO • co • • • • * inCN. 00 o o <T UO tH tH o tH<T I i— 1 1—1 o CN CO 1 i— 1 i •

CTv t— 1 1 tH 00

I"- CN i—1 CTv <r CTv vo o 'do uo CO (Tv CO i—1 00 tH <r in o co

. UO • CO • • • • • voi CTv O o • uo vO tH tH o CN<r 1 t—1 tH o CN CO 1 ■—1 •

CTv t—1 1 i—1 00 <r

s~\a Oo O r~\a rH d i—1 tHCJ ✓“N CO 1 co 1d CJ n5o rH Pv /\ o H 4-1t—1 a CO CO CO CO tH 3 Ov_y 1 i s—-< 4-1

d d a u P* aId Xd \y ID \y fD "D ID

<onJ

4-1o4-1v ^cr

coPmPmO

►>1QJO4QJU4-4UQJ,CtuO•Ho4-1

CO4-1m•flco

iiV4>

QJrH3OQJi—IO£doj4-1rtrHOW•HQJ4-1

O4-1

4-1OOJCXCQQJ

,34-1•H&dQJ4->HOo,cuuQJHnJco4-14-i•H,03CO

I—I cdo• H£QJ,3oQJ,3H ie

mg

posi

tive

.

5.4 Conclusions.

For second-row element nuclear shielding calculations, the

agreement with experiment is much less satisfactory than for those

of first-row elements. This is most probably a reflection of the

influence of the parameterizations used. The parameters employed

in the present work for second-row element nuclear shielding calculations

are taken from the CNDO/2 m e t h o d ^ ^ which is used for the study of

ground state properties such as charge distribution and dipole moment

etc. Nuclear shielding is a second-order molecular property, it depends

upon a satisfactory estimate of the excited electronic states. The

satisfaction in the shielding calculations in order to get a good

correlation between the calculated and experimental data of chemical

shifts is not a feature of Pople's GIAO-MO-SOS method only, but the

eigenfunctions and eigenvalues which depend upon the choices of the

parameterizations. This is confirmed by comparison of the results from

CNDO/2, INDO, CNDO/S and INDO/S calculations of the first-row elements(40,41,

As mentioned before, the calculated nuclear shielding results are-3 -3obtained as a function of the values of <.r >_ and <r The value3p 3d

-3 -3of <r > 2^ is smaller than that of < r for 5 times (Appendix F ) .

This will cause some effects in the prediction of the influence of any-33d orbital contribution. Probably, the variation of <r > 3 3 * independent

-3from <r should be introduced. This may improve the prediction of

the 3d orbital influence in the total shielding and correlation between

the calculated and experimental values of chemical shifts.

If we can solve the problem of the too small calculated values of

<r ^>3p when compared to the experimental d ata^"^ ^-59) dependence

of < r ^>3^ on <r ^>3p> probably we can reach a possible mechanism to

explain the phenomenon in the experimental observations, in which the back donation has been both p r o p o s e d ^ ^ ^ and opposed^^"^ .

CHAPTER SIX

CALCULATIONS OF SHIELDING FOR SOME TRANSITION METALS


So far most of the theoretical work on the shielding of heavy

nuclei has been performed by using semi-empirical calculations such

» of t:(194)

as those of Griffith and O r g e l ^ ^ ^ for the first series of transition

metal elements and those for platinum by Pesek and Mason

Generally, heavy nuclei have a large range of chemical shifts,4 dabout 10 ppm, the diamagnetic contribution, d , is often assumed to

be not very different from the free-atom value, so that the chemical

shift variations are ascribed solely to changes of the paramagnetic

contribution, P . For the first-series of transition metal elements,

Beach and G r a y ^ ^ \ Fenske and D e K o c k ^ ^ ^ , N a k a n o ^ ^ ^ , Freeman,(191) (192) (19cMurray and Richards , Kamimura and Yamasaki, Yajima and Fujiwara

(T Rfi)based upon the semi-empirical calculations of Griffith and Orgel ,-3considered only 3d orbitals and fixed the value of <r as from the

^ (188-190) , _ „ _ _ . _ (191-193)free atom for vanadium and from Co_0. for cobalt3 4(194)calculations. Pesek and Mason have performed some shielding(194)calculations of platinum. They considered only the paramagnetic

-3contribution from 5d orbitals. The value of <r > _ J was treated as a5dconstant for all of the molecules considered. The transition energy

was treated as a constant value for the platinum calculations. For

the first-series transition metal elements, Vanadium and cobalt, the

transition energy used in the calculations of the paramagnetic contri

bution of the shielding for each compound was based upon the experimental (188—193)data , i.e. the wavelengths of the first absorption band.

In the present work, we performed calculations of nuclear shielding

by using Pople's GIA0-M0-S0S method, including 4p electron contribution-3to the paramagnetic contribution and varying the values of <r >^d and

-3<r > for vanadium and cobalt. In the case of platinum, we include

a 6p electron contribution to the paramagnetic term and varying the-3 -3 .values of < r a n d < r by using the equations given m Appendix F

and include the transition energies calculated from equation (3.23).

6.2 Vanadium shieldings.

6 .2.1 Introduction.

Vanadium-51 is a favourable nucleus for observations because of

its natural abundance of 99.76% and its high detection sensitivity^^"^.

The vanadium nucleus has spin = and thus a quadrupolemoment hence its

signal widths are highly sensitive to electric field gradients at the(188) (189)nucleus. Beach and Gray and Fenske and DeKock applied MO

theory to estimate the chemical shifts in Vanadium compounds by using(188 189)equation (3.14). They , showed that the vanadium shielding

calculations are understood in terms of the variations of molecular

parameters the energy separation of the highest occupied and lowest

unoccupied MO*s (A e ), an orbital reduction factor (k* ) and the value -3of < r > , in turn, these parameters are related to the paramagnetic

term, c> , of the overall shielding, Cj,tota-^ fry equation (6 .1 )

.total ,d 1 -3. , f2d = d Const. <r >3(J- k' (6.1)

This equation has been used by N a k a n o ^ ^ ^ for vanadium complexes on

the basis of SCF-MO considerations. According N a k a n o ^ ^ ^ , the value-3 -3of <r > 0 , = 2.0755 a.u. for the free-atom was used in his calculations. 3d

-3Generally, the molecular value of < r > will be different from

that in the free-atom and it will vary from compound to compound. There

fore, the calculations which are reported in section 6.2.3 are primarily

directed towards an explanation of the relative shielding of vanadium

bonded to various ligands. Nuclear shieldings are calculated by INDO

parameterization scheme. The method of obtaining the conformations

used in the shielding calculations are discussed in section 6 .2 .2 .

6.2.2 Molecular conformations used in the shielding calculations.

The molecular conformations used were obtained from standard

bond lengths and a n g l e s ^ ^ ’^ ^ , standard conformation data, or by

analogy with similar types of compounds. Often combinations of these

procedures were needed in order to calculate the coordinates. The

conformations used are discussed below.

VOCl^ were initially obtained by Palmer^"^^. The V-0 bond

length is 1.56 A°, V-Cl is 2.12 A° and angle C1VC1 is 111.2°.

~3 (198)(VO ) was obtained by Qurashi and Barns . The conformation

was assumed to be tetrahedral with the V-0 bond length is 1.86 A°.

+1 (199)(VO^) was obtained by Anderson . The conformation was

assumed to be linear with V-0 bond length is 2.03 A°.

+1((Cp-C,_H,_)2V(C0)2) , which has been determined by Anderson

et a l ^ ^ \ was initially assumed to be standard configuration for

Cp-V(CO)^. The V-C bond length for (Cp-C^H^part is 2.268 A° and

V-C bond length for V(C0)^ part is 1.97 A° and C-0 is 1.13 A°.

(V(CN)^) \ which has been determined by Levenson and T o w n s ^ ^ \

was initially assumed to be of standard configuration for V(CN) part of

(Cp-V(CO)^(CN)) \ The V-C bond length is 2.1457 A° and C-N bond length

is 1.450 A°. The configuration for Cp-V(CO)^ part was assumed by using

the conformation of Cp-V(CO)^.

The conformation of (V(CO)^(PF^)) ^ was determined by analogy

with the crystal structure of (Cp-V(CO)2(Ph2P(CH3 )2PPh2)) which was

determined by Rehder et a l ^ ° ^ . The bond lengths of V-P is 2.443 A°,


In the present work, the results of vanadium shieldings have

been calculated for a variety of compounds. The results of chemical

shifts, with respect to VOCl^, are compared with the experimental £224 255)data ’ and other theoretical calculations where available.

The results of vanadium shieldings and chemical shifts are reported

in Table 6.2.1. Also in order to assess the importance of the non-local-3 -3contributions and the values of < r , and <r >. , we have included3d 4p

them in Table 6.2.1 for further discussion.

However, from Table 6.2.1, the calculated shieldings and chemical

shifts of vanadium compounds do not agree very well with the experimental

data, it is clear that the dominant change of the shieldings is due to

the local paramagnetic contribution. The calculated values of the

diamagnetic contribution, with an average value of 1752.73 ppm, are in(203)agreement with that for the neutral atom obtained by Dickinson with

the Hartree-Fock approximation. The diamagnetic contribution, which

consists of local and non-local t e r m s ^ ^ ^ , is practically invariant to

change in the coordination of the vanadium. This also has been shown on

the basis of SCF-MO calculations for some vanadium c o m p o u n d s 205).

The variation in the local diamagnetic term, cj^CIoc) is within 8 ppm,

i.e. 0.5% for the vanadium environments considered, whereas changes in

the local paramagnetic term for 3d orbitals, (j^(3d-loc.), account almost

entirely for the vanadium shielding differences. The value of the local

paramagnetic term for 4p orbitals, d^(4p-loc.), is small when compared

to the local paramagnetic contribution from 3d orbitals. The value of

c/Vnon-loc.) is found to be negligible in all cases. In general, the

TABL

E 6.

2.1

The

resu

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and

chem

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comp

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wi

th

expe

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data

for

vana

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-51

compounds^^* 2

55)^

rpPaX ,-

536 in

vaini•v o

<r00inrH1•v

d o o o o <r ov-/ co <r vo <r CM<o m

iini

[V1

i—! 1

ini—i i

o rrl o 00 in o\a o VO OV o in 00c$ • • • • • •O o oo VO <r o p v

00 1—1 vo ov <r<o I I—I

1i iH

1✓p vo p <r <r o inP CO <r p ov <r CMo • • • • • •4-) CF\ 00 vo oo o pv

M3 in 00 00 vO 1—1r> in VO vo vo vo p.

<—( «—i rH 1—1 i—1 I—1/P in(—1 <f p <r CM o\p uo CM o i—1 CM CM

> • • • • • •%

CM <r o i—I CO o

/P inO CM CM m t—1 ooo 00 00 i—i VO H HfH • • • • • •v_x 00 vO OV CM 00 ooD, 00 Ov VO CM Ov COt) r—1

11 1 <—1

11 1

O <f CM in 00 00H VO Ov 00 P <r OO1 r—1 i—1 vo OV CMa • • • • • •<r O i—1 i—i 1—1 o oa 1 1 i 1 1 1

a<r rH VO H P VO rH/\ O o i—1 o O Oco O o O o O Ot • • • • • •

O/ sO

o o o o o o

t—( vO CO vo co ov CM1 VO 1—1 <* 1—I i—i OvTd • • • • • •co 00 in p 1—1 P CMv-/ 00 o\ vo CM CSV 00Pf\0 I—1

1i 1 rH

11 1

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r> r—t i—i

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<rCO /p ✓—\ CM O ✓p

i—1 <5 CM ✓P o•d O O O 53 co oP o > > O /—Ndoa6oo

> v-/ -V_><r

s~\Oo>

oCJ>1ao

>1ao

No. I-l CM 00 in vO

in p 00 o o avo CO o p CM voin in VO vo P oviH1

•HI

ai

ai

ai

ai

VO m in <r CO ovo co OV CM in <r• • « * • •co in vo vo in ooPv1 i

UOin

uo a VOI

CM CM a a CM <ro\ CM OV VO CO CO •• • • • • 0 dP. in CM oo <r 00 >vf rH a i—i UO CO • HVO VO o uo in vo •Ui—I rH a a a a •HCO

Oco «H p OV o Ov aCM 00 <r VO in <r• • * • • • CUo O 00 CM CM CM

a

o VO OV VO <r m op. CO o CM o co P• . . • • • Q)Pv i—1 VO P o o do vt <r CO o CM d 1(—I rH p CM CM a 0)1 1 i 1 1 i Pao rH o o o CM ,P00 VO o o p 00 CuO<r CO <r o o <r •H. • • • • • Xio O o o o oi I i 1 1 i o

•pCOp

o P o 1—1 a p a1—1 o a o o o *Ho o o o o o X• ■ ■ • • • COo o o o o o •v

aa

CM OV o VO p p oCM ov p o ov 00 om « • . . >Pv o m p ov ov

o <r <r CO ov a o1—1 1

i—iio

pi

CM1

ai

i—ii

ppo

CO CO co P in p dCM co CM a co a00 co co in <r co COm • . • • • d

o o o o o o p

p00 rH 1—1 a U0 ov •Ho O a a o o« « . • » •o o

1o o o

1o to

dCOCO

H p CM p a a d00 CM <r 00 OV a p• • • • • • ain VO uo p i—i VO Xin in m UO uo dPv p p p p prH i—i

CM00

aavy

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a

1—1 ■—i a

, 6,

are

/P CM a 1 i CO00 /—\ *. tVp p

a O CM f-s /—s aa o /-s co co • Hv_y O 53 a Xoo > O r3 O a CO/P 1 v / v_/ v-/o a > U0 in UO i—io a 1 /P r\ ✓P nJv_ i a o O O O> w a o CJ> O • H1 P i v-/ v_/ £a nJ w > > > do P

H ov_x Xo

p 00 Ov Oi—1 11 12 *

ppm

o> O

ga ,a,

wTd33oa6oo

n O

OOOOCM

CO.34-i•H,3COaJo•Hgcu,3oTdcu4Jasi—i3 O i—I3a

•HTSCt)33><ugoCO

4-iOCO ■P 4-4 • Hr3CO

I—IajO•HgCU,3o

•HTdrt3aS>H34->3cug•H3cuaXcucu,34-J4-1 O .4-Jo

r—IPM

Tdo,3-3CUgoQS3CU,34-J

£Tdcu4-JaSi—i3Oi-1cSoCOcu3i—IccJ>cu,34-J

4-J CO 3

• 3340aS

i—I34-J3cug•r*43CUa oXcu

CM

33340• r“t

shielding trends are not well reproduced by the calculated results

presented in Table 6.2.1. The trends are good for just a particular

group but not overall.

By using Pople's GIAO-MO-SOS method as described in Chapter 3,

the shielding differences in the compounds are almost entirely accounted

for by the changes in the local paramagnetic term which depends upon-3 -3the valence shell 3d and 4p orbitals, < r >_, and <r >, , and transition3d 4p

energies. Since the nuclear shielding depends upon a satisfactiry -3 -3estimate of <r a n d < r > and transition energies in the paramag

netic contribution. Hence it seems probable, at least in part, that

the lack of success in calculating this NMR property by the INDO para

meterization scheme could be attributed to the MO parameter sets in the

parameterization for all of the series of compounds considered.

Table 6.2.2 shows the variation of the average weighted value of

transition energies which is obtained from each compound by weighting

the energy in proportion to the size of the corresponding contribution

to the local paramagnetic term. Closer consideration of the various

transition energies for each compound we found that various transition

energies contribute to the paramagnetic term of the vanadium shielding

significantly. Therefore a linear correlation between the vanadium

shielding and average weighted value of the transition energies is not

anticipated in the present work. Also, the large variation observed-3m these average weighted value of transition energies and <r > ^ and

-3<r imply that it is unreasonable to use an average excitation energy-3 -3and the fixed values of < r >^ and <r > m interpreting the vanadium

shieldings and chemical shifts of various species considered in the

present study.

-3From Table 6.2.1, although we can see that the variation of < r > ^

Table 6.2.2 The average weighted value of transition energies of

compounds in Table 6.2.1.

No. compound calculated transition energy (e.v.)

1 V0C13 16.21937

2 (vV 3 7.47133

3 (v o / 1 18.75974

4 (v(co)4(cn)2)2~a 22.61926

5 (Cp-V(C0)3(CN))"1 24.94093

6 Cp-V(C0)4 24.15571

7 Cp-V(C0)3(PF3) 22.80882

8 Trans-Cp-V(CO)2(PF3 )2 24.30367

9 Cis -Cp-V(C0)2(PF3 )2 24.04369

10 (v(co)5(nh3 ))_1 22.52678

11 (v(co)5(cn))"1 22.49583

12 Cv(C0 )5(pf3 ))_1 22.80882

-3and <r >, for the vanadium atom vary from compound to compound 4P-3depending upon the type of the substituent. The values of <r

are too small, by about 5 to 6 times, when compared to the values for-3 (190) -3the corresponding free-atom values, 2.076 a.u. and 2.565 a.u. ,

the Relativistic Dirac-Fock expectation value^"*^. The values of-3 -3<r > which depend upon the values of <r >^ are too small. These

-3 -3small values of <r >_ , and <r >. caused the calculated values of3d 4pthe paramagnetic contributions of the 3d and 4p orbitals to be too small.

-3For compounds number 1 to 3, the calculated values of <r are about

20% of the free-atom value^"^^ and the Relativistic Dirac-Fock expecta-(158) —3tion value . If we include the factor of the small values of <r

-3a n d < r in the shielding calculations for compounds number 1 to 3

which have the variation of the total shielding less than that obtained

experimentally by about 80%, we see that we can have approximately the

same magnitude of the variation of shielding calculations as in the

experimental data. For compounds number 4 to 6 , which are in different

chemical environments from compounds number 1 to 3, eventhough including-3 -3the factor of the small values of < r , and < r >. m the shielding3d 4p

calculations but we still can not reproduce the amount of magnitude of

the variation of shielding calculations as the same in the experimental

data. This is probably, at least in part, due to the over-estimate of

the values of the transition energies in the paramagnetic contribution

in compounds number 4 to 6 . For compounds number 10 to 12 which are

also in different chemical environments from compounds number 1 to 3

and 4 to 6 , show the same trend of chemical shifts as the experimental

data but the magnitude of the variation of chemical shifts is also too

small when compared to the experimental data. This can probably be

explained as for compounds number 4 to 6 but not for compounds number

7 to 9 which the trend is in the opposite direction to the experimental

data. So from Table 6.2.1, the overall agreement of the variation of

the magnitude of the calculated shieldings and the experimental chemical

shifts is not good.

6.3 Cobalt shieldings.

6.3.1 Introduction.

59Cobalt occurs in nature only as the isotope Co . Cobalt-59 has

recieved considerable attention from the early days of NMR spectroscopy

and is now a very useful method of investigating diamagnetic cobalt7 —28compounds. It has nuclear spin I = and a quadrupolemoment of 0.4x10

—2 (195)cm . The quadrupolemoment makes cobalt-59 linewidths sensitive

to electric field gradients at cobalt and hence to the symmetry about

the cobalt atom.

The interpretation of cobalt chemical shifts has been first

performed by using semi-empirical calculations by Griffith and O r g e l ^ ^ ^ .

T h e y ^ * ^ showed that for octahedral complexes of cobalt(lll), the

paramagnetic term in the shielding could be calculated semi-empirically

using wavefunctions derived from a crystal field description of the

complex to evaluate the orbital angular momentum induced by the magnetic

field and spectroscopically determine values of the electronic excitation

energy forAE . The total shielding is evaluated and obtained from

equation (6.2.1). Agreement between the calculated and observed chemical

shift values of Proctor and Y u ^ ^ ^ is reasonable. Freeman, Murray and

R i c h a r d s ^ ^ ^ extended this work to a greater number of cobalt complexes

(14 compounds) and showed that a linear relation holds between the NMR

frequencies and the wavelengths of the first absorption band which they(192) -3assume corresponds toAE. According to Kamimura , < r > is found

-3 3+to be 5.6 a.u. from the observed value of the chemical shift of Co-3m Co00. . This empirical value of < r is about 20% reduced from3 4 3d

-3the free-ion value of 6.7 a.u. which has been calculated using Hartree-3 + f 207^Fock wavefunctions for a 3d electron of Co ion . Therefore the

3f3d function of the Co ion in c°3°^ i-s radially expanded relative to

that of the free ion.

(193)However, it has been found that the calculated shielding

values are not linearly related to the first absorption band values for

all compounds. Hence it is suggested that the simple treatment of

Griffith and O r g e l ^ ^ ^ is not completely satisfactory-, and consequently

the molecular orbital treatment in which electrons are considered to

move in the combined field of all the nuclei in the molecular has been

applied to the whole system of cobalt complexes in the present work.

Shieldings are calculated by INDO parameters. The chemical_3

shifts of cobalt, refered to (Co(CN)^) , have been obtained for 16

compounds. The method of obtaining the conformations used in the

shielding calculations are discussed in section 6.3.2.


The molecular conformations used in the shielding calculations

were obtained from the standard bond lengths and a n g l e s ^ ^ * ' ^ ^ ,

standard configuration data, or by analogy with similar types of

compounds. Often combinations of these procedures are needed in

order to calculate the conformations.

The crystal structure of (Co(NH.)^(Co(CN),) has been determined3 6 6( 208 )by three-dimentional x-ray data by Iwata and Saito . The coordina

tion around the metal atoms is regular octahedral. The Co-N distance

is 1.972 and Co-C is 1.894 A°. The C-N distance is 1.572 A°. The NH3(47)group conformation was assumed to be of standard conformation

The crystal structure of (Co(N02 )6“3 has been determined by ( 209)Driel and Verweel . The coordination around the metal atom is

regular octahedral. The Co-N distance is 2.03 A° and N-0 is 1.1 A°.. (47)The NO2 group conformation was assumed to be of standard conformation

The crystal structure of Trans-bis(dimethyl glyoximato) dimer' (210) cobalt(III) Bromide which has been determined by Heeg and Elder ,

+1was initially assumed to be of standard configuration for (Co(DMGH)2(NHg^2

The Co-N bond length of the NH^ group is 1.960 A° and the DMGH group it

is 1.893 A°. The conformation of DMGH part was employed from Heeg and

Elder(210) ^ group conformation was assumed to be of standardt_- (47)configuration

The crystal structure of (Co(CNC-H_)_)(C10.) which has been6 5 5 4determined by Brown, Greig and R a y m o n d ^ ^ ^ was initially assumed to

+1be of standard configuration for Co(CNCH3) part of (Co(DMGH)2(CHg)(CNCH2))

The Co-C bond distance is 1.850 A°, C-N is 1.16 A° and N-C is 1.354 A°.

The configuration for Co(DMGH)2 part was assumed by using the conforma

tion of (Co(DMGH)2(NH3)2)+ 1 .


The cobalt shieldings of some cobalt(IIl) compounds are summarized

in Table 6.3.1. Since the range of the chemical shifts of cobalt(IIl)4 .compounds is of order of 10 ppm, one is inclined to believe that the

main contribution to the chemical shift is the paramagnetic term, while

the diamagnetic contribution plays only a minor role in affecting the

total shielding. While the absolute magnitude of the diamagnetic termin

is substantial withvan overall average value of 2078.43 ppm which is(203)good agreement with those obtained by Dickinson in the Hartree-Fock

approximation, the variation in the diamagnetic term from compound to

compound is negligible in comparison with the large variation of the

total shielding calculations. This arises from the fact that the diamag

netic term is largely determined by the core electrons whose wavefunctions

are practically uninfluenced by chemical bonding, and calculations for

atoms from the first-row of the periodic table when the core electron

comprise a large fraction of the total show this to be the case. The

diamagnetic and paramagnetic shielding contributions of some cobalt(IIl)

compounds, have been calculated using equations (3.29) to (3.32) described

in Chapter 3.

In Table 6.3.1, a number of different compounds used for referencing_ 3

cobalt(III) shielding data are listed. (Co(CN)^) is the reference which

has been adopted by the greatest number of workers in the field. The

agreement between the calculated shielding differences and the experimental

data is demonstrated in Figure 6.3.1. The overall agreement with a standard

deviation of 1834.53 ppm and correlation coefficient of 0.83. The slightly

poor correlation coefficient and the large value of the standard deviation

arise, at least in part, from the fact that the conformations employed for

these compounds in the present calculations do not represent their real

conformations in solution. The experimental chemical shifts are spread

over the range of 0 to 10000 ppm, whereas the corresponding calculated

shielding differences are concentrated between the range of 0 to 2400 ppm.-3This is probably, at least in part, due to the too small values o f < r > ^

-3and <r >, as shown m Table 6.3.1.4p

Consideration of the results given in Table 6.3.1 reveals that our

calculations have reproduced some features of the shieldings of cobalt(III)

compounds. In agreement with experimental data, the successive replacement

of substituents results in a decrease in total shielding (i.e. chemical

shifts become more positive) when one of the cyanide groups in the hexa-

cyanide cobalt(III) compounds is replaced by a different group. It is

to be noted that, for compounds number 1 to 4, the agreement between the

ininc m

<rCMCM

COTdG3o(X60 ao nin1+JrHClJXOoGo

<4-1

nJ4-3aJTd

nJ4-3g0)6•Hgci)cuXCl).C4-3•H>TdCDClrtCU&Oo

4->afxj<13

c_y<o

aJOv-/O

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oo

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calculated shielding differences and those from experimental data is

fairly good.

As mentioned before, the shielding differences of compounds are

almost entirely accounted for by changes in the paramagnetic contribution

which, as shown in equation (3.35), depends upon the valence shell 3d-3 -3and 4p electron contributions, <r a n d < r > ^ , and transition energies.

From Table 6.3.1, the valence shell 3d and 4p orbital contributions and -3 -3<r > 2^ a n d < r > vary from compound to compound. The increase in the

valence shell 3d and 4p orbital contributions tend to increase the values

of the paramagnetic term. This occurs with simultaneous increments in -3 -3<r >OJ and < r >. because of a net contraction of the 3d and 4p electron 3d 4p

orbitals. Thus the total shielding decreases.

Table 6.3.3 shows the variation of the average weighted value of

transition energies which is obtained from each compound by weighting the

energy in proportion to the size of the corresponding contribution to

local paramagnetic term for compounds considered in Table 6.3.1. Comparison

between the calculated average weighted value of transition energies and(213-217)the experimental first absorption band which they assumed

correspond t o A E are shown in Figure 6.3.3. From a close consideration

of the various transitions which contribute to the local paramagnetic term

for each compound, we find that various transitions make substantial

contributions to the cobalt shieldings. Therefore a linear correlation

between cobalt shieldings and the average weighted value of transition

energies is not anticipated for the cobalt environments considered in

the present work. Also the large variation observed in these average

weighted values of transition energies implies that it is unreasonable

to use an averaged transition energy in interpreting the cobalt shieldings

of the various species considered in the present work.


compounds in Table 6.3.1

No. compoundTransition energy (e.v.)

calculated experimental^

1 (Co(CN)6 )"3 45.90328 39.910058

2 (co(cn)5(no2 ))"3 25.99258 -

3 (Co(cn)5(h2o ))"2 25.49651 -

4 (co(cn)5(oh))“3 24.83833 -

5 (co(dmgh)2(ch3 )(cnch3 ))+1 31.95439 -

6 (co(dmgh)2(ch3)(h2o)) 32.16943 -

7 (co(dmgh)2(nh3)2 )+1 28.04940 -

8 Cis-(Co(NH3)4(N02)2)+1 23.72991 27.899857

9 (Co(N02)6 )"3 27.46000 25.656466

10 (Co(nh3)5(no2 ))+2 25.41298 27.069431

11 (Co(NH3 )6 )+3 26.92388 26.090271, 36. 5 6 3 5 6 3 ^

12 (co(nh3)5(h2o ))+3 27.42815 -

13 (Co(NH3)5(N)3 )+2 40.09328 -

14 (co(nh3 )5(no3 ))+2 26.03778 -

15 (co(nh3)5(co3 ))+1 24.99128 25.297028, 3 3 .675661(b)

16 (Co(nh3 )5ci)+2 26.30814 23.202369

a : from references number 213 to 217

b : from references number 218 to 219

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Figure 6.3.3 Plot of the experimental first absorption band values

against the average weighted value of transition energies

calculated by INDO method for some cobalt compounds.

1O

8O 10o 1115

O

16O

From Table 6.3.3, the predicted average weighted values of the

transition energies compare reasonably with the experimental first

absorption band and is shown in Figure 6.3.3. The slightly difference

between the calculated average weighted value of transition energies and, 2^2 21A 216)the first absorption band which Birada and Pujar ’ * , Dharmatti

and K a n e k a r ^ ^ ^ , and Martin and W h i t e ^ ^ ^ assumed corresponds to the

average transition energy is probably due to the conformations considered.

_3From Table 6.3.1, although we can see that the variation'of < r- 3and <r > for the cobalt atom from compound to compound, is more reasonable

than a fixed value for the various kind of compounds considered, the-3 -3calculated values of <r > and < r > for the cobalt atom are too small

-3 -3 (192) -3when compared to the value of <r >_. = 5.6 a.u. in Co00, , 6.7 a.u.3d 3 4(207) -3for the free-atom and 5.876 a.u. obtained by Relativistic Dirac-Fock

(158) —3 —3calculations . A small value of < r and < r > causes a small

value for the paramagnetic contribution and thus a small variation in the

total shielding when compared with those obtained experimentally.

Table 6.3.2 shows the calculated shieldings and chemical shifts

compared with the experimental chemical shifts by means of equation (6 .3 .1 )

cjP(expt.) = a tfP(3d-cont.)ca^ + b tfP(4p-cont)ca^

= <jP (3d-cont.) + (jP (4p-cont.) .. (6.3.1)Cell • Cell •

where (jP(3d-cont.)ca^ and cjP(4p-cont.) - are the calculated values of

the paramagnetic contributions for the 3d and 4p orbitals, respectively.

"a" ans "b" in equation (6.3.1) are the integral product of tfP(3d-cont.)

and (jP(4p-cont), respectively, which are treated as least-squares parameters.

The INDO shielding results for cobalt after multiple regression

with the experimental values are presented in Table 6.3.2. The agreement

between the calculated and experimental chemical shifts is demonstrated in

Figure 6.3.2, with standard deviation of 1859.19 ppm and correlation

coefficient 0.83. From the cobalt paramagnetic contributions after

multiple regression, we obtained values of "a" = 4.56 and "b" = 5.04,

respectively for cjP(3d-cont.) and (jP(4p-cont.) The larger values for the

paramagnetic contributions appear to be due to an increase in the values-3 -3 . -3 -3of < r a n d < r The increase m the values of < r a n d < r

-3obtained by multiple regression shows that the value of < r > of the-3 -3cobalt atom in (Co(CN),) increases from 1.07 a.u. , as shown in Tableo

-3 -36.3.1, to 4.64 a.u. which is in good agreement with 4.8 a.u. obtained

from experimental d a t a ^ ^ \

6.4 Platinum shieldings.

6.4.1 Introduction.

The only naturally occuring isotope of platinum with nuclear 1 195spin I ~ ^ Ft, with natural abundance 33.8%, magnetic moment

1.0398, magnetogyric ratio 5.7505 and relative sensitivity is-3 (220)9.94x10 . The sensitivity of platinum is low, compare it with

1 113 (221) 195H and Cd. Proctor and Yu reported the Pt chemical shift

resonance and confirmed that the moment is positive. There were two (222 223)separated reports ' illustrating the ability to determine the

platinum chemical shifts by double resonance methods. Latter Kidd andC 2 2 A )Goodfellow reviewed experimental platinum chemical shift data by

using Fourier transformation method.

(194)Pesek and Mason showed that the shielding of platinum,

according to Ramsey's t h e o r y ^ * c a n be expressed by the sum of

contributions, the largest of which are the diamagnetic term, c/*, and

the paramagnetic term, 0^, as shown in equation (3.14). T h e y ^ ^ ^

considered the paramagnetic term to arise from 5d orbitals only. The-3values of < r > a n d A E are treated as constants for all of the molecule

considered.

The calculations reported here are primarily directed towards

an explanation of the relative shielding of the platinum atom by considering-3the contribution of 6p, as well as 5d orbitals, the variation o f < r > ^

-3a n d < r > calculated from equations m Appendix F, and appropriate

transition energies by means of the Sum-Over-States (SOS) procedure,

equation (3.23). Chemical shieldings and chemical shifts are obtained for// n \

15 compounds by means of IND0/5R wavefunctions . The method of obtaining

the conformations used in the shielding calculations are discussed in

section 6.4.2.


The molecular conformations were obtained from standard bond

lengths and a n g l e s ^ ^ ’^ ^ , standard configuration data, or by analogy

with similar types of compounds. Often combinations of these procedures

were needed in order to calculate the coordinates of molecules consedered.

(PtCl^CC^H^)) \ an x-ray analysis of Zeise*s salt hydrate

K(PtCl0(C0H.) *H_0 was under taken by Wunderlich and M e l l o r ^ ^ ^ . It 3 2 4 2appears that the platinum and the three chlorine atoms are co-planar.

The Pt-Cl bond, which is trans to the expected position of the ethylene

is a little longer (2.38 A°) than the other two Pt-Cl bonds (2.33 A°).

The C-C separation is roughly 1.5 A°, the C-C axis is perpendicular to

the plane of the PtCl^. The crystal and molecular structure of Zeise*s

salt has been redetermined with a diffractometer data by Hamilton(226) (225)et al . I t was found as earlier .that the C-C bond is nearly

perpendicular to and symmetrical about the platinum coordination plane.

The C-C bond length (1.373 A°) is slightly longer than in free ethylene.

The methylene molecule deviates from planarity, the hybridization of3the carbon atoms tending slightly toward sp . The Pt-Cl bond trans

to ethylene is slightly longer than the two cis Pt-Cl bonds (2.357 A°

and 2.305 A°) which is the conformation used in the shielding calculation

of (PtClQ(C0H.))_ 1 .3 2 4_2(PtCl^) , the structure of K^PtCl^ has been determined by Mais

et a l ^ ^ \ The Pt-Cl distance is 2.316 A° after correlation for

thermal motion, in square planar complex of Pt(II) by crystallographic

data.

(PtClÎttKCH^^)) ^ was initially assumed from a standard

configuration of Pt(C2H^)NH(CHg)2Cl-2 which has been determined by ( 228")Alderman et al . The complex examined is unstable to x-ray, which

limited the work to two-dimentional studies only. The interatomic

distances of Pt-Cl are 2.30 and 2.33 A°, Pt-N is 2.02 A°. The N H C C H ^ ^

structure was introduced by using standard bond lengths and a n g l e s ^ 196) +2(Pt(NHg)^) , the structure consists of discrete planar of

( 2 2 9 )this ion was reported by Shandies et al in (Pt(NH_).)n(Reo0_(CN)o)3 4 2 2 3 8by crystallographic data. Tetra amine platinum(II) ion along the

z-axis are perpendicular to each other. The NH^ hydrogens are involved

in weak hydrogen bonding with the cyanide ion nitrogens and the terminal'

rhenium oxygens. The platinum atoms are each bonded in a square planar

to four amine groups with nearly equal Pt-N distances, 2.051 A°, the

NPtN angle, the only one not fixed by symmetry is 89.2°. The same

ion in (Pt(NH0).)(PtCl.) shows a Pt-N distance of 2.06 A° (230)^3 4 4(PtCl^COH^)) ^ was initially obtained FtCl^ part from (PtCl^) ^

and OH2 from standard bond lengths and a n g l e s ^ ^ * ^ ^ by assuming

Pt-0 bond length is 1.96 A° (231)^

(PtCl^(CO)) ^ was assumed to be planar and linear for trans-(232) -1Cl-Pt-C-0 . The PtClg part was obtained from (PtCl^) and C-0

part from standard bond lengths and a n g l e s ^ ^ * ^ ^ .—2 (233) 19(PtFr ) has been observed by Matwiyoff et al . The F6

NMR spectral splitting pattern to compounds of the type Mn(PtF^)m is_2indicative of non-equivalence of the Pt-F bonds in the anion (PtF^) ,

the four Pt-F bonds of the four equatorial F atoms should differ from_2the bonds of the two apical F atoms. The distortion in the (PtF^)

hexagonal is not caused by steric factors, but mainly by intrinsic

reasons, for which the partly covalent nature of the Pt-F bond is

r e s p o n s i b l e ^ " ^ . Wilhelm et a l ^ " ^ found from various complexes

of (PtF,.) ^ that the Pt-F distance varies from 1.88 to 1.90 A°.6The crystal structure consists of nearly square planar (Pt(CN)^)

/ 236 )groups was found stacked parallel, forming linear Pt atom chains

The Pt-Pt distance of 3.478 A° in the chain. The (Pt(CN)^) 2 groups-2are nearly square planar. The bond distances on (Pt(CN)^) group

are Pt-C 1.986 A° and C-N 1.159 A° with the bond angles PtCN 178.5°

in K_Pt(CN) *3H 0.2 4 2_2(Pt(CN),) is assumed to be of a standard hexagonal configuration 6

with bond distances of Pt-C 1.986 A° and C-N 1.159 A° and bond angle

PtCN 178.5^236,237\

_2(PtCl,) is assumed to be determined from a standard configura- 6(44) otion with Pt-Cl bond lengths of 2.29 A .

-2(PtCNO^)^) is assumed to be determined from a standard configu

r a t i o n ^ ^ with Pt-N, N-0 are 2.02 and 1.22 A°, and bond angles of PtNO

and 0N0 are 117° and 125°, respectively.


The outcome of the platinum shielding calculations are reported

in Table 6.4.1. An overall average value of 9458.62 ppm is found for

the local diamagnetic term. This is in agreement with the value(339)9395.58 ppm obtained from experimental data for the free-atom and

from a Hartree-Fock computation of the internal diamagnetic field for the3 (203)free-atom, 9x10 ppm

Only a few theoretical reports have appeared for platinum^ ^ . (194,240,241) (194,240,241)chemical shifts m the literature . They rely

on the average excitation energy approximation to evaluate the paramagnetic

contribution to the total shielding.

( A 2 )The IND0/5R parameterization scheme , which is parameterized

for the reproduction of bond lengths and angles, is shown to predict

dipolemoments and metal-ligand-bond force constants for platinum compounds

TABLE

6.4.1

The

resu

lts

of INDO

calc

ulat

ions

of

nuclear

shie

ldin

gs,t

f,

and

chem

ical

sh

ifts

, 6','

comp

ared

with

expe

rime

ntal

data

for

plat

inum

-195

co

mpou

nds^

.

pCl, I—I VOX OV VOQi o 1—1 <rw«o 1—1 ,—i

s~\ o VO in1—1 o t—1 mnJ • • •o o t-M<r av«o CM

1vO

CO CMP in r. oo • • •P <r VOov <r o■O co VO r^

00 00

H ,—i ,—iC o i—iv-/ • • •

a, o o ot> 1 i

OV co COo 00 00 00o • • •1—1 CO o co

VO CM ma, o 00

1—1 1

1 !—1 1

o I—1 OV l—li—1 in 00J OV VO Ova, • • •vO i—i 00

CO 1—1 1—1Pft> 1 1 1

aVO CO Ov r.A o i—i00 CM i—J CMI • • .PV o o o

/'“N CM r- ino OV i—i 001—1 * • •1 i—1 CM 00

X3 CO O COm o 00|—1 1 i—i1 I

m in o VOA 00 <r 00oo 00 00 00I • • «PV r—I 1—1 1—1

/~sr-M CM r.C co i—1 CMv^ • • •o o o'D 1 J 1

<r oO 00 OV COO • • •i—1 r—1 <r o

VO VO VOto <r <r O'X) crv Ov

i7~\/~so

OV

T? i CMCJ cm PC 1d /—s /~\o VO CO < 1CU i—l 1—1 1—1£ O c_> oo P P po CM PM PMv_/ v. v_>

No. I—1 CM CO

<r co 1—1 oCO o in l—lVO 00 o1—1 i—i i—i CM

<r CM o 0000 00 i. vO• • • •VO vO r^00 CM 00 co1—1 1

CO1

CM1

<ri

1—1 crv m<r CO CM CM• • • •<r <r m in00 CM 00 com VO 0000 CO 00 00

in in i—i 00in crv in oo• • • •o i—i o

io

o 00 o CO<j l—l co• • • •«—i co OV i—i00 'd- CO001 1 I

vo1

l—l Ov o 0000 CO o oOV 00 CM• ' • • •CM CM aCM1

1 I r~1 1

VO CO CMr>.i—i i—i i—i i—i• • • .o o o o

CM <r o CMVO CM ov i—l• • . »00 o I—I i—lin <r r^ CM00i 1 1

VO1

<r m<r CO CO co00 00 00 00• • • •l—l 1—1 ,—i i—i

I—1 VO in COl—l o o 1—1• • • •ol

o1

oi

oI

oo co coCO VO in 00• • • •m in m inVO VO vo vo<r <r <r vd-crv crv ovi

"V-sovii CM t—s

r-N /~ \ OfCO s~\

CO PC Oftc o KCO v^ CJ)

i o p~| '—^04 53 ar~\ v_> u<r CO CO

■—i i—1 l—l Vw>o o CJ Ofp p p 1—1PM PM PM ov- w '— / p

PM

<f LO v£)

00 l—l co <fVO CM r- CO m1— 1 vO O CMCM CM CM co co

i— 1 in CM CM OVOV i— i VO m« • • • •00 i— i i—i mCO i—i inCM1 i— iI <r

icoI CO

i

00 CM ov OV VO<r r—l o CO. « . • •VO 00 m ov COVO co o inVO in 0000 00 00 00 00m VO 00 VO <rin CM CO I-l co• • • « •oi

O1 i—ii

l— l coICO 1^ in CM <rCO 00 r—1 CO o• * • . •co 1— 1 OV 00 oov OV CM in r—l

i001 VO1 1 I-".1

<r vf VOCM OV mco r—l o r- CM• • • • •o 00 co <r co

l— l1

l— l1 r— l1 i l— l1

CM o CM OV VO<r r-i vo VOiH CM i—i i— l l— l• • • • •

O O o o o

m <r m in ovOV o in• . • • •CM co VO CO VO

00 r—l in OVI 00I VO1 I VO1

co >d- in oOV m co CO co00 oo 00 00• • • • •

i—i i— i 1—1 i—i i— i

i—i 1—1 inCM CM r—l o o• . • • •o o o1 o

1oi

in 00 CO CO oi—i <r 00 co 00• • • • •o o m vo VOVO VO VO vo<r <r <r <r vd-OV ov av

i

7“v

OVi

ov

■ + r - v co i

CM CM <r tc 7-V/—N /*—\ o ^■S<r CM 53 Os~\ CO o COC M CO V w v—/

O p- 1, co co Cfi—i r—l l— l'— ■> v_^ o O oP p p P pP M P M P M P m P Mv_> v_^ v_>

00 OV O i—I CM«—I i—I i—I

OV in vQ<r CM CM00 COco <r

I

r"- r. vOin 00 OV• « •VO COi—i <r 00coCM 1 I

o <r coo <r in• • •CO <r i—i00 <r 00vo r. i—iin 00 OV

vo 00 r-~»i—l o o• • •o o CM

O i—1 ino r—l in• • •l'- OV

OV CM voVO CMCO I 1ICO Ov OVOv O inl— l VO in

m • •o oCO i—i CMCM1 I 1

o CO in<r CO Hr—l co• • •i—i o o

r—l o Ov00 in OV* • •VO ov 00VO o <f<r CMco1

I 1

<rin oi—i 00 Ov• • *co i—i r—l

CO vo CMCM o i—l• • .O o O

1

i— 1 1—1 <rvo <r i— i• • •OV i—i OVr'- <rCO <r <rOv ov av

I lCM vVO

CM<r-N

<r i/ —N CM

!3 zO o v£v> V-V PmP p PPM PM PM

CO <r in1—1 i—i i—i

Chem

ical

sh

ifts

, 6

, are

ex

pres

sed

with

resp

ect

to (P

tClA

)2”,

shifts

to high

freq

uenc

y are

po

siti

ve.

experimentalchemicalshifts

ppm

x 10

O 13

012O11

0100 9

0 3

02

0 31 2- 1

calculated chemical shifts

ppm

Figure 6.4.1 Plot of the experimental platinum chemical shifts of some

platinum compounds against the values calculated by the

INDO method.

(42,244)considered . The method is promising as a means of predicting

ground state properties generally in such compounds. It is apparent

that the agreement between the calculated and experimental values of

the shielding of platinum in these compounds is not good. A good ground

state property is important for the diamagnetic contribution, Cjd, to

the shielding calculation.

In the present calculations, the shielding differences of compounds

considered in Table 6.4.1 are almost entirely accounted for by the changes

in the paramagnetic contribution, , which, as shown in equation (3.35),

depends upon the unperturbed LCAO coefficients of the valence shell 5d-3 -3and 6p orbitals, < r and < r , and transition energies. Since the

shielding is a second-order molecular property, it depends upon a

satisfactory estimate of excited electronic states. Hence it seems probable

that the lack of success in shielding calculations by the INDO/5R wave-

functions could be, at least in part, attributed to a poor account of the

requisite excited electronic states.

From Table 6.4.1, the paramagnetic contributions under consideration-3 -3 -3 -3involve <r and <r >, . The calculated values of < r , and < r >-^5d 6p 5d 6p

of the platinum atom in the compounds considered are too small when

compared to the values obtained by the Relativistic Dirac-Fock procedure,-3 -3<r > CJ = 12.254 a.u. . The small values and variations of the local 5d

paramagnetic contribution are due to the too small contributions of 5d

and 6p orbitals.

As can be seen, the results presented in Table 6.4.1 and the

correlation between the calculated and experimental values of the chemical

shifts for those platinum compounds as shown in Figure 6.4.1, the trend in

the shielding differences is not reproduced. The variations of the shielding

differences are too small and not in agreement with the experimental data.


compounds in Table 6.4.1.

No. compound calculated transition energy (e.v.)a

1 (ptci6)"2 17.80598

2 (PtCl3(H20 ))"1 18.93678

3 (PtCl^)"2 16.61002

4 (PtCl^)"1 energy not converge

5 (PtCl3(NCCH3))"1 21.84598

6 (PtCl3(NH(CH3 )2 ))"1 21.27449

7 (PtCl3(NCC(CH3 )3 ))_1 20.79246

8 (Pt(N02)4 )"2 24.38074

9 (Pt(NH3 )4 )+2 24.11983

10 (PtCl3 (C2H4 ))"1 19.64631

11 (PtCl3(CNCH3 ))‘'1 21.04249

12 ( P t C l ^ O )"1 19.80775

13 (Pt(CN)6 )“2 energy not converge

14 (Pt(CN)4 )” 2 22.97996

15 (PtF,)"26 16.88781

a : the AEE approximation used 49.5777 e.v.^2^ ' 2^2^

One reason for this may be due to the parameters used in the calculations

that are not suitable for the excited state property calculations.

6.5 Conclusions.

In this chapter, a consideration of INDO calculations overall,

the correlations obtained for vanadium, cobalt and platinum compounds

are not good. This is probably due to the parameters used which employed

the parameter sets, as shown in Appendix B, for the reproduction of ground

state properties.

An extension to the present work would be to optimize the geometry

individually, this is suggested for two reasons. One is to make it possible

to compare the conformational calculations with the experimental data, the

basis of this being the minimization of the total conformation energy. The

second is to examine in particular the relationships of the metal atoms and

the ligands. The use of the geometry optimization provides a good criterion

for the determination of conformation. The use of standard configuration

data as input suffers from the defect that the data is obtained from gaseous

or solid state phase. Often combinations of the procedures with standard

bond lengths and angles are needed in order to calculate the conformations,

whereas the NMR spectra are obtained from liquid samples. Standard bond

lengths and angles, while being extremely useful approximations, should be

recognized as such and ideally be used only to obtain initial conformations

to which geometry optimizations are applied.

Another useful extension of the work in this chapter involves the

variation in shieldings with the variation of conformations. Results could

be obtained for calculated shieldings as a function of bond length and

angle. Comparison of these with the experimental values of chemical shifts

would be a further step towards the accurate determination of conformation.

General conclusions and suggestions for further developments.

The general theory of nuclear magnetic shielding was first

developed by Ramsey,by using second-order perturbation theory in 1950.

However, a lack of knowledge of excited states and problems associated

with the gauge of the vector potential describing the magnetic field

has severely limited the value of this approach for particular applications.

Latter in 1962 Pople demonstrated that difficulties associated with the

gauge can be eliminated by using an approach in which each molecular

orbital is composed of a linear combination of gauge invariant atomic

orbitals (GIAO).

In the present study, Pople's method has been employed throughout

in our calculations for some first-row, second-row and transition metal

elements in the periodic table. From our calculations, it is clearly

demonstrated that, for the elements considered here, the value of the

local diamagnetic contribution remains roughly constant for each nucleus

as its chemical environment changes. The variation in the local diamagnetic

contribution is in most cases less than 2% of the observed shielding

differences experienced by various nuclei in different electronic environments.

It is worthwhile mentioning that the magnitude of the local diamagnetic

contribution to the shielding is found to be insensitive to the choice of

the wavefunctions under consideration. Thus the shielding differences,

observed for anyone of these nuclei in chemically different environments,

are due predominantly to changes in the local paramagnetic contributions.

In most cases considered here, contributions arising from non-local

diamagnetic and non-local paramagnetic contributions are found to be

negligible. The present study has shown that the ability of Pople's

GIA0-M0-S0S method to satisfactorily account for the observed shielding

is critically dependent upon the choice of parameters employed, especially

for the first-row and second-row nuclei shielding calculations in Chapters

4 and 5, respectively. The successful application of this method depends

to a large extent on the accuracy of the calculated singlet transition

energies as well as the charge distributions.

In the present study for some first-row elements, Pople's GIAO-MO-

SOS method in conjunction with the INDO/S wavefunctions, which include an

account of one-centre exchange integrals, appear to have a significant effect

on the calculated contributions from the molecular excited states compared

with those obtained by means of the CNDO/S parameterization scheme.

For second-row elements, Pople’s GIAO-MO-SOS method, in conjunction

with the CNDO/S wavefunctions, is shown to be more capable of accounting

for the observed chemical shifts in a variety of different electronic

environments of molecules than is the INDO/S approach. By including the

one-centre exchange integrals appropriate for the INDO/S modification, the

calculated shieldings show no significant improvement such as that found

for first-row elements. This is probably due to the neglect of mixed

(neither Coulomb nor exchange) integrals in the INDO/S calculations and

the parameter set employed in the INDO/S calculations. This was the CNDO/2

parameter set, as were used in the CNDO/S calculations, Appendix B.

In dealing with transition metal elements, only the INDO parameteri

zation has been employed in conjunction with Pople's GIAO-MO-SOS method.

The average weighted value of the transition energies contributing to the

local paramagnetic contribution are reported for most of the transition

metal element nuclei considered. It is demonstrated that large variations

occur in the transition energies even for very closely related molecules.

Thus we consider that the use of an average excitation energy (AEE)

approximation in the interpretation of shielding data for transition metal

nuclei is unreasonable.

Since most of the theoretical estimates of nuclear shielding

are based upon an isolated molecule as a model, it seems unrealistic to

expect the exact reproduction of experimental data which are usually

reported for liquid samples and are susceptible to medium effects. In

the present work for some first-row nuclei, the "solvaton" model shows that

the local paramagnetic contributions to the shielding vary with a variation

of dielectric constant of the medium. Various results obtained from the

"solvaton" model together with information from hydrogen-bonding calcula

tions for some model compounds, can be useful in providing some qualitative

interpretation guidelines about the experimental chemical shifts and the

conformation of the molecules considered.

In conclusion, it may be stated that Pople*s GIAO-MO-SOS model

proves successful in accounting for various species of the shielding of

various kind of nuclei depending upon the proper parameterization scheme,

e.g. CNDO/2, INDO, CNDO/S and INDO/S calculations. Although the present

calculations by no means give quantitative agreement with experiment,

neverthless the present approach represents a considerable improvement

in previous semi-empirical calculations and lends encouragement for further

developments in shielding calculations.

For heavy nuclei, such as transition metal elements, non-relativis-

tically parameterized INDO calculations within the Pople* s GIAO-MO-SOS

perturbation framework are unable to provide a satisfactory account of the

shielding calculations, as is however possible for first-row nuclei. So it

seems, that it is probably necessary, to include a relativistic parameteri

zation of the GIAO-MO-SOS shielding calculations for heavy nuclei in order

to investigate the ability of the present approach to reproduce the experi

mental chemical shifts of heavy nuclei.

Appendix A.

In calculations on molecular structure by the molecular

orbital (MO) method in linear combinations of atomic orbitals

(LCAO) approximation, the molecular wavefunctions are built from

AO’s (Atomic Orbitals). The calculations of physical and chemical

quantities finally reduces to the evaluation of a great many integrals

over these A O ’s. Whereas formulae for many of these integrals canC35 2 4-5 2 -49)be found in the literature " , others are lacking; and it

was considered worthwhile to undertake a systematic study for

the rest of the integrals.

The calculations follow closely those of Mulliken and co-

workers^ 5 \ the notation of these authors,as modified by Roothaan

and Rudenberg is followed throughout this calculation. The

A O ’s have the general form

^nlm = Rnl(r) Ylm(0’ (A‘1)

where the atomic orbitals are given in the form of Slater Type atomic

orbitals (STO) / 250-1

Atomic Complete cartesian Radial factor Angular factororbital £orm " n l W Ylm(e,<f>)

Is (t3/Tl)5e'5r (4t3)2e'?r Y002s (?3/37r)2re ^r (4?5/3)2re'?r Y002Px (?S/3TT)2xe‘Cr (4t5/3)5re'cr > 1 1 + Yi-i^

2py (?S/3ir)Jye'cr (4t5/3)^re-?r i > n " Yi-i)2p z (?5/3Tr)2ze'cr (4t5/3)2re‘?r Y103s (2?7/4STr)2r2e‘?r (8?7/45)2r2e'cr Yoo

Atomic Complete cartesian Radial factor Angular factororbital fom *nbn « n l W Ybn<6’«

3Px (2;7/15TT)^rxe'cr (8?7/45)Me'cr , > 11 + Yl-P3Py (2?7/15tt) 2rye (8c7/45)Me-?r i t > l l - Yl-A3PZ (2?7/15tt) srze (8c7/45)2r2e'?r Y103dz2 (C7/18ir)2(3z2-r2)e_?r (8i;7/45)2r2e_?r Y203dxz (2?7/3tt) 5xze (8?7/45)2r2e'?r , > 2 1 + Y2-l3dyz (2c7/3ir)^yze'?r (8?7/4S)2r2e"?r ' Y2-P

3d 2 2 x -y (C7/6TT)J(x2-y2)e'cr (8c7/4S)Me‘?r , > 2 2 + Y2-23dxy (2c7/3ir)2xye"Cr (8?7/45)Me'?r i» >22 " Y2-A

where *00

' > 1 1 + Y i-i^

i > n - Y i-i^ Y,10

_ > 2= (| )2 Sine Cos<(>

= C f) ® Sine Sin*

= ( > 2 c°s<(>

Y20 ■ <16,, > 2 1 + Y2-l

1 t 1= 2

i, >21 ” Y2-l1 R 1 = f— )2 Mir

, > 2 2 + Y2-2 = C15 1*>6 ttj

i . / f l 22 ~ X2-2) = c15

> (A.2)

in2( • 2,

The computation is best effected after transforming from polar

coordinates of the two atoms to spheroidal coordinates £>ri,<J) given by

5 = (r„ + r, )/Ra ~b- n = (ra - rb)/R(f) = (j) - <f>T

(A.3)

aof an electron in an AO of atom a or b. The coordinate £ ranges from

1 to 00, and T) from -1 to +1

For any given AO pair, we can obtain overlap integral values

for various pairs of atoms, each for various interatomic distance R.

To accomplish this, the best procedure is first to set up for each

AO pair a single master formula expressed in terms of suitable

parameters depending on the orbital exponent, £, of the two AO's and

on R . For this purpose, the two parameters p and t are defined as

follows

(A.4)

The integrals over E, and r) may be evaluated by making use of the

following mathematical relations:

k+lAk (p) =

Bk(pt) =

f 1. r r

£ e p ^ d£ = e p £ ( k!/p^(k-c+l)'-) (A. 5)1 C=1 f+l i

n e"P dn-1

j. k+l k+l V r r= e“pt I (k!/(pt)c(k-c+l)0 - ept I ((“1) ki/(pt) (k”C+l)')

5=1 5=1(A.6 )

and we have Bk C°) = 2/(k + 1 ) for k even,and = 0 for k odd.

The explicit expressions for the overlap integrals of interest are

5(ls’3dz2) = -T5ZT- {(Ao V A4 V + ^ i V S V + 3(A2Bo-AoV

+ 3(A4B2"A4B2)}

C3S’1S> = {(A4 V AoV + 2(A3Br AlB3)}

(3s,2s) = 48)/3q {<-A5Bo+A4B1-2A3B2_2A2B3+A1B4+AoB5')}7 5 fL52 c2 R

(3s'2pz> = { V Bo-2V ' V 2V V - Bl(A5-2A3) + B4(2A2-Ao)}

7 7•2 -r 2Ea 5b R 7 ,

(3s ’3 dz2) = - fyB?5 W 3V V - “ i ( V B5) + 3A2(3W

.7 „7+ 6A,(B,-B_) + 3A.(B -3B.) - 6A [.(B1-B„) - A,(B -3B '3 1 5 4 o 2 5 1 3 6 o 2

32 t R ( )= '35 576/5 r Ao + A2( ^°r t le same tyPe atoms

Px’V = {A5(V V - V W + V W " V W+ A j C b ^ ) - ao(b5-b3 )}

= (3p ,2p )7 J i•2 y-25 5 ^ 5 ^(3V ls) = "S/XT tV W + A1(B2+B4) - V A2+V + B3(Ao+A2)}I I

(3V 2s) = T5OTT"{B1A5 + Bo \ - 2B3A3 - 2A2B2 + BSA1 + Vo}Z 5

(3pz'2pz> = f e vVVV + V W - V W - V W }7 7 V7 7

(3PZ’3S) = %07T-{AoB1 + V W - V B1+2V + 2A3(W+ A_(2B,+BJ + A.CB.-B,) - i B J 7 3 2 3 5 1 4 6 0 5 )

^ 2 jr 2 g(3dz2’ls) = J W - { (AoB4-A4Bo) ' A(A1B3-A3 B1) + 3(A2Bo-AoB2)

7 , + 3(A4B2-A 2B4)}£ 2 5 2 6

(3dz2'2s) = ■ "9^6R {Ao(3B3-B5) - V 3B2-5V - A2(3B1+4B3-3Bs), 5 + A3(3Bo^B2-3B4) + A4 (5Br 3B3) - A ^ - S B ^ }

(3dz2’2pz) = ^ 9 S 7 2 ^ f A o(3B2-B4) - V W + V 3W7 7 + A3 (B1+ 3B5) - A4 (Bo+B2) + A 5 (Br 3B3 )}

C3V ’3S) = % T/5-7{~A°(3EA'B6) + 6V V V + 3V 3W- 6A_(Bn -B^) + 3A.(B -3B_) + dA.-CB.-B,)3 1 5 4 o 2 5 1 3

7 7 - A 6(B0-3B )}W R 7 ,

(3dz2,3pz) = - I 9 2 7 I 5 \Ao (3B3-B5) - A ^ S B ^ B ^ - B g ) - A ^ B ^ B ^ B g )

+ A,(3B -B.+B.-3B,) + A,(2B,+B_+3BC)3 o 2 4 6 4 1 3 5- A 5(B0-2B2-3B4 ) + A 6 (Br 3B3)}

_ 32 ( )~ UTS I W I 5 l " 3Al + 16A3 ~ 7A5 / f o r the same type o f atoms

£ 2 2 ^(3dxz’3Px) = -I927T- { - V W + A1(B2-2B4+B6) + A2(B1+B3-2B5)

' A3(V B2-VB6) " A4(2Br B3-B5) + A5(Bo-2B2+B4) + a 6 (Bi-b3)}

16 xJ |= -jq - Y 9 ~/5 ) A i ” + j :for the same type of atomE (3d ,3p )yz y

7 7

? a R71152

c7 r7 f5040 I7 7

_ R7192

. c7 r7 f—

< -A (9B0-6B- +B> ) + 3A_(3B - B + 2 B . ) ( o 2 4 6 2 O ' 4 6- 3A (2B -Bn+3B^) + A^(B -6Bn+9B,)> 4 o 2 6 6 o 2 4 /

atom

48= (3d ,3d )yz yz

7 7£-2 ^2 y

(3dxy>3V = -Tran" { -Ao(V 2W + A2(V 3B4+2B6) - A4(2V 3B2+V+ A6 (Bo- 2B2+B4 )}

- ^ {""ô + ^2 ~ ^^4 + ^ 3} °r tle same type of5040atom

Appendix C .

One-centre integrals.

The one-centre core terms, U of the one electron Hamiltonianyymatrix are given by

Uuu = (W| ' I 72 ^ (C,1)rA

where the first term is the electron kinetic energy and the second

term is the electron potential enefgy in the field of the core of

the atom to which <j) belongs.yWhile these integrals could be calculated from atomic ...

(5A 251)orbitals and then corrected by core-pseudo potentials ’ ,the CNDO and INDO methods relate the core integrals to parameters

obtained from atomic spectroscopy.

To do this, the average energy of the atomic configuration. . .(78 )is considered

E(s^ pm dn) = £ u + £ (pairs) interaction energy1 (C.2)

where 1 , m and n are the number of s, p and d electrons respectively,

in the configuration, and the interaction energy of the possible

pairs is

s, s = F°(s,s)

p,p = F°(p,p)

d, d = F°(d,d)

s,p = F°(s,p)

s, d = F°(s,d)

P, d = F°(p,d)

| 5 F 2(p,p)

| 3 F 2(d,d) - | 3 F4 (d,d)

1 1 ( C -3) | G (s,p)

y Q G 2(s,d)

y 5 G 1(p,d) - | 0 G 3 (p,d)

, . 1 m ,n , (251)then for the configuration s p d , we have

ECs1 pm dn) = 1 U gs + m Upp + n U dd + \ 1(1-1) F°(s,s)

+ m(m-l) F°(p,p) - -|5 F 2(p,p)

+ i n(n-l) F°(d,d) - | 3 F2(d,d) + FA(d,d)

+ 1m F°(s,p) - | G1(s,p)

+ In F°(s,d) - G 2(s,d)

+ nm F°(p,d) - y 5 G1(p,d) - -|0 G 3 (p,d)

(42 )Blair showed the general relationship of U to the

ionization potential energy, I and electron affinity, A.

If y is an orbital of the valence shell 1 of atom X, then

1where 1^ and are the configurationally averaged ionization

potential and electron affinity of subshell 1 of atom X.

1. If 1 is an occupied subshell in the ground state of the

neutral atom X, then h = 1 and 1^ is observed by direct

removal of an electron from 1 .

2. However, if 1 is unoccupied in the neutral ground state,

then h represents the highest occupied subshell and an

electron is promoted from h to 1 prior to observation

of IJL •

3. If 1 has a vacancy for at least one electron in thef I

neutral ground state of X, then 1 = 1 and A^ is observed

by direct addition of an electron to subshell 1 .

4. However, if 1 is full in the neutral ground state, thenII

1 represents the lowest unfilled subshell and an electronII

is promoted from 1 to 1 prior to observation of .

« tis the occupancy of subshell 1 in the neutral ground state of X.

Y^x* is the spherically averaged energy of interaction of an electront

in subshell 1 with an electron in subshell 1 . If the CNDO

option is being employed, then Y ^ t is the simple coulomb

interaction energy. If the INDO option is being employed,

then Y-q , is the average interaction energy, including exchange

terms, as formulated by Slater^23^

■ I h i + Aj) = Uw + J.Nj1 Yn ' - |(Ylh + Y u ■ Yu")

For Na - 4 ( 1 + A2 s s

- 2 ( + Ap

- I ( Fd + A d

For Mg - ■=■ ( I + A & 2 s s

- 2 ( ^ + Ap

- I ( :d + A d

For atoms B - F and A1 - Cl

- 4 ( 1 + A2 p p

“ 2 Id + A d

= U ss

= UPP

= U dd

= U ss

= U PP

= U dd

= U ss

= UPP

^ F° also for H and Li

also for Li♦ - i c 1

+ J F° - I o g2

+ I F° " 1 2 Gl alS° f°r Be

+ 4 F° - 4 G1 also for Be2 4+ 3 ° . 3 2

2 F 20 G

+ (za - I )f° - I (za' ! )g1

+ (ZA- |)F° - iG1 + fg(ZA- | ) F 2

U dd + (ZA- -|)F° - ^ G 2

Appendix D.

One-centre two-electron integrals.

The general expressions in the INDO method for the unrestricted

Fock matrix elements are shown by equations (2.47)-(2.48) and (2.56)-

(2.58).

For a basis set of s and p orbitals, many of the one-centre

integrals vanish by symmetry, leaving those of the form ( y y |y y ) , ( y y |w ) ,

and (y v |y v ) . Since there is only one atomic orbital of each of the

s, p , p , and p types in the basis set, all the one-centre, off- x y zdiagonal core elements, H , vanish as they do in the CNDO method.yV

The non-vanishing one-centre integrals for an s, p basis set,(78making use of the notation of Slater and assuming that the s and

p orbitals have the same exponent, are specified by

(y y |y y ) = (y y |v v ) = Coulomb in te g r a l , y , v on A

1. (ssjss) = (ss|pxPx ) = (ss|pypy ) = (ss|pzP z) = F°(s,s) = F°(s,p)

= F°(p,p) = yAA

2 - = (pypy |pypy ) = (pzpzlpzpz) = F°(p’p) +

3 - (pxpx lpypy ) = ^ P j P z V = (pypy lpzpz) = F°(p’p) “

(y v |y v ) = exchange in te g r a l , y , v on A

2 2making use of the fact that G = F

1 14. (spx lspx ) = (spy |spy ) = (spz|spz) = ^G (s,p)

5. Cpxpy |pxpy ) = (PXPZ |PXPZ) = (pypz lpypz) = | 5F 2(p,p)

For a basis set containing'd orbitals (spd basis set),4fortunately many of the 9 possible one-centre integrals vanish by

symmetry; again leaving those in the form (yy|yy), (yy|w) and

(yv|yv) plus a number of mixed or hybrid integrals of the form

(yy|Xo), (yv|Xa) and (yv|ya).

For the method to be rationally invariant these mixed integrals

(mixed as they are neither Coulomb nor exchange integrals) must be

included in the calculation of the one-centre Fock matrix elements,

neglecting these mixed integrals are not serious. Thus, at the loss

of rotational invariance, we neglect these mixed integrals. This

greatly simplifies the evaluation of the Fock elements, and yield

the same expressions for an s, p, d basis set as for an s, p basis

set.

If the s and p orbitals are again assumed to have the same

exponent, then the s, p Coulomb, equations(D.1 )-(D.3 ), and exchange,

equations (D.4 )-(D.5 ), integrals are the same for an s, p, d basis,

and the remaining integrals are specified by

but Clack (252) L u g h ^ ' ^ f o u n d that the errors introduced in

Coulomb integrals, (yy|yy) and (yy|vv), y, v on A

1' (pxpxl dz2dz2) = (pypyldz2dz2) = F°(P’d) ~ fijF P.d)

3 5 F 2(p,d)

= ( p p | d 2 2d 2 2) = (p p Id d ) = (p p Id d )y y x -y x -y y y 1 xy xy y y 1 xz xz= (p p Id d ) = (p p |d d ) = F°(p,d) +*z z 1 xz xz *z*z' yz yz 35

10. (pzPz |dz2dz2) = F°(p,d) + | 5F2(p,d)

1 1 . (d 2d 2 |d 2d 2) = (d 2 2d 2 2 1 d 2 2d 2 2) = (d d 1 d d )z z 1 z z x -y x -y x -y x -y xy x y 1 xy xy= ( d d | d d ) = ( d d | d d ) = F°(d,d) + ^QF 2(d,d) + ^ r F A (d,d) xz x z 1 xz xz yz y z 1 yz yz 49 441

12. (d 2d 2|d 2 2d 2 2) = (d 2d 2|d d ) = F°(d,d) - -^F 2(d,d)z z 1 x -y x -y z z 1 xy xy 49+ _6 _4r . .v

441 d,d^

13. (d 2d 2|d d ) = (d 2d 2|d d ) = F°(d,d) + -^F 2(d,d) - |£-FA(d,d)z z 1 xz xz z z 1 yz yz 49 441

14. (d 2 2d 2 2 d d ) = F°(d,d) + ^ F 2(d,d) - ■^TF4 (d,d)x - y x - y x y x y 49 441

15. (d 2 2d 2 21 d d ) = (d 2 2d 2 21 d d ) = (d d Id d )x -y x -y ' xz xz x -y x -y ' yz yz xz xz* yz yz

= (V x y l dx A z > = < V W ' W = F°(d’d) " f 9 r2(d’d)

- s f J A d . d )

.exchange integrals, (yv|yv), y, v on A

2 2 4 4making use of the fact that G = F and G = F

16. (sd 2 |sd 2) = (sd Isd ) = (sd Isd ) = (sd 2 2|sd 2 2)z 1 z xz1 xz yz1 yz x -y 1 x -y= (sd [sd ) = - b 2(s,d) xy1 xy 5

17. (p d 2|p d 2) = (p d 2|p d 2) = i=G 1(p,d) + ^ = G 3 (p,d)*x z 1 x z y z '■ y z 15 245

18. (p d |p d ) = (p d |p d ) = (p d 2 2 |p d 2 2)x yz1 x yz y xz'*y xz z x -y '*z x -y= (p d |p d ) = -i^G3 (p,d) z xy1 *z xy 245

19. (p d 2 2| p d 2 2) = (p d | p d ) = (p d | p d )x x -y 1 x x -y x x y ' x xy x xz' x xz= (p d 2 2 |p d 2 2) = (p d |p d ) = (p d |p d )y x -y 1 y x -y y xy'*y xy y yz1 y yz= ( p d | p d ) = ( p d | p d ) = ■^=G1 (p,d) + -^=G3 (p,d) z xz1 rz xz z yz1 z yz 15 245

20. (pzdz2lpzdz2) = J!p1(v>d') +

21.

22.

23.

24.

(d 2d 2 21 d 2d 2 2) = (d 2d | d 2d ) = ^ F 2(d,d) + i^-F4 (d,d)z x - y ' z x - y z x y ' z x y 49 441

(d 2d Id 2d ) = (d 2d | d 2d ) = ~ F 2(d,d) + ^ - F ^ C d ^ ) z xz1 z xz z y z 1 z yz 49 441

(d 2 2d Id 2 2d ) = -^-F^Cd^)x -y x y 1 x -y xy 441

(d 2 2d Id 2 2d ) = (d 2 2d Id 2 2d ) = (d d |d d )v x -y x z 1 x -y xz x -y y z 1 x -y yz xy xz' xy xz= (d d Id d ) = (d d Id d ) = I^F^d.d) + | j-F4(d,d)xy yz* xy yz xz yz* xz yz 49 441

30 _4,

Appendix E.

Some values of angular momentum matrix elements for p and d orbitals.

LX IV Ip >y Iv Id 2>1 z |d > 1 xz |d > 1 yz |d 2 2> 1 x -y |d >xy

X 1 0 0 0 0 0 0 0 0

1 y I V |dz2> Id >xz |d > yz | d 2 2> ' x -y ■ v

 1 X 1 p >y 1V |d 2> |d >1 Z 1 XZ |d >yz I d 2 2> 1 x -y ‘V

< P x» 0 i 0 0 0 0 0 0

< p y' -1 0 0 0 0 0 0 0

A N 0 0 0 0 0 0 0 0

< dz21 0 0 0 0 0 0 0 0

< dxz< 0 0 0 0 0 1 0 0< d 1yz' 0 0 0 0 -1 0 0 02 2 1 x -y 0 0 0 0 0 0 0 2< d |x y 1 0 0 0 0 0 0 -2 0

Appendix F.

Integrals required in equations (3.29) and (3.30) may be

written in the form

<<j> I r"m (r^6 D - r r0)|<J) > , where m=0 and 3 (F.l)ry1 ■ a3 a 3 1 y

These integrals vanish unless a = Non-vanishing integrals are

evaluated using Slater Type atomic Orbitals (STO) and can be

expressed in terms of integrals over radial and angular spherical

harmonic functions such that

<cf) I r m(r - r) U >Yy 1 v cr 1yy r_ln r2 r2 dr

7T 2 t t _2’ ' (i - -| )(Ylm(e,<|.))2 Sine de d<f> r o o

Au Bym a (F .2)

wherem = f 0&M)2 r2 ~-m r2 r2 dr

and

C2ri+2-m) 2m~2 Zum-2

(2nJtt 2tt 2 r

na. (F .3)

O 0a - -f X^Ce.-fO) Sine de d*

(F.4)the values of the integrals By for s, p and d orbitals are given

BuX

ByyBy

s Px py Pz d 2 z dxz dyz d 2 2 x-y d.

2 2 4 4 16 4 6 4 43 5 5 5 21 7 7 7 72 4 2 4 16 6 4 4 43 5 5 5 71 7 7 7 72 4 4 2 10 4 4 6 63 5 5 5 21 7 7 7 7

xy

Integrals required in equations (3.31) and (3.32) may be written in the form

r'm U,/ = | ( I ^ i W )2 r_m r2 dr8

TT 7T2(yjm^,^)) Sine de d<J> (F.5)

0 0For a normalised spherical harmonic function, Yljê,^), the second integral in equation ( F.5) is equal to 1.

Thus<<$> | r"m U > = AyYu 1 1 Mi m+2

(F .6)

using equation (p.3 ) for equation ( F.6), the following expressions-3 -3 -1obtained for. the integrals <r >n^, <r >n^ and <r

<r'3>npZ_n£

Lna0J(F. 7)

<r"3>nd 15nd

Lna0 J(F.8)

<r_1>Z__yn2a

(F. 9)

are

Appendix G.

The evaluation of the molecular Coulomb and exchange integrals.

When performing a calculation of nuclear shielding, we are

concerned with the excitation of an electron from an occupied

orbital "i" to a virtual (unoccupied) orbital "j". For a closed-

shell ground state, such a transition gives rise to excited singlet

and triplet configurations. The energy corresponding to singlet

transition is given by

By expressing the molecular orbitals (indicies i, j, k, 1) as

LCAO's (indicies a, 3, y, 6 for atomic orbitals) then

Mae) (G.l)

where e- and e- are the eigenvalues of orbitals j and i respectively

are the molecular Coulomb and exchange integrals

J. .ij (iiljj) (G.2)

K. . (G.3)

where the integral

(ij |kl) ♦i(1) (1) t 3- ^ ( 2 ) ^ ( 2 ) dT1dx2 CG.4)12

(ij|kl) = ^ 6ciacj e V i « (0lB|Y6)(G.5)

Using the CNDO approximatation, the integrals reduce to

(ij|kl) = ^ CiaCjaCkyClY F°(a,Y) (G.6)

since (ag|Y«) = (cat | YY) CG.7)

Under the INDO approximation, where the one-centre exchange integrals

are no longer neglected, then

Cij|ki) = ^ c - ^ ^ c o a l r r )

+ I f A W i s M l r V * * 6*110*0 * (G-8)y*6

where a e A, M B, y 0 C and 6 e D.

The first term in this expression can be further separated into

the F°(a>y) terms which CNDO supplies and the non-F°(a,y) Coulomb

terms, A(aa|yy) which INDO supplies. The equation (G.8 ) can then

be written in terms of the CNDO expression plus a correction for INDO

(ulUOiNEO W I k b CNDO + E CiaCjaCkYClYA('aaMY) ot,y

+ 7 C. C.0C. C-, r.Afa|31y6) ^ la j B ky 16 p 1 [ J

y*6 (G. 9)

where a, 3, Y , <5 0 A.

For spd basis set, this expression is expanded using the

Coulomb and exchange integrals.

Then

(ij|klM NDO = I CiaCj aCkyClyA (aa I a ,ycorrection

+ ^ g CiaCjeCkYC16A aeM 6)y =£6

(G.10)

4_25 F (P,P) ( Ci2Cj2Ck2C12 + Ci3C j3Ck3C13 + Ci4Cj4Ck4C14 )

+ (|5 F 2(d ,d) + “ j F4(d,d)] ( ci5cj5ck5c15 + c.6c.6ck6c16

+ Ci7Cj7Ck7C17 + Ci8C.8Ck8Clg H- C.9CjgCk9C19 )

2 2 — - F 25 (P.P) ( (Ci2C j2Ck3C13 + Ci3C j3Ck2C12

(Ci3C j3Ck4C14 + Ci4Cj4Ck3C13

- § 5 F2(p,d) ( (Ci2c j2ck5c15 + ci5c_.5ck2c12

+ — F2 35

35

(p,d) [ (Ci4Cj4Ck5C15 + Ci5Cj5Ck4C14

F (p,d) ((Ci4Cj4Ck8C18 + Ci8Cj8Ck4C14

+ (ci2C j2Ck7C17 + Ci7C j7Ck2C12 + | g F 2(p,d) ( (C.4C j4Ck6C16 + C.6C.6Ck4C14

+ (Ci3Cj3Ck8C18 + Ci8Cj8Ck3C13

+ (Ci3C j3Ck7C17 + Ci7C j7Ck3C13

+ (“Ci2C j2Ck8C18 + C i8C j8Ck2C12

- i h F2(d*d) + FM d , d ) } ( (Ci5c.5ck9c19 + c i9c.9ck5c15)

+ (Ci5Ck5Ck8C18 + Ci8Cj8Ck5CX5) )

+ (|s FM d , d ) - Ifj F4(d,d)][ (Ci5C j5Ck6C 16 + C.6C.6Ck5C15)

+ (Ci5C j5Ck7C17 + Ci7Cj7Ck5C15) )

+ (C.0C._C. .c.. i2 j2 k4 14

)+ (Ci3C j3Ck5C15 +

)+ <Ci3C j3Ck6Cj6 +

+ (Ci4C j4Ck9C19 +

+ (Ci4C j4Ck7C17 +

+ (Ci3Cj3Ck9C19 +

+ <'Ci2C j2Ck9C19 +

+ (Ci2Cj2Ck6Cj6 +

c. , c ..c. _cn.)i4 j4 k2 12

Ci5C j5Ck3C13))

c. .c .,c. _cn _)16 j6 k3 13C.QC.QC./Cl/)1 i9 j9 k4 14 >

Ci7C j7Ck4C14)

Ci9C j9Ck3C13)

Ci9C j9Ck2C12)

Ci6Cj6Ck 2C1 2 ) 1

+ ( 49 F (d’d " 441 F (d>d))( ^Ci8Cj8Ck9C19 + Ci9Cj9Ck8C18') 1

- I k p2(d>d) + TOT F'(d’d)K (Ci6Cj6Ck9C19 + Ci9Cj9Ck6C16>+ (ci 7cj?ck 9cig + cigc jgck 7c17)

+ (Ci6Cj6Ck8C18 + Ci8Cj8Ck6C16)+ (ci7cj7ckgc lg + ci8c j8ck 7c17)

+ (Ci6Cj6Ck7C17 + Ci7Cj7Ck6C16) )

+ I g M s . p ) [ ( c n c j2 + ci2cjl)(cklc12 + ck2Cll) + (Cilcj3 + ci3c j]L>

(CklC13 + Ck3CU > + (CilCj4 + Ci4Cjl)(CklC14 + Ck4Cl l d

+ 3 G2(P’d) ( (CilCj5 + Ci5Cjl)(CklC15 + M s V + (CilCj6 + Ci6Cjl) (CkXC16 + Ck6Cll) + (CilCj7 + Ci7Cjl)(CklC17 + Ck7Cll)

+ (CilCj8 + Ci8Cjl)(CklC18 + Ck8Cll> + (CilCj9 + Ci9Cjl>(CklC19 + Ck9Cll) ]

+ § 5 F 2(P>p)[(ci2c j3 + ci3c.2 )(ck 2c13 + ck3c12) + (C.2C.4 + c.4cj2)

(Ck2C14 + Ck4C12} + (Ci3Cj4 + Ci4Cj3)(Ck3C14 + Ck4C13) 1f ( I 5 e M p . d ) f H 3 G 3 (p,d)][ (c.2cj5 + c.5c j2)(ck 2c15 + ck 5c.12

+ (C._C._ + C.^C.-XC. QCnc. + C. ,-C., i3 j5 i5 j3 k3 15 k5 13

+ ( i G1 (p,d) + “ 5 G3 (p,d)][ (C.2Cjg + C.9C.2)(Ck2Cig + CkgCk9 12

+ (Ci2C j8 + Ci8C j 2 ^ Ck2C18 + Ck8C12

+ ( C i2C j 6 + C i6C j2)(Ck2C1 6 + 'Ck6C12

+ <-Ci3C j9 + Ci9C j3')<‘Ck3C19 + Ck9°13

+ (Ci3C j8 + Ci8Cj3)(Ck3C18 + Ck8C13

+ (Ci3C j7 + Ci7C i3K C k3C17 + Ck7C13+ (c. .c.. + C.QC..)(C. ,c10 + c. 0Cn .i4 j8 18 j4 k4 18 k 8 14+ (C..C._ + C. _C )(C. ,C. _ + C. -X., i4 j7 i7 j4 k4 17 k7 14

1

]+ f s G3 (p,d) ( ( C .2C .7 + c .7c j2)(ck2c17 + ck 7c 1 2 ) + (c.3cj6 + c .6c j 3 )

(Ck3C16 + Ck6C13) + (ci4c j9 + Ci9C j 4 ^ Ck4C19 + >

+ (Ci4C j8 + Ci8C j4)(Ck4C18 + Ck8C14) 1

'k9 14‘

+ G1(p,d) + G 3(p,d) j ( (C.4C j5 + Ci5C j4)(Ck4C15 + Ck5C14) ]

+ I k F2(p’d) + H i FA(P ’d ) ) ( (Cx5Cj9 + Ci9C j5)(Ck5CX9 + Ck9CX5)

+ (Ci5Cj8 + Ci8Cj5)(Ck5CX8 + C k8CX5) )

+ (1_ F2(d,d) + ^ F4(d,d)] ((C.5C.6 + C.6C.5)(Ck5C16 + Ck6C15)

+ (C i 5C j 7 + Ci7°j5)CCk5CX7 + Ck7CX5^ 1

+ 44X F ^d ’d ( <-Ci8C j9 + Ci9C j8')<-Ck8CX9 + Ck9CX8^]

+ F 2(d,d) + |2- F4(d,d) ) [(C.6C ,g + C.9C j6)(Ck6C19 + C ^ )

+ (C. ,c._ + C.7C_)(C. ,c__ + c. 7c_ , ) l16 j7 i7 j6 -k6 17 k7 16 >

where the abbreviation used is

1 = s, 2. = p , 3 = p , 4 = p , 5 = d 2, 6 = d , 7 = d , 8 = d 2 2,* x y z z xz yz x -y

Appendix H .

The Slater-Condon parameters^ ^ F°, G 1 , F^, G^, F^ are

two-electron integrals involving the radial parts of the atomic

orbitals. The integrals F° (or Y ^ ) are evaluated theoretically1 2 3 Afrom Slater atomic orbitals. The values for G , F , G and F

are fitted semi-empirically to give best fits with experimental

energy levels, and are given in the table below (in eV)

1 2 o /Element G F G F

Carbon 7.2877 4.7259 - -

Nitrogen 9.4134 5.9592 - -

Oxygen 11.8128 7.2476 - -

Fluorine 14.4808 8.5910 - -

Silicon 4.8110 2.2615 2.8071 1.4749

Phosphorus 3.4493 2.9464 2.0591 1.9215

Sulphur 3.0743 4.5360 1.8353 2.9583

Chlorine 2.8634 5.2758 1.7094 3.4408

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J.C. Slater, in "Quantum theory of Atomic Structure" Vol. 1,

McGraw-Hill, Newyork (1960), pp. 324-325.

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