OBJECTIVES OF THE WORK AND COMPUTATIONAL...

OBJECTIVES OF THE WORK ANDCOMPUTATIONAL METHODOLOGY

Aneesh. M.H “ A theoretical study on the regioselectivity of electrophilicreactions of heterosubstituted allyl systems ” Thesis. Department of Chemistry, University of Calicut, 2012

66

CHAPTER II

OBJECTIVES OF THE WORK AND

COMPUTATIONAL METHODOLOGY

ContentsContentsContentsContents

2.1 The Scope of the Present Work

2.2 A Brief Review of Earlier Theoretical Works on Allyl Systems and

Objectives of the Present Work

2.2.1 Major observations in the previous works

2.2.2 Shortcomings of the previous study on allyl systems

2.2.3 Objectives, model chemistry & plan of execution of the work

2.2.4 Justification of the model chemistry used in the study

2.3 Computational Methodology

2.3.1 In brief

2.3.2 Density functional theory

2.3.2.1 Introduction

2.3.2.2 The Hohenberg and Kohn theorems

2.3.2.3 The Kohn-Sham approach

2.3.2.4 DFT vs HF

2.3.2.5 Exchange-correlation functionals

The local density methods

Local spin density approximation (LSDA) method

Generalized gradient approximation (GGA)

Meta-GGA functionals

Hybrid methods

2.3.3 Conceptual DFT and reactivity descriptors


2.3.3.2 Chemical reactivity descriptors

2.3.3.3 Global descriptors

Electronic chemical potential & electronegativity

Chemical hardness and softness

Global electrophilicity index

2.3.3.4 Local reactivity descriptors

Fukui function

Local softness

Local philicity

2.3.3.5 Major principles in the theory of chemical reactivity

67

Maximum hardness principle

Minimum polarisability principle

Electronegativity equalization principle

2.3.3.6 Concluding remarks and some generalisations

regarding DFT based reactivity descriptors

2.3.4. Software used in the work

2.3.4.1 Gaussian 03

2.3.4.2 Chemcraft

2.3.4.3 Gaussview

2.3.5 Brief descriptions of the principles involved in the

various calculations

2.3.5.1 Optimization techniques

2.3.5.2 Population analyses

Mulliken population analysis (MPA)

Natural population analysis (NPA)

CHELPG charges

Merz-Singh-Kollman (MK) scheme

2.3.5.3 Characterization of the transition states

2.3.6 Solvation models


2.3.6.2 Self consistent reaction field (SCRF) methods

Onsager reaction field model

Tomasis polarized continuum model (PCM)

The isodensity PCM (IPCM) model

Self-consistent isodensity polarized continuum

model (SCF-IPCM)

Appendix 2A: Geometry and relative energy of allyl systems reported in the

literature

Appendix 2B: Ratio of α:γ products in the reactions of 3,3-dichloro propene

with some carbonyl compounds in the presence of LDA and

pot-tert-butoxide

REFERENCES

68

CHAPTER II

OBJECTIVES OF THE WORK AND

COMPUTATIONAL METHODOLOGY

2.1 The Scope of the Present Work

Allyl anion (1) and the corresponding allyl metals (2) are stabilized by

resonance. Considerable further stabilization arises when these systems

contain one or more hetero substituent(s) X. Unsymmetrically substituted

allyl anions (3) and allyl metals (4) herein after called allyl systems are

ambident nucleophiles, which can react with electrophiles at two sites (α or

γ), invoking the question of regioselectivity. Regioselectivity can be defined

as the preferential reaction at one (or more) possible sites in a molecule

resulting in the preferential formation of one (or more) structural isomers.

One of the goals of synthetic organic chemists is to increase selectivity –

ability to do chemical transformations at specific sites in a molecule without

affecting the rest of the molecule.

The regioselectivity of reactions involving the allyl systems is both of

considerable synthetic importance and of theoretical interest.1

-

M+

M+ = Li+, Na+, K+

-

1 2

-

X

-

M+

X

α γ α γ

3 4

X = F, Cl, Br, OH, SH, SeH, NH2, PH2, AsH2

Figure 2.1: Allyl systems

The allyl systems can be conveniently classified into different groups

(reference 1 gives a comprehensive review)

• Allyl systems stabilized by one heteroatom [C=C-C-X]

• Allyl systems stabilized by two identical gem-hetero-atoms [C=C-CX2]

69

• Allyl systems stabilized by two different gem-hetero-atoms [C=C-CXY]

• Allyl systems stabilized by two non vicinal-hetero-atoms [XC=C-CY]

• Allyl systems stabilized by three hetero-atoms [XYC=C-C-Z]1

Some of them belonging to the first category have been shown in table 2.1 for

a quick review.

Table 2.1: Examples of allyl systems stabilized by one heteroatom

Hetero substituent

containing Substituent X in C=C-C-X Name of the system

Halogen

-F Allyl fluorides

-Cl Allyl chlorides

-Br Allyl bromides

Oxygen

-OR 1-alkoxy-2-propenes

-OSiR3 Silyl allyl ethers

-O-CO-NR2 2-alkoxycarbamates

Sulphur

-SH Allyl mercaptans

-SR Allyl sulphides

-SOR Allyl sulphoxides

-SO2R Allyl sulphones

Nitrogen

-NR2 Allyl ammines

-NO2 3-nitroprop-1-enes

-N=C N- allyl imines

-N(CH2)4 1- allyl pyrolidines

Phosphrous

-PO(OR)2 Allyl phosphonates

-P(O)R2 Allyl phosphine oxide

-PR2 Allyl phosphines

-P(O)(NR)2 Allyl phosphonamides

Silicon -SiR3 Ally trialkylsilanes

Whether an α or γ product is formed when these allyl systems react with

electrophiles, depends on various factors such as nature of the hetero

substituent(s), nature of electrophiles, nature of the counter ion and the

reaction conditions including solvent and temperature. There have been

numerous theoretical investigations on the regioselectivity of allyl systems.

However, there is still no general concept to describe the regioselectivity of

the reactions of electrophiles with these compounds. Fundamental to the

understanding and control of regioselectivity is knowledge of the structure of

70

these allyl systems both in the gas phase and in solution. Further, to the best

of our knowledge no work has been done to describe the regioselectivity of

these compounds using DFT based reactivity indices (described in section

2.3.3.2). The present work aims to characterize the regioselectivity

preferences of hetero substituted allyl systems in terms of DFT based

reactivity indices.

2.2. A Brief Review of Earlier Theoretical Works on Allyl Systems and

Objectives of the Present Work

2.2.1. Major observations in the previous works

Early computations on the allyl anion suggested that its geometrical structure

slightly departs from planarity due to tiny pyramidalization of the terminal

CH2 groups.2 But later studies by Tonachini and Canepa showed that a

completely planar structure is the only stable isomer for the allyl anion.3 With

RHF/3-21+G geometry optimization and MP2/6-31+G single point energy

calculation, they also investigated the structures and regioselectivity

preferences of 1-fluropropenide (monofluroallyl anion), 1,1-difluropropenide

(gem- difluroallyl anion) and the corresponding lithiated species. The stable

isomers reported by them are depicted in appendix 2A (Figures 2A.1 to 2A.6)

and major findings regarding the structures and regioselectivity of these allyl

systems are summarized in tables 2.2 & 2.3 (see the first two rows in each

table). The chloro and bromo substituted allyl systems were not included in

this work and no reference about the attacking electrophile is made in their

report. Moreover, Li+ was the only counter ion considered in their work. The

effect of Na+ and K

+ ions on the structures and regioselectivity of fluoro

substituted allyl systems had not been included not only in this work but, to

the best of our knowledge, in any of the later theoretical studies.

71

Table 2.2: Summary of previous theoretical works on substituted allyl anions

a The respective figures are reproduced in appendix 2A.

b Polarization of the HOMO: This is

expressed in terms of a parameter P(χ) defined for each atom in the molecule. Its value is given by

the sum of the squares of the HOMO atomic orbital coefficients belonging to that atom. c

Group

charges: given by the sums of the net Mulliken atomic charges for different groups in the allyl

system, CHX or CX2 (α), and CH2 (γ).

Substit

uent(s)

X No

. o

f

sta

ble

iso

mer

s

Description of the

isomersa

Model

Chemistry

for geometry

optimization

Conclusions regarding

selectivity and their

basis (shown in

parenthesis)

F 3

Syn non-planar, anti

non-planar & anti-

planar. Syn-planar is

reported as a transition

state

RHF/3-21+G

No special preference

for α or γ (polarization

of the HOMOb and

group chargesc)

F,F- (gem-

difluoro)

2 Non-planar & planar RHF/3-21+G

Strong α selectivity

(polarization of the

HOMO and group

charges)

Cl 2

Syn-planar & anti-

planar.

No non-planar isomers were reported

RHF/3-

21+G*

γ for hard electrophiles

(electrostatic potential

maps) & less selective

for soft electrophiles

(polarization of the

HOMO)

Cl,Cl (gem-

dichloro) 2 Non-planar & planar

RHF/3-

21+G*

α selectivity


maps & polarization of

the HOMO)

F 1 Non-planar HF/6-31+G* No prediction of

regioselectivity

OH 1 Non-planar HF/6-31+G* No prediction of

regioselectivity

NH2 1 Nitrogen in the plane

of the allyl skeleton. HF/6-31+G*

No prediction of

regioselectivity

72

Table 2.3: Summary of previous theoretical works on substituted allyl metals

a The figures of stable isomers are reproduced in appendix 2A.

b No diffuse or polarization function

on the metal

Substitu

ent(s)

X Co

un

ter

ion

(M+)

No

. o

f

sta

ble

iso

mer

s

Description of the

isomersa

Model

chemistry for

geometry

optimization

Conclusions regarding

selectivity and their basis

(shown in parenthesis)

F

Li

4

K-syn external, K-

syn internal, K- anti

and Y-syn. No isomer in which Li is at a bridging position.

RHF/3-21+G

The first three isomers - α

selectivity & the Y-syn

isomer - γ selectivity

(polarization of the HOMO

and group charges).

No special preferences since relative energy of the isomers were very close.

F,F-

(gem-

difluoro)

Li

3

K-external (Li close

to syn F), K-external

(Li close to anti F)

& Y. No internal isomer was observed

RHF/3-21+G

α selectivity

( polarization of the HOMO,

group charges, structure of

the anticipated TS &

relative stability of different

isomers).

Cl

Li

2

Syn-external & syn-

internal. No bridged isomers;

bRHF/3-

21+G*

γ selectivity for hard

electrophiles (electrostatic

potential maps); α

selectivity for soft

electrophiles (

polarization of the HOMO)

Cl,Cl-

(gem-

dichloro)

Li

2

External & internal.

The stability of the two isomers are comparable

bRHF/3-

21+G*

α selectivity (polarization of

the HOMO) & γ selectivity


maps)

Cl Na 1 Syn- external. No bridged isomer

bRHF/3-

21+G*

γ selectivity for hard

electrophiles (electrostatic

potentials) & α selectivity

for soft electrophiles

(polarization of the HOMO)

Cl,Cl-

(gem-

dichloro)

Na

2

External & internal.

No bridged isomer

bRHF/3-

21+G*

α selectivity (electrostatic

potential maps &


Cl K 1 Syn- internal bRHF/3-

21+G*

Strong γ selectivity for

hard & soft electrophiles

(electrostatic potentials &


Cl,Cl-

(gem-

dichloro)

K

2 External & internal.

No bridged isomer.

bRHF/3-

21+G*

α selectivity (electrostatic

potential maps &


73

Most of the later studies concentrated on the chloroallyl systems. In a

combined experimental and theoretical investigation, Angeletti and co-

workers studied the effect of lithium complexation by 12-Crown-4 on the

regioselectivity of the attack of gem-dichloroallyl lithium on some carbonyl

compounds.4 They experimentally observed that substituted benzaldehydes

and acetophenone preferentially attack the γ position of gem-dichloroallyl

lithium, but show significantly increased α selectivity if 12-Crown-4 is

present. Theoretical computations at the RHF level using the minimal basis

set STO-3G* had been done on the free anion and on lithiated species in order

to provide probable reasons for the above experimental observations. The

possibility of two reasons (deaggregation of dimers of allyl systems present in

the tetrahydrofuran (THF) solution and weakening of the ionic C-Li bond for

the role of 12-Crown-4) were examined. However, the computations have

been done at a low level of theory (HF/STO-3G*). Therefore, there may be

changes in both the structural and electronic features if a higher level of

theory is employed.

In another experimental work, Canepa et al. studied the effect of cation (Li+ &

K+) in the regioselectivity control of chloro allyl systems by carrying out two

sets of reactions of 3,3-dichloropropene with substituted benzaldehydes.5 In

the first set, these reactions were carried out in the presence of lithium

diisopropylamide (LDA) and mostly γ selectivity was observed. But a shift in

regioselectivity from predominantly γ to exclusive α was observed when these

reactions were carried out in presence of both LDA and potassium tertiary

butoxide (see table 2B.1 in appendix 2B). It was argued that the counter ion

might play a more important role in the regiochemical control in connection

with its participation in the transition structure. The electrophile may attack

that carbon to which the counter ion is more tightly bound. This assumption

was made based on a computational study by Schleyer for a model reaction of

formaldehyde with the monomer of methyl lithium, as well as its dimer.6 In

74

the proposed reaction pathway the carbonyl oxygen strongly interacts with Li

before the formation of the new C-C bond providing a four-centre transition

structure. The same research group substantiated these hypotheses through a

theoretical investigation on chloroallyl systems (both monochloro and

dichloro) where, Li, Na and K acted as counter ions, using the ab initio model

chemistry HF/3-21+G*.7 The structures of stable isomers reported by them

are reproduced in appendix 2A (Figures 2A.7 to 2A.14) and major

conclusions regarding regioselectivity are described in tables 2.2 & 2.3 (see

the rows 3 & 4 of table 2.2 and rows 3 to 8 of table 2.3). Three major features

- polarization of the HOMO of the allyl systems, electrostatic potential maps

and the position of the counterion with respect to the allyl backbone - of the

different stable structures were compared before arriving at conclusions

regarding regioselectivity. Unfortunately, the transition structures of the α and

γ reaction pathways for the attack of carbonyl or other electrophiles are not

included in these investigations. Instead, conclusions are made based on the

transition structure of a model reaction between formaldehyde and methyl

lithium. Moreover, there is no mention regarding whether the reaction is

kinetically controlled or thermodynamically controlled. The computations

were carried out using the split valence basis set 3-21G augmented with sp

functions on carbon and chlorine atoms and with polarization functions on

chlorine. This basis set does not contain neither diffuse nor d functions on the

cations. Since the binding of the metal ion with the allyl framework and the

relative energy between the various possible isomers were regarded to be the

key factors deciding the regioselectivity, we feel that inclusion of polarization

and diffuse functions on the metal ions and use of a theory addressing

electron correlation effects are absolutely essential in these investigations.

Canepa and Tonachini theoretically investigated the monomer-dimer

equilibrium in lithium and sodium gem-difluoro and gem-dichloro allyl and

corresponding methyl systems.8,9

Their thermochemical calculations

75

suggested the following. Lithium and sodium gem-difluoro and gem-dichloro

allyl systems have a tendency to dimerize in the gas phase. Moreover, the

difluoro systems showed a greater inclination than the corresponding dichloro

systems towards dimerisation. However, in the presence of a solvent, the

difluoro systems exhibited an increased tendency to dimerize while the

dichloro systems preferred to be monomeric.

The same authors also investigated the addition reactions of (1,1-difluroallyl)

lithium, (1,1-dichloroallyl) lithium & potassium and the corresponding free

anions with formaldehyde at the HF and MP2 levels of theory.10

The two

competing pathways leading from the critical electrostatic σ-complexes to the

α and γ addition products had also been studied. In (1,1-difluroallyl)lithium,

(1,1-dichloroallyl)potassium and both free anions, the α-pathway was sharply

preferred. In contrast, for (1,1-dichloroallyl)lithium, the difference between

two activation energies is smaller and in favour of the γ-pathway. The

transition states in their report (see figures 1 to 5 in appendix 2C) were

characterized only by one technique - diagonalization of the analytically

calculated Hessian matrix and looking for one imaginary frequency.

However, it is generally accepted that simply getting one imaginary frequency

does not guarantee that we have found the correct TS (the one that connects

the reactants and products of interest). The imaginary frequency must

correspond to the reaction coordinate. This should be made clear by the

animation of the imaginary frequency and through an intrinsic reaction

coordinate (IRC) calculation (see section 2.3.5.3). Unfortunately, these

methods were not employed in the work by Canepa. In this context, for the

sake of checking whether the reported structures are true transition states or

not, some of them are reproduced and various characterization techniques

(IRC calculation and animation of the imaginary frequency using the software

CHEMCRAFT) are performed. Surprisingly, it is found that most of the

reported α transition structures are not true ones (details included in chapter

76

4). Moreover, the highest level of theory employed in this work for geometry

optimization was MP2/3-21G* (though single point calculations were

performed at the MP2/6-31+G(d) level). These findings prompted us to take

up an investigation of the reaction between 1,1-difluoropropenide and

formaldehyde at two levels of theory: one at the HF/3-21G* used by Canepa

and another at the B3LYP/6-31+G(d).

2.2.2. Shortcomings of the previous study on allyl systems

A detailed survey of the literature and some test calculations on selected

reactions revealed the following limitations of earlier theoretical works

• To the best of our knowledge, no attempt has been made to characterize

heterosubstituted allyl systems and their regioselectivity preferences as

indicated by density functional theory based reactivity descriptors.

• In most of the works mentioned above and summarized in tables 2.2 &

2.3, the prediction of regioselectivity is made on the basis of arbitrary

concepts like group charges.

• Further, no attempt is made to distinguish between the kinetic and

thermodynamic control in most of these studies.

• The effect of solvent on the energy and geometry of stable isomers was

not included in any of the earlier works.

• Transition states of specific reactions (reactions with either alkyl halides

or aldehydes) are not included in any of the study involving mono

substituted allyl systems. Nevertheless, in some cases, speculations of

probable transition states were made on the basis of the reaction between

formaldehyde and methyl lithium rather than actually locating the

transition states. Based on such model reactions it was anticipated that the

counter ion may be substantially involved in the transition structures for

the addition reaction with carbonyl compounds. It was also concluded that

77

the carbonyl attack will be favoured on that carbon to which the counter

ion is closer. Consequently, the identification of transition states for the

alternate α and γ attack for the addition reaction between mono-halogen

substituted allyl systems and formaldehyde is still an unsolved problem.

• To the best of our knowledge and belief, no systematic theoretical work

has been undertaken for analyzing the regioselectivity preferences of

mono substituted allyl systems with alkyl halides.

• Some of the reported transition structures for the reaction between

dihalogen substituted allyl systems and formaldehyde were found to be

wrong ones based on IRC calculations.

• Earlier studies on allyl systems didn’t employ adequate model chemistry

(electron correlation methods using moderate level basis sets) for

geometry optimization (though single point calculations at the MP2 level

were performed in many cases).

• The regioselectivity predictions were more or less based on the HSAB

principle,11

the Mulliken group charges and the effect of polarization of

HOMO of the allyl unit.12

• Effect of substituents on the regioselectivity was not systematically

addressed in the earlier works.

2.2.3. Objectives, model chemistry & plan of execution of the work

In the context discussed in section 2.2.2., we undertook the present work

which tries to address most of the above listed limitations. The present work

include the following,

1. Evaluation of the global and local electrophilicity descriptors of hetero

substituted (both mono and di substituted) allyl systems with a view to

systematically characterizing the regioselectivity preferences. The effects of

substituents and counter ion on the global and local electrophilicity

descriptors of these allyl systems will also be investigated. Different sets of

78

substituents are considered for this purpose. They include, halogens (F, Cl,

Br), 16th

group substituents (OH, SH & SeH), 15th

group substituents (NH2,

PH2 & AsH2), deactivating groups (like CN, NC, NO2 & ONO) and activating

groups (like OCH3 & SCH3). In each case, the effect of Li+, Na

+, K

+ as

counter ions will also be investigated. Such a study involves the following

computational calculations

• Geometry optimization of all substituted allyl systems

mentioned above

• Evaluation of global and local electrophilicity descriptors of

these systems

2. Detailed investigation of two classes of reactions covering all the aspects of

their regioselectivity preferences such as the thermodynamic control, the

activation control etc. The gas phase reactions of mono-halogen substituted

allyl systems with methyl chloride (scheme 2.1) and with formaldehyde

(scheme 2.2), were selected for this purpose. These reactions are considered

as theoretical models for testing the performance of Density functional theory

(DFT) based reactivity descriptors.

-

M+

X

α γ

+

M+ = Li+, Na+, K+

X = F, Cl, Br

CH3Cl

XX

α γα γ

OR + MCl

α product γ product

Scheme 2.1

-

M+

X

α γ

+

M+ = Li+, Na+, K+

X = F, Cl, Br

HCHO

X

MOH2C α γ

α product

OR

X

α γ

γ product

CH2OM

Scheme 2.2

79

Such a study involves the following computational calculations

• Geometry optimization of the allyl systems to locate the stable

isomers at a higher level of theory than used in the previous works.

• Investigation of the effect of solvent on the energy and geometry of

stable isomers of these allyl systems using solvation models.

• Geometry optimization of α and γ products with a view to identifying

all the possible isomers and hence the thermodynamic product of the

above reactions.

• Transition state optimization for finding out the transition states for

the alternate α and γ products and hence the kinetic product.

• IRC calculations for characterizing the transition states.

• Computation of atomic charges using population analyses (such as

NPA, CHelpG and MKS) other than the Mulliken population analysis

(which was the only one used in earlier studies).

3. The addition reaction between some of the dihalogen substituted allyl

systems and formaldehyde (scheme 2.3) will be simulated at two different

levels of theory - one at the HF/3-21+G* used by Canepa and another at the

present level.

-

M+

X

α γ

+

M+ = Li+, K+

X = F, Cl,

HCHO

X γ

α product

OR

X

α γ

γ product

CH2OM

X

MOH2C

X

α X

Scheme 2.3

The major objective of this portion of the work is to find out the correct

transition states in place of the wrongly identified transition structures in the

earlier study and look for any difference in the conclusion made therein.

4. Characterization of the regioselectivity preferences of the sulphur stabilized

allyl lithium compounds, 1-thiophenylallyllithium figure 2.2 (a), its

80

sulphoxide figure 2.2 (b) and its sulphone figure 2.2 (c) using DFT based

reactivity descriptors.

-

Li+

SPh

-

Li+

PhSO

-

Li+

PhSO2

Figure 2.2: sulphur stabilized allyllithium compounds

(a) (b) (c)

One of the major hurdles before getting on with a computational research

project is the selection of suitable model chemistry. It is always extremely

difficult to select the right one from a plethora of many. Two things had to be

kept in mind while selecting one – computational cost and accuracy. We

decided to go with a moderate level model chemistry B3LYP/6-31+G(d)

whose selection is justified in section 2.2.4.

2.2.4. Justification of the model chemistry used in the study

The selection of good model chemistry is one of the most important tasks in

any computational research work. Since one of the major objectives of the

present work is to characterize the regioselectivity preferences of

heterosubstituted allyl systems in terms of DFT based reactivity descriptors,

the choice of the method was made with relative ease: the method should be a

DFT method. DFT methods are classified into different categories based on

the exchange-correlation functionals used (see section 2.5 for a detailed

discussion). Regarding the exchange-correlation functional to be used for the

work, the choice was most obvious. The B3LYP method was chosen since it

is the most widely and successfully used exchange-correlation functional in

dealing with a large variety of molecules (both organic and inorganic). In fact,

the B3LYP functional (which is a hybrid GGA) is largely responsible for

DFT becoming one of the most popular tools in computational chemistry. It

has been well-established that B3LYP functional is better for main-group

81

chemistry than for transition metals. In the present work, only main group

elements are involved and hence we believe that the choice is always a good

one.

The choice of the basis set was made after examining several previous studies

done on similar systems which demanded results of almost quantitative

accuracy. Barbour and Karty used B3LYP/6-31+G* level of theory for

successfully calculating the resonance energies of allyl cation and allyl

anion13

. A brief account of their work is given in the following section.

Resonance energy in allyl anion: To evaluate the resonance energy in allyl

anion, the following two reactions were considered (scheme 2.4).

H2C

HC

CH3 H2C

HC

CH2- + H+

(2.1)

H3C

H2

C

CH3+ H+

(2.2)H3C

H2

C

CH2-

Scheme 2.4

Enthalpy change of reaction 2.1 gives the acidity of propene [(∆acidHº)propene]

and that of reaction 2.2 gives the acidity of propane [(∆acidHº)propane].

[(∆acidHº)propene] is greater than [(∆acidHº)propane] and the difference is taken to

be due to resonance energy in allyl anion, based on the following arguments.

The difference in acidity between propene and propane is taken to be mainly

due to resonance and inductive effects provided by the vinyl group [CH2=CH-]

in allyl anion. This is because of the fact that the CH3-CH2 group in propane

and propyl anion does not contribute significantly via induction (there are no

significantly electronegative atoms) and does not contribute via resonance.

Thus the acidity difference between propene and propane is attributed to the

sum of resonance and inductive effects provided by the vinyl groups in allyl

anion and propene [equation 2.1]. Given that resonance is not active in

82

propene, the total resonance contribution toward the acidity enhancement of

propene over propane is taken to be the resonance energy in allyl anion.

Similarly, the resonance energy in allyl cation is computed as the difference

in hydride abstraction enthalpies (∆HAHº) of the following reactions (scheme

2.5).

H2C

HC

CH3 H2C

HC

CH2+ + H-

(2.3)

H3C

H2

C

CH3+ H-

(2.4)H3C

H2

C

CH2+

Scheme 2.5

The difference in hydride abstraction enthalpies between propene and propane

is taken to be due mainly to resonance stabilization in allyl cation.

∆E = [(∆HAHº)propene] - [(∆HAHº)propane] (2.5)(2.5)(2.5)(2.5) The acidities and hydride abstraction enthalpies calculated at the B3LYP level

using 6-31+G(d) basis set are in good agreement with available experimental

values as well as previous DFT calculations14,15

. Some numerical values are

collected in table from the work of Barbour of Karty. The resonance energy

calculated are in reasonably good agreement with that calculated by others16

,

demonstrating that there is significant resonance stabilization in both the

cation and the anion of about same magnitude.

Table 2.4: Acidity & hydride abstraction enthalpy of propane and propene

Molecule

Acidity (kcal/mol) Hydride abstraction enthalpy

AM1 B3LYP/6-

31+G(d) Expt. AM1

B3LYP/6-

31+G(d) Expt.

Propane 336.2 414.3a 415.6 317.6 309.0 307.1

Propene 335.9 387.6b 390.0 305.2 293.8 291.4

aWhen 6-311+G** was used the value computed becomes 414.4 (No significant improvement).

bThe value becomes 384.5 when B3LYP/6-311++G (2df, 2pd)//B3LYP/6-31+G(d) model

chemistry was used (the value deviates more from experimental value).

83

The study was extended to the vinylogues of propene and propane (n=1 to 3)

and good results were obtained. In an early study17

, Schleyer, has proved that

relatively simple 4-31+G basis set (which include a set of diffuse functions)

give good results with allyl anions. In this work, proton affinity of allyl anion

computed at the MP2/4-31+G//HF/4-31+G level of theory is compared with

experimental values. Proton affinity of allyl anion was determined

experimentally by two independent works: one by Oakes and Ellison18

(using

photoelectron spectroscopic measurements) and another by Mc. Kay et. al19

.

(using flowing afterglow methods). The value was found to be of the order of

391+1 kcal/mol. Schleyer exactly reproduced the result with single point

MP2/4-31+G calculations on the geometry optimized at the HF/4-31+G level.

This work pointed towards the importance of including a set of diffuse

functions for modeling the behavior of outer electrons (which are not strongly

bound) in anions. The above conclusion is based on the fact that the earlier

calculations on anions which did not include diffuse functions gave

unsatisfactory results.

In another work, on the structure of allyl anion Schleyer concludes that the C-

C bond length in allyl anion is unlikely to differ by more than 0.02 Aº from

the values calculated using 6-31+G* basis set20

. In order to justify the use of

6-31+G* basis set in substituted allyl anions we refer to the work of Kass. et.

al21

. They did MP2/6-31+G* single point calculations on the geometry

optimized at the HF/6-31+G* level of theory on a series of 1-substituted allyl

anions. The computed and experimental acidities reported in their work is

summarized in table 2.5, to show the effectiveness of 6-31+G* basis set in

fairly reproducing the chemistry in these kinds of systems.

84

Table 2.5: Calculated & experimental acidity values of 1-substituted allyl anions

Substituent Calculated Acidity (kcal/mol) Experimental Acidity

(kcal/mol) MP2/6-31+G*//HF/6-31+G*

-H (propene) 388.7 390.0

-NH2 383.2 390+4

-OH 383.2 390+4

-F 383.5 390+4

Gas phase experiments carried out with a variable temperature flowing

afterglow apparatus showed that 3-flouropropene, 3-hydroxypropene and 3-

amino propene have identical acidities. The computed results justified the

observation and reasonable agreement has been observed with the

experimental values.

In another work, on the structure of sulphur stabilized allyllithium compounds

in solution Anders and coworkers also used the B3LYP/6-31+G(d) model

chemistry22

. They used computations at the B3LYP/6-31+G(d) level and

experimental investigations (NMR and cryoscopic measurements) for the

structural assignments in solution for a series of three sulphur stabilized

allyllithium compounds (see figure 2.1). This work establishes the utility of 6-

31+G* basis set in dealing with sulphur stabilized allyl systems. Moreover,

we wanted to examine the regioselectivity preferences of these compounds

based on the DFT based reactivity descriptors. So we opted to stick on to the

same level of theory in our work.

All the works cited above, point to the fact that the 6-31+G* basis set is quite

useful in satisfactorily modeling the behavior of electrons in allyl anions and

substituted allyl systems. Thus, we selected the model chemistry B3LYP/6-

31+G(d) for our work.

85

2.3. Computational Methodology

2.3.1. In brief

In recent years, theoretical methods based on DFT have emerged as an

alternative to traditional ab initio methods in the study of structure and

reactivity of chemical systems (section 1.7). A unique combination of a

computational method and a basis set is known as model chemistry (see

section 1.9). In the present study, geometry optimizations have been carried

out using the B3LYP/6-31+G(d) model chemistry. The optimizations were

performed using the Berny analytical gradient optimization method.23

The

stationary points were characterized by frequency calculations at the same

level in order to verify that minima on the potential energy surfaces have no

imaginary frequency. Atomic charges were evaluated by various population

schemes such as Mulliken population analysis (MPA), natural population

analysis (NPA) and electrostatic methods (CHelpG and MKS). The solvent

effects have been studied at B3LYP/6-31+G(d) level as single point

calculations on the stable gas phase geometries. The method used was the

self-consistent reaction field (SCRF)24

based on the polarizable continuum

model (PCM).25

Activation energy of different reactions considered were

computed through transition state optimization (keyword: OPT=TS) using the

same model chemistry. Intrinsic Reaction Coordinate (IRC) analysis was used

to characterize the optimized transition states. The condensed to atom fukui

functions (fk-) for electrophilic attack and global electrophilicity descriptors

have been calculated from single point calculations at the B3LYP/6-31+G(d)

level of theory using the module developed by Contreras et. al.26

All

calculations were carried out using the Gaussian 03 suite of programs.27

Since the present work largely uses DFT and reactivity descriptors based on

DFT as the basic tools, it is important and useful to discuss the principles

involved in and the evolution of DFT and Conceptual DFT in the present

86

chapter. In addition, brief descriptions of the software used and of various

techniques employed in the different calculations should also be discussed.

The remaining sections of the present chapter deal with these aspects.

2.3.2. Density functional theory

2.3.2.1. Introduction

The foundation of DFT can be traced back to one basic question (see section

1.7). Is it necessary to solve the Schrödinger equation and determine the 3N

dimensional wave function in order to compute the ground state energy? The

Hamiltonian operator (equation 1.4) consists of a single electron and bi

electronic interactions – i.e., operators that involve on the coordinates of one

or two electrons only. Therefore in the non-relativistic treatment, the total

energy depends only on averages involving no more than two electrons at a

time (equations 1.10 to 1.13). In a sense, the wavefunction of a many electron

molecule contains more information than is needed and is lacking a direct

physical significance. This has prompted a search for functions that involve

fewer variables than the wavefunction and that can be used to calculate the

energy and other properties. The above argument suggests that there exists

some function of two electrons which we could use instead of the N- electron

wave function. Consequently, we need to introduce a new mathematical

construct which is more general than a wave function and of which a wave

function is a special case. Such a construct is the density matrix28. Density

matrix in some sense describes the degree to which individual basis functions

contribute to the many electron wavefunction. 29

Therefore it is concluded that

the diagonal elements of the first (a function of the spatial coordinates of one

electron) and second order (a function of the spatial coordinates of two

electrons) density matrices (P1 and P2) completely determine the total

energy.30

Energy in terms of P1 and P2 can be represented as (see HF equations

1.10 to 1.13)

87

= −∇2 +−

+ 1 (. )

This appears to vastly simplify the task in hand. The solution of the full

Schrodinger equation for Ψ is not required; it is sufficient to determine P1 and

P2 and the problem in a space of 3N coordinates has been reduced to a

problem in six dimensional space. Unfortunately, no convenient principle

(analogous to the variation principle used to calculate the wavefunctions) has

been developed that would allow direct calculation of these density matrices

without first requiring calculation of the wavefunction. The observation which

underpins DFT is that we do not even require P2 to find the total energy. The

g.s. energy is completely determined by the first order density matrix, P1 – the

charge density. DFT explicitly recognizes that non-relativistic systems differ

only by their potential and supplies a prescription for dealing with the

universal operators T and V.

2.3.2.2. The Hohenberg and Kohn theorems

The entire DFT is built upon the two basic theorems suggested by Hohenberg

and Kohn.31

The Hohenberg and Kohn theorem-1: The ground state wavefunction, Ψ0, is a

unique functional of the ground state electron density, ρ0(r) i.e. Ψ0=

Ψ[ρ0(r)]. As a consequence, the ground state expectation value of any

observable, A, is a functional of ρ0(r). i.e.

⟨ ⟩ = "Ψ[ρ&(r)]) *)Ψ[ρ&(r)]+(. ,)where, * is the quantum mechanical operator corresponding to the observable

A

88

The Hohenberg and Kohn theorem-2: This theorem introduces the variational

principle into DFT; for a trial density, ρ(r) such that ρ(r) ≥ 0, and ∫ ρ(r) dr = N,

(where N is the total number of electrons)

&[ρ&()] ≤ [ρ(r)](. .)i.e. the energy calculated using the trial electron density is never going to be

less than the actual ground state energy of the system.

An elegant method of enforcing constraints during an optimization is the

Lagrange method. Suppose that the function to be optimized depends on a

number of variables

f (x1,x2…,xn) and the constraint condition can always be written as another

function,

g (x1,x2…,xn) = 0; the Lagrange function, L, can be defined as the original

function minus a constant times the constraint function.

i.e., L (x1,x2…,xn, λ) = f (x1,x2…,xn) – λ g (x1,x2…,xn). (2.9)(2.9)(2.9)(2.9)If there are more than one constraints, one additional multiplier is added for

each constraint. The optimization is then performed on the Lagrange function

by requiring that the gradient with respect to x and λ variables is equal to

zero. In many cases the multipliers can be given a physical interpretation at

the end.32

Applying the Lagrange method to the minimization of [ρ(r)] subject to the

constraint ∫ ρ(r) dr - N = 0, the fundamental statement of DFT can be written

as,

012[3(4)] − 5(∫ 3(4)74– 9): = ;(. <;) where,δ stands for the functional derivative (i.e. the derivative with respect to

a function). If the exact electron density is known, then the cusps in ρ(r) will

provide the positions of the nuclei. The slope of ρ(r) at the nucleus A, must

obey

89

>∂ρ(@A)∂r BCD&

= −2ZFρ(0)(. <<) (due to E B Wilson 1965) where ρ denotes the spherical average of the

density.

Suppose one gives to an observer a visualization of the function ρ(r) and tells

him that this function corresponds to the ground state electron density of a

molecule, the first HK theorem then states that this function corresponds to

unique number of electrons (N) and constellation of nuclei (i.e. number,

charge and position).Thus the charges at the nuclei are known and hence the

Hamiltonian, as it is completely defined by the nuclear charges and position.

2.3.2.3 The Kohn-Sham approach

HK theorem does not tell us the form of the functional dependence of energy

on the density; it confirms only that such a functional exists. (i.e. how to

calculate E0 from ρ or how to find ρ without first finding Ψ). A solution to this

fundamental issue was given by the Kohn- Sham approach.33

Kohn–Sham

density functional theory (DFT) has become one of the most popular tools in

electronic-structure theory due to its excellent performance-to-cost ratio as

compared with correlated wave function theory.

The Kohn-Sham Approach does not exclusively work in terms of the particle

density, but brings a special kind of wave functions (single particle orbitals)

back into the game. Here, a set of one electron equations are derived from

which, in theory, the electron density, ρ(r), could be obtained. As a

consequence, DFT then looks formally like a single particle theory although

many body effects are still included via the so-called exchange – correlation

functional.

As detailed in section 1.4.1, the electronic energy (by the HF method) of a

molecule can be conveniently written as a sum of three terms (see equation

90

1.14). If we assume that the total energy is a functional of electron density the

same equation can be written in terms of ρ ρ(r) abbreviated as ρ as,

HIJ[ρ] = KI[ρ] + LMI[ρ] + LII[ρ](. <)In equation ( ), potential energy due to nuclear-electronic coulombic attraction

is trivial (see second term in equation 1.11).

LMI[ρ] = LNMIρ(r)dr(. <P)where,

LNMI =−

(. <Q)

The other two functionals are unknown. If good approximations to these

functionals could be found, direct minimization of the energy would be

possible. The KS scheme gives approximations to these functionals. They

introduced a fictitious system of N non-interacting electrons to be described

by a single determinant wave function in ‘N’ orbitals ‘R’ (similar to Slater

determinant; see equation 1.8). The fictitious system of non-interacting

electrons is assumed to have the same ground state density as that of the real

system where the electrons do interact.

In this system,

E[ρ] = Ts[ρ] + VMI[ρ] + VH[ρ] + Exc [ρ] (2.15)(2.15)(2.15)(2.15) TU[ρ] =

∫ V(@W)V(@X)@WX drdr(. <)

YZ[ρ] =R(1)−∇2 R(1)[

dr(. <,)

Exc [ρ] = T[ρ] - Ts[ρ] + Vee[ρ] - VH[ρ] (2.18)(2.18)(2.18)(2.18)

91

Where, T[ρ] - Ts[ρ] is the error made in using a non-interacting K.E. and

Vee[ρ] - VH[ρ] is the error made in treating the electron- electron interaction

classically. Kohn and Sham also proved that the total electron density is given

by the expression

ρ =|R|(. <^)[

_

Exc is called the exchange correlation (XC) functional. Thus, KS theory

permits the K.E. to be computed as the expectation value of the K.E. operator

over the KS single determinant, avoiding the tricky issue of determining the

K.E. as a functional of the density. Writing the functional explicitly in terms

of the density built from non-interacting orbitals and applying the variational

theorem one can find that the orbitals, which minimize the energy, satisfy the

one electron equations

−∇2 −

+ρ() + ab(1)R(1) = ε,efR(1)(. ;)

Where, `ab = 02gh[V]0V is the functional derivative of the XC functional. The

above equation can be abbreviated as

ijef(1)R(1) = k,efR(1)(. <)This equation is called the Kohn – Sham equation and has the same structure

as the HF equation (see equation 1.15). Here ijef is called the Kohn-Sham

operator and its only difference with the HF operator (in equation 1.15) is that

the non-local exchange potential lNm(n)is replaced by the local exchange-

correlation potential vxc .

92

2.3.2.4. DFT vs HF

HF is a deliberately approximate theory, whose development was in part

motivated by the inability to solve the relevant equations exactly, while DFT

is an exact theory, but the relevant equations must be solved approximately

because a key operator has an unknown form. Although exact DFT is

variational, this is not true once approximations for Exc are adopted. DFT

optimizes an electron density while MO theory optimizes a wave function. So

to determine a particular molecular property using DFT, we need to know

how that property depends on the density, while to determine the same

property using a wave function, we need to know the correct quantum

mechanical operator. The KS orbitals are not an approximation to the wave

function. They have no physical significance other than in allowing the exact

ρ to be calculated. In practice, the shapes of KS orbitals tend to be remarkably

similar to HF MOs, and they can be quite useful in qualitative analysis of

chemical properties.

2.3.2.5. Exchange-correlation functionals

The accuracy of a DFT calculation depends upon the quality of the exchange–

correlation (XC) functional. The past two decades have seen remarkable

progress in the development and validation of XC density functionals. In

principle, the XC functional not only accounts for the difference between the

classical and quantum mechanical electron-electron repulsion, but it also

includes the difference in kinetic energy between the fictitious non-interacting

system and the real system (equation). However, in practice, most modern

functionals either ignore the latter or incorporate some empirical parameters

which introduce the required kinetic energy correction. The functional

dependence of Exc on the electron density is usually expressed as an

interaction between the electron density,ρ, and an energy density,kpq[ρ].ρ is

a per unit volume density and kpq[ρ] is a per particle density.

93

Epq[ρ] = ρkpq [ρ]dr(. )

The energy density is always treated as a sum of individual exchange and

correlation contributions.

kpq = kp + kq(. P) Depending on the ways in which approximations tokpandkq are arrived at

the XC functionals can be classified into mainly three classes: the local

density methods, the Generalized Gradient methods and Hybrid methods.

The Local Density Methods: In these methods it is assumed that the density

locally can be treated as a uniform electron gas. The concept of uniform

electron gas was introduced by Thomas and Fermi in an attempt to express

energy as a functional of electron density (much earlier to the invention of the

KS formalism of DFT). It is a fictitious system (also called Jellium) of an

infinite number of electrons moving in an infinite volume of space that is

characterized by a uniformly distributed positive charge (i.e. the positive

charge is not particulate in nature, as it is when represented by nuclei). Such a

system has a constant non-zero densityuρ = vw = xyz|x~. The exchange

energy for a uniform electron gas is given by the Dirac formula and is used as

the exchange functional in a Local Spin Density Approximation (LSDA)

method.

a = −aρ (. Q)ka = −aρ (2.25)(2.25)(2.25)(2.25)

Where,a = u~

(. )

94

In the more general case, where α and β spin densities are not equal, Local

Spin Density Approximation (LSDA) is used. In LSDA, exchange energy is

given as the sum of α and β spin densities raised to 4 ⁄ 3 power.

af = −2 a(ρ + ρ )(. ,)kaf = −2 a(ρ + ρ )(. .)

For closed shell systems LSDA is equal to LDA and since this is the most

common case, LDA is often used interchangeably with LSDA. The

Xαmethod proposed by Slater in 195134 is an LDA method where thecorrelationenergyisneglectedandtheexchangetermisgivenas

k ¡f = −32 αaρ (. ^)Where, α = 1 in the original Xαmethodbutavalueof¾hasbeenshownto give better agreement for atomic and molecular systems. Thecorrelation energy of a uniform electron gas has been determined byMonte Carlo methods for a number of different densities. One of theearlierkq functionalisduetoVosko,WilkandNusair,commonlyknownasVWNfunctional.35LSDAcalculationthatemploysacombinationofSlaterexchangeandtheVWN correlation is sometimes referred to as SVWN method. AnothermodifiedformofcorrelationfunctionalwasgivenbyPerdewandWang.36kb®¯° = −2xρ(1 + αx) ln u1 +

±(Wp²XpX²³p³²´p´)~(2.30)(2.30)(2.30)(2.30)where,a,α, β, β, β&βaresuitableconstants.ItisobviousthattheXCfunctionals are very complex and it is very difficult to have a firstprinciples analysis here. So the actual functional form of the otherfunctionalswon’tbeincludedinthissection.Instead,thebasicprinciplesinvolved in thedevelopment of these functionalswill bediscussed in ageneralway.Themeritsanddemeritsofthedifferentfunctionalsarealso

95

included. The LSDA, in general, underestimates the exchange energy by ~ 10% and over estimates correlation energy by a factor close to 2. Despite the simplicity of the fundamental assumptions, LSDA methods are often found to provide results with accuracy similar to that obtained by the HF method. Although LSDA gives surprisingly accurate predictions for solid-

state physics, it is not a useful model for chemistry due to its severe over-

binding of chemical bonds and underestimation of barrier heights.

Generalized gradient approximation (GGA): Such methods make

improvements over LSDA by considering a non-uniform electron gas. They

do so by making the exchange and correlation energies dependent not only on

the electron density but also on its derivatives. Perdew and Wang proposed PW8637 and Becke proposed B or B8838 exchange functionals modifying ka. Another exchange functional in this category is due to Becke and Roussel (BR).39 Perdew and Wang had also proposed another exchange functional to be used along with the PW91 correlation functional.40 It should be noted that several of the proposed functionals violate fundamental restrictions, such as predicting correlation energies for one electron systems (for example, P86 and PW91) or failing to have the exchange energy cancel the coulomb self-repulsion. One functional which does not have these problems is developed by Becke and is known as B95.41 There have been various functional forms proposed for the correlation energy. The most popular among them is the LYP functional developed by Lee, Yang and Parr.42 Note that it is not a correction to LSDA but is designed to compute the full correlation energy. Meta-GGA functional: Functionals that depend explicitly on the Laplacian of

spin density (∇ρ) or on the local kinetic energy density, τ, are referred to as

meta-GGA functionals. The functional form is

96

ab = Ãkab(Ã, |∇Ã|, ∇Ã, Ä)(. P<) where,

Ä = 12|∇R|

(. P) Hybrid methods: Imagine that one could control the extent of electron-

electron interactions in a many electron system. That is, imagine a switch that

would be smoothly converting the non-interacting KS reference system to the

real interacting system. Using the Hellmann-Feynman theorem, it can be

shown that the exchange-correlation energy can be computed as

ab = < Æ(Ç)& | ab(Ç)|Æ(Ç) > Ç(. PP)

where, Ç,describes the extent of inter-electronic interaction, ranging from

zero to exact; Ç = 0 for non-interacting electrons; Ç = 1 for real system. In

the crudest approximation (taking `ab to be linear in x) the integral is given as

the average of the values at the two end points (i.e., at Ç = 0 and Ç = 1

ab ≈ 12 < Æ(0)| ab(0)|Æ(0) > +12 < Æ(1)| ab(1)|Æ(1) > (. PQ) In the Ç = 0 limit, the electrons are non-interacting and consequently there is

no correlation energy only exchange energy. Furthermore, since the exact

wavefunction in this case is a single Slater determinant composed of KS

orbitals (see eqn. 1.8 where each R; a KS orbital), the exchange energy is

exactly that given by the HF theory (eqn. 1.13). If the KS orbitals are identical

to the HF orbitals, the “exact” exchange is precisely the exchange energy

calculated by the HF method. The last term in eqn. (2) is still unknown. In the

Half-and-Half (H+H) method43

that term is defined as the exchange-

correlation functional in LSDA

97

abU²U = 12aÊaËbÌ + 12 (af + bf)(. PÍ)Since the GGA methods give a substantial improvement over LDA, a

generalized version of Half-and-Half method may be defined by writing the

exchange energy as a suitable combination of LSDA, exact exchange and

gradient correction term. The correlation energy may be similarly taken as the

LSDA formula plus a gradient correction term

abÎ = (1 − x)af + xaÊaËbÌ + Ï∆aÎÑÑ + bf + y∆bÒÒ (2.36)(2.36)(2.36)(2.36)Where a, b and c parameters are determined by fitting to experimental data

and depend on the form chosen for bÒÒ typical values are a≈0.2, b≈0.7 and

c≈0.8.

Models which include exact exchange are often called hybrid methods. Becke

3 parameter functional and Adiabatic Connection Model (ACM) are examples

of such hybrid methods. The B3LYP method is defined as

abÎÓ® = (1 − x)af + xaUÔ + Ï∆aÎ + (1 − y)bf + y∆bÓ®

and B3PW91 is defined as

abÎ®¯° = (1 − x)af + xaUÔ + Ï∆aÎ + bf + y∆b®¯°

in both a, b and c parameters have the same values a=0.2, b=0.72 and

c=0.81.

Table 2.6: Basis of classification of XC functionals

Family Dependencies

Hybrid exact exchange, |∇Ã|,Ã

Meta-GGA ∇Ã, Ä

GGA |∇Ã| LDA Ã

98

Table 2.7: Some of the XC functionals and the type to which they belong

Density Functional Exchange / Correlation Type

Xα Exchange LDA

VWN Correlation LSDA

PW86 Exchange GGA

B or B88 Exchange GGA

BR Exchange GGA

PW91 Exchange and Correlation GGA

LYP Correlation GGA

P86 Correlation GGA

B95 Correlation GGA

B3 Exchange Hybrid

B3LYP Exchange and Correlation Hybrid GGA

BLYP Exchange and Correlation GGA

BP86 Exchange and Correlation GGA

BPW91 Exchange and Correlation GGA

B3P86 Exchange and Correlation Hybrid GGA

B3PW91 Exchange and Correlation Hybrid GGA

PBE44

Exchange and Correlation GGA

B9845

Exchange and Correlation Hybrid GGA

PBEh or PBEO46

Exchange and Correlation Hybrid GGA

TPSSh47

Exchange and Correlation Hybrid meta GGA

BMK48


MO5-2X49


A review of earlier XC functionals can be found elsewhere.50

The authors

show how such functionals can be derived in a systematic fashion via a

perturbation expansion, utilizing the KS system as a non-interacting reference

system. A simple and systematic approach to the generation of XC

functionals in DFT by linear least squares fitting to accurate thermochemical

99

reference data like G2 data set is summarized.51

DFT yields accurate total

energies of atoms and molecules. However, most of them do not reproduce

the well known R-6

behavior characterizing the van der Waals (vdw)

interaction (interaction of two widely separated neutral fragments). The true

correlation energy functional must include the vdw interaction. The

conventional LDA and GGA are essentially local; i.e., the `ab (r), at a point r

is determined by the density and its low-order gradients at the same point r.

The description of long range forces such as the vdw interaction requires fully

non-local functionals. There have been several suggestions for constructing

XC functionals yielding the vdw interaction.52

Of all modern functionals B3LYP has proven to be the most popular. The

B3LYP functional which is a hybrid GGA is largely responsible for DFT

becoming one of the most popular tools in computational chemistry. But

some of the shortcomings reported with this functional include its inability to

describe van der Walls complexes bound by medium range interactions, its

unreliable performance for transition metal chemistry and its systematic

underestimation of barrier heights. Zhao and Truhlar developed a series of

new generation density functionals which attempt to overcome the above

mentioned shortcomings: starting with the MPW1K functional53

(a hybrid

GGA) in 2000, upto the MO6-class developed recently.54

They have also

developed some database for systematically checking the validity of newly

developed density functionals. TMAE955

(a data base of bond energies in nine

transition metal dimers) and MLBE2156

(a metal-ligand database) are to name

a few. Strengths and weaknesses of different types of exchange-correlation

functionals are also reviewed in their report.

Over the years, different kinds of XC functionals have been reported and

extensively tested. Some of them withstood the test of time and still survive

(the best example is the B3LYP) while many of them were discarded. The

search is on and the ultimate aim is to get to that magical XC functional

100

which is exact. If we somehow reach there (chances are rare since there is no

way of systematically improving the XC functionals), then quantum

mechanics of multi electron species will be no more approximate. That will

be the greatest revolution in the history of quantum mechanics after the theory

itself.

2.3.3. Conceptual DFT and reactivity descriptors


Density functional theory has revolutionized the evolution of quantum

chemistry during the past 20 years. Born out of the basic idea that the electron

density, ρ(r), at each point r determines the ground state properties of atomic

or molecular system, DFT has grown into a full-fledged quantum mechanical

technique through the works of Hohenberg, Kohn and Sham as detailed in

section 2.3.1. A central paradigm in ab initio quantum chemistry is the

structure-properties-wavefunction triangle, structure and properties further

determining reactivity. This has evolved into structure-properties-electron

density triangle upon the introduction of DFT (figure 2.3).

Structure

Properties Wavefunction

Structure

Properties Electron density

DFT

Figure 2.3: Evolution of the central paradigm upon the introduction of DFT

DFT as a theory and tool for calculating molecular energetics and properties

has been termed by Parr and Yang “computational DFT”.57

Later on,

theoreticians were able to give sharp quantitative definitions for chemical

concepts such as electronegativity, chemical potential, chemical hardness,

chemical softness etc. This step initiated the formulation of a theory of

chemical reactivity and paved way for a new branch of DFT, namely,

101

“conceptual DFT”. “Conceptual DFT” concentrates on the extraction of

chemically relevant concepts and principles of DFT.58

2.3.3.2. Chemical reactivity descriptors

Relations between the structure of molecules and their reactivity constitute a

fundamental problem of modern chemistry. The term chemical reactivity is

treated as a set of quantitative parameters of possible reaction centers in a

molecule with respect to different reagents and reaction types. These

quantitative parameters are usually called reactivity indices (RI). Thus a

reactivity descriptor or reactivity index is some scalar quantity characterizing

the ability of a molecule as a whole (global descriptor) or its particular

fragment (local descriptor) to undergo a chemical reaction in general or a

certain kind of reactions. Historically, the first calculated RI were charges on

atoms and free valence indices. Probable directions of electrophilic and

nucleophilic reactions were qualitatively estimated on the basis of the

calculated charge distribution over the corresponding atoms. Electronegativity

(Õ), chemical potential (Ö), chemical hardness (×), global softness (S) and

global electrophilicity (ω) are some of the commonly used global descriptors.

Local reactivity descriptors include the Fukui function, local softness, local

philicity etc. Definitions, significance and methods of evaluation of these

descriptors will be outlined in the following sections.

2.3.3.3. Global descriptors

Electronic chemical potential and electronegativity

The abstract Lagrangian multiplier, µ, in the basic equation of DFT, δ E [ρ] -

µ (∫ ρ(r) dr – N) = 0, has been identified as the electronic chemical potential

(or negative of electronegativity) by Parr.59

Ö = ØÙÙÚÛÜ = −Õ(. P,)

102

Thus Ö is the derivative of the energy of the atom or molecule with respect to

its number of electrons at constant external potential (i.e. identical nuclear

charges and positions). The name electronic chemical potential was given in

analogy with the thermodynamic potential, ÖÝÞÊCß = uàÒàá~®,Ý. Chemical

potential can be regarded as a measure of the escaping tendency of electrons

from a species in its ground state. Fundamental problems like the derivative

discontinuity at integral values of N arise while implementing these sharp

definitions. The derivative discontinuity is most easily rationalized using the

MO description: when one adds an electron to a molecule, it goes into the

LUMO while when one removes an electron from a molecule, it comes from

the HOMO. Consequently, in systems with non-degenerate ground states (so

that HOMO and LUMO have distinct energies), the chemical effects of

adding electrons to the system are not simply the opposite of subtracting

electrons from the system. This is because electron addition and electron

removal are associated with different orbitals and consequently, give rise to

qualitatively different chemical effects. This causes derivative discontinuity.

Owing to this, the derivative will be different if taken from the right or left

side resulting in two possible values for ÖxÕ.

Õâ = −uàãàv~Ü (2.38)(2.38)(2.38)(2.38)when the derivative is taken as N decreases from N0 to N0 – δ

Õ² = −uàãàv~Ü (2.39)(2.39)(2.39)(2.39)when the derivative is taken as N increases from N0 to N0 + δ

ÕâxÕ² correspond to the response of the energy of the system to

electrophilic (dN < 0) and to nucleophilic (dN > 0) perturbations respectively.

Within the finite difference approach, the electronegativity is calculated as the

average of the left and right hand side derivatives.

Õâ ≈ (vväâ) − (vvä) ≈ å, ℎçnzn|xnzèzçnxé (2.40)(2.40)(2.40)(2.40)

103

Õ² ≈ (vvä) − (vvä²) ≈ , ℎççéçyzxêênnë (2.41)(2.41)(2.41)(2.41) Õ ≈ ìí²ìî

≈ ï² (2.42)(2.42)(2.42)(2.42)

The energy of species withÚ& − 1 & Ú& + 1 electrons should be evaluated

under the frozen orbitals approach (i.e. there should not be any geometry

change while adding or removing electrons). The above argument gives a

strong theoretical support for the Mulliken’s empirical definition of

electronegativity60

(see figure 2.4).

Figure 2.4: Atomic or molecular energy (E) vs number of electrons (N) at constsnt

external potential – the modern definition of electronegativity

According to the Koopman’s theorem,61

the energy of the HOMO should

represent the ionization energy and that of the LUMO the electron affinity for

a closed shell species. Thus, I and A in the above expression can be replaced

by the HOMO and LUMO energy respectively giving a working equation for

electronegativity in terms of Frontier Molecular Orbitals (FMO).

Õ ≈ ðñòóò²ðôõóò (2.43)(2.43)(2.43)(2.43)This offers a systematic way to calculate electonegativity values for atoms,

functional groups, clusters and molecules (the evaluation was impossible

using the earlier existing scales of electronegativity). For a system with non-

integer number of electrons adding a fraction of an electron will increase the

occupation number of the partially occupied MO(s) while removing fraction

of an electron will decrease the occupation number of the same partially

104

occupied MO (s). In such a case, the chemical effect of adding and removing

electrons are exactly opposite because the HOMO and LUMO are the same.

Thus there is no derivative discontinuity in this case.

Various other quantities representing the response of system’s energy to

perturbation in its number of electrons and/or its external potential were also

defined.

Chemical hardness and softness: The concepts of chemical hardness and

softness were introduced in the early 1960s by Pearson, in connection with

the study of generalized Lewis acid-base reactions62

.

A + :B A

Lewis acid

(e- pair acceptor)

Lewis base

(e- pair donor)

B

Scheme 2.6

On the basis of a variety of experimental data, Pearson presented a

classification of Lewis acids in two groups (class a & class b) starting from

the classification of the donor atoms of the Lewis bases in terms of increasing

electronegativity:

As < P < Se < S ~ I ~ C < Br < Cl < N < O < F

The criterion used was that Lewis acids of ‘class a’ would form stabler

complexes with donor atoms to the right of the series, whereas, those of ‘class

b’ would preferably interact with donor atoms to the left.

‘class a’ acids:- the acceptor atoms positively charged and having small

volume (H+, Li

+, Na

+, Mg

2+ etc.)

‘class b’ acids:- acceptor atoms with low positive charge and having greater

volume (Cs+, Cu

+ etc.)

This classification turns out to be based on polarizability, leading to the

classification of bases as ‘hard’ and ‘soft’.

105

Hard bases :- low polarizability (NH3, H2O, F- etc.)

Soft bases :- high polarizability (H-, R

-, R2S etc.)

Pearson stated his HSAB principle as hard acids preferably interact with hard

bases and soft acids with soft bases. Detailed reviews of Pearson’s HSAB

principle can be found elsewhere.63

However, the classification of a new acid

or base is not always so obvious and the inclusion of a compound on a

hardness or softness scale is not straightforward. The lack of a sharp

definition, just as was the case with Pauling’s electronegativity, is again

causing the difficulty.

The lack of sharp definitions for these quantities prevailed until Parr and

Pearson identified chemical hardness as the second derivative of the energy

with respect to the number of electrons at fixed external potential.64

Chemical hardness, × = uàXãàvX~Ü. (2.44)(2.44)(2.44)(2.44)

It represents the resistance of a system to change its number of electrons.

Based on the finite difference method,

× ≈ kö÷øù − kúùøù2 (. QÍ) The above equation indicates that hardness is related to the energy gap

between occupied and unoccupied orbitals of a molecule. Derivative

discontinuity problems similar to those described for the electronegativity will

be encountered. To tackle such problems different methods had been

suggested in literature. Komorowski’s approach is to take as the hardness the

average of the neutral and negatively charged atom or the neutral and

positively charged atom respectively for acidic and basic hardness.65

Alternatively, Chattaraj and co-workers had proposed three types of hardness

for electrophilic, nucleophilic and radical attack (in analogy with

equations…..)66

Chemical hardness is related to other atomic or molecular properties. Global

softness, S, is one among them and is defined as the reciprocal of hardness

106

(equation.). It is regarded as a measure of the polarisability of the system.

Various studies relating atomic polarisability and softness confirm this view.

Global softness, S = η

(2.46)(2.46)(2.46)(2.46) Global electrophilicity index: The allyl systems selected for the present study

act as nucleophiles in their reaction with aldehydes or alkyl halides.

Nucleophiles can also be regarded as Lewis bases or reducing agents.

Therefore there is a strong connection between electrophile-nucleophile

chemistry, acid-base chemistry and oxidation-reduction chemistry. The study

of polar processes involving the interaction of electrophiles and nucleophiles

may be significantly facilitated if reliable scales of nucleophilicity and

electrophilicity are available. The utility of such global reactivity scales is of

great importance to answer some fundamental questions in chemistry such as

reaction feasibility (whether or not a reaction will take place) or

intermolecular selectivity (which one will be more reactive). There have been

numerous attempts to classify atoms, molecules and charged species within

empirical scales of electrophilicity and nucleophilicity and some good

reviews can found elsewhere.67

Maynard et al. based on their study68

of

reaction rates of some proteins with several electrophilic agents qualitatively

suggested that electronegativity squared divided by hardness is a measure of

the electrophilic power of a ligand, i.e. its capacity to “soak up” electrons.

Inspired by this work, Parr et al. have done a theoretical study in which they

considered an electrophilic ligand immersed in an idealized zero-temperature

free-electron sea of zero chemical potential. By theoretically studying the

electron transfer ∆N from the electron sea to the ligand (until the chemical

potentials of the ligand and the sea become equal), they suggested ω = µXη

(where, µ is the electronic chemical potential and η is the chemical hardness)

as the measure of the electrophilicity of the ligand.69

In view of the analogy

between the above equation and the equation for power in classical electricity

107

(W=V2/R), ω is suggested as a sort of electrophilic power. ω is called the

global electrophilicity index and it measures the stabilization in energy when

the system acquires an additional electronic charge ∆N from the environment.

Unlike the other empirical scales of electrophilicity and nucleophilicity, the

global electrophilicity index can be regarded as an absolute scale in the sense

that the hierarchy of electrophilicity is built up from the electronic structure of

molecules independent of the nucleophilic partner, which is replaced by an

unspecified environment viewed as a sea of electrons. It may be noted that a

global nucleophilicity index is not necessary because in comparison a system

with lower ω will be more nucleophilic in character. Ever since its

introduction, the global electrophilicity index has been successfully used for

the characterization of a wide variety of electrophiles and nucleophiles.70

Under the finite difference approximation and Koopman’s theorem, ω can be

found out using the relation,

ω ≈ (å + )8(å − ) ≈ (küý + kUýý)8(küý − kUýý)(. Q,)

2.3.3.4. Local reactivity descriptors

Global reactivity descriptors such as electronegativity, chemical potential,

hardness and global electrophilicity index are defined for the system as a

whole. To describe the site selectivity within a molecule (i.e. intra-molecular

reactivity) local descriptors of reactivity have also been proposed.

Fukui function: The Fukui function, f(r), representing the change in electron

density ρ (r) at a given point r, when the total number of electrons is changed,

is by far the most important local reactivity index.71

108

FukuiFunction, ê() = u∂ρ(r)∂N ~` (. Q.)

It can also be defined as the functional derivative of chemical potential with

respect to the external potential, `, at constant N (see definition of Ö) .

ê() = ØþÖþ`Ûv = > þþ`ÙÚB = > þÙÚþ`B(. Q^) The Fukui function f(r) plays a key role in linking frontier MO theory and the

HSAB principle. The above definition of fukui function suffers from the

derivative discontinuity problem at integral number of electrons of atoms and

molecules (see section..)72

, leading to the introduction of both right and left

hand side derivatives.

ê²() = u∂ρ(r)∂N ~Ü+ ≈ ρN+1(r)− ρN(r) ≈ ρLUMO(r) (for a nucleophilic attack

provoking an electron increase in the system). (2.50)(2.50)(2.50)(2.50)êâ() = u∂ρ(r)∂N ~Ü

− ≈ ρN(r)− ρN−1(r) ≈ ρHOMO(r) (for an electrophilic attack

provoking an electron decrease in the system). (2.51)(2.51)(2.51)(2.51)ê&() = î(C)²í(C)

(for radical attack). (2.52)(2.52)(2.52)(2.52)The approximate values in the expressions are results of a finite difference

method.

In the above expressions ρ(r), ρ

²(r)andρâ(r) correspond to the

electron density of the neutral molecule, the cation and the anion respectively.

Sometimes it becomes difficult to analyze site selectivity using these local ‘r’

dependent quantities and one usually prefers to assign a numerical value of a

quantity to an atom or a fragment of a molecule instead of assigning a number

to a point in space. To tackle this problem, the related condensed to atom

Fukui function for the atomic site ‘k’ of the molecule (fkα) has been

introduced. This is based on the idea of integrating the Fukui function over

atomic regions, similar to the procedure followed in population analysis

109

techniques. A three dimensional function is reduced to a number during the

process of condensing. The way to condense a function is arbitrary, as far as

the definition of an atom in a molecule is arbitrary. Therefore one can expect

various different definitions, all giving reasonable results as long as one ask

for trends and tendencies in a family of molecules and not for absolute values.

Different methods had been suggested for the evaluation of condensed Fukui

functions. One of the earlier methods was suggested by Yang and Mortier73

based on a finite difference method coupled with frozen orbital calculations

(i.e. geometry of the molecule is fixed during cation and anion formation). In

their method, the associated electron densities (in equations.) were replaced

by the respective electron populations (i.e. atomic charges).

ê² = qk(N+1) − qk(N) (for nucleophilic attack) (2.53) (2.53) (2.53) (2.53) êâ = qk(N) − qk(N−1) (for electrophilic attack) and (2.54)(2.54)(2.54)(2.54)

ê& = î−í

2 (for radical attack) (2.55)(2.55)(2.55)(2.55) In this method, three different calculations need to be carried out (for the

molecule and also for the cation and anion). There are a number of factors

which influence the calculated condensed to atom Fukui functions using the

above scheme. The choice of the treatment levels (i.e. model chemistry) and

the population scheme (used to condense the electron density) influence the

calculated values of these indices. The main problem in this type of

calculation arises due to the spin multiplicity of the electronic state which is

usually different for the neutral molecule, the anion and the cation. Different

population schemes have their own merits and demerits in handling this

computational difficulty. Some of the studies74

point to the effectiveness of

the Hirshfeld partitioning scheme75

over others like Mulliken and NPA

schemes.

An exact definition for the Fukui function, has been proposed by Senet76

within the Kohn-Sham theory as

110

êZ() = |R()|(. Í)

where α = negative for electrophilic attack →ê is HOMO → HOMO electron

density and

α = positive for nucleophilic attack →ê is LUMO → LUMO electron

density.

The above equation was obtained under the condition that the KS potential is

kept constant. This condition entails not only the external potential is frozen,

but also the electron repulsion, namely, the Hartree potential and the

exchange-correlation potential, are kept fixed in the presence of deriving the

electron density with respect to the number of particles.

Alternatively, Contreras et.al77

have developed a simple scheme of calculation

for direct evaluation of regional Fukui functions in molecules, without

resorting to additional calculations involving ionic species of different spin

multiplicity and arrived at a condensed to site Fukui function given by

ê =ê∈

(. Í,)where

ê = |y| + y y,

(. Í.) where is the overlap integral between the basis functions.

These equations are completely equivalent to that derived by Komorowski et

al.78

from a quite different approach based on the group analysis of atoms in

molecules. This simple formalism has been tested for several benchmark

model reactions that are well documented.79

Methods avoiding population analysis have also been put forward for

condensing the Fukui functions. Four such ways to condense Fukui function

are compared in a recent paper.80

In these methods, instead of population

analysis a numerical integration of the Fukui function over an a priori defined

111

region of the space Ωk is done. The whole space divided into various regions

Ωk and the condensed Fukui function at region k will be

ê = Ω ê() (. Í^) The way to divide the space into various regions in principle is arbitrary and

should be selected judiciously (for example, regions which define an atom, a

bond, a lone pair etc.). The division into different regions is based on the

topological analysis of different scalar functions such as the electron density,

the electron localization function and the fukui function itself. The merits and

demerits of each of them are compared in the paper by Fuentealba et al.71

More recently, a method based on perturbations in the molecular external

potential, which avoids differentiating with respect to the electron number has

been suggested by Ayers et. al.81

In the present work we adopted the method suggested by Contreras et al.68 for

the evaluation of condensed to atom Fukui functions. In fact, they have

developed a module for the calculation of such regional Fukui functions by

taking the FMO coefficients through the Mul Pop link in the Gaussian

Package and performing a single point calculation at the same level at which

the geometry of the molecule has been optimized. We are thankful to

Contreras and his group for providing the module for our calculations.

Local softness: The Fukui function clearly contains relative information

about the reactivity of different region in a given molecule (i.e. a good index

for comparing the intra-molecular reactivity). When comparing different

regions in different molecules, another descriptor called the local softness

s(r) turns out to be more effective. A detailed review can be found

elsewhere.82

This quantity was introduced by Yang and Parr in 198583

and is

defined as

|() = >∂ρ(r)∂Ö BÜ(. ;)

112

It is a local analogue of the Global softness defined by the equation

= >∂Ú∂ÖBÜ (. <)By applying the chain rule, it can be written as the product of the total

softness and the Fukui function,

|() = >∂ρ(r)∂Ö BÜ= >∂ρ(r)∂N B

Ü>∂Ú∂ÖBÜ = ê()(. )

This indicates that ê() redistributes the global softness among the different

parts of the molecule and that s(r) integrates to S

= |() (. P) Though, s(r) and f(r) contain the same information on the relative site

selectivity within a single molecule, s(r), in view of the information about the

total molecular softness, is more suited for intermolecular reactivity

sequences. Owing to the derivative discontinuity two types of local softness

can be identified |²()x|â(). The corresponding condensed to atom

local softness values are given by | = ê ( being positive for nucleophilic

attack and negative for electrophilic attack and k is the atomic site). The

individual values |² and |â might be influenced by basis set limitations and

thus insufficiently take into account electron correlation effects. To overcome

these difficulty two new indices called relative nucleophilicity and relative

electrophilicity were defined by Roy et al.84

çéxn`çyéçzèℎnénynë = |â|² (. Q) Relativeelectrophilicity = |²|â (. Í)

Local philicity: Another powerful local descriptor, called local philicity or

philicity, encompassing all the information about the global and local

descriptors discussed above, has been suggested by Chattaraj et al.85

It is

based on the following arguments. When two molecules react, which one will

113

behave as an electrophile (nucleophile), will depend on which has a higher

(lower) electrophilicity index? This global trend originates from the local

behavior of the molecules or precisely the atomic site that is prone to

electrophilic (nucleophilic) attack. Considering the existence of a local

electrophilicity index say, ω(r) that varies from point to point in an atom, a

molecule or an ion, we may define it in reference to the global electrophilicity

index (ω) as

ω = ω(r)dr(. ) The normalization condition of the local Fukui function leads to the equation

f(r)dr = 1(. ,)ω = ω f(r)dr = ωf(r)dr = ω(r)dr(. .)

Thus the local descriptor ω(r) is argued to be equal to ωf(r) (the product of

the global electrophilicity and the Fukui function). This is called the philicity.

Corresponding condensed to atom philicity for nucleophilic, electrophilic and

radical attacks will given by

ω² = ωê²fornucleophilicattack(. ^)

ωâ = ωêâforelectrophilicattack(. ,;)

ω& = ω

² − ωâ

2 êzxnyxéxxy(. ,<) Local philicity provides the information given by the Fukui function, but the

converse is not true because the Fukui function does not have information

about the global electrophilicity index. Since, the global electrophilicity of

two different molecules are different, best sites of two different molecules for

a given reaction can be explained only in terms of the philicity and not the

Fukui function. Moreover, since philicity is based on an absolute scale, there

is no need of an additional nucleophilicity index.

114

2.3.3.5. Major principles in respect of theory of chemical reactivity

In the way described above, Conceptual DFT gained importance over the last

20 years by sharply defining many of the concepts known for a long time in

chemistry. The various reactivity descriptors offer chemists ways of

quantitatively characterizing the reactivity of different classes of reactions. It

is also to be noted that the physical consistency of these reactivity descriptors

is justified on the basis of traditional views on the relations between the

electronic structure of a molecule and its reactivity and empirical principles

such as hard-soft acid-base principle (HSABP), the maximum hardness

principle (MHP), the minimum polarisability principle (MPP) and the

electronegativity equalization principle (EEP).

Maximum hardness principle: Pearson has suggested that “there seems to be

a rule of nature that molecules arrange their electronic structure so as to be as

hard as possible”.86

This principle has been proved by Parr and Chattaraj.87

But

the proof has been questioned by Sebastian in a subsequent paper.88

In spite of

this, the principle is still being used for describing reactivity in many systems.

Minimum polarisability principle: “the natural direction of evolution of any

system is toward a state of minimum polarizability”.89

Thus, hardness

measures the stability and softness (polarisability) measures the reactivity.

Electronegativity equalization principle: Upon molecule formation, atoms

(or more general arbitrary portions of space of the reactants) with initially

different electronegativities [χi0

(i=1,2,3…..M)] combine in such a way that

their “atoms-in molecule” electronegativities are equal.90

The corresponding

value is termed molecular electronegativity χM. Thus during molecule

formation electron transfer take place from atoms with lower electronegativity

to those with higher electronegativity, the later reducing their χ value, the

former increasing it.

115

χ10, χ 2

0……. χ M

0 χ 1 = χ 2 =……. = χ M

isolated atoms

molecule formation

Physical foundations of the HSABP, MHP, MPP and EEP acquire a formal

mathematical support in terms of DFT.48,91

Atomic and molecular properties

as energy derivatives with respect to N (number of electrons and ` (external

potential) can be conveniently illustrated using the Nalewajski’s Sensitivity

Analysis (scheme 2.7).

= (Ú, `)

Electronegativity Ö = uàãàv~Ü = −Õ Ã() = uãÜ~v

Electrondensity

uàXãàvX~Ü u Xã

àÜàv~ = u XãàvàÜ~

× = − uà àv~w u

w~v = uv~Ü

f ê()inêynz

zêç|| = ∫ |() nℎ|() = ê()Úözyxézêç||

Scheme 2.7: Nalewajski’s Sensitivity Analysis: atomic and molecular properties

as energy derivatives with respect to N and `

2.3.3.6. Concluding remarks and some generalizations regarding DFT

based reactivity descriptors

Chemical reactions are mainly adjustment of valence electrons among the

reactant orbitals. Fukui proposed his frontier orbital theory (FOT) which

allows a chemical reaction to be understood in terms of HOMO and LUMO

only. If the HOMO of the electron donor and the LUMO of the electron

acceptor have the same shape and phase, then electron transfer from the

116

HOMO of the first molecule to the LUMO of the second can occur, often

forming a bond between the reagents.92

A primary limitation of FOT is that it

presupposes the validity of the orbital model and thus fails to incorporate the

effects of electron correlation or orbital relaxation. This motivated the

definition of the Fukui function in the context of DFT. The Fukui function not

only captures the essence of classical FOT, but also includes both electron

correlation and orbital relaxation.93

Orbital relaxation can be defined as the

change in orbital shape that accompanies the addition or removal of electrons

from the system. Condensed Fukui functions at each atomic site in a molecule

also carry this advantage.

When two reactants A and B approach each other, the energy change (up to

second order) may be written as94

∆E = ∆Ecovalent + ∆Eelectrostatic + ∆Epolarization (2.72)(2.72)(2.72)(2.72) Which of the terms in the above equation contributes more to the total ∆E is

decided by the nature of the interacting species. Some of the general

principles which are being used and still being debated are the following

• If both A and B are soft species (large in size with a low charge), the nuclear

charge is adequately screened by the core electrons and the two soft species

will mainly interact via frontier orbitals. ∆Ecovalent dominates in ∆E and

electron transfer between the species controls the reaction. Such reactions

are termed as electron-transfer-controlled or Frontier-orbital-controlled

reactions.

• If both A and B are hard species (small size and high charge), the core

orbitals are not just “spectators” and the interaction follow “through space”

interactions rather than HOMO-LUMO interactions. ∆Eelectrostatic in ∆E

predominates95 and the charges on each atom will decide the course of the

reaction. Such reactions are termed as charge-controlled or electrostatic-

controlled reactions.

117

• It has also been shown that for the interaction between a hard and a soft

species the reactivity is generally very low and it cannot be identified as a

charge-/frontier-controlled reaction, justifying the HSAB principle.

• Li and Evans96

by modeling the softness kernel comprising a local and a

nonlocal part and following an earlier work of Berkowitz,97

have shown that

for frontier-orbital controlled reactions (soft-soft interactions) the

maximum Fukui function site is preferred, while, for charge-controlled

reactions (hard-hard interactions) the minimum Fukui function site is

preferred.

• In the article by Chattaraj98

it is highlighted that the Fukui function is not

the proper descriptor of the hard-hard interactions since they are not

frontier-controlled. Possible other descriptors for these interactions include

the molecular electrostatic potential (MEP- which comprises potentials due

to all the nuclei and electrons in a molecule, calculated at every points in

space)99

, local hardness and the nuclear Fukui function.100

An appropriate

local descriptor for analyzing hard-hard interactions could have been the

local hardness which, however, cannot be defined in an unambiguous

way.101

• Even if it is considered that a minimum Fukui function84

(or equivalently

local softness) site corresponds to maximum local hardness the highest

reactivity/selectivity of this is counter to MHP and MPP.

• Recently, a new general-purpose reactivity indicator has been derived by

Ayers et al. 102

It is claimed that this indicator can describe all classes of

reactions including the reactions that are neither charge- nor frontier-orbital-

controlled. The ‘minimum Fukui function rule’ for the hard reagents also

emerges naturally from their analysis.

In conclusion, the global HSAB principle and the frontier orbital theory

properly augmented by Klopman’s ideas are adequate in explaining both

reactivity and selectivity. Soft-soft interactions are frontier-controlled and

118

predominantly covalent in nature, and the site with the maximum value of the

Fukui function would be preferred in these reactions whereas hard-hard

interactions are charge-controlled and predominantly ionic in nature and for

these reactions the preferred site is that which contains maximum net charge

that may coincide in certain cases with the site associated with the minimum

value of the Fukui function.

One of the major objectives of the present work is to characterize the

regioselectivity preferences of hetero-substituted allyl systems in the light of

the above indices and principles.

2.3.4. Software used in the work

Quantum mechanical calculations are implemented on a computer with the

help of suitable software. A large number of quantum chemistry packages are

available in the market. In this section, the different software used in the

present work and their capabilities are summarized.

2.3.4.1. Gaussian 03

Gaussian is a very high-end quantum chemical software package, available

commercially through Gaussian, Inc. Gaussian is the most powerful software

available to educators and student researchers through the North Carolina

High School Computational Chemistry server. Currently, Gaussian 09 (G09)

is available. The “09” and “03” refers to the respective years – 2009 and 2003

– in which the software were published. The name Gaussian comes from the

use of the Gaussian Type Orbitals that Gaussian’s originator, John Pople,

used extensively to create a large number of basis sets to overcome the

computational difficulties that arose from the use of Slater Type Orbitals see

section (1.8). Gaussian is considered to be the industry standard in the area

of molecular modeling and computational chemistry. Gaussian is capable of

running all of the major methods in molecular modeling, including molecular

119

mechanics; ab initio; semi-empirical; and density functional theory (DFT). It

is probably best known for its robustness in running ab initio and DFT

calculations. Gaussian also does several post HF methods (like MPn, CI, CC

and their variants) and compound methods such as CBS and Gn see section

(1.4.3.5)

Gaussian Keywords: Like many computational chemistry codes, Gaussian

uses a keyword system. Keywords are short and typically cryptic instructions

to the software that describe what the user wishes to do.

There are four types of keywords in Gaussian:

1. Method: this is an indication of the theory that is requested. For example,

the keyword HF is used for requesting a Hartree-Fock calculation and B3LYP

for a DFT calculation using the XC functional B3LYP.

2. Basis Set: this keyword specifies the basis set to be used in the calculation.

For example, the Pople-style basis set, 6-31+G (d) is used in the present

work.

3. Job Type: the keyword which specifies the type of calculation to be done.

Some of the representative keywords are

a. SP: for single point energy (energy at an already specified geometry), b.

OPT: for optimizing a given geometry, c. FREQ: for computing the

vibrational frequencies.

4. Properties: the keyword which is used to evaluate specific molecular

properties. Some examples:

a. POP=FULL: this requests that all of the molecular orbitals, and a

description of how the electrons are distributed among those orbitals, be

printed in their entirety.

b. AIM: Atoms In Molecules, a keyword that calculates bond order for a

given molecule.

c. NMR: this keyword generates a Nuclear Magnetic Resonance (NMR) scan

of the specified molecule.

120

An appropriate combination of these four types of keywords is specified in

the route section of a Gaussian input file prior to a calculation. A list of such

combinations and description of their meanings is given in table 2.8

Table 2.8: Examples of Gaussian keywords used in the present work

Combination of key words Description

B3LYP/6-31+G(d) OPT FREQ

POP=REG

requests geometry optimization followed by frequency

calculation using the DFT method B3LYP and 6-

31+G(d)basis set. POP=REG asks the program to list

only the upper five occupied and lower five unoccupied

MO coefficients

B3LYP/6-31+G(d) SCRF(IPCM)

SCF=Tight GEOM=ALLCHECK

Requests a single point energy calculation (model

chemistry B3LYP/6-31+G(d)in presence of a solvent

(the dielectric constant to be specified in the molecule

specification section) on the geometry of a molecule

stored in the check point file (during an optimization

step). The solvation model used will be the Isodensity

Polarized Continuum model (IPCM)

B3LYP/6-31+G(d)

SCRF(SCIPCM) OPT

SCF=TIGHT

GEOM=ALLCHECK

Requests a an optimization (model chemistry

B3LYP/6-31+G(d)in presence of a solvent (the

dielectric constant to be specified in the molecule

specification section) on the geometry of a molecule

stored in the check point file (during an optimization

step in the absence of a solvent). The solvation model

used will be the Self-Consistent Isodensity Polarized

Continuum model

B3LYP/6-31+G(d) SCF=Tight

pop=NPA geom=allcheckpoint

Pop= NPA requests atomic charges to be computed

using the natural population analysis


pop=MK geom=allcheckpoint

Pop=MK requests atomic charges to be computed using

the Merz-Singh-Kollman scheme

121


pop=CHelpG

geom=allcheckpoint

Pop=CHelpG requests atomic charges to be computed

using CHELPG (= CHarges from ELectrostatic

Potentials using a Grid based method) scheme

HF/3-21+G*

Opt(TS,CalcFC,NoEigenTest,Ma

xCycle=100) Freq

Transition state optimization at the HF/3-21+G* level

of theory. Maximum number of steps is set as 100

through the MaxCycle option.

HF/3-21+G* IRC=

(RCFC,MAXPOINTS=10)

GUESS=READ

GEOM=ALLCHECK

This calculation type keyword requests that a reaction

path be followed by integrating the intrinsic reaction

coordinate (IRC) The initial geometry (in the

checkpoint file) is that of the transition state, and the

path can be followed in one or both directions from that

point. Maximum points to be evaluated on either side

of TS is set as 10.

B3LYP/6-31+G(d) SP

IOP(3/33=1) SCF=TIGHT

Pop=Full geom=allcheckpoint

The combination of key words used to evaluate the

condensed to atom fukui functions and other local and

global electrophilicity descriptors.

2.3.4.2. Chemcraft

Chemcraft103

is a graphical program for working with quantum chemistry

computations. It is a convenient tool for visualization of computed results and

preparing new jobs for the calculation. Chemcraft is mainly developed as a

graphical user interface for Gamess (US version and the PCGamess) and

Gaussian program packages. Chemcraft does not perform its own

calculations, but can significantly facilitate the use of widespread quantum

chemistry packages.

The main capabilities of the program include:

- Visualization of Gamess, Gaussian, NWChem, ADF, Molpro, Dalton,

Jaguar, Orca output files: representation of individual geometries from the file

(optimized structure, geometry at each optimization step, etc), animation of

vibrational modes, graphical representation of gradient (forces on nucleus),

122

visualization of molecular orbitals in the form of isosurfaces or colored

planes, visualization of vibrational or electronic (e.g. TDDFT) spectra,

possibility to show SCF convergence graph;

- Different tools for constructing molecules and modifying molecular

geometry: using standard molecular fragments, "dragging" atoms or

fragments on the molecule's image, utility for setting a point group, and other

possibilities;

- Producing publication-ready images of molecules in customizable display

modes, containing required designations (labels, lines, etc);

- Some additional utilities for preparing input files: visual construction of Z-

matrixes, automatic generation of input files with non-standard basis sets,

converting MOs read from an output file into the format of input file;

- Provides very detailed structured visualization of output files, based on

dividing a file into separate elements and presenting them in hierarchical

multi-level list; this feature allows one to easily analyze complicated files,

such as scan jobs, IRC jobs, or multi-job calculations;

- IRC graphs can be viewed and copied to other file formats;

- In addition to Gaussian output files, Chemcraft can read Formatted

Checkpoint files (.fch), extracting molecular structure and orbitals from the

file. For visualization of molecular orbitals and other properties, Gaussian

Cube files can be also read.

2.3.4.3 Gaussview

GaussView is another Graphical User Interface that helps us prepare input for

submission to Gaussian and permits us to graphically examine the output.

GaussView makes using Gaussian 03 simple and straightforward by

providing three benefits:

- Sketch in molecules using its advanced 3D Structure Builder, or load in

molecules from standard files.

123

- Set up and submit Gaussian 03 jobs right from the interface, and monitor

their progress as they run.

- Examine calculation results graphically via state-of-the-art visualization

features: display molecular orbitals and other surfaces, view spectra,

animate normal modes, geometry optimizations and reaction paths.

GaussView supports all Gaussian 03 features, and it includes graphical

facilities for generating keywords and options, molecule specifications and

other input sections for even the most advanced calculation types.

2.3.5. Brief descriptions of the principles involved in various calculations

2.3.5.1. Optimization techniques

Many problems in computational chemistry can be formulated as an

optimization of a multidimensional function (see section 1.8 for the

optimization of exponents of basis sets). Optimization is a general term for

finding stationary points of a function, i.e., points where the first derivative of

the function is zero. In the majority of cases, the desired stationary point is a

minimum, i.e., all the second derivatives should be positive. In some cases the

desired point is a first order saddle point (as in a transition state). Here, the

second derivative is negative in one, and positive in all other dimensions. One

of the examples of optimization, which is an essential part of all quantum

chemical applications, is the geometry optimization. Here energy optimization

as a function of nuclear coordinates is carried out. The simplest approach for

minimizing a function would be to step one variable at a time until the

function has reached a minimum, and then switch to another variable.

However, as the variables are not independent, several cycles through the

whole set are necessary for finding a minimum. It is now commonly agreed

that an efficient optimization method for quantum chemical applications

should be able to compute the gradient, g (first derivative of the function with

124

respect to all variables) and if possible, the Hessian (second derivative matrix)

analytically (i.e., directly) and not as numerical differentiation by stepping the

variables. There are three classes of commonly used optimization methods for

finding minima: Steepest Descent (SD) methods, Conjugate Gradient (CG)

methods and Newton-Raphson (NR) methods. The advantages and

disadvantages of these methods are detailed elsewhere104

.

Gaussian program uses the Berny optimization algorithm for geometry

optimization. This employs an NR method in redundant internal coordinates.

The way the energy of a molecule varies as a function of molecular

coordinates (see section1.4) is specified by its PES. A potential energy

surface is a mathematical relationship linking molecular structure and the

resultant energy. For a diatomic molecule, it is a two dimensional plot with

the internuclear separation on the X-axis and the energy on the Y-axis. For

larger systems, the surface has as many dimensions as there are degrees of

freedom within the molecule. At both minima and saddle points, the first

derivative of the energy, known as the gradient is zero. Since the gradient is

the negative of the forces, the forces are also zero at such point.

F = − up~ = 0 (2.73(2.73(2.73(2.73))))

A point on the PES where the forces are zero is called a stationary point. A

geometry optimization begins at the molecular structure specified as its input,

and steps along the PES. It computes the energy and the gradient at that point,

and then determines how far and in which direction to make the next step.

The gradient indicates the direction along the surface in which the energy

decreases most rapidly from the current point as well as steepness of that step.

Most optimization algorithms also estimate or compute the value of the

second derivative of the energy with respect to the molecular coordinates,

updating the matrix of force constants (known as Hessian). Therefore

constants specify the curvature of the surface at that point, which provides

125

additional information useful for determining the next step. An optimization

is complete when it has converged: i.e., when the forces are zero and some

other conditions are met. The convergence criteria used by Gaussian are the

following.105

• The forces must be essentially zero. Specifically the maximum component

of the force must be below the cut off value of 0.00045

• The root mean square of the forces must be below the cut off value of

0.0003

• The calculated displacement of the next step must be smaller than the

defined cut off value of 0.0018

• The root mean square of the displacement for the next step must also be

below the cut off value of 0.0012

The presence of four distinct convergence criteria prevents a premature

identification of the minimum. Quantum chemical geometry optimization

methods evolved rapidly over the past three decades. A major developmental

milestone included the analytical gradients of the potential energy and the

methods based on them, such as quasi-Newton methods and their

modifications. Hessian update techniques allowed information to be collected

for the potential energy surface, which accelerated the optimization process.

2.3.5.2. Population analyses

Once the accurate wavefunction of a molecule has been computed, in

principle all the properties could be derived. But, in practice, converting

molecular wavefunction to molecular properties is not often straight forward

and is one of the challenges still to be completely resolved in computational

chemistry. Although the quantum mechanical description of a molecule is in

terms of positive nuclei surrounded by a cloud of negative electrons,

chemistry is still formulated as ‘atoms’ held together by ‘bonds’. This raises

questions such as; given a wavefunction how can we define an atom and its

126

associated electron population or how do we determine whether two atoms

are bonded? Atomic charge is an example of a property often used for

discussing structural and reactivity differences. There are three commonly

used methods for assigning a charge to a given atom.106

1. Partitioning the wavefunction in terms of the basis functions; this is the

method employed in MPA

2. Fitting schemes; population analyses based on the electrostatic potential

use this technique.

3. Partitioning the wavefunction based on the wavefunction itself; Atoms in

Molecules (AIM) method107

by Bader makes use of this technique.

Mulliken population analysis (MPA): A partitioning scheme based on the

use of density and overlap matrices of allocating the electrons of a molecular

entity in some fractional manner among its various parts (atoms, bonds,

orbitals). MPA is arbitrary and strongly dependent on the particular basis set

employed. However, comparison of population analyses for a series of

molecules is useful for a quantitative description of intra-molecular

interactions, chemical reactivity and structural regularities.108

Natural population analysis (NPA): The analysis of the electron density

distribution in a molecular system based on the orthonormal natural atomic

orbitals. Natural populations, ( ) are the occupancies of the natural atomic

orbitals (see section 1.8.2.11) in an atom A (where i is the number of natural

orbitals). These rigorously satisfy the Pauli exclusion principle: 0 < ( ) < 2.

The population of an atom n (A) is the sum of natural populations

( ) =( )

(. ,Q) A distinguished feature of the NPA method is that it largely resolves the basis

set dependence problem encountered in the Mulliken population analysis

method.109

127

CHELPG charges: In the CHELPG (= CHarges from ELectrostatic

Potentials using a Grid based method) scheme110

by Breneman and Wiberg

atomic charges are fitted to reproduce the molecular electrostatic potential

(MEP) at a number of points around the molecule. As a first step of the fitting

procedure, the MEP is calculated at a number of gridpoints spaced 3.0 pm

apart and distributed regularly in a cube. The dimensions of the cube are

chosen such that the molecule is located at the center of the cube, adding 28.0

pm headspace between the molecule and the end of the box in all three

dimensions. All points falling inside the van-der-Waals radius of the molecule

are discarded from the fitting procedure. After evaluating the MEP at all valid

grid points, atomic charges are derived that reproduce the MEP in the most

optimum way. The only additional constraint in the fitting procedure is that

the sum of all atomic charges equals that of the overall charge of the system.

Merz-Singh-Kollman (MK) scheme: In the Merz-Singh-Kollman (MK)

scheme by U.C. Singh and P.A. Kollman111

atomic charges are fitted to

reproduce the molecular electrostatic potential (MEP) at a number of points

around the molecule. As a first step of the fitting procedure, the MEP is

calculated at a number of grid points located on several layers around the

molecule. The layers are constructed as an overlay of van der Waals spheres

around each atom. All points located inside the van der Waals volume are

discarded. Best results are achieved by sampling points not too close to the

van der Waals surface and the van der Waals radii are therefore modified

through scaling factors. The smallest layer is obtained by scaling all radii with

a factor of 1.4. The default MK scheme then adds three more layers

constructed with scaling factors of 1.6, 1.8, and 2.0. After evaluating the MEP

at all valid grid points located on all four layers, atomic charges are derived

that reproduce the MEP as closely as possible. The only additional constraint

in the fitting procedure is that the sum of all atomic charges equals that of the

overall charge of the system.

128

2.3.5.3. Characterization of the transition states

A minimum on the PES has all the normal mode force constants (all the

eigenvalues of the Hessian) positive; for each vibrational mode there is a

restoring force, like that of a spring. As the atoms execute the motion, the

force pulls and slows them till they move in the opposite direction; each

vibration is periodic, over and over. The species corresponding to the

minimum sits in a well and vibrates forever (or until it acquires enough

energy to react). For a transition state, however, one of the vibrations that falls

along the reaction coordinate is different from others. Motion of the atoms

corresponding to this mode takes the transition state toward the product or

toward the reactant without restoring force. This one ‘vibration’ is not a

periodic motion but rather takes the species through the transition state

geometry on a one-way journey. Now, the force constant is the first derivative

of the gradient or slope (the derivative of the first derivative); along the

reaction coordinate the surface slopes downward, so the force constant of this

mode is negative.

A transition state (a first order saddle-point) has one and only one negative

normal mode force constant (one negative eigen value of the Hessian). Since

a frequency calculation involves taking the square root of force constant, and

the square root of a negative number is an imaginary number, a transition

state has one imaginary frequency, corresponding to the reaction coordinate.

In general, an nth

order saddle-point (an nth

order hilltop) has ‘n’ negative

normal mode force constants and so ‘n’ imaginary frequencies, corresponding

to motion from stationary point of some kind to another.

A stationary point could of course be characterized just from the negative

force constants, but the mass weighting requires much less time than

calculating the force constants, and the frequencies themselves are often

wanted anyway, e.g., for comparison with experiment. In practice one usually

checks the nature of stationary point by calculating the frequencies and seeing

129

how many imaginary frequencies are present. A minimum has none, a

transition state, one and a hill top more than one.

If one is seeking a particular transition state, the criteria to be utilized are;

1. It should look right. The structure of a transition state should lie

somewhere between that of the reactants and the products.

2. It must have one and only one imaginary frequency. [Some programs

indicate this as a negative frequency, e.g., -1900 cm-1

instead of the correct

1900 i]

3. The imaginary frequency must correspond to the reaction coordinate. This

is usually clear from the animation of the frequency.

There are two standard ways of characterizing a T.S,

Successfully completing a transition structure optimization does not guarantee

that you have found the right transition structure: the one that connects the

reactants and products of interest. One way to determine what minima a

transition structure connects is to examine the normal mode corresponding to

the imaginary frequency, determining whether or not the motion tends to

deform the transition structure as expected.

An IRC calculation examines the reaction path leading down from a transition

structure on a potential energy surface. Such a calculation starts at the saddle

point and follows the path in both directions from the transition state,

optimizing the geometry of the molecular system at each point along the path.

In this way, an IRC calculation definitively connects two minima on the

potential energy surface by a path which passes through the transition state

between them.

When studying a reaction, the reaction path connects the reactants and the

products through the transition state. Note that two minima on a potential

energy surface may have more than one reaction path connecting them,

corresponding to different transition structures through which the reaction

130

passes. Reaction path computations allow us to verify that a given transition

structure actually connects the starting and ending structures that we think it

does. Once this fact is confirmed, we can compute the activation energy for

the reaction by comparing the (zero-point corrected) energies of the reactants

and the transition state. An IRC calculation begins at a transition structure and

steps along the reaction path a fixed number of times in each direction, toward

the two minima that it connects. However, in most cases, it will not step all

the way to the minimum on either side of the path.112

2.3.6. Solvation models


Solvation effects are essential components of all liquid state chemistry and it

is impossible to understand liquid phase organic, biological or inorganic

chemistry without including them. Electronic structure methods are aimed at

solving the Schrodinger equation for a single or a few molecules, infinitely

removed from all other molecules. Physically, this corresponds to the

situation in the gas phase under low pressure(vacuum). Experimentally,

however, the majority of chemical reactions are carried out in solution. Most

reactions are both qualitatively and quantitatively different under gas and

solution phase conditions, especially those involving ions or polar species.

There are various methods for treating solvation, ranging from a detailed

description at the molecular level to reaction field models where the solvent is

modeled as a continuous medium.113

If a detailed description at the molecular

level is desired, we have to surround the solute with a large number of solvent

molecules and do a quantum mechanical calculation on the resulting system.

But this is computationally demanding and almost impossible for most

systems. Therefore, we go for what is known as continuum solvation models.

The assumption underlying continuum solvation models is that one may

remove the huge number of individual solvent molecules from the model, as

131

long as one modifies the space those molecules used to occupy so that

modeled as a continuous medium, it has properties consistent with those of

solvent itself.

2.3.6.2. Self Consistent Reaction Field (SCRF) Methods

Such methods model the solvent as a continuum of uniform dielectric

constant (the reaction field) and the solute is placed into a cavity within the

solvent There are four different types based on how they define the cavity and

the reaction field

1. Onsager reaction field model114

: In this model, the system occupies a fixed

spherical cavity of radius a0 within the solvent field. A dipole in the molecule

will induce a dipole in the medium, and the electric field applied by the

solvent dipole will in turn interact with the molecular dipole, leading to net

stabilization.

Keyword SCRF=dipole; the dielectric constant (є) and the molecular volume

(a0) of the solvent should be given as input parameters. Gaussian also includes

a facility for estimating molecular volumes for these types of calculations. An

energy calculation run with the volume keyword will produce an estimate

value for a0. Molecules having a net dipole moment, µ=0, will not exhibit

solvent effects in the Onsager model.

2. Tomasis Polarized Continuum model (PCM)115

: PCM defines the cavity

as the union of a series of interlocking atomic spheres. The effect of

polarization of the solvent continuum is represented numerically – it is

computed by numerical integration.

Keyword SCRF= PCM; required input are є and number of points or spheres.

3. The Isodensity PCM (IPCM) : IPCM, defines the cavity as an isodensity

surface of the molecule. This isodensity is determined by an iterative process

in which an SCF cycle is performed and converged using the current

isodensity cavity. The resultant wavefunction is then used to compute an

132

updated isodensity surface, and the cycle is repeated until the cavity shape no

longer changes upon completion of the SCF. An isodensity surface is a very

natural, intuitive shape for the cavity since it corresponds to the reactive shape

of the molecule to as great a degree as is possible (rather than being a simpler

pre-defined shape such as sphere or a set of overlapping spheres).

Keyword SCRF= IPCM; required input is only the dielectric constant, є

4. Self-Consistent Isodensity Polarized Continuum model (SCF-IPCM)116:

It takes into account the coupling between the cavity defined as an isosurface

and the electron density. It includes the effect of solvation in the solution of

the SCF problem. This procedure solves for the electron density which

minimizes the energy, including the solvation energy. The effects of solvation

are folded into the iterative SCF computation rather than comprising an extra

step afterwards. SCF-IPCM thus accounts for the full coupling between the

cavity and the electron density and includes coupling terms that IPCM

neglects.

Keyword SCRF= SCIPCM; required input is the dielectric constant, є and

keep a blank line after the molecule specification.

A useful way of analyzing the frequency data (gas phase and solvated) is to

compute the frequency shifts on going from the gas phase to solution

(∆ν) = (ν)solvated – (ν)gas phase ((((2.752.752.752.75)))) (∆ν)expt = [(v)solvated – (v)gas phase]expt (2.76(2.76(2.76(2.76)))) The gas phase is delightful in its simplicity. Of course, one can carry out

accurate gas phase calculations and then make broad generalizations about

how we might expect a surrounding condensed phase to affect the results.

In the present work the isodensity PCM (IPCM) model has been used to

study the effect of solvent on the energy of allyl systems.

133

Appendix 2A

Geometry and relative energy of allyl systems reported in the literature

Figure 2A.1: Allyl anion

K anti (2.44)K Syn (0.00) anti planar (3.91)Syn planar (1.70)

Figure 2A.2: mono fluoro allyl anion

K (0.00) planar (16.90)

Figure 2A.3: difluoro allyl anion

134

Figure 2A.4: Allyl lithium

K syn 'external' (0.00) K syn 'internal' (0.54) K anti(1.11) Y syn (3.39)

Figure 2A.5:Fluoroallyl lithium

K 'external' (0.00) Y (19.21)K 'external' (1.43)

Figure 2A.6: Difluoroallyl lithium

Figure 2A.1 to 2A.6 – Structures taken from the report ‘G. Tonachini; C. Canepa, “An ab

initio Theoretical Study of the Structure and Stability of 1-fluoropropenide and 1,1-

difluoropropenide and of the corresponding Monomeric Lithiated Species”,Tetrahedron, 45,

5163-5174 (1989). Original labels used by the authors are retained in the above figures for

convenience. The numbers in parentheses indicate the relative energies of the isomers.

135

Figure 2A.7: Monochloro allyl anion

Syn planar (0.00) Anti planar (2.35)

Pyramidal (0.00) Planar (4.11)

Figure 2A.8: Dichloro allyl anion

Syn external (1.00)Syn internal (0.00)

Figure 2A.9: Monochloro allyl lithium

136

External (0.00) Internal (3.67)

Figure 2A.10: Dichloro allyl lithium

Syn (0.00)

Figure 2A.11: Monochloro allyl sodium

External (0.00) Internal (6.50)Figure 2A.12: Dichloro allyl sodium

137

Syn (0.00)

Figure 2A.13: Monochloro allyl potassium

External (0.00) Internal(5.70)

Figure 2A.14: Dichloro allyl potassium

Figure 2A.7 to 2A.14 – Structures taken from the report ‘C. Canepa; G. Tonachini; P.

Venturello, “Regioselectivity in lithium, sodium and potassium chloroallyl systems. An ab initio-

theoretical and experimental study”,Tetrahedron, 47, 8739-8752 (1991). Original labels used by

the authors are retained in the above figures for convenience. The numbers in parentheses

indicate the relative energies of the isomers.

Figure 2A.15: Gem-dichloro allyl anion and corresponding lithiated species at HF/STO-3G* level

Figure 2A.15 – Structures taken from the report ‘E. Angeletti; R. Baima; C. Canepa; I. Degani;

G. Tonachini; P. Venturello, “Effect of lithium complexation by 12-Crown-4 on the regioselectivity of

the attackof gem-dichloroallyl-lithium on some carbonyl compounds”, Tetrahedron, 45, 7827-7834

(1989).

138

Appendix 2B

Table 2 B.1: Ratio of α:γ products in the reactions of 3,3-dichloropropene with

some carbonyl compounds in the presence of LDA & in the presence of LDA and

pot-tert-butoxidea

a Data from the report C. Canepa; S. Cobianco; I. Degani; A. Gatti; P. Venturello, “Effect of

the cation in the regioselectivity control in reactions of 3,3-dichloroallyl metals with

substituted benzaldehydes”,Tetrahedron, 47, 1485-1494 (1991).

Carbonyl compound

Reaction in presence of

LDA alone

(here Li acts as the

counter ion) α: γ

Reaction in presence of LDA

and pot-tert-butoxide (here

K acts as the counter ion)

α:γ

C6H5-CHO 15:85 100:1

o-ClC6H4-CHO 1:100 100:1

p-ClC6H4-CHO 1:100 100:1

o-MeOC6H4-CHO 1:100 100:1

p-MeOC6H4-CHO 23:77 100:1

o-MeC6H4-CHO 1:100 100:1

p-MeC6H4-CHO 1:100 100:1

139

Appendix 2C

140

Figure 1 to 5 – Structures taken from the report ‘C. Canepa; G. Tonachini, “Regioselectivity

patterns featured by formaldehyde in the electrophilic addition to gem-difluro and gem-

dichloroallylsystems: An ab initio theoretical study” J. Org. Chem. 61, 7066-7076 (1996).’

141

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7827-7834 (1989).

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[6] E. Kaufmann; P. v. R. Schleyer; K. N. Houk; Y. D. Wu, “ab initio mechanisms for the addition of methyllithium, lithium hydride and their dimers to formaldehyde” J. Am. Chem. Soc., 107, 5560-5562 (1985).

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