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
2.3.3.1 Introduction
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
2.3.6.1 Introduction
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
(electrostatic potential
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
(electrostatic potential
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 &
polarization of the HOMO)
Cl K 1 Syn- internal bRHF/3-
21+G*
Strong γ selectivity for
hard & soft electrophiles
(electrostatic potentials &
polarization of the HOMO)
Cl,Cl-
(gem-
dichloro)
K
2 External & internal.
No bridged isomer.
bRHF/3-
21+G*
α selectivity (electrostatic
potential maps &
polarization of the HOMO)
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
abή¯° = (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
Exchange and Correlation Hybrid meta GGA
MO5-2X49
Exchange and Correlation Hybrid meta GGA
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
2.3.3.1. Introduction
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
B3LYP/6-31+G(d) SCF=Tight
pop=MK geom=allcheckpoint
Pop=MK requests atomic charges to be computed using
the Merz-Singh-Kollman scheme
121
B3LYP/6-31+G(d) SCF=Tight
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
2.3.6.1. Introduction
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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