CHAPTER-1
1.1
INTRODUCTION:
The performance of explosive depends on several physical properties including oxygen
balance, positive heat of formation, high density and thermo stability Over the decade, aza
heterocyclic nitramino urea compounds have attained prominence in the explosive chemistry
because of their inherent high moleculer density, increased stability and insensitivity coupled
with high performance or favorable
compromise between these two properties. Another
important aspects of the cyclic nitrounea compounds is the cost effective and mach simpler
synthesis and scale – up strategies from the technically readily available and cheap starting
materials Some of the important compounds as reported here in have shown renewed interests
with unique energetic capability, the most ability as well as impart and friction sensitivity.
Among which 2-oxo-1,3,5- Trinitro – 1,3,5,triagza cyclohexane have ( K-6 or keto- RDX) is
reported to be one the most powerful inexpensive new energetic compound with 4% more energy
than HNX.
If is therefore, considered worthwhile to undertake computational studies on structural
parameter of K- 56, TNPDU, DINGU, TNIGU – TNTNZ and K-6 ect.
Table1
High energy molecular sytems selected ( aza heterocyclic nitramino urea compounds)
O
O
-
N
O
O
+
N
+
O
-
N
O
O
N
N
+
O
N
+
N
-
-
N
N
+
N
+
N
-
O
-
N
O
O
O
O
O
O
O
O
+
N
N
N
-
-
N
+
O
1, 3, 4, 6-tetranitrotetrahydroimidazo[4,5d]imidazole-2,5(1H,3H)-dione
TNIGU (1,3,5,7 tetranitro 2,4,6,8 tetramino)
O
1,3,4,7-tetranitrooctahydro-2H-imidazo[4,5b]pyrazin-2-one
Tetranitrotetraazabicyclonanone
O
-
O
O
+
N
O
H
N
+
N
N
O
N
-
N
+
+
-
N
H
+
N
O
O
N
O
N
N
O
N
-
O
O
O
O
O
O
N
+
N
-
O
-
2,4,6,8-tetranitro-2,4,6,8tetraazabicyclo[3.3.1]nonane-3,7-dione
1,4-dinitrotetrahydroimidazo[4,5-d]imidazole2,5(1H,3H)-dione
DINGU
Tetranitropropanediurea
O
-
O
O
+
+
N
O
O
N
N
N
O
-
-
O
+
O
+
N
N
N
N
N
N
O
O
N
+
O
-
N
O
1,3,5-trinitro-1,3,5-triazinan-2-one
K--6
O
-
+
+
N
N
O
O
-
O
1,3,4,7-tetranitrooctahydro-2H-imidazo[4,5b]pyrazin-2-one
O
O
N
-
O
+
N
N
+
O
-
N
O
N
O
-
N
N
+
N
O
+
O
-
O
1,3,4,6-tetranitrohexahydroimidazo[4,5-d]imidazol-2(1H)-one
(2,3,4,5 tetrazabicyclanone)
The results of Explosive and Ballistic parameters of these molecules have been presented
in table 2
Table 2 Explosive and Ballistic parameters of high energy molecules
Molecule
ISPD
C*
(s-gmcm-3)
(ms-1)
K-6
499.9
1605
TNPDU
504.3
DINGU
Mw
Tf
VOD
C-J pressure
(C)
(ms-1)
(kbar)
29.5
3456
8815
349
1574
3.05
3287
8770
365
419.5
1365
25.79
2448
8672
326.7
TINGU
567.7
1540
33.2
3440
10526
504.7
K56
495.7
1640
26.3
3233
8691
351
*Ballistic parameters computed using the NASA-CEC 71 code.
a) ISPD: Specific impulse,
b) C*: Characteristic velocity (from Computational Chemistry, Grants and Richards),
c) Mw: mean molecular wt. of exhaust gases,
Other explosive/ballistic parameterse) Tf: Gas phase heat of formation
f) VOD: Velocity of detonation
g) C-J pressure: Chapmann Jouget detonation pressure; parameters f and g are obtained
from the Becker Kistiakowsky-Wilson (BKW) equation of state.
Outline of Hartree- Fock Theory and Density Functional Theory:
Ab initio Hartree–fock calculations using 6-31G (d, p) basis set. The parameters might
further commensurate to better understandings of this structure-properties relationally and
ultimate applications in propellant.
Quantum mechanics is about electronic processes and is capable of giving descriptions of
energies and distributions of the electrons which determine chemical structures and processes of
arbitrary accuracy with the limitations which are technological rather than theoretical. We start
with the Schrödinger equation, which is a second order and first-degree differential equation1 and
an exact solution to the Schrödinger equation leads to prediction of the behavior of matter at
microscopic level. In practice, however, such an exact solution is not possible for systems
having more than one electron. Within a decade of its inception, physicists and chemists devised
methods that can solve, in principle, the time-independent Schrödinger equation for many
electron systems, albeit with drastic approximations and the errors associated with them. Thus
the practical applications of the Schrödinger equation seems to be restricted which necessitates
formulation of approximate methods for many electrons. Advent of computers and the
availability of a variety of software have made it possible to deal with the complex mathematical
problems. Until the late 1970’s, limitations of computer memory, hard disk storage and the CPU
speed constituted a major bottleneck to explore the applications of ab initio electronic structure
methods. In the recent years the advances in supercomputers, or the vector/parallel machines,
powerful workstations and desktop computers have widened the scope for running the electronic
structure programs for the systems up to nearly 100 atoms or so. With the aid of quantum
chemical (QC) tools, one can predict not only the structure and charge distribution but also the
accompanying properties related to the spectroscopy and thermodynamics. Thus a variety of
properties encompassing the equilibrium molecular geometry, total energy, Frontier orbitals,
electron affinity, multipole moments, polarizabilities, electronic, vibrational and rotational
transitions, nuclear and electron-paramagnetic shielding constants etc can be derived for the
isolated molecules and small clusters as well.
In order to model the chemical properties one may employ different methods based on (a)
Molecular mechanics (MM) (b) semi-empirical quantum chemical methods (c) ab-initio quantum
chemical methods (d) the density functional methods. On the other hand the molecular dynamics
and Monte Carlo based methods can be explored to investigate the macroscopic properties of a
system. The MM methods2-4 employ the treatment of classical mechanics, wherein the total
energy is assumed to be function of nuclear coordinates solely, turns out to be the least expensive
computational method, while it can handle chemical systems such as proteins, enzymes and other
macromolecular systems, containing tens of thousands of atoms. However, there is a
compromise in the accuracy of predicted results of such calculations, which are dependent on the
force fields (FFs) used. On the other hand, ab initio methods provide a mathematical description
of the chemical system by solving the Schrödinger equation from first principles, without
employing any empirical data except the universal constants. Hence, the results obtained from
these sophisticated ab initio methods are known to be accurate and generally are in good
agreement with the experimental results5.
By far, the Hartree-Fock (HF) formulation that uses, in its simplest form, a singledeterminantal form of wavefunction subjected to the variational principle as discussed later, is
the most commonly used ab initio method. It, however, should be remarked here that in the HF
theory the correlation between electron pair with opposite spins is not accounted.
This
necessitates the use of post-Hartree-Fock methods and bridge the gap between HF-level and
electronic energies determined from the experiments. The methods like configuration interaction
(CI) and coupled-cluster (CC) are some of the examples of such high-quality ab initio methods,
which provide accurate energies and molecular properties. Due to its high computational cost,
these methods can be employed for systems with utmost a few tens of atoms. The second order
correction term in the Møller-Plesset perturbation theory, (MP2) has been widely recognized as
an attractive post-HF alternative. The expensive computational requirements, however, restrict
their use to only relatively small or medium-sized molecules. On the other hand for dealing with
the large molecular systems different approximations and parameterization schemes has led to a
plethora of methods termed as the semi-empirical methods.6,7 These semi-empirical methods
have proved useful for the molecular systems containing a few thousands of atoms.
The Hohenberg-Kohn8 and Kohn-Sham9 formalisms, formally have shown that the
physical observables of a system in its ground state can be expressed as functionals of the
electron density (ED) and led to the density functional theory10 (DFT) which has become quite
successful and popular in the recent times. DFT-based methods promise to provide reliable
predictions of the structure and properties of chemical systems with lesser computational effort
than that required for post-HF methods. The computational cost of the HF and DFT level
calculations on medium/large-sized molecules, practically scales as O(N3), N being the number
of basis functions used in the calculations. An outline of the quantum chemical methods used in
the present thesis is given below.
1.2 Quantum Chemical Methods
In all these methods, the electronic structure of matter is described in terms of Schrödinger
equation. The term ab-initio simply suggest that the chemical phenomenon are explained in
terms of fundamental physical constants such as Planck’s constant, velocity of light, mass of
electrons and nuclei. Other than the determination of these constants, ab-initio methods are
independent of experiments. Since most of the chemical phenomena are due to the timedependent interactions, one may write time-independent Schrödinger equation, which written in
its simplest form is
H = E
(1.1)
where H is the Hamiltonian operator comprising of the nuclear and electronic kinetic energy
operators and the potential energy operators corresponding to the nuclear-nuclear, nuclearelectron and electron-electron interactions. The many-particle wavefunction  describes the
system, while E is the energy eigenvalue of the system.
In 1927, Born and Oppenheimer (BO) proposed an approximation11 that led to a major
leap towards a practical solution to the Schrödinger equation. The BO approximation is based on
the notion that the nuclear and electronic motions take place at different time scales, the latter
being much smaller than the former. Hence, an electron in motion sees relatively static nuclei,
while a moving nucleus feels averaged electronic motion. It has been found that the calculations
beyond BO approximation lead to an error of approximately 0.3 %. This enables one to separate
the electronic and nuclear Hamiltonians and the corresponding wavefunctions as well. The
molecular wavefunction can then be represented as a product of electronic and nuclear
counterparts.
{ri };{R A }  Φ elec {ri };{R A }Φ nuc {R A }
(1.2)
Here, {ri} and {RA} are the positions of electrons and nuclei, respectively. The eigenvalue
equation for the electronic case is then written as
H elec  elec {ri };{R A }  E elec  elec {ri };{R A }
(1.3)
The electronic Hamiltonian depends explicitly on the electronic coordinates and parametrically
on nuclear coordinates. Since the chemical phenomena occur primarily due to various
interactions between electrons of the systems, explicit treatment of the electronic Hamiltonian
generally suffices to investigate chemical properties and reactions. However, probing the
vibrational, rotational and translational motions of molecules would further require an explicit
treatment using the nuclear Hamiltonian.
The system of atomic units (a.u.), sometimes referred as Hartree units, has been
conveniently adopted as a convention in QM wherein the mass of an electron, Planck’s constant,
charge of a proton, the length corresponding to radius of first Bohr orbit in hydrogen atom and
4 times the permittivity in free space (40) are all set to unity and the unit of energy in a.u.
numerically turns out to be one-half the energy of a hydrogen atom in its ground state (GS). The
use of atomic units leads to a simplified and convenient electronic Hamiltonian (excluding the
internuclear repulsion) as shown below.
N
N M
N N
ZA
1
1
H elec    i2  
 
i 1 2
i 1 A 1 ri  R A
i 1 ji ri  rj
(1.4)
The first term is the electronic kinetic energy operator summed over the number of electrons, N.
The second term represents coulombic attraction between electrons {i} and M nuclei {A}, while
{ZA} are the nuclear charges. The last term attributes to the Coulombic repulsion between
electrons. Once the wavefunction  is obtained by solving the Schrödinger equation, any
experimental observable can be computed as the expectation value of appropriate operator,
   . In the case of total electronic energy, the operator is electronic Hamiltonian.
1.2.1 Hartree-Fock Theory
The acceptable solutions of Eq (1.4) need to be necessarily well-behaved, i. e.
wavefunction is finite, single valued, continuous, quadratically integrable and obeying the
appropriate boundary conditions. Thus

*
(r1 , r2 ......, rN ) (r1 , r2 ......, rN )d 3 r1d 3 r2 ...d 3 rN  1
(1.5)
Initial attempts by Hartree12 to devise such an approximate wavefunction of many-electron
systems lead to the Hartree product (HP) function, which is a product of the one-electron
functions and normally referred as orbitals. In this scheme, the spatial distribution of electrons
has been defined in terms of spatial orbitals. These spatial orbitals are functions of the position
vectors of electrons. To define an electron completely, in addition to the spatial coordinates the
spin of electron is also required. Thus the notion of electronic spin is introduced in the oneelectron function (orbital) by means of the spin functions () and () that correspond to the
up and down spin electrons, respectively. This leads to the spin orbitals, {  j } defined below.
χ 2i1 (x)  ψ iα (r)()
χ 2i (x)  ψβi (r)()
(1.6)
, where each electron is defined in terms of combined spatial and spin coordinates, x. Thus, the
HP function can be written as
 HP (x1 , x 2 ,....., x N )   i (x1 ) j (x 2 )..... k (x N )
(1.7)
Hartree (1928) proposed an iterative ‘self –consistent field’ (SCF) method. In the first step of the
SCF process, one guesses the wavefunction Ψ for all of the occupied MOs and uses these to
construct the necessary one-electron operators h. Solution of each differential equation provides
a new set of Ψ, presumably different from the initial guess. So, the one-electron Hamiltonians
are formed a new using these presumably more accurate Ψ to determine each necessary ρ, and
the process is repeated to obtain a still better set of Ψ. At some point, the difference between a
newly determined set and the immediately preceding set falls below some threshold criterion,
and we refer to the final set of Ψ as the converged SCF orbitals.
The HP function is an independent-electron wavefunction and do not satisfy the
antisymmetry principle. This requirement within the orbital picture is equivalent to the Pauli’s
exclusion principle,13 which states that no two electrons of an atom shall have identical value of
all the four quantum numbers viz. n, l, m and s. Slater14 and Fock15 independently proposed that
an antisymmetrized sum of all the permutations of HP functions would solve this problem for
many-electron systems.
( x1 , x 2 ,...x i ,...x j ,...x N )  ( x1 , x 2 ,...x j ,...x i ,...x N )
(1.8)
This is achieved by using a determinantal form of function, later came to be known as the Slater
determinant. Thus, an N-electron wavefunction within the HF formulation can be written as
χ i ( x 1 ) χ j ( x 1 ) ... χ N ( x 1 )
Ψ HF ( x 1 , x 2 ,...., x N ) 
The pre-factor 1
1 χ i ( x 2 ) χ j ( x 2 )... χ N ( x 2 )
. ... .
N! .
χ i ( x N ) χ j ( x N )... χ N ( x N )
(1.9)
N! ensures a normalized representation of the wavefunction.
The spin
orbitals are denoted as ’s, while x1, x2...etc. represent the combined spatial and spin coordinates
of the respective electrons. The normalized Slater determinant can also be represented in a
shorter notation as
Ψ(x 1 , x 2 ....., x N )  χ i χ j .....χ k
(1.10)
In this notation (cf. Eq.1.11), it is presumed that electrons 1,2...etc. sequentially occupy the spin
orbitals.
Fock proposed the extension of Hartree’s SCF procedure to Slater determinantal wave functions.
Just as with Hartree product orbitals, the HF MOs can be individually determined as
eigenfunctions of a set of one-electron operators, but now the interaction of each electron with
the static field of all of the other electrons includes exchange effects on the Coulomb repulsion.
Later, Roothaan16 and Hall17 formulated matrix algebraic equations that permitted HF
calculations to be carried out using a basis set representation for the MOs. This formalism
described for closed shell systems (i.e., all electrons spin paired, two per occupied orbital) with
wave functions represented as a single Slater determinant is called ‘Restricted Hartree-Fock’
(RHF).
The one-electron Fock operator is defined for each electron i as
1
f i   i2 
2
nuclei
r
k
Zk
 ViHF {j}
(1.11)
ik
where the first two terms represent the one-electron operator, called the core-Hamiltonian and is
a sum of kinetic energy and nuclear-electron attraction energy operators while the final term, the
HF potential, is 2Ji – Ki, and the Ji and Ki are the Coulomb and exchange operators. The
Coulomb operator gives the electron-electron repulsion energy and the exchange operator
provides lowering of energy due to correlation between electrons of same spin, an artifact of the
determinantal wavefunction. MOs are determined using the Roothaan approach by solving the
secular equation with in a given set of N basis functions
F11 – ES11
F12 –ES12
…
F1N –ES1N
F21 – ES21
F22 –ES22
…
F2N –ES2N
…
…
…
…
FN1 – ESN1
FN2 –ESN2
…
FNN –ESNN
(1.12)
to find its various roots Ej. The values for the matrix elements F and S are computed explicitly.
Matrix elements S are the overlap matrix elements. For a general matrix element Fμν
Fμ ν
1
 μ  2 ν
2
 Z
nuclei
k
k
μ
1
ν 
rk
P
λσ
λσ
1
[(μ( λσ)  (μ λ νσ )]
2
(1.13)
The notation <μ│g│ν > where g is some operator which takes basis function φν as its argument,
implies a so called one electron integral of the form
μgν =
 φ (g φ ) dr
μ
ν
(1.14)
Thus for the first term in equation 1.13, g involves the Laplacian operator and for the second
term g is the distance operator to a particular nucleus. The notation (μν│λσ) also implies a
specific integration, in this case
(μν│λσ) =
 φ
μ
1
(1) φ ν (1) φ λ (2) φ λ (2) φ σ (2) dr1dr2
2
(1.15)
where φμ and φν represent the probability density of one electron and φλ and φσ the other. The
exchange integrals (μλ│νσ) are preceded by a factor of ½ because they are limited to electrons of
the same spin while Coulomb interactions are present for any combinations of spins.
The final sum in equation 1.13 weights the various so-called ‘four index integrals’ by elements
of the density matrix’ P. This matrix in some sense describes the degree to which invidual basis
functions contribute to the many-electron wave function, and thus how energetically important
the Coulomb and exchange integrals should be (i.e. if a basis function fails to contribute in a
significant way to any occupied MO, clearly integrals involving that basis function should be of
no energetic importance). The elements of P are computed as
occupied
Pλ σ  2
a
i
λi
aσi
(1.16)
where the coefficients aζi specify the (normalized) contribution of basis function ζ to MO i and
the factor of two appears because with RHF theory we are considering only singlet wave
functions in which all orbitals are doubly occupied.
The process of solving the HF secular determinant to find orbital energies and coefficients is
characterized by some paradox present in the Hartree formalism. That is, we need to know the
orbital coefficients to form the density matrix that is used in the Fock matrix elements, but the
purpose of solving the secular equation is to determine those orbital coefficients. So, just as in
the Hartree model, the HF method follows a SCF procedure, where first we guess the orbital
coefficients (e.g. from an effective Hamiltonian method) and then we iterate to convergence.
Hartree-Fock theory as constructed using the Roothaan approach suffers from certain limitations
which stems from the one-electron nature of Fock operator. Other than exchange, all electron
correlation is ignored. Furthermore, from a practical standpoint, HF theory posed some very
challenging technical problems to early computational chemists. One problem was choice of
basis set. The LCAO approach using hydrogenic orbitals remains attractive in principle however,
this basis set requires numerical solution of the four index integrals appearing in the Fock matric
elements and that is a very tedious process. Moreover, the number of four index integrals is
daunting. Since each index runs over the total number of basis functions, there are on principle
N4 total integrals to be evaluated, and this quartic scaling behavior with respect to basis set size
proves to be a bottleneck in HF theory applied to essentially any molecules.
These problems initiated the computational chemists to progress in two different directions.
Some of these undertook the path of considering some approximations that may be introduced to
simplify their solution, and possibly at the same time improve their accuracy (by some sort of
parametrization to reproduce key experimental quantities). This underlies the motivation for so
called ‘ semiempirical’ MO theories. Other group of computational chemists view HF theory as a
stepping stone on the way to exact solution of the Schrodinger equation. HF theory provides a
very well defined energy, one which can be converged in the limit of an infinite basis set, and the
difference between that converged energy and reality is the electron correlation energy. It was
anticipated that developing the technology to achieve the HF limit with no further
approximations would not only permit the evaluation of the chemical utility of the HF limit, but
also probably facilitate moving on from that base camp to the Schrodinger equation summit.
1.2.1a Basis Sets:
Within the LCAO-MO approximation, MOs are expanded as linear combinations of AOs.
However, since it is not possible to obtain the exact AOs for many-electron atoms, Slater type
orbitals18 (STOs) were used to mimic AOs in the early days19-20. Due to the difficulty in
computing two- and other multi-center integrals using STOs, Boys21 suggested the use of
standard Gaussian functions24 centered on atoms (GTO). A Cartesian Gaussian function used in
electronic structure calculations has the form:
2
g(α( l, m, n; x, y, z)  N l m n (x  x A ) l (y  y A ) m (z  z A ) n e -α| r - rA |
(1.17)
where,  is the orbital exponent and l, m, n are the powers of Cartesian components x, y and z
respectively.
The center of a Gaussian function is denoted by rA= (xA, yA, zA).
The
computational advantage of GTOs over STOs is primarily due to the Gaussian product
theorem,22 viz. the product of two GTOs is also a Gaussian function centered at the weighted
midpoint of the two functions. In addition, the resulting integrals can be evaluated analytically.
However, for the sake of computational convenience, it is a common practice23 to bunch together
a set of GTOs with fixed coefficients, di (cf. Eq. 1.38). Such a linear combination is termed the
contracted GTO (CGTO).
g CGTO (l, m, n, x, y, z) 
d
i
g(α i , l, m, n, x, y, z)
(1.18)
The
i
orbital exponent and contraction coefficients are determined from appropriate atomic
calculations. There are varieties of Gaussian basis sets available24 now, which have been
continuously improved over the years. The single-zeta Gaussian basis sets, also known, as
minimal basis set are the simplest GTOs. It consists of one CGTO per AO in shell and provides
a rather crude description of the electronic system. The STO-3G basis set25, which has been used
extensively during the early days of molecular quantum mechanics, is the most popular singlezeta basis set. For reliable qualitative results, however, it is essential to use double-zeta basis
functions that contain two CGTOs per AO in shell. State-of-art calculations in recent years make
use of triple-zeta basis functions. In addition, the extended basis sets26 include basis function of
higher angular momentum than those present in the basis. This ensures adequate representation
of polarization of the electronic charge density. Similarly, it is essential to add the diffuse
functions to the basis set when the regions of low ED in the system are of great interest. There is
also a wide range of customized basis sets. The most frequently used being the Pople’s 27 split
valence basis set (multiple CGTOs for valence shells only) with a varying number of polarization
functions such as 6-31G(d,p), 6-311++G(2d,2p) etc., Dunning28-29 basis set and the correlation
consistent30 valence double and triple zeta basis functions (cc-pvdz, cc-pvtz).
1.2.2 Post Hartree-Fock Methods
The electron in a molecule exhibits a correlated motion, i.e., each electron moves in a
way that avoids the instantaneous positions of the other electrons. This is called the coulomb
correlation. In the HF method the motion of electrons with opposite spins are not correlated
i.e. the instantaneous probability of finding two electrons having opposite spins and
occupying the same space is non-zero. This results in a higher total energy at the HF level.
In addition, the quantities computed from HF calculations are known to have errors to
various degrees. For example with the HF theory, the bond lengths of organic compounds
are too small, the orbital ordering of N2 is wrong, it predicts longer metal- bonds and high
heats of formation to name a few. The H2 homolytic cleavage is a well-known case for the
failure of the HF theory. Though UHF calculations explain such a homolytic cleavage, the
potential energy surfaces constructed using UHF are unrealistic. Such factors focus the need
for further accurate methods that would produce results in coherence with the experimental
counterparts. The post-HF methods try to obtain the correlation energy Ecorr, defined31 as the
difference between the exact ab initio energy and exact (complete basis) HF energy, viz.
E corr  ε 0  E 0
(1.19)
where, ε 0 is the exact eigenvalue of Helec and E0 the “best” HF energy with the basis set
extrapolated to completeness. The two popular approaches that try to compute Ecorr are the
configuration interaction32-35 (CI) and many body perturbation theory36-37 (MBPT) methods.
The next two Sections provide a summary of these methods.
1.2.3 Configuration Interaction
There are a number of ways in which correlation effects can be incorporated into an ab initio
molecular orbital calculation. A popular approach is configuration interaction (CI), in which
excited states are included in the description of an electronic state. A configuration interaction
wave function is a multiple-determinant wave function constructed by starting with the HF wave
function and making new determinants by promoting electrons from the occupied to unoccupied
orbitals. The HF wavefunction is used as the reference determinant and the energy is minimized
variationally with respect to the determinant expansion coefficients.
The complete CI wavefunction is a linear combination of Slater determinants with all
the permutations of electron occupancies expanded as shown below.
rs
 0  c 0 0   c ar ar   c ab
abrs 
ar
a b
r s
c
a  bc
r s  t
rst
abc
rst
abc
 ....
(1.20)
The first term in Eq. (1.48) represents the Slater determinant corresponding to the HF
wavefunction and rest of the terms constitute singly, doubly, triply... n-tuply excited
determinants with appropriate expansion coefficients. The indices a,b,r,s, etc. signify the
occupied and virtual orbitals involved in the electron excitations. It is a convention to use the
indices a, b, c... for occupied orbitals and r, s, t... for the virtual ones.
Configuration interaction calculations can be very accurate, but the cost in CPU time is very high
(N8 time complexity or worse). The number of excitations used to make each determinant
classifies configuration interaction calculations. If only one electron has been moved from each
determinant, it is called a configuration interaction single-excitation (CIS) calculation. CIS
calculations give an approximation to the excited states of the molecule, but do not change the
ground-state energy. Single-and double excitation (CISD) calculations yield a ground-state
energy that has been corrected for correlation. Triple-excitation (CISDT) and quadrupleexcitation (CISDTQ) calculations are done only when very-high-accuracy results are desired.
The configuration interaction calculation with all possible excitations is called a full CI. The full
CI calculation using an infinitely large basis set will give an exact quantum mechanical result.
However, full CI calculations are very rarely done due to the immense amount of computer
power required. CI results can vary a little bit from one software program to another for openshell molecules. This is because of the HF reference state being used. Some programs, such as
Gaussian, use a UHF reference state. Other programs, such as MOLPRO and MOLCAS, use a
ROHF reference state. The difference in results is generally fairly small and becomes smaller
with higher-order calculations. In the limit of a full CI, there is no difference.
1.2.4 Perturbation Theory
Moller and Plesset proposed an alternative way to tackle the problem of electron
correlation. Their method is based upon Rayleigh-Schrodinger perturbation theory, 38-41 in which
the true Hamiltonian operator H is expressed as the sum of a ‘zeroth order’ Hamiltonian H0 (for
which a set of molecular orbitals can be obtained) and a perturbation U:
H = H0 + U
(1.21)
The eigenfunctions of the true Hamiltonian operator are ψi with corresponding energies Ei. The
eigenfunctions of the zeroth order Hamiltonian are written ψi(0) with energies
Ei(0). The
ground state wavefunction is thus ψ0(0) with energy E0(0). To improve the eigenfunctions and
eigenvalues of H0 we can write the true Hamiltonian as
H= H0 + λU
(1.22)
λ is a parameter that can vary between 0 and 1; when λ is zero then H is equal to the zeroth-order
Hamiltonian but when λ is one then H equals its true value. The eigen functions ψi and
eigenvalues Ei of H are then expressed in powers of λ
ψ i  ψ i(0)  λψ i(1)  λ 2 ψ i(2) ... 
λ ψ
n
(n)
i
(1.23)
λ E
(1.24)
n 0
E i  E i(0)  λE i(1)  λ 2 E i(2)  ... 
n
(n)
i
n 0
Ei(1) is the first order correction to the energy, Ei(2) is the second order correction and so on.
These energies can be calculated from the eigenfunctions as follows.

E i(0)  ψ i(0) H 0 ψ i(0) dτ
(1.25)
E i(1) 
ψ
E i(2) 

E i(3) 
ψ
U ψ i(0) dτ
(1.26)
ψ i(0) U ψ i(1) dτ
(1.27)
(0)
i
U ψ i(2) dτ
(0)
i
(1.28)
To determine the corrections to the energy it is therefore necessary to determine the
wavefunctions to a given order. In Moller-Plesset perturbation theory42 the unperturbed
Hamiltonian H0 is the sum of the one-electron Fock operator for the N electron
N
H0 

N
fi 
i 1

i 1
 core
H 


N

j1

( Ji  Ki )


(1.29)
The Hartree wavefunction, ψ0(0) is an eigenfunction of H0 and the corresponding zeroth order
energy E0(0) is equal to the sum of the orbital energies for the occupied molecular orbitals:
occupied
E
(0)
0

ε
(1.30)
i
i 1
In order to calculate higher order wavefunctions we need to establish the form of the
perturbation, U . This is the difference between the ‘real’ Hamiltonian H and the zeroth order
Hamiltonian, Ho,
The Hamiltonian is equal to the sum of the nuclear attraction terms and electron repulsion terms:
N
H
 (H
i 1
N
core
)
N
r
1
i  1 j  i 1 i j
Hence the perturbatioin U is given by
(1.31)
 (J  K )
N
U
N
i  1 j  i 1
1

ri j
N
i
(1.32)
i
j1
The first order energy E0(1) is given by
E
(1)
0
1

2
N
N
i 1
j1
[( ii
jj ) - ( ij ij )]
(1.33)
The sum of the zeroth order and first order energies thus corresponds to the Hartree-Fock
energies
N
E E
(0)
0
(1)
0

1

εi 
2
i 1
N
N
[( ii
jj )  ( ij ij )]
(1.34)
i 1 j 1
To obtain an improvement on the Hartree-Fock energy it is therefore necessary to use MollerPlesset perturbation theory to atleast second order. This level of theory is referred to as MP2 and
involves the integral
ψ
(0)
0
U ψ (1)
0 dτ
The higher order wavefunction ψ0(1) is expressed as linear combinations of solutions to the zeroth
–order Hamiltonian:
ψ (1)
0 
c
(1)
j
ψ (0)
j
(1.35)
j
The ψj(0) in eq 1.35 will include single, double etc. excitations obtained by promoting electrons
into virtual orbitals obtained from a Hartree-Fock calculation.
The second order energy is given by
occupied
E (2)
0 
virtual
  
i
ji
a
ba

 1 
dτ1dτ 2 χ i (1) χ j (2)   [χ a (1) χ b (2)  χ b (1) χ a (2)
 r12 
εa  εb  εi  ε j
(1.36)
These integrals will be non zero only for double excitations, according to the Brillouin theorem.
The advantage of many-body perturbation theory is that it is size independent, unlike
configuration interaction- even when a truncated expansion is used. Moller-Plesset, however, is
not variational and can sometimes give energies that are lower than the ‘true energy’. MollerPlesset calculations are computationally intensive and so their use is often restricted to single
point calculations at a geometry obtained using a lower level of theory. They are at present the
most popular way to incorporate electron correlation in molecular quantum mechanical
calculations, especially at the MP2 level.
Other high level correlated methods that yield accurate correlation energies are the
coupled cluster
(CC)
43-44
and multi-configuration self consistent field (MCSCF)45-46
methods. These methods, however, scales as (non-iterative) O(N7) and cannot be employed
to systems with more than ~10 atoms. Similarly, the CASSCF (complete active space SCF) a
variant of MCSCF, has found practical utility only for small molecules.
Unlike these
rigorous ab initio methods, density functional theory (DFT) offers a substantial improvement
over HF energy without consuming much additional computation time.
1.2.5 Density Functional Theory
Density functional study endows the single particle density (r), the status of a basic
entity and justifies its usage as a fundamental quantity on a rigorous, general rational footing.
There are several advantages over the conventional wavefunction approach to quantum
mechanics. First, the wavefunction for an N-elelctron system is a function of 3N spartial coordinates whereas rhe density (r) is dependent only on three independent spartial coordinates (or four, if the spin is included). Second, the density (r) is an observable, subject
to a measurement experimentally while the many-particle wavefunction is an intangible
entity. Third, the density is a very conventional parameter for a collective description of
many-electron system wherein single particle co-ordinates lose their indensity.
Thomas47 and Fermi48,49 (TF) developed a statistical model, wherein the total electronic
energy of a system was expressed as a functional of the charge density distribution, (r).
The kinetic energy functional, TTF[] derived by them is
TTF [ ρ ] 

3
(3 π 2 ) 2 / 3 d 3 r ρ 5 / 3 (r )
10
(1.37)
The total TF energy is obtained by adding the contributions due to Hartree (Coulomb)
potential to Eq.(1.37) They assumed the local homogeneity of the inhomogeneous density
distribution. Though the TF model has been augmented with corrections in the later years,9,50
it remained as a simple qualitative model until the formulation of coveted Hohenberg-Kohn
(HK) theorems.8 The HK theorem laid the foundation of the DFT. In this theory, the electron
density plays the role of basic variable. The HK theorems may be stated as: (i) The external
potential is determined, within an additive constant, by the ground state ED. (ii) The energy
due to any normalized, non-negative trial density that satisfies certain conditions is
variational. It was also shown that
E 0  Eρ(r)
(1.38)
where, E0 is the exact energy. In DFT, the electronic energy functional is expressed in terms
of the (cf. Eq. 1.39) contributions due to kinetic energy T, external potential V and the
electron-electron interaction energy U.
E[ρ]  T[]  V[]  U[]
(1.39)
However, Eq. (1.39) has relevance only if there exists an external potential, V for a given
trial (r).
Levy51-53 overcame this problem of V-representability noticed by Larsson54
through a constrained-search approach.
Despite these developments, a practical solution was elusive since the exact form of
functionals T and U were not known. Kohn and Sham9 (KS) introduced the concept of
orbitals into DFT in 1965. Since the GS density of non-interacting system may be treated as
the GS density of completely interacting system, the KS formalism gives unique set of
orbitals for a given density. Applying the HK variational principle with respect to the KS
orbitals yields the following canonical KS orbital equations, analogous to the HF equations:
 1 2

    Veff (r )  ψ i (r )  ε i ψ i (r )
 2

(1.40)
It should be noted here that the effective potential operator Veff is independent of the index of
electron and is much simpler than the effective one-electron potential in the Fock operator.
The effective potential (cf. Eq. 1.40) is a sum of external potential Vext, the electron-electron
Coulomb potential and the exchange-correlation potential Vxc as given below.

Veff (r)  Vext (r)  d 3 r'
ρ(r' )
 Vxc (r)
r  r'
(1.41)
The exchange correlation potential, Vxc is a functional derivative of the form
Vx c (r ) 
δEX C  ρ(r ) 
δρ(r )
(1.42)
Eq. (1.40) is a set of nonlinear equations much like the Roothaan-Hall equations and is to be
solved iteratively by SCF procedure. Within KS formalism, the total electronic energy of the
system is computed as
N
E tot[ ρ ] 
 ε  2  d r d r'
1
i
i
3
3
ρ(r) ρ(r' )
 E X C[ ρ ]  d3 r VX C ρ(r)
r  r'

(1.43)
where the first term is the sum of orbital energies and the last term is due to derivative of the
KS exchange correlation functional.
Even the HK and KS formalisms do not lead to the exact form of exchange-correlation
functional EXC[], much of it is left to a systematic search and guesswork. Thus a variety of
exchange-correlation functionals exist in literature,55,56 which are generally integrals of some
function of ED. The Slater’s X method57,58 is considered the predecessor of modern DFT
methods. Slater replaced the exact exchange potential term in HF equation by a statistical
one given by
 3

VXα (r )  6 α  ρ(r )
4π

1/ 3
(1.44)
The value of =1 is prescribed in the original formulation by Slater. The local density
approximation (LDA) has been made in deriving Eq. (1.44) Gáspár59 and later, Kohn and
Sham9 suggested that  in the coefficient should hold a value of 2/3 instead of 1, i.e.  times
Slater’s potential and hence the method gained a generic name, X. This had been the most
successful method for decades until newer functionals within KS formalism were introduced.
A notable difference came with the generalized gradient approximation82 (GGA) where, in
addition to the density values, the functionals are dependent on the gradient of densities. The
GGA functionals have the general form
EXC 
 f( ρ , ρ ,  ρ
α
β
α
. ρ α ,  ρ α . ρ β ,  ρ β . ρ β ) d 3 r
(1.45)
The most widely used exchange functionals are Slater’s X57, B8860 (Becke’s 1988
functional that includes Slater’s exchange with gradient corrections), having the form as
88
E Becke
 E LDA
XC
XC  γ
where x  ρ
4 / 3  ρ

ρ4 / 3x 2
d3 r
1
(1  6 γsinh x)
(1.46)
and γ is a parameter chosen to fit the exchange energy of inert gas atoms
(0.0042 a.u. as defined by Becke). Similarly, there exit local and gradient-corrected
correlation functionals. Amongst the other functionals, available are VWN and VWN5 by
Vosko, Wilk and Nusair (1980), P86 and PW91 by Perdew61 and Wang (1992) etc.
There also exits another type of functional, which offers some improvement over the
corresponding pure DFT functional. This includes a mixture of HF and DFT exchange along
with DFT correlation. The popular functional BLYP is obtained by coupling of Becke’s
generalized gradient corrected exchange functional with the gradient corrected correlation
functional of Lee, Yang and Parr. Becke was the first to use this methodology (1993) and he
proposed B3PW91 functional, bearing the following form.
B88
E B3PW91
 E(1  a)E LSDA
 aE HF
 E cLSDA  cE PW91
xc
x
x  bΔΔ x
(1.47)
Where a, b, c have optimized values of 0.20, 0.72 and 0.1 a.u. respectively. The name
indicated the implementation of three-parameters scheme as well as use of Becke and PW91
functional.
Yet another popular hybrid functional is B3LYP62, Becke-3 parameters non-local exchange
functional60, with the non-local correlation functional of Lee63 et al., functional having the
form:
B88
E B3LYP
 E(1  a)E LSDA
 aE HF
 E cVWN3  (1  c)E cLSDA  cE cLYP
xc
x
x  bΔΔ x
(1.48)
Since the KS-scheme involves a variational procedure to arrive at the KS orbitals, it is
expected that the energies from this formalism would be variationally bound. However, the
exchange-correlation functionals have a strong parametric dependence and hence practically
the energies obtained from KS-based methods are not variational. Recently, Becke64,65 has
benchmarked his exchange-correlation functionals against the G1 (The Gaussian-1 theory66 is
an elaborate procedure for obtaining highly accurate energies through a series of calculations
at predefined levels and basis sets) database generated by Pople and coworkers. A mean
absolute error in the heats of atomization of 55 test molecules turns out to be 2.4 kcal mol –1.
Apart from the total electronic energy, there are certain other molecular properties rigorously
defined within the DFT. Parr9 defined the electronic chemical potential of a system as the
first derivative of energy (cf. Eq. 1.68) with respect to the number of electrons N and
identified it with the negative of its electronegativity.
 E 
μ

  N v
(1.49)
The curvature of energy with respect to the number of electrons is also related rigorously in
the hard-soft acid-base (HSAB) principle and the hardness is given by
1  2 E 

η  
2   N 2  v
(1.50)
These two quantities are being extensively used in the recent years.
In summary, DFT provides an economical alternative for treating molecules at correlated
level of theory. However, disadvantage of DFT is the exchange-correlation functional,
which cannot be derived rigorously from first principles. As of today, DFT remains an
attractive alternative to more rigorous methods, such as CI.
1.3 Molecular Properties
While the energy is undoubtedly the fundamental quantity, chemists usually characterize
molecules by other properties, for example, the dipole moment or the molecular structure. The
ability to accurately calculate these properties is one of the major strengths of modern electronic
structure theory. These calculations are made possible by the fact that the properties are
responses of the molecule to external parameters such as the nuclear coordinates, applied electric
and magnetic fields, etc. These parameters become variables on which a potential energy surface
is mapped out. Therefore analytic derivatives of the energy with respect to these variables yield
the familiar molecular properties. The derivatives with respect to nuclear positions give the
nuclear forces, which allow rapid minimization of the energy with respect to nuclear coordinates,
providing the molecular structure. Second derivatives with respect to nuclear position reveal the
force constants, allowing harmonic frequencies to be calculated.67 These derivatives also allow
the classification of stationary points, greatly facilitating the location of transition structures
(which will be first order saddle points).
The various derivatives with respect to electric field, magnetic field and nuclear spin allow
determination of a range of properties, including: electric polarizability, infrared intensities,
magnetic susceptibility, chemical shielding, spin-spin coupling, Raman intensities and
hyperpolarizabilities. What is important is that they all result from derivatives of the energy, and
thus fast evaluation of the molecular energy is highly desired. The density-based properties that
have been mainly employed in the present work can be further classified as point-dependent and
point-independent properties. The point-dependent properties include scalar fields such as ED,
electrostatic potential (ESP) V(r), Laplacian of density 2, density in momentum space and
vectors fields such as the gradients of ED () and ESP ( V, electric field). The point-
independent (overall) properties are dipole moment () and electronic energy (E). Among the
above-mentioned properties, ED and ESP provide insights for the reactivity and related
applications in chemistry.
1.3.1 Molecular Electron Density
Among various molecular properties of chemical interest, the molecular electron density (MED)
distribution, ρ(r), has been found to be of great conceptual value. An analysis of the total
electron density distribution ρ(r) with the aid of its gradient vector field  ρ(r ) and its Laplace
distribution  2 ρ(r ) leads to meaningful definition of an atom in a molecules and chemical
bonding. Utilizing these definitions one can determine bond order, π character, bond polarity,
and bent bond character of covalent bonds from the properties of ρ(r) in the bonding region.
MED can be extracted from the corresponding many-particle wave function Ψ (x, x2, … xN) as
ρ(r )  N
 Ψ(x, x ,..., x
2
2
N
) d 3 r2 ... d 3 rN
(1.51)
σ
the summation here runs over all spin coordinates, integration over all but one spatial coordinates
(x stands for position and spin) and N is the total number of electrons. The electron density ρ(r)
does have a probabilistic interpretation. The probability that an electron is found in an
infinitesimally small volume element d3r around r is proportional to ρ(r)d3r.
As pointed out earlier within the Hartree-Fock (HF) framework wherein the wavefunction Ψ is
expressed in the form of a Slater determinant constructed from the MO’s, ρ(r) assumes the form
ρ(r) 
P
μν
φμ (r) φ*ν (r)
(1.52)
μν
here P stands for the charge density-bond order matrix. The above equation is valid in the
general case as well i.e. within a CI/MC-SCF or a similar post-HF theory incorporating
correlation effects. Appropriate charge density-bond order then has to be employed for
computing the corresponding ρ(r).
Much of the chemical meaning derived from the analysis of MED is due to the pioneering work
by Bader68 and coworkers. Since MED is a scalar field, to analyze and understand its properties,
it is useful to have the aid of 3D-graphics visualization tools. In general, MED is evaluated over
a 3D-grid encompassing the molecule and the analyses are carried out in terms of iso-valued
contour and surface plots or textured planes. Alternatively, one can locate the topography
(Greek: mapping of a place) of MED to analyze its silent features. Bader68 pioneered the
topological concepts to study ED and its Laplacian.
Consider a 3D-scalar field f. In each direction, f can be a local maximum or minimum.
Thus, it engenders four types of topological features: maxima, minima and two types of saddle
points. Any point in space where all the first-order partial derivatives of a function f vanish
(f/xi=0 for i=1,3) is termed as a critical point (CP). The topological features of point, P is
characterized in terms of the rank and signature of its Hessian matrix, defined as
H ij 
 2f
x i x j
(1.53)
P
The rank is defined as the number of non-zero eigenvalues of the Hessian matrix and signature,
as algebraic sum of the signs of non-zero eigenvalues. Thus a CP is characterized as (R, )
where, R is the rank and  the signature of CP of f. Hence, any 3D scalar function can have four
types of non-degenerate CPs viz., (3, +3) corresponding to a local minimum, (3, +1) and (3, 1)
that represent saddle points on the surface and finally, a (3, -3) that represents local maximum.
However, not all these features are necessarily exhibited by the density of every molecule.68
A pseudo maximum, which obeys Kato’s cusp condition69 [(r)/r|r=0 = 2Z(0)] is
observed at every nuclear position in atomic and molecular ED’s. Any two atoms that fall within
a moderate distance, indicating a strong or weak bonding interaction between them, exhibit a (3,
1) saddle point. A set of bonded (strong or weak) atoms forming a ring possess a (3, +1) CP at
the center of the ring, while a closed cage formed by bonded atoms has a minimum (3, +3) at its
center. However, chemists are generally interested in the bond critical point (BCP) characterized
as (3, 1). A relation between bond length and the bond order had earlier been established by
Coulson.70
Bader71,72 and coworkers proposed an exponential relation between the density at
the bcp, b and the bond order for a series of hydrocarbons. A similar analysis was later carried
out by Boyd et al.73 for a wide range of bond types. Further, the curvature at b gives insight
into the nature of bond and explains features of -bond, bent bond, etc. The bonds can also be
characterized in terms of the ellipticity74  given by
  [1 /  2  1]
(1.54)
where, 1 and 2 are the principal eigenvalues of the Hessian of ED at bcp, the former being
higher in magnitude than the latter. Bader and co-workers68 employed ED to build a rigorous
QM basis for the definition of an atom in a molecule (AIM). The AIM approach has hence been
widely employed to partition MED and other derived properties into atomic components.
ρ(r) is used to construct difference or deformation electrondensities delta ρ(r) as
∆ρ(r) = ρ(molecule) – ρ(promolecule).
The density of the promolecule is obtained by a superposition of atomic densities where the
atoms are considered to be a) in the ground state b) neutral c) noninteracting
d) located
in positions which they adopt in molecule. Analysis of difference density offers a simple
description of chemical bonding : Accumulation of electron density in the bonding are, indicated
by a positive difference (deformation) density (delta ρ >0), acts as an electrostatic glue that keeps
the atomic nuclei together and leads to chemical bonding.
1.3.3 Population Analysis
A very old concept in chemistry is to associate molecular polarity with charge build up or
depletion on the individual atoms. Part of the driving force for this conceit is that it allows one to
conveniently ignore the wave character of the electrons and deal only with the pleasantly more
particulate atoms, these atoms reflecting electronic distribution by the degree to which they carry
positive or negative charge. The partial atomic charges can be computed using well-defined
quantum mechanical operators. However there is no universally agreed best procedure for
computing partial atomic charges, since the partial atomic charges are used in different ways
within the context of different quantitative and qualitative models in chemistry. Thus, many
methodologies have been promulgated for computing partial charges, a few of which have been
discussed here.
A direct partitioning of the molecular wave function into atomic contributions following
some arbitrary, orbital-based scheme has proposed by Mulliken86,87 and this method of
population analysis now bears his name. In this scheme, the electrons are divided amongst the
atoms according to the degree to which the different atomic AO basis functions contribute to the
overall wavefunction. The total number of electrons as given by expression
electrons
  ψ (r ) ψ (r ) d r
N
j
j
j
j
j
(1.59)
j
when expanded into its AO basis set
electrons

 c
j
r,s
jr
 r (r j ) c js  s (r j ) d r j
(1.60)
electrons


   c   c
j
2
jr
jr
r s
r

c j s S r s 

(1.61)
where r and s index AO basis functions φ, cjr is the coefficient of basis function r in MO j, and S
is the overlap matrix element.
Thus the total number of electrons may be divided into two sums, one including only squares of
single AO basis functions, and the other including products of two different AO basis functions.
Electrons associated with only a single basis function belongs entirely to the atom on which that
basis function resides. Electrons shared between basis functions may be divided evenly between
the two atoms on which basis functions r and s reside. The atomic population Nk can then be
calculated as
electrons
Nk 


 c2j r 
c j r c j s Sr s 


rk,sk
 rk

 
j

(1.62)
Cases where r and s are different AO basis function both residing on k drop out because the
orthonormality of the atomic basis functions on a single center makes all the corresponding
overlap integrals zero. The Mulliken partial atomic charge is then defined as
qk= Zk - Nk
(1.63)
where Z is the nuclear charge.
With minimal or small split valence basis sets, Mulliken charges tend to be reasonably intuitive.
Analysis of changes in charge as a function of substitution or geometric change tends to be the
best use of Mulliken charges. Mulliken partial charges prove to be sensitive to the basis set size,
so that comparisons of partial charges from different levels of theory are insignificant. Moreover,
with very complete basis sets, Mulliken charges have a tendency to become unphysically large.
One of the observables from which charges are derived is the electrostatic potential (ESP). The
molecule-molecule interactions at short as well as long range can be modeled through partial
atomic charges calculated using electrostatic potential. All ESP charge-fitting schemes involve
determining atomic partial charges qk that, when used as a multipole expansion according to
nuclei
VESP (r) 
 r-r
qk
(1.64)
k
k
minimize the difference between VESP and the correct VMEP calculated according to
nuclei
VMEP (r) 
 r-r
k
Zk
k

 ψ(r' )
1
ψ(r' ) d r'
r - r'
(1.65)
Typical algorithms select a large number of points spaced evenly on a cubic grid surrounding the
van der Waals surface of the molecule.
Central to any population analysis is the idea of unimpaired validity, that a good way to describe
a molecule is to divide it into atoms and examine how these differ from the free atom. For a
straightforward dissection of molecular charge, the partitioning scheme advanced by Politzer and
Harris has two advantages: (i) It makes the charge on each atom an unambiguous property of the
molecular charge distribution, independent of the mathematical formalism used to derive this
distribution. (ii) When applied to the promolecule, made up of overlapping ground state atoms
prior to any charge migration between or within these atoms, it yields neutral atomic charges.
The atomic fragmentation thus proposed by Politzer and Harris face some shortcomings as if to
say they are bounded somewhat artificially to an array of partition planes. While yielding
acceptable atomic charges they are ill-suited to the definition of other properties that are needed
for a more complete specification of the molecular charge distribution. These shortcomings have
been remedied in case of Hirshfeld partitioning scheme.88 Here, in the promolecule the total
density is a sum of well-defined contributions from all the constituent atoms. The atomic
composition of the promolecule density is mimicked while apportioning the actual molecular
density among the several atoms. Accordingly the molecular density at each point is divided
among the atoms of the molecule in proportion to their respective contributions to the
promolecule density at that point.
Algebraically, the promolecule density at point r is given by
ρ pro (r ) 
ρ
at
i
(r )
(1.66)
i
where the functions ρ iat are suitably positioned, spherically averaged ground state atomic
densities. For each atom a sharing function defined as
w i (r)  ρiat (r) ρpro (r)
(1.67)
specifies its relative share in the promolecule density at r. Functions wi(r) are all positive and
their sum equals one everywhere. The charge density of the bonded atom i as given by
ρib . a . (r)  w i (r) ρmol (r)
(1.68) has
ρmol as the actual molecular density. Thus an overlapping, continuous bonded-atom distributions
thus obtained fully retain the two advantages as those of Politzer and Harris. It produces welldefined atomic fragments that differ from the free atoms only to the degree that the molecule
itself differs from a superposition of free atom densities.
The atomic deformation density then can be obtained by subtracting the density of the free atom
from that of the bonded atom as
δρi (r)  ρib . a (r)  ρiat (r)
(1.69)
Now, the total electronic charge in our bonded atom is given by

Q i   ρ ib . a (r) dν
(1.70)
Adding the nuclear charge Zi gives the net atomic charge
qi = Qi +Zi
(1.71)
In practice the integrand ρ ib . a varies too steeply for easy numerical integration; thus it is generally
more convenient to integrate the atomic deformation density, which yields directly

qi   δρi (r) dν
(1.72)
1.3.7 Self Consistent Reaction Field (SCRF) methods
Methods for evaluating the effect of a solvent may broadly be divided into two types: those
describing the individual solvent molecules and those, which treat the solvent as a continuous
medium.91 The latter one is responsible for “macroscopic” or “long range” effects showing
screening of charges (solvent polarization) responsible for generating a (macroscopic) dielectric
constant different from 1. There are several methods for modeling the long-range solvation
usually employing the concept of a “reaction field” in some way or another.
One of these methods the Self-Consistent Reaction Field (SCRF) model considers the solvent as
a uniform polarizable with a dielectric constant of ε, with the solute M placed in a suitable
shaped hole in the medium.92 Creation of cavity in the medium costs energy, i.e. this is a
destabilization, while dispersion interactions between the solvent and solute add stabilization.
The electric charge distribution of M will furthermore polarize the medium, which in turns acts
back on the molecule, thereby producing an electrostatic stabilization. The solvation (free)
energy may thus be written as
ΔGsolvation= ΔGcavity + ΔGdispersion + ΔGelectrostatic
(1.104)
The energy required to create the cavity and the stabilization due to the van der Waals
interactions between the solute and solvent is usually assumed to be proportional to the surface
area. Thus the corresponding energy terms may be taken simply as being proportional to the total
surface area, or parametrized by having a constant ξ specific for each atom type, with the ξ
parameters being determined by fitting to experimental solvation data.
atoms
ΔGcavity + ΔGdispersion =
ξ S
i i
(1.105)
i
The shape of the cavity thus becomes important. The simplest shape of the cavity is a sphere or
an ellipsoid, which makes the calculation of the electrostatic interaction between M and the
dielectric medium easy. More realistic models employ molecular shaped cavities. Taking the
atomic radius as a suitable factor times a van der Waals radius defines a van der Waals surface
which may have small “pockets” where no solvent can enter. A more better descriptor may be
the surface traced out by a spherical particle of a given radius rolling on the van der Waals
surface denoted as solvent accessible surface (SAS).
The charge density on the surface of the hole, σ(rs), is given by standard electrostatics in terms of
the dielectric constant, ε, and the electric field perpendicular to the surface, F, generated by the
charge distribution within the cavity.
4πεσ(rs) = (ε-1)F(rs)
(1.106)
For spherical or ellipsoidal cavities, eqn 106 can be solved analytically, but for molecular shaped
surface, it needs to be done numerically. Once σ(rs) is determined, the associated potential is
added as an extra term to the Hamilton operator.
H= H0 +Vσ
(1.107)
The potential from the surface charge Vσ is given by the molecular charge distribution,
Vσ (r ) 

σ (rS )
d rS
r  rS
(1.108)
which also enters the Hamiltonian and thus influences the molecular wave function. The
procedure is therefore iterative.
The simplest reaction field model is a spherical cavity, where only the net charge and dipole
moment of the molecule are taken into account. For a net charge q in a cavity of radius a, the
difference in energy between vacuum and a medium with a dielectric constant of ε is given by
the Born model93
2
 1 q
ΔG el (q)   1  
 ε  2a
(1.109)
using a set of partial atomic charges is often called the generalized Born model. The spherical
cavity, dipole only, SCRF model is known as Onsagar model.94 The Kirkwood model95 refers to
a general multipole expansion while the ellipsoidal cavity gives the Kirkwood-Westheimer
model.96 The Polarizable continuum Model (PCM) employs a van der Waals surface type cavity,
a detailed description of the electrostatic potential, and parametrizes the cavity/dispersion
contributions based on the surface area.
Experimental:
If is thus, considered worthwhile to undertake computational studies on structural parameter of
K- 56, TNPDU, DINGU, TNIGU – TNTNZ and K-6 ect. The optimized geometric structures
have been presented below.
Conclusion:
1.4 High Energy Molecules:
The relationships of structure and properties of high energy molecules, particularly their
sensitivity to shock or impact, in terms of electronic structure parameters like bond order,
hydrogen bonding, dipole moment etc., have provided a means to design and synthesize the
molecules with improved stability97-105 and high explosive performance. Such strained systems
with high nitrogen content engender high mass density and large heat of formation, which
facilitates their use in propellants and explosives. Quantum-mechanical methods have recently
been explored to study the decomposition pathways of cyclotrimethylenetrinitramine (RDX) or
cyclotetramethylene-tetranitramine (HMX) since the rupture of C-NO2 bond is a key step in the
decomposition of nitroamines.102-105
The work presented here is a preliminary step to understand the explosive nature of theIn
order to gain deeper understanding
of systems we will take up the calculations of charge
distributions in terms of molecular electrostatic potential and molecular electron density
topography in further studies.
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