12

CHAPTER 2

Computational Methodology

The study of the kinetics of a chemical reaction requires knowledge of the potential

energy surface (PES) for nuclear motion. For bimolecular reaction, the PES should cover

the range of geometries from separated reactants through the strong interaction region

(TS) and on to the products. Once the PES or potential energy profile is known, kinetic

modeling is necessary to calculate the reaction rate constant. Here we briefly give an

account of the electronic structure methods used for the development of potential energy

profile and the theoretical procedures adopted for kinetic modeling.

Quantum chemical calculations provide a straight forward picture of the geometry at

the bottom of the minimum, experimental observations pertain instead to a dynamic

average.1 Now, quantum chemical methods are generally classified into two types: ab

initio methods, that use only fundamental constants of physics as parameters and semi-

empirical methods that contain one or more adjustable parameters. In this work, we have

used various ab initio [Hartree-Fock (HF), perturbation method (MP2) for including

electron-correlation and high level model G2(MP2) method], and density functional

theory (DFT) based electronic structure theories for geometry optimizations (both

stationary and saddle points) and calculating thermochemical parameters. The input data

13

required for kinetic modeling can also be obtained from this electronic structure

calculation.

2.1. The Hartree-Fock Method:

Hartree–Fock (HF) method is an approximate method for the determination of the

ground-state wave function and ground-state energy of a quantum many-body system in

computational physics and chemistry. This theory is fundamental to electronic structure

theory, applicable to atoms as well as molecules. It is the basis of molecular orbital (MO)

theory, which posits that each electron's motion can be described by a single-particle

function (orbital) which does not depend explicitly on the instantaneous motions of the

other electrons.

The HF method assumes that the exact N-body electronic wave function of the system

can be approximated by a single Slater determinant.2 The ubiquity of orbital concepts in

chemistry is a testimony to the predictive power and intuitive appeal of Hartree-Fock MO

theory. However, it is important to remember that these orbitals are mathematical

constructs which only approximate reality. Only for one-electron systems (hydrogen

atom or other one-electron systems, like He+), orbitals are exact eigen functions of the

full electronic Hamiltonian. As long as we are content to consider molecules near their

equilibrium geometry, Hartree-Fock theory often provides a good starting point for more

elaborate theoretical methods which are better approximations to the electronic

Schrödinger equation (e.g., many-body perturbation theory, single-reference

configuration interaction).

14

The Hartree-Fock method consists of approximating the N-electron wave function by

an antisymmetrized product of N one-electron wave functions χi (xi). This product is

referred to as a Slater determinant, ΦSD:

𝜓 ≈ 𝛷𝑆𝐷 =1

𝑁!

𝜒1(𝑥1) 𝜒2 𝑥1 ⋯ 𝜒𝑁 𝑥1

𝜒1 𝑥2 𝜒2 𝑥2 ⋯ 𝜒𝑁 𝑥2 ⋮

𝜒1 𝑥𝑁 ⋮

𝜒2 𝑥𝑁 ⋯⋮

𝜒𝑁 𝑥𝑁

(2.1)

The one-electron functions χi (xi) are called spin orbitals, and are composed of a spatial

orbital фi (r) and one of the two spin functions, α(s) or β(s).

χ (x) = ф(r)σ(s), σ = α,β. (2.2)

The energy minimization using determinantal wave function leads to an equation of the

form: f(1)χi(1) = εi χi(1)

The Fock operator 𝑓 is an effective one-electron operator and is defined as

𝑓 = −1

2𝛻𝑖

2 − 𝑍𝐴

𝑟𝑖𝐴

𝑀

𝐴

+ 𝑉𝐻𝐹 𝑖 (2.3)

The first two terms are the kinetic energy and the potential energy due to the electron-

nucleus attraction. VHF(i) is the Hartree-Fock potential. It is the average repulsive

potential experienced by the ith

electron due to the remaining N-1 electrons. VHF consists

of two components:

𝑉𝐻𝐹 𝑥1 = (𝐽 𝑗

𝑁

𝑗

𝑥1 − 𝐾 𝑗 𝑥1 ) (2.4)

15

where 𝐽 and 𝐾 are the Coulomb and Exchange operator and are defined as:

𝐽 𝑗 𝑥1 = 𝜒𝑗 (𝑥2) 2 1

𝑟12𝑑𝑥2 (2.5)

𝐾 𝑗 𝑥1 𝜒𝑖 𝑥1 = 𝜒𝑗 𝑥2 1

𝑟12𝜒𝑖 𝑥2 𝑑𝑥2𝜒𝑗 𝑥1 (2.6)

The Coulomb operator represents the potential that an electron at position x1 experiences

due to the average charge distribution of another electron in spin orbital χj. 𝜒𝑗 (𝑥2) 2𝑑𝑥2

represents the probability that the electron is within the volume element 𝑑𝑥2. Thus, the

Coulomb repulsion corresponding to a particle distance between the reference electron at

x1 and another one at position x2 is weighted by the probability that the other electron is

at this point in space. The coulomb operator and the corresponding potential are called

local. The Exchange operator 𝐾 has no classical interpretation and is defined through its

effect when operating on a spin orbital. 𝐾 𝑗 (𝑥1) leads to an exchange of the variables in

the two spin orbitals. The exchange operator and the corresponding exchange potential

are called non-local operator.

2.2. Møller-Plesset Theory:

Møller–Plesset perturbation theory (MP)3 is one of several quantum chemistry post-

Hartree–Fock ab initio methods in the field of computational chemistry. It improves on

the Hartree–Fock method by adding electron correlation effects by means of Rayleigh–

Schrödinger perturbation theory (RS-PT),2 usually to second (MP2),

4 third (MP3)

5,6 or

fourth (MP4)7 order.

16

The electron correlation energy is defined as the difference between the exact non-

relativistic energy of the system and the HF energy at the limit of infinite basis set.

In RS-PT, we consider an unperturbed Hamiltonian operator Ĥ0, to which is added a

small (often external) perturbation :

(2.7)

where λ is an arbitrary real parameter. In MP theory the zeroth-order wave function is an

exact eigenfunction of the Fock operator, which thus serves as the unperturbed operator.

The perturbation is the correlation potential involving the instantaneous effect of the

electrons on each other.

In RS-PT the perturbed wave function and perturbed energy are expressed as a power

series in λ:

,

Substitution of these series into the time-independent Schrödinger equation gives a new

equation

(2.8)

17

Equating the factors of λk in this equation gives an k

th-order perturbation equation (k =

0,1,2, ..., n).

The MP-energy corrections are obtained from RSPT with the perturbation (correlation

potential) as:

(2.9)

where the normalized Slater determinant Φ0 is the lowest eigenfunction of the Fock

operator

(2.10)

Here N is the number of electrons of the molecule under consideration, H is the usual

electronic Hamiltonian, f(1) is the one-electron Fock operator, and εi is the orbital energy

belonging to the doubly-occupied spatial orbital υi. The shifted Fock operator

serves as the unperturbed (zeroth-order) operator.

The Slater determinant Φ0 being an eigenfunction of F, it follows readily that

(2.11)

so that the zeroth-order energy is the expectation value of H with respect to Φ0, i.e., the

Hartree-Fock energy:

18

(2.12)

Since the first-order MP energy correction for electron correlation

(2.13)

is obviously zero, the lowest-order MP correlation energy appears in second order. This

result is the Møller-Plesset theorem3 which states that ‗the correlation potential does not

contribute in first-order to the exact electronic energy‘.

In order to obtain the MP2 formula for a closed-shell molecule, the second order RS-

PT formula is written on the basis of doubly-excited Slater determinants. Singly-excited

Slater determinants do not contribute because of the Brillouin theorem. After application of

the Slater-Condon rules for the simplification of N-electron matrix elements with Slater

determinants in bra and ket and integrating out spin, the MP2 energy correction becomes

………………….(2.14)

where υi and υj are canonical occupied orbitals and υa and υb are canonical virtual orbitals.

The quantities εi, εj, εa, and εb are the corresponding orbital energies. Clearly, through

second-order in the correlation potential, the total electronic energy is given by the

Hartree-Fock energy plus the second-order MP correction: E ≈ EHF + EMP2. The solution of

19

the zeroth-order MP equation gives the HF energy. The first non-vanishing perturbation

correction beyond the HF treatment is the second-order energy.

The Second (MP2)4, third (MP3)

5,6, and fourth (MP4)

7 order Møller–Plesset

calculations are standard levels used in calculating small systems and are implemented in

many computational chemistry codes. Higher level MP calculations, generally only MP5,8

are possible in some codes. However, they are rarely used because of their huge

computational cost.

For open shell molecules, MPn-theory can directly be applied only to unrestricted HF

reference functions (since ROHF states are not in general eigenvectors of the Fock

operator). However, the resulting energies often suffer from severe spin contamination,

leading to large errors.

Gaussian-2 (G2)9 theory is based on ab initio molecular orbital theory for the

computation of total energies of molecules at their equilibrium geometries. This theory

uses the 6-311G(d,p) basis set and corrections for several basis set extensions. Treatment

of correlation is by Møller-Plesset (MP) perturbation theory and quadratic configuration

interaction (QCISD). The final total energies obtained in G2 theory are effectively at the

QCISD(T)/6-311G(3df,2p) level, making certain assumptions about additivity of the

corrections. The G2(MP2) method uses MP2 for basis set extensions, and is nearly as

accurate as the full G2 method at substantially reduced computational cost. The total

G2(MP2) energy is given by:9

E0 = E[QCISD(T)/-311G(d,p)] +∆MP2 + HLC + E(ZPE)

20

where HLC is the higher level correction, which is given as HLC = Anβ - Bnα, where A =

4.81 mhartree, B = 0.19 mhartree, and nα and nβ are the number of α and β valence

electrons, respectively, with nα ≥ nβ and ∆MP2 = E[MP2/6-311+G(3df,2p)] – E[MP2/6-

311G(d,p)].

2.3. Density Functional Theory:

Density Functional Theory (DFT),10,11

is another quantum mechanical method used in

physics and chemistry to investigate the electronic structure (principally the ground state)

of many-body systems, in particular atoms, molecules, and the condensed phases. With

this theory, the properties of a many-electron system can be determined by using

functionals, i.e. functions of another function, which in this case is the spatially

dependent electron density. Hence the name density functional theory comes from the use

of functionals of the electron density [(r)]. DFT is among the most popular and versatile

methods available in condensed-matter physics, computational physics, and

computational chemistry. Since DFT uses electron density as the key variable, unlike

wave function based methods the complexity of the problem, in principle, does not

increase in DFT with the increase in electron number.

In the last few years, methods based on density functional theory have gained steadily

in popularity. The best DFT methods achieve significantly greater accuracy than Hartree-

Fock theory at only a modest increase in cost (far less than MP2 for medium size and

larger molecular systems). They do so by including some of the effects of electron

correlation much less expensively than traditional correlation methods. DFT has been

21

very popular for calculations in solid state physics since the 1970s. However, DFT was

not considered accurate enough for calculations in quantum chemistry until the 1990s,

when the approximations used in the theory were greatly refined to better model the

exchange and correlation interactions. In many cases the results of DFT calculations for

solid-state systems agree quite satisfactorily with experimental data. Computational costs

are relatively low when compared to traditional methods, such as Hartree-Fock theory

and its descendants based on the complicated many-electron wave function.

Although density functional theory has its conceptual roots in the Thomas-Fermi

model, DFT was put on a firm theoretical footing by the two Hohenberg-Kohn theorems

(H-K)12

which demonstrated the existence of a unique functional which determines the

ground state energy and density exactly. The theorem does not provide the form of this

functional i.e; the exact form of universal energy functional is unknown and extension to

excited state is no obvious.

Following on the work of Kohn and Sham, the approximate functional employed by

current DFT methods partition the electronic energy into several terms;

E = ET + E

V + E

J +E

XC (2.15)

Where ET is the kinetic energy term arising from the motion of the electrons, E

V

includes term describing the potential of the nuclear-electron attraction and of the

repulsion between pairs of nuclei, EJ is the electron-electron repulsion-term or coulomb

self-interaction of the electron-density, and EXC

is the exchange-correlation term and

includes the remaining part of the electron-electron interactions.

22

ET + E

V + E

J correspond to the classical energy of the charge distribution ρ(r). The

EXC

term accounts for the remaining terms in energy:

• The exchange energy arising from the antisymmetry of the quantum mechanical

wave function.

• Dynamic correlation in motion of the individual electrons.

Kohn and Sham (1965)10

realized that things would be considerably simpler in DFT if

only the Hamiltonian operator were one for a non-interacting system of electrons (Kohn

and Sham 1965). Such a Hamiltonian can be expressed as a sum of one-electron

operators, has eigenfunctions that are Slater determinants of the individual one-electron

eigenfunctions, and has eigenvalues that are simply the sum of the one-electron

eigenvalues. The crucial bit of cleverness, then, is to take as a starting point a fictitious

system of non-interacting electrons that have for their overall ground-state density the

same density as some real system of interest. In the usual fashion to find the orbitals χ

that minimize E, we find that they satisfy the pseudoeigenvalue equations

𝑕𝑖𝐾𝑆𝜒𝑖 = 휀𝑖𝜒𝑖 (2.16)

where the Kohn-Sham (KS) one-electron operator is defined as

𝑕𝑖𝐾𝑆 = −

1

2𝛻𝑖

2 − 𝑍𝑘

𝑟𝑖 − 𝑟𝑘

𝑛𝑢𝑐𝑙𝑒𝑖

𝑘

+ 𝜌 𝑟′

𝑟𝑖 − 𝑟′ 𝑑𝑟 ′ + 𝑉𝑋𝐶 (2.17)

and 𝑉𝑋𝐶 =𝛿𝐸𝑋𝐶

𝛿𝜌 (2.18)

Vxc is a so-called functional derivative.

23

Traditional functionals: A variety of functionals have been defined, generally

distinguished by the way that they treat the exchange and the correlation functional;

• Local and exchange functionals involve only the values of the electron-spin

densities. Slater and Xα are well known local exchange functionals, and the local spin

density treatment of Vosko, Wilk and Nusair (VWN)13

is a widely used local correlation

functional.

The local exchange is virtually always defined as follows:

rdEX

LDA

33/4

3/1

4

3

2

3

(2.19)

where ρ is a function of vector r. This form was developed to reproduce the exchange

energy of a uniform gas. However, it has a weakness in describing molecular systems.

• Gradient-Corrected functionals involve both the values of the electron spin

densities and their gradients. Such functionals are also sometimes referred to as non-

local. A popular gradient-corrected exchange functional is one proposed by Becke in

1988; a widely used gradient corrected correlation functional is LYP functional of Lee,

Yang and Parr.14

Perdew has also proposed some important gradient-corrected correlation

functional, known as Perdew 86 and Perdew-Wang 91.14

The functional form of gradient-correlated exchange functional based on the LDA

exchange functional is given as:

rd

x

x

EEX

LDA

X

Becke

3

1

23/4

88 sinh61

(2.20)

24

where x = ρ-4/3│ ρ│.

ν is a parameter chosen to fit the known exchange energies of the inert gas atoms, and

Becke defines its value as 0.0042 Hatrees. As the above equation makes it clear, Becke‘s

functional is defined as a correlation to the local LDA exchange functional, and it

succeeds in remedying many of the LDA functional‘s deficiencies.

Hybrid functionals: There are also several hybrids functional which define the exchange

functionals as a linear combination of Hartree-Fock, local and gradient-corrected

exchange terms; this exchange functional is then combined with a local and/or gradient-

corrected correlation functionals. The best known of this hybrid functionals is Becke‘s

three-parameter formulation; hybrid functionals based on it are available in Gaussian via

B3LYP15,16

and B3PW9117

keywords. Becke style hybrid functionals have proven to be

superior to the traditional functionals defined so far.13,18-26

Becke-style three parameter functional may be defined via the following expression:

EB3LYPXC

= ELDAX

+ C0 (EHFX- ELDA

X) + Cx ∆EB88

X EVWN3

X + CC (ELYP

C- EVWN3

C) (2.21)

Here, the parameter C0 allows any mixture of Hartree-Fock and LDA local exchange

to be used. In addition, Becke‘s gradient correction to LDA exchange is also included,

scaled by the parameter CX. Similarly, the VWN3 local correlation functional is used, and

it may be optically corrected by the LYP correlation correction via the parameter CC. In

the B3LYP functional, the parameters values are those specified by Becke, which he

determined by fitting to the atomization energies, ionization potentials, proton affinities

and first-row atomic energies in the G1 molecule set: C0=0.20, CX=0.72 and CC=0.81.

25

The DFT based theory used here are BB1K27

and MPWB1K28

methods. These are

hybrid meta functional methods and their functional form is given as:

𝐸𝑋𝐶𝑕𝑦𝑏

=𝑋

100𝐸𝑋

𝐻𝐹 + 1 −𝑋

100 𝐸𝑋

𝐷𝐹𝑇 + 𝐸𝐶𝐷𝐹𝑇 (2.22)

where EXHF

is the nonlocal Hartree-Fock exchange energy, X is the percentage of

Hartree-Fock exchange in the hybrid functional, EXDFT

is the local exchange energy, and

ECDFT

is the local DFT correlation energy. The value of X at BB1K is 42 and at

MPWB1K is 44. These methods are specially parameterized for kinetic modeling against

different data set.

2.4. Kinetic Modeling

Here we discuss briefly the methods used for rate constant calculations for all the

bimolecular reactions studied in this work. As stated in the introduction (Chapter-1), our

objective was to develop a simple kinetic model capable of producing reasonably accurate

results. We have therefore adopted the simple statistical method of Transition State Theory

(TST) for our study.

2.4.1. Rate Constant Calculation:

In the entire thesis, rate constant have been calculated using the conventional

transition state theory (CTST).

26

The theory of reactions rates that was published almost simultaneously by Henry

Eyring29

and by M. G. Evans and M. Polanyi30

in 1935 is referred to as conventional

transition state theory (CTST).31

The great value of CTST is that the resulting rate

equation, although simple, provides a framework in terms of which even quite

complicated reactions can be understood in a qualitative way. The main assumptions of

CTST are as follows:

Assumption 1. Molecular systems that have surmounted the col or saddle point of

potential energy surface in the direction of products cannot turn back and form reactant

molecules again.

Assumption 2. The energy distribution among the reactant molecules is in accordance

with the Maxwell-Boltzmann distribution. Furthermore, it is assumed that even when the

whole system is not at equilibrium, the concentration of these activated complexes that

are becoming products can also be calculated using equilibrium theory.

Assumption 3. It is permissible to separate the motion of the system over the col from the

other motions associated with the activated complex.

Assumption 4. A chemical reaction can be satisfactorily treated in terms of classical

motion over the barrier, quantum effects being ignored.

In addition, CTST involves the same assumptions and approximations that are made in

the calculation of equilibrium constants using statistical mechanics. Usually, these are not

serious, and corrections for them can be made. Therefore, understanding of CTST

requires some knowledge of how equilibrium constants are treated by the methods of

27

statistical mechanics. According to statistical mechanics, the molecular equilibrium

constant for a reaction

aA + bB yY + zZ

is given by: 𝐾𝑐 =𝑞𝑌

𝑦𝑞𝑍

𝑧

𝑞𝐴𝑎𝑞𝐵

𝑏 𝑒−∆𝐸0

𝑅𝑇 (2.23)

where the q‘s are the partition functions per unit volume. The energy E0 is the molar

energy change at the absolute zero when ‗a‘ mol of A reacts with ‗b‘ mol of B to form the

products, all substances being in their standard states.

Let us consider the reaction:

A + B [AB]#

P (2.24)

where A and B are the reactants, [AB]# is the activated complex [or transition state (TS)]

and P is the product. Then, the rate of reaction, ν, at temperature T is given by the

expression:

RTE

BA

AB eqq

qBA

/#

#0]][[

(2.25)

The rate constant is defined by k ν / [A][B],

Therefore, the rate equation for a bimolecular reaction, derived by the methods of

conventional transition-state theory is,32,33

28

RTE

BA

ABB eqq

q

h

Tkk

/#

#0

(2.26)

where kB is the Boltzmann constant, h is Planck‘s constant and E0# is the barrier height.

The barrier height is estimated from the difference of energy between TS and reactants

including zero point energy.

Fig. 2.1. General potential energy diagram

The total partition function q for a molecule is defined by

𝑞 ≡ 𝑔𝑖𝑒−휀𝑖 𝑘𝐵𝑇

𝑖 (2.27)

The energy εi is the energy of the ith

state relative to the zero-point energy, and gi is the

degeneracy, that is, the number of energy states corresponding to the ith

level. Usually, it

is assumed that the various types of energy- electronic (ei), vibrational (νi), rotational (ri)

and translational (ti) - are independent of one another. The total energy corresponding to

the ith

energy state is thus expressed as the sum of the different types:

εi = ei + νi + ri + ti (2.28)

29

The four energies on the right-hand side represent the four types of energy corresponding

to the ith

state. The partition function becomes

𝑞 = 𝑔𝑒𝑖𝑒−𝑒𝑖 𝑘𝐵𝑇 𝑔𝜈𝑖𝑒

−𝜈𝑖 𝑘𝐵𝑇 𝑔𝑟𝑖𝑒−𝑟𝑖 𝑘𝐵𝑇 𝑔𝑡𝑖𝑒

−𝑡𝑖 𝑘𝐵𝑇 (2.29)

𝑖

the gi having factorized as well as the exponential terms. This equation may be written as

q = qeqνqrqt (2.30)

where qe, qν, qr and qt are separate partition functions, each referring to one type of

energy. Thus, the partition function has been factorized, so that each term may be

evaluated separately.

Each of the Partition Functions were evaluated from the following expressions:34

𝑞𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑖𝑜𝑛 = 2𝜋𝑀𝑘𝐵𝑇 3 2

𝑕3𝑉

𝑞𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 = 8𝜋2 𝜍 П𝑗

2𝜋𝐼𝑗𝑘𝐵𝑇 1 2

𝑕

𝑞𝑣𝑖𝑏𝑟𝑎𝑡𝑖𝑜𝑛 = П𝑗 1 − 𝑒−𝑕𝜈𝑗 𝑘𝐵𝑇 −1

(𝑗 = 1,3𝑁 − 6 𝑜𝑟 3𝑁 − 5)

𝑞𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑖𝑐 = 𝑒−∆𝐸𝑗 𝑘𝐵𝑇

𝑗

𝑞𝑡𝑜𝑡𝑎𝑙 = 𝑞𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑖𝑜𝑛 𝑞𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 𝑞𝑣𝑖𝑏𝑟𝑎𝑡𝑖𝑜𝑛 𝑞𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑖𝑐

30

Here M is the total molecular mass, Ij stands for the moment of inertia of the j-th

rotational mode, is the symmetry number and j is vibrational frequency for the j-th

normal mode. The qtranslation is evaluated in per unit volume.

2.5. Tunneling:

Quantum tunneling refers to the quantum mechanical phenomenon where a particle

tunnels through a barrier that it classically could not surmount because its total kinetic

energy is lower than the potential energy of the barrier. Quantum tunneling is a

consequence of the wave-particle duality of matter and is often explained using the

Heisenberg uncertainty principle. This tunneling plays an essential role in several

physical phenomena, including radioactive decay, and has important applications to

modern devices such as flash memory, the tunneling diode, and the scanning tunneling

microscope.35

In fact, tunneling is the rapidly declining tail on a wave function which penetrates

into a classically forbidden region of a barrier, if the barrier were of finite thickness,

emerge from the remote side and become wave-like again, representing the tunneling of

particles through the barrier. The tunneling probability depends upon the height and

shape of the barrier and most importantly on the mass of the penetrating particle. It is

most important for lighter particles. Therefore one must include a correction for quantum

mechanical barrier penetration () while computing chemical reaction rates. The

correction factor can be interpreted as

= quantum mechanical rate/classical mechanical rate

31

We have estimated the value of by using Eckart‘s one-dimensional unsymmetric

potential energy function.36

In this method, first the reaction path through TS is fitted in a potential function

(2.31)

where y = -exp(2πx/L) and A and B are two parameters that depend upon forward and

reverse barrier heights, x is the variable dimension and L is a characteristic length

depending upon shape of the reaction barrier.36

Then Γ(T) is estimated by numerically

integrating the tunneling probability with energy E, (E), for this potential function over

all possible values of energy and divided by the classical probability37

(2.32)

We calculate (E) by approximating the form of the real barrier by an unsymmetrical

Eckart potential function as described in Eqn. (2.31)

(2.33)

where a, b, and d are related to the forward (E0#) and reverse barrier (V2) and to the

imaginary frequency (*) associated with the transition state. The value of Γ at different

temperature is estimated from the numerical integration of Eqn. (2.32) and by using our

program Tunnel.37

The TST equation including tunneling correction is then expressed as

RTE

BA

ABB eqq

q

h

TkTTk

/#

#0)()(

(2.34)

2)1(1)(

y

By

y

AyyV

dba

dbaE

2cosh)(2cosh

2cosh)(2cosh1)(

0

//)/()()(

#

TkEdEeeT b

TkETkE bb

32

2.6. Hindered Rotor Correction:

Generally, harmonic oscillator (HO)-rigid rotator approximation is made for

calculating vibrational-rotational partition functions mainly because geometrical

structures and harmonic vibrational frequencies can easily be obtained from

straightforward electronic structure calculations. However, in many cases the leading

deviations from HO behaviour may be attributed to low frequency torsional motions

about single bonds. Such modes generally show a hindered rotation transition from HO

behaviour at low temperature to free internal rotation at high temperature. Proper

treatment of these low frequency modes is essential for getting accurate rate constant and

thermochemical parameters.38,39

Treating internal rotation is especially important for

transition states for OH addition and H-abstraction reactions where apart from other

modes OH rotation may have to be treated as hindered rotor.40

We have used the simple method prescribed by Truhlar et al.39

for estimating the free

rotor or hindered rotor partition function depending upon the barrier height (V0) for

rotation and temperature (T). The free-rotor partition function is estimated from the

standard classical result

2/1)2( IkTq FR

(2.35)

where k is Boltzmann‘s constant, ћ is Planck‘s constant divided by 2, T is temperature, I

is the effective moment of inertia for torsional motion and is the effective symmetry

number. The hindered rotor partition function can be estimated approximately using the

33

expression suggested by Truhlar by interpolating between the harmonic oscillator (kT <<

V0) and free rotor (kT >> V0) limits,

)/tanh( .. clHOFRqHOHIN qqqq (2.36)

where qHO.q

and qHO.cl

are the quantum and classical HO partition function for the mode

being treated as internal rotation.

The moment of inertia (I) has been calculated using our program HINROT. First the two

units of rotation (R1 and R2) and the rotation axis are identified from the normal modes of

vibration, and then IR1 and IR2 are calculated using the standard formula. Finally, like

reduced mass, the effective moment of inertia for the internal rotation is obtained from38

)( 21

21

RR

RR

II

III

2.7. Thermochemical Parameters:

The thermodynamic properties of molecules are of fundamental interest in physics,

chemistry and biology. The thermochemical parameters like standard reaction enthalpy

(∆rH) and Gibbs free energy (∆rG) at a temperature T were estimated from the difference

of H and G values of products and reactants at that temperature:

∆𝑟𝐻 = 𝐸0 + 𝐻𝑐𝑜𝑟𝑟

𝑃𝑟𝑜𝑑

− 𝐸0 + 𝐻𝑐𝑜𝑟𝑟

𝑅𝑒𝑎𝑐

(2.37)

∆𝑟𝐺 𝑇 = 𝐸0 + 𝐺𝑐𝑜𝑟𝑟

𝑃𝑟𝑜𝑑

− 𝐸0 + 𝐺𝑐𝑜𝑟𝑟

𝑅𝑒𝑎𝑐

(2.38)

34

where E0 is the total electronic energy including ZPE and Hcorr and Gcorr are the factors to

be added to E0 for getting enthalpy and Gibbs free energy, respectively, at a temperature

T for taking into account the contribution of translation, rotation and vibrational motion

of a molecule. The Hcorr and Gcorr are defined as:

𝐻𝑐𝑜𝑟𝑟 = 𝐻𝑡𝑟𝑎𝑛𝑠 + 𝐻𝑟𝑜𝑡 + 𝐻𝑣𝑖𝑏 + 𝑅𝑇 (2.39)

𝐺𝑐𝑜𝑟𝑟 = 𝐻𝑐𝑜𝑟𝑟 − 𝑇. 𝑆𝑡𝑜𝑡𝑎𝑙 (2.40)

where entropy,

𝑆𝑡𝑜𝑡𝑎𝑙 = 𝑆𝑡𝑟𝑎𝑛𝑠 + 𝑆𝑟𝑜𝑡 + 𝑆𝑣𝑖𝑏 + 𝑆𝑒𝑙 (2.41)

The ∆rH(T) (as well as ∆rG(T)) is comparable to the standard thermodynamic expression

based on heats of formation because number of each elements remain same before and

after the reaction,

∆𝑟𝐻0 𝑇 = ∆𝑓

𝑃𝑟𝑜𝑑

𝐻0 𝑇 − ∆𝑓

𝑅𝑒𝑎𝑐

𝐻0 𝑇 (2.42)

Similarly, the ∆rE is estimated from the energy difference of products and reactants

including zero point energy.

All the electronic structure calculations were performed using Gaussian 03 programme

suite,41

whereas rate constant calculations were done using our own program Birate.

35

References:

1. P, -O, Åstrand, G. Karlström, A. Engdahl and B. Nalendra, Novel model for

calculating the intermolecular part of the infrared spectrum for molecular complexes,

J. Chem. Phys. 1995, 102, 3534.

2. A. Szabo and N. S. Ostlund, Modern Quantum Chemistry, Dover, New York, 1996.

3. C. Møller, M.S. Plesset, "Note on an Approximation Treatment for Many-Electron

Systems" Phys. Rev. 1934, 46, 618.

4. H.G. Martin; J.A. Pople, M.J. Frisch, "MP2 energy evaluation by direct methods"

Chem. Phys. Lett. 1988, 153, 503.

5. J.A. Pople, R. Seeger, R. Krishnan, "Variational configuration interaction methods

and comparison with perturbation theory" (abstract) Int. J. of Quan. Chem. 1977, 12,

149.

6. J.A. Pople, J.S. Binkley, R. Seeger, "Theoretical models incorporating electron

correlation" Int. J. of Quan. Chem. 1976, 10, 1.

7. R. Krishnan, J.A. Pople, "Approximate fourth-order perturbation theory of the

electron correlation energy" Int. J. Quan. Chem. 1978, 14, 91.

8. R. Krishnan, J.A. Pople, E.S. Replogle, H.G. Martin, "Fifth order Moller-Plesset

perturbation theory: comparison of existing correlation methods and implementation

of new methods correct to fifth order" J. Phys. Chem. 1990, 94, 5579.

9. L.A. Curtiss, K. Raghavachari, J.A. Pople, J. Chem. Phys. 1993, 98, 1293.

10. R.G. Parr, W.T. Yang, Density-Functional Theory of Atoms and Molecules; Oxford

University Press: New York 1989.

36

11. R.M. Dreizler, E.K.U. Gross, Density-Functional Theory; Springer; Berlin, 1990.

12. P. Hohenberg, W. Kohn "Inhomogeneous electron gas". Phys. Rev. 1964, 136, 864.

13. S.H. Vosko, L. Wilk and M. Nusair, ―Accurate spin-dependent electron liquid

correlation energies for local spin density calculations: a critical analysis,‖ Canndian

J. Phys. 1980, 58, 1200.

14. C.J. Cramer, Essentials of Computational Chemistry, Theories and Models, 2nd

Edn;.

John Wiley & Sons Ltd, England, 2004.

15. K. Kim and K.D. Jordan (1994), "Comparison of Density Functional and MP2

Calculations on the Water Monomer and Dimer" J. Phys. Chem. 98 (40): 10089–

10094.

16. P.J. Stephens, F.J. Devlin, C. F. Chabalowski and M.J. Frisch (1994), "Ab Initio

Calculation of Vibrational Absorption and Circular Dichroism Spectra Using Density

Functional Force Fields" J. Phys. Chem. 98 (45): 11623–11627.

17. D.A. McQuarrie Statistical Mechanics, 1st Indian Ed., VIVA Books, New Delhi,

2003.

18. W. Kohn and L.J. Sham, ―Self-Consistent Equations Including Exchange and

Correlation Effects,‖ Physical Review, 1965, 140, A1133.

19. B. Miehlich, A. Savin, H. Stoll and H. Preuss, Chem. Phys. Lett. 1989, 157, 200.

20. C. Lee, W. Yang and R.G. Parr, ―Development of the Colle-Salvetti correlation-

energy formula into a functional of the electron density,‖ Phys. Rev. B, 1988, 37, 785.

21. A.D. Becke, Phys. Rev. A, 1988, 38, 3098.

37

22. A.D. Becke, J. Chem. Phys. 1993, 98, 1372.

23. A.D. Becke, ―Density-functional thermochemistry. III. The role of exact exchange,‖ J.

Chem. Phys. 1993, 98, 5648.

24. J.P. Perdew and Y. Wang, ―Accurate and Simple Analytic Representation of the

Electron gas Correlation energy,‖ Phy. Rev. B, 1992, 45, 13244.

25. D.R. Salahub and M.C. Zerner, eds., The Challenge of d and f Electrons (ACS,

Washington D. C., 1989).

26. G.W. trucks and M.J. Frisch, ―Rotational Invariance Properties of Pruned Grids for

numerical integration,‖ in preparation (1996).

27. Y. Zhao, D.G. Truhlar, J. Phys. Chem. A 2004, 108, 6908.

28. Y. Zhao, B.J. Lynch, D.G. Truhlar, J. Phys. Chem. A 2004, 108, 2715.

29. H. Eyring, J. Chem. Phys., 1935, 3, 107.

30. M.G. Evans, M. Polanyi, Trans. Faraday Soc. 1935, 31, 875.

31. Transition-state theory is the expression now recommended by IUPAC [V. Gold,

Pure. Appl. Chem. 1979, 51, 1725; K.J. Laidler, Pure. Appl. Chem. 1981, 53, 753].

The expression ―theory of absolute reaction rates‖ and ―activated-complex theory‖

also have been used.

32. H. Eyring, J. Chem. Phys., 1935, 3,107. A more comprehensive treatment can be

found in W.F.K. Wynne-Jones and H. Eyring, J. Chem. Phys., 1935, 3, 492. This

article is reproduced in full in M.H. Back and K.J. Laidler (Eds.), Selected Readings

in Chemical Kinetics, Pergamon, Oxford, 1967.

33. M.G. Evans, M. Polanyi, Trans. Faraday Soc., 1935, 31, 875; 1937, 33, 448.

38

34. P. Atkins, J.D. Paula, Atkins’ Physical Chemistry, 8th

Edn.; Oxford University Press,

2006

35. J. Taylor, Modern Physics, 234. Prentice Hall, 2004.

36. H.S. Johnston, J. Heicklen, J. Phys. Chem. 1962, 66, 532.

37. The transmission coefficient (T) was estimated from our own Fortran program.

Numerical integration was performed using trapezoidal algorithm with a step-size of

0.000025 and 800000 steps.

38. D.G. Truhlar, J. Comput. Chem. 1991, 2, 266.

39. Y. Chuang, D.G. Truhlar, J. Chem. Phys. 2000, 112, 1221.

40. P.Y. Ayala, H.B. Schlegel, J. Chem. Phys. 1998, 108, 2314.

41. M.J. Frisch et al., Gaussian 03, Rev.C.01, Gaussian Inc., Wallingford, CT, 2004.