Summer Research Fellowship Programme of India's Science Academies 2017
DFT study of HNCCN
+
molecule detected in interstellar
medium
Anju M
Central University of Tamil Nadu
Guided by
Dr. T. J. Dhilip Kumar
Department of Chemistry, Indian Institute of Technology Ropar, Rupnagar, 140001 Punjab India
1. Introduction
Computational chemistry is a branch of chemistry which uses methods of theoretical chemistry
incorporated into computer software which assists in solving chemical problems. Computational
chemistry simulates chemical structures and reactions based on fundamental laws of physics.
It allows chemists to study chemical phenomenon by running calculations on computers rather than
by examining reactions and compounds experimentally. Some methods can be used to model not
only stable molecules, but also short-lived, unstable intermediates and transition states. In this way
information can be provided about molecules and reactions which is impossible to obtain through
experimental observations.
Therefore, computational chemistry is an independent research area and a vital adjunct to
experimental studies.
2. An overview of computational chemistry
There are two main broad areas within computational chemistry involving in the area of
characterisation of structure of molecules and their reactivity: molecular mechanics and electron
structure theory. They both perform same basic calculations:
1. Computing the energy of a particular molecular structure.
2. Performing geometry optimizations to locate the nearest lowest energy possessing molecular
structure in close proximity to the one specified as the starting structure in the input.
Geometry optimizations depend primarily on gradient of energy the first derivative of
energy with respect to the atomic positions.
3. Computing vibrational frequencies of molecules resulting from interatomic motions within
the molecule. Frequencies depend on the second derivative of energy with respect to atomic
position.
2.1. Molecular mechanics
This simulation uses laws of classical physics to predict the structure and property of molecules.
The methods of molecular mechanics are found in MM3, hyperChem, Quanta, Alchemy. Each
method in molecular mechanics is characterized by its particular Force Field. A Force Field is a
vector field which denotes force experienced by each particle at a point.
Another important point about molecular mechanics is that they do not explicitly treat electrons in
the simulations instead they consider interactions of nuclei. They inexplicitly consider electronic
effects in computations using parametrization. Since they don’t treat electrons explicitly, the
computation is comparatively inexpensive.
Limitations
They cannot compute problems involving electronic effects such as bond breaking.
There is no general force field for all sets of molecules. Therefore each force field yield good
result only for a limited set of molecules which was related to parametrizations.
2.2. Electronic structure methods
These computations use laws of quantum mechanics instead of laws of classical physics.
Quantum mechanics states that energy and other related properties can be found by solving
Schrodinger equation.
Schrödinger equation: ĤΨ(r:R) = E(R)Ψ(r:R) for fixed set of locations R of the nuclei. The
electronic wavefunction Ψ depends on the electronic coordinates r and relates to or are in terms of
R. E(R) is the electronic energy.
There are two major classes of electronic structure theory
I. Semi-empirical methods such as AM1, PM3, PM6 use parameters derived from experimental
data to simplify the computation. They solve an approximate form of Schrödinger equations
which depends on the parameters available for the chemical system under consideration.
II. Ab initio method unlike molecular mechanics or semi-empirical methods do not use any
experimental parameters for computations. Instead the simulations are solely based on laws
of quantum mechanics.
Semi-empirical and ab initio methods differ in the trade off made between computational cost and
accuracy of result. Semi-empirical calculations are relatively inexpensive and provide reasonable
qualitative simulations about molecular systems and fairly accurate predictions of energies and
structures for the system.
In contrast, ab initio computations provide high quality results for a broad range of systems. They
are not limited to any specific class of system. Modern ab initio programs can handle jobs of up to
a few hundred atom.
3. Born-Oppenheimer approximation
The Born Oppenheimer approximation is one of the first approximations done to simplify the
Schrödinger equation, solving which is the ultimate goal. The approximation considers that since
the mass of the nucleus is considerably greater than the mass of electron, the nuclei move very
slowly with respect to the electrons and the electrons react almost instantaneously to the changes is
nuclear position. Thus, the electron distribution on any system depends on the position of the nuclei
and not on the velocity of the nuclei or it can be said that the Born-oppenheimer approximation
takes electronic motion occurring in a field of fixed nuclei.
The Born-Oppenheimer approximation allows to construct an electronic Hamiltonian (Ĥelec) which
neglects the kinetic energy term for the nuclei. This Hamiltonian is used in the Schrödinger equation
describing the motion of electrons in the field of fixed nuclei.
ĤelecΨelec(r:R) = Eeff(R)Ψelec(r:R)
Solving this equation for the electronic wavefunction provides the effective nuclear potential
function (Eeff). Eeff is one of the terms used to describe nuclear Hamiltonian (Ĥnucl). This
Hamiltonian is used in the Schrödinger equation for nuclear motion describing the vibrational,
rotational and translational states of nuclei. Solving nuclear Schrödinger equation is required to
obtain the vibrational states of the molecule.
4. Approximate methods
Schrödinger equation cannot be solved exactly for any atom or molecule more complicated than
hydrogen atom but fortunately there are approximate methods that can be used to solve the
Schrödinger equation to almost any accuracy required.
The two most widely used approximation methods are perturbation theory and variational method.
4.1. Perturbation theory
Perturbation theory expresses the solution to one problem in terms of another problem that has been
solved previously.
Schrödinger equation can be solved exactly for systems like hydrogen atom, harmonic oscillator,
rigid rotator and it turs out that most of the system cannot be solved exactly. The classic examples
being helium atom and anharmonic oscillator.
The basic idea behind perturbation theory is that the total Hamiltonian of the systems whose
solutions cannot be solved exactly is brought into two parts. One for which the Schrödinger equation
can be solved exactly and an additional term, whose presence prevents an exact solution.
Ĥ
= Ĥ
(0)
+ Ĥ
(1)
The first term is called the unperturbed Hamiltonian Ĥ
(0)
and the additional term the perturbation
Ĥ
(1)
.
The aim is to generate the wavefunctions and energy of the perturbed system with the prior
knowledge of the unperturbed system. If the perturbation terms are small, then the solution to the
complete perturbed problem should be close to the solution of the unperturbed system.
The first order correction to E
(0)
is:
E = E
(0)
+ Ψ
(0)*
Ĥ
(1)
Ψ
(0)
+ higher order terms where
dτ is the volume element.
4.2. Variational principle
It provides an upper bound to the ground state energy of a system. This method is more reliable or
more useful because unlike perturbation theory this method doesn’t need a prerequisite of a similar
problem solved.
Let the ground state energy of a system be satisfied by Schrödinger equation
ĤΨ = EΨ
Multiplying by Ψ
(0)*
and integrating it over all space, we get
Considering that Ψ
(0)
is not normalised the denominator is not equal to unity.
The main step is to guess a trial function ɸ for Ψ
(0)
and to optimize it.
The corresponding energy after using the trial function is
By variational principle Eɸ >= E0. That is an upper bound on E0 by using a trial function is found.
The closer ɸ is to Ψ
(0)
the closer Eɸ will be to E0.
is minimized by differentiating with respect to α where ɸ is dependent on some arbitrary
parameters α called variational parameters and the result is set equal to zero to find the value of α.
The value of α is substituted in Eɸ so that the minimum value of Eɸ is found.
Trial functions with more than one parameter can produce better result
Trial functions which are linearly dependent on the variational parameter leads to secular
determinant which when solved gives a quadratic equation.
The quadratic secular equation given two values of E, out of which the smallest value is
taken.
If the trial function having linear dependence with variational parameters are orthonormal,
then it simplifies the secular determinant.
5. Hartree Fock method
Hartree Fock (HF) is one of the least expensive ab initio method and is typically used to solve the
time independent Schrödinger equation for multi electron system. This method solves equations
non-linearly or called as iteration and hence it is named as “self-consistent method”. HF method
involves minimization of Rayleigh ratio to get the HF equation.
Certain approximations considered in the HF method are:
i. Born-Oppenheimer approximation is inherently incorporated.
ii. All the operators are assumed to be completely non-relativistic.
iii. HF method do not explicitly consider electron correlation instead electron-electron
repulsion is treated in an average way. Each electron is considered to be moving in an
electrostatic field of nuclei and the average field of n-1 electrons. Hence HF method will
not give accurate results for chemical phenomenon largely dependent on electronic effects
such as bond breaking.
iv. It is assumed that it is enough to describe each energy eigen functions by a single slater
determinant.
The last two approximations can lead to a considerable deviation from accurate results which give
rise to many “post-Hartree Fock methods”.
6. Density functional theory
Density functional theory (DFT) is a method which are similar to ab initio methods in many ways
but they are called as the third class of electronic structure methods. DFT methods give more
accurate results when compared with HF method and computationally almost as expensive as HF.
DFT methods are more accurate because they include the effects of electron correlation (the fact
that electrons in a system react to one another’s motion and attempt to keep out of one another’s
way) in their model. DFT methods include electron correlation effects through general functionals.
A Functional is defined as a function of a function in mathematics. Here functionals are function of
electron density which itself is a function of coordinates in space. DFT functionals separate the
electron energy into several components like the kinetic energy, electron-nuclear interaction, the
Coulomb repulsion and an exchange-correlation term all of which are computed separately.
7. Basis sets
A basis set is a function or a set of function that numerically represents molecular orbitals which in
turn combine to approximate the total electronic wavefunction. The two general categories of basis
sets are:
1. Minimal basis set: such basis set contain minimum number of basis function required for each
atom in a system. Such basis sets describes only basic aspects of orbitals. Minimal basis sets
use fixed-size type atomic orbitals.
Orbitals can be numerically represented in two ways. They are called as slater type orbitals (STO)
and Gaussian type orbitals (GTO).
STO = N*e
(-αr)
GTO = N*e
(-αr²)
where N is the normalisation constant, α is the orbital exponent and r is the radius.
The only difference between STO and GTO is the power of r. Though STO are seen to be more
accurate than a similar number of GTO, they are not mathematically convenient to use. Hence GTOs
are preferred over STO even though large number of functions are required. All basis set equations
in the form STO-NG (where N represents the number of GTOs combined to approximate the STO)
are considered to be minimal basis sets.
2. Extended basis set: They come up with much more detailed description of the orbitals and
consider higher orbitals of the molecule and account for the shape and size of molecular charge
distribution. There are different types of basis sets coming under the category of extended basis
set. Some of them are:
i. Split valence basis set: The first thing that can be done to increase the size of the basis
set so as to get better results is to increase the number of basis function per atom.
Double split valence basis set such as 6-31G, 3-21G have two basis function for each
valence orbital. For example H and C atom are represented as
H: 1s, 1sꞌ
C: 1s, 2s, 2sꞌ, 2px, 2py, 2pz, 2pxꞌ, 2pyꞌ, 2pzꞌ
Where the prime and unprimed orbitals differ in size.
ii. Polarized basis set: Split valence basis set allows change in orbital size but no change
can be made in its shape. Polarized basis set can modify shape as well by adding orbitals
with angular momentum more than required for the atom’s ground state description.
For instance, polarized basis set add d function to C atom, f functions to transition
metals and sometimes p functions to H atom. An example of a polarized basis set is 6-
31G(d) or written as 6-31G*. It indicates 6-31G basis set with d function added to heavy
atoms.
iii. Diffuse functions: Diffuse functions denoted with + sign along with the basis set is
important for molecules with lone pair, anions, system in excited state, system with
significant negative charge etc. Diffuse functions are large size versions of s and p type
functions. 6-31G+(d) is 6-31G(d) basis set with diffuse function added to heavy atoms.
iv. Correlation consistent basis set: These are one of the most widely used basis sets
designed for converging Post-HartreeFock calculations systematically to the complete
basis set limit using empirical extrapolation techniques. For first- and second-row
atoms, the basis sets are cc-pVNZ where N=D, T, Q, 5, 6, (D=double, T=triples, etc.).
The 'cc-p', stands for 'correlation-consistent polarized' and the 'V' indicates they are
valence-only basis sets. They include successively larger shells of polarization
(correlating) functions (d, f, g, etc.).
8. Open and closed shells
Open and closed shell models describes how electron spin is to be handled also called as restricted
and unrestricted respectively. For closed shell molecules, having an even number of electrons are
divided into pairs of opposite spin and by default a spin restricted model is used. This simulation
force each pair of electron into a single spatial orbital. For open shell molecules, having unequal
number of spin up (α) and spin down (β) electrons or system in excited state an unrestricted spin
model is used. Here calculations use separate spatial orbital for α and β electrons and hence more
time consuming. It is also possible to define spin restricted open shell models where all the paired
electrons are forced into one spatial orbital and the α or β electron is given a separate single spatial
orbital. The keyword used for spin restricted open shell model is RO and they are more inexpensive
when compared to unrestricted model.
9. Gaussian software
Gaussian is a software capable of computing chemical phenomenon through mathematical
simulations and serves computational chemists. Gaussian can predict properties and features of
molecular system like:
Molecular energies and structures
Energies and structures of transition state
Reaction pathway
Vibrational frequencies of systems
Bond and reaction energies
Molecular orbitals
Atomic charge and electrostatic potentials
Multipole moments
IR and Raman spectrum
NMR properties
Thermochemical properties
Potential energy curve
Substituent effects
Reaction mechanism
Computations can be carried out in gas phase or in solution, ground state or in excited state, in
solvent mediums, at different thermodynamic conditions.
A typical Gaussian input contains route section which has keywords describing the job, tittle of the
job if required and molecular specification. Molecular specification includes the first line with
charge and multiplicity and in second row z matrix (internal coordinates) or Cartesian coordinates.
Example: Z matrix of H
2
O molecule
O
H 1 R1
H 1 R2 A1
Variables
R1 = 0.9669
R2 = 0.9669
A1 = 107.724