Computational Chemistry
Semester II M.Sc Chemistry
M.G University, Kottayam
AB INITIO METHODS
Aby Jimson
Dept Of Chemistry
St. Stephen’s College
Uzhavoor, Kottayam
Ab initio methods in computational
chemistry
"Ab Initio” means “from the beginning
This name is given to computations which are derived directly from theoretical principles,
with no inclusion of experimental data
Most of the time this is referring to an approximate quantum mechanical calculation.
The approximations made are usually mathematical approximations, such as using a
simpler functional form for a function or getting an approximate solution to a differential
equation.
The most common type of ab initio calculation is the ‘Hartree Fock’ calculation in which
the primary approximation is called the central field approximation
A review of Hartree Fock Calculations
THE WAY TO HF CALCULATIONS
Hartree and
Hydrogen Variation &
SWE 1 D box 3 D box Hartree Fock
Atom Perturbation
Methods
Self Consistent Field Concept
An important unsolved problem in quantum mechanics is how to
deal with indistinguishable, interacting particles like electrons
If particles interact, that interaction must be in the Hamiltonian.
So until we know where the particles are, we can’t write down the
Hamiltonian, but until we know the Hamiltonian, we can’t tell where
the particles are.
SCF is an iterative method used to calculate the molecular orbitals
with maximum possible accuracy
Hartree’s Method
According to Hartree's self-consistent-field (SCF) model of the atom, the motion of each electron in the effective field of
the N-1 others is governed by a one-particle Schrödinger equation.
It follows that the electrons are independent, and interact only via the mean-field Coulomb potential
the electrons feel the averaged field of all the other electrons in the system.
We assume the wavefunction can be written as a Hartree product:
Ψ(𝑟1 + 𝑟2 ) = Ψ1 𝑟1 Ψ2 𝑟2
The individual one-electron wavefunctions, Ψ1 &Ψ2 are called molecular orbitals.
This form of the wavefunction does not allow for instantaneous interactions of the electrons.
Instead, the electrons feel the averaged field of all the other electrons in the system.
The Hartree form of the wavefunction is sometimes called the independent electron approximation.
Pauli's Exclusion Principle
Total wavefunction must be antisymmetric with respect to the
interchange of electron coordinates
The Pauli Principle is a consequence of antisymmetry
The Hartree wavefunction is not antisymmetric
Fock modified the Hatrees Equation by adding the concept of
antisymmetry and formulated the Hartree- Fock Equation
Hartree Fock Equation
The antisymmetrized wavefunction is called the Hartree-Fock
wavefunction.
It can be written as a Slater determinant
This ensures the electrons are indistinguishable and are therefore
associated with every orbital
The HF wavefunction is an antisymmetric wavefunction written in
terms of the one-electron Molecular Orbitals.
The Schrodinger wave equation 𝐻Ψ
= 𝐸Ψ
The Hamiltonian is made up of energy terms
=𝑻
𝑯 𝒏 + 𝑻
𝒆 + 𝑽
𝒏𝒏 + 𝑽
𝒏𝒆 + 𝑽
𝒆𝒆
HF theory is the simplest wavefunction-based method
It relies on the following approximations:
The Born-Oppenheimer approximation
The independent electron approximation
The Linear Combination of Atomic Orbitals Approximation (LCAO)
Central Field Approximation
The Coulombic electron-electron repulsion is not specifically taken
into account. It's net effect is included in the calculation
This is a variational calculation, meaning that the approximate
energies calculated are all equal to or greater than the exact energy
The energies from HF calculations are always greater than the exact
energy and tend to a limiting value called the Hartree Fock limit.
The second approximation in HF calculations is that the wave
function must be described by some functional form, which is only
known exactly for a few one electron systems.
Because of this approximation, most HF calculations give a
computed energy greater than the Hartree Fock limit
A number of types of calculations begin with a HF calculation then
correct for the explicit electron-electron repulsion, referred to as
correlation.
Eg -Moller-Plesset perturbation theory, Configuration Interaction
(CI) etc
The first step in computational chemistry is the calculation of the
molecular orbitals (MOs) for a given molecule.
If we can calculate the MOs for a molecule, then we can know lots
of things about the molecule, including
energy
electron density
electrostatic potential
transition state
frequency
A molecular-orbital theory calculation is a mathematical expression
of an electron in a molecule.
Although there are many types of molecular-orbital functions, here
we will only look at the Slater Type Orbitals (STOs) and the Gaussian
Type Orbitals (GTOs).
STO & GTO
STOs require more calculating, which takes tremendous amounts of
time, however their calculations have been found to be more
accurate than GTOs. On the other hand, GTOs, although less
accurate, are much faster to calculate than STOs.
By adding several GTOs, we can mimic the STOs accuracy. As the
number of GTOs used increased, the better they were able to
model the STO equation.
When using GTOs to model STOs, the new equations are given a
new name. They are identified as STO-nG equations where n is a
constant that represents the number of GTOs used. For instance, two
common equations are the STO-3G and the STO-6G in which 3 and
6 GTOs are used respectively.
HF calculations overview
Advantages Disadvantages
The Born-Oppenheimer approximation Electronic Correlation
The independent electron approximation Consequence: Calculated energy is always
higher than true energy
Central Field Theory The calculated value can only reach a
minimum energy up to Hartree Fock Limit,
which is in turn higher than the true energy
The Linear Combination of Atomic Orbitals The integrodifferential equations are thereby
approximation (LCAO) transformed into
algebraic equations (Roothaan's equations) for
the expansion coefficients
Roothaan equations
The Roothaan equations are strictly the equations for a closed-shell
Restricted Hartree-Fock
Numerically much easier than integro-differential equations
