Haripriya.
Asst. professor
STC Ranni
An Outline of What Computational
Chemistry Is All About
You can calculate molecular geometries, rates and equilibria, spectra,and other
physical properties. The tools of Computational chemistry are molecular
mechanics, ab 1nitio, semi empirical and density functional methods, and
molecular dynamics. Computational chemistry is widely used in the
pharmaceutical industry to explore the interactions of potential with
biomolecules, for example by docking a candidate drug into the active site of an
enzyme. It is also used to investigate the properties of solids (e.g. plastics) in
materials science.
What You Can Do with Computational Chemistry
Computational chemistry (also called molecular modelling; the two terms mean
about the same thing) is a set of techniques for investigating chemical problems
on a computer. Questions commonly investigated computationally are
Molecular geometry: the shapes of molecules - bond lengths, angles and
dihedrals
Energies of molecules and transition states: this tells us which isomer is
favoured at equilibrium, and (from transition state and reactant energies) how
fast a reaction should go.
Chemical reactivity: for example, knowing where the electrons are concerted
(nucleophilic sites) and where they want to go (electrophilic sites) enables us to
predict where various kinds of reagents will attack a molecule.
IR, UV and NMR spectra: these can be calculated, and if the molecule is
unknown'. Someone trying to make it knows what to look for.
The interaction of a substrate with an enzyme: seeing how a molecule
fits into the active Site of an enzyme is one approach to designing better
drugs.
The physical properties of substances: these depend on the properties of
individual molecules and on how the molecules interact in the bulk
material al. For example, the strength and melting point of a polymer (e.g.
Plastic) depend on how well the molecules tit together and on how
Strong the forces between them are. People who investigate things like this
work in the field of material sciences.
The Tools of Computational Chemistry
Molecular mechanics- based on a ball-and-springs model of molecules
lf we know the normal spring lengths and the angles between them, and
how much energy it takes to stretch and bend the springs, we can
calculate the energy of a given collection of balls and springs,
i.e. of a given molecule; changing the geometry until the lowest energy
is found enables us to do a geometry optimization, i.e. to calculate a
geometry for the molecule. Molecular mechanics is fast: a fairly large
molecule like a steroid (e.g. Cholesterol, C27H460) can be optimized in
seconds on a good personal computer.
Ab initio methods-(ab initio, Latin: "from the start, i.e. from first
principles") are based on the Schrodinger equation. This is one of the
fundamental equations of modern physics and describes, how the electrons in a
molecule behave. The ab initio method solves the Schrodinger equation for a
molecule and gives us an energy and wave function. The wave function is a
mathematical function that can be used to calculate the electron distribution .Ab
initio methods are based on approximate solutions of the Schrodinger equation
without appeal to fitting to experirnent .The Schrodinger equation cannot be
solved exactly for any molecule with more than one electron. Thus
approximations are used. An ab initio calculation is based only on basic
physical theory (quantum mechanics) and is in this sense "from first principles”
Semi empirical methods - based on approximate solutions of the
Schrodinger equation with appeal to fitting to experiment (1.e. using
parameterization) Semi empirical calculations are, like ab initio, based
the Schrodinger equation. However. more approximations are made in solving
it, and the very complicated integrals that must be calculated in the ab initio
method are not actually evaluated in semi empirical calculations: instead, the
programme draws a library of integrals that was compiled by finding the best
fit of some calculated entity like geometry or energy to the experimental values.
This plugging of experimental values into a mathematical procedure to get the
best calculated values is called parameterization. It is the mixing the theory and
experiment that, makes the method semi empirical: it is based on the
Schrodinger equation. Semi empirical calculations are slower than molecular
mechanics but much faster than ab initio calculations. Semi empirical
calculations take roughly100 times as long as molecular mechanics
calculations, and ab initio calculations take roughly 100-1,000 times as
long as semi empirical. A semi empirical geometry optimization on a
steroid might take seconds on a PC.
Density functional theory (DFT) methods based on approximate solutions of
the Schrodinger equation, bypassing the wave function that is a central feature
of ab initio and semi empirical methods. Density functional calculations (DFT
calculations, density functional theory) are, like ab initio and semi empirical
calculations, based on the Schrodinger equation. However, unlike the other two
methods, DFT does not calculate a conventional wave function, but rather
derives the electron distribution (electron density function) directly. A
functional is a mathematical entity related to a function. Density functional
calculations are usually faster than ab initio, but slower than semi empirical.
DFT is relatively new (serious DFT computational chemistry goes back to the
1980s, while computational chemistry with the ab initio and semi empirical
approaches was being done in the 1960s).
Molecular dynamics calculations apply the laws of motion to molecules.-
Thus one can simulate the motion of an enzyme as it changes shape on
binding to a substrate, or the motion of a swarm of water molecules around a
molecule of solute. Molecular dynamics methods study molecules in motion..
Semi empirical methods, which are much faster than ab initio or even DFT, can
be applied to fairly large molecules (e.g. Cholesterol, C27H460), while molecular
mechanics will calculate geometries and energies of very large molecules such
as proteins and nucleic acids; however, molecular mechanics does not give
information on electronic properties. Computational chemistry is widely used in
the pharmaceutical industry to explore the interactions of potential drugs with
biomolecules, for example by cocking a candidate drug into the active site of an
enzyme. lt is also used to investigate the properties or Solids (e.g. plastics) in
material Science.
Computational Chemistry is the culmination of the view that chemistry is best
understood as the manifestation of the behaviour of atoms and molecules and
that these are real entities rather than merely convenient intellectual molecules.
Computational Chemistry allows one to calculate molecular geometry,
reactivity, spectra and other properties.
Very large biological molecules like proteins or DNA are studied mainly with
molecular mechanics because other methods will take too long. Key portions of
a large molecule like an active site of an enzyme, can be studied with semi
empirical or even Ab initio method. Moderately large molecules like steroid can
be studied with semi empirical calculations or if one is willing to invest time,
with Ab initio calculations. Novel molecules with unusual structures are
best investigated with Ab initio or DFT calculations, since the parameterization
inherent in molecular mechanics or semi empirical methods make them
unreliable for molecules that are different from those used in the
parameterization.
The energies of the molecules can be calculated using molecular mechanics,
semi empirical, Ab initio or Density Functional method. The choice of method
depends very much on the particular problem. Reactivity which depends largely
on electron distribution must usually be studied with a quantum mechanical
method (semi empirical, Ab Initio or DFT). Spectra are most reliably calculated
by Ab initio or DFT methods and some molecular mechanics programs
fairly good IR spectra.
Docking a molecule into the active site of an enzyme to see how it fits is an
extremely important application of computational chemistry. This work is
usually done with molecular mechanics because of the large molecules
involved. The results of such docking experiments serve as
guide to designing better drugs.
Computational chemistry is valuable in studying the properties of materials.
Semiconductors Super conductors, plastics, ceramics have all been investigated
with the aid of computational chemistry.
What are the scopes of computational chemistry?
Computational chemistry is a set of technique for investigating chemical
process on computer. The following are the scopes of computational chemistry.
The shapes of the molecules, bond lengths, bond angles and dihedral angles can
be investigated.
It helps in calculating the energies of molecules and its transition states which
aids to infer which isomer is favoured at equilibrium and how fast a reaction
would take place.
Computational chemistry calculations help us to predict where various kinds of
reagent will attack a substrate. The simulations help us to know the interaction
of a substrate with an enzyme. Thus computational chemistry serves as a guide
to designing better drugs, molecules that will interact better with the designed
enzymes, etc.
Semiconductors, superconductors, plastics, ceramics, etc. have been
investigated with the aid of computational chemistry.
It is fairly cheap and fast.
Computational techniques are generally employed before starting an
experimental project to ensure its success.
The Born-Oppenheimer Approximation
Born and Oppenheimer showed in 1927 that to a very good approximation the
nuclei in a molecule are stationary with respect to the electrons.
Mathematically, the approximation states that the Schrodinger equation for a
molecule may be separated into an electronic and a nuclear equation.
consequences are: (1) to calculate the energy of a molecule, it is enough to solve
the electronic Schrodinger equation and then add the electronic energy to the
inter nuclear repulsion to get the total internal energy, (2) a molecule has a
shape. The nuclei see the electrons as a smeared-out cloud of negative charge
which binds them in fixed relative positions and defines the surface of the
molecule.
Because of the rapid motion of the electrons compared to the nuclei the
"permanent" geometric parameters of the molecule are the nuclear coordinates.
The energy (and the other properties) of a molecule is a function of the electron
coordinates φ( x, y ,z )of each electron) but depends only parametrically on the
nuclear coordinates, i.e. for each geometry (1, 2, 3.) there is a particular energy.
A simple expression for the electronic Hamiltonian to a molecule with 2n
electrons and µ atomic nuclei (the nucleus has Z µ charge) gives
(the electronic Hamiltonian has kinetic energy term, e-nuclear attraction term,
e-e repulsion tem)
The variation theorem states that the energy calculated from the Schrodinger
equation must be greater than or equal to the true ground-state energy of the
molecule.
The Concept of the Potential Energy Surface
The potential energy surface (PES) is a central concept in computational
chemistry. A PES is the relationship - mathematical or graphical - between the
energy of a molecule (or a collection of n molecules) and its geometry. The
Born-Oppenheimer approximation says that in a molecule the nuclei are
essentially stationary compared to the electrons. This is one of the cornerstones
of computational chemistry.Because it makes the concept of molecular shape
(geometry) meaningful, makes possible the concept of a PES, and simplifies the
application of the Schrodinger equation to molecules by allowing us to focus on
the electronic energy and add in the nuclear repulsion energy later. Geometry
optimization and transition state optimization are explained.
Consider a diatomic molecule AB. In some ways a molecule behaves like balls
(atoms) held together by springs (chemical bonds). If we take a macroscopic
balls-and-spring model of our diatomic molecule in its normal geometry (the
equilibrium geometry), grasp the "Atoms” and distort the model by stretching or
compressing the bonds, We increase the potential energy of the molecular
model. The Stretched or compressed spring possesses energy, by definition,
since we moved a force through a distance to distort it. Since the model is
Motionless while we hold it at the new geometry, this energy is not kinetic and
but potential. The graph of potential energy against bond length is an example
of a potential energy surface. A line is a one-dimensional “surface".
Near the equilibrium bond length qe the potential Energy vs bond length curve
for a macroscopic balls-and-spring model or a real molecule is described by a
quadratic equation. For a Simple harmonic oscillator E=1/2 k (q-q e) where k is
the force constant of the spring). However, the potential energy deviates from
the quadratic curve as we move away from qe. The deviations from Molecular
reality represented by an harmonicity
The potential energy surface for a diatomic molecule.
The potential energy increases if the bond length q is stretched or compressed
away from it s equilibrium value q e..The potential energy at qe(zero distortion of
bond length) is zero.
The Concept of the Potential Energy Surface
The potential energy surface (PES) is a central concept in computational
chemistry. A PES is the relationship - mathematical or graphical -
between the energy of a molecule ( or a collection of molecules) and its
geometry. The Born-Oppenheimer approximation says that in a
molecule the nuclei are essentially stationary compared to the electrons.
This is one of the cornerstone of computational chemistry because it
makes the concept of molecular shape (geometry) meaningful, makes
possible the concept of a PES, and simplifies the application
of the Schrodinger equation to molecules by allowing us to focus on
the electronic energy and add in the nuclear repulsion energy later; this
third point, very important in practical molecular computations.
Geometry optimization and transition state optimization are explained.
Consider a diatomic molecule AB. In some ways a molecule behaves like balls
(atoms) held together by springs (chemical bonds) we take a macroscopic
balls-and-spring model of diatomic molecule.
A diatomic molecule AB has only one geometric parameter for us to vary, the
bond length q AB. Suppose we have a molecule with more than one geometric
parameter, for example water: the geometry is defined by two bond lengths and
a bond angle. The PES for this triatomic molecule is a graph of E versus two
geometric parameters, q1= the O-H bond length and q2= H-O-H bond angle
Consider a triatomic molecule of lower symmetry, such as hypoflurous acid
HOF. This has three geometric parameters, the H-O & O-F length and the
H-O-F angle. To construct a PES for HOF analogous to H2O would require to
plot E vs q 1, q 2, q 3. The HOF PES is a 3-D "surface of more than two
dimensions in 4-D space: it is a hypersurface and potential surfaces are
sometimes called potential energy hypersurfaces. In AB diatomic molecule
PES the minimum potential energy geometry is the point at which
d E / d q= 0. On the H20 PES the minimum energy geometry is defined by the
point Pm, corresponding to the equilibrium values of q l and q 2, at this point
d E /d q l = d E/ d q 2= 0.
Pm corresponds to the minimum energy geometry of water molecule
Consider the interconversion of HCN and HNC, through a transition state.
HCN is the global minimum and HNC is the local minimum in the PES. The
lowest energy path connecting the two minima is called the intrinsic reaction
coordinate (the reaction path). The three species of interest, HNC, HCN & the
transition state linking these two, are called stationary points. A stationary point
on a PES is a point at which the surface is flat, i.e. parallel to the horizontal line
corresponding to the one of the geometric parameter.
Mathematically, a stationary point is one at which the first derivative of the
potential energy with respect to each geometric parameter is zero.
∂𝐸 ∂𝐸
∂𝑞1
= ∂𝑞2
=…=0
Stationary points that correspond to actual molecules with a finite lifetime (in
contrast to the transition states, which exist only for an instant), like HCN or
HNC, are minima, or energy minima: each occupies the lowest-energy point in
its region of the PES, and any small change in the geometry increases the
energy, as indicated in Figure. HCN is a global minimum, since it is the lowest-
energy minimum on the whole PES, while HNC is a relative minimum, a
minimum compared only to nearby points on the surface. The lowest-energy
pathway linking the two minima, the reaction coordinate or intrinsic reaction
coordinate (IRC) is the path that would be followed by a molecule in going
from one minimum to another should it acquire just enough energy to overcome
the activation barrier, pass through the transition state, and reach the other
minimum.
The transition state linking the two minima represents a maximum along the
direction of the IRC, but along all other directions it is a minimum. This is a
Characteristic of a saddle-shaped surface, and the transition state is called a
saddle point .The saddle point lies at the " center " of the saddle-shaped
region and is a stationary point, since the PES at that point is parallel to the
plane defined by the geometry parameter axes.we can see that a marble placed
(precisely) there will balance.
Mathematically, minima & saddle points differ in that although both are
stationary points (they have zero first order derivatives), a minimum is a
minimum in all directions , but a saddle point is a maximum along the reaction
coordinate & minimum in all other directions.
Minima and maxima can be distinguished by their second derivatives. We
can write
along the reaction coordinates.
The distinction is sometimes made between a transition state and a transition
state structure. Strictly speaking, a transition state is a thermodynamic concept.
Since equilibrium constants are determined by free energy differences, the
transition structure, within the strict use of the term, is a free energy maximum
along the reaction coordinate. This species is also often also called an activated
complex. A transition structure, in strict usage, is the saddle point on a
theoretically calculated PES. .
A potential energy surface is a plot of the energy of a collection of nuclei and
electrons against the geometric coordinates of the nuclei- essentially a plot of
molecular energy versus molecular geometry.
Geometry optimization
The characterization of a stationary point on a PES, and calculating its
geometry and energy, is a geometry optimization. The stationary point of
interest might be a minimum or a transition state. Locating a minimum is called
an energy minimization, and locating a transition state is referred to as a
transition state optimization. Geometry optimizations are done by starting with
an input structure that resemble the desired stationary point and submitting this
structure to a computer software that changes the geometry until it has found a
stationary point.
To characterize the structure as a minimum or saddle point, vibrational
frequencies are calculated to obtain IR spectra, and to obtain zero point
energies. The frequency calculation is done using the same method that was
used to optimize the stationary point. For a minimum in the PES all the
vibrational frequencies are positive (the Force constant is positive- the second
derivative is the force constant). For a transition state the criteria are: (1) The
structure at a transition state should lie somewhere between that of
the reactants and the products, (2) It must have one and only one imaginary
frequency (a negative frequency): (3) The imaginary frequency must correspond
to the reaction coordinate (4) The energy or the transition state must be higher
than that of the two species it connects.
Ab initio method
The term 'Ab-initio' is the Latin for from the beginning this name is given to
computations that are derived directly from theoretical principles with no
inclusions of experimental data.
Ab initio calculations depend on solving the Schrodinger equation the nature of
the necessary approximations determines the level of the calculation.
Schrodinger equation could not be solved exactly for any molecule with more
than one electron. Thus approximations are used the ab initio calculation is
based only on physical theory (quantum mechanics) if is in this sense it starts
"from first principles"
In the simplest approach, the Hartree Fock method, the total molecular wave
function φ is approximated as a slater determinant composed of occupier spin
orbital's. To use there in practical calculations, the spatial orbitals are
approximated as a linear combination of basis functions. That is
φ𝐻𝑃 = φ1 φ2…… φ𝑁 Such a wave function is called Hartree wave function.
Strengths: Ab initio calculation are based on the Schrodinger equation without
empirical adjustments φ this ensure that they will give correct answers
provided the approximations needed to obtain numerical result 'to solve the
Schrodinger equation are not several for the problem at hand A consequence
absence of empirical parameter is that ab initio calculations can be performed
for any kind of molecular species rather than only species for which empirical
parameters are available. One of the approximation Characteristic in all ab
initio method is the introduction of the basis set.
Uses: This method is used to calculate equilibrium geometry, energy,
vibrational frequency, Spectra, IE, EA, dipole moment etc.
Disadvantages: These calculation are slow, they need more computer
resources. This increase with the level of calculation. This can be over comed
by increasing computer power.
SCF method
HSCF is the most important Ab initio method. It is based on Central field
approximation method i.e. columbic e- e- repulsion is taken into account by
integrating the repulsion term. This gives the average effect of repulsion not
explicit repulsion interaction. It is a single e- approximation method used in
multi e- systems.
In SCF method a trial wave function is written for a molecule as the product of
one-electron wave functions (atomic orbitals). This function is called a Hartree
product. Ψ0 = ψ0(1) ψ0(2) ψ0(3)…………… ψ0(n)
Here ψ0 is a function of the coordinates of all the electrons in the
molecule,ψ0(1) is a function of the coordinates of electron 1, ψ0(2) is a function
of the coordinates of electron 2, etc., the one-electron functions etc. are called
atomic orbitals. The initial guess, Ψ0 is our zeroth approximation to the true
total
wavefunction Ψ.
First a wavefunction is solved for electron 1 a one-electron Schrodinger
equation in which the electron-electron repulsion comes from electron 1 and an
average electrostatic field due to all other electrons.
Solving this equation gives Ψ1 (1) and improved version of Ψ0 (1). Next solve
for electron 2 a one electron Schrodinger equation with electron 2 moving in an
average field due to all other electrons. Likewise the solving the Schrodinger
equation is continued till electron n. This completes first cycle of calculations
and gives,
Ψ1 = ψ1(1) ψ1(2) ψ1(3)…………… ψ1(n). The cycle is repeated until the field
of cycle k is essentially the same as that of cycle k-1, i.e.it is consistent with the
previous field. Hence the Hartree procedure is called the self-consistent-field
(SCF) procedure.
These defects of the Hartree SCF method were corrected by Fock and by Slater
in 1930, and Slater devised a simple way to construct a total wave function
from one-electron functions (i.e. orbitals) such that will be antisymmetric to
electron switching. Hartree's iterative, average-field approach supplemented
with electron spin and antisymmetry leads to the HF equations.
In the SCF method, introduced by Hartree, the total polycentric wave function
for the system is obtained by the cyclic or the iterative procedure as Ψk = ψk(1)
ψk(2) ψk(3)…………… ψk (n).The wave function is self consistent in the sense
that the kth iteration and( k-1) th iteration provides essentially the same result Ψk
is called Hartree product wave function.
H-F Equation
Slater wave function is a determinant whose elements are the functions of 1
electron wave function & spin orbitals.
Slater determinant for 4electron closed shell system
Ψ=
1
ψ1 (1)α1 ψ1 (1)β1 ψ2 (1)α1 ψ1 (2)α2 ψ1 (2)β2 ψ2 (2)α2 ψ1 (3)α3
4
ψ1 (4)α4 ψ1 (4)β4 ψ2 (4)α4 ψ2(4)β4
Hartree iterative average field approach with electron spin And antisymmetry
leads to HF equation.
𝐹 ψ𝑖 = 𝐸𝑖 ψ𝑖
I=1,2,3……n
Ψi= ith spin orbital
F = operator called Fock operator, is the effective HF Hamiltonian
Ei =orbital energy of ith spin orbital.
The Hamiltonian for 2n electron system is given by
2𝑛 2𝑛 2𝑛
^ 1 2 𝑧 1
𝐻 =− 2
∑ ∇𝑗=1 − ∑ 𝑟𝑗
+ ∑ ∑ 𝑟𝑖𝑗
𝑗=1 𝑗=1 𝑖=1 𝑗>𝑖
The first term = electronic kinetic energy
Second term = nucleus- electron attraction potential energy
Third term = electron-electron repulsion potential energy
Assuming the wave function to be normalised & the energy could be
determined using variational theorem.
* ^
𝐸 = ∫ ψ 𝐻 ψ 𝑑τ
^
Substituting the ψ & 𝐻 and after much algebraic manipulations we get
𝑛 𝑛 𝑛
𝐸 = 2 ∑ 𝐻𝑖𝑖 + ∑ ∑ (2 𝐽𝑖𝑗 − 𝐾𝑖𝑗)
𝑖=1 𝑖=1 𝑗=1
This equation gives the electronic E of 2n electron systems
𝐻𝑖𝑖 𝐽𝑖𝑗 𝐾𝑖𝑗 are integrals
H ii – electronic energy of a single electron moving simply under the attraction
of the nucleus with all other electrons stripped away.
Jij - columbic integral representing repulsion between i &j th electron
Kij – exchange integral – arises only in the exchange of electrons.
