INTRODUCTION TO COMPUTATIONAL CHEMISTRY

REMYA L Assistant Professor in Chemistry

Sree Narayana College Nattika

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Computational chemistry

• Use of computers to help solving chemical problems

Theoretically determining properties of molecules

Computer experiments need models and theories to describe the laws of nature with the language of mathematics

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Why its rapidly growing?

Some of the almost limitless properties that can be calculated with computational chemistry are:

• Equilibrium and transition-state structures

• dipole and quadrapole moments and polarizabilities

• Vibrational frequencies, IR and Raman Spectra

• NMR spectra

• Electronic excitations and UV spectra

• Reaction rates and cross sections

• thermochemical data

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Benefits

• Schrödinger Equation is exact for simple systems

Rigid Rotor, Harmonic Oscillator, Particle in a Box, Hydrogen Atom

• For many electron atoms/molecules, we need to make some

simplifying approximations and solve it numerically.

• It is possible to get very accurate results, but the “cost” of the

calculation increases with the accuracy of the calculation and

the size of the system.

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Electronic energy of molecules

Three parts to solve the Schrödinger equation for molecules - To separate the electronic part of Schrodinger equation

Born-Oppenheimer Approximation – The expansion of the many-electron wave function in terms of

Slater determinants.

The Method – Representation of Slater determinants by molecular orbitals,

which are linear combinations of atomic-like-orbital functions.

The basis set

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Born-Oppenheimer Approximation

To a high degree of accuracy we can separate

electron and nuclear motion:

(R,r)= el(r;R) N(R)

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The ‘Method’

For systems involving more than 1 electron- the electron-electron interaction is the culprit

After the B-O approximation, the next important approximation is the

expansion of in a basis of Slater determinants:

Ѱ= 1

√𝑛!ϕ1 1 α 1 ϕ1 2 β 2 ϕ2 3 α 3 ϕ2 4 β 4 … .ϕ𝑛/2(𝑛)β(𝑛)

– / are spin-functions (spin-up/spin-down)

– i are spatial functions ( molecular orbitals)

– i and i are called spin-orbitals

– Slater determinant gives proper anti-symmetry (Pauli Principle)

H el

2

2me

i

2

i

Z e

2

rii

e

2

riji j

j

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Computational Mehods

Classical Mechanics

Motion of macroscopic objects

Newton’s law

Quantum Mechanics

Motion of microscopic objects

Schrödinger Equation

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Computational methods

Molecular Mechanics (~ 10,000 atoms)

Does not use Schrödinger's equation

Uses Newton’s laws of motion

Semi-empirical Methods (~ 1,000 atoms)

Uses experimentally derived empirical parameters

Extensive simplifications of Schrödinger's equation

Density Functional Theory (~ 100 atoms)

Uses Schrödinger's equation

Functional of electron density

Ab initio Methods (~ 50 atoms)

Uses Schrödinger's equation

Computationally demanding

Listed in the order from least to most accurate

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Molecular Mechanics

• Molecular mechanics or force field calculations are based on a model that treats a

molecule as a collection of atoms held together by bonds governed by a set of

classical mechanical potential (ball and spring).

• It is easy to calculate the energy of a collection of atoms and bonds, if one knows the

equilibrium bond lengths, angles between them and the energy cost of stretching and

bending of bonds.

• Thus, by varying the geometry of a molecule, one can calculate the structure of the

molecule corresponding to the minimum energy (geometry optimization).

Force fields can be defined as a set of those parameters which are used to define a molecule in molecular mechanics method.

Force Field = bond stretching + angle bending + electrostatic interaction +…

The method used in representing a force field using various parameters is called “parametrization”

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Molecular Mechanics Benefits

quite fast to find the minimum energy structure; a molecule of 100 atoms can be optimized in seconds on a standard PC.

It can be used for molecules as large as enzymes.

It can handle systems of more than thousand atoms.

Drawbacks

This method relies on force-field with embedded empirical parameters obtained mainly from experiment.

It is computationally least intensive and useful with limited computer resources.

This technique does not provide information on electronic structure of a molecule.

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Semi-empirical methods

Based on Schrodinger equation but parameterized with experimental

values.

Only valence electrons considered: core is treated by reducing nuclear

charge or by adding special core functions.

Only a minimum basis is used (one AO basis function per real AO in atom)

Some two-electron and sometimes one electron integrals are neglected to

speed up the computation

Some empirical parameters are inserted to make up for the neglected

integrals

It can handle systems of 100 atoms easily.

It is computationally less demanding.

Somewhat more robust than force fields because at least based on QM

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Density Functional Theory

Density Functional Theory provides that alternate method by reformulating

the many-electron problem in terms of its density.

In wave functional approach, a large number of orbitals in terms of 3N

coordinates of N electrons are involved, but in density-functional theory, only one variable–the density of electrons–in terms of only 3 coordinates involved.

For a given particle- particle interaction, the ground-state density ρ(r) of a

system gives the ground- state wavefunction ψ uniquely.

(Hohenberg-Kohn theorem)

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Density Functional Theory

Some of the challenges are:

• to develop excited-state theory

• to explore fundamental aspects of theory to make it more

accurate

• to develop functionals for strongly-correlated systems.

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Ab initio methods

Ab initio (Latin: „from the beginning‟) calculations are based on the

Schrodinger equation which describes the behaviour of electrons in a

molecule.

This method solves Schrodinger equation for a molecule that provides

electron distribution and energy of the system considering the values of the

fundamental constants and the atomic numbers of the atoms present.

Ab inito calculations are computationally demanding; needs good computer

resource and applicable for small size molecules.

It is mathematically rigorous and useful for a broad range of systems with

no input from experiment.

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Hartree-Fock Approximation

Hartree-Fock is the basic ab initio model.

Start with a reasonable set of Slater type orbitals,ф‟s.

A variational method (energy for approximate will always be higher than energy of the true )

Uses self-consistent field (SCF) procedure

Finds the optimal set of molecular orbitals

Each electron only sees average repulsion of the remaining

electrons (no instantaneous interactions).

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Two ways to compute molecular orbitals using HF.....

RHF (restricted) or UHF (unrestricted)

RHF uses the same orbital spatial function for electrons in the

same pair

(good for species with paired electrons, no spin contamination)

UHF uses a separate orbital for each electron, even if they are

paired

(used for ions, excited states, radicals, etc.)

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Accuracy of Hartree-Fock Calculations

• Hartree-Fock wavefunctions typically recover almost 99% of the total electronic energy.

• As chemistry is primarily interested in energy differences, not total energies, Hartree-Fock calculations usually provide at least qualitative accuracy in this respect.

• Bond lengths, bond angles, vibrational force constants, thermochemistry, ... can generally be predicted qualitatively with HF theory.

• With more electrons this gets worse.

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Post- HF Methods

Introduces correction for electron repulsion.

Adding the electron correlation factor to the HF method.

Correlated Methods

1. Configuration Interaction

2. Møller-Plesset (MP) Perturbation

3. Coupled Cluster

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Configuration Interaction

Treat the different Slater determinants that can be formed from

any occupation of HF orbitals to themselves be a basis set to

be used to create an improved many-electron wave function.

The bigger the CI matrix, the more electron correlation can be

captured.

The CI matrix can be made bigger either by increasing basis-

set size or by adding more highly excited configurations.

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Møller-Plesset (MP) Perturbation

Here, we start with a simple system and gradually turn on an

additional “perturbing” Hamiltonian, representing a weak

disturbance to the system.

The basic idea is to expand the energy and wave functions of

the perturbed system in powers of the small potential as

MP0,MP1,MP2....

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Coupled Cluster

Coupled cluster proceeds from the idea that accouting for the

interaction of one electron with more than a single other

electron is unlikely to be important. Thus, to the extent that

“many-electron” interactions are important, it will be through

simultaneous pair interactions,

“disconnected clusters”

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The Basis Set

• An approximate wave function, Eg. Slater determinant

• It is made from molecular orbitals which are themselves approximated by atomic orbitals(LCAO)

LCAO coefficients of the AO’s, cki, determined in SCF procedure

The basis functions, i, are atom-centered functions that mimic solutions of the H-atom (s orbitals, p orbitals,...)

List of all basis functions used in a calculation is called Basis Set.

i ckik

k

M

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Slater Type Orbitals

• Differ from Hydrogen like orbitals only in the radial part.

• Most of the required integrals needed in the course of the SCF

procedure must be calculated numerically.

• Time consuming and computational cost is high.

• In poly atomic molecule having different atomic

centres,calculation cannot be performed efficiently.

,n,l ,m(r,,) NYl,m (,)rn1

er

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Gaussian Type Orbitals

• Obtained from quantum calculations on atoms

• STO‟s are approximated as linear combination of GTO‟s.

• The main difference is that the variable r in the exponential

function is squared.

lx + ly + lz = l and determines type of orbitals (l=1 is a p...)

,l x ,ly ,lz(x,y,z) Nx

lx yly z

lzer 2

Gaussian type function

Slater type function

-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1

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Types of Basis Sets

• Minimal Basis set

• Multi zeta Basis set

• Split valence Basis set

• Polarized Basis set

• Diffused Basis set

• Contracted Basis set

• Correlation consistent Basis set

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Minimal Basis set

• This basis set consists of one function each for the core

orbitals and valence orbitals (whether occupied or not).

Hydrogen 1s = one basis function;

Fluorine 1s + 2s +2px + 2py + 2pz = five basis functions.

• Often called an “STO-nG” basis - combination of 'n' Gaussians to mimic an STO

For example, “STO-3G” is short-hand for a minimal basis set in which each basis function is a contraction of three primitive

Gaussians.

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Multi zeta Basis set

• These have multiple number of functions for each orbital.

Double-zeta: Two basis functions for each AO

Triple-zeta: Three basis functions for each AO ... and etc. for quadruple-zeta (QZ), 5Z, 6Z, ...

Having

different-sized functions allows the orbital to get bigger or

smaller when other atoms approach it

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Split valence Basis set

• Developed by Pople, so called Pople type Basis set • A split-valence basis uses only one basis function for each core

AO, and a larger basis for the valence AO‟s.

For example, “6-31G” basis set for fluorine:

1s orbital described by 6 primitive Gaussians contracted to one basis function,

One set of 2s and 2p orbitals described by contraction of 3 primitive Gaussians,

One set of 2s and 2p orbitals described by 1primitive Gaussian.

ie, 1 function for the core + 2 functions each for the valence 2s, 2px, 2py and 2pz orbitals .

'6-311G' - one basis function each for the core orbitals and three basis functions each for the valence orbitals with contractions of 6,3 ,1 and 1 primitives

respectively.

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Polarized Basis set

• Account for the polarisation of atoms that occurs when

forming chemical bonds.

• Usually p functions are used to polarise s electrons,d functions to polarise p electrons and f functions to polarise d electrons.

Eg: 6-31G(d) (or 6-31G*) - a 6-31G basis set to which a set of d polarization functions added to heavier atoms (non hydrogen atoms). 6-31G(d,p) (or 6-31G**) - a 6-31G basis set to which a set of d polarization functions added to heavier atoms and a set of p functions on hydrogen atom.

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Diffused Basis set

• Describe the electron density away from the nucleus

( for anions and weakly bonded molecules).

• They are indicated by a + symbol in the notation.

Eg. 6-31++G(d) - includes a set of polarisation functions on heavy atoms and hydrogen.

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Contracted Basis set

• An energy optimized basis set which gives a good description

of the outer part of the wave function.

• ie, the basis being the energetically required inner wave

functions and as an addition to this, adding functions like

diffuse, polarization etc to account for the valence electrons.

• Pople type, diffused, polarized basis sets etc. belongs to this

category.

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Correlation consistent Basis set • These basis set are geared to the recovery of correlation energy of the

valance electrons.

• They are designed so that the functions which produce similar amount of correlation energy are included in the same stage of calculation of parameters.

ie, the value of second ‘d’ orbital function and first ‘f ’ orbital function have almost the same energy, hence when they are included,they should be included together for a calculation.

• They are generally known as „cc‟ wave functions and written as cc-Basis sets.

• They are generally used for d and f orbitals.

Eg: cc-pVDZ cc - correlation-consistent basis meaning that the functions were optimized for best performance with correlated calculations. p - polarization functions are included on all atoms. VDZ - valence double zeta, meaning that the valence orbitals are described by two contractions.

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Molecule Specification

All computational chemistry packages require the specification of the geometry of the molecule of interest in some format.

Z-matrix

Cartesian coordinates

Unique coordinates

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Z-matrix

Each atom of the molecule is specified in terms of the internal coordinates of distance, angle and dihedral angle

To write the Z-matrix the following steps may be followed.

1. Draw the structure of the molecule.

2. Number all the atoms sequentially from 1.

3. Each atom is specified in a separate line.

4. For the first atom only the symbol is needed.

5. The second line consists of atom symbol,1,distance w.r.to atom 1 in angstrom

units.

6. The third line consists of: atom-symbol, connectivity, distance, connectivity,

angle

7. The fourth and all remaining atoms require a distance, an angle and a

dihedral angle w.r.to previously defined atoms.

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Z Matrix for Hydrogen

1

2 H1 H2 1 r2 Variables: r2 = 0.71 Ie, atom H2 is connected to input atom H1at a distance r2.

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Z Matrix for Water

O1

H2 1 r2

H3 1 r3 2 a3

Variables: r2= 0.9687 r3= 0.9687 a3= 104.00

In sequence, this says: Atom H2 is connected to input atom O1

at a distance r2. Atom H3 is connected to input atom O1 at a

distance r3 and is forming with atom O1 and H2 the angle a3.

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Z Matrix for Formaldehyde

C1

O2 1 r2

H3 1 r3 2 a3

H4 1 r4 2 a4 3 d4

Variables: r2 = 1.182, r3 = 1.102, a3 = 122.178, r4 = 1.102, a4 = 122.178, a5 = 180

Central atom is C1, O as second which is connected to C1 by a bond r2. H3 is

connected by atom 1 by a bond r3 and by an angle a3(ahco). For atom 4, we have to

include two dihedral angles to define its orientation in space which is the angle

between the planes 4-1-2 and 3-1-2. If the 4th atom is in the same plane of the atoms

1-2-3 and in the same side of the bond 1-2, the dihedral angle should be zero. Since it

is in the other side, dihedral angle is 180.

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Z Matrix for Ammonia

N1

H2 1 r2

H3 1 r3 2 a3

H4 1 r4 2 a4 3 d4

r2=1.0, r3=1.0, r4=1.0, a3=110., a4=110., d4=120.

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Z Matrix for Methanol

C1

O2 1 r2

H3 2 r3 1 a3

H4 1 r4 2 a4 3 d4

H5 1 r5 2 a5 4 d5

H6 1 r5 2 a5 4 -d5

Variables:

oc2 1.400

ho3 0.950

hoc3 109.471

hc4 1.089

hco4 109.471

hc5 1.089

hco5 109.471

dih5 120.000

dih4 0.000 S 2 30.0

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Cartesian Coordinates

• The geometry specification of a molecule in Cartesian coordinates consists of the name of the atom, atomic number and x, y, z coordinates of each atom specified in separate lines.

• The name of the atom may be up to 10 characters long or its symbol. The atom is identified by its atomic number. The atomic number, x ,y and z coordinates must be given with a decimal point.

eg. N2 The first atom is placed at the origin. The second atom may be placed at say z axis and so the bond length can be given as the z coordinate. The x and y coordinates are 0.0 N 7.0 0.0 0.0 0.0

N 7.0 0.0 0.0 1.1

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Unique Coordinates

• Here the Cartesian coordinates of the symmetry unique atoms

only are specified. Using the given point group of the molecule

the software will automatically generate all the atoms during

program run.

For eg. Methane with point group Td .

• The unique atoms are the carbon atom at the centre of the

coordinate system and one hydrogen atom.Thus the molecule

specification is

Td

C 6.0 0.0 0.0 0.0

H 1.0 0.62 0.62 0.62

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Gaussian programme input method

RFERENCE:

GAUSSIAN 09W TUTORIAL

AN INTRODUCTION TO COMPUTATIONAL CHEMISTRY USING G09W AND

AVOGADRO SOFTWARE

Anna Tomberg

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