Abinitio Tech

7/31/2019 Abinitio Tech

1/14

Molecular Modeling: HartreeMolecular Modeling: Hartree

Fock MethodFock Method


2/14

overviewoverview

Molecular Models

Empirical/Molecular Modeling Semi Empirical Ab Initio/DFT

Neglect ElectronsNeglect Core Electrons

Approximate/parameterize HF IntegralsFull Accounting of Electrons


3/14

Full Quantum MethodsFull Quantum Methods


4/14

Hartree-FockHartree-Fock


5/14

The Hamiltonian will be separableThe Hamiltonian will be separable

resulting in Hartree productresulting in Hartree product

Hartree product fails to followHartree product fails to follow

antisymmetry principle forantisymmetry principle for

FermionsFermions

Instead of (spatial orbital )Instead of (spatial orbital )

define (spin orbital) , wheredefine (spin orbital) , where

So we getSo we get

Is ???Is ???

Hartree-FockHartree-Fock


6/14

Slater determinantSlater determinant

ConsiderConsider Set we getSet we get

Satisfies Anti SymmetrySatisfies Anti Symmetry

Generalizing using determinantsGeneralizing using determinants

Also implies that each electron belongsAlso implies that each electron belongs

to every orbital (indistinguishability ofto every orbital (indistinguishability of

)()( 11 NijNji rrrrrrrr =


7/14

Variational PrincipleVariational Principle

Variational PrincipleVariational Principle: The: The

best wavefunction is the onebest wavefunction is the one

with thewith the lowest energylowest energy Solutions are obtained bySolutions are obtained by

expanding the orbitals in theexpanding the orbitals in the

basis set asbasis set as

A linear algebra problem canA linear algebra problem canbe solved by iterative techniquebe solved by iterative technique

like iterative diagonalizationlike iterative diagonalization

etc. and ground stateetc. and ground state

wavefunction may be calculatedwavefunction may be calculated


8/14

What is Density?What is Density?

Density providesDensity provides

us informationus information

about howabout howsomething(s)something(s)

is(are)is(are)

distributed/spreaddistributed/spread

about a givenabout a givenspacespace

For a chemicalFor a chemical

system thesystem the

electron densityelectron density

*


9/14

FunctionFunction

A function maps a set of numbersA function maps a set of numbers

to another set of numbersto another set of numbers Ex. F(X)=XEx. F(X)=X

1

2

34

1

2

34

F(X)=Y


10/14

FunctionalFunctional

A function of a functionA function of a function

A function which maps a set ofA function which maps a set of

functions to a set of numbersfunctions to a set of numbers Ex.Ex. FF(A(X),B(X),C(X),.)=y(A(X),B(X),C(X),.)=y

A(X)

B(X)

C(X)

D(X)

2013F

1

2

3

4


11/14

FunctionalFunctional

A function which maps a set ofA function which maps a set of

functions to a set of numbersfunctions to a set of numbers Ex. Energy is a functional of the waveEx. Energy is a functional of the wave

functionfunction

=*(

vr) H(

vr)d

vr

*(vr)(

vr)d

vraverage /expectationvalueof observableE


12/14

GoalGoal

Energy

Potential

Density


13/14

DFTDFT

Replaces 3N spatial coordinate andReplaces 3N spatial coordinate and

N-spin coordinate wave functionN-spin coordinate wave function

with functionalwith functional Reduces # integrationsReduces # integrations

Simplifies computationSimplifies computation

Quantum Monte Carlo methodQuantum Monte Carlo method

based on statistical Monte Carlobased on statistical Monte Carlo

methodmethod


14/14

referencesreferences

An Introduction to Hartree-An Introduction to Hartree-

Fock Molecular Orbital TheoryFock Molecular Orbital Theory

--C. David SherrillC. David Sherrill Techniques and ApplicationsTechniques and Applications

of Quantum Monte Carloof Quantum Monte Carlo

-- Paul RichardPaul RichardCharles KentCharles Kent

Documents

Abinitio Tech