Abinitio Tech
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Transcript of Abinitio Tech
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Molecular Modeling: HartreeMolecular Modeling: Hartree
Fock MethodFock Method
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overviewoverview
Molecular Models
Empirical/Molecular Modeling Semi Empirical Ab Initio/DFT
Neglect ElectronsNeglect Core Electrons
Approximate/parameterize HF IntegralsFull Accounting of Electrons
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Full Quantum MethodsFull Quantum Methods
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Hartree-FockHartree-Fock
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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
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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 =
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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
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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
*
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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
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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
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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
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GoalGoal
Energy
Potential
Density
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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
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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