Molecular Modeling: Density Molecular Modeling: Density Functional TheoryFunctional Theory

C372Introduction to Cheminformatics II

Kelsey Forsythe

RecallRecall

Molecular Models

Empirical/Molecular Modeling Semi Empirical Ab Initio/DFT

Neglect ElectronsNeglect Core Electrons

Approximate/parameterize HF IntegralsFull Accounting of Electrons

Full Quantum MethodsFull Quantum Methods

Quantum Methods

Wavefunction Density Function

Hartree-Fock DFT

MP2, CI

Basis Set MethodsBasis Set Methods

NN44 dependence on # electrons dependence on # electrons Does not account for direct e-e Does not account for direct e-e

correlation (communication)correlation (communication) Perturbation theory (Moller-Plesset Perturbation theory (Moller-Plesset

methods)methods) Configuration interactionConfiguration interaction

Configuration Interaction Configuration Interaction (CI)(CI)

∑=

Φ+Φ=ΨN

i

HFii

HF aa1

00

unoccupied

occupied

Moller Plesset (MP)Moller Plesset (MP)

Hartree-Fock Hartree-Fock close to Full close to Full HamiltonianHamiltonian

VHH HFexact λ+= 0

Perturbation

DFTDFT

Replaces 3N spatial coordinate Replaces 3N spatial coordinate and N-spin coordinate wave and N-spin coordinate wave function with functionalfunction with functional Reduces # integrationsReduces # integrations Simplifies computations?Simplifies computations?

What is Density?What is Density?

Density provides us Density provides us information about information about how something(s) how something(s) is(are) is(are) distributed/spread distributed/spread about a given spaceabout a given space

For a chemical system For a chemical system the electron density the electron density tells us where the tells us where the electrons are likely to electrons are likely to exist (e.g. allyl)exist (e.g. allyl)

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qr = n j *j

∑ a jr2

1111 ad∫ =Ψ τφ

What is Density?What is Density?

Allyl Cation: Allyl Cation:

ΨΨ∝ *ρ

What is Density?What is Density?

For a chemical system the For a chemical system the electron density tells us where electron density tells us where the electrons are likely to existthe electrons are likely to exist

Sum over all space gives total # Sum over all space gives total # electronselectrons

Nxdx

x

xdxdxddsxxxNx NN

=

=∞→

Ψ=

∫

∫ ∫

11

1

321

2

211

)(

0)(

...)...,(....)(

vv

v

vvvvvvv

ρ

ρ

ρ

Probability of finding any electron within dx1 while other electrons are elsewhere

∑ =i

iq 2Allyl cation

FunctionFunction

A function maps a set of numbers A function maps a set of numbers to another set of numbersto another set of numbers Ex. F(X)=XEx. F(X)=X

1234

1234

F(X)=Y

What’s a Functional?What’s a Functional?

A function of a functionA function of a function How does it differ from simple How does it differ from simple

function? function?

FunctionalFunctional

A function which maps a set of A 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),….)=X(A(X),B(X),C(X),….)=X

A(X)B(X)C(X)D(X)

2013F1234

FunctionalFunctional

A function which maps a set of A function which maps a set of functions to a set of numbersfunctions to a set of numbers Ex. Energy is a functional of the Ex. Energy is a functional of the

wave functionwave function

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< E >=Ψ*(

v r ) ˆ H Ψ(

v r )d

v r ∫

Ψ*(v r )Ψ(

v r )d

v r ∫

≡ average /expectation value of observable E

Goal?Goal?

Energy

Potential

Density

How now brown cow?

Energy From Density?Energy From Density?Classical ApproachClassical Approach

Nuclear-Electron InteractionNuclear-Electron Interaction

Electron-Electron InteractionElectron-Electron Interaction

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Vne =Zk

v r −

v r k

∫k

nuclei

∑ ρ(v r )d

v r

Ve1e2=

1

2

ρ(v r 1)ρ(

v r 2)

v r 1 −

v r 2

dv r 1d

v r 2∫∫ Quantal Effects:

Exchange?Correlation?

Energy From Density?Energy From Density?

Electron-Electron InteractionElectron-Electron Interaction

);()(

2

1)()(

2

1

)()(

2

1

2121

21121

21

21

2121

21

21

21

∫∫∫∫

∫∫

−+

−=

++−

=

rdrdrr

rrhrrdrd

rrrr

V

nCorrelatioExchangerdrdrrrr

V

ee

ee

vvvv

vvvvv

vvvv

vvvvvv

ρρρ

ρρ

Exchange & Correlations

Hole function

Energy From Density?Energy From Density?

Kinetic EnergyKinetic Energy Thomas-Fermi’s uniform metallic Thomas-Fermi’s uniform metallic

electron gaselectron gas

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Te ≈ Tueg =3

10(3π 2)2 / 3 ρ 5 / 3(

v r )d

v r ∫

Hohenberg-KohnHohenberg-Kohn

Existence TheoremExistence Theorem

Variational TheoremVariational Theorem

BUTBUT Don’t know how to guess density formDon’t know how to guess density form Don’t want to have to calculate wavefunctionDon’t want to have to calculate wavefunction

exactguessguessguess EE

E

≥∴Ψ→

→

ρ

ρ

Energy Functional Existence Energy Functional Existence (Hohenberg-Kohn (1965))(Hohenberg-Kohn (1965))

For a given system of non-For a given system of non-interacting electron in the interacting electron in the presence of an external field presence of an external field (nuclei) there exists:(nuclei) there exists:[ ]

[ ] [ ] rdrFrdrVrE

ts

rdVTrF

ne

vvvvv

vv

)()()(

..

)̂ˆ()( *

ρρρ

ρ

+=

Ψ+Ψ≡

∫

∫

What is this?

Functional Form?Functional Form?

For a given system of non-For a given system of non-interacting electrons in the interacting electrons in the presence of an external field presence of an external field (nuclei):(nuclei):

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F ρ(v r )[ ] ≡ Ψ*∫ ( ˆ T + ˆ V )Ψd

v r

≡ (Fkinetic + Fpotential )

What is this?

Kohn-Sham Self Consistent Kohn-Sham Self Consistent field Methodologyfield Methodology

For a given system of non-For a given system of non-interacting electrons in the interacting electrons in the presence of an external field presence of an external field assume they have a density of assume they have a density of some some realreal system or system or replaced by replaced by

∑=

→

−

iii

realreal

realeractingnon

rcr )()(

int

φχ

χρ

ρρ

HF-like

Functional Form?Functional Form?

HF-like?HF-like?

ni=non-ni=non-interactinginteracting

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F ρ(v r )[ ] ≡ Ψ*∫ ( ˆ T + ˆ V )Ψd

v r

≡ (Fkinetic

?1 2 3 + Fcoulomb

Poisson1 2 3 + Fexchange + Fcorrelation

?1 2 4 4 4 3 4 4 4

)

Fkinetic = Tni + ΔT

F? ρ(v r )[ ] ≡ DFT input

What is this?

DFT ProcedureDFT Procedure Guess electron densityGuess electron density Choose basisChoose basis Calculate KS-integrals for TCalculate KS-integrals for Tnini and V and Vnene using basis using basis Calculate remaining integrals usingCalculate remaining integrals using Solve matrix equations (just as in HF-SCF) Solve matrix equations (just as in HF-SCF) Calculate new electron density ( )Calculate new electron density ( ) Repeat to error tolerance until difference betweenRepeat to error tolerance until difference between minimizedminimized

0ρ

0ρ

newρ

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ρ0 and ρ new

DFT Challenge( )DFT Challenge( ) Determining the Determining the

form of the form of the exchange-correlation exchange-correlation functionalfunctional LDA-Local Density LDA-Local Density

ApproximationApproximation Uniform electron gasUniform electron gas

Becke Exchange Becke Exchange Correction (1988)Correction (1988) Asymptotic correctionAsymptotic correction

Lee-Yang-Parr(1988)Lee-Yang-Parr(1988) Correlation correctionCorrelation correction

0ρ

321 )(3/1 ρρ FCFF correx +=+

QM-MC simulationsCeperly and Alder(1980)

DFT-SummaDFT-Summa

Exact (by construction)!Exact (by construction)! Includes CorrelationIncludes Correlation Includes ExchangeIncludes Exchange

Approximate (by application)Approximate (by application)

NOT variational as a result (E<ENOT variational as a result (E<Eexactexact))

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F ρ(v r )[ ] ≡ Ψ*∫ ( ˆ T + ˆ V )Ψd

v r

F ρ(v r )[ ] ≡ DFT input ≡ unknown

DFT-SummaDFT-Summa

Does not describe:Does not describe: Dispersion Forces (due to LDA)Dispersion Forces (due to LDA) DynamicsDynamics

No phasesNo phases Transition probabilitiesTransition probabilities No resonance and interference Includes No resonance and interference Includes

ExchangeExchange

ScalingScaling

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N 3 vs N 4 (ab initio)