lecture on dtf

8/11/2019 Lecture on DTF

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Lecture 8:Introduction to Density

Functional Theory

Marie Curie Tutorial Series: Modeling BiomoleculesDecember 6-11, 2004

Mark Tuckerman

Dept. of Chemistry

and Courant Institute of Mathematical Science

100 Washington Square East

New York University, New York, NY 10003


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Background 1920s: Introduction of the Thomas-Fermi model.

1964: Hohenberg-Kohn paper proving existence of exact DF.

1965: Kohn-Sham scheme introduced.

1970s and early 80s: LDA. DFT becomes useful. 1985: Incorporation of DFT into molecular dynamics (Car-Parrinello)

(Now one of PRLs top 10 cited papers).

1988: Becke and LYP functionals. DFT useful for some chemistry. 1998: Nobel prize awarded to Walter Kohn in chemistry for

development of DFT.


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1 1( , ,..., , )N Ns s r r


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e e ee eN H T V V= + +

External Potential:


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Total Molecular Hamiltonian:

e N NN H H T V= + +2

1

,

1

2

| |

N

N II I

NI J

NN

I J I I J

T M

Z ZV

=

>

=

=

R R

Born-Oppenheimer Approximation:

[ ]1 0 1

0

(x ,..., x ; ) ( ) (x ,..., x ; )

[ ] ( , ) ( , )

e ee ee eN N N

N NN

T V V E

T V E t i t t

+ + =

+ + =

R R R

R R

x ,i i is= r


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Hohenberg-Kohn Theorem

Two systems with the same numberNe of electrons have the sameTe + Vee. Hence, they are distinguished only by Ven.

Knowledge of |0> determines Ven.

Let V be the set of external potentials such solution of

yields a non=degenerate ground state |0>.

Collect all such ground state wavefunctions into a set . Eachelement of this set is associated with a Hamiltonian determined by the external

potential.

There exists a 1:1 mapping Csuch that

C :V

[ ] 0e e ee eN H T V V E = + + =


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0 0 =

( ) 0 0 0 (2)e ee eN T V V E + + =


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( ) 0 0 0e ee eN T V V E + + =


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Hohenberg-Kohn Theorem (part II)

Given an antisymmetric ground state wavefunction from the set

, theground-state density is given by

1

2

2 1 2 2( ) ( , , , ,..., , )e e eNe

e N N N

s s

n N d d s s s= r r r r r r

Knowledge of n(r) is sufficient to determine |>

LetNbe the set of ground state densities obtained from Ne-electron groundstate wavefunctions in . Then, there exists a 1:1 mapping

D-1 : N D : N

The formula for n(r) shows that D exists, however, showing that D

-1

existsIs less trivial.


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Proof that D-1 exists

0 0 0 0 0e e ee eN E H T V V = = + +


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[ ]0 0 0 ( ) ( ) ( ) (2)ext ext E E d n V V < r r r r


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(CD)-1 : N V

0 0 0 0 0[ ] [ ] [ ]n O n O n =

The theorems are generalizable to degenerate

ground states!


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The energy functional

The energy expectation value is of particular importance

0 0 0 0 0[ ] [ ] [ ]en H n E n =

From the variational principle, for |> in :

0 0e eH H

Thus,

0[ ] [ ] [ ] [ ]en H n E n E n =

Therefore, E[n0] can be determined by a minimization procedure:

0 ( )[ ] min [ ]nE n E n= r N


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0 0

0 0

0 0

0 0

0 0 0

0 0

( ) ( ) ( )

n e ee eN n e ee eN

n e ee n ext e ee

n e ee n e ee

T V V T V V

T V d n V T V d n

T V T V

+ + + +

+ + + +

+ +

r r r r 0 ( )extVr r


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( ) min [ ] ( ) ( )ext

nF n d n V = + r r r r


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*

2 1 2 2 1 2 2

{ }

( , , , ,..., , ) ( , , , ,e e ee N N N

s

N d d s s s s s= r r r r r r r( , ) ..., , )e eN Ns r r r

The Kohn Sham Form lation


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The Kohn-Sham Formulation

Central assertion of KS formulation: Consider a system of NeNon-interacting electrons subject to an external potential VKS. ItIs possible to choose this potential such that the ground state densityOf the non-interacting system is the same as that of an interactingSystem subject to a particular external potential Vext.

A non-interacting system is separable and, therefore, described by a setof single-particle orbitals i(r,s), i=1,,Ne, such that the wave function is

given by a Slater determinant:

1 1 1

1(x ,..., x ) det[ (x ) (x )]

!e e eN N N

eN

=

The density is given by 2

1

( ) (x)eN

i i j ij

i s

n =

= =r

The kinetic energy is given by

* 2

1

1(x) (x)

2

eN

s i i

i s

T d =

= r


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KS( )( )

( )

cext

EnV V dn

= + +

rr r

r r r


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/ 22

1

1

( ) ( )2

eN

s i iiT ==

r r

S i l lt f DFT


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Some simple results from DFT

Ebarrier(DFT) = 3.6 kcal/mol

Ebarrier(MP4) = 4.1 kcal/mol

G t f th t t d th l di


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Geometry of the protonated methanol dimer

2.39

MP2 6-311G (2d,2p) 2.38

Results methanol


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Results methanol

Dimer dissociation curve of a neutral dimer

Expt.: -3.2 kcal/mol

Lecture Summary


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Lecture Summary

Density functional theory is an exact reformulation of many-bodyquantum mechanics in terms of the probability density rather than

the wave function

The ground-state energy can be obtained by minimization of the

energy functional E[n]. All we know about the functional is that

it exists, however, its form is unknown.

Kohn-Sham reformulation in terms of single-particle orbitals helps

in the development of approximations and is the form used in

current density functional calculations today.

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