Lecture 9: Advanced DFT concepts: The Exchange-correlation functional and time-dependent DFT Marie...

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Lecture 9: Advanced DFT concepts: The Exchange-correlation functional and time-dependent DFT Marie Curie Tutorial Series: Modeling Biomolecules December 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

Transcript of Lecture 9: Advanced DFT concepts: The Exchange-correlation functional and time-dependent DFT Marie...

Lecture 9:Advanced DFT concepts: The

Exchange-correlation functional and time-dependent DFT

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

Mark TuckermanDept. of Chemistry

and Courant Institute of Mathematical Science100 Washington Square East

New York University, New York, NY 10003

Ψ0λ is the ground state wavefunction of a Hamiltonian

H=T+λW+V. Note that when λ=0. V=VKS and when λ=1, V=Vext.

λ λ

λ λ

0 ( ; ( ))1[ ] ( )

2xc

g nE n d n d

r r r

r r rr r

1( )n r

Where fxc can be obtained by integrating over r’ with aChange of variables s=r-r’.

2

[ ] [ ] ( [ ] [ ])

( ) ( )

SICxc xc i xc i

i

i i

E n E n J n E n

n

r r

Gradient Corrections

Extend the LDA form to include density gradients:

2[ ] ( ) ( ( ), ( ), ( ))xc xcE n d n f n n n r r r r r

Example: Becke, Phys. Rev. A (1988)

24/3 4 /3

1

( )[ ] ( ) ( )

1 6 sinh ( )x xE n C d n d n

rr r r r

r

4/3

( )( )

( )

n

n

rr

r

Functional form chosen to have the correct asymptotic behavior:

1 1 ( ) ( ) lim ( ) ( )

2r

x x xrE d n e e n e

r

r r r r r

Motivation for TDDFT

• Photoexcitation processes• Atomic and nuclear scattering• Dynamical response of inhomogeneous metallic systems.

The time-dependent Hamiltonian

Consider an electronic system with a Hamiltonian of the form:

( ) ( )e eeH t T V V t

Where V(t) is a time-dependent one-body operator.

Our interest is in the solution of the time-dependent Schrödinger equation:

0 0

( ) ( ) ( )

( )

H t t i tt

t

Let be the set of time-dependent potentials associated and let be the set of densities associated with time-dependent solutionsof the Schrödinger equation. There exists a map G such that

G :

The Hohenberg-Kohn Theorem

Since V(t) is a one-body operator:

( ) ( ) ( ) ( , ) ( , )extt V t t d n t V t r r r

Assume the potential can be expanded in a Taylor series:

00

1( , ) ( )( )

!k

ext kk

V t v t tk

r r

Suppose there are two potentials such that

( , ) ( , ) ( )ext extV t V t c t r r

Then, there exists some minimum value of k such that

0

( ) ( ) ( , ) ( , ) constk

k k ext extk t tv v V t V t

t

r r r r

The Hohenberg-Kohn Theorem

For time-dependent systems, we need to show that both thedensity n(r,t) and the current density j(r,t) are different for the twodifferent potentials, where the continuity equation is satisfied:

( , ) ( , ) 0n t tt

r j r

For any operator O(t), we can show that:

( ) ( ) ( ) ( ) ( ) [ ( ), ( )] ( )d

i t O t t t i O t O t H t tdt t

2

2 1{ }

* *2 1 1 1 1

{ }

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

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

e e

e e e e e

N Ns

N N N N Ns

n t d d s t

t d d s t s t s t s t

r r r r

j r r r r r r r

The Hohenberg-Kohn Theorem

From equation of motion, we can show that

0

0 0 0( , ) ( , ) ( , ) ( , ) ( , )ext extt ti t t in t V t V t

t

j r j r r r r

And, in general, for the minimal value of k alluded to above:

0 0

1

0( , ) ( , ) ( , ) ( , ) ( , ) 0k k

ext extt t t ti t t in t i V t V t

t t

j r j r r r r

Hence, even if j and j’ are different initially, they will differ for times just laterthan t0.

The Hohenberg-Kohn Theorem

For the density, since

( , ) ( , ) ( , ) ( , ) 0n t n t t tt

r r j r j r

0 0

2

0( , ) ( , ) ( , ) ( , ) ( , ) 0k k

ext extt t t tn t n t n t V t V t

t t

r r r r r

It follows that:

Therefore, even if n and n’ are initially the same, they will differ for times justlater than t0.

[ ]( ) | ( ) [ ]( ) [ ]( )n t O t n t O n t

Hence, any observable can be written as a functional of n and afunction of t.

Actions in quantum mechanics and DFT

Consider the action integral:

0

' ( ') ( ') ( ')'

t

tA dt t i H t t

t

Schrödinger equation results requiring that the action be stationary according to:

0( )

A

t

Hence, if we view A as a functional of the density,

0

[ ] ' [ ]( ') ( ') [ ]( ')'

t

tA n dt n t i H t n t

t

0

0

[ ] [ ] ' ( , ) ( , )

[ ] ' [ ]( ') [ ]( ')'

t

extt

t

eet

A n B n dt d n t V t

B n dt n t i T V n tt

r r r

Hohenberg-Kohn and KS schemes

Hohenberg-Kohn:

( , ) 0( , ) ( , ) ext

A BV t

n t n t

rr r

Kohn-Sham formulation: Introduce a non-interacting system with effectivepotential VKS(r,t) that gives the same time-dependent density as theinteracting system. For a non-interacting system, introduce single-particleorbitals ψi(r,t) such that the density is given by

2

1

( , ) ( , )eN

ii

n t t

r r

KS action:

0

1 ( , ') ( ', ')[ ] ' ( ) ( ') ( , ') ( , ') ' [ ]

' 2 '

t

KS s ext xct

n t n tA n dt t i T t d n t V t d d A n

t

r rr r r r r

r r

21( , ) ( , ) ( , )

2

( ', )( , ) ( , ) '

' ( , )

i KS i

xcKS ext

i t V t tt

An tV t V t d

n t

r r r

rr r r

r r r

Time-dependent Kohn-Sham equations

From :/ ( , ) 0KSA n t r

Adiabatic LDA/GGA:

0

[ ] ' ( , ) ( ( , ), ( , ))t

xc xctA n dt d n t f n t n t r r r r

Linear response solution for the density

Strategy: Solve the Liouville equation for the density matrix to linear order.

0( ) ( )H t H V t

Quantum Liouville equation for the density operator ρ(t):

( ) [ ( ), ( )]i t H t tt

Time-dependent density:

( , ) ( ) ( ) ( )n t t t t r

( ) ( ) ( ) ( , ) ( , )extt V t t d n t V t r r r

Linear response solution for the density

Write the density operator as:

0( ) ( )t t

To linear order, we have

0 0( ) [ , ( )] [ ( ), ]t i H t i V tt

Solution:

0 0

0

( ') ( ')0( ) ' [ ( '), ]

t iH t t iH t t

tt i dt e V t e

Linear response solution for the density

To linear order:

00 0( , ) ( ) ( , , , ) ( , ')

t

exttn t t d dt t t V t r r r r r

Where the Fourier transform of the response kernel is:

0 0 0 0 0 0 0 0

0 0

( ) ( ) ( ) ( )( , , )

( ) ( )m m m m

m m mE E i E E i

r r r rr r

Hence, poles of the response kernel are the electronic excitation energies.

from Appel, Gross and Burke, PRL 93, 043005 (2003).

Lecture Summary• Adiabatic connection formula provides a rigorous theory of the

exchange-correlation functional and is the starting point of many approximations.

• Generalization of density functional theory to time-dependent systems is possible through generalization of the Hohenberg-Kohn theorem.

• In linear response theory, the response kernel (or its poles) is the object of interest as it yields the excitation energies.