Fundamentals of DFTR. WentzcovitchU of MinnesotaVLab Tutorial

• Hohemberg-Kohn and Kohn-Sham theorems

• Self-consistency cycle

• Extensions of DFT

BO approximation

Ir r

intVeT

- Basic equations for interacting electrons and nuclei Ions (RI ) + electrons (ri )

222 2 22 2

, ,

1ˆ2 2 2

I JIion i I

i i j i I I i Ie i I I I Ji j

Z Z eZ eeH

m r R M R Rr r

intˆ ˆ ˆ ˆ ˆtot ext ion ion ion ionH T V V E H E

IRR

ionTextV ion ionE

22ˆ ˆ ( )

2ion I totI I

H H RM

ˆ

|

el el

tot ion ionel el

HE R E

This is the quantity calculatedby total energy codes.

Pseudopotentials

NucleusCore electronsValence electrons

V(r)

1.0

0.5

0.0

-0.5

0

Radial distance (a.u.)

rRl (

r)

1 2 3 4 5

3s orbital of Si

Real atom

Pseudoatom

r

Ion potential

Pseudopotential

1/2 Bond length

BO approximation• Born-Oppenheimer approximation (1927) Ions (RI ) + electrons (ri )

2 2

2 2( )

2 totII I

E R R RM R

2

( )2

I Jtot I J

I J

Z ZeE R E R

R R

( )totI

I

E RF

R

( )tot

lmlm

E R

22( )1

det 0tot

I JI J

E R

R RM M

IRR

Molecular dynamics Lattice dynamics

forces stresses phonons

Electronic Density Functional Theory (DFT) (T = 0 K)

• Hohemberg and Kohn (1964). Exact theory of many-body systems.

3int

ˆˆ ˆ( ) ( ) ( )

|

el el

tot ion ion ext ion ionel el

HE R E T V d rV r n r E

DFT1Theorem I: For any system of interacting particles in an external potential Vext(r), the potential Vext(r) is determined uniquely, except for a constant, by the ground state electronic density n0(r).Theorem II: A universal functional for the energy E[n] in terms of the density n(r) can be defined, valid for any external potential Vext(r). For any particular Vext(r), the exact ground state energy is the global minimum value of this functional, and the density n(r), that minimizes the functional is the ground state density n0(r).

• Proof of theorem I

Assume Vext(1)(r) and Vext

(2)(r) differ by more than a constant and produce the same n(r). Vext

(1)(r) and Vext(2)(r) produce H(1) and H(2) ,

which have different ground state wavefunctions, Ψ(1) and Ψ(2)

which are hypothesized to have the same charge density n(r). It follows that

Then

and

Adding both which is an absurd!

(1) (1) (1) (1) (2) (1) (2)ˆ ˆE H H

(2) (1) (2) (2) (2) (2) (2) (1) (2) (2)ˆ ˆ ˆ ˆH H H H

(2) 3 (1) (2)0( ) ( ) ( )ext extE d r V r V r n r

(1) (2) 3 (1) (2)0( ) ( ) ( )ext extE E d r V r V r n r

(2) (1) 3 (2) (1)0( ) ( ) ( )ext extE E d r V r V r n r

(2) (1) (1) (2)E E E E Hohemberg and Kohn, Phys. Rev. 136, B864 (1964)

• Proof of theorem II

Each Vext(r) has its Ψ(R) and n(r). Therefore the energy Eel(r) can be viewed as a functional of the density.

Consider

and a different n(2)(r) corresponding to a different

It follows that (1) (1) (1) (1) (2) (1) (2)ˆ ˆE H H

(1) (1) (1) (1) (1)ˆHKE E n H

int[ ] [ ] [ ] ( ) ( )HK ext ion ionE n T n E n drV r n r E

[ ] ( ) ( )HK ext ion ionF n drV r n r E (1) ( )extV r

(2) ( )extV r

Hohemberg and Kohn, Phys. Rev. 136, B864 (1964)

The Kohn-Sham Ansatz

3int[ ] ( ) ( )extE n T n E n d rV r n r

[ ] [ ] [ ] ( ) ( ) [ ]Hartree ext xcE n T n E n drV r n r E n

Replacing one problem with another…(auxiliary and tractable non-interacting system)

• Kohn and Sham(1965)

Hohemberg-Kohn functional:

How to find n?

i

ii pmnT 2

2

1][

)()()( rrrni

ii

' ( ') ( )( )

'Hartree

dr n r n rE r

r r

Kohn and Sham, Phys. Rev. 140, A1133 (1965)

• Kohn-Sham equations: (one electron equation)

)()()()()(2

22

rrrVrVrVm iiixcHartreeext

'

)('

)(

][)(

rr

rndr

rn

nErV Hartree

Hartree

)(

][)(

rn

nErV xc

xc

With εis as Lagrange multipliers associated with the orthonormalization constraint and

and

dft2

Minimizing E[n] expressed in terms of the non-interacting system w.r.t. Ψs, while constraining Ψs to be orthogonal:

,|i j i j

• Exchange correlation energy and potential: By separating out the independent particle kinetic energy and the long range Hartree term, the remaining exchange correlation functional Exc[n] can reasonably be approximated as a local or nearly local functional of the density.

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

( ) ( )xc xc

xc xc

E n n rV r n r n r

n r n r

with and

• Local density approximation (LDA) uses εxc[n] calculagted exactly for the homogeneous electron system

( ) ([ ], )xc xcE n drn r n r

Quantum Monte Carlo by Ceperley and Alder, 1980

• Generalized gradient approximation (GGA) includes density gradients in εxc[n,n’]

• Meaning of the eigenvalues and eigenfunctions:• Eigenvalues and eigenfunctions have only mathematical meaning

in the KS approach. However, they are useful quantities and often have good correspondence to experimental excitation energies and real charge densities. There is, however, one important formal identity

• These eigenvalues and eigenfunctions are used for more accurate

calculations of total energies and excitation energy.

• The Hohemberg-Kohn-Sham functional concerns only ground state

properties.

• The Kohn-Sham equations must be solved self-consistently

ii

dE

dn

Self consistency cycle

0 ( )inn r

0[ ]inV n

( ) ( ) ( ) ( )2

2 ( ) ( ) ( ) ( ) ( )2

i i i iin in out outext Hartree xc i i iV r V r V r r r

m

( )ioutn r

[ ]ioutV n

1[ ] [ ] [ ]i i iin in outV n V n V n

1i i ( ) ( )i i

out inn r n runtil

Extensions of the HKS functional• Spin density functional theory

The HK theorem can be generalized to several types of particles. The most important example is given by spin polarized systems.

( ) ( ) ( )n r n r n r ( ) ( ) ( )s r n r n r

[ , ]HKE E n s2

2 ( , ) ( , ) ( ) ( ) ( )2 ext Hartree xc i i iV r s V r s V r r rm

• Finite T and ensemble density functional theory

The HK theorem has been generalized to finite temperatures.

This is the Mermin functional. This is an even stronger generalization of density functional.

[ , ] [ , ]HK elF n T E n T T S

[ ln (1 ) ln ]B i i ii

S k f f f

1

1 expi

i

B el

f

k T

( ) ( ) ( )ii i

i

n r f r r

D. Mermim, Phys. Rev. 137, A1441 (1965)

Wentzcovitch, Martins, Allen, PRB 1991

Use of the Mermin functional is recommended in the study of metals. Even at 300 K, statesabove the Fermi level are partially occupied.It helps tremendously one to achieve self-consistency. (It stops electrons from “jumping” from occupied to empty states in one step of the cycle to the next.)

This was a simulation of liquid metallic Li at P=0 GPa. The quantity that is conservedwhen the energy levels are occupied according to the Fermi-Dirac distribution is the Mermin free energy, F[n,T].

Dissociation phase boundary

Umemoto, Wentzcovitch, AllenScience, 2006

Few references:-Theory of the Inhomogeneous electron gas, ed. byS. Lundquist and N. March, Plenum (1983).- Density-Functional Theory of Atoms and Molecules, R. Parr and W. Yang, International Series of Monographs on Chemistry, Oxford Press (1989).- A Chemist’s Guide to Density Fucntional Theory, W. Koch, M. C. Holthause, Wiley-VCH (2002).

Much more ahead…