Introduction to Orbital-Free

Density-Functional TheoryRalf Gehrke

FHI Berlin, February 8th 2005

OutlineBasics of functional derivativesI Principles of Orbital-free Density-Functional Theory• basics of Density-Functional Theory• motivation for OF-DFT• basics of the numerical implementationII Approximations to the kinetic-energy functional• The Thomas-Fermi Approximation• The von-Weizsäcker term• Linear-Response Theory• Combination of TF and vW• The Conventional Gradient Expansion• Average-Density Approximation

Basics of functional derivativesfunction of multiple argument functional

[ ] [ ] ( ) ( )∫=−+= xyxy

FdxyFyyFF δδδδδ∑ ∂

∂=

ii

i

dyyFdF

NyyF ,,0 L ( )[ ]xyF

iy ( )xynumber number

analogous

( ) ( )00

xyxyFF δ

δδδ =i

i

dyyFdF∂∂

= y(x)

xx0

( )0xyδ

xxi

yi

idy

Basics of functional derivativessome rules

[ ] [ ] [ ]yFyyFyF −+= δδ

( )( ) ( )

( )

( )

( )xy

yfdxyfdxyyfdx

xyFyy

yfyf

∫ ∫∫ ∂∂

=−+=

+∂∂

+

δδ

δδδδ

2O

43421

( )[ ] ( )( )∫= xyfdxxyF [ ]( ) y

fxyyF

∂∂

=δδ

„Proof“:

[ ]( ) ( ) ( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂∂

∂−

∂∂∂∂

−∂∂∂

∂−

∂∂

=∂∇∂

∇−∂∂

=332211 ///r xy

fdxd

xyf

dxd

xyf

dxd

yf

yf

yf

yyFr

δδ

( )[ ] ( ) ( )( )∫ ′= xyxyfdxxyF , [ ]( ) y

fdxd

yf

xyyF

′∂∂

−∂∂

=δδ

Basics of functional derivativessome rules

[ ] ( )∫ ′+′+=+ yyyyfdxyyF δδδ ,

( ) [ ]∫ ∫∫

∫

′′∂

∂+

∂∂

+=⎟⎟⎠

⎞⎜⎜⎝

⎛ ′′∂

∂+

∂∂

+=

⎟⎟⎠

⎞⎜⎜⎝

⎛′∂

∂−

43421y

yf

dxddx

yyfdxy

yfdxyFy

yfy

yfyfdx

δ

δδδδ

( )( )0⎯⎯ →⎯ ±∞→xxy

[ ]( )

( )∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛′∂

∂−

∂∂

= xyyf

dxd

yfdxF

xyyF

δδ

δδ

4434421

„Proof“:

Basics of functional derivativessome rules

some examples

[ ]( ) [ ]( )

( )r-rrr

rr, 3

rr

rr

rr

′′

′== ∫ρ

δρρδρ dJVH

( )[ ]( )[ ]xxyGF , ( ) [ ]( )[ ]( )( )xy

xyGxyG

Fxdxy

Fδ

δδ

δδδ ′

′′= ∫

,,

„chain rule“

[ ] ( ) ( )∫∫ ′

′′=

r-rrrrr

21 33

rr

rrrr ρρρ ddJ

[ ]( ) ( ) r-r

1rr

2

rrrr ′=

′δρδρρδ J

Basics of DFTFirst Hohenberg-Kohn-Theorem

[ ]ρST

Energy of groundstate is a mere function of the electron density ρ

[ ] [ ] ( ) ( )rr r ext3 rrr

ρρρ ∫+= VdFE

[ ] [ ] [ ] [ ] [ ] [ ]ρρρρρρ XCSee EJTETF ++=+=

[ ] ( ) ( )∫∫ ′

′′=

r-rrrrr

21 33

rr

rrrr ρρρ ddJ

[ ]ρXCE

classical coulomb interaction

kinetic energy of a non-interacting system with density ρexchange-correlation energy

Basics of DFTSecond Hohenberg-Kohn-Theorem

(Variational Principle)

„Euler-Lagrange-Equation“

( ) [ ] ( )( )( ) 0rrr

3 =−− ∫ NdErr

r ρµρδρδ

µ chemical potential (negative of first ionization energy of system)

[ ]( ) µ

δρρδ=

rrE

⇒

• What is the kinetic energy ?

• orbitals are solutions of single-particle Schrödinger-equations

(Kohn-Sham-equations)• effective potential has to be determined self-consistently

Basics of DFT

ijji δϕϕ = ( ) ( )rr1

0

2 rrρϕ =∑

−

=

q

ii

iiiV ϕεϕ =⎟⎠⎞

⎜⎝⎛ +∆− KS

eff21

[ ]( ) [ ] [ ]extXCHext

XCKSeff r, VVVVEJV ++=++=

δρρδ

δρρδρ

r

[ ]ρST

iϕ

→ introduction of a reference-system of non-interacting particles

∑ ∆−=i

iiT ϕϕ21

S⇒

• Introduction of orbitals in DFT is not desirable→ matrix diagonalization scales cubically with basis size → many data have to be stored during computation

• the goal is to get rid of KS-orbitals and to express each energy contribution in terms of the electron charge density

• high accuracy for is required since is of the same order as (virial theorem) ↔ is much smaller than⇒ many schemes for approximating might be inappropriate for

Motivation for OF-DFT

[ ]ρST⇒ a functional has to be found

[ ]ρST[ ]ρST

[ ]ρST

[ ]ρE[ ]ρE[ ]ρXCE

[ ]ρXCE

Basics of numerical implementation

Once an appropriate approximation for exists, how can the totalenergy of the groundstate be calculated?

two different approaches:

• direct minimization of the total energy

• self-consistent calculation

[ ]ρST[ ]ρE

Basics of numerical implementation

direct minimization• energy has to be minimized under the constrained of constant

particle numbers

• variable substitution to ensure positivity of electron density ρ

• minimization of Π with steepest descend, conjugate-gradients, etc.

[ ] [ ] ( )( )∫ −−=Π NdE rr3 rrρµρρ

[ ]( ) 0r

=Πr

δϕρδ

( ) ( )rrrr

ρϕ =

( ) ( ) [ ]( ) ( )r

1 rrr

rr

rr

n

nnϕδϕ

ρδτϕϕ Π−=+

Basics of numerical implementation

self-consistent calculation• add and subtract kinetic energy of a bosonic system

• insert total energy expression in Euler-Lagrange-equation

[ ] [ ] [ ] [ ] [ ] ( ) ( ) [ ]ρρρρρρρ Bext3

XCSB rr r TVdEJTTE −++++= ∫rrr

[ ] ( ) ( )∫ ∆−= rrr21 3

Brrr

ρρρ dT

[ ]( )

( )( )r

r21

rB r

r

Lrϕϕ

δρρδ ∆

−==T

integration by parts and chain rule ⇒

Basics of numerical implementation

self-consistent calculation• Euler-Lagrange-equation is transfered to a Kohn-Sham-like

equation with an additional potential term

• same solution methods can be used as in the case of the KS-scheme, but only with one orbital ⇒ no orthogonalization

µϕϕ =H

( ) ( )rr2ˆ

extXCH rrδρδ

δρδ BS TTVVVH −++++

∆−=

( )( ) ( ) ( ) µ

δρδ

δρδ

ϕϕ

=−++++∆

−rrr

r21

extXCH rrr

rBS TTVVV[ ]

( ) µδρ

ρδ=

rrE

⇒

( ) ( )r4rHrr

kV πρ−=∆

[ ]( )r,XCr

kV ρ [ ]( ) ( ) [ ]( ) ( ) [ ]( )r,ˆ

r,ˆ

r,ˆ BS rr

rr

rkkk r

Tr

TG ρδρδρ

δρδρ −=

extXCHˆˆˆˆ

2ˆ VGVVH ++++

∆−=

µϕϕ =H

21

~ ϕρ =+k ( )kkk f ρρρρ ,,,~011 L++ =

self-consistency achieved?

calculate new electron density

solve poisson-equation

calculate potentials

create pseudo-hamiltonian

solve

yes

no

1+→ kkερρ <−+ kk 1

OF-scheme

ερρ <−+ kk 1

self-consistency achieved?

KS-scheme

( ) ( )r4rHrr

kV πρ−=∆

[ ]( )r,XCr

kV ρ

extXCHˆˆˆ

2ˆ VVVH +++

∆−=

1,,0 ,ˆ −== qiH iii Lϕεϕ

∑−

=+ =

1

0

21

~ q

iik ϕρ ( )kkk f ρρρρ ,,,~

011 L++ =

calculate new electron density

solve poisson-equation

calculate exchange-correlation potential

create hamiltonian

solve

yes

no

k : interation indexq : number of KS-orbitals

1+→ kk

The Thomas-Fermi approximation

• consider a system, consisting of non-interacting, free electrons

• introducing periodic boundary conditions

∑∆−=i

iH21ˆ ( ) ( )rkexprk

rrr iC=ϕ ( )( )const.r 2k == Cr

ρ

zyxiL

nki

ii ,, ,2==

π

Vkkk zyx

33 8k π

∆∆∆=∆r

( ) 3/123 ρπ=Fk

The Thomas-Fermi approximation

• kinetic energy can be calculated exactly

• kinetic energy density may be used to approximate the kinetic energy of a non-homogenous system with sufficiently slowly varying electron density

52

2 k

8 k

k1

22

5

20

42

2

0

33

k

233

k

2F

kkkk kVkdkVkdVkkTFFFF

πππ==≈∆

∆== ∫∫∑∑

rrr

( ) 3/223/5 3103 , πρ =⋅⋅= CCVT

3/5ρCVTt == ( )[ ]∫= r r3 rr

ρtdT

VTt /=

The Thomas-Fermi approximation

• by construction, the TF approximation is correct in the limit of ahomogenous electron gas

• TF is correct in the limit of infinite nuclear charge

flaws• infinite charge density at the nucleus• bad total energies compared to Kohn-Sham • algebraical decay of charge density instead of

exponential• no binding of atoms to form molecules or solids• no shell structure in atoms

( )∞→Z

( )6−∝ r( )( )rµ22exp −−∝

The von-Weizsäcker term• originally, von-Weizsäcker derived intuitively a correction to the

Thomas-Fermi approximation to describe the kinetic energy of core particles to explain mass defects

• the von-Weizsäcker term is exact for one-orbital systems (e.g. bosonic systems, one or two electrons)

( )∫ ∇= ρρ , r3vW tdT

r

vWTF TTT +=

The von-Weizsäcker termderivation

• for a one-orbital system, the kinetic energy can be calculated exactly

( ) ( )∫ ∆−= rrr21 *3

vWrrr

ϕϕdT

( ) ( )∫∫ ∇=∇=2323 rr

21rr

21 rrrr

ρϕ dd

( )( ) ( )rrrr

ggfgf ∇∂∂

=∇

( )( )∫

∇=

rr

r81

23

vW r

rr

ρρ

dT

( )( )

( )rr2

1rr

rr

ρρ

ρ ∇=∇

( )( )0r ⎯⎯ →⎯ ∞→r

rϕintegration by parts

improvements compared to TF• exponential decay of electronic density• finite charge at the nucleus

flaws• in a homogenous system , but for a one-orbital system

⇒ does not reproduce the two limits

attempts to improve the bridge between the two extremes:

• experiences indicated that the prefactor of the vW-Term might be too large

The von-Weizsäcker term

0TF ≠T0vW =T

( ) vWTFS TTNGT +=

vWTFS TTT +=

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+−−−=

322

311

21 11N

A

N

ANG NN δδempirical parametersnumber of electrons

Linear-Response TheoryWhy Linear-Response Theory?

• relationship between Linear Response and functional derivative of can be derived

• correct Linear Response is important to describe charge oscillations in solids

definition• small change in potential causes a first-order change of the

charge density

( ) ( )( )( )

( )∫ ′′′=

′

rr-r rr

rr

3 r43421rrrr

r

r

δνχδρ

δνδρ

d ( ) ( ) ( )qqqrrr

δνχδρ =

fourier transformation

ST

Linear-Response TheoryThe Linear-Response Function

( ) ( )( )

( )( ) ( )q

1rrˆ

rrˆq1 rr

r

r

rr

χδρδν

δρδνχ =⎟⎟

⎠

⎞⎜⎜⎝

⎛′

⇒⎟⎟⎠

⎞⎜⎜⎝

⎛′

= FF

„inversion“ theorem of functional derivatives:

( ) ( )( )

( )( ) ( ) ( )

( )( )

∫∫′

′′′=

′′′

=′′′321r

rrrr

r

r

r

rrrr

rr r-r

33

rrr-r r

rr

rr rr-r

f

ddδρδνχ

δρδν

δνδρδ

fourier transformation:

Linear-Response TheoryLinear-Response in the DFT scheme

eff2ˆ VH +

∆−= effS ETE +=

( ) ( ) ( ) ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

′−=

egroundstat

S2

rrˆ

q1

rrrδρδρ

δχ

TF

( )reff

eff δρδEV =

( ) ( ) ( )( )( ) ( ) ( ) ( )rrrrrrr

S2

S2

eff2

eff

′−=

′−

=′

=′ δρδρ

δδρδρ

δδρδρ

δδρδ TTEEV

( ) const.r

== µδρδr

E

unknown !

( ) ( )( )

( )qTF

f

Lindhard

r-r

S2 1

rrˆ

0χδρδρ

δ

ρ

−=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

′

′4434421

rr

rr

Linear-Response TheoryThe Free-Electron-Gas limit of the LR-function

2ˆ

0∆

−=HF

2

2F

Lindhard 2 ,

11ln

41

21

kqk

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−+−

+−= ηηη

ηη

πχ

homogenous system

due to symmetry...

Linear-Response Theory

( ) ( )3

F2cosrlimr

rkr

∝∞→δρ

weak logarithmicsingularity

fourier transformation

„Friedel oscillations“

⇒ singularity is important to describe the physics of a solid properly

Combination of TF and vW• goal is to put the combination of TF and vW on a physical basis by

considering the linear-response

• consider a nearly homogeneous electron gas with small fluctuations

• kinetic energy may be expanded around the average density

vWTFS TTT λ+=

( ) 0r r3 =∆∫rr

ρd

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

( )

( ) ( )rrrr

r r21r

r r

Lindhard

00TF

1ˆ

S2

33

const.

S30SS ′∆∆

′′+∆+≈ ∫∫∫

⎟⎟⎠

⎞⎜⎜⎝

⎛−

rr

44 344 21

rrrrr

43421

rr

321ρρ

δρδρδρ

δρδρρ

χ

ρρ

qF

T

TddTdTT

⇒ linear term vanishes

( ) ( )rr 0rr

ρρρ ∆+=

Combination of TF and vW

⎪⎪⎩

⎪⎪⎨

⎧−

−≈

31

3

1

2

2

2F

LindhardK

L

η

η

πχ k

⎪⎩

⎪⎨⎧ −

−≈+

−= 3

131

311

2

2

2F

22F

vWTF

λη

λη

πληπχ λ

kk 91=λ

1=λ

, small η

, large η ⇒

, small η ⇒

, large η

• Kompaneets, Pavlovskii (1957)• Kirzhnits (1957)• Le Couteur (1964)• Stoddart, Beattie, March (1970)

F2kq

=

Combination of TF and vW• valid for long wavelength perturbation

→ e.g. appropriate for impurity problems where long wavelength components of potential are dominant

• valid for short wavelength perturbation→ e.g. appropriate for perfect lattices

flaws• bad estimation of total energy:

→ overestimation, → underestimation

(interpolation leads to )

• still no shell structures in atoms

1=λ

1=λ

5/1=λ

9/1=λ

9/1=λ

Basic Idea• include higher order gradient corrections to the kinetic energy

functional by expanding the Lindhard function to higher orders

Conventional Gradient Expansion

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−−≈ L

1531

42

2F

Lindhardηη

πχ k

[ ] ( )( ) ( ) ( )

∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛ ∇+

∇∇−

∇= 4

4

3

22

2

22313

3224 389 r

35401

ρρ

ρρρ

ρρρ

πρ

rdT

[ ] L=ρ6T (extremely complicated...)(Hodges, 1973)

(Murphy, 1981)

L+++= 420 TTTT

TFT vW91 T

Conventional Gradient Expansionflaws

• not suitable for isolated systems with exponential decay of charge density→ kinetic energy potential diverges for order four and higher→ T diverges for order six

⇒ in self-consistent calculations, charge density shows wrong decay behaviour

• linear response is wrong, since expansion does only converge for

δρδT

1<η

⇒ CGE is of no practical use

Average-Density ApproximationBasic idea

• include non-local effects to the kinetic energy functional

• correct linear-response behaviour is enforced by appropriate chose of the weight function

[ ] [ ] [ ] ( ) ( )( )∫++−= r~r r58

53

03

vWTFSrrr

ρρρρρ tdTTT

[ ] ( ) ( )( )∫= rr r 03

TFrrr

ρρρ tdT

( ) ( ) ( )qTF

Lindhard

!S

2 1rr

ˆ0

χδρδρδ

ρ

−=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

′rr

( ) ( ) ( )∫ ′Ω′′= r,rr rr~ 3 rrrrrρρ d

( )r,r ′Ωrr

example: (Garcia-Gonzales et al., PRA 54, 1897 (1996) )

Average-Density Approximation

( ) ( )r,rr,rrrrr ′Ω≠′Ω

( ) ( )r,rr,rrrrr ′Ω=′Ω

Kohn-Shamsymmetric ADAnon-symmetric ADATFλvW

⇒ non-local functionals can improve numerical results significantly

(Garcia-Gonzales et al., PRA 54, 1897 (1996) )

Kr

Summary

• goal is to get rid of the Kohn-Sham-orbitals to reduce the computational time

• trade-off are less acurate results due to the necessity of approximating the kinetic energy functional

• state-of-the-art are non-local functionals which improve the results of the classical functionals significantly

Literature• Wang, Y.A., Carter, E.A. 2000, „Orbital-free kinetic energy density functional theory“, Theoretical

Methods in Condensed Phase Chemistry, S.D. Schwartz, Ed. Kluwer, 117-184• P. Garcia-Gonzales, J.E. Alvarellos and E. Chacon, „Kinetic-energy density functional: Atoms and

shell structure“, Phys. Rev. A, 54, 1897 (1996)• M. Levy, J.P. Perdew and V. Sahni, „Exact differential equation for the density and ionization

energy of a many-particle system“, Phys. Rev. A, 30, 2745 (1984)• C.H. Hodges, „Quantum Corrections to the Thomas-Fermi Approximation-The Kirzhnits Methods“,

Can. J. Phys., 54, 1428 (1973)• R.G. Parr, S. Liu, A.A. Kugler and A. Nagy, „Some identities in density-functional theory“,

Phys. Rev. A, 52, 969 (1995)• Y. Wang and R.G. Parr, „Construction of exact Kohn-Sham orbitals from a given electron density“,

Phys. Rev. A, 47, R1591 (1993)• T. Gal and A. Nagy, „A method to get an analytical expression for the non-interacting kinetic

energy density functional“, J. Mol. Struc. 501-501, 167 (2000)