Post on 09-Jun-2020
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)