Exact exchange(spin current) density-functional methodsfor magnetic properties, spin-orbit effects,
and excited states
DFT with orbital-dependent functionals
A. GorlingUniversitat Erlangen-Nurnberg
F. Della SalaW. Hieringer
Y.-H. KimS. Rohra
M. Weimer
Deutsche Forschungsgemeinschaft
Overview• Why are we interested in new types of functionals in DFT?
What is wrong with standard density-functional methods?
• Reconsidering the KS formalism
• Orbital-dependent functionals and their derivatives, the OEP approach– Examples of applications
• Spin-current density-functional theory– Formalism– Spin-orbit effects in semiconductors
• Time-dependent density-functional methods
– Reconsidering the basic formalism– Shortcomings of present TDDFT methods– Exact-exchange TDDFT
• Concluding remarks
Shortcomings of standard Kohn-Sham methods
Shortcomings due to approximations of exchange-correlation functionals
• Static correlation not well described(Description of bond breaking problematic)
• Van-der-Waals interactions not included
• Limited applicability of time-dependent DFT methods– Excitations into states with Rydberg character not described– Charge-transfer excitations not described– Polarizabilities of chain-like conjugated molecules strongly overestimated
• Qualitatively wrong orbital and eigenvalue spectradue to unphysical Coulomb self-interactions– Calculation of response properties, e.g., excitation energies or NMR data, impaired– Anions often not accessible– Interpretation of chemical bonding and reactivity impaired– Orbitals or one-particle states do not represent those
underlying the thinking of chemists and physicists?
Orbital-dependent functionals can help to solve these problems
Reconsidering of the Kohn-Sham formalismRelation between ground state electron density ρ0, real and KS wave functionsΨ0 and Φ0, and real and KS Hamiltonian operators with potentials vext and vs
HK HK←− ←− −→ −→T + vs −→ Φ0 −→ ρ0←− Ψ0←− T +Vee+ vext
Relations between ρ0, vext, vs, Ψ0, and Φ0 are circularEach quantity determines any single one of the others
Thomas-Fermi methodvext −→ ρ0 −→ E0[ρ0]
Conventional view on conventional KS methods
vext −→ T + vext+ u[ρ0]+ vxc[ρ0] −→ φj −→ ρ0 −→ E0[ρ0]↑ ↑ /−−−−−−−−−−−−−−−−−−−−−
Orbitals often considered as auxiliary quantities to generate density without phy-sical meaning. However, KS approach works because of orbital-dependent func-tional Ts[φj] for noninteracting kinetic energy.
Kohn-Sham methods with orbital-dependent functionals
Ts[φj], Exc[φj]
vext −→ T + vext+ u[ρ0]+ vxc[φj] −→ φj −→ ρ0 −→ U [ρ0],∫
dr vext(r) ρ0(r) / /−−−− /−−−−−−−−−−−−−−−−
−−−−−−−−−→/
Examples for orbital-dependent functionals
Exchange energy
Ex = −occ.∑a,b
∫drdr′
φa(r′)φb(r′)φb(r)φa(r)|r′ − r|
meta-GGA functionals
Exc =∫
dr εxc([ρ,∇ρ,∇2ρ, τ ]; r) ρ(r) with τ(r) =12
occ.∑a
[∇φa(r)]2
Orbital-dependent functional F [φj] are implicit density functionals F [φj[ρ]]
Alternative view on the Kohn-Sham formalism
• KS system is meaningful model system associated with real electron system,electron density mediates this association
• KS potential vs defines physical situation as well as it does vext
• Depending on situation quantities like E0 or Exc are considered as
– functionals E0[vs] or Exc[vs] of vs
– as functionals E0[ρ0] or Exc[ρ0] of ρ0
If obtained by KS approach free of Coulomb self-interactions then
• KS eigenvalues approximate ionization energies
• KS eigenvalue differences approximate excitation energies and band gaps
• KS orbitals, eigenvalues correspond to accepted chemical, physical pictures
Derivatives of orbital-dependent functionals, the OEP equation
Exchange-correlation potential vxc(r) =δExc[φj]
δρ(r)
Integral equation for vxc by taking derivative δExcδvs(r)
in two ways∫dr′
δExc
δρ(r′)δρ(r′)δvs(r)
=∫
dr′occ.∑a
δExc
δφa(r′)δφa(r′)δvs(r)∫
dr′ Xs(r, r′) vxc(r′) = t(r)
KS response function Xs(r, r′) = 4∑occ.
a
∑unocc.s
φa(r)φs(r)φs(r′)φa(r′)εa−εs
Perturbation theory yields δφa(r′)δvs(r)
=∑
j 6=a φj(r′)φj(r)φa(r)
εa−εj
Derivative of Exc[φj] yields δExcδφa(r′)
Numerically stable procedures for solids with plane-wave basis setsNumerical instabilities for molecules with Gaussian basis sets
(Effective) Exact exchange methods
occ.∑a
unocc.∑s
φa(r)φs(r) 〈φs|vx|φa〉εa − εs
=occ.∑a
unocc.∑s
φa(r)φs(r) 〈φs|vNLx |φa〉
εa − εs
Instead of evaluating density-functional for vx([ρ]; r), e.g. vLDAx ([ρ]; r) = c ρ1/3(r),
integral equation for vx(r) has to be solved.
Localized Hartree-Fock method
Approximation of average magnitudes of eigenvalue differences ∆ = |εa − εs|
vx(r) = 2∑
a
φa(r) [vNLx φa](r)
ρ(r)+ 2
occ.∑a,b
(a,b) 6=(N,N)
φa(r)φb(r)ρ(r)
〈φb|vx − vNLx |φa〉
The localized Hartree-Fock method requires only occupied orbitals and isefficient and numerically stable
Rydberg orbitals of ethene
LHF
+12A π(2p)
-12A -10.22 eV
+12A π(3p)
-12A -2.03 eV
+12A π(4p)
-12A -1.02 eV
BLYP
+12A π(2p)
-12A -6.58 eV
+2500A π(3p)
-2500A +0.0001 eV
HF
+12A π(2p)
-12A -10.23 eV
+2500A π(3p)
-2500A +0.0001 eV
Adsorption of tetralactam macrocycles on gold(111)
Investigation of electronic situation of adsorptionfor interpretation of scanning tunneling microscopy data
Calculation of STM images via energy-filtered electron densities
Response propertiesMore accurate response properties with exact treatment of KS exchange
Example: NMR chemical shifts for organo-iron-complexes
Fe(CO)5 Ferrocene Fe(CO)3(C4H6) CpFe(CO)2CH3 Fe(CO)4C2H3CN
57Iron NMR chemical shifts in ppm relative to Fe(CO)5
System BP86a BPW91b,c LHFb Exp.c
Ferrocen 166 657 969 1532Fe(CO)3(C4H6) -123 -105 12 4CpFe(CO)2CH3 – 336 594 684Fe(CO)4C2H3CN 90 120 170 303
aSOS-IGLO, M. Buhl et al., J. Comp. Chem. 20, 91 (1999). c GIAO.c as cited in: M. Buhl et al., Chem. Phys. Lett. 267, 251 (1997).
Band gaps of semiconductors
EXX band gaps agree withexperimental band gaps
Cohesive energies of semiconductors
Accurate cohesive energies withcombination of exact exchangeand GGA correlation
EXX-bandstructure of polyacetylene
Band gaps
LDA: 0.76 eVEXX: 1.62 eVGW: 2.1 eVExp: ≈ 2 eV
Current Spin-Density-Functional Theory
Treatment of magnetic effects (magnetic fields, magnetism etc.)
HK HK←− ←− −→ −→T+ vs+ Hmag
s −→ Φ0 −→ ρ0, j0, σ0←− Ψ0←− T+Vee+ vext+ Hmag
Vignale, Rasolt 87/88
Problem: lack of good current-depend functionals
Possible solution: exact treatment of exchange
Preliminary work on EXX-CSDFT with approximation of colinear spin
Helbig, Kurth, Gross
Spin-Current Density-Functional Theory IHamiltonian operator of real electron system
H = T + Vee + vext + Hmag + HSO
= T + Vee +N∑
i=1
[vext(ri) + pi·A(ri) + A(ri)·pi + 1
2A(ri)·A(ri)
+12 σ ·B(ri) + 1
4c2 σ ·[(∇vext)(ri)× pi]]
= T + Vee +∫
dr ΣT V(r) J(r)
Σ0= 1 Σ1= σx Σ2= σy Σ3= σz
J0(r) =N∑
i=1
δ(r− ri) Jν(r)=(
12
) N∑i=1
pν,i δ(r− ri) + δ(r− ri) pν,i
V(r) =
V00(r) V01(r) V02(r) V03(r)
V10(r) V11(r) V12(r) V13(r)
V20(r) V21(r) V22(r) V23(r)
V30(r) V31(r) V32(r) V33(r)
=
vext(r)+
A2(r)2 Ax(r) Ay(r) Az(r)
Bx(r)2 0 −vext,z(r)
4c2vext,y(r)
4c2
By(r)2
vext,z(r)4c2 0 −vext,x(r)
4c2
Bz(r)2 −vext,y(r)
4c2vext,x(r)
4c2 0
Spin-Current Density-Functional Theory IIHamiltonian operator of real electron system
H = T + Vee +∫
dr ΣT V(r) J(r)
Total energy
E = T + Vee +∑µν
∫dr Vµν(r) ρµν(r) = T + Vee +
∫dr VT (r) ρ(r)
V(r) treated as matrix or as vector with superindex µν
Spin current density ρ(r) (treated as vector)Density ρ00, spin density ρµ0, current density ρ0ν, spin current density ρµν
Constrained search, Hohenberg-Kohn functional
F [ρ] = MinΨ→ρ(r)
〈Ψ|T + Vee|Ψ〉 −→ Ψ[ρ]
Spin-Current Density-Functional Theory III
Kohn-Sham Hamiltonian operator
H = T +∫
dr ΣT Vs(r) J(r)
Kohn-Sham potential
Vs(r) = V(r) + U(r) + Vxc(r)
U(r) + Vxc(r) =
u(r) + Vxc,00(r) Vxc,01(r) Vxc,02(r) Vxc,03(r)
Vxc,10(r) Vxc,11(r) Vxc,12(r) Vxc,13(r)
Vxc,20(r) Vxc,21(r) Vxc,22(r) Vxc,23(r)
Vxc,30(r) Vxc,31(r) Vxc,32(r) Vxc,33(r)
Constrained search, noninteracting kinetic energy
Ts[ρ] = MinΨ→ρ(r)
〈Ψ|T |Ψ〉 −→ Φ[ρ]
Spin-Current Density-Functional Theory IV
HK HK← ← → →T+
∫dr ΣTVs(r)J(r)→ Φ0→ ρ0← Ψ0← T+Vee+
∫dr ΣTV(r)J(r)
Energy functionals
Coulomb energy: U [ρ] =∫drdr′ ρ(r)ρ(r′)
|r−r′|
Exchange energy: Ex[ρ] = 〈Ψ[ρ]|Vee|Ψ[ρ]〉 − U [ρ]
Correlation energy: Ec[ρ] = F [ρ]− Ts[ρ]− U [ρ]− Ex[ρ]
Potentials
Vx,µν(r) = δExδρµν(r) Vc,µν(r) = δEc
δρµν(r)
In the formalism that is actually applied components of ρ0 are omitted
Spin-current density-functionals required
Here, exact treatment of exchange: EXX spin-current DFT
Exact exchange spin-current density-functional theoryIntegral equation for vx,µν by taking derivative δEx
δVs,κλ(r) in two ways∑µν
∫dr′
δEx
δρµν(r′)δρµν(r′)δVs,κλ(r)
=occ.∑a
∫dr′
δEx
δφa(r′)δφa(r′)
δVs,κλ(r)
Spin current EXX equation∫dr′ Xs(r, r′) Vx(r′) = t(r)
KS response function
Xµν,κλs (r, r′) =
occ.∑a
unocc.∑s
〈φa|σµJν(r)|φs〉〈φs|σκJλ(r′)|φa〉εa − εs
Right hand side t(r)
tµν(r) =occ.∑a
unocc.∑s
〈φa|σµJν(r)|φs〉〈φs|vNLx |φa〉
εa − εs
16-component vectors Vx(r) and t(r), 16x16-dimensional matrix Xs(r, r′)
Plane wave implementation of EXX-SCDFT
Introduction of plane-wave basis set turns spin current EXX equation∫dr′ Xs(r, r′) Vx(r′) = t(r)
into matrix equation
Xs Vx = t
Vx and t consist, at first, of 16 subvectors , Xs of 16x16 submatrices
Transformation of Vx, Xs, and t on transversal and longitudinal contributions
Removal of subvectors in Vx and t and of rows and columns of submatrices inXs yields, e.g., pure current DFT, pure spin DFT etc.
Spin-orbit-dependent pseudopotentials from relativistic atomic OEP program
Hock, Engel 98
Test of pseudopotentials and of implementation of fields
Oxygen orbital eigenvalues with bare pseudopotential in magnetic field
Zeemann splitting: ∆E = β gJ B mJ
Paschen-Back splitting: ∆E = β B (m` + 2ms)
Example of oxygen atom in magnetic field
Eigenvalues of oxygen KS orbitals sm`,ms and pm`,ms
Spin-orbit effects in semiconductor band structuresBandstructures of Germanium without and with spin-orbit
Spin-orbit splitting energies in [meV] (preliminary results)
Germanium
∆(Γ7v − Γ8v) ∆(Γ6c − Γ8c)
Spin currrent 258.097 173.255
Spin current+SO 258.084 173.221
Experiment 297 200a
aMadelung, Semiconductor Data Handbook
Silicon
∆(Γ25v)
Spin currrent 42.481
Spin current+SO 42.587
Experiment 44.1a
Spin-current densities and potentials for Germanium
ρ12 = ρ23 = −ρ13 = −ρ21 = −ρ32 = ρ31
ρ0µ = ρµ0 = ρµµ = 0 for µ = 1, 2, 3
Summary spin-current density-functional theory
• Treatment of magnetic and spin effects
• Spin-orbit can be included
• Flexible choice of considered interactions
• Exchange can be handled exactly, no approximate exchange functional needed
• Promising first results
Time-dependent DFTElectronic system with periodic perturbation w(ω, r) cos(ωt)eεt, ε→ 0+
T + Vee + vext+ w(ω) cos(ωt)
Linear response ρL(ω, r) of electron density
w(ω, r) cos(ωt)→ ρL(ω, r) cos(ωt)
Kohn-Sham Hamilton operatorOriginal Kohn-Sham state: Φ(−∞) = Φ0
T + vext + u + vxc +[w(ω) + wuxc(ω)
]cos(ωt) +O
T + vs + ws(ω) cos(ωt) + O
Linear response ρ(ω, r)ρL(ω) = Xs(ω) ws(ω)
ρL(ω) = Xs(ω)[w(ω) + Fuxc(ω)ρL(ω)
][1−Xs(ω)Fuxc(ω)
]ρL(ω) = Xs(ω) w(ω)
Poles of response function: excitation frequencies
Density-matrix-based coupled Kohn-Sham equation IResponse of density as product of occupied and unoccupied orbitals
ρL(ω, r) =∫
dr′χs(ω, r, r′) ws(ω, r′)
=occ∑a
unocc∑s
φa(r)φs(r)4εsa 〈φs|ws(ω)|φa〉
ω2 − ε2sa
=occ∑a
unocc∑s
φa(r)φs(r) xas
Substitution in density-matrix based coupled KS equation
ρL(ω, r) =∫
dr′χs(ω, r, r′)[w(ω, r′) +
∫dr′′ fu,xc(ω, r′, r′′)ρL(ω, r′′)
]yields∑
as
φa(r) φs(r)xas(ω)
=∑as
φa(r)φs(r)4εsa
ω2 − ε2sa
[was(ω) +
∑bt
Kas,bt xbt(ω)]
Density-matrix-based coupled Kohn-Sham equation IISufficient and necessary [JCP 110, 2785 (1999)] condition for density-based
CKS equation to hold is that density-matrix-based CKS equation holds
x(ω) = λ(ω)[w(ω) + K(ω)x(ω)]
[1− λ(ω)K(ω)]x(ω) = λ(ω)w(ω)
with
was(ω) =∫
drw(ω, r)φa(r)φs(r) λbt,as(ω) = δbt,as4εsa
ω2 − ε2sa
Kas,bt(ω) =∫
dr∫
dr′φa(r)φs(r)fu,xc(ω, r, r′)φb(r′)φt(r′)
Derivation of time dependent DFT without invoking the quantum mechanical orthe Keldysch formalism possible [IJQC 69, 265 (1998)]
Density-matrix-based coupled Kohn-Sham equation III
[1− λ(ω)K(ω)]x(ω) = λ(ω)w(ω)
Multiplication with λ−1(ω) = (1/4) ε−12 [ω21− ε2] ε−
12 yields
(1/4) ε−12
[ω21− ε2 − 4ε
12K(ω)ε
12
]ε−
12 x(ω) = w(ω)
With adiabatic approximation K = K(ω = 0) and spectral representation
M = ε2 + 4ε12 K ε
12 =
∑i
xi ω2i xT
i
of symmetric, positive definite matrix M
Excitation energies from eigenvalues of M
Absorption spectra of solids and molecules
Molecules
Exact excitation energies for only a few transitionsIdeal for density-matrix-based methods[ε 2 + 4ε 1/2 K ε 1/2
]xi = ω2
i xi
Iterative eigensolvers for determination of excitation energies ωi
Solids
Involved one-particle states: occupied bands x unoccupied bands x k-pointsExact excitation energies depend on k-points and are meaningless
Oscillator strengths for large number of transitions requiredIdeal for density-based methods considering frequencies ω + iη
X(ω + iη) =[1−Xs(ω + iη)Fuxc(ω + iη)
]−1Xs(ω)
Absorption spectrum from imaginary part of response function X
Shortcomings of present time-dependent DFT methods
• Charge-transfer excitations not correctly described
• Polarizability of chain-like conjugated molecules strongly overestimated
• Two-electron or generally multiple-electron excitations not included
[ε2 + 4ε
12 K ε
12
]= ω2
i xi
Kas,bt(ω) =∫
dr∫
dr′φa(r)φs(r)fu,xc(ω, r, r′)φb(r′)φt(r′)
TDDFT’Charge-transfer’-problemalso in symmetric systems
Time-dependent DFT with exact exchange kernel
Integral equation for exact frequency-dependent exchange kernel fx(ω, r1, r2)∫dr3dr4 Xs(ω, r1, r3) fx(ω, r3, r4) Xs(ω, r4, r2) = h(ω, r1, r2)
The function h(ω, r1, r2) depends on KS orbitals and eigenvalue differences
Implementation for solids using plane-wave basis sets leads to promising results
Function hx, matrix Hx
Hx,GG′(q;ω) =
− 2Ω
∑ask
∑btk′
[〈ak|e−i(q+G)·r|sk+q〉〈sk+q; bk′|tk′+q; ak〉〈tk′+q|ei(q+G′)·r|bk′〉(εak − εsk+q + ω + iη)(εbk′ − εtk′+q + ω + iη)
]− 2
Ω
∑absk
[ 〈ak|e−i(q+G)·r|sk+q〉〈bk|vNLx − vx|ak〉〈sk+q|ei(q+G′)·r|bk〉
(εak − εsk+q + ω + iη)(εbk − εsk+q + ω + iη)
]+
2Ω
∑astk
[ 〈ak|e−i(q+G)·r|sk+q〉〈sk+q|vNLx − vx|tk+q〉〈tk+q|ei(q+G′)·r|ak〉
(εak − εsk+q + ω + iη)(εak − εtk+q + ω + iη)
]+ ...
Absorption spectrum of silicon
Realistic spectrum only if exchange is treated exactlyDescription of excitonic effects with DFT methods possible
Concluding remarks
State- and orbital-dependent functionals lead to new generation of DFT methods
• More information accessible than through electron density alone
• (Effective) exact exchange methods– Problem of Coulomb self-interactions solved– Improved orbital and eigenvalues spectra corresponding
to basic chemical and physical concepts; improved response properties
• Exact exchange spin-current DFT– Access to magnetic properties– Inclusion of spin-orbit effects
• Exact exchange TDDFT– Includes frequency-dependent kernel– Might solve some of the present shortcomings of TDDFT
• New KS approaches based on state-dependent functionals– DFT beyond the Hohenberg-Kohn theorem
• Formally as well as technically close relations totraditional quantum chemistry and many-body methods
• Development of new functionals for correlation desirableFurther information: www.chemie.uni-erlangen.de/pctc/