dft with orbital-dependent functionals · 2009-03-04 · exact exchange (spin current)...
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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
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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
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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
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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.
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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]]
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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
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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
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(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
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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
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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
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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).
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Band gaps of semiconductors
EXX band gaps agree withexperimental band gaps
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Cohesive energies of semiconductors
Accurate cohesive energies withcombination of exact exchangeand GGA correlation
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EXX-bandstructure of polyacetylene
Band gaps
LDA: 0.76 eVEXX: 1.62 eVGW: 2.1 eVExp: ≈ 2 eV
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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
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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
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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]
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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]
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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
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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′)
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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
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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)
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Example of oxygen atom in magnetic field
Eigenvalues of oxygen KS orbitals sm`,ms and pm`,ms
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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
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Spin-current densities and potentials for Germanium
ρ12 = ρ23 = −ρ13 = −ρ21 = −ρ32 = ρ31
ρ0µ = ρµ0 = ρµµ = 0 for µ = 1, 2, 3
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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
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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
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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)]
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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)]
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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
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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
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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′)
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TDDFT’Charge-transfer’-problemalso in symmetric systems
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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
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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)
]+ ...
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Absorption spectrum of silicon
Realistic spectrum only if exchange is treated exactlyDescription of excitonic effects with DFT methods possible
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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/