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/