Exact exchange (spin current) density-functional methods for magnetic properties, spin-orbit effects, and excited states

Size: px

Start display at page:

Download "Exact exchange (spin current) density-functional methods for magnetic properties, spin-orbit effects, and excited states"

Wesley Blair
5 years ago
Views:

1 Exact exchange (spin current) density-functional methods for magnetic properties, spin-orbit effects, and excited states DFT with orbital-dependent functionals A. Görling Universität Erlangen-Nürnberg F. Della Sala W. Hieringer Y.-H. Kim S. Rohra M. Weimer Deutsche Forschungsgemeinschaft

2 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

3 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 spectra due 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

4 Reconsidering of the Kohn-Sham formalism Relation between ground state electron density ρ 0, real and KS wave functions Ψ 0 and Φ 0, and real and KS Hamiltonian operators with potentials ˆv ext and ˆv s HK HK ˆT + ˆv s Φ 0 ρ 0 Ψ 0 ˆT + ˆV ee + ˆv ext Relations between ρ 0, ˆv ext, ˆv s, Ψ 0, and Φ 0 are circular Each quantity determines any single one of the others Thomas-Fermi method ˆv ext ρ 0 E 0 [ρ 0 ] Conventional view on conventional KS methods ˆv ext ˆT + ˆv ext + û[ρ 0 ]+ ˆv xc [ρ 0 ] φ j ρ 0 E 0 [ρ 0 ] / Orbitals often considered as auxiliary quantities to generate density without physical meaning. However, KS approach works because of orbital-dependent functional T s [{φ j }] for noninteracting kinetic energy.

5 Kohn-Sham methods with orbital-dependent functionals T s [{φ j }], E xc [{φ j }] ˆv ext ˆT / + ˆv ext + û[ρ 0 ]+ ˆv xc [φ j ] φ j ρ 0 U[ρ 0 ], dr v ext (r) ρ 0 (r) / / / Examples for orbital-dependent functionals E x = occ. a,b Exchange energy drdr φ a(r )φ b (r )φ b (r)φ a (r) r r E xc = meta-gga functionals dr ɛ xc ([ρ, ρ, 2 ρ, τ]; r) ρ(r) with τ(r) = 1 2 occ. a [ φ a (r)] 2 Orbital-dependent functional F [{φ j }] are implicit density functionals F [{φ j [ρ]}]

6 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 v s defines physical situation as well as it does v ext Depending on situation quantities like E 0 or E xc are considered as functionals E 0 [v s ] or E xc [v s ] of v s as functionals E 0 [ρ 0 ] or E xc [ρ 0 ] of ρ 0 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

7 Derivatives of orbital-dependent functionals, the OEP equation Exchange-correlation potential v xc (r) = δe xc[{φ j }] δρ(r) Integral equation for v xc by taking derivative δe xc δv s (r) dr δe xc δρ(r ) δρ(r ) δv s (r) = occ. dr δe xc δφ a a (r ) dr X s (r, r ) v xc (r ) = t(r) in two ways δφ a (r ) δv s (r) KS response function X s (r, r ) = 4 occ. a unocc. s Perturbation theory yields δφ a(r ) δv s (r) = j a φ j(r ) φ j(r)φ a (r) ε a ε j φ a (r)φ s (r)φ s (r )φ a (r ) ε a ε s Derivative of E xc [{φ j }] yields δe xc δφ a (r ) Numerically stable procedures for solids with plane-wave basis sets Numerical instabilities for molecules with Gaussian basis sets

8 (Effective) Exact exchange methods occ. a unocc. s φ a (r)φ s (r) φ s ˆv x φ a ε a ε s = occ. a unocc. s φ a (r)φ s (r) φ s ˆv NL x φ a ε a ε s Instead of evaluating density-functional for v x ([ρ]; r), e.g. vx LDA ([ρ]; r) = c ρ 1/3 (r), integral equation for v x (r) has to be solved. Localized Hartree-Fock method Approximation of average magnitudes of eigenvalue differences = ε a ε s v x (r) = 2 a φ a (r) [ˆv x NL φ a ](r) ρ(r) + 2 occ. a,b (a,b) (N,N) φ a (r)φ b (r) ρ(r) φ b ˆv x ˆv NL x φ a The localized Hartree-Fock method requires only occupied orbitals and is efficient and numerically stable

9 Rydberg orbitals of ethene +12Å π(2p) +12Å π(3p) +12Å π(4p) LHF -12Å ev -12Å ev -12Å ev +12Å π(2p) +2500Å π(3p) +12Å π(2p) +2500Å π(3p) BLYP HF -12Å ev -2500Å ev -12Å ev -2500Å ev

10 Adsorption of tetralactam macrocycles on gold(111) Investigation of electronic situation of adsorption for interpretation of scanning tunneling microscopy data Calculation of STM images via energy-filtered electron densities

Response properties More accurate response properties with exact treatment of KS exchange

CpFe(CO) 2 CH 3 Fe(CO) 4 C 2 H 3 CN 57 Iron NMR chemical shifts in ppm relative to Fe(CO) 5

c Ferrocen 166 657 969 1532 Fe(CO) 3 (C 4 H 6 ) -123-105 12 4 CpFe(CO) 2 CH 3 336 594 684

11 Response properties More accurate response properties with exact treatment of KS exchange Example: NMR chemical shifts for organo-iron-complexes Fe(CO) 5 Ferrocene Fe(CO) 3 (C 4 H 6 ) CpFe(CO) 2 CH 3 Fe(CO) 4 C 2 H 3 CN 57 Iron NMR chemical shifts in ppm relative to Fe(CO) 5 System BP86 a BPW91 b,c LHF b Exp. c Ferrocen Fe(CO) 3 (C 4 H 6 ) CpFe(CO) 2 CH Fe(CO) 4 C 2 H 3 CN a SOS-IGLO, M. Bühl et al., J. Comp. Chem. 20, 91 (1999). c GIAO. c as cited in: M. Bühl et al., Chem. Phys. Lett. 267, 251 (1997).

12 Band gaps of semiconductors EXX band gaps agree with experimental band gaps

13 Cohesive energies of semiconductors Accurate cohesive energies with combination of exact exchange and GGA correlation

14 EXX-bandstructure of polyacetylene Band gaps LDA: 0.76 ev EXX: 1.62 ev GW: 2.1 ev Exp: 2 ev

15 Current Spin-Density-Functional Theory Treatment of magnetic effects (magnetic fields, magnetism etc.) ˆT+ ˆv s + Ĥmag s HK HK Φ 0 ρ 0, j 0, σ 0 Ψ 0 ˆT+ ˆV ee + ˆv ext + Ĥmag 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

16 Spin-Current Density-Functional Theory I Hamiltonian operator of real electron system Ĥ = ˆT + ˆV ee + ˆv ext + Ĥmag + ĤSO = ˆT + ˆV N [ ee + v ext (r i ) + p i A(r i ) + A(r i ) p i A(r i) A(r i ) i=1 ] σ B(r i) + 1 σ [( v 4c 2 ext )(r i ) p i ] = ˆT + ˆV ee + dr Σ T V(r) Ĵ(r) Σ 0 = 1 Σ 1 = σ x Σ 2 = σ y Σ 3 = σ z N ( ) 1 N Ĵ 0 (r) = δ(r r i ) Ĵ ν (r)= p ν,i δ(r r i ) + δ(r r i ) p ν,i 2 i=1 i=1 V 00 (r) V 01 (r) V 02 (r) V 03 (r) v ext (r)+ A2 (r) 2 A x (r) A y (r) A z (r) V(r) = V 10 (r) V 11 (r) V 12 (r) V 13 (r) V 20 (r) V 21 (r) V 22 (r) V 23 (r) = B x (r) 2 0 v ext,z(r) v ext,y (r) 4c 2 4c 2 B y (r) v ext,z (r) 2 0 v ext,x(r) 4c 2 4c 2 V 30 (r) V 31 (r) V 32 (r) V 33 (r) B z (r) 2 v ext,y(r) 4c 2 v ext,x (r) 4c 2 0

17 Spin-Current Density-Functional Theory II Hamiltonian operator of real electron system Ĥ = ˆT + ˆV ee + dr Σ T V(r) Ĵ(r) E = T + V ee + µν Total energy dr V µν (r) ρ µν (r) = T + V ee + dr V T (r) ρ(r) V(r) treated as matrix or as vector with superindex µν Spin current density ρ(r) (treated as vector) Density ρ 00, spin density ρ µ0, current density ρ 0ν, spin current density ρ µν Constrained search, Hohenberg-Kohn functional F [ρ] = Min Ψ ˆT + ˆV ee Ψ Ψ[ρ] Ψ ρ(r)

18 Spin-Current Density-Functional Theory III Kohn-Sham Hamiltonian operator Ĥ = ˆT + dr Σ T V s (r) Ĵ(r) Kohn-Sham potential V s (r) = V(r) + U(r) + V xc (r) u(r) + V xc,00 (r) V xc,01 (r) V xc,02 (r) V xc,03 (r) U(r) + V xc (r) = V xc,10 (r) V xc,11 (r) V xc,12 (r) V xc,13 (r) V xc,20 (r) V xc,21 (r) V xc,22 (r) V xc,23 (r) V xc,30 (r) V xc,31 (r) V xc,32 (r) V xc,33 (r) Constrained search, noninteracting kinetic energy T s [ρ] = Min Ψ ˆT Ψ Φ[ρ] Ψ ρ(r)

19 Spin-Current Density-Functional Theory IV HK HK ˆT + dr Σ T V s (r)ĵ(r) Φ 0 ρ 0 Ψ 0 ˆT + ˆV ee + dr Σ T V(r)Ĵ(r) Energy functionals Coulomb energy: U[ρ] = drdr ρ(r)ρ(r ) r r Exchange energy: Correlation energy: E x [ρ] = Ψ[ρ] ˆV ee Ψ[ρ] U[ρ] E c [ρ] = F [ρ] T s [ρ] U[ρ] E x [ρ] Potentials V x,µν (r) = δe x δρ µν (r) V c,µν (r) = δe c δρ µν (r) In the formalism that is actually applied components of ρ 0 are omitted Spin-current density-functionals required Here, exact treatment of exchange: EXX spin-current DFT

20 Exact exchange spin-current density-functional theory Integral equation for v x,µν by taking derivative µν dr δe x δρ µν (r ) δρ µν (r occ. ) δv s,κλ (r) = a dr δe x δv s,κλ (r) δe x δφ a (r ) in two ways δφ a (r ) δv s,κλ (r) dr Spin current EXX equation X s (r, r ) V x (r ) = t(r) X µν,κλ s (r, r ) = t µν (r) = occ. a occ. a KS response function unocc. s φ a σ µ Ĵ ν (r) φ s φ s σ κ Ĵ λ (r ) φ a ε a ε s Right hand side t(r) unocc. s φ a σ µ Ĵ ν (r) φ s φ s ˆv NL x φ a ε a ε s 16-component vectors V x (r) and t(r), 16x16-dimensional matrix X s (r, r )

21 Plane wave implementation of EXX-SCDFT Introduction of plane-wave basis set turns spin current EXX equation dr X s (r, r ) V x (r ) = t(r) into matrix equation X s V x = t V x and t consist, at first, of 16 subvectors, X s of 16x16 submatrices Transformation of V x, X s, and t on transversal and longitudinal contributions Removal of subvectors in V x and t and of rows and columns of submatrices in X s yields, e.g., pure current DFT, pure spin DFT etc. Spin-orbit-dependent pseudopotentials from relativistic atomic OEP program Höck, Engel 98

22 Test of pseudopotentials and of implementation of fields Oxygen orbital eigenvalues with bare pseudopotential in magnetic field Zeemann splitting: E = β g J B m J Paschen-Back splitting: E = β B (m l + 2m s )

23 Example of oxygen atom in magnetic field Eigenvalues of oxygen KS orbitals s ml,m s and p ml,m s

24 Spin-orbit effects in semiconductor band structures Bandstructures of Germanium without and with spin-orbit Spin-orbit splitting energies in [mev] (preliminary results) Germanium (Γ 7v Γ 8v ) (Γ 6c Γ 8c ) Silicon (Γ 25v ) Spin currrent Spin current+so Experiment a a Madelung, Semiconductor Data Handbook Spin currrent Spin current+so Experiment 44.1 a

25 Spin-current densities and potentials for Germanium ρ 12 = ρ 23 = ρ 13 = ρ 21 = ρ 32 = ρ 31 ρ 0µ = ρ µ0 = ρ µµ = 0 for µ = 1, 2, 3

26 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

27 Time-dependent DFT Electronic system with periodic perturbation w(ω, r) cos(ωt)e εt, ε 0 + ˆT + ˆV ee + ˆv ext + ŵ(ω) cos(ωt) Linear response ρ L (ω, r) of electron density w(ω, r) cos(ωt) ρ L (ω, r) cos(ωt) Kohn-Sham Hamilton operator Original Kohn-Sham state: Φ( ) = Φ 0 ˆT + ˆv ext + û + ˆv xc + [ ŵ(ω) + ŵ uxc (ω) ] cos(ωt) + O ˆT + ˆv s + ŵ s (ω) cos(ωt) + O Linear response ρ(ω, r) ρ L (ω) = X s (ω) w s (ω) ρ L (ω) = X s (ω) [ w(ω) + F uxc (ω)ρ L (ω) ] [ 1 Xs (ω)f uxc (ω) ] ρ L (ω) = X s (ω) w(ω) Poles of response function: excitation frequencies

28 Density-matrix-based coupled Kohn-Sham equation I Response of density as product of occupied and unoccupied orbitals ρ L (ω, r) = ρ L (ω, r) = = = occ a occ a dr χ s (ω, r, r ) w s (ω, r ) unocc s unocc s φ a (r)φ s (r) φ a (r)φ s (r) x as 4ɛ sa φ s ŵ s (ω) φ a ω 2 ɛ 2 sa Substitution in density-matrix based coupled KS equation dr χ s (ω, r, r ) [ w(ω, r ) + dr f u,xc (ω, r, r )ρ L (ω, r ) ] as φ a (r) φ s (r)x as (ω) = as φ a (r)φ s (r) yields 4ɛ sa ω 2 ɛ 2 sa [ w as (ω) + bt ] K as,bt x bt (ω)

29 Density-matrix-based coupled Kohn-Sham equation II Sufficient 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) w as (ω) = with dr w(ω, r)φ a (r)φ s (r) λ bt,as (ω) = δ bt,as 4ɛ sa ω 2 ɛ 2 sa K as,bt (ω) = dr dr φ a (r)φ s (r)f u,xc (ω, r, r )φ b (r )φ t (r ) Derivation of time dependent DFT without invoking the quantum mechanical or the Keldysch formalism possible [IJQC 69, 265 (1998)]

30 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) ɛ 1 2 [ω 2 1 ɛ 2 ] ɛ 1 2 yields ] (1/4) ɛ 1 2 [ω 2 1 ɛ 2 4ɛ K(ω)ɛ 2 ɛ 1 2 x(ω) = w(ω) With adiabatic approximation K = K(ω = 0) and spectral representation M = ɛ 2 + 4ɛ K ɛ 2 = x i ωi 2 x T i of symmetric, positive definite matrix M Excitation energies from eigenvalues of M i

31 Absorption spectra of solids and molecules Molecules Exact excitation energies for only a few transitions Ideal for density-matrix-based methods [ ɛ 2 + 4ɛ 1/2 K ɛ 1/2] x i = ω 2 i x i Iterative eigensolvers for determination of excitation energies ω i Solids Involved one-particle states: occupied bands x unoccupied bands x k-points Exact excitation energies depend on k-points and are meaningless Oscillator strengths for large number of transitions required Ideal for density-based methods considering frequencies ω + iη X(ω + iη) = [ 1 X s (ω + iη)f uxc (ω + iη) ] 1 Xs (ω) Absorption spectrum from imaginary part of response function X

32 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 [ɛ 2 + 4ɛ 1 2 K ɛ 1 2 ] = ω 2 i x i K as,bt (ω) = dr dr φ a (r)φ s (r)f u,xc (ω, r, r )φ b (r )φ t (r )

33 TDDFT Charge-transfer -problem also in symmetric systems

34 Time-dependent DFT with exact exchange kernel Integral equation for exact frequency-dependent exchange kernel f x (ω, r 1, r 2 ) dr 3 dr 4 X s (ω, r 1, r 3 ) f x (ω, r 3, r 4 ) X s (ω, r 4, r 2 ) = h(ω, r 1, r 2 ) The function h(ω, r 1, r 2 ) depends on KS orbitals and eigenvalue differences Implementation for solids using plane-wave basis sets leads to promising results

35 Function h x, matrix H x H x,gg (q; ω) = 2 Ω 2 Ω + 2 Ω +... [ ak e i(q+g) r sk+q sk+q; bk tk +q; ak tk +q e i(q+g ) r bk ] (ɛ ak ɛ sk+q + ω + iη)(ɛ bk ɛ tk +q + ω + iη) ask btk absk astk [ ak e i(q+g) r sk+q bk v NL x v x ak sk+q e i(q+g ) r bk ] (ɛ ak ɛ sk+q + ω + iη)(ɛ bk ɛ sk+q + ω + iη) [ ak e i(q+g) r sk+q sk+q v x NL v x tk+q tk+q e i(q+g ) r ak ] (ɛ ak ɛ sk+q + ω + iη)(ɛ ak ɛ tk+q + ω + iη)

36 Absorption spectrum of silicon Realistic spectrum only if exchange is treated exactly Description of excitonic effects with DFT methods possible

37 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 to traditional quantum chemistry and many-body methods Development of new functionals for correlation desirable Further information:

A new generation of density-functional methods based on the adiabatic-connection fluctuation-dissipation theorem

A new generation of density-functional methods based on the adiabatic-connection fluctuation-dissipation theorem Andreas Görling, Patrick Bleiziffer, Daniel Schmidtel, and Andreas Heßelmann Lehrstuhl für