Exchange-Correlation Functionals

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1 Exchange-Correlation Functionals E.K.U. Gross Max-Planck Institute of Microstructure Physics Halle (Saale)

2 The Hamiltonian of Condensed Matter (atoms, molecules, clusters, solids)

3 Hamiltonian for the complete system of N e electrons with coordinates r r and N n nuclei with coordinates ( R R ) R ( ) r N e N n Ĥ = Tˆ n (R) Ŵnn (R) Tˆ e(r) Ŵee(r) Vˆ en (R, r) with Tˆ n = N n ν= M ν ν Tˆ e = N e i= i m Ŵ nn = N n R Z µ Z ν R µ µ, ν ν µ ν Ŵ ee = N e j,k j k r j r k Vˆ en = N N e n r j= ν= j Z ν R ν

4 Hamiltonian for the complete system of N e electrons with coordinates r r and N n nuclei with coordinates ( R R ) R ( ) r N e N n Ĥ = Tˆ n (R) Ŵnn (R) Tˆ e(r) Ŵee(r) Vˆ en (R, r) with Tˆ n = N n ν= M ν ν Tˆ e = N e i= i m Ŵ nn = N n R Z µ Z ν R µ µ, ν ν µ ν Ŵ ee = N e j,k j k r j r k Vˆ en = N N e n r j= ν= j Z ν R ν Stationary Schrödinger equation Ĥ Ψ ( r, R) = EΨ( r, R)

5 Hamiltonian for the complete system of N e electrons with coordinates r r and N n nuclei with coordinates ( R R ) R ( ) r N e N n Ĥ = Tˆ n (R) Ŵnn (R) Tˆ e(r) Ŵee(r) Vˆ en (R, r) with Tˆ n = N n ν= M ν ν Tˆ e = N e i= i m Ŵ nn = N n R Z µ Z ν R µ µ, ν ν µ ν Ŵ ee = N e j,k j k r j r k Vˆ en = N N e n r j= ν= j Z ν R ν Time-dependent Schrödinger equation i Ψ( r, R, t) = H( r, R) V ( r, R, t) laser ψ r, R, t t V laser ( ) ( ) N e N n ( r, R, t) = rj Z R ν ν E f ( t) cosωt j = ν =

6 Born-Oppenheimer approximation Neglect nuclear kinetic energy in full Hamiltonian and solve ( T ˆ (r) W ˆ (r) W ˆ ˆ ) ( ) ( ) nn (R) V (r,r) Φ BO BO BO r = ( R) e ee en R ΦR r for each fixed nuclear configuration R. Make adiabatic ansatz for the complete molecular wave function: Ψ BO ( ) BO ( ) BO r, R = Φ r χ ( R) R and find best χ BO by minimizing <Ψ BO H Ψ BO > w.r.t. χ BO :

7 Nuclear equation ˆ ˆ T (R) W (R) BO n nn A ( R) (-i υ) υ Mυ BO υ ( R) Φ BO BO BO ( r) Tˆ ( R) Φ ( r) dr χ ( R) Eχ ( R) BO * R n R = Berry connection A γ BO υ BO BO * BO ( R) ΦR ( r) (-i υ )ΦR ( r) = dr BO ( C) = A C ( R) dr is a geometric phase corresponding to interesting topological features of molecules BO ( R) expansion of to second order around equilibrium positions yields phonon spectrum

8 Focus on the electronic-structure problem (in Born-Oppenheimer approximation): ( T ˆ (r) W ˆ (r) W ˆ ˆ ) ( ) ( ) nn (R) V (r,r) Φ BO BO r = BO ( R) e ee en R ΦR r for fixed nuclear configuration R. This is still an exponentially hard problem!

9 Why don t we just solve the MB SE Example: Oxygen atom (8 electrons) Ψ ( r,, ) r 8 depends on 4 coordinates rough table of the wavefunction 0 entries per coordinate: 0 4 entries byte per entry: 0 4 bytes bytes per DVD: 0 4 DVDs 0 g per DVD: 0 5 g of DVDs = 0 9 t of DVDs

10 Two fundamentally different classes of ab-initio approaches: Wave function approaches -- Quantum Monte Carlo -- Configuration interaction -- Tensor product decomposition Functional Theories

11 Two fundamentally different classes of ab-initio approaches: Wave function approaches -- Quantum Monte Carlo -- Configuration interaction -- Tensor product decomposition Functional Theories Write total energy as functional of a simpler quantity and minimize

12 Motivation Functional Theories MBPT RDMFT DFT G(r, r', t t' ) γ (r,r ') = G(r,r ',0 ) ρ( r) = γ(r, r)

13 Motivation Functional Theories G(r, MBPT r', t t' Functional: Φ xc [G] or Σ xc [G] ) RDMFT γ (r,r ') = G(r,r ',0 ) Functional: E xc [γ] DFT ρ( r) = γ(r, Functional: E xc [ρ] or v xc [ρ] r)

14 Motivation Functional Theories MBPT RDMFT DFT G(r, r', t t' ) Functional: Φ xc [G] or Σ xc [G] easy (e.g. GW) γ (r,r ') = G(r,r ',0 ) Functional: E xc [γ] difficult ρ( r) = γ(r, r) Functional: E xc [ρ] or v xc [ρ] very difficult

15 Motivation Functional Theories MBPT RDMFT DFT G(r, r', t t' ) Functional: Φ xc [G] or Σ xc [G] easy (e.g. GW) numerically heavy γ (r,r ') = G(r,r ',0 ) Functional: E xc [γ] difficult moderate ρ( r) = γ(r, r) Functional: E xc [ρ] or v xc [ρ] very difficult light

16 ESSENCE OF DENSITY-FUNTIONAL THEORY Every observable quantity of a quantum system can be calculated from the density of the system ALONE The density of particles interacting with each other can be calculated as the density of an auxiliary system of non-interacting particles

ESSENCE OF DENSITY-FUNTIONAL THEORY Every observable quantity of a quantum system can be calculated from the density of the system ALONE The density of particles interacting

17 ESSENCE OF DENSITY-FUNTIONAL THEORY Every observable quantity of a quantum system can be calculated from the density of the system ALONE The density of particles interacting with each other can be calculated as the density of an auxiliary system of non-interacting particles Hohenberg-Kohn theorem (964) Kohn-Sham theorem (965) (for the ground state)

18 DFT papers per year Jul DFT: 5th, A Theory 07 Full of Holes, Aurora Pribram-Jones, David A. Gross, KITSKieron Burke, Annual Review of Physical Chemistry (04). 8

19 HOHENBERG-KOHN THEOREM. v(r) ρ(r) one-to-one correspondence between external potentials v(r) and ground-state densities ρ(r). In other words: v(r) is a functional of ρ(r). Variational principle Given a particular system characterized by the external potential v 0 (r). There exists a functional, E HK [ρ], such that the solution of the Euler-Lagrange equation yields the exact ground-state energy E 0 and ground-state density ρ 0 (r) of this system 3. E HK [ρ] = F [ρ] ρ(r) v 0 (r) d 3 r F[ρ] is UNIVERSAL. δ δρ ( r) E [ ρ] 0 HK =

20 What is a FUNCTIONAL? ρ(r) E[ρ] functional R set of functions set of real numbers Generalization: v r [ ρ] = v[ ρ]( r) functional depending parametrically on r r...r N [ ρ] = ψ[ ρ]( r... r ) ψ or on ( r... ) N r N

21 Explicit algorithm to construct the HK map v s for non-interacting particles ρ h ( m v s (r) ) ϕ i = i ϕ i Σ ϕ i * i N Σ ϕ i *( )ϕ i v s (r)ρ(r) = Σ i ϕ i (r) i = h m v s (r) = Σ ( i ϕ i (r) ϕ i * (- h ρ(r) ) ϕ i ) m Iterative procedure N i = N i = ρ 0 (r) given (e.g. from experiment) Start with an initial guess for v s (r) (e.g. LDA potential) h m solve ( v s (r) ) ϕ i = i ϕ i N v new s (r) = Σ ( i ϕ i (r) h ϕ i * (- ρ ) ϕ i ) 0 ( r) i = m solve SE with v new s and iterate, keeping ρ 0 (r) fixed Consequence: The orbitals are functionals of the density: ϕ i [ρ]

22 KOHN-SHAM EQUATIONS Rewrite HK functional: E HK [ρ] = T S [ρ] ρ(r) v 0 (r) d 3 r E H [ρ] Ε xc [ρ] [ ] [ ]( )( ) [ ]( ) 3 where TS ρ = d r φj ρ r / φj ρ r is the kinetic energy functional of non-interacting particles δ δρ ( r) E [ ρ] 0 HK = yields the Kohn-Sham equations: ( / v ( ) ( ) ( )) 0 r v H[ ρ ] r v xc[ ρ] r φ j( r) = jφ j( r) The orbitals from these equations yield the true density of the interacting system Walter Kohn: The KS equations are an exactification of the Hartree mean-field equation E xc [ρ] is a universal functional of the density which, in practice, needs to be approximated.

23 Approximations to the exchange-correlation functional

24 Local density approximation (LDA) Exc [ ] ρ = E = xc smallest part of total energy Nature's glue [ ] simplest approximation : E ρ 0 Hartree approach xc Result: lattice constants and bonding distances much too large (0%-50%) LDA (Kohn and Sham, 965) LDA 3 unif Exc ρ = drρ r εxc ρ r unif xc ( ) [ ] ( ) ( ) ( ) ε ρ xc energy per particle of a uniform electron gas of density ρ (known from quantum Monte-Carlo and many-body theory) Result: decent lattice constants, phonons, surface energies of metals

25 Quantity Atomic & molecular ground state energies Typical deviation (from expt) < 0.5 % Molecular equilibrium distances < 5 % Band structure of metals, Fermi surfaces few % Lattice constants < %

26 Quantity Atomic & molecular ground state energies Typical deviation (from expt) < 0.5 % Molecular equilibrium distances < 5 % Band structure of metals, Fermi surfaces few % Lattice constants < % Systematic error of LDA: Molecular atomisation energies too large and bond lengths and lattice constants too small

27 One would expect the LDA to be good only for weakly inhomogeneous systems, i.e., systems whose density satisfies: ρ << kf = 3 π ρ ρ ( ) 3 and ρ << ktf = 4(3 ρ / π ) ρ 6 Why is the LDA good also for strongly inhomogeneous systems? Answer: Satisfaction of many exact constraints (features of exact xc fctl) LDA LDA [ ] 3 3 nxc Exc ρ = drρ( r) dr' r' r nxc ( r,r ') Important constraints: ( ) 3 d r'nx r,r ' = are satisfied in LDA ( r,r ') coupling-constant-averaged xc hole density ( ) 3 d r'nc r,r ' = 0 nxc = nx nc nx 0

28 Generalized Gradient Approximations (GGA) GGA 3 Exc ρ = drf ρ r, ρ r ( ) [ ] ( ) ( ) Langreth, Mehl (983), Becke (986), Perdew, Wang (988) PBE: Perdew, Burke, Ernzerhof (996) Construction principle: Satisfaction of exact constraints (important lesson from LDA and from gradient expansion of E xc ) Results: GGAs reduce the LDA error in the atomisation energy significantly (but not completely) while LDA bond lengths are over-corrected (i.e. are in GGA too large compared with expt)

29 Detailed study of molecules (atomization energies) B. G. Johnson, P. M. W. Gill, J. A. Pople, J. Chem. Phys. 97, 7847 (99) 3 molecules (all neutral diatomics from first-row atoms only and H ) Atomization energies (kcal/mol) from: E E VWN c mean deviation from experiment mean absolute deviation B x E E B x LYP c HF for comparison: MP -.4.4

30 LIMITATIONS OF LDA/GGA Not free from spurious self-interactions: KS potential decays more rapidly than r - for finite systems Consequences: Dispersion forces cannot be described W int (R) e -R (rather than R -6 ) no Rydberg series negative atomic ions not bound ionization potentials (if calculated from highest occupied orbital energy) too small band gaps too small: G E gap (LDA/GGA) 0.5 E gap (expt) Energy-structure dilemma of GGAs atomisation energies too large bond lengths too large (no GGA known that gets both correct!!) Wrong ground state for strongly correlated solids, e.g. CoO, La CuO 4 predicted as metals

31 Meta Generalized Gradient Approximations (MGGA) MGGA 3 MGGA Exc ρ = d r ρ(r) εxc ρ(r), ρ(r), τ(r) [ ] ( ) occup τ = ψ ( r) ( r) ασ, ασ, Ts n = drτ r [ ] 3 ( ) Result: Solves energy-structure dilemma of GGAs

32 Jacob s ladder of xc functionals (John Perdew) heaven (exact functional) RPA-like ρ, ρ, τ, occupied & unoccupied KS orbitals energies hybrid ρ, ρ, τ, occupied orbitals MGGA GGA LDA ρ, ρ, τ ρ, ρ ρ E = 0 earth (Hartree ) xc

33 Hybrid functionals

34 Adiabatic Connection Formula H H N ( λ ) = T v ( r ) λ i i = i,k i k = ri N λ ( λ=) = T v ( ) nuc ri i= i,k= i i k = Hamiltonian of fully interacting system e e N N r r r k k 0 λ Choose v λ (r) such that for each λ the ground- state density satisfies ρ λ (r) = ρ λ= (r) Hence v λ=0 (r) = v KS (r) v λ= (r) = v nuc (r) Solve many-body Schroedinger eq for each λ, yielding Ψ λ

35 xc [ ] λ W [ ] λ = ψλ Vee ψλ EH ρ 0 E ρ = dλw[ ρ] 0 0 ee 0 H = HF exchange HF = Ex { ϕi} [ ] W = ψ V ψ E ρ W Exact representation of E xc [ρ]: = E (Correlation energy in nd-order Görling-Levy ' GL 0 c perturbation theory) Becke (JCP 993): W = λ a b λ E = ( W W ) E W BHH HF DFA xc 0 x GGA

36 E = 0.5E 0.75E E PBE0 HF PBE PBE xc x x c E = 0.E 0.8E 0.7E 0.8E 0.5E B3LYP HF LDA B88 LYP VWN xc x x x c c Another way of constructing hybrids: Range separation (Savin, Baer, Kronik) ( µ ) ( µ ) erf rij erfc r = r r r ij ij ij long range ij short range Treat long-range part by wavefunction theory, HF Treat short-range part by density functional approximation (GGA) For solids, it makes sense to do it the other way around (Heyd, Scuseria, Ernzerhof): E = ae ( µ ) ( a)e ( µ ) E ( µ ) E HSE HF,SR PBE,SR PBE,LR PBE xc x x x c

37 5th-rung functionals (using unoccupied KS orbitals) E MP c = occup unoccup ij ab ij ab i j a b ε ε ε ε RPA E [G ] and beyond RPA-functionals, e.g. plus TDDFT RPA c KS E c E xc = dλ du π 0 0 d 3 r d 3 r' e r r' { χ } ( λ ) ( r, r';iu) ρ( r) δ( r r' ) with the response function χ (λ) (r,r';ω) corresponding to H(λ). χ (λ) (r,r';ω) can be obtained from linear-response TDDFT χ ( ) ( λ λ ) ( λ) = χ s χ s λw Clb f xc χ

38 Given an orbital functional for E xc (and hence for E tot ), how can one use it in practice?

39 Non-selfconsistently: post-lda, post-gga, or post-hf Self-consistently, the Kohn-Sham way, yielding the Optimized-Effective Potential (OEP) procedure. This determines the variationally best local potential (i.e. that local potential whose orbitals minimize the given total-energy functional) Self-consistently, the Generalised Kohn-Sham (GKS) way. This determines the variationally best orbitals (not restricted to come from a local potential). The resuting GKS potential can be non-local Self-consistently, using the Gidopoulos variational principle. This avoids the variational collapse of PT-derived functionals by finding that local potential reproducing the density of a given approximate (derived, e.g., from PT) many-body wave function N.I. Gidopoulos, Phys. Rev. A 83, 04050(R), (0) Self-consistently, using the Sham-Schlüter-way (specifically for functionals coming from an approximate selfenergy Σ[G KS ]). This yields that local potential whose density reproduces the density of G.

40 From: A.J. Cohen, P. Mori-Sanchez, W. Yang, Chem. Rev., 89 (0)

41 Apply HK theorem to non-interacting particles ρ given v s = v s [ρ] ( v s [ρ](r))ϕ i (r) = i ϕ i (r) ϕ i = ϕ i [ρ] i = i [ρ] consequence: Any orbital functional, E xc [ϕ,ϕ ], is an (implicit) density functional provided that the orbitals come from a local (i.e., multiplicative) potential. optimized effective potential KS xc potential OEP δ v ( r) = E [ ϕ... ϕ ] xc xc N δρ( r) OEP 3 3 δe δϕ δv xc s v ( r) = d r' d r'' xc j δϕ δv δρ act with χ KS on equation: j ( r' ) s ( r' ) j ( r'' ) ( r'' ) ( r) χ KS - (r'',r) ( r' ) c.c. ( r' ) ( r) OPM 3 3 δe δϕ xc j KS ( r, r' ) v ( r' ) d r' = d r' xc δϕ δv χ OPM integral equation j j s c.c. known functional of {ϕ j }

42 OEP integral equation N σ d 3 r ( V xc,σ (r') u xc,iσ (r'))k iσ (r,r')ϕ iσ (r)ϕ iσ *(r') c.c. = 0 i = where K iσ (r,r') = k = k i ϕ kσ *(r)ϕ kσ (r') kσ iσ δe and u xc,iσ (r) := xc [ϕ σ...] ϕ iσ *(r) δϕ iσ (r) to be solved simultaneously with KS equation: ( ) v ( ) ( r) ( / v ) 0 r H[ ρ ] r vx c ϕ j( r) = jϕ j( r) By contrast, the GKS equations have a non-local potential: dr'v r,r' r' r ( ) ( ) ( ) ψ i GKS ψ i =εψ i i

43 From: A.J. Cohen, P. Mori-Sanchez, W. Yang, Chem. Rev., 89 (0)

44 Comparison OEP vs GKS: GKS is numerically much simpler than OEP Total-energy-related quantities very similar Single-particle gap close to experiment (for some hybrids) when used in GKS way See: J.P. Perdew, W. Yang, K. Burke, Z. Yang, E.K.U. Gross, M. Scheffler, G.E. Scuseria, T.M. Henderson, I. Ying Zhang, A. Ruzsinszky, H. Peng, J. Sun, E. Grushin, A. Goerling, PNAS 4, 80 (07).

45 Comparison OEP vs GKS: GKS is numerically much simpler than OEP Total-energy-related quantities very similar Single-particle gap close to experiment (for some hybrids) when used in GKS way See: J.P. Perdew, W. Yang, K. Burke, Z. Yang, E.K.U. Gross, M. Scheffler, G.E. Scuseria, T.M. Henderson, I. Ying Zhang, A. Ruzsinszky, H. Peng, J. Sun, E. Grushin, A. Goerling, PNAS 4, 80 (07). Ionization energy I, electron affinity A, fundamental gap G=I A and band gap g=ε LU -ε HO of an infinite linear chain of H molecules, evaluated by extrapolation from finite chains

46 DFT description of quantum phases: Magnetism and Superconductivity

47 MAGNETIC SYSTEMS Quantity of interest: Spin magnetization density m(r) (order parameter of magnetism) In principle, Hohenberg-Kohn theorem guarantees that m(r) is a functional of the density: m(r) = m[ρ](r). In practice, good approximations for the functional m[ρ] are not known. Strategy: Include m(r) as basic variable, i.e. as an additional density, in the formalism (in addition to the ordinary density ρ(r)).

48 Start from fully interacting Hamiltonian with Zeeman term: ˆ ˆ ˆ ˆ ( ) ( ) 3 ( ) ( ) 3 H T V v,b ee r v r dr m r B r dr = ρ ˆ m(r) ˆ (r) ˆ (r) =µ 0 ψα σαβψβ αβ ˆ HK theorem ρ v(r),b(r) [ ] - (r),m(r) total energy: E v,b [ ρ, m] = F[ ρ,m] d r( v(r) ρ(r) B(r) m(r) ) 3 universal

49 KS scheme For simplicity: 0 B(r) = 0 Br ( ), 0 m(r) = 0 m( r) m j j j [ v( r) v ( r) v xc (r) ] ± µ [ B( r) B xc (r)] ϕ ( r) = ϕ ( r) H o ± ± ± v xc [ρ,m] = δe xc [ρ,m]/δ ρ B xc [ρ,m] = δe xc [ρ,m]/δ m ρ (r) = ρ (r) ρ - (r), m (r) = ρ (r) - ρ - (r), ρ ± = Σ ϕ j ± B 0 limit These equations do not reduce to the original KS equations for B 0 if, in this limit, the system is magnetic, i.e. has a finite m(r).

DENSITY-FUNTIONAL THEORY OF THE SUPERCONDUCTING STATE BASIC IDEA: Include order parameter, χ, characterising superconductivity as additional density L.

50 DENSITY-FUNTIONAL THEORY OF THE SUPERCONDUCTING STATE BASIC IDEA: Include order parameter, χ, characterising superconductivity as additional density L.N. Oliveira, E.K.U.G., W. Kohn, PRL 60, 430 (988) Include N-body density, Γ, of the nuclei as additional density T. Kreibich, E.K.U.G., PRL 86, 984 (00)

51 General (model-independent) characterization of superconductors: Off-diagonal long-range order of the -body density matrix: ρ ( ) ( ) ( ) xx', yy' = ψˆ x' ψˆ ( x) ψˆ ( y) ψˆ ( y' ) X Y ψˆ ( ) x' ψˆ ( x) ψˆ ( y) ψˆ ( y' ) x x' y χ ( x, x' ) χ( y, y' ) y' χ ( r,r' ) = ψˆ ( r) ˆ ( r' ) ψ order parameter of the N-S phase transition

52 Hamiltonian Ĥ e = Tˆ e Ŵ ee ˆρ ( r) v( r)d r - d r d r' ( ˆχ ( r, r' ) *(r,r') H.c. )

53 ANALOGY S N F B P χ m x x proximity effect

54 Hamiltonian Ĥ Ĥ e = Tˆ e Ŵ ee = Nn Tˆ n d R ˆΓ(R W(R) n ) ˆρ ( r) v( r)d r - d r d r' ( ˆχ ( r, r' ) *(r,r') H.c. ) Ĥ = Ĥ Ĥ e n Û en 3 densities: ρ ( r) = ψˆ ( r) ψˆ ( r) χ(r, r') = Γ(R) = σ= φˆ ψˆ σ ( r) ψˆ ( r' ) σ electron density order parameter ( R ) φˆ ( R ) φˆ ( R ) φˆ ( ) R diagonal of nuclear N n -body density matrix

55 Hohenberg-Kohn theorem for superconductors - [v(r), (r,r ),W(R)] [ρ(r),χ(r,r ),Γ(R)] Densities in thermal equilibrium at finite temperature

56 Electronic KS equation 3 µ v s [ρ,χ,γ](r) u( r) s [ρ,χ,γ](r,r ) v( r' ) d r' = Eu( r) 3 s *[ρ,χ,γ](r,r ) u( r' ) d r' µ v s [ρ,χ,γ](r) v( r) = Ev( r) Nuclear KS equation N n α= α M α W s [ρ,χ,γ](r) ψ(r) = Eψ(R) 3 KS potentials: v s s W s No approximation yet! Exactification of BdG mean-field eqs. KS theorem: There exist functionals v s [ρ,χ,γ], s [ρ,χ,γ], W s [ρ,χ,γ], such that the above equations reproduce the exact densities of the interacting system

57 Electronic KS equation 3 µ v s [ρ,χ,γ](r) u( r) s [ρ,χ,γ](r,r ) v( r' ) d r' = Eu( r) 3 s *[ρ,χ,γ](r,r ) u( r' ) d r' µ v s [ρ,χ,γ](r) v( r) = Ev( r) Nuclear KS equation N n α= α M α W s [ρ,χ,γ](r) ψ(r) = Eψ(R) Solved in harmonic approximation 3 KS potentials: v s s W s No approximation yet! Exactification of BdG mean-field eqs. KS theorem: There exist functionals v s [ρ,χ,γ], s [ρ,χ,γ], W s [ρ,χ,γ], such that the above equations reproduce the exact densities of the interacting system

58 Ĥ = Ĥ o Ĥ CONSTRUCTION OF APPROXIMATE F xc : Ĥ o = σ d ψˆ 3 r (r) σ µ d 3 r' ψˆ [ ψ ψ ] * ˆ (r) ˆ (r') (r, r') H.c. v s (r, R ) o s σ (r)d develop diagrammatic many-body perturbation theory on the basis of the H o -propagators: G s normal electron propagator (in superconducting state) 3 r q Ω q bˆ q bˆ q 3 F s F s * D s anomalous electron propagators phonon propagator Immediate consequence: ph F xc = F xc el F xc all diagrams containing D s all others diagrams

59 Phononic contributions First order in phonon propagator: F ph xc [ n, χ, Γ] = = ij ij dωα dωα * i j Fij( Ω) E E F ( Ω) ij ( µ )( ) i j µ EiE j i j ( I(E, E, Ω) I(E, E, Ω) ) ( µ )( ) i j µ EiE j i I(E i j, E j I(E, Ω) i i j, E j, Ω) Input to ph F xc : Full k,k resolved Eliashberg function α F nk,n'k' ( ) λq Ω = g δ( Ω Ω ) nk,n'k' λq λq Calculated with Quantum Espresso code

60 Purely electronic contributions F ee [ ρ, χ] = xc F GGA xc [ρ] RPA-screened electron-electron interaction Crucial point: NO ADJUSTABLE PARAMETERS

61 Full Green fctn KS Green fctn Ab-initio prediction of critical temperatures of superconductors

Time-dependent density functional theory

Time-dependent density functional theory E.K.U. Gross Max-Planck Institute for Microstructure Physics OUTLINE LECTURE I Phenomena to be described by TDDFT Some generalities on functional theories LECTURE