Introduction to real-space, linear-response, and time-dependent methods Heiko Appel Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods July 19, 2011 1 / 62 Outline Linear Response in DFT I Response functions I Casida equation I Sternheimer equation Real-space representation and real-time propagation I Real-space representation for wavefunctions and Hamiltonians I Time-propagation schemes I Optimal control of electronic motion Time-evolution of open quantum systems I Stochastic Schr¨odingerequations, master equations I Stochastic current DFT I Stochastic quantum molecular dynamics Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods July 19, 2011 2 / 62 Motivation Where is electron dynamics important? I Electron-hole pair creation in solar cells I Photosynthesis and energy transfer in light-harvesting antenna complexes I Quantum computing (e.g. electronic transitions in ultracold atoms) I Molecular electronics, quantum transport Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods July 19, 2011 3 / 62 Time-dependent density-functional theory I One-to-one correspondence of time-dependent densities and potentials v(r; t) 1−!1 ρ(r; t) For fixed inital states, the time-dependent density determines uniquely the time-dependent external potential and hence all physical observables. I Time-dependent Kohn-Sham system The time-dependent density of an interacting many-electron system can be calculated as density N X 2 ρ(r; t) = j'j (r; r)j j=1 of an auxiliary non-interacting Kohn-Sham system 2r2 i @ ' (r; t) = − ~ + v [ρ](r; t) ' (r; t) ~ t j 2m S j with a local multiplicative potential Z ρ(r0; t) v [ρ(r0; t0)](r; t)= v(r; t) + d3r0 + v [ρ(r0; t0)](r; t) S jr − r0j xc E. Runge, and E.K.U. Gross, Phys. Rev. Lett. 52, 997 (1984). Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods July 19, 2011 4 / 62 Linear Response Theory I Hamiltonian H^ (t) = H^0 + Θ(t − t0)v1(r; t) I Initial condition: for times t < t0 the system is in the ground-state of the unperturbed Hamiltonian H^0 with potential v0 and density ρ0(r) I For times t > t0, switch on perturbation v1(r; t). Leads to time-dependent density ρ(r; t) = ρ0(r) + δρ(rt) I Functional Taylor expansion of ρ[v](r; t) around v0: ρ[v](r; t) = ρ[v0 + v1](r; t) = ρ[v0](r; t) Z δρ[v](rt) + v (r0t0)d3r0dt0 0 0 1 δv(r t ) v0 ZZ δ2ρ[v](rt) + v (r0t0)v (r00t00)d3r0dt0d3r00dt00 0 0 00 00 1 1 δv(r t )δv(r t ) v0 + ::: Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods July 19, 2011 5 / 62 Computing Linear Response Different ways to compute first order response in DFT I Response functions, Casida equation I (frequency-dependent) perturbation theory, Sternheimer equation I real-time propagation with weak external perturbation Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods July 19, 2011 6 / 62 Response functions I Functional Taylor expansion of ρ[v](r; t) around external potential v0: Z δρ[v](rt) ρ[v + v ](r; t) = ρ[v ](r) + v (r0t0)d3r0dt0 + ::: 0 1 0 0 0 1 δv(r t ) v0 I Density-density response function of interacting system δρ[v](rt) χ(rt; r0t0) := 0 0 δv(r t ) v0 I Response of non-interacting Kohn-Sham system: Z δρ[v ](rt) ρ[v + v ](r; t) = ρ[v ](r) + S v (r0t0)d3r0dt0 + ::: S;0 S;1 S;0 0 0 S δvS (r t ) v0 I Density-density response function of time-dependent Kohn-Sham system δρ [v ](rt) χ (rt; r0t0) := S S S 0 0 δvS (r t ) vS;0 Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods July 19, 2011 7 / 62 Derivation of response equation I Definition of time-dependent xc potential vxc(rt) = vKS (rt) − vext(rt) − vH (rt) I Take functional derivative δv (rt) δv (rt) δv (rt) δ(t − t0) xc = KS − ext − δρ(r0t0) δρ(r0t0) δρ(r0t0) jr − r0j 0 0 −1 0 0 −1 0 0 0 0 fxc(rt; r t )= χS (rt; r t ) − χ (rt; r t ) − Wc(rt; r t ) I Act with reponse functions from left and right −1 −1 χS ·j Wc + fxc = χS − χ j· χ χS (Wc + fxc)χ = χ − χS I Dyson-type equation for response functions χ = χS + χS (Wc + fxc)χ Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods July 19, 2011 8 / 62 First order density response I Exact density response to first order ρ1 = χv1 = χS v1 + χS (Wc + fxc)ρ1 I In integral notation Z 3 0 0 0 0 h 0 0 ρ1(rt) = d r dt χS (rt; r t ) v1(r t ) Z 3 00 00 0 0 00 00 0 0 00 00 00 00 i + d r dt (Wc(r t ; r t ) + fxc(r t ; r t ))ρ1(r t ) I For practical application: iterative solution with approximate kernel fxc δv [ρ](r0t0) f (r0t0; r00t00) = xc xc 00 00 δρ(r t ) ρ0 Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods July 19, 2011 9 / 62 Lehmann representation of linear response function I Exact many-body eigenstates H^ (t = t0)jmi = Emjmi I Lehmann representation of linear response function: 0 0 0 X h0jρ^(r)jmihmjρ^(r)j0i h0jρ^(r )jmihmjρ^(r )j0i χ(r; r ; !) = lim − η!0+ ! − (Em − E0) + iη ! + (Em − E0) + iη m Neutral excitation energies are poles of the linear response function! I Exact linear density response to perturbation v1(!) ρ1(!) =χ ^(!)v1(!) Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods July 19, 2011 10 / 62 Excitation energies I Dyson-type equation for response functions in frequency space [1^ − χ^S (!)(W^ c + f^xc(!))]ρ1(!) = χS v1(!) I ρ1(!) has poles for exact excitation energies Ωj ρ1(!) ! 1 for ! ! Ωj I On the other hand, rhs χS v1(!) stays finite for ! ! Ωj hence the eigenvalues of the integral operator [1^ − χ^S (!)(W^ c + f^xc(!))]ξ(!) = λ(!)ξ(!) vanish, λ(!) ! 0 for ! ! Ωj . I Determines rigorously the exact excitation energies [1^ − χ^S (Ωj )(W^ c + f^xc(Ωj ))]ξ(Ωj ) = 0 Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods July 19, 2011 11 / 62 Casida equation I (Non-linear) eigenvalue equation for excitation energies 2 ΩFj = !j Fj with 2 p p Ωiaσ;jbτ = δσ,τ δi;j δa;b(a − i) + 2 (a − i)Kiaσ;jbτ (b − j ) and Z Z h 1 i K (!)= d3r d3r0φ (r)φ (r) + f (r; r0;!) φ (r)φ (r) iaσ;jbτ iσ jσ jr − r0j xc kτ lτ I Eigenvalues !j are exact vertical excitation energies I Eigenvectors can be used to compute oscillator strength I Drawback: need occupied and unoccupied orbitals Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods July 19, 2011 12 / 62 Adiabatic approximation I Adiabatic approximation: evaluate static Kohn-Sham potential at time-dependent density adiab static DFT vxc [ρ](rt) := vxc [ρ(t)](rt) I Example: adiabatic LDA ALDA LDA 1=3 vxc [ρ](rt) := vxc (ρ(t)) = −αρ(r; t) + ::: I Exchange-correlation kernel δvALDA[ρ](rt) @vALDA f ALDA(rt; r0t0) = xc = δ(t − t0)δ(r − r0) xc xc 0 0 δρ(r t ) @ρ(r) ρ0(r) @2ehom = δ(t − t0)δ(r − r0) xc 2 @n ρ0(r) Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods July 19, 2011 13 / 62 Failures of the adiabatic approximation in linear response I H2 dissociation is incorrect 1 + 1 + R!1 E( Σu ) − E( Σg ) −! 0 (in ALDA) Gritsenko, van Gisbergen, Grling, Baerends, JCP 113, 8478 (2000). I sometimes problematic close to conical intersections I response of long chains strongly overestimated Champagne et al., JCP 109, 10489 (1998) and 110, 11664 (1999). I in periodic solids fxc(q; !; ρ) = c(ρ), whereas for insulators, exact q!0 2 fxc −! 1=q divergent I charge transfer excitations not properly described Dreuw et al., JCP 119, 2943 (2003). Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods July 19, 2011 14 / 62 Sternheimer equation Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods July 19, 2011 15 / 62 Sternheimer equation I Perturbed Hamiltonian and states (zero frequency) (H^0 + λH1 + :::)( 0 + λ 1 + :::) = (E0 + λE1 + :::)( 0 + λ 1 + :::) I Expand and keep terms to first order in λ 2 H^0 0 + λH1 0 + λH0 1 = E0 0 + λE0 1 + λE1 0 + O(λ ) I Use H^0 0 = E0 0 (H^0 − E0) 1 = −(H^1 − E1) 0; Sternheimer equation Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods July 19, 2011 16 / 62 Sternheimer equation in TDDFT I (Weak) monochromatic perturbation v1(r; t) = λri cos(!t) I Expand time-dependent Kohn-Sham wavefunctions in powers of λ (0) (1) m(r; t) = exp(−i(m + λm )t)× n 1 o (0)(r) + λ[exp(i!t) (1)(r;!) + exp(−i!t) (1)(r; −!)] m 2 m m I Insert in time-dependent Kohn-Sham equation and keep terms up to first order in λ Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods July 19, 2011 17 / 62 Sternheimer equation in DFT I Frequency-dependent response (self-consistent solution!) h (0) i (1) (1) (0) H^ − j ± ! + iη (r; ±!) = H^ (±!) (r); with first-oder frequency-dependent perturbation Z ρ (r;!) Z H^ (1)(!) = v(r) + 1 d3r0 + f (r; r0;!)ρ (r0;!)d3r0 jr − r0j xc 1 and first-order density response occ.