Introduction to linear-response, and time-dependent density-functional theory 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 August 12, 2013 1 / 44 Motivation Where is electron dynamics important? I Electron-hole pair creation and exciton propagation 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 I ... Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 2 / 44 Motivation - Decoherence in Quantum Mechanics Today's Google frontpage: 126. birthday of Erwin Schr¨odinger Schr¨odingercat state 1 jΨi = p (a(t)j Ai + d(t)j Di) 2 Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 3 / 44 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 Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 4 / 44 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 ext jr − r0j xc 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. 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 August 12, 2013 5 / 44 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. E. Runge, and E.K.U. Gross, Phys. Rev. Lett. 52, 997 (1984). 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 ext jr − r0j xc Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 5 / 44 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 + ::: 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) Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 6 / 44 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 + ::: 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) Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 6 / 44 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 August 12, 2013 6 / 44 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 August 12, 2013 7 / 44 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 0 0 0 ≡ Θ(t − t )h0j[^ρ(r; t)H ; ρ^(r ; t )H ]j0i 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 August 12, 2013 8 / 44 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 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 ) Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 9 / 44 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 August 12, 2013 9 / 44 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 August 12, 2013 10 / 44 I Neutral excitation energies are poles of the linear response function! 0 0 0 X h0jρ^(r)H jmihmjρ^(r )H j0i h0jρ^(r )H jmihmjρ^(r)H j0i χ(r; r ; !) = lim − η!0+ ! − (Em − E0) + iη ! + (Em − E0) + iη m I Exact linear density response to perturbation v1(!) ρ1(!) =χ ^(!)v1(!) I Relation to two-body Green's function i2G(2)(r; t; ; r0; t0; r; t; ; r0; t0) = χ(r; t; ; r0; t0) + ρ(r)ρ(r0) I Current-current response function: 0 0 0 0 Πα,β (r; t; ; r ; t ) = Θ(t − t )h0j[^jα(r; t)H ; ^jβ (r; t )H ]j0i Lehmann representation of linear response function I Exact many-body eigenstates H^ (t = t0)jmi = Emjmi I Lehmann representation of linear density-density response function: 0 0 0 0 0 χ(r; t; ; r ; t ) = Θ(t − t )h0j[^ρ(r; t)H ; ρ^(r ; t )H ]j0i Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 11 / 44 I Exact linear density response to perturbation v1(!) ρ1(!) =χ ^(!)v1(!) I Relation to two-body Green's function i2G(2)(r; t; ; r0; t0; r; t; ; r0; t0) = χ(r; t; ; r0; t0) + ρ(r)ρ(r0) I Current-current response function: 0 0 0 0 Πα,β (r; t; ; r ; t ) = Θ(t − t )h0j[^jα(r; t)H ; ^jβ (r; t )H ]j0i Lehmann representation of linear response function I Exact many-body eigenstates H^ (t = t0)jmi = Emjmi I Lehmann representation of linear density-density response function: 0 0 0 0 0 χ(r; t; ; r ; t ) = Θ(t − t )h0j[^ρ(r; t)H ; ρ^(r ; t )H ]j0i I Neutral excitation energies are poles of the linear response function! 0 0 0 X h0jρ^(r)H jmihmjρ^(r )H j0i h0jρ^(r )H jmihmjρ^(r)H j0i χ(r; r ; !) = lim − η!0+ ! − (Em − E0) + iη ! + (Em − E0) + iη m Heiko Appel (Fritz-Haber-Institut der MPG)