Introduction to linear-response, and time-dependent density-functional theory. Heiko Appel

Transcription

1 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, / 44

2 Motivation Where is electron dynamics important? Electron-hole pair creation and exciton propagation in solar cells Photosynthesis and energy transfer in light-harvesting antenna complexes Quantum computing (e.g. electronic transitions in ultracold atoms) Molecular electronics, quantum transport... Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

3 Motivation - Decoherence in Quantum Mechanics Today s Google frontpage: 126. birthday of Erwin Schrödinger Schrödinger cat state Ψ = 1 2 (a(t) ψ A + d(t) ψ D ) Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

4 Outline Linear Response in DFT Response functions Casida equation Sternheimer equation Real-space representation and real-time propagation Real-space representation for wavefunctions and Hamiltonians Time-propagation schemes Optimal control of electronic motion Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

5 Time-dependent density-functional theory 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, / 44

6 Time-dependent density-functional theory 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). Time-dependent Kohn-Sham system The time-dependent density of an interacting many-electron system can be calculated as density N ρ(r, t) = ϕ j(r, r) 2 of an auxiliary non-interacting Kohn-Sham system ) i tϕ j(r, t) = ( vs[ρ](r, t) ϕ j(r, t) 2m with a local multiplicative potential v S[ρ(r, t )](r, t) = v ext(r, t) + j=1 ρ(r, t) r r d3 r + v xc[ρ(r, t )](r, t) Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

7 Linear Response Theory Hamiltonian Ĥ(t) = Ĥ0 + Θ(t t0)v1(r, t) Initial condition: for times t < t 0 the system is in the ground-state of the unperturbed Hamiltonian Ĥ0 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, / 44

8 Linear Response Theory Hamiltonian Ĥ(t) = Ĥ0 + Θ(t t0)v1(r, t) Initial condition: for times t < t 0 the system is in the ground-state of the unperturbed Hamiltonian Ĥ0 with potential v0 and density ρ0(r) For times t > t 0, switch on perturbation v 1(r, t): leads to time-dependent density ρ(r, t) = ρ 0(r) + δρ(rt) Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

9 Linear Response Theory Hamiltonian Ĥ(t) = Ĥ0 + Θ(t t0)v1(r, t) Initial condition: for times t < t 0 the system is in the ground-state of the unperturbed Hamiltonian Ĥ0 with potential v0 and density ρ0(r) For times t > t 0, switch on perturbation v 1(r, t): leads to time-dependent density ρ(r, t) = ρ 0(r) + δρ(rt) Functional Taylor expansion of ρ[v](r, t) around v 0: ρ[v](r, t) = ρ[v 0 + v 1](r, t) = ρ[v 0](r, t) δρ[v](rt) + v 1(r t )d 3 r dt δv(r t ) v0 δ 2 ρ[v](rt) + v 1(r t )v 1(r t )d 3 r dt d 3 r dt δv(r t )δv(r t ) v Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

10 Computing Linear Response Different ways to compute first order response in DFT Response functions, Casida equation (frequency-dependent) perturbation theory, Sternheimer equation 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, / 44

11 Response functions Functional Taylor expansion of ρ[v](r, t) around external potential v 0: δρ[v](rt) ρ[v 0 + v 1](r, t) = ρ[v 0](r) + v 1(r t )d 3 r dt +... δv(r t ) v0 Density-density response function of interacting system χ(rt, r t ) := δρ[v](rt) δv(r t ) v0 Θ(t t ) 0 [ˆρ(r, t) H, ˆρ(r, t ) H] 0 Response of non-interacting Kohn-Sham system: δρ[vs](rt) ρ[v S,0 + v S,1](r, t) = ρ[v S,0](r) + v S(r t )d 3 r dt +... δv S(r t ) v0 Density-density response function of time-dependent Kohn-Sham system χ S(rt, r t ) := δρs[vs](rt) δv S(r t ) vs,0 Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

12 Derivation of response equation Definition of time-dependent xc potential v xc(rt) = v KS(rt) v ext(rt) v H(rt) Take functional derivative δv xc(rt) δρ(r t ) = δvks(rt) δρ(r t ) δvext(rt) δρ(r t ) δ(t t ) r r f xc(rt, r t ) := χ 1 S (rt, r t ) χ 1 (rt, r t ) W c(rt, r t ) Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

13 Derivation of response equation Definition of time-dependent xc potential v xc(rt) = v KS(rt) v ext(rt) v H(rt) Take functional derivative δv xc(rt) δρ(r t ) = δvks(rt) δρ(r t ) δvext(rt) δρ(r t ) δ(t t ) r r f xc(rt, r t ) := χ 1 S (rt, r t ) χ 1 (rt, r t ) W c(rt, r t ) Act with reponse functions from left and right χ S W c + f xc = χ 1 S χ 1 χ χ S(W c + f xc)χ = χ χ S Dyson-type equation for response functions χ = χ S + χ S(W c + f xc)χ Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

14 First order density response Exact density response to first order In integral notation ρ 1(rt) = + ρ 1 = χv 1 = χ Sv 1 + χ S(W c + f xc)ρ 1 [ d 3 r dt χ S(rt, r t ) v 1(r t ) ] d 3 r dt (W c(r t, r t ) + f xc(r t, r t ))ρ 1(r t ) For practical application: iterative solution with approximate kernel f xc f xc(r t, r t ) = δvxc[ρ](r t ) δρ(r t ) ρ0 Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

15 Lehmann representation of linear response function Exact many-body eigenstates Ĥ(t = t 0) m = E m m Lehmann representation of linear density-density response function: χ(r, t;, r, t ) = Θ(t t ) 0 [ˆρ(r, t) H, ˆρ(r, t ) H] 0 Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

16 Lehmann representation of linear response function Exact many-body eigenstates Ĥ(t = t 0) m = E m m Lehmann representation of linear density-density response function: χ(r, t;, r, t ) = Θ(t t ) 0 [ˆρ(r, t) H, ˆρ(r, t ) H] 0 Neutral excitation energies are poles of the linear response function! ( 0 ˆρ(r)H m m ˆρ(r χ(r, r ) H 0 ; ω) = lim 0 ˆρ(r ) ) H m m ˆρ(r) H 0 η 0 + ω (E m E m 0) + iη ω + (E m E 0) + iη Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

17 Lehmann representation of linear response function Exact many-body eigenstates Ĥ(t = t 0) m = E m m Lehmann representation of linear density-density response function: χ(r, t;, r, t ) = Θ(t t ) 0 [ˆρ(r, t) H, ˆρ(r, t ) H] 0 Neutral excitation energies are poles of the linear response function! ( 0 ˆρ(r)H m m ˆρ(r χ(r, r ) H 0 ; ω) = lim 0 ˆρ(r ) ) H m m ˆρ(r) H 0 η 0 + ω (E m E m 0) + iη ω + (E m E 0) + iη Exact linear density response to perturbation v 1(ω) ρ 1(ω) = ˆχ(ω)v 1(ω) Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

18 Lehmann representation of linear response function Exact many-body eigenstates Ĥ(t = t 0) m = E m m Lehmann representation of linear density-density response function: χ(r, t;, r, t ) = Θ(t t ) 0 [ˆρ(r, t) H, ˆρ(r, t ) H] 0 Neutral excitation energies are poles of the linear response function! ( 0 ˆρ(r)H m m ˆρ(r χ(r, r ) H 0 ; ω) = lim 0 ˆρ(r ) ) H m m ˆρ(r) H 0 η 0 + ω (E m E m 0) + iη ω + (E m E 0) + iη Exact linear density response to perturbation v 1(ω) ρ 1(ω) = ˆχ(ω)v 1(ω) Relation to two-body Green s function i 2 G (2) (r, t;, r, t, r, t;, r, t ) = χ(r, t;, r, t ) + ρ(r)ρ(r ) Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

19 Lehmann representation of linear response function Exact many-body eigenstates Ĥ(t = t 0) m = E m m Lehmann representation of linear density-density response function: χ(r, t;, r, t ) = Θ(t t ) 0 [ˆρ(r, t) H, ˆρ(r, t ) H] 0 Neutral excitation energies are poles of the linear response function! ( 0 ˆρ(r)H m m ˆρ(r χ(r, r ) H 0 ; ω) = lim 0 ˆρ(r ) ) H m m ˆρ(r) H 0 η 0 + ω (E m E m 0) + iη ω + (E m E 0) + iη Exact linear density response to perturbation v 1(ω) ρ 1(ω) = ˆχ(ω)v 1(ω) Relation to two-body Green s function i 2 G (2) (r, t;, r, t, r, t;, r, t ) = χ(r, t;, r, t ) + ρ(r)ρ(r ) Current-current response function: Π α,β (r, t;, r, t ) = Θ(t t ) 0 [ĵ α(r, t) H, ĵ β (r, t ) H] 0 Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

20 Excitation energies Dyson-type equation for response functions in frequency space [ˆ1 ˆχ S(ω)(Ŵc + ˆf xc(ω))]ρ 1(ω) = χ Sv 1(ω) ρ 1(ω) has poles for exact excitation energies Ω j ρ 1(ω) for ω Ω j On the other hand, rhs χ Sv 1(ω) stays finite for ω Ω j hence the eigenvalues of the integral operator [ˆ1 ˆχ S(ω)(Ŵc + ˆf xc(ω))]ξ(ω) = λ(ω)ξ(ω) vanish, λ(ω) 0 for ω Ω j. Determines rigorously the exact excitation energies [ˆ1 ˆχ S(Ω j)(ŵc + ˆf xc(ω j))]ξ(ω j) = 0 Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

21 Casida equation (Non-linear) eigenvalue equation for excitation energies ΩF j = ω 2 j F j with Ω iaσ,jbτ = δ σ,τ δ i,jδ a,b (ɛ a ɛ i) (ɛ a ɛ i)k iaσ,jbτ (ɛb ɛ j) and K iaσ,jbτ (ω) = d 3 r [ d 3 r 1 ] φ iσ(r)φ jσ(r) r r + fxc(r, r, ω) φ kτ (r)φ lτ (r) Eigenvalues ω j are exact vertical excitation energies Eigenvectors can be used to compute oscillator strength Drawback: need occupied and unoccupied orbitals Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

22 Adiabatic approximation Adiabatic approximation: evaluate static Kohn-Sham potential at time-dependent density v adiab xc [ρ](rt) := v static DFT xc [ρ(t)](rt) Example: adiabatic LDA vxc ALDA [ρ](rt) := vxc LDA (ρ(t)) = αρ(r, t) 1/ Exchange-correlation kernel f ALDA xc (rt, r t ) = δvalda xc [ρ](rt) δρ(r t ) = δ(t t )δ(r r ) valda xc ρ(r) = δ(t t )δ(r r ) 2 e hom xc n 2 ρ0 (r) ρ0 (r) Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

23 Failures of the adiabatic approximation in linear response H 2 dissociation is incorrect E( 1 Σ + u ) E( 1 Σ + g ) R 0 (in ALDA) Gritsenko, van Gisbergen, Grling, Baerends, JCP 113, 8478 (2000). sometimes problematic close to conical intersections response of long chains strongly overestimated Champagne et al., JCP 109, (1998) and 110, (1999). in periodic solids f xc(q, ω, ρ) = c(ρ), whereas for insulators, q 0 1/q 2 divergent f exact xc 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 August 12, / 44

24 Relation of TDHF eigenvalue problem to RPA, CIS and drccd energies RPA equation ( A B B A ) ( X Y ) = ( X Y ) ω RPA correlation energy Ec RPA = 1 T r(ω A) 2 CIS correlation energy from Tamm-Dancoff approximation TDA: B = 0 Ec CIS = 1 T r( ω A) 2 Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

25 Relation of TDHF eigenvalue problem to RPA, CIS and drccd energies RPA equation ( A B B A ) ( X Y ) = ( X Y ) ω RPA correlation energy Ec RPA = 1 T r(ω A) 2 CIS correlation energy from Tamm-Dancoff approximation TDA: B = 0 Ec CIS = 1 T r( ω A) 2 Keeping only particle-hole ring contractions, yiels matrix Ricatti equation for CCD cluster amplitudes B + AT + TA + TBT = 0, t ab ij = T ia,jb Correlation energy in direct ring Coupled Cluster Doubles (drccd) E drccd c = 1 2 T r(bt) Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

26 Relation of TDHF eigenvalue problem to RPA, CIS and drccd energies RPA equation ( A B B A ) ( X Y ) = ( X Y ) ω Multiplication of RPA equation with X 1 from right yields A + BT = XωX 1, where T := YX 1 Taking trace yields correlation energies 2E drccd c = T r(bt) = T r(xωx 1 A) = T r(ω A) = 2E RPA c Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

27 Relation of TDHF eigenvalue problem to RPA, CIS and drccd energies RPA equation ( A B B A ) ( X Y ) = ( X Y ) ω Multiplication of RPA equation with X 1 from right yields A + BT = XωX 1, where T := YX 1 Taking trace yields correlation energies 2E drccd c = T r(bt) = T r(xωx 1 A) = T r(ω A) = 2E RPA c Multiplication of RPA equation with (T, 1) from left and X 1 from right ( ) ( ) ( ) A B 1 1 (T, 1) B A YX 1 = (T, 1) YX 1 XωX 1 Expanding yields drccd Ricatti equation B + AT + TA + TBT = 0 T := YX 1 satisfies drccd amplitude equation G. Scuseria, et. al. J. Chem. Phys. 129, (2008). Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

28 Computing Linear Response Different ways to compute first order response in DFT Response functions, Casida equation (frequency-dependent) perturbation theory, Sternheimer equation 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, / 44

29 Sternheimer equation Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

30 Sternheimer equation Perturbed Hamiltonian and states (zero frequency) (Ĥ0 + λh1 +...)(ψ0 + λψ1 +...) = (E0 + λe1 +...)(ψ0 + λψ1 +...) Expand and keep terms to first order in λ Ĥ 0ψ 0 + λh 1ψ 0 + λh 0ψ 1 = E 0ψ 0 + λe 0ψ 1 + λe 1ψ 0 + O(λ 2 ) Use Ĥ0ψ0 = E0ψ0 (Ĥ0 E0)ψ1 = (Ĥ1 E1)ψ0, Sternheimer equation Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

31 Sternheimer equation in TDDFT (Weak) monochromatic perturbation v 1(r, t) = λr i cos(ωt) Expand time-dependent Kohn-Sham wavefunctions in powers of λ ψ m(r, t) = exp( i(ɛ (0) m + λɛ (1) m )t) { ψ m (0) (r) + 1 } 2 λ[exp(iωt)ψ(1) m (r, ω) + exp( iωt)ψ m (1) (r, ω)] 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 August 12, / 44

32 Sternheimer equation in DFT Frequency-dependent response (self-consistent solution!) [Ĥ(0) ɛ j ± ω + iη] ψ (1) (r, ±ω) = Ĥ(1) (±ω)ψ (0) (r), with first-oder frequency-dependent perturbation ρ1(r, Ĥ (1) ω) (ω) = v(r) + r r d3 r + f xc(r, r, ω)ρ 1(r, ω)d 3 r and first-order density response occ. { } ρ 1(r, ±ω) = [ψ (0) (r)] ψ (1) (r, ω) + [ψ (1) (r, ω)] ψ (0) (r) m Main advantages Only occupied states need to be considered Scales as N 2, where N is the number of atoms (Non-)Linear system of equations. Can be solved with standard solvers Disadvantage Converges slowly close to a resonance Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

33 Different types of perturbations The response equations can be used for different types of perturbations Electric perturbations v(r) = r i Response contains information about polarizabilities, absorption, fluoresence, etc. Magnetic perturbations v(r) = L i Response contains e.g. NMR signals, etc. Atomic displacements v(r) = v(r) R i Response contains e.g. phonons, etc. Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

34 Outline Linear Response in DFT Response functions Casida equation Sternheimer equation Real-space representation and real-time propagation Real-space representation for wavefunctions and Hamiltonians Time-propagation schemes Optimal control of electronic motion Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

35 Real-space grids Simulation volumes: sphere, cylinder, parallelepiped Minimal mesh: spheres around atoms, filled with uniform mesh of grid points Typically zero boundary condition, absorbing boundary, optical potential Finite-difference representation ( stencils ) for the Laplacian/kinetic energy Pseudopotentials Domain-parallelization Domain A Domain B Ghost Points of Domain B Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

36 Real-space grids Example: five-point finite difference Laplacian in 2D 1 2 ψ 2m x 1 1 [ 2 2m h ψ 2m y 1 1 [ 2 2m h 2 Stencil notation for kinetic energy 1 1 2m h 2 Leads to sparse matrices ] ψ(i 1, j) + 2ψ(i, j) ψ(i + 1, j) ] ψ(i, j 1) + 2ψ(i, j) ψ(i, j + 1) ψ(i, j) Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

37 Real-space grids Size of Hamiltonian matrix can easily reach 10 7 x10 7 Basic operation Ĥψ sparse matrix vector operations Sparse solvers Conjugate gradients Krylov subspace/lanczos methods Davidson or Jacobi-Davidson algorithm Multigrid methods Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

38 Real-time evolution for the time-dependent Kohn-Sham system Time-dependent Kohn-Sham equations i tϕ j(r, t) = ( 2 2 2m v S[ρ(r, t )](r, t) = v(r, t) + ρ(r, t) = N ϕ j(r, r) 2 j=1 ) + vs[ρ](r, t) ϕ j(r, t) ρ(r, t) r r d3 r + v xc[ρ(r, t )](r, t) Initial value problem ϕ j(r, t) = ϕ (0) j (r) Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

39 Real-time evolution for the time-dependent Kohn-Sham system Time-dependent Kohn-Sham equations i tϕ j(r, t) = ( 2 2 2m v S[ρ(r, t )](r, t) = v(r, t) + ρ(r, t) = N ϕ j(r, r) 2 j=1 ) + vs[ρ](r, t) ϕ j(r, t) ρ(r, t) r r d3 r + v xc[ρ(r, t )](r, t) Initial value problem ϕ j(r, t) = ϕ (0) j (r) Time-evolution operator Û(t, t0) ϕ j(r, t) = Û(t, t0)ϕj(r, t0) Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

40 Properties of Û(t, t 0) Û(t, t 0) is a non-linear operator The propagator is unitary Û = Û 1 In the absence of magnetic fields the propagator is time-reversal symmetric Û 1 (t, t 0) = Û(t0, t) Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

41 Properties of Û(t, t 0) Û(t, t 0) is a non-linear operator The propagator is unitary Û = Û 1 In the absence of magnetic fields the propagator is time-reversal symmetric Û 1 (t, t 0) = Û(t0, t) Equation of motion for the propagator i t Û(t, t 0) = Ĥ(t)Û(t, t0), Û(t 0, t 0 ) = ˆ1 Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

42 Properties of Û(t, t 0) Û(t, t 0) is a non-linear operator The propagator is unitary Û = Û 1 In the absence of magnetic fields the propagator is time-reversal symmetric Û 1 (t, t 0) = Û(t0, t) Equation of motion for the propagator i t Û(t, t 0) = Ĥ(t)Û(t, t0), Û(t 0, t 0 ) = ˆ1 Representation in integral form Û(t, t 0) = ˆ1 i t t 0 dτĥ(τ)û(τ, t0) Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

43 Properties of Û(t, t 0) Û(t, t 0) is a non-linear operator The propagator is unitary Û = Û 1 In the absence of magnetic fields the propagator is time-reversal symmetric Û 1 (t, t 0) = Û(t0, t) Equation of motion for the propagator i t Û(t, t 0) = Ĥ(t)Û(t, t0), Û(t 0, t 0 ) = ˆ1 Representation in integral form Û(t, t 0) = ˆ1 i t t 0 dτĥ(τ)û(τ, t0) Iterated solution of integral equation - time-ordered exponential ( i) n t t t Û(t, t 0) = dt 1 dt 2... dt n ˆT [ Ĥ(t 1)Ĥ(t2)... Ĥ(tn)] n! t 0 t 0 t 0 n=0 = ˆT exp( i t t 0 dτĥ(τ)) Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

44 Real-time evolution - Short-time propagation Group property of exact propagator Û(t 1, t 2) = Û(t1, t3)û(t3, t2) Split propagation step in small short-time propagation intervals Û(t, t 0) = N 1 j=1 Û(t j, t j + t j) Why is this a good idea? If we want to resolve frequencies up to ω max, the time-step should be no larger than 1/ω max The time-dependence of the Hamiltonian is small over a short-time interval The norm of the time-ordered exponential is proportional to t. Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

45 Real-time evolution - Magnus expansion Time-ordered evolution operator Û(t, t 0) = ( i) n n=0 n! = ˆT exp( i t dt 1 t 0 t t t 0 dτĥ(τ)) t 0 dt 2... t dt n ˆT [ Ĥ(t 1)Ĥ(t2)... Ĥ(tn)] t 0 Magnus expansion Û(t + t, t) = exp (ˆΩ1 + ˆΩ 2 + ˆΩ ) 3 + Magnus operators ˆΩ 1 = i ˆΩ 2 = t+ t t t+ t τ1. t t Ĥ(τ)dτ [Ĥ(τ1), Ĥ(τ2)]dτ2dτ1 Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

46 Real-time evolution - Magnus expansion Second-order Magnus propagator - Exponential midpoint rule ) Û (2) (t + t, t) = exp (ˆΩ1 + O( t 3 ) ˆΩ 1 = iĥ(t + t/2) + O( t3 ). Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

47 Real-time evolution - Magnus expansion Second-order Magnus propagator - Exponential midpoint rule ) Û (2) (t + t, t) = exp (ˆΩ1 + O( t 3 ) ˆΩ 1 = iĥ(t + t/2) + O( t3 ). Fourth-order Magnus propagator Û (4) (t + t, t) = exp ˆΩ 1 = (ˆΩ1 + Ω 2 ) + O( t 5 ) i(ĥ(τ1) + Ĥ(τ2)) t 2 + O( t5 ). 3 t 2 ˆΩ 2 = i[ĥ(τ1), Ĥ(τ2)] + O( t 5 ). 12 τ 1,2 = t + ( ± 6 ) t Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

48 Real-time evolution - Crank-Nicholson/Cayley propagator Padé approximation of exponential, e.g. lowest order (Crank-Nicholson) 1 iĥ t/2 exp( iĥ t) 1 + iĥ t/2 Need only action of operator on a state vector Ψ(t + t) = 1 iĥ t/2 Ψ(t) 1 + iĥ t/2 (Non-)Linear system of equations at each time-step (1 + iĥ t/2) Ψ(t + t) = (1 iĥ t/2) Ψ(t) Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

49 Real-time evolution - Operator splitting methods Typically, the Hamiltonian has the form Ĥ = ˆT + ˆV ˆT is diagonal in momentum space, ˆV in position space Baker-Campbell-Hausdorff relation eâe ˆB = exp(â + ˆB [Â, ˆB] +...) Split-Operator exp( i t( ˆT + ˆV )) exp( i t ˆT /2) exp( i t ˆV ) exp( i t ˆT /2) Use FFT to switch between momentum space and real-space. Higher-order splittings possible, but require more FFTs Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

50 Real-time evolution - Enforced time reversal symmetry Enforced time-reversal symmetry exp(+i t t Ĥ(t + t)) Ψ(t + t) = exp( i 2 2 Ĥ(t)) Ψ(t) Propagator with time-reversal symmetry Û ETRS (t + t, t) = exp( i t t Ĥ(t + t)) exp( i 2 2 Ĥ(t)) Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

51 Real-time evolution - Matrix exponential Û CN (t + t, t) = 1 iĥ t/2 1 + ( i tĥ(t iĥ t/2 ) Û EM (t + t, t) = exp + t/2) Û SO (t + t, t) = exp( i t ˆT /2)exp( i t ˆV )exp( i t ˆT /2) Û ETRS (t + t, t) = exp( i t t Ĥ(t + t))exp( i 2 2 Ĥ(t))... Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

52 Real-time evolution - Matrix exponential Û CN (t + t, t) = 1 iĥ t/2 1 + ( i tĥ(t iĥ t/2 ) Û EM (t + t, t) = exp + t/2) Û SO (t + t, t) = exp( i t ˆT /2)exp( i t ˆV )exp( i t ˆT /2) Û ETRS (t + t, t) = exp( i t t Ĥ(t + t))exp( i 2 2 Ĥ(t))... Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

53 Real-time evolution - Matrix exponential C. Moler and C. Van Loan, Nineteen Dubious Ways to Compute the Exponential of A Matrix, SIAM Review 20, 801 (1978) C. Moler and C. Van Loan, Nineteen Dubious Ways to Compute the Exponential of A Matrix, Twenty-Five Years Later, SIAM Review 45, 3 (2003) Task: Compute exponential of operator/matrix Taylor series Chebyshev polynomials Padé approximations Scaling and squaring Ordinary differential equation methods Matrix decomposition methods Splitting methods Task: Compute eâv for given v Taylor series Chebyshev rational approximation Lanczos-Krylov subspace projection Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

54 Real-time evolution - Movie time Proton scattering of fast proton with ethene Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

55 Octopus code Octopus: real-space, real-time TDDFT code, available under GPL Page (Parsec: real-space, real-time code using similar concepts) libxc: Exchange-Correlation library, available under LGPL (used by many codes: Abinit, APE, AtomPAW, Atomistix ToolKit, BigDFT, DP, ERKALE, GPAW, Elk, exciting, octopus, Yambo) Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

56 Optimal control theory Control of ring current in a quantum ring external potential density profile Optimal Control of Quantum Rings by Terahertz Laser Pulses, E. Räsänen, et. al, Phys. Rev. Lett. 98, (2007). Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

57 Optimal control theory Goal: find optimal laser pulse ɛ(t) that drives the system to a desired state Φ f maximize overlap functional J 1[Ψ] = Ψ(T ) Φ f 2. constrain laser intensity T J 2[ɛ] = α 0 ɛ 2 (t) dt. 0 Lagrange multiplier density to ensure evolution with TDSE J 3[Ψ, χ, ɛ] = 2 Im T 0 (i ) χ(t) t Ĥ(t) Ψ(t) dt, Find maximum of J 1[Ψ] + J 2[ɛ] + J 3[Ψ, χ, ɛ] Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

58 Optimal control theory First variation of the functional δj = δ ΨJ + δ χj + δ ɛj = 0 Control equations δ ΨJ = 0 : δ χj = 0 : ) (i t Ĥ(t) χ(t) = 0, χ(t ) = Φ f Φ f Ψ(T ) ) (i t Ĥ(t) Ψ(t) = 0, Ψ(0) = Φ i, δ ɛj = 0 : α 0 ɛ(t) = Im χ(t) ˆµ Ψ(t). Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44

59 Optimal control theory Optimal laser pulse and level population Optimal Control of Quantum Rings by Terahertz Laser Pulses, E. Räsänen, et. al, Phys. Rev. Lett. 98, (2007). Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, / 44