Intro to Real-Space, Linear-Response, and TD Methods August 12, 2013 1 / 44 Motivation

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 |Ψi = √ (a(t)|ψAi + d(t)|ψDi) 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) = |ϕj (r, r)| j=1 of an auxiliary non-interacting Kohn-Sham system 2∇2 i ∂ ϕ (r, t) = − ~ + v [ρ](r, t) ϕ (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 |r − r0| 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) = |ϕj (r, r)| j=1 of an auxiliary non-interacting Kohn-Sham system 2∇2 i ∂ ϕ (r, t) = − ~ + v [ρ](r, t) ϕ (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 |r − r0| 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 Hˆ (t) = Hˆ0 + Θ(t − t0)v1(r, t)

I Initial condition: for times t < t0 the system is in the ground-state of the unperturbed Hamiltonian Hˆ0 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 Hˆ (t) = Hˆ0 + Θ(t − t0)v1(r, t)

I Initial condition: for times t < t0 the system is in the ground-state of the unperturbed Hamiltonian Hˆ0 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 Hˆ (t) = Hˆ0 + Θ(t − t0)v1(r, t)

I Initial condition: for times t < t0 the system is in the ground-state of the unperturbed Hamiltonian Hˆ0 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 )h0|[ˆρ(r, t)H , ρˆ(r , t )H ]|0i

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 ·| Wc + fxc = χS − χ |· 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) |r − r0|

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) |r − r0|

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 ·| Wc + fxc = χS − χ |· 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

I Exact linear density response to perturbation v1(ω)

ρ1(ω) =χ ˆ(ω)v1(ω)

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 )h0|[ˆjα(r, t)H , ˆjβ (r, t )H ]|0i

Lehmann representation of linear response function

I Exact many-body eigenstates

Hˆ (t = t0)|mi = Em|mi

I Lehmann representation of linear density-density response function: 0 0 0 0 0 χ(r, t; , r , t ) = Θ(t − t )h0|[ˆρ(r, t)H , ρˆ(r , t )H ]|0i

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(ω)

ρ1(ω) =χ ˆ(ω)v1(ω)

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 )h0|[ˆjα(r, t)H , ˆjβ (r, t )H ]|0i

Lehmann representation of linear response function

I Exact many-body eigenstates

Hˆ (t = t0)|mi = Em|mi

I Lehmann representation of linear density-density response function: 0 0 0 0 0 χ(r, t; , r , t ) = Θ(t − t )h0|[ˆρ(r, t)H , ρˆ(r , t )H ]|0i

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 11 / 44 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 )h0|[ˆjα(r, t)H , ˆjβ (r, t )H ]|0i

Lehmann representation of linear response function

I Exact many-body eigenstates

Hˆ (t = t0)|mi = Em|mi

I Lehmann representation of linear density-density response function: 0 0 0 0 0 χ(r, t; , r , t ) = Θ(t − t )h0|[ˆρ(r, t)H , ρˆ(r , t )H ]|0i

I Exact linear density response to perturbation v1(ω)

ρ1(ω) =χ ˆ(ω)v1(ω)

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 11 / 44 I Current-current response function: 0 0 0 0 Πα,β (r, t; , r , t ) = Θ(t − t )h0|[ˆjα(r, t)H , ˆjβ (r, t )H ]|0i

Lehmann representation of linear response function

I Exact many-body eigenstates

Hˆ (t = t0)|mi = Em|mi

I Lehmann representation of linear density-density response function: 0 0 0 0 0 χ(r, t; , r , t ) = Θ(t − t )h0|[ˆρ(r, t)H , ρˆ(r , t )H ]|0i

I Exact linear density response to perturbation v1(ω)

ρ1(ω) =χ ˆ(ω)v1(ω)

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)

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 11 / 44 Lehmann representation of linear response function

I Exact many-body eigenstates

Hˆ (t = t0)|mi = Em|mi

I Lehmann representation of linear density-density response function: 0 0 0 0 0 χ(r, t; , r , t ) = Θ(t − t )h0|[ˆρ(r, t)H , ρˆ(r , t )H ]|0i

I Exact linear density response to perturbation v1(ω)

ρ1(ω) =χ ˆ(ω)v1(ω)

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 )h0|[ˆjα(r, t)H , ˆjβ (r, t )H ]|0i

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 11 / 44 Excitation energies

I Dyson-type equation for response functions in frequency space

[1ˆ − χˆS (ω)(Wˆ c + fˆxc(ω))]ρ1(ω) = χS v1(ω)

I ρ1(ω) has poles for exact excitation energies Ωj

ρ1(ω) → ∞ for ω → Ωj

I On the other hand, rhs χS v1(ω) stays ﬁnite for ω → Ωj hence the eigenvalues of the integral operator

[1ˆ − χˆS (ω)(Wˆ c + fˆxc(ω))]ξ(ω) = λ(ω)ξ(ω)

vanish, λ(ω) → 0 for ω → Ωj .

I Determines rigorously the exact excitation energies

[1ˆ − χˆS (Ωj )(Wˆ c + fˆxc(Ωj ))]ξ(Ωj ) = 0

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 12 / 44 Casida equation

I (Non-linear) eigenvalue equation for excitation energies

2 ΩFj = ωj 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 (ω)= d3r d3r0φ (r)φ (r) + f (r, r0, ω) φ (r)φ (r) iaσ,jbτ iσ jσ |r − r0| xc kτ lτ

I Eigenvalues ωj 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 August 12, 2013 13 / 44 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 August 12, 2013 14 / 44 Failures of the adiabatic approximation in linear response

I H2 dissociation is incorrect

1 + 1 + R→∞ 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, ω, 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 August 12, 2013 15 / 44 I Keeping only particle-hole ring contractions, yiels matrix Ricatti equation for CCD cluster amplitudes

ab B + AT + TA + TBT = 0, tij = Tia,jb

I Correlation energy in direct ring Coupled Cluster Doubles (drCCD) 1 EdrCCD = T r(BT) c 2

Relation of TDHF eigenvalue problem to RPA, CIS and drCCD energies

I RPA equation AB X X = ω −B −A Y Y

I RPA correlation energy 1 ERPA = T r(ω − A) c 2

I CIS correlation energy from Tamm-Dancoﬀ approximation TDA: B = 0 1 ECIS = T r(ω˜ − A) c 2

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 16 / 44 Relation of TDHF eigenvalue problem to RPA, CIS and drCCD energies

I RPA equation AB X X = ω −B −A Y Y

I RPA correlation energy 1 ERPA = T r(ω − A) c 2

I CIS correlation energy from Tamm-Dancoﬀ approximation TDA: B = 0 1 ECIS = T r(ω˜ − A) c 2

I Keeping only particle-hole ring contractions, yiels matrix Ricatti equation for CCD cluster amplitudes

ab B + AT + TA + TBT = 0, tij = Tia,jb

I Correlation energy in direct ring Coupled Cluster Doubles (drCCD) 1 EdrCCD = T r(BT) c 2

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 16 / 44 −1 I Multiplication of RPA equation with (T, −1) from left and X from right

AB 1 1 (T, −1) = (T, −1) XωX−1 −B −A YX−1 YX−1

I 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, 231101 (2008).

Relation of TDHF eigenvalue problem to RPA, CIS and drCCD energies

I RPA equation AB X X = ω −B −A Y Y

−1 I Multiplication of RPA equation with X from right yields

A + BT = XωX−1, where T := YX−1

I Taking trace yields correlation energies

drCCD −1 RPA 2Ec = T r(BT) = T r(XωX − A) = T r(ω − A) = 2Ec

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 17 / 44 Relation of TDHF eigenvalue problem to RPA, CIS and drCCD energies

I RPA equation AB X X = ω −B −A Y Y

−1 I Multiplication of RPA equation with X from right yields

A + BT = XωX−1, where T := YX−1

I Taking trace yields correlation energies

drCCD −1 RPA 2Ec = T r(BT) = T r(XωX − A) = T r(ω − A) = 2Ec

−1 I Multiplication of RPA equation with (T, −1) from left and X from right

AB 1 1 (T, −1) = (T, −1) XωX−1 −B −A YX−1 YX−1

I 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, 231101 (2008).

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 17 / 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 18 / 44 Sternheimer equation

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 19 / 44 Sternheimer equation

I Perturbed Hamiltonian and states (zero frequency)

(Hˆ0 + λH1 + ...)(ψ0 + λψ1 + ...) = (E0 + λE1 + ...)(ψ0 + λψ1 + ...)

I Expand and keep terms to ﬁrst order in λ

2 Hˆ0ψ0 + λH1ψ0 + λH0ψ1 = E0ψ0 + λE0ψ1 + λE1ψ0 + O(λ )

I Use Hˆ0ψ0 = E0ψ0

(Hˆ0 − E0)ψ1 = −(Hˆ1 − E1)ψ0, Sternheimer equation

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 20 / 44 Sternheimer equation in TDDFT

I (Weak) monochromatic perturbation

v1(r, t) = λri cos(ωt)

I Expand time-dependent Kohn-Sham wavefunctions in powers of λ

(0) (1) ψm(r, t) = exp(−i(m + λm )t)× n 1 o ψ(0)(r) + λ[exp(iωt)ψ(1)(r, ω) + exp(−iωt)ψ(1)(r, −ω)] 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 August 12, 2013 21 / 44 Sternheimer equation in DFT

I Frequency-dependent response (self-consistent solution!)

h (0) i (1) (1) (0) Hˆ − j ± ω + iη ψ (r, ±ω) = Hˆ (±ω)ψ (r),

with ﬁrst-oder frequency-dependent perturbation Z ρ (r, ω) Z Hˆ (1)(ω) = v(r) + 1 d3r0 + f (r, r0, ω)ρ (r0, ω)d3r0 |r − r0| xc 1 and ﬁrst-order density response

occ. 0 X n (0) ∗ (1) (1) ∗ (0) o ρ1(r , ±ω) = [ψ (r)] ψ (r, ω) + [ψ (r, −ω)] ψ (r) m

I Main advantages

I Only occupied states need to be considered 2 I Scales as N , where N is the number of atoms I (Non-)Linear system of equations. Can be solved with standard solvers

I Disadvantage

I Converges slowly close to a resonance

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 22 / 44 Diﬀerent types of perturbations The response equations can be used for diﬀerent types of perturbations

I Electric perturbations

v(r) = ri

Response contains information about polarizabilities, absorption, ﬂuoresence, etc.

I Magnetic perturbations

v(r) = Li

Response contains e.g. NMR signals, etc.

I Atomic displacements ∂v(r) v(r) = ∂Ri Response contains e.g. phonons, etc.

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 23 / 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 24 / 44 Real-space grids

I Simulation volumes: sphere, cylinder, parallelepiped

I Minimal mesh: spheres around atoms, ﬁlled with uniform mesh of grid points

I Typically zero boundary condition, absorbing boundary, optical potential

I Finite-diﬀerence representation (”stencils”) for the Laplacian/kinetic energy

I Pseudopotentials

I 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, 2013 25 / 44 Real-space grids

I Example: five-point finite difference Laplacian in 2D

1 ∂2ψ 1 1 h i − ≈ − ψ(i − 1, j) + 2ψ(i, j) − ψ(i + 1, j) 2m ∂x2 2m h2 1 ∂2ψ 1 1 h i − ≈ − ψ(i, j − 1) + 2ψ(i, j) − ψ(i, j + 1) 2m ∂y2 2m h2

I Stencil notation for kinetic energy

 −1  1 1 −1 4 −1 ψ(i, j) 2m h2   −1

I Leads to sparse matrices

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 26 / 44 Real-space grids

7 7 I Size of Hamiltonian matrix can easily reach 10 x10

I Basic operation Hψˆ −→ sparse matrix vector operations

I Sparse solvers

I Conjugate gradients

I Krylov subspace/Lanczos methods

I Davidson or Jacobi-Davidson algorithm

I Multigrid methods

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 27 / 44 I Time-evolution operator Uˆ(t, t0)

ϕj (r, t) = Uˆ(t, t0)ϕj (r, t0)

Real-time evolution for the time-dependent Kohn-Sham system

I Time-dependent Kohn-Sham equations

2∇2 i ∂ ϕ (r, t) = − ~ + v [ρ](r, t) ϕ (r, t) ~ t j 2m S j Z ρ(r0, t) v [ρ(r0, t0)](r, t) = v(r, t) + d3r0 + v [ρ(r0, t0)](r, t) S |r − r0| xc N X 2 ρ(r, t) = |ϕj (r, r)| j=1

I Initial value problem

(0) ϕj (r, t) = ϕj (r)

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 28 / 44 Real-time evolution for the time-dependent Kohn-Sham system

I Time-dependent Kohn-Sham equations

2∇2 i ∂ ϕ (r, t) = − ~ + v [ρ](r, t) ϕ (r, t) ~ t j 2m S j Z ρ(r0, t) v [ρ(r0, t0)](r, t) = v(r, t) + d3r0 + v [ρ(r0, t0)](r, t) S |r − r0| xc N X 2 ρ(r, t) = |ϕj (r, r)| j=1

I Initial value problem

(0) ϕj (r, t) = ϕj (r)

I Time-evolution operator Uˆ(t, t0)

ϕj (r, t) = Uˆ(t, t0)ϕj (r, t0)

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 28 / 44 I Equation of motion for the propagator ˆ ˆ ˆ ˆ ˆ i~∂tU(t, t0) = H(t)U(t, t0), U(t0, t0) = 1

I Representation in integral form Z t Uˆ(t, t0) = 1ˆ − i dτHˆ (τ)Uˆ(τ, t0) t0

I Iterated solution of integral equation - time-ordered exponential ∞ X (−i)n Z t Z t Z t Uˆ(t, t ) = dt dt ... dt Tˆ[Hˆ (t )Hˆ (t ) ... Hˆ (t )] 0 n! 1 2 n 1 2 n n=0 t0 t0 t0 Z t = Tˆ exp(−i dτHˆ (τ)) t0

Properties of Uˆ(t, t0)

I Uˆ(t, t0) is a non-linear operator † −1 I The propagator is unitary Uˆ = Uˆ I In the absence of magnetic ﬁelds the propagator is time-reversal symmetric −1 Uˆ (t, t0) = Uˆ(t0, t)

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 29 / 44 I Representation in integral form Z t Uˆ(t, t0) = 1ˆ − i dτHˆ (τ)Uˆ(τ, t0) t0

Properties of Uˆ(t, t0)

I Uˆ(t, t0) is a non-linear operator † −1 I The propagator is unitary Uˆ = Uˆ I In the absence of magnetic ﬁelds the propagator is time-reversal symmetric −1 Uˆ (t, t0) = Uˆ(t0, t)

I Equation of motion for the propagator ˆ ˆ ˆ ˆ ˆ i~∂tU(t, t0) = H(t)U(t, t0), U(t0, t0) = 1

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 29 / 44 I Iterated solution of integral equation - time-ordered exponential ∞ X (−i)n Z t Z t Z t Uˆ(t, t ) = dt dt ... dt Tˆ[Hˆ (t )Hˆ (t ) ... Hˆ (t )] 0 n! 1 2 n 1 2 n n=0 t0 t0 t0 Z t = Tˆ exp(−i dτHˆ (τ)) t0

Properties of Uˆ(t, t0)

I Uˆ(t, t0) is a non-linear operator † −1 I The propagator is unitary Uˆ = Uˆ I In the absence of magnetic ﬁelds the propagator is time-reversal symmetric −1 Uˆ (t, t0) = Uˆ(t0, t)

I Equation of motion for the propagator ˆ ˆ ˆ ˆ ˆ i~∂tU(t, t0) = H(t)U(t, t0), U(t0, t0) = 1

I Representation in integral form Z t Uˆ(t, t0) = 1ˆ − i dτHˆ (τ)Uˆ(τ, t0) t0

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 29 / 44 Properties of Uˆ(t, t0)

I Uˆ(t, t0) is a non-linear operator † −1 I The propagator is unitary Uˆ = Uˆ I In the absence of magnetic ﬁelds the propagator is time-reversal symmetric −1 Uˆ (t, t0) = Uˆ(t0, t)

I Equation of motion for the propagator ˆ ˆ ˆ ˆ ˆ i~∂tU(t, t0) = H(t)U(t, t0), U(t0, t0) = 1

I Representation in integral form Z t Uˆ(t, t0) = 1ˆ − i dτHˆ (τ)Uˆ(τ, t0) t0

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 29 / 44 Real-time evolution - Short-time propagation

I Group property of exact propagator

Uˆ(t1, t2) = Uˆ(t1, t3)Uˆ(t3, t2)

I Split propagation step in small short-time propagation intervals

N−1 Y Uˆ(t, t0) = Uˆ(tj , tj + ∆tj ) j=1

I Why is this a good idea?

I If we want to resolve frequencies up to ωmax, the time-step should be no larger than ≈ 1/ωmax

I The time-dependence of the Hamiltonian is small over a short-time interval

I 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, 2013 30 / 44 Real-time evolution - Magnus expansion

I Time-ordered evolution operator

∞ X (−i)n Z t Z t Z t Uˆ(t, t ) = dt dt ... dt Tˆ[Hˆ (t )Hˆ (t ) ... Hˆ (t )] 0 n! 1 2 n 1 2 n n=0 t0 t0 t0 Z t = Tˆ exp(−i dτHˆ (τ)) t0

I Magnus expansion Uˆ(t + ∆t, t) = exp Ωˆ 1 + Ωˆ 2 + Ωˆ 3 + ···

I Magnus operators

Z t+∆t Ωˆ 1 = − i Hˆ (τ)dτ t Z t+∆t Z τ1 Ωˆ 2 = [Hˆ (τ1), Hˆ (τ2)]dτ2dτ1 t t . .

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 31 / 44 I Fourth-order Magnus propagator

(4) 5 Uˆ (t + ∆t, t) = exp Ωˆ 1 + Ω2 + O(∆t ) ∆t Ωˆ = −i(Hˆ (τ ) + Hˆ (τ )) + O(∆t5). 1 1 2 2 √ 3∆t2 Ωˆ = −i[Hˆ (τ ), Hˆ (τ )] + O(∆t5). 2 1 2 12 √ 1 3 τ = t + ( ± )∆t 1,2 2 6

Real-time evolution - Magnus expansion

I Second-order Magnus propagator - Exponential midpoint rule

(2) 3 Uˆ (t + ∆t, t) = exp Ωˆ 1 + O(∆t )

3 Ωˆ 1 = −iHˆ (t + ∆t/2) + O(∆t ).

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 32 / 44 Real-time evolution - Magnus expansion

I Second-order Magnus propagator - Exponential midpoint rule

(2) 3 Uˆ (t + ∆t, t) = exp Ωˆ 1 + O(∆t )

3 Ωˆ 1 = −iHˆ (t + ∆t/2) + O(∆t ).

I Fourth-order Magnus propagator

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 32 / 44 Real-time evolution - Crank-Nicholson/Cayley propagator

I Pad´eapproximation of exponential, e.g. lowest order (Crank-Nicholson)

1 − iHˆ ∆t/2 exp(−iHˆ ∆t) ≈ 1 + iHˆ ∆t/2

I Need only action of operator on a state vector

1 − iHˆ ∆t/2 |Ψ(t + ∆t)i = |Ψ(t)i 1 + iHˆ ∆t/2

I (Non-)Linear system of equations at each time-step

(1 + iHˆ ∆t/2)|Ψ(t + ∆t)i = (1 − iHˆ ∆t/2)|Ψ(t)i

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 33 / 44 Real-time evolution - Operator splitting methods

I Typically, the Hamiltonian has the form Hˆ = Tˆ + Vˆ

I Tˆ is diagonal in momentum space, Vˆ in position space

I Baker-Campbell-Hausdorﬀ relation

ˆ ˆ 1 eAeB = exp(Aˆ + Bˆ + [A,ˆ Bˆ] + ...) 2

I Split-Operator

exp(−i∆t(Tˆ + Vˆ )) ≈ exp(−i∆tT/ˆ 2) exp(−i∆tVˆ ) exp(−i∆tT/ˆ 2)

Use FFT to switch between momentum space and real-space.

I 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, 2013 34 / 44 Real-time evolution - Enforced time reversal symmetry

I Enforced time-reversal symmetry ∆t ∆t exp(+i Hˆ (t + ∆t))|Ψ(t + ∆t)i = exp(−i Hˆ (t))|Ψ(t)i 2 2

I Propagator with time-reversal symmetry

∆t ∆t Uˆ ETRS(t + ∆t, t) = exp(−i Hˆ (t + ∆t)) exp(−i Hˆ (t)) 2 2

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 35 / 44 Real-time evolution - Matrix exponential

1 − iHˆ ∆t/2 Uˆ CN(t + ∆t, t) = 1 + iHˆ ∆t/2 Uˆ EM(t + ∆t, t) = exp −i∆tHˆ (t + ∆t/2)

Uˆ SO(t + ∆t, t) = exp(−i∆tT/ˆ 2)exp(−i∆tVˆ )exp(−i∆tT/ˆ 2) ∆t ∆t Uˆ ETRS(t + ∆t, t) = exp(−i Hˆ (t + ∆t))exp(−i Hˆ (t)) 2 2 ...

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 36 / 44 Real-time evolution - Matrix exponential

1 − iHˆ ∆t/2 Uˆ CN(t + ∆t, t) = 1 + iHˆ ∆t/2 Uˆ EM(t + ∆t, t) = exp −i∆tHˆ (t + ∆t/2)

Uˆ SO(t + ∆t, t) = exp(−i∆tT/ˆ 2)exp(−i∆tVˆ )exp(−i∆tT/ˆ 2) ∆t ∆t Uˆ ETRS(t + ∆t, t) = exp(−i Hˆ (t + ∆t))exp(−i Hˆ (t)) 2 2 ...

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 37 / 44 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

I Taylor series

I Chebyshev polynomials

I Pad´eapproximations

I Scaling and squaring

I Ordinary diﬀerential equation methods

I Matrix decomposition methods

I Splitting methods

Task: Compute eAˆv for given v

I Taylor series

I Chebyshev rational approximation

I Lanczos-Krylov subspace projection

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 38 / 44 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, 2013 39 / 44 Octopus code

I Octopus: real-space, real-time TDDFT code, available under GPL http://tddft.org/programs/octopus/wiki/index.php/Main Page (Parsec: real-space, real-time code using similar concepts)

I libxc: Exchange-Correlation library, available under LGPL (used by many codes: Abinit, APE, AtomPAW, Atomistix ToolKit, BigDFT, DP, ERKALE, GPAW, Elk, exciting, octopus, Yambo) http://tddft.org/programs/octopus/wiki/index.php/Libxc

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 40 / 44 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. R¨as¨anen,et. al, Phys. Rev. Lett. 98, 157404 (2007).

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 41 / 44 Optimal control theory

Goal: ﬁnd optimal laser pulse (t) that drives the system to a desired state Φf

I maximize overlap functional

2 J1[Ψ] = | h Ψ(T ) | Φf i | .

I constrain laser intensity Z T 2 J2[] = −α0 (t) dt. 0

I Lagrange multiplier density to ensure evolution with TDSE Z TD E ˆ J3[Ψ, χ, ] = −2 Im χ(t) i∂t − H(t) Ψ(t) dt, 0

Find maximum of J1[Ψ] + J2[] + J3[Ψ, χ, ]

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 42 / 44 Optimal control theory

I First variation of the functional

δJ = δΨJ + δχJ + δJ = 0

I Control equations δΨJ = 0 : i∂t − Hˆ (t) | χ(t) i = 0, | χ(T ) i = | Φf i h Φf | Ψ(T ) i δχJ = 0 : i∂t − Hˆ (t) | Ψ(t) i = 0, | Ψ(0) i = | Φi i ,

δJ = 0 : α0 (t) = −Im hχ(t)|µˆ|Ψ(t)i.

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 43 / 44 Optimal control theory

Optimal laser pulse and level population

Optimal Control of Quantum Rings by Terahertz Laser Pulses, E. R¨as¨anen,et. al, Phys. Rev. Lett. 98, 157404 (2007).

Heiko Appel (Fritz-Haber-Institut der MPG) Intro to real-space, linear-response, and TD methods August 12, 2013 44 / 44