THEORETICAL METHODS TO TREAT CORRELATED ELECTRON AND NUCLEAR - PowerPoint PPT Presentation

THEORETICAL METHODS TO TREAT CORRELATED ELECTRON AND NUCLEAR DYNAMICS FOR CLOSED AND OPEN QUANTUM SYSTEMS Peter Saalfrank I. Andrianov, F. Bouakline, T. Klamroth, S. Klinkusch, P. Krause, U. Lorenz, F. L¨ uder, M. Nest, J.C. Tremblay University of Potsdam, Germany

REAL-TIME DYNAMICS: FEMTOCHEMISTRY Zewail et al. , 1990’s femtosecond chemistry: 1 fs = 10 − 15 s nuclear (atomic) motions

REAL-TIME DYNAMICS: ATTOPHYSICS Corkum, Krausz, . . . , > 2000 attosecond physics: 1 as = 10 − 18 s electronic motions

THIS TALK IS ABOUT . . . ➊ Electron dynamics (mostly light-driven) • Methods – Wavefunction-based: TD-CI, TD-CASSCF (=MCTDHF) – Open-system density matrix based: ρ -TDCI • Some applications – Response to laser pulses – Correlation and its control ➋ Nuclear dynamics (mostly for system-bath problems) • Methods – Wave-function based: MCTDH – Open-system density matrix based: Lindblad approach • Application – Vibrational dynamics and relaxation

LASER-DRIVEN ELECTRON DYNAMICS

ELECTRON MOTION IN MOLECULES: LASERS • Electronic wavepackets (and control) dissociation of D + 2 • HHG, orbital tomography HOMO of N 2 Kling et al. , Science 312 , 264 (2006) Corkum et al. , Nature 432 , 867 (2004)

LASERS AND ELECTRON DYNAMICS: METHODS • The N-electron time-dependent Schr¨ odinger equation � � h∂ Ψ( x 1 , . . . , x N , t ) ˆ i ¯ = H el ( x 1 , . . . , x N ) − ˆ µE ( t ) Ψ( x 1 , . . . , x N , t ) ∂t K • Solution techniques r • One-electron approaches AS=(4,5) • Single-determinant methods N/2 a ........ – TD-HF: Ψ( t ) = ψ 0 ( t ) 2 Ψ( t ) = ψ KS – TD-DFT: ( t ) 0 1 • Multi-determinant methods Ψ( t ) = C 0 ( t ) ψ 0 + � a + � ar C r a ( t ) ψ r ab,rs C rs ab ( t ) ψ rs – TD-CI: ab + · · · – TD-CASSCF: Ψ( t ) = C 0 ( t ) ψ 0 ( t ) + � a ( t ) + � ar C r a ( t ) ψ r ab,rs C rs ab ( t ) ψ rs ab ( t ) + · · · TD-CI: TD-CIS, TD-CIS(D), TD-CISD, . . . TD-CISD ·· N=Full-CI (FCI) TD-CASSCF(N,M): TD-CASSCF (N,N/2) = TD-HF, . . . , TD-CASSCF(N,K) =FCI

EXAMPLE: GROUND STATES FROM TD-CASSCF • Dirac-Frenkel variational principle: C ( t ) , φ n ( t ) • Imaginary-time propagation: TD-CASSCF(6,K) 1D jellium model Molecules: LCAO-MO Li 2 , 6-31G ∗ , N = 6 d=100 a 0 , N = 6, K orbitals -2 -403.5 TD-CASSCF (6,3) CAS (6,3) TD-CASSCF (6,4) -2.1 energy (hartree) CAS (6,4) TD-CASSCF (6,5) CAS (6,5) -404 TD-CASSCF (6,6) CAS (6,6) -2.2 TD-CASSCF (6,7) CAS (6,7) energy (eV) -404.5 -2.3 HF -2.4 -405 -2.5 FCI 0 1 2 3 4 -405.5 imaginary time (fs) 0 0.2 0.4 0.6 0.8 1 time (fs) Convergence to Full-CI M. Nest, T. Klamroth, PS, JCP 122 , 124102 (2005) M. Nest, JTCC 6 , 653 (2007)

EXCITED STATES FROM TD-CASSCF • Excited states by real-time propagation via FT of autocorrelation function via FT of dipole moment � � h � n | ˆ C ∗ n C n e − iE n t/ ¯ h C ∗ n C m e i ( E n − E m ) t/ ¯ � Ψ(0) | Ψ( t ) � = � ˆ µ � ( t ) = µ | m � n n,m 10 1 2, 3 0 4 x-polarized pulse dipole - x z-polarized pulse |FT(< Ψ (t) | Ψ (0)>)| intensity (arb. units) dipole - z 1 1 2, 3 4 0.1 0.01 -9 -8.9 -8.8 -8.7 -8.6 -8.5 -8.4 -8.3 -8.2 0 0.05 0.1 0.15 0.2 0.25 0.3 energy (hartree) excitation energy (hartree) LiH molecule, TD-CASSCF(4,4)/6-31G ∗ M. Nest, R. Padmanaban, PS, JCP 126 , 214106 (2007)

EXCITED STATES FROM TD-CASSCF • Excited states by real-time propagation 1 2, 3 4 dipole - x intensity (arb. units) dipole - z Performance of dipole method (LiH) n = 1 n = 2,3 n = 4 35 35 30 30 Exc. energy = 3.30455 eV Exc. energy = 4.33232 eV Exc. energy = 7.08095 eV 0 0.05 0.1 0.15 0.2 0.25 0.3 25 25 excitation energy (hartree) Error (meV) Error (meV) 20 20 15 15 10 10 5 5 0 0 -5 -5 S ) D ) ) ) S ) D ) ) ) S ) D ) ) ) D 3 4 5 D 3 4 5 D 3 4 5 I S I S I S , , , , , , , , , C C C ( ( ( 4 4 4 4 4 4 4 4 4 S I S I S I C C C ( ( ( ( ( ( ( ( ( I I I C C C M. Nest, R. Padmanaban, PS, JCP 126 , 214106 (2007) • Also: Pulsed laser-driven real-time dynamics F. Remacle, M. Nest, R.D. Levine, PRL 99 , 183902 (2007)

RESPONSE TO LASER PULSES

A SIMPLE EXAMPLE: THE H 2 MOLECULE • The potential curves H H y x z E(t) h , E(t) ν

A SIMPLE EXAMPLE: THE H 2 MOLECULE • TD-CISD (=FCI) treatment: aug-cc-pV5Z; | 0 � → | 1 � laser excitation sin 2 π pulses E z ( t ) = E 0 sin 2 ( πt/ 2 σ ) cos( ω 10 t ) with FWHM σ “long pulse”: σ = 1000 ¯ h/E h “short pulse”: σ = 50 ¯ h/E h µ z (t) µ z ( t ) 1.5 1 1 0.5 0 . 5 0 -0.5 0 -1 -1.5 − 0 . 5 − 1 0.1 0.05 0 500 1000 1500 2000 2500 0 . 004 0 E z (t) 0 -0.05 0 . 002 t 500 -0.1 0 1000 E z ( t ) 1500 2000 − 0 . 002 t 2500 − 0 . 004 1 . 0 1 . 0 Population P i Population P i 0 . 5 0 . 5 0 . 0 0 . 0 1 . 86 1 . 395 0 . 93 0 . 465 0 . 0 1 . 86 1 . 395 0 . 93 0 . 465 0 . 0 E [ E h ] E [ E h ] single-photon, state-to-state multi-photon, wavepacket

LINEAR RESPONSE: POLARIZABILITY OF H 2 • Strategy: Apply E q = E 0 q sin 2 ( πt/ 2 σ ) cos( ωt ) ⇒ µ ind = = α qq ′ E q ′ q • Dynamic: ω � = 0 Kennlinien for H 2 • Static: ω = 0 TD-CISD a Stat. QC b Exp. α � 6.3989 6.303 6.3970 α ⊥ 4.5845 4.913 4.5749 a aug-cc-pVQZ; b FCI/aug-cc-pVQZ µ 2 z, 0 n ω n 0 � SOS: α zz = 2 ω 2 n 0 − ω 2 n � =0

NONLINEAR RESPONSE: HIGHER HARMONICS E ( t ) , µ ind ( t ) − → µ ind ( ω ) , E ( ω ) → FT − • H 2 : Higher harmonics 1HG: polarizability α zz ( − ω, ω ) 3HG: 2nd hyperpolariz. γ zzzz ( − 3 ω, ω, ω, ω ) 5HG: 4th hyperpolarizability . . . crossed fields: elements, e.g. β xyz only odd P. Krause, T. Klamroth, PS, JCP 127 , 034107 (2007)

NONLINEAR RESPONSE: HIGHER HARMONICS • H 2 HHG: The role of diffuse functions HHG cutoff region requires diffuse functions E. Luppi, M. Head-Gordon, Mol. Phys. 110 , 909 (2012)

INCLUSION OF IONIZATION • Ionization in TD-CI E n → E n − i 2Γ n • Polarizability H 2 , bound → bound/unbound transitions TD-CIS/cc-pVTZ σ = 2000 ¯ h/E h S. Klinkusch, PS, T. Klamroth, JCP 131 , 114304 (2009)

INCLUSION OF DISSIPATION: ρ -TDCI • Liouville-von Neumann equation for laser-driven electrons perturbation � ∂ ˆ � ∂ ˆ ∂t = − i ρ ρ h [ ˆ H el − ˆ µE ( t ) , ˆ ρ ] + ¯ ∂t energy relaxation D � �� � � �� � < system coupling, H sb dissipation < system, H s < bath, H dephasing b • Lindblad dissipation, CI eigenstate basis: “ ρ -TDCI” Populations: Diagonal elements of system density operator ˆ ρ n N dρ nn [ − i V Γ � = h [ V np ( t ) ρ pn − ρ np V pn ( t )] + (Γ p → n ρ pp − Γ n → p ρ nn )] n−>m mn dt ¯ p m dipole coupling V mn ( t ) = − µ mn E ( t ) energy relaxation rates Γ n → m dephasing enters ˙ ρ mn via dephasing rates γ mn

INCLUSION OF IONIZATION AND DISSIPATION • The ρ -TD-CI method, and inclusion of ionization ∂ ˆ ∂t = − i ρ �� � � LvN equation H el − i ˆ ˆ W − ˆ µE ( t ) , ˆ ρ + L D ˆ ρ h ¯ • Excitation of H 2 , bound → bound transition 1 |0> | 0 � → | 1 � → | 2 � → | 5 � |1> |2> 0.8 |5> σ 1 , σ 2 , σ 3 = 500 ¯ h/E h |9> Norm TD-CIS(D)/aug-ccpVQZ Population 0.6 0.4 0.9 |11> |12> 0.2 0.8 Free |10> with dissipation 0 0.7 |9> |7> |6> |8> 0.8 0.6 |5> µ 15,z |3> |4> 0.6 0.5 Population |2> E (E h ) |1> 0.4 0.4 µ 01,z 0.3 0.2 with ionization All µ 03,x µ 04,y 0.2 0 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 50 0.1 Time (fs) Time (fs) 0 |0> J.C. Tremblay, S. Klinkusch, T. Klamroth, PS, JCP 134 , 044311 (2011)

TIME-DEPENDENT ELECTRON CORRELATION

TIME-DEPENDENT CORRELATION • Time-dependent correlation energy sin 2 pulse, 3fs, E 0 = 0 . 01, ω = 0 . 15 LiH, TD-CASSCF(4,n)/6-311++G(2df,2p) -7.96 TD-HF (4,2) energy (E h ) -7.98 TD-CASSCF (4,4) Li H + 0.0185 -8 -8.02 E corr ( t ) = E ( t ) − E HF ( t ) -0.015 correlation energy correlation energy (E h ) + 0.0067 -0.02 -0.025 0 1 2 3 4 5 6 time (fs) M. Nest, PS, unpublished

TIME-DEPENDENT CORRELATION • Time-dependent correlation energy sin 2 pulse, 3fs, E 0 = 0 . 025, ω = 0 . 15 LiH, TD-CASSCF(4,n)/6-311++G(2df,2p) Li H E corr ( t ) > 0 !! Nest, PS, unpublished

ELECTRON CORRELATION: OTHER MEASURES • One-electron entropy S and “quantum impurity” C � � � γ 2 � C = 1 − 1 S = − k B Tr γ ln γ N Tr � d 1 d 1 ′ χ ∗ i (1) γ (1 , 1 ′ ) χ j (1 ′ ) γ ij = 1-density matrix (HF orbital basis) • H 2 minimal basis, dynamics of a Hartree-Fock state 1 � , | ψ 2¯ g states | 0 � , | 1 � from determinants ψ HF = | 1¯ 1 � = | 2¯ • Full-CI 1 Σ + 2 2 � 1¯ | 0 � = cos( β/ 2) | 1¯ 1 � + sin( β/ 2) | 2¯ 2 � energy E 0 | 1 � = − sin( β/ 2) | 1¯ 1 � + cos( β/ 2) | 2¯ 2 � energy E 1 • Dynamics of an initial Hartree-Fock state ψ (0) = ψ HF = cos( β/ 2) | 0 � − sin( β/ 2) | 1 � h � � ψ ( t ) = e − iE 1 t/ ¯ cos( β/ 2) e iω 10 t | 0 � − sin( β/ 2) | 1 � ω 10 = ( E 1 − E 0 ) / ¯ h