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

nuclear (atomic) motions femtosecond chemistry: 1 fs = 10−15s

Zewail et al., 1990’s

REAL-TIME DYNAMICS: FEMTOCHEMISTRY

electronic motions attosecond physics: 1 as = 10−18s

Corkum, Krausz, . . . , > 2000

REAL-TIME DYNAMICS: ATTOPHYSICS

• Methods

– Wave-function based: MCTDH – Open-system density matrix based: Lindblad approach

• Application

– Vibrational dynamics and relaxation ➋ Nuclear dynamics (mostly for system-bath problems)

• 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 ➊ Electron dynamics (mostly light-driven) THIS TALK IS ABOUT . . .

LASER-DRIVEN ELECTRON DYNAMICS

Corkum et al., Nature 432, 867 (2004) HOMO of N2

• HHG, orbital tomography

Kling et al., Science 312, 264 (2006) dissociation of D+

2

• Electronic wavepackets (and control)

ELECTRON MOTION IN MOLECULES: LASERS

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

........ 1 2 K AS=(4,5) N/2

a r

• One-electron approaches
• Single-determinant methods

– TD-HF: Ψ(t) = ψ0(t) – TD-DFT: Ψ(t) = ψKS (t)

• Multi-determinant methods

– TD-CI: Ψ(t) = C0(t)ψ0 +

ar Cr a(t)ψr a + ab,rs Crs ab(t)ψrs ab + · · ·

– TD-CASSCF: Ψ(t) = C0(t)ψ0(t) +

ar Cr a(t)ψr a(t) + ab,rs Crs ab(t)ψrs ab(t) + · · ·

• Solution techniques

i¯ h∂Ψ(x1, . . . , xN, t) ∂t =

• ˆ

Hel(x1, . . . , xN) − ˆ µE(t)

• Ψ(x1, . . . , xN, t)
• The N-electron time-dependent Schr¨
• dinger equation

LASERS AND ELECTRON DYNAMICS: METHODS

• M. Nest, JTCC 6, 653 (2007)
• M. Nest, T. Klamroth, PS, JCP 122, 124102 (2005)

0.2 0.4 0.6 0.8 1

time (fs)

• 405.5
• 405
• 404.5
• 404
• 403.5

energy (eV)

CAS (6,3) CAS (6,4) CAS (6,5) CAS (6,6) CAS (6,7)

FCI HF

Molecules: LCAO-MO

Li2, 6-31G∗, N = 6

Convergence to Full-CI

1 2 3 4

imaginary time (fs)

• 2.5
• 2.4
• 2.3
• 2.2
• 2.1
• 2

energy (hartree)

TD-CASSCF (6,3) TD-CASSCF (6,4) TD-CASSCF (6,5) TD-CASSCF (6,6) TD-CASSCF (6,7)

1D jellium model

d=100 a0, N = 6, K orbitals

• Imaginary-time propagation: TD-CASSCF(6,K)
• Dirac-Frenkel variational principle: C(t), φn(t)

EXAMPLE: GROUND STATES FROM TD-CASSCF

• M. Nest, R. Padmanaban, PS, JCP 126, 214106 (2007)

LiH molecule, TD-CASSCF(4,4)/6-31G∗

0.05 0.1 0.15 0.2 0.25 0.3 excitation energy (hartree) intensity (arb. units)

1 2, 3 4 dipole - x dipole - z

via FT of dipole moment ˆ µ(t) =

• n,m

C∗

nCm ei(En−Em)t/¯ h n|ˆ

µ|m

• 9
• 8.9
• 8.8
• 8.7
• 8.6
• 8.5
• 8.4
• 8.3 -8.2

energy (hartree) 0.01 0.1 1 10 |FT(< Ψ(t) | Ψ(0)>)|

x-polarized pulse z-polarized pulse 1 2, 3 4

via FT of autocorrelation function Ψ(0)|Ψ(t) =

• n

C∗

nCn e−iEnt/¯ h

• Excited states by real-time propagation

EXCITED STATES FROM TD-CASSCF

• F. Remacle, M. Nest, R.D. Levine, PRL 99, 183902 (2007)
• Also: Pulsed laser-driven real-time dynamics
• M. Nest, R. Padmanaban, PS, JCP 126, 214106 (2007)

0.05 0.1 0.15 0.2 0.25 0.3 excitation energy (hartree) intensity (arb. units)

1 2, 3 4 dipole - x dipole - z

Performance of dipole method (LiH)

C I S C I S ( D ) C I S D ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 )

• 5

5 10 15 20 25 30 35 Error (meV)

C I S C I S ( D ) C I S D ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) C I S C I S ( D ) C I S D ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 )

• 5

5 10 15 20 25 30 35 Error (meV)

• Exc. energy = 3.30455 eV
• Exc. energy = 4.33232 eV
• Exc. energy = 7.08095 eV

n = 1 n = 2,3 n = 4

• Excited states by real-time propagation

EXCITED STATES FROM TD-CASSCF

RESPONSE TO LASER PULSES

h , E(t) ν

z x y

H H

E(t)

• The potential curves

A SIMPLE EXAMPLE: THE H2 MOLECULE

“long pulse”: σ = 1000 ¯ h/Eh “short pulse”: σ = 50 ¯ h/Eh

µz(t) t Ez(t)

1 0.5 −0.5 −1 0.002 0.004 −0.002 −0.004 2500 500 1000 1500 2000 0 500 1000 1500 2000 2500 t

• 0.1
• 0.05

0.05 0.1 Ez(t)

• 1.5
• 1
• 0.5

0.5 1 1.5 µz(t)

0.0 0.5 1.0 Population Pi E [Eh] 0.465 0.93 1.395 1.86 0.0 0.0 0.5 1.0 Population Pi E [Eh] 0.465 0.93 1.395 1.86 0.0

single-photon, state-to-state multi-photon, wavepacket sin2 π pulses Ez(t) = E0 sin2 (πt/2σ) cos(ω10t) with FWHM σ

• TD-CISD (=FCI) treatment: aug-cc-pV5Z; |0 → |1 laser excitation

A SIMPLE EXAMPLE: THE H2 MOLECULE

SOS: αzz = 2

• n=0

µ2

z,0n ωn0

ω2

n0 − ω2

• Dynamic: ω = 0

TD-CISDa Exp.

• Stat. QCb

α 6.3989 6.303 6.3970 α⊥ 4.5845 4.913 4.5749

a aug-cc-pVQZ; b FCI/aug-cc-pVQZ

• Static: ω = 0

Kennlinien for H2

Apply Eq = E0q sin2(πt/2σ) cos(ωt) = ⇒ µind

q

= αqq′Eq′

• Strategy:

LINEAR RESPONSE: POLARIZABILITY OF H2

• P. Krause, T. Klamroth, PS, JCP 127, 034107 (2007)

crossed fields: elements, e.g. βxyz

1HG: polarizability αzz(−ω, ω) 3HG: 2nd hyperpolariz. γzzzz(−3ω, ω, ω, ω) 5HG: 4th hyperpolarizability . . .

• nly odd
• H2: Higher harmonics

E(t), µind(t) − → FT − → µind(ω), E(ω) NONLINEAR RESPONSE: HIGHER HARMONICS

• E. Luppi, M. Head-Gordon, Mol. Phys. 110, 909 (2012)

HHG cutoff region requires diffuse functions

• H2 HHG: The role of diffuse functions

NONLINEAR RESPONSE: HIGHER HARMONICS

• S. Klinkusch, PS, T. Klamroth, JCP 131, 114304 (2009)

TD-CIS/cc-pVTZ σ = 2000 ¯ h/Eh

• Polarizability H2, bound → bound/unbound transitions

En → En − i 2Γn

• Ionization in TD-CI

INCLUSION OF IONIZATION

Γ V

mn n−>m n m dρnn dt =

N

• p

[− i ¯ h [Vnp(t)ρpn − ρnpVpn(t)] + (Γp→nρpp − Γn→pρnn)] dipole coupling Vmn(t) = −µmnE(t) energy relaxation rates Γn→m dephasing enters ˙ ρmn via dephasing rates γmn Populations: Diagonal elements of system density operator ˆ ρ

• Lindblad dissipation, CI eigenstate basis: “ρ-TDCI”

∂ ˆ ρ ∂t = −i ¯ h[ ˆ Hel − ˆ µE(t), ˆ ρ]

• system

+ ∂ ˆ ρ ∂t

• D

dissipation

system, H bath, H

sb

coupling, H energy relaxation dephasing

s b

perturbation

< < <

• Liouville-von Neumann equation for laser-driven electrons

INCLUSION OF DISSIPATION: ρ-TDCI

J.C. Tremblay, S. Klinkusch, T. Klamroth, PS, JCP 134, 044311 (2011)

|0 → |1 → |2 → |5 σ1, σ2, σ3 = 500 ¯ h/Eh TD-CIS(D)/aug-ccpVQZ

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 E (Eh) µ15,z µ01,z µ03,x µ04,y |0> |1> |2> |3> |4> |5> |6> |7> |8> |9> |10> |11> |12>

0.2 0.4 0.6 0.8 1 Population Free with dissipation |0> |1> |2> |5> |9> Norm 0.2 0.4 0.6 0.8 5 10 15 20 25 30 35 40 45 Population Time (fs) with ionization 5 10 15 20 25 30 35 40 45 50 Time (fs) All

• Excitation of H2, bound → bound transition

∂ ˆ ρ ∂t = −i ¯ h

• ˆ

Hel − i ˆ W

• − ˆ

µE(t), ˆ ρ

• + LDˆ

ρ LvN equation

• The ρ-TD-CI method, and inclusion of ionization

INCLUSION OF IONIZATION AND DISSIPATION

TIME-DEPENDENT ELECTRON CORRELATION

• 8.02
• 8
• 7.98
• 7.96

energy (Eh) 1 2 3 4 5 6 time (fs)

• 0.025
• 0.02
• 0.015

correlation energy (Eh) TD-HF (4,2) TD-CASSCF (4,4) correlation energy + 0.0185 + 0.0067

• M. Nest, PS, unpublished

Ecorr(t) = E(t) − EHF(t)

Li H sin2 pulse, 3fs, E0 = 0.01, ω = 0.15 LiH, TD-CASSCF(4,n)/6-311++G(2df,2p)

• Time-dependent correlation energy

TIME-DEPENDENT CORRELATION

Nest, PS, unpublished

Ecorr(t) > 0 !!

Li H sin2 pulse, 3fs, E0 = 0.025, ω = 0.15 LiH, TD-CASSCF(4,n)/6-311++G(2df,2p)

• Time-dependent correlation energy

TIME-DEPENDENT CORRELATION

• Full-CI 1Σ+

g states |0, |1 from determinants ψHF = |1¯

1, |ψ2¯

2 1¯ 1 = |2¯

2 |0 = cos(β/2) |1¯ 1 + sin(β/2) |2¯ 2 energy E0 |1 = − sin(β/2) |1¯ 1 + cos(β/2) |2¯ 2 energy E1

• Dynamics of an initial Hartree-Fock state

ψ(0) = ψHF = cos(β/2)|0 − sin(β/2)|1 ψ(t) = e−iE1t/¯

h

cos(β/2)eiω10t|0 − sin(β/2)|1

• ω10 = (E1 − E0)/¯

h

• H2 minimal basis, dynamics of a Hartree-Fock state

γij =

• d1 d1′ χ∗

i(1) γ(1, 1′) χj(1′)

1-density matrix (HF orbital basis)

C = 1 − 1

N Tr

• γ2

S = −kB Tr

• γlnγ
• One-electron entropy S and “quantum impurity” C

ELECTRON CORRELATION: OTHER MEASURES

• scillation with period

T =

2π¯ h E1−E2

ultrafast buildup of electron correlation

• Example: TD-CID/STO-3G, R=1.4 a0

k1 = cos4(β/2) + sin4(β/2) k2 = 2 sin2(β/2) cos2(β/2) b(t) = k2 cos(2πt/T)

C(t) = 1 −

• (k1 − b(t))2 + (k2 + b(t))2

S/kB = −2

• (k1 − b(t))ln(k1 − b(t)) + (k2 + b(t))ln(k2 + b(t))
• H2, minimal basis: Dynamics of a HF state

CORRELATION-DRIVEN ELECTRON DYNAMICS

ultrafast buildup of electron correlation

• ne-electron entropy S

S = −kB Tr

• γlnγ
• quantum impurity

C = 1 − 1

2 Tr

• γ2

(=“correlation”)

10 20 30 40

t [h/(2πEh)]

0.0 0.1 0.2 0.3 0.4 0.5

entropy S [kB]

0.00 0.04 0.08 0.12 0.16 0.20

quantum impurity QI

Klinkusch, Klamroth, PS, unpublished

TD-CISD/6-31G*: no field, ψ(0) = Hartree-Fock ground state

• H2 molecule: More than two states

CORRELATION-DRIVEN ELECTRON DYNAMICS

creation of HF state? attosecond dynamics

0.5 1 1.5 2

propagation time [fs]

0.5 1 1.5 2 2.5

entropy [a.u.]

N2 CO NO

+

CN

• HCCH

0.05 0.1 0.15 0.2 0.5 1 1.5 2 2.5

N2 CO NO

+

CN

• HCCH

Beyvers, Nest, Klamroth, Klinkusch, PS, unpublished

small molecules, TD-CIS/6-31G*: no field, ψ(0) = Hartree-Fock ground state

• Correlation-driven electron dynamics: Other molecules

CORRELATION-DRIVEN ELECTRON DYNAMICS

how to stabilize the low-entropy state? partial success

• 0.01

0.00 0.01 Ex [Eh/(ea0)]

(a)

• 0.01

0.00 0.01 Ey [Eh/(ea0)]

(b)

0.965 0.970 0.975 0.980 0.985 P

HF

(c)

200 400 600 800 time t [h/(2πEh)] 0.15 0.20 0.25 0.30 0.35 0.40 S [kB]

tf

(d)

10 20 30 40 50

OCT iterations η

0.981 0.982 0.983 0.984 0.985 0.986 0.987 0.988 0.989 0.990

HF state population

<O

^ >

(e) Klamroth, Klinkusch, PS, unpublished

H2, TD-CISD/cc-pVQZ with field, ψ(0) = CISD ground state (PHF = 0.982, S=0.23 kB)

• Application of Optimal Control Theory

ATTEMPTS TO BUILD A HF STATE

E(t) = − 1 ¯ hα ImΦ(t)|ˆ µ|Ψ(t)

• Calculate field to self-consistency

i¯ h ∂ ∂t|Φ(t) =

• ˆ

Hel − ˆ µE(t)

• |Φ(t)

backward from t = tf ,|Φ(tf) = ˆ O|Ψ(tf)

• Lagrange function Φ(t): Backward propagation

J = Ψ(tf)| ˆ O|Ψ(tf) − α tf |E(t)|2dt − tf dtΦ(t)| ∂ ∂t + i ¯ h ˆ Hel(t)Ψ(t) + c.c. ˆ O = target operator; α = penalty

• Maximize constrained target functional:

i¯ h ∂ ∂t|Ψ = ˆ Hel(t)|Ψ forward from t = 0,|Ψ(0) = |Ψ0 ˆ Hel(t) = ˆ Hel − ˆ µE(t)

• Time-dependent Schr¨
• dinger equation:

OPTIMAL CONTROL THEORY

• M. Nest, M. Ludwig, I. Ulusoy, T. Klamroth, PS,

JCP 138, 164108 (2013)

complicated, indirect, ionizing

make approximate HF state ψHF ∼

n=0,5,...,25 Cnψn

from correlated ground state ψ(0) = ψ0

• Control Strategy

T = 40 as, “breathing” electrons

TD-CISD/QZ basis, ψ(0) = ψHF

0.00 0.01 0.02 0.03 0.04 0.05

correlation C(t)

1.40 1.42 1.44 1.46 1.48

<r12> (a0) 50 100 150

time (as)

0.000 0.004 0.008 0.012

HF populations

2s 2p 3s

• He: HF state dynamics

CONTROLLING CORRELATION IN ATOMS

3-pulse strategy (15 fs) works

make approximate HF state ψHF ∼

n=0,13,25 Cnψn

from correlated ground state ψ(0) = ψ0

• Optimal control for Mg atom
• M. Nest, M. Ludwig, I. Ulusoy, T. Klamroth, PS, JCP 138, 164108 (2013)

(ns2) → long T

TD-CISD, ψ(0) = ψHF

• Other atoms: Be, Mg

CONTROLLING CORRELATION IN ATOMS

• Test of apprpximate methods, e.g. TD-DFT
• Treatment of ionization, nuclear motion
• Time-dependent Coupled Cluster
• Outlook
• Ultrafast dynamics (and control)
• WF-based alternatives to TDDFT

– systematically improvable – correct asymptotics – multi-determinant effects

• Findings
• Electron dynamics in real time
• TD-CI, ρ-TD-CI, TD-CASSCF
• Response
• Time-dependent correlation
• Summary

SUMMARY AND OUTLOOK: ELECTRONS

NUCLEAR (ATOM) DYNAMICS

standard, MCTDH (“exact”), TDSCF (approximation), . . .

• Methods

∂Ψ(s, q1, . . . , qM, t) ∂t = −i ¯ h ˆ HΨ(s, q1, . . . , qM, t)

• The time-dependent Schr¨
• dinger equation

ˆ H =

• ˆ

Hs(s) − ˆ µE(t)

• system

+ ˆ Hsb(s, q1, . . . , qM)

• system-bath

+ ˆ Hb(q1, . . . , qM)

• bath

direct

Bath System

Phase Energy

E(t)

indirect

• The system-bath Hamiltonian

FULL SYSTEM-BATH DYNAMICS

• Rates Γ, γ: Perturbation theory, non-perturbative

dρmn dt = − i ¯ h  (Em − En) +

N

• p

[Vmp(t)ρpn − ρmpVpn(t)]   −γmn ρmn

• dephasing

Coherences: dρnn dt =

N

• p

− i ¯ h [Vnp(t)ρpn − ρnpVpn(t)]

• system-field

+

N

• p

[Γp→nρpp − Γn→pρnn]

• dissipation

Populations:

• Lindblad in system eigenstate representation: ˆ

Ckl =

• Γk→l|lk|

∂ ˆ ρs ∂t = −i ¯ h[ ˆ Hs − ˆ µE(t), ˆ ρs]

• system

+ ∂ ˆ ρs ∂t

• D
• system-bath

direct

Bath System

Phase Energy

E(t)

indirect

• Open-system density matrix theory

REDUCED DYNAMICS

1 force field: D. Brenner, PRB 42, 9458 (1990); NMA: I. Andrianov, PS, JCP 124, 034710 (2006)

500 1000 1500 2000 2500

ω (cm

• 1)

ρ (arb. units)

Si-H stretch, 2037 Si-H bend, 637

normal mode analysis (Nat=180, FF1)

• Vibrational state density

The bath states (nr, nφ) The system

• The model

ˆ H = ˆ T + V (r, φ)

• ˆ

Hs

+

M

• i=1

λi(r, φ) qi

• 1-phonon

+1 2

M

• i,j=1

Λij(r, φ) qiqj

• 2-phonon
• ˆ

Hsb

+

M

• i=1

ˆ p2

i

2mi + miω2

i

2 q2

i

• ˆ

Hb

• A “system-bath” model for H on Si(100)

H:Si(100): VIBRATIONAL RELAXATION

matrix theory Lindblad density

• Decay mechanism
• stretch mode: τvib = Γ−1

1→0 = ns

• bending mode: ps
• Γn→m ≈ τvib−1 n δm,n−1: ∆n = −1

ideal: HO, bilinear coupling

• Lifetimes (T=0)

M=534

H:Si(100): GOLDEN RULE AND RDM THEORY

CPL 433, 91 (2006), JPC C 111, 5432 (2007)

• Half-life times T1/2 of (0,1): Golden Rule: 0.94 ps, TDSCF: 0.92 ps

TDSCF (M=534) MCTDH (M=50 oscillators)

• Relaxation of the bending mode: MCTDH and TDSCF
• Solve i¯

h∂Ψ

∂t = ˆ

HΨ by MCTDH or TDSCF for F=M+2 DOF H:Si(100): NON-PERTURBATIVE, FULL DYNAMICS

G.K. Paramonov, PS et al., PRB 75, 045405 (2007)

• F. L¨

uder, M. Nest, PS, TCA 127, 183 (2010)

τvib goes up with T

MCTDH (M=20) Random Phase Thermal Wavefunction Method

• Full: MCTDH treatment

Si-H bending mode, ω0 = ωφ, tf = 1 ps Vmn(t) = −µmn E0 sin2 πt tf

• cos(ω0t)

mode-selective, not state selective

• IR excitation by π-pulses: RDM

TEMPERATURE: REDUCED & FULL DYNAMICS

3 Lorenz, PS, JCP 140, 044106 (2014) 2 Manthe, Huarte-Larranaga, CPL 127, 349 (2001) 1 Nest, Kosloff, JCP 127, 134711 (2007) 0.05 0.1 0.15 0.2 0.25 100 200 300 400 500 Pstick T (K)) rAvec rAvec (+1 SPF) rAvec (+2 SPF) rAvec (+3 SPF) rSPF rSPF (+1 SPF) rSPF (+2 SPF)

and 20 bath oscillators Morse oscillator

• Example: Atom sticking at surface3

randomize single-particle functions ψ(i)(xi) =

ni(−1)αni ϕ(i) ni (α= random integer)

Ψ(x1, . . . , xF) = ψ(1)(x1) · · · ψ(F)(xF)

• “rSPF” method2:

randomize coefficients Aj1···jF (replace by random phases eiθ ( θ ∈ [0, 2π]) Ψ(x1, . . . , xF) =

n1

• j1=1

· · ·

nF

• jF=1

Aj1···jF

F

• k=1

φ(k)

jk (xk)

• “rAvec” method1:

THERMAL WAVEFUNCTIONS AND MCTDH

Nest, Meyer, JCP 119, 24 (2003); Bouakline et al., JPC A 116, 11118 (2012)

• resonant vs. non-resonant baths: ωb, ωs
• reduced vs. full dynamics
• scaling of “vibrational lifetimes” with v
• Questions

relaxation

V(s) s surface 1 2 M

3 2 1

• Ohmic bath ωi = i ∆ω = iωf/M
• coupling constant ci = i
• 2mimsΓ∆ω3/π

1/2, damping parameter Γ

• non-linear coupling function f(s) = (1 − e−αs)/α −

→ s for s → 0 ˆ H = − ¯ h2 2ms d2 ds2 + D[1 − e−αs]2

• ˆ

Hs

− f(s)

M

• i=1

ciqi

• ˆ

Hsb

+

M

• i=1

ˆ p2

i

2mi + miω2

i

2 q2

i

• ˆ

Hb

• A 1D “system-bath” model vibrational relaxation

A SIMPLE 1D SYSTEM-BATH MODEL

complete vs. incomplete Rabi oscillations resonant: single bath oscillators dominate dynamics

Bouakline, L¨ uder, Martinazzo, PS, J. Phys. Chem. A 116, 11118 (2012)

non-resonant bath

0.5 ρ11 ρ00 trace (ρ

2)

entropy 0.5 1

populations

full bath 7 oscillators 2 oscillators

200 400 600 800 1000 time (fs)

0.5 1

populations

system bath mode 24 bath mode 25

(a) (b) (c)

v = 1 v = 0

resonant bath

0.5

populations

full bath 3 oscillators 1 oscillator 0.5

populations

system bath mode 20 bath mode 21 bath mode 22

200 400 600 800 1000 time (fs)

0.5

populations

c21=9.70d-3 au c21=4.85d-3 au c21=2.42d-3 au

v = 1

(a) (b) (c)

v = 0 v = 1

MCTDH (full) calculation, M=40, Γ = (500 fs)−1

• Resonant vs. non-resonant bath

RESULTS

Bouakline, L¨ uder, Martinazzo, PS, J. Phys. Chem. A 116, 11118 (2012)

HO scaling, bilinear: τ(v → v − 1) = τ(1 → 0)/v Golden Rule:

Γi→f = 2∆ω ¯ h |i|f(s)|f|2 msΓ

M

• b=1

ωb δ(ωb − ωi,f)

agreement full and reduced non-monotonic scaling in real system

100 200 300

half-life time (fs)

MCTDH scaling law coupling

5 10 15 20

initial state v0

100 200 300

half-life time (fs)

MCTDH Golden Rule

(a) (b)

MCTDH vs. Golden Rule; M=40, Γ = (500 fs)−1; HO system vs. Morse oscillator

• “Half-lifetime” scaling, “full” vs. “reduced” dynamics

RESULTS

4 Zhang et al., JCP 122, 091101 (2005) 3 Burghardt et al., JCP 111, 1927 (1999) 2 Martinazzo et al., JCP 125, 194102 (2006) 1 Meyer, Manthe, Cederbaum: CPL 165, 73 (1990)

• G-MCTDH3, CC-TDSCF4, . . .

|Ψ =

• α

Cα |ξα

• DVR states subsystem

|Φα

• bath

Local Coherent State Approximation “diagonal approximation to MCTDH”

• LCSA2

Ψ(x1, . . . , xF, t) =

F

• k=1

ϕκ(xk, t) single-configuration approximation

• TDSCF

Variants: Mode combination, ML-MCTDH (Thoss, Wang), . . . Ψ(x1, . . . , xF) =

n1

• j1=1

· · ·

nF

• jF=1

Aj1···jF

F

• k=1

φ(k)

jk (xk)

• MCTDH1 and variants thereof

LARGE BATHS WITH WAVEFUNCTIONS

Martinazzo et al., JCP 125, 194102 (2006)

recurrence time τ = 2π/∆ω

Decay of v = 1, Γ = (500 fs)−1, ∆ω = ωc/M

• ML-MCTDH

Scaling behaviour

MCTDH and LCSA, 1D+M=50, Γ = (50 fs)−1

• LCSA

LARGE BATHS, LONG-TIME DYNAMICS

• Redfield and non-Markovian theories
• Non-Markovian measures
• Light-induced processes
• Outlook
• “Easy” and “real” Hamiltonians
• Anharmonicity matters
• Findings
• System-bath models
• MCTDH and variants
• Lindblad open-system density matrix
• Vibrational relaxation
• Summary

SUMMARY AND OUTLOOK: NUCLEI

CORRELATION MATTERS SUMMARY AND OUTLOOK

• Deutsche Forschungsgemeinschaft

SFB 450, SFB 658, SPP 1145, UniCat, Sa 547/7-11

• FCI
• BMBF
• AvH
• . . . the sponsors:
• . . . the group:

THANKS TO . . .