methods of quantum dynamics and simulation of pump-probe ... · methods of quantum dynamics and...

1

Christoph Meier

Laboratoire Collisions, Agrégats, RéactivitéUniversité Paul Sabatier,

Toulouse

Cargese, August 2008

Methods of quantum dynamics and simulation of pump-probe spectra

2

I. Introduction – Laser / molecule interaction-- electron / nuclei separation-- coupled channel equations-- simulation of pump probe spectra-- numerical implementation

II. Methods of wave packet propagation-- standard methods: SOD, Crank-Nicholson, FFT-SO-- methods for high dimensional problems: TD-SCF, MCTDH

III. Outlook and further developments-- 6D quantum wp: vibrational predissociation of 4 atom complex-- Local control and within a mixed qu/classical propagation scheme

Methods of quantum dynamics and simulation of pump-probe spectra

3

Context: Femtosecond laser interaction with atomic and molecular systems

experiment-----theory

classical MD

wave packet propagation

pulse shaped IRexcitation

dynamics subject to environments

dynamics of excited states

IVR, electronic / vibrational relaxation, predissociation...

I. IntroductionI. Introduction

4I. Introduction

We want to have access at:

• vibration, geometrical rearrangement• breaking and formation of bonds• isomerisation• IVR• transitions between electronic states:IC, ISC

• vibrational, electronic relaxation

timescales: femtosecond: 10-15 spicosecond: 10-12 s

Aim: observation of elementary processes of chemical reactivity at the atomic scale in real time

Principle of pump-probe experiments:

Two time-delayed ultrashort laser pulses

Pump pulse: triggers molecular dynamicsProbe pulse: probes quantum state

at different delay times

Femtosecond spectroscopy in molecular systems

5I. Introduction

• description of the system

• interaction with (complex-shaped) laser pulses

• time dependent approach

• quantum wave packet dynamics

• weak fields: time dependent perturbation theory

• classical dynamics

Numerical simulatioins:

experiment

Electronic structure

calculations

Model interactions, SE

Force fields

6

• dipole interaction Hamiltonian Eμ ⋅−= )(sHH ∑=α

ααrrrμ qen ),(

• electronic/nuclear separation:

• electronic structure calculation

∑=Ψi

neinien tt );(),(),,( rrrrr ϕχ

);()();()(neininei

e VH rrrrr ϕϕ =

);();( nejneiij rrμrrμ ϕϕ=

( )Ψ⋅−=Ψ )()( tHi s Eμ&

∑∑∑ −=i

neinii

neinis

ineini tttHti );(),()();(),();(),( )( rrrμErrrrrr ϕχϕχϕχ&

)(22

12

2

2)( ),(2 e

nn

mence em

e

nm

s HVHnn

n +∇−=++= ∑∑∑ rrPPalso:

NT

∑∑∑ −+∇−=i

niijnjnji

neininnejn

mnj tttVttin

),()(),()();(),();(),( 22

1 rμErrrrrrrr χχϕχϕχ&

I. Introduction – Laser / molecule interaction

),(2

2

2)( 2

ence em

e

nm

s VHn

n rrPP ++= ∑∑• system Hamiltonian:

7I. Introduction – Laser / molecule interaction

∑∑∑ −+∇−=i

niijninji

neininnejn

mnj tttVttin

),()(),()();(),();(),( 22

1 rμErrrrrrrr χχϕχϕχ&

),();();(),();();( 122

1 tt ninneinnejmninmneinej nnrrrrrrrrrr χϕϕχϕϕ ∇∇+∇

),();();( 22

1 tnineinnejmnrrrrr χϕϕ ∇+

ijδ

Properties of :);();( neinnej rrrr ϕϕ ∇

0);();();();();();( =∇+∇=∇ neinnejneinejnneinejn rrrrrrrrrrrr ϕϕϕϕϕϕ

0);();();();( =∇+∇= neinnejnejnnei rrrrrrrr ϕϕϕϕ

);();();();( nejnneineinnej rrrrrrrr ϕϕϕϕ ∇−=∇⇒

0);();( =∇ neinnei rrrr ϕϕand specifically :

( ) ( ) ∑∑ −∑ ∇∇++=≠ i

niijji

ninneinnejmnjnjnnj ttttVTtin n

),()(),();();(),()(),( 1 rμErrrrrrrr χχϕϕχχ&

)(naijV non-adiabatic couplingsjH

8

( ) ( )∑≠

−+−=∂∂

jiniij

naijnjjjjnj ttVttHt

ti ),()(),()(ˆ),( )( rμErμEr χχχ

Vibrational excitation, IR

Electronic excitationvis-UV

Nuclear dynamics in excited states

non-adiabatic couplings

Non-radiativetransition

Nuclear dynamics ion the electronic ground state

reminder:

nuclear wave

function on el. surface

=),( tnj rχ)( njV r

initial wave function

excitation laser

dynamics


Expression valid for high intensities

9

Nuclear dynamics in excited states

Non-radiativetransition

Nuclear dynamics ion the electroonic ground state

• Aim pump probe: choose two laser pulses with variable delay to study the combined electronic / nuclear dynamics

• Aim coherent control: choose (complex–shaped) laser pulses to inducea predifined dynamics (wrt a specific exit channel)


10

ggggggt Hi χχχ Eμ−=∂ ˆ

Nuclear dynamics in electronic ground state

Nuclear dynamics in electronically excited states

)0()( 0

')ˆ(

g

dtHi

g

t

g

Tet χχ∫

=−− Eμ

gegeeet

egegggt

Hi

Hi

χχχ

χχχ

E μ

E μ

−=∂

−=∂

ˆ

ˆpert th..

( ) ')0()( 'ˆ

0

)'(ˆ dteeit gtHi

t

egttHi

ege χχ −−−∫= Eμ

)()( )ˆ( tett gtHi

gg χδχ δEμ−−=+ )()()(

ˆˆ tetitett gtHi

egetHi

ege χδχδχ δδ −− +=+ μE


)()( )(ˆ tett ittHi

ii χδχ δ−=+ “quantum propagator”In both cases:

( ) ( )∑≠

−+−=∂∂

jiniij

naijnjjjjnj ttVttHt

ti ),()(),()(ˆ),( )( rμErμEr χχχ

Vibrational excitation, IR

Electronic excitationvis-UV

non-adiabatic couplings

11

Methods of wave packet propagations have a wide range of applications, well beyond laser pulse interactions with atomic / molecular systems

e.g: • atomic and molecular physics:

[see also: D. J. Tannor, Introduction to Quantum Mechanics: A Time Dependent Perspective (University Science Press, Sausalito, 2006)]

-- elastic / inelastic collisions, reactive scattering -- spectroscopy via correlation functions -- explicit time dependent perturbations, laser interactions,

coherent control

• Optics: Maxwell’s equation • ....

II. Methods of wave packet propagation

12

Example: propagation of electromagnetic waves in nanostructures

EBHD×∇−=

×∇=

cc

&

&

ψψ Hi =&⎟⎠

⎞⎜⎝

⎛×∇−

×∇= 0

0μ

εc

c

ii

H⎟⎠⎞⎜

⎝⎛= HEψ

ED ε=HB μ=

• can be scaled to be hermitian• Can be formulated for dispersive media • Application: light transmission through nanostructured apertures

[A. G. Borisov, S. V. Shabanov, J. Comp. Phys. 209 643 (2005)]


13

)()( )(ˆ tett ittHi

ii χδχ δ−=+

)(tUi

[ ] 1)()( =appi

appi UU †

norm should be conserved

• propagation not exact, discretisation error, should always be converged for →t 0δ

quantum propagator

)()()()()()( )()( tttUtUtttt eeeapp

ieapp

iee χχχχδχδχ ==++


14II. Methods of wave packet propagation

iterative global

- second order differening- (implicit) Cayley- Split-Operator- short iterative Lanczos

- time-dependent self consistent field (TD-SCF)

- Multiconfiguration time dependent Hartree (MCTDH)

- Lanczos- Chebycheff

propagators

Can also be used fore the propagation of density matrices


• derivation:

• method:

• characteristics:

-- storage: -- operations:-- stablility:

( ) )(ˆ1)( ttHitt χχ Δ−=Δ+

( ) )(ˆ1)( ttHitt χχ Δ+=Δ−

)(2)()( ttiHtttt χχχ Δ−Δ−=Δ+

)(ˆ2)()( ttHitttt χχχ Δ−=Δ−−Δ+

)(),( ttt Δ−χχ)(ˆ tHχ

max1

Et <Δ

Second order differencing (SOD):


• derivation:

• method:


-- storage: -- operations:-- norm-conserving-- symetric wrt.

)()(ˆˆ

22 tette HiHi tt

χχΔΔ −=Δ+

ξχχ +=Δ+ )()( ttt

ξχ ),(t)(ˆ tHχ

( ) ( ) )(ˆ1)(ˆ1 22 tHittHi tt χχ ΔΔ −=Δ++

( ) ( ) )(ˆ1ˆ1)( 2

1

2 tHiHitt tt χχ Δ−

Δ −+=Δ+problem: inversion

( )2)()(ˆ)()( tttHtittt χχχχ +Δ+Δ−=−Δ+

( ))(2ˆ2 tHi t χξξ +−= Δ

tt Δ−→Δ

Cayley (Crank-Nicholson) :


• derivation:

• Fourier representation:

• method:


-- storage: -- operations:-- stable, norm-conserving-- symetric wrt

χ

)()()( 22ˆ teeetett VitTiViHti tt

χχχΔΔ −Δ−−Δ− ≈=Δ+

)()(ˆnVTH

nrr += ∂

∂

( ) ( ) ⎟⎠

⎞⎜⎝

⎛=Δ+ ∑∑

−

=

−−

=

⎟⎠⎞⎜

⎝⎛Δ−− ΔΔ

1

0'

)(211

0

21)( )()( 2'2

22

2

N

n

rViiN

N

j

ktiiN

rVi teeeeett nt

NjnjmN

jnn

t

χχ ππh

Vi t

e 2Δ− tTie Δ− Vi t

e 2Δ−1−FFT FFT

)()( 21

2 teeett ViFFTtTiFFTVi tt

χχΔ−Δ −Δ−− ⎯⎯⎯←⎯⎯ ⎯←=Δ+

multiplications, FFT, no !

tt Δ−→Δ[M. D. Feit, J. A. Fleck, A. Steiger, J. Comput. Phys. 47, 412 (1982) ]

FFT-Split Operator (FFT-SO) :

χH


avantages désavantages

SOD • flexible: any representation• H time dependent

• small timestep

Crank-Nicolson(implicit)

• flexible: any representation• H time dependent• unitary

• iteration at every timestep

FFT-SO • FFT: efficient• no matrix-vector multiplication• H time dependent• unitary

• equidistant grids• H : no cross-terms

Overview / comparison:

19

N=416000GbCPU time:10 days


3q

Approximations

• approximate dynamics: TD-SCF, MCTDH• dynamics in reduced dimensionality • symmetry, periodicity• harmonic approximation• classical mechanics (trajectoires)• mixed quantum/classical dynamics

N=316 MbCPU time: 1sec

• N particles, 3N-6 internal DoF, (10 points/basis fcts per DoF)• calculation of spectrum / dynamics • exponential scaling

N~10~1040 GbCPU time: ~1020 years

Quantum wave packet for high-dimensional problems:

20II. Methods of wave packet propagation: TD-SCF

• TD-SCF: suppose wf can be described by a product

• Rem: this decomposition is not unique ! constraints

• Schrödinger equation becomes:

• Schrödinger eq. in 2D is replaced by 2 coupled 1D equations

• Dynamics in one DoF is determined by the dynamic mean field over the other DoF

• even if H is time independent, and are time-dependent !

• can be used for up to ~100 DoF !

• disavantage: approximate, error hard to estimate

),(),()(),,( )()( tytxtatyx yx φφχ =

consider 2 DoF for the presentation of the basic idea, but the usefulness lies in the extension to many DoF

0)()()()( == yyxx φφφφ &&

)()( taHtai =&

( ) ),(),( )()()( txHHtxi xxx φφ −=&

( ) ),(),( )()()( tyHHtyi yyy φφ −=&

)()()()( yxyx HH φφφφ=)()()( yyx HH φφ=)()()( xxy HH φφ=

Other formulations exist

)( xH)( yH

21II. Methods of wave packet propagation: MCTDH

• MCTDH: multi-configurationalexpansion of the wavefunction

Rem: this is not unique !constraints

• Schrödinger equation becomes: -- coupled equations for the -- for and , become complete:

MCDTH becomes an exact standard method become time independent

• flexible, any representation is possible, different representations for different DoF possible

• gain in storage requirements: for Nb and Mb basis fcts/grid pointsN* Nb+M* Mb + N*M vs. Nb*Mbgenerally: N<<Nb M<<Mb,

∑∑= =

=N

n

M

m

ym

xnnm tytxtatyx

1 1

)()( ),(),()(),,( φφχ

')(

')(

nnx

nx

n δφφ =&'

)('

)(mm

ym

ym δφφ =&

),(),,(),( )()( tytxta ym

xnnm φφ

),(),,( )()( tytx ym

xn φφ∞→N ∞→M

),(),,( )()( tytx ym

xn φφ

U. Manthe, H.-D. Meyer, L. S. Cederbaum, J. Chem. Phys. 97, 3199 (1992); H.-D. Meyer, U. Manthe, L.S. Cederbaum, Chem. Phys. Letters 165, 73 (1990).


22II. Methods of wave packet propagation: TD-SCF

• MCTDH: multi-configurationalexpansion of the wavefunction

Rem: this is not unique !constraints

• Schrödinger becomes:

∑∑= =

=N

n

M

m

ym

xnnm tytxtatyx

1 1

)()( ),(),()(),,( φφχ

')(

')(

nnx

nx

n δφφ =&

∑∑= =

=N

n

M

mnmmnmnmn taHtai

1 1'''' )()(&

( )∑∑==

−=N

n

xn

xnn

xN

n

xn

x HPinn

1

)()(',

)(

1

)()( 1'

φφρ &

∑∑= =

=N

n

M

m

xn

ym

ym

xnmnmn HH

1 1

)()()('

)(''' φφφφ

')(

')(

mmy

my

m δφφ =&

( )∑∑==

−=M

m

ym

ymm

ym

m

ym

y HPimm

1

)()(',

)(

1

)()( 1'

φφρ &

)()('

1 1'

*''

)('

ym

ym

M

m

M

mnmmn

xnn HaaH φφ∑∑

= =

=

)()('

1 1'

*''

)('

xn

xn

N

n

N

nnmmn

ymm HaaH φφ∑∑

= =

=

∑=

=N

n

xn

xn

xP1

)()()( φφ

∑=

=M

m

ym

ym

yP1

)()()( φφ

∑=

=M

mnmmn

xnn aa

1

*'

)('ρ

∑=

=N

nnmnm

ymm aa

1'

*)('ρ



References:

Méthodes de propagation de paquets d’ondes:

C. Cerjan, N umerical Grid methods and their application to the Schrödinger equation, Kluwer, 1993.C. Leforestier, R. H. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G.Jolicard,W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, R. Kosloff J. Comp. Phys. 94, 59 (1991)R. Kosloff, J. Chem. Phys. 92, 2087 (1988)

Multiconfiguration time-dependent Hartree:

M. H. Beck, A. Jäckle, G. A. Worth, H.-D. Meyer, Phys. Reports, 324, 1 (2000); U. Manthe, H.-D. Meyer, L. S. Cederbaum, J. Chem. Phys. 97, 3199 (1992); H.-D. Meyer, U. Manthe, L.S. Cederbaum, Chem. Phys. Letters 165, 73 (1990).

• quantum wave packet methods • errors: representation errors, errors in the calculation of the propagator• avantages / disadvantages• feasible for moderate number of DoF: 1-3• for more than 3 DoF: MCTDH, else: approximations

24

Full-dimensional quantum treatment 6D

Vibrational predissociation of I2Ne2

Dynamical processes of more complex systems-- More degrees of freedom-- Non-isolated systems (effects of environment, decoherence…)

Control by complex shaped laser pulses, spectral regions IR, UV,XUVHigher intensity

r

R

R1

2

2

1θ

φ

I

Ne

NeI

θ

pulse shaped IR excitation Hb-CO

Local control schemeMixed qu/cl dynamics

III. Outlook + future developments

25III. Outlook + future developments

• model system to study the sovlant-solute interactions • weak van der Waals interaction:

Challenges:• zero-point energy effects important • 1 quantum of vibration of I2 is enough to dissociate one rare gas atom • strongly correlated dynamics of the two Ne atoms

r

R

R1

2

2

1θ

φ

I

Ne

NeI

θ

Quantum treatment in `full dimensionality’

Exemple: vibrational predissociation of I2Ne2

[C. Meier, U. Manthe, J. Chem. Phys. 115, 5477 (2001) ]

• reference calculation for approximate schemes • processus studied:

vibrational predissociation of : I2(B,n=21)Ne2 I2(B,n=20)Ne + NeI2(B,n=19)Ne + Ne


r

R

R1

2

2

1θ

φ

I

Ne

NeI

θ

( ) ( ) ( )( )2

2

2222

2

2

2

2

2 sin1

sin1

21

21

2,1

1211

21

2121 sin),,,,,(φθθθμμμμ θφθθ∂∂

∂∂

=∂∂

∂∂ +++−+−= ∑

iiiiiIiiiI iRri

iRRrr RrRRrH

),,,,,( 2121,,,,, 2121

2 φθθφθθλγ

λγγλ RRrVCRR

++ ∑=

∂∂∂

Satellite coordonnées (non-orthogonal):distance I-I(centre of mass I2 )-Neangle ( )angle between the I-I-Ne planes

r21, RR21,θθ

φ2,1, Rr


=),,,,,,( 2121 tRRr φθθψ

∑ ∑= =

1

1

6

6

654321611 1

)6(2

)5(1

)4(2

)3(1

)2()1( ),(),(),(),(),(),(N

n

N

nnnnnnnnn ttttRtRtra φχθχθχχχχLL

wave packet propagation using the MCTDH scheme in 6 D

coordinate Ni (SPF)

nbr. of basis fct.

representation

5 49 fct. propres de I220 384 FFT

20 384 FFT

4 80 Legendre DVR

4 80 Legendre DVR

6 192 FFT

r1R

1θ

φ2θ

2R

number of configurations: 192000Hamiltonian matrix in standard method: ~1013 x 1013

28III. Outlook + future developmentsen

ergi

e

21, RR

ν=21

ν=20

ν=19

asymptotic

‘IVR’

‘direct’ν=19

ν=20

• life time of the resonance: I2(B,n=21)Ne2 : ~55 ps

process: I2(B,ν=21) Ne2 I2(B, ν=20) Ne + Ne: IVRprocess: I2(B,ν=21) Ne2 I2(B,ν=19) Ne + Ne: direct

• dynamical process requires QM and full dimensionality !


Why ?

• CO vibration as local probe of protein environment

• studies of vibrational relaxation: relaxation pathways – flow of energy

• inducing conformal changes ?“ground state chemistry”

aim: • depositing as much energy in the CO-

stretch as possible :conformal changes ? ground state dissociation ?energy relaxation pathways

• exciting single vibrational states or coherent superpositions of statessubsequent measures of stateresolved relaxation times, decoherence: sensitive measure of environment / protein dynamics

Exemple: Multiphoton IR excitation of HbCO with shaped pulses


anharmonic quantum oscillator CO

heme complex ~30 atomsFeP(Im)-CO

sub-unit of protein ~2200 atomsmodel: [1]

CO FeP(Im) protein

close environment:DFT (B3LYP)metal-ligandpropertiesπ-backbonding [2]

dipole moment

protein: Charmm

modifies FePentity

creates fluct. electric fields

[future work: full protein, solvation]

[1] M. D. Fayer, Annu. Rev. Phys. Chem, 52, (2001)J.R. Hill et al. J. Phys. Chem. 98, 11213 (1996),K. A. Merchant et al, JACS, 125, 13804 (2003)

D.E. Sagnella, J.E Straub, Biophys. J. 77, 70 (1999)[2] M. C. Heitz, C. M., J. Chem. Phys. 123, 044504 (2005)

31

( ) ),())()()(,(),(),( 1121

2

2

tqtEtEqqVtqi protNNqt ψμψrr

Lr

L +++=∂∂

−∂∂ rrrr

III. Outlook + future developments

),( tqquantum:ψ

Schrödinger equation

),((t) 11

Nmi qVii

rrr r L&& ∇−= no back-reaction: force is evaluated at CO equilibrium position

[include backreaction future]

Newtons equations(t)ir

2 step simulation: 1. Charmm with CO fixed2. multiple quantum wave packet calculations

with fluctuations in potential + dipole orientation

Mixed quantum-classical dynamics

:q CO stretch within FeP(Im)-CO complex

classical:

)(),(),()( 110 tEqqHqH NNf

rL

rL rrrr μ++≈

q


),( tqiψmultiple WP propagations with different

density matrix i

N

iiN ψψρ ∑

=

=1

1

( )observables ∑=

==N

iiiN AAtrA

1

1 ψψρ

[ ]( )ρ00 , HHtr

dtHd

=local control for 0HA =

fH

[ ]∑=

−≈N

iiiiN tHtttfitE

10

1 )(|,|)()(cos)()( ψμψθ

Observables, density matrices and Local control

[ ]∑=

≈N

iiiiN tHttitE

10

1 )(|,|)()(cos)( ψμψθ

then we have:

If one chooses:

00 ≥dtHd External field ensures heating of specific mode:

Local control scheme within Q/C

[1] R. Kosloff, S. A. Rice, P. Gaspard, S. Tersigni, D. Tannor Chem. Phys. 139, 201 (1989)M. Sugawara Y. Fujimura, J. Chem. Phys. 100, 5646 (1994)P. Gross, H. Singh, H. Rabitz, K. Mease, G. M. Huang, Phys. Rev A. 47, 4593 (1993), and lots more...


• simulations: 1000 mixed qu/classical runs for isotropically oriented sample

[ ]∑=

±≈N

iiiiN tHttfitE

10

1 )(|,|)(cos)()( ψμψθ

chosen to ensure pulse lengths of: 1ps, 1.5 ps, 2ps, 2.5 psand constant intensity: 1 μJ @ 40 μm (parameters of IR pulse shape experiments of M. Joffre, LOB, Paris [1])

)(tf

• Wigner plots of LCT pulses • comparison with non-fluctuating system, isotropic

ener

gy [c

m-1

]

0

1000

2000

3000

4000

5000

6000

1.0ps 1.5ps 2.0ps 2.5ps

[1] C. Ventalon et al, PNAS, 101, 13216 (2004)


• future + + :

-- development of mixed quantum / classical approaches

-- application to realistic systems

-- control in dissipative environments

-- combination of MCTDH and classical mechanics

• funding: ANR French ministry: post doc position

methods of quantum dynamics and simulation of pump-probe ... · methods of quantum dynamics and...

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