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Femtochemistry: A theoretical overview. VII – Methods in quantum dynamics. Mario Barbatti [email protected] This lecture can be downloaded at http://homepage.univie.ac.at/mario.barbatti/femtochem.html lecture7.ppt.

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slide1

Femtochemistry: A theoretical overview

VII – Methods in quantum dynamics

Mario Barbatti

[email protected]

This lecture can be downloaded at

http://homepage.univie.ac.at/mario.barbatti/femtochem.html

lecture7.ppt

slide2

Multiply by yi at left and integrate in the electronic coordinates

Time dependent Schrödinger equation for the nuclei

Quantum dynamics

nuclear wave function

slide3

Wave-packet propagation

Expand nuclear wave function in a basis of nk functions fk for each coordinate Rk:

Wavepacket program:

http://page.mi.fu-berlin.de/burkhard/WavePacket/

Kosloff, J. Phys. Chem. 92, 2087 (1988)

slide5

Multiconfiguration time-dependent Hartree (MCTDH) method

Variational method leads to equations of motion for A and f.

Conventional: 1 – 4 degrees of freedom

MCTDH: 4 -12 degrees of freedom

Heidelberg MCDTH program:

http://www.pci.uni-heidelberg.de/tc/usr/mctdh/

Meyer and Worth, Theor. Chem. Acc. 109, 251 (2003)

slide6

UV absorption spectrum of pyrazine (24 degrees of freedom!)

Exp.

MCTDH

Raab, et al. J. Chem. Phys. 110, 936 (1999)

slide7

Multiple spawning dynamics

Nuclear wavefunction is expanded in a Gaussian basis set centered at the classical trajectory

Note that the number of gaussian functions (Ni) depends on time.

Ben-Nun, Quenneville, Martínez, J. Phys. Chem. A 104, 5161 (2000)

slide8

and

with

Multiple spawning dynamics

slide9

Multiple spawning dynamics

The non-adiabatic coupling within Hij is computed and monitored along the trajectory.

When it become larger than some pre-define threshold, new gaussian functions are created (spawned).

slide10

Multiple spawning dynamics

Saddle point approximation

slide11

S0

S1

slide12

QM (FOMO-AM1)/MM

Virshup et al. J. Phys. Chem. B 113, 3280 (2009)

slide13

E

E1

t’

R’

t

R

Global nature of the nuclear wavefunction

slide14

E

E1

t’

R’

t

R

Global nature of the nuclear wavefunction

slide15

Within this approximation, the nuclear wavefunction is local: it does not depend on the wavefunction values at other positions of the space

  • This opens two possibilities:
  • On-the-fly approaches (global PESs are no more needed)
  • Classical independent trajectories approximations

However, because of the non-adiabatic coupling between different electronic states, the problem cannot be reduced to the Newton’s equations

We use, therefore, Mixed Quantum-Classical Dynamics approaches (MQCD)

slide16

y

x

Mixed Quantum-Classical Dynamics

Is the classical position

slide17

Nuclear motion:

Mixed Quantum-Classical Dynamics

Is the classical position

Multiply at left by d(R-Rc)1/2 and integrate in R.

slide18

With

(adiabatic basis)

which solves:

Prove!

Remember we assumed

Weighted average over all gradients

Ehrenfest (Average Field) Dynamics

Meyer and Miller, J. Chem. Phys. 70, 3214 (1979)

slide20

Ehrenfest dynamics fails for dissociation

Energy

pp*

10 fs

ns*

ps*

cs

N-H dissociation

slide21

0.57

0.43

Landau-Zener predicts the populations at t = ∞

In this case, Ehrenfest dynamics would predict the dissociation on a surface that is ~half/half S0 and S1, which does not correspond to the truth.

slide22

Decoherence

The problem with the Ehrenfest dynamics is the lack of decoherence.

The non-diagonal terms should quickly go to zero because of the coupling among the several degrees of freedom.

The approximation ci(R(t)) ~ ci(t) does not describe this behavior adequately.

Ad hoc corrections may be imposed.

Zhu, Jasper and Truhlar, J. Chem. Phys. 120, 5543 (2004)

slide23

Mixed Quantum-Classical Dynamics

Is the classical position

Nuclear motion:

slide24

Surface hopping dynamics

Dynamics runs always on a single surface (diabatic or adiabatic).

Every time step a stochastic algorithm decides based on the non-adiabatic transition probabilities on which surface the molecule will stay.

The wavepacket information is recovered repeating the procedure for a large number of independent trajectories.

Because the dynamics runs on a single surface, the decoherence problem is largely reduced (but not eliminated).

Tully, J. Chem. Phys. 93, 1061 (1990)

slide25

E

tci

t

0

slide26

Comparison between methods

Tully, Faraday Discuss. 110, 407 (1998).

surface-hopping (diabatic)

surface-hopping (adiabatic)

mean-field

wave-packet

Landau-Zener

Burant and Tully, JCP 112, 6097 (2000)

slide27

Ethylene

Barbatti, Granucci, Persico, Lischka, CPL401, 276 (2005).

Butatriene cation

Worth, hunt and Robb, JPCA 127, 621 (2003).

Oscillation patterns are not necessarily quantum interferences

slide28

Hierarchy of methods

Quantum

Wave packet (MCTDH)

Multiple spawning

MQCD dynamics

Classical

slide29

Next lecture:

  • Implementation of surface hopping dynamics

Contact

[email protected]

This lecture can be downloaded at

http://homepage.univie.ac.at/mario.barbatti/femtochem.html

lecture7.ppt

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