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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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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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Multiply by yi at left and integrate in the electronic coordinates

Time dependent Schrödinger equation for the nuclei

Quantum dynamics

nuclear wave function


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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)


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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)


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

Exp.

MCTDH

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


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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)


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and

with

Multiple spawning dynamics


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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).


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Multiple spawning dynamics

Saddle point approximation


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S0

S1


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QM (FOMO-AM1)/MM

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


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E

E1

t’

R’

t

R

Global nature of the nuclear wavefunction


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E

E1

t’

R’

t

R

Global nature of the nuclear wavefunction


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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)


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y local: it does not depend on the wavefunction values at other positions of the space

x

Mixed Quantum-Classical Dynamics

Is the classical position


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Nuclear motion: local: it does not depend on the wavefunction values at other positions of the space

Mixed Quantum-Classical Dynamics

Is the classical position

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


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With local: it does not depend on the wavefunction values at other positions of the space

(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)


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Average surface local: it does not depend on the wavefunction values at other positions of the space

tci

E

t

0


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Ehrenfest dynamics fails for dissociation local: it does not depend on the wavefunction values at other positions of the space

Energy

pp*

10 fs

ns*

ps*

cs

N-H dissociation


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0.57 local: it does not depend on the wavefunction values at other positions of the space

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.


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Decoherence local: it does not depend on the wavefunction values at other positions of the space

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)


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Mixed Quantum-Classical Dynamics local: it does not depend on the wavefunction values at other positions of the space

Is the classical position

Nuclear motion:


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Surface hopping dynamics local: it does not depend on the wavefunction values at other positions of the space

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)


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E local: it does not depend on the wavefunction values at other positions of the space

tci

t

0


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Comparison between methods local: it does not depend on the wavefunction values at other positions of the space

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)


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Ethylene local: it does not depend on the wavefunction values at other positions of the space

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

Butatriene cation

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

Oscillation patterns are not necessarily quantum interferences


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Hierarchy of methods local: it does not depend on the wavefunction values at other positions of the space

Quantum

Wave packet (MCTDH)

Multiple spawning

MQCD dynamics

Classical


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  • Next lecture: local: it does not depend on the wavefunction values at other positions of the space

  • 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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