Time Domain nonadiabatic dynamics of NO2Time Domain nonadiabatic dynamics of NO2

International Symposium on Molecular Spectroscopy 62nd Meeting

June 18-22, 2007

Michaël SANREY

Laboratoire de Spectrométrie Physique(CNRS UMR 5588)Université Joseph Fourier Grenoble 1, France

Scientific context Scientific context

Why is NO2 an interesting molecule from the physical point of view?

• it displays a conical intersection between its two lowest electronic states

this implies strong vibronic couplings and complex optical spectra.

nonadiabatic dynamics (in time domain), during which transfer between electronic populations occurs.

This is the topic of the investigations reported here…

Previous theoretical studies dealing with time-dependent dynamics of NO2

U.Manthe and H.Koppel, J. Chem. Phys. 93, 1658 (1990)

Using a vibronic coupling constant 2250 cm-1, they concluded that the adiabatic representation was well adapted to the understanding of time-domain dynamics.

S. Mahapatra, H. Koppel and L.S. Cederbaum, J. Chem. Phys. 110 , 5691 (1999)S. Mahapatra et al., Chem. Phys. 259, 211 (2000)F. Santoro and C. Petrongolo, J. Chem. 110, 4419 (1999) F. Santoro et al., Chem. Phys. 259, 193 (2000).

is certainly of the order of several hundreds of cm-1 (we used 330 cm-

1). This leads to a richer behaviour of time-dependent observables.

The effective hamiltonian for NO2 The effective hamiltonian for NO2

3

1 ji...

ijiijiig nnxnH

M. Joyeux, R. Jost and M. Lombardi, J. Chem. Phys., 119, 5923 (2003)

The Hamiltonian is written in a diabatic basis :

where

and a diabatic coupling

Vibrational normal modes : symmetric stretch, mode 1

bending, mode 2

antisymmetric stretch, mode 3

The parameters of the hamiltonian were fitted against the first 307 experimentally observed levels.

gc

ce

HH

HHH

ji

jiijii

ie ''n'nx'n'EH3

10

3qHc

Quantum wave packet dynamics : example and physical remarksQuantum wave packet dynamics : example and physical remarks

Probability densities for wave packet launched on He.Excited electronic state component at the top and ground electronic component at the bottom.

• A component quickly appears on the ground electronic surface in the intersection zone.

• Each component evolves quasi-independently up to 40 fs.

• Components of the wave packet transferred to the ground electronic surface at different times interfer.

Dynamics occurs as if there were an effective intersection zone.

Question: How to define this intersection zone?- Use exact propagation of quantum wave packets launched on each electronic surface at t and around the intersection in phase space.- Calculate of time-dependant observables. - Characterise the intersection zone through quasiclassical approximations and comparison with quantum results.

Classical mappingClassical mapping

The principle G. Stock and M. Thoss, Phys. Rev. Lett. 78, 578 (1997)

• Mapping consists in transforming the discrete electronic degrees of freedom into continuous ones. Ex: (Qg ,Pg ) is the set of electronic coordinates associated with the ground state.

• Use of quantification rules in the classical mapping formulation with

Wave packet propagation within the mapping approximation

• We propagate swarms of classical trajectories that at t are normally distributed around the center of the quantum wave packet.

• This enables to calculate electronic populations But it is impossible to obtain probability densities on each electronic surface. We have to develop a different model to calculate such complex time-dependant quantities.

cegegggee HQQPPHIHIH

21

21 22

21

iii QPI

Diabatic surface hopping modelDiabatic surface hopping model

Interaction zone

• Hamilton equations applied to the mapping approximation

• The quasi-stationary phase condition

defines the interaction zone.

Classical wave packet propagation

• Each trajectory has a constant probability to jump to the other (e,g) state only when it is located in the interaction region defined above. Hopping rates were fitted against mapping results (rougher approximation than TFS model !).

eggeceg IIH

dtdI

dt

dI sin2

ege

g

g

ecegeg I

I

II

HHHdtd

cos

ceg HHH 2

Quasiclassical electronic populations compared with quantum onesQuasiclassical electronic populations compared with quantum ones

Time evolution of the populations in the initial electronic state for quantum wave packets and swarms of 20000 classical trajectories launched on He and Hg.

For the wave packet initially located on He :

• Good agreement between classical and quantum results.

•This regime lasts less than 300 fs.

For the wave packet initially located on Hg:

• Good agreement and reproduction of the slight and regular increase of the population during the plateaux phases.

Continuous transfer due to the fact that the excited component remains in the intersection zone.

Quantum vs classical «hopping» resultsQuantum vs classical «hopping» results

Very good agreement

Probability densities on both electronic surfaces for a wave packet launched on Hg.Left: quantum results. Right: Diabatic surface hopping results.

Wave packet dynamics of NO2 at longer times : state of artWave packet dynamics of NO2 at longer times : state of art

- Existence of a quasi-classical regime up to a few hundreds of fs.

- At longer times, a purely quantum mechanical regime sets in, which cannot be described by any quasi-classical analysis.

Slow periodic oscillations in Pe(t) were obtained with both our effective Hamiltonian and realistic ab initio surfaces ( Mahapatra et al.).

Mahapatra points out that « a precise explanation for the time scale of the oscillations cannot be given at present ».

Recent pump-probe experiments on NO2 close to its first dissociation limit exhibit a slow NO+ signal with an oscillatory component with period 600-800 fs ( 40-55 cm-1).

A. T. J. B. Eppink et al, J. Chem. Phys. 121, 7776 (2004)

N. T. Form et al., Phys. Chem. Phys. 8, 2925 (2006)

Questions:

• Can we analyse precisely the origin of the slow oscillations in Pe(t) ?• Can we reproduce the oscillations observed in time resolved experiments?

Time evolution of the electronic populationTime evolution of the electronic population

Time evolution of the excited electronic state population for wave packets initially centered around p1=p2=p3=q1=q3=0 and q2=3.0 (dashed line) or q2=0.5 (solid line) on the A2B2

electronic state

•Oscillations in Pe(t) originate from differences between energies of eigenstates.

• Pairs of eigenstates involved in fluctuations of Pe(t) necessarily :

- are populated at t.

- result from resonant vibronic couplings between at least 2 harmonic vectors of each electronic state.

j

j

gjjj

j

ejjj

jjjtiEc

tiEc

tiEct

exp

exp

exp

tEEccccctP kjek

ej

j kj

*kjk

*j

ejj

e

cos22

Time evolution of the autocorrelation functionTime evolution of the autocorrelation function

Time evolution of the squared modulus of the autocorrelation function |A(t)|2 for the same wave packets as previously.

• Experimental signals are often related to the autocorrelation function – we suppose it is indeed the case here.

• White lines represent the signal smoothed with a gaussian window of width 90 fs to take the temporal width of the laser into account.

slow-periodic oscillations are also observed in the autocorrelation signal.

the pairs of eigenstates involved in the oscillations of the autocorrelation function don’t necessarily result from vibronic couplings.

j kj

kjkjj tEEcccttA cos2022422

Spectral analysisSpectral analysis

Squared modulus of the Fourier transform of Pe(t) and |A(t)|2 for the wave packet launched around

13800 cm-1

Slow oscillations in Pe(t) and |A(t)|2 originate from the same lines.

It is no longer the case when the wp is excited around 20800 cm-1

low-frequency oscillations of |A(t)|2 don’t reflect IVER at these energies.

• lines p-u are due to [0,v2’,0]/[1,v2’-2,0] pairs, where v2’ goes from 12(u) to 17(p) .

Oscillations are due to detuning from exact 1:2 resonance.

Squared modulus of the Fourier transform of Pe(t) and |A(t)|2 for the wave packet launched around 20800 cm-1

Relation to time resolved experimentsRelation to time resolved experiments

Time evolution of the squared modulus of the autocorrelation function |A(t)|2 smoothed with a gaussian window of width 90 fs, for the wave packets already mentionned.

Conclusion

In our model, the low-frequency oscillatory signal for wave packets created by quasi-vertical excitation of the vibronic ground state (20800 cm-1) is thefingerprint of the detuning from exact 1:2 resonance between the bend and thesymmetric stretch.

• Detuning from 1:2 resonance critically depends on the anharmonicity x’22.

• Recent experimental data indicate that x’22 3cm-1 (this is the value assumed in our model).

Detuning = 2(’2+x’22(2v’2-1))-’1

= 55cm-1 (if x’22 3cm-1)

= 217 cm-1 (if x’22 0 cm-1)

BibliographyBibliography

I thank my PHD supervisor, Marc Joyeux, and all others members of the team : Maurice Lombardi, Remi Jost, Sahin Buyukdagli.

Related articles : An effective model for the X2A1-A

2B2 conical intersection in NO2

M. Joyeux, R. Jost and M. LombardiJournal of Chemical Physics, 119, 5923-5932 (2003)

Quantum mechanical and quasiclassical investigation of the time domain nonadiabatic dynamics of NO2 close to the bottom of the X2A1-A

2B2 conical intersection

M. Sanrey and M. JoyeuxJournal of Chemical Physics, 125, 014304 (1-8) (2006)

Slow periodic oscillations in time domain dynamics of NO2

M. Sanrey and M. JoyeuxJournal of Chemical Physics, 126, 074301 (1-8)  (2007)