1. Time Domain nonadiabatic dynamics of NO 2 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

2. Scientific context Why is NO 2 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 NO 2 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.

3. The effective hamiltonian for NO 2 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.

4. Quantum wave packet dynamics : example and physical remarks Probability densities for wave packet launched on H e. 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.

5. Classical 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: (Q g,P g ) 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.

6. Diabatic 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 !).

7. Quasiclassical 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 H e and H g. For the wave packet initially located on H e : Good agreement between classical and quantum results. This regime lasts less than 300 fs. For the wave packet initially located on H g : 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.

8. Quantum vs classical «hopping» results  Very good agreement Probability densities on both electronic surfaces for a wave packet launched on H g. Left: quantum results. Right: Diabatic surface hopping results.

9. Wave packet dynamics of NO 2 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 P e (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 NO 2 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 P e (t) ? Can we reproduce the oscillations observed in time resolved experiments?

10. Time evolution of the electronic population Time evolution of the excited electronic state population for wave packets initially centered around p 1 =p 2 =p 3 =q 1 =q 3 =0 and q 2 =3.0 (dashed line) or q 2 =0.5 (solid line) on the A 2 B 2 electronic state Oscillations in P e (t) originate from differences between energies of eigenstates. Pairs of eigenstates involved in fluctuations of P e (t) necessarily : - are populated at t . - result from resonant vibronic couplings between at least 2 harmonic vectors of each electronic state.

11. Time 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.

12. Spectral analysis Squared modulus of the Fourier transform of P e (t) and |A(t)| 2 for the wave packet launched around 13800 cm -1  Slow oscillations in P e (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,v 2 ’,0]/[1,v 2 ’-2,0] pairs, where v 2 ’ goes from 12(u) to 17(p).  Oscillations are due to detuning from exact 1:2 resonance. Squared modulus of the Fourier transform of P e (t) and |A(t)| 2 for the wave packet launched around 20800 cm -1

13. Relation 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 the fingerprint of the detuning from exact 1:2 resonance between the bend and the symmetric 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 )

14. Bibliography 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 X 2 A 1 -A 2 B 2 conical intersection in NO 2 M. Joyeux, R. Jost and M. Lombardi Journal of Chemical Physics, 119, 5923-5932 (2003) Quantum mechanical and quasiclassical investigation of the time domain nonadiabatic dynamics of NO 2 close to the bottom of the X 2 A 1 -A 2 B 2 conical intersection M. Sanrey and M. Joyeux Journal of Chemical Physics, 125, 014304 (1-8) (2006) Slow periodic oscillations in time domain dynamics of NO 2 M. Sanrey and M. Joyeux Journal of Chemical Physics, 126, 074301 (1-8) (2007)