Attosecond electronic and nuclear quantum photodynamics of ozone: time-dependent Dyson orbitals and dipole
A. Perveaux, D. Lauvergnat, B. Lasorne, F. Gatti, M. A. Robb, G. J. Halász, Á. Vibók
AAttosecond electronic and nuclear quantum photodynamics ofozone: time-dependent Dyson orbitals and dipole
A. Perveaux, , D. Lauvergnat, B. Lasorne, F.Gatti , M. A. Robb, G. J. Halász, and Á. Vibók CTMM, Institut Charles Gerhardt Montpellier,Université Montpellier 2, F-34095 Montpellier, France Laboratoire de Chimie Physique, Bâtiment 349, CNRS, UMR8000,Orsay, F-91405; Univ Paris-Sud, Orsay, F-91405; France Imperial College London, Department of Chemistry, London SW7 2AZ, UK Department of Information Technology, University of Debrecen,H-4010 Debrecen, PO Box 12, Hungary and Department of Theoretical Physics, University of Debrecen,H-4010 Debrecen, PO Box 5, Hungary ∗ A nonadiabatic scheme for the description of the coupled electron and nuclearmotions in the ozone molecule was proposed recently (PRA, ,023425, (2013)). Aninitial coherent nonstationary state was prepared as a superposition of the groundstate and the excited Hartley band. In this situation neither the electrons nor thenuclei are in a stationary state. The multiconfiguration time dependent Hartreemethod was used to solve the coupled nuclear quantum dynamics in the framework ofthe adiabatic separation of the time-dependent Schrödinger equation. The resultingwave packet shows an oscillation of the electron density between the two chemicalbonds. As a first step for probing the electronic motion we computed the time-dependent molecular dipole and the Dyson orbitals. The latter play an importantrole in the explanation of the photoelectron angular distribution. Calculations ofthe Dyson orbitals are presented both for the time-independent as well as the time-dependent situations. We limited our description of the electronic motion to theFranck-Condon region only due to the localization of the nuclear wave packets aroundthis point during the first 5–6 fs. a r X i v : . [ phy s i c s . a t m - c l u s ] J a n I. INTRODUCTION
In recent years, essential effort has been made to develop different attosecond techniques,see, e.g., [1–3] and further references therein. These techniques are based on an appropriateconstruction of single ultrashort pulses or trains of such pulses, which allows one to take real-time snapshots of ultrafast transformations of matter. Pump-probe experimental techniquesusing these ultra short laser pulses [3–8] have made possible to control complex molecularprocesses. Experimentalists can excite or ionize atoms with a controlled few-cycle laserfield and then probe them through the spectrally resolved absorption of an attosecond XUVpulse. It is the simplest experimental technique used to study ultrafast electronic dynamics.During this process one is able to fully image the electronic quantum motion and determinethe degree of coherence in the studied system [9–13]. However, the real challenge now ishow to transfer this technique to molecules.Despite the fact that the electron dynamics in molecules almost always are stronglycoupled to nuclear dynamics in case of diatomics with only one and two electrons the properdescription of the nuclear-electron dynamics is satisfactorily solved. In this case the totaltime-dependent Schrödinger equations (TDSE) can be solved numerically including boththe electronic and nuclear degrees of freedom explicitly [14–24]. Another branch of methodstreat the electron dynamics very accurately, while keeping fixed the nuclear geometry. Thisis a fairly reasonable assumption if the masses of the nuclei are large. In this situationthe motion of the electrons takes place due to the pump of a nonstationary electronic stateduring a period of time that is much shorter than the period of the vibrational motion of thenuclei [25–31]. The really challenging task is to create an electronic wave packet in a neutralfew-atom system and then describe precisely its coupled electron nuclear dynamics. Suchsystems are really large enough not to use the advantage of the solution of the TDSE withoutseparation, but they are not large enough to use the rigid nuclear geometry assumption. Inthis case the difference between the nuclear vibration period and the period of the motionof the electrons are not always very significant.Recently, we have investigated the ozone molecule and proposed a new scheme for thedescription of the coupled electron and nuclear motion [32, 33]. An initial coherent non- ∗ Electronic address: [email protected] stationary state was prepared by a sub-femtosecond pulse [33]. It is a superposition of theground state X and the excited weakly-bound state B of the Hartley band [34–37]. In thissituation neither the electrons nor the nuclei are in a stationary state, and we used quantumdynamics simulations. The nuclear wave packets, the electronic populations, the relativeelectronic coherence between the ground X and B electronic states and the electron wavepacket dynamics were analyzed. The time evolution of the electronic motion was plottedin the Franck-Condon (FC) region only due to the localization of the nuclear wave packetaround this point during the first − fs.The purpose of this paper is to go further in the photodynamics simulation of this system.We now present the time-dependent dipole moment [38] and investigate the Dyson orbitals[39–43]. While the former one is useful to visualize the exciton migration as the electrondensity shows a fast oscillation between the two chemical bonds, the Dyson orbitals areknown to be central in the explanation of the photoelectron angular distribution. Boththe time-independent as well as the time-dependent Dyson orbitals are calculated and theirproperties are discussed.In Sec. II. we give a short description of the chosen theoretical approaches. The relevantexpressions and formalism will also be presented here. The results and the discussions willbe presented in Sec. III. Section IV. provides conclusions. II. METHODS AND FORMALISM
The adiabatic partition formalism (beyond Born-Oppenheimer [44]) assumes the totalmolecular wave function Ψ tot ( (cid:126)r el , (cid:126)R, t ) as a sum of products of electronic wave functions, ψ kel ( (cid:126)r el ; (cid:126)R ) , and nuclear wave packets, Ψ knuc ( (cid:126)R, t ) : Ψ tot ( (cid:126)r el , (cid:126)R, t ) = n (cid:88) k =1 Ψ knuc ( (cid:126)R, t ) ψ kel ( (cid:126)r el ; (cid:126)R ) . (1)Here k denotes the k − th adiabatic electronic state, (cid:126)r el and (cid:126)R are the electronic and thenuclear coordinates, respectively. We are interested in solving the coupled evolution of thenuclear wave packets, Ψ knuc ( (cid:126)R, t ) , by inserting the product ansatz (1) into the time-dependentSchrödinger equation of the full molecular Hamiltonian. The electronic wave function obeysthe time-independent Schrödinger equation for the electronic Hamiltonian H el ( (cid:126)R ) H el ( (cid:126)R ) ψ lel ( (cid:126)r el ; (cid:126)R ) = V l ( (cid:126)R ) ψ lel ( (cid:126)r el ; (cid:126)R ) , (2)and integrating over the electronic coordinates we can also obtain the coupled nuclearSchrödinger equations: i (cid:126) ∂∂t Ψ knuc ( (cid:126)R, t ) = (cid:88) l =1 ,n H k,l ( (cid:126)R )Ψ lnuc ( (cid:126)R, t ) . (3)Here H k,l is the matrix element of the vibronic Hamiltonian, which reads, e.g., for n = 2 , H = T nuc + V k K ∗ k,l K k,l T nuc + V l , (4)where T nuc is the nuclear kinetic energy, V k ( k = 1 , ...n ) is the k − th adiabatic potentialenergy and K k,l with k (cid:54) = l is the coupling term between the ( k, l ) − th electronic states,which contains the light-matter interaction, − (cid:126)µ k,l · −→ E ( t ) (electric dipole approximation),where (cid:126)E ( t ) is an external field resonant between the k − th and the l − th states and (cid:126)µ k,l isthe (cid:126)R − dependent transition dipole moment.One has to solve the electronic and nuclear Schrödinger equations Eqs. (2-3) to obtainthe potential energy surfaces and the electronic and nuclear wave functions. The electronicstructure calculations are performed by the MOLPRO code [45] and a development versionof the GAUSSIAN program package [46]. As for the nuclear dynamics calculations themulticonfiguration time-dependent Hartree method (MCTDH) method [47–50] was used.The MCTDH nuclear wave packets, Ψ knuc ( (cid:126)R, t ) , contain all the information about therelative phases between the electronic states. Using the interaction picture, Ψ knuc ( (cid:126)R, t ) canequally be written as: Ψ knuc ( (cid:126)R, t ) = exp( − iV k ( (cid:126)R ) t/ (cid:126) ) a k ( (cid:126)R, t ) . (5)Here, V k ( (cid:126)R ) is the potential energy of the k − th state. The first part of this wave functionis the phase factor, ( exp( − iV k ( (cid:126)R ) t/ (cid:126) ) ), of the k − th state, which oscillates very fast.From the electronic and nuclear wave functions the total density matrix of the molecule,the electronic populations on the different states and the electronic relative coherence be-tween the different electronic states have been calculated [32, 33].In the present case we have two states (the ground X and the Hartley B states), and theelectronic wave packet of the neutral molecule can be written as follows Ψ tot ( (cid:126)r el , (cid:126)R, t ) = Ψ Xnuc ( (cid:126)R, t ) ψ Xel ( (cid:126)r el ; (cid:126)R ) + Ψ Bnuc ( (cid:126)R, t ) ψ Bel ( (cid:126)r el ; (cid:126)R ) . (6)Applying this formula, the total time-dependent dipole of the molecule reads as (cid:126)µ tot ( (cid:126)R, t ) = (cid:68) Ψ tot ( (cid:126)R, t ) | (cid:126)µ tot | Ψ tot ( (cid:126)R, t ) (cid:69) = (cid:88) k,l = X,B Ψ k ∗ nuc ( (cid:126)R, t )Ψ lnuc ( (cid:126)R, t ) (cid:126)µ k,l ( (cid:126)R ) (7)here (cid:126)µ k,l is the transition dipole moment between the k − th and l − th electronic states.Probing the electronic wave packet will lead to ionize the ozone molecule. The Dysonorbitals correspond to the molecular orbitals of the neutral molecule from which an electronhas been removed where the cation relaxation is accounted for. They can be computedas one-electron transition amplitudes between the N-electron neutral and (N-1) - electroncationic states: Φ Dcat ( (cid:126)r ; (cid:126)R ) = √ N ˆ d(cid:126)r ...d(cid:126)r N − ψ Nel,neut ( (cid:126)r , ...(cid:126)r N = (cid:126)r ; (cid:126)R ) ψ N − el,cat ( (cid:126)r , ...(cid:126)r N − ; (cid:126)R ) . (8)Applying the occupation number representation the Dyson orbitals can also be expressedin the molecular orbitals of the neutral molecule Φ Dcat ( (cid:126)r ; (cid:126)R ) = (cid:88) k ϕ neutk ( (cid:126)r ) (cid:68) ψ N − el,cat ( (cid:126)R ) | ˆ a k ψ Nel,neut ( (cid:126)R ) (cid:69) . (9)Here ˆ a k is the operator which removes an electron from the molecular orbital ϕ k .We now define the time-dependent Dyson orbitals. These orbitals may be useful forinstance when the neutral molecule is excited by an ultrashort laser pulse creating a coherentsuperposition of the different stationary states in the neutral molecule that will be probedin the next step by sudden XUV ionization. We focus here on such a situation. Φ Dcat,i ( (cid:126)r ; (cid:126)R, τ ) = √ N ˆ d(cid:126)r ...d(cid:126)r N − ψ Nel,neut ( (cid:126)r , ...(cid:126)r N = (cid:126)r ; (cid:126)R, τ ) ψ N − el,cat,i ( (cid:126)r , ...(cid:126)r N − ; (cid:126)R )= (cid:88) k Ψ knuc ( (cid:126)R, τ )Φ Dcat,i ( (cid:126)r ; (cid:126)R ) , (10)here τ is the time when the ionization takes place. i denotes the different cation chan-nels. At a given (cid:126)R , ψ Nel,neut ( (cid:126)r , ...(cid:126)r N ; (cid:126)R, τ ) is the electronic wave packet, which is a coherentsuperposition of the ground (X) and the Hartley (B) states, and reads as ψ Nel,neut ( (cid:126)r el ; (cid:126)R, τ ) = Ψ tot ( (cid:126)r el , (cid:126)R, τ ) = Ψ Xnuc ( (cid:126)R, τ ) ψ Xel ( (cid:126)r el ; (cid:126)R ) + Ψ Bnuc ( (cid:126)R, τ ) ψ Bel ( (cid:126)r el ; (cid:126)R ) . (11)From this quantity one can form the time-dependent density of the Dyson orbitals at theFC geometry ρ Φ Dcat ( (cid:126)r ; (cid:126)R, τ ) = (cid:88) k,l = X,B Ψ k ∗ nuc ( (cid:126)R, τ )Ψ lnuc ( (cid:126)R, τ )Φ D,k ∗ cat ( (cid:126)r ; (cid:126)R )Φ D,lcat ( (cid:126)r ; (cid:126)R ) . (12)This density can be written for each cation channel as well ρ Φ Dcat,i ( (cid:126)r ; (cid:126)R, τ ) = | Ψ Xnuc ( (cid:126)R, τ ) | ρ Φ D,Xcat,i ( (cid:126)r ; (cid:126)R )+ | Ψ Bnuc ( (cid:126)R, τ ) | ρ Φ D,Bcat,i ( (cid:126)r ; (cid:126)R )+2 Re Ψ X ∗ nuc ( (cid:126)R, τ )Ψ Bnuc ( (cid:126)R, τ )Φ D,X ∗ cat,i ( (cid:126)r ; (cid:126)R )Φ D,Bcat,i ( (cid:126)r ; (cid:126)R ) (13)where index i runs over each of cation channel.Let us define the local norm of the Dyson orbitals at the FC geometry, which is approxi-mately proportional to the ionization probability (similar to a Franck-Condon factor in theimpulsive picture) (cid:68) Φ Dcat ( (cid:126)r ; (cid:126)R, τ ) | Φ Dcat ( (cid:126)r ; (cid:126)R, τ ) (cid:69) = (cid:88) k,l = X,B Ψ k ∗ nuc ( (cid:126)R, τ )Ψ lnuc ( (cid:126)R, τ ) (cid:68) Φ D,kcat ( (cid:126)r ; (cid:126)R, τ ) | Φ D,lcat ( (cid:126)r ; (cid:126)R, τ (cid:69) . (14)For a certain cation channel this quantity reads as: (cid:68) Φ Dcat,i ( (cid:126)r ; (cid:126)R, τ ) | Φ Dcat,i ( (cid:126)r ; (cid:126)R, τ ) (cid:69) = | Ψ Xnuc ( (cid:126)R, τ ) | (cid:68) Φ D,Xcat,i ( (cid:126)r ; (cid:126)R, τ ) | Φ D,Xcat,i ( (cid:126)r ; (cid:126)R, τ (cid:69) + | Ψ Bnuc ( (cid:126)R, τ ) | (cid:68) Φ D,Bcat,i ( (cid:126)r ; (cid:126)R, τ ) | Φ D,Bcat,i ( (cid:126)r ; (cid:126)R, τ (cid:69) +2 Re Ψ X ∗ nuc ( (cid:126)R, τ )Ψ Bnuc ( (cid:126)R, τ ) (cid:68) Φ D,X ∗ cat,i ( (cid:126)r ; (cid:126)R ) | Φ D,Bcat,i ( (cid:126)r ; (cid:126)R ) (cid:69) . (15)These expressions and equations will serve as our working formulae in the next section. III. RESULTS AND DISCUSSION
Figure 1 shows the total time-dependent dipole Eq. (7) of the molecule. In the presentsituation the time-dependent dipole is created after the interaction of the molecule with the -150-100-50050100150 M o l ec u l a r d i po l e / a . u . -5 0 5 10 15 20 25 Time / fs
Figure 1: Time-dependent molecular dipole along the y-axis.Symmetry of the cation state Excitation energy at the MRCI level [eV] Experimental value [eV] A B A pump laser field [38]. Its oscillatory behaviour can be considered as a direct consequenceof the coherent superposition of the X and B electronic states. This quantity has a similartime evolution to the electronic relative coherence, which illustrates exciton migration. TheFC point of the ozone molecule has C v symmetry and therefore only the y -component ( B )of the transition dipole between the ground state X ( A ) and Hartley B ( B ) is nonzero.The only effective polarization of the electric field is y .Probing the electronic wave packet Eq. (11) will lead to ionize the neutral molecule.We focus here on the ionization threshold of the spectrum (fastest electrons). The threelowest cationic states ( A , B , and A ) must be considered together because they areenergetically close with respect to the expected bandwith of the probe pulse. We havecalculated the energies of the different electronic states of the cation at the MRCI level oftheory and compared them to the experimental values [51]. Reasonably good agreement hasbeen obtained for each electronic state. Results are presented in Table 1.We now present the Dyson orbitals. They are very illustrative as they are one-electronwave functions that represent the probability amplitude of the electron that is removedfrom the neutral molecule. They also play an important role in the interpretation of thephotoelectron angular distribution as well. Namely, they are approximately proportional Figure 2: Time-independent Dyson orbitals between the two different electronic states (X, B) ofthe neutral and the three different channels ( A , B and A ) of the cation. The most importantmolecular orbital components of the Dyson orbitals are also shown. to the latter, as their norms give some contributions to the photoionization intensity. Inthe present situation we have three ionization channels for the ground (X) and also for theHartley (B) states of the neutral molecule. These six different time-independent Dysonorbitals are calculated according to Eqs. (8 - 9) and are shown in Figure 2. The MRCImethod with aug-cc-pvqz basis set was used in the numerical simulations.As already discussed the Dyson orbitals can also be written as linear combinations of themolecular orbitals of the neutral molecule Eq. (8). As it can be seen in Fig. 2 they followthe Koopman’s like rule. Hence, the Dyson orbitals for each cation channel are constructedmainly from one or two neutral’s orbitals. Below we give the most important molecularorbitals contributions to the different type of time-independent Dyson orbitals calculatedfor the present situation: ( A )Φ DX − A = − . ϕ + 0 . ϕ ;( A )Φ DX − A = − . ϕ ;( B )Φ DX − B = 0 . ϕ − . ϕ ; (16) ( B )Φ DB − A = − . ϕ − . ϕ ;( B )Φ DB − A = 0 . ϕ − . ϕ ;( A )Φ DB − B = − . ϕ + 0 . ϕ . The symmetries of the Dyson orbitals are provided by the symmetries of the cation statesand the symmetries of the X and B states of the neutral.Our initial wave function of the neutral ozone molecule is a wave packet, Eq. (6), whichis a superposition of two electronic states (X and B). Therefore, the Dyson orbitals are thesuperpositions of Dyson orbitals from each electronic states of the neutral to the cation.Thus, we are going to have three time-dependent Dyson orbitals Eq. (10); one for eachcation channel ( A , B and A ).In Figures (3-5) the time-dependent densities of the Dyson orbitals are presented. Theyare calculated according to Eqs. (10,12,13). It can be noticed that the time-dependentdensities of the Dyson orbitals are quite similar to the graphical representation of the time-independent Dyson orbitals between the X state of the neutral and an appropriate electronicstate of the cation. This is due to the fact that the population on the ground state is alwaysmore pronounced than in the excited state (in our case it is the Hartley (B) band). Thetime-dependent densities of the Dyson orbitals oscillate in time with the same period as thedipole moment of the neutral wave packet which is the period of the coherence. Therefore,there are three different stages in the time evolution of the time-dependent densities of theDyson orbitals. The first period is between [ − . ; ] fs when the laser light is on andthere is significant relative electronic coherence between the X and B states. In the secondperiod [ . ; ] fs the laser pulse is off and there is no coherence between the differentelectronic states. Finally, the third period is [ ; ] fs when the pump pulse is still off butthe electronic coherence reappears between the ground X and the Hartley B states of theozone molecule. The oscillations of the time-dependent densities are the most pronouncedin the first interval as the relative electronic coherence is the most significant here. In the0 -0.3fs 0.1fs 0.5fs 1.4fs 1fs 1.8fs 4.3fs 5.5fs 5.8fs 7.3fs 8.8fs 9.2fs 9.6fs 10fs 10.4fs Figure 3: Time-dependent densities of the Dyson orbitals belong to A cation channel. second region there is practically no motion of the time-dependent densities due to a lackof coherence. As for the third stage, the revival of electronic coherence induces a revival ofmotion in the time-dependent densities but with much less amplitude than the first interval.The latter is due to the fact that the coherence in the field free situation is less prominentthan in presence of the light field. Although all three regions were properly included in thenumerical simulations the time evolution of the time-dependent densities in the snapshots(Figs. (3-5)) are only presented between the period of [ − . ; . ] fs.We have also calculated the local norms of the Dyson orbitals Eqs. (14,15). These quan-tities are approximately proportional to the ionization probabilities. Probability densitiesat the FC geometry, are shown in Figure 6. They can be considered as "local norms", orrather local weights, with values higher than 1. At first sight, the striking feature on thepictures (see Figs. 6a - 6b) is that the ground X state is the essential component of theDyson orbital’s norm. This is consistent with the previous finding that the population inthe ground X state is more pronounced than that one on the Hartley B state. We can alsonotice, that the cation channel that presents more coherence ( B ) is at the same time theone that provides less total probability. And the reverse is also true, namely that the onepresenting less coherence ( A ) is the one presenting more total probability. It is partlyexplained by the fact that the more there is population on the ground X state, the lessthere is on the Hartley B state, therefore the less there is electronic coherence between thesetwo (neutral and cation) excited states (see Fig. 6c). On the other hand the more thereis population on the ground X state the more the probability of the time-dependent Dyson1 -0.3fs 0.1fs 0.5fs 1fs 1.4fs 1.8fs 4.3fs 5.5fs 5.8fs 7.3fs 8.8fs 9.2fs 9.6fs 10fs 10.4fs Figure 4: Time-dependent densities of the Dyson orbitals belong to B cation channel. -0.3fs 0.1fs 0.5fs 1.4fs 1fs 1.8fs 4.3fs 5.5fs 5.8fs 7.3fs 8.8fs 9.2fs 9.6fs 10fs 10.4fs Figure 5: Time-dependent densities of the Dyson orbitals belong to A cation channel. P r ob a b ilit y -5 0 5 10 15 20 25 Time / fs B A A (a) P r ob a b ilit y -5 0 5 10 15 20 25 Time / fs B A A (b) P r ob a b ilit y -5 0 5 10 15 20 25 Time / fs B B B A B A (c) -2 10 -20 -1 10 -20
01 10 -20 -20 P r ob a b ilit y -5 0 5 10 15 20 25 Time / fs XB B XB A XB A (d) Figure 6: Probability density of the time-dependent Dyson orbitals between the different states ofthe neutral molecule and the cation as a function of time [fs]. (a): Total probability density ofthe time-dependent Dyson orbitals as a function of time; B channel (unmarked); A channel(marked with square); A channel (marked with circle). The total probability density can beobtained as a sum of the probability densities on sub figures (b) and (c). (b): Probability densityof the time-dependent Dyson orbitals as a function of time between the X state of the neutral andthe different states of the cation; B channel (unmarked), A channel (marked with square); A channel (marked with circle). (c): Probability density of the time-dependent Dyson orbitals as afunction of time between the B state of the neutral and the different states of the cation; B channel(unmarked), A channel (marked with square); A channel (marked with circle). (d): Probabilitydensity of the relative electronic coherence of the X and B states of the neutral with the B cationchannel (unmarked), A cation channel (marked with square); A cation channel (marked withcircle) as a function of time. orbitals increases.3 IV. CONCLUSIONS
We have started to develop a complex theoretical description of the coupled electron andnuclear motion in the ozone molecule on the attosecond time scale [32, 33]. An initial coher-ent nonstationary state was created as a coherent superposition of the ground X and excitedHartley B states. In this situation we were able to induce attosecond electron dynamics inthe neutral molecule. The MCTDH approach was used to solve the dynamical Schrödingerequation for the nuclei in the framework of the time-dependent adiabatic partition includingthe light-matter interaction (electric dipole approximation). Based on this dynamical sim-ulation the description of the time evolution of the electronic motion is limited only to theFranck-Condon region due to the localization of the nuclear wave packet around this pointduring the first − fs.Applying the nuclear wave packet we have determined the total density matrix of themolecule and from it were able to calculate the electronic populations and the relativeelectronic coherence between the ground X and B electronic states [33]. In order to calculatethe excited-state differential charge density at the FC point we used the total molecular wavepacket which is a coherent mixture of multiple electronic states, whereby the time-dependentcoefficients are the nuclear wave packets. As a results, an oscillation of the electronic cloudbetween the two chemical bonds was observed with a . fs period of time [33].Going further in our photodynamics description we calculated the time-dependent dipolemoment and the Dyson orbitals. They are very useful to visualize the exciton migrationwithin the molecule as the electron density oscillates between the two chemical bonds. Inaddition, the Dyson orbitals are known to be important in the explanation of the pho-toelectron angular distribution. We calculated both the time-independent as well as thetime-dependent Dyson orbitals and discussed their properties for different situations. Cor-responding experiments are in progress [52]. Acknowledgements
The authors would like to thank R. Kienberger , M. Jobst and F. Krausz, for supportand for fruitful discussions. We acknowledge R. Schinke for providing the potential energysurfaces and the transition dipole moment and H.-D. Meyer for fruitful discussions. The au-4thors also acknowledge the TÁMOP 4.2.4. A/2-11-1-2012-0001 project. Á.V. acknowledgesthe OTKA (NN103251). Financial support by the CNRS-MTA is greatfully acknowledged. [1] Corkum P B and Krausz F 2007 Nature Phys. . . [24] von den Hoff P, Siemering R, Kowalewski M and de Vivie-Riedle R 2012 IEEE Journal ofSelected Topic in Quantum Electronics, . [48] Manthe U, Meyer H D and Cederbaum L S 1992 J. Chem. Phys.
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