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(field-free σ$_{g}$ and σ$_{u}$ states). All pathways involve three- photon resonance of the ground state to the first ex- cited electronic state of D + 2 , L$_{0}$. In one pathway, L$_{1}$ in Fig. 1, the ion’s internuclear distance increases in a dissociative state until reaching the critical values ( R$_{c}$ ), where the second ionization takes place with high proba- bility and the Coulomb explosion occurs. The final KER of the nuclear fragments will include the kinetic energy gained during dissociation and also kinetic energy of the Coulomb explosion (1 /R$_{c}$ ). In another possible pathway, L$_{2}$ in Fig. 1, the second ionization may take place directly at shorter internuclear distances for the shorter wave- lengths. In this case the three-photon excitation to the σ$_{u}$ state is only a virtual one, with ionization yield being strongly enhanced by the presence of the intermediate resonance. The fragments would gain their full kinetic energy on the Coulomb potential, and that energy will be very close to that of the first discussed pathway, L$_{1}$, since the Coulomb and σ$_{u}$ potentials are nearly parallel to each other. The present experimental data are not suf- ficient to determine which of these two pathways may be the real pathway or near to reality [3]. Another possible pathway, a net-two-photon pathway (three-photon exci- tation followed by subsequent de-excitation at the one- photon resonance), would result in lower kinetic energy fragments that is inconsistent with the experimental re- sults [3, 8]. 

To explore the details of the Litvinyuk’s experiment and the real pathway, it is necessary to set up a sim- ulation that allows us to declare the details of the dissociation-ionization pathway of D + 2 . The dynamics of H$_{2}$ (D$_{2}$) and H + 2 (D + 2 ) exposed to an intense laser field are very complicated because it involves electron and nuclear dynamics simultaneously with different time scales that results in a complex dissociation-ionization process. In intense laser field, electron dynamics occur in attosecond time scales and nuclear dynamics, i.e. vi- bration and rotation, takes place in femtosecond and pi- cosecond time scales respectively. It is sometimes possi- ble based on Born Oppenheimer approximation (BOA) to investigate the nuclei and electron dynamics, sepa- rately. This approach is extensively used to investigate electronic dynamics of molecules in intense laser fields [17, 20, 21]. But this assumption fails when both nuclei and electron dynamics are important and considerable. This situation happens with intense $_{∼}$10-100 femtosec- ond laser pulses, when the duration of the laser pulse is comparable to the vibrational period of molecules [3– 8]. In these conditions, a complex simulation based on the solving of the time dependent Schr¨ odinger equation (TDSE) beyond the BOA would be required for the com- plete description of molecular dynamics. For a molecule with two or more electrons, this task is very far from present available computer ability, even without consid- eration of the nuclear dynamics. The rigorous approach is feasible only for linear molecules aligned with the elec- tric laser field with only one electron. In this condition, it is possible to consider the full dimensional electron dy- 

namics and also the vibrational nuclear dynamics in the solution of the TDSE for H + 2 (D + 2 ) [9, 10]. + 

In this work, we have performed a simulation for D 2 beyond the BOA and also without the soft-core approxi- mation [11, 12], i.e., by the rigorous solution of the TDSE for the full dimensional electron dynamics of the aligned D + 2 with the electric laser field. This paper is organized as follows: the details of the numerical implementation and the simulation setup are presented in Sec. II. Results and discussions appear in Sec. III. Section IV contains observations and concluding remarks. Throughout this article we use the atomic units unless stated otherwise. 

## II. NUMERICAL SOLUTION OF THE TDSE 

Time dependent Schr¨ odinger equation in the cylindri- cal polar coordinate system for H + 2 (D + 2 ) located in the laser field parallel to the internuclear axis in atomic unit (au) can be expressed as 

<img src='content_image/433683.jpg'>

where the Hamiltonian for this system, H ( z, ρ, R, t ), is given by [13–15] 

<img src='content_image/433684.jpg'>

<img src='content_image/433685.jpg'>

with E$_{0}$ being the laser peak amplitude, ω = 2 πν the an- gular frequency, and finally f ( t ) the laser pulse envelope which is set as 

where τ$_{p}$ is the full width at half maximum (FWHM) duration of the Gaussian shape of the laser pulse, with τ$_{p}$ =25 cycles in this work. For the time discretization of the TDSE, a propagator derived from split-operator methods has been used. This propagator is unitary and is obtained by combining the classical split operator and the Crank-Nicholson method [16]. The details of our cal- culations are described in our previous reports [9, 15, 17]. In this simulation, the time step is set to δt = 0 . 02 and δt$_{R}$ = 0 . 2 for the electron and nuclei time propagation respectively and τ$_{p}$ =25 cycles ($_{∼}$40 fs). The differen- tial operators in Eq. (2) are discretized by the eleven- point difference formulae which have tenth-order accu- racies [17]. To solve the above TDSE numerically, we 

<img src='content_image/433686.jpg'>

