Theoretical studies of high-power laser ionization of molecules in the tunneling region

In this paper, we extend our previous works on the generalization of Keldysh's theory to the photoionization processes of molecules. In particular, we include the Franck-Condon factors into our photoionization rate formulas which are based on the use of the molecular orbital theory to describe the electronic degrees of freedom. The inclusion of Franck-Condon factors leads to the proper treatment of the molecular vibrational degrees of freedom. All of our formulas consist of the preexponential and exponential factors, and have explicit laser frequency dependence in the same manner as the original atomic Keldysh theory. The latter fact facilitates the exploration of the laser frequency dependence of the photoionization rate, which is more advantageous than the popular Ammosov-Delone-Krainov formulas. As a result, our analytical expressions turn out to be quite instructive to deduce physical meanings of the photoionization processes of molecules. As an illustrative example, we have applied our formulas to the photoionization process of H-2 molecules and found that our formulas reproduce the numerical results reported in the literature quite well. Without the Franck-Condon factors, our formulas cannot fit the numerical results well, which implies the importance of including properly the Franck-Condon factors for the tunneling photoionization processes of molecules. The results also indicate that the exponential factors which depend on the nuclear equilibrium state play a key role in determining the photoionization rates of the spatially aligned molecules. Comparing the Condon and non-Condon approximations shows that the Condon approximation is usually appropriate for the case of the laser polarization perpendicular to the molecular axis, while it is not necessarily true for the parallel case. Our theoretical results are also applied to analyze the experimental data of Urbain [Phys. Rev. Lett. 92, 163004 (2004)] for the photoionization process of H-2 molecules.