Semiclassical Ehrenfest theory

Another basic theory of nonadiabatic transitions is the semiclassical Ehrenfest theory (SET). Although it can cope with multidimensional nonadiabatic electronic-state mixing, it inevitably produces a nuclear path that runs on an averaged potential energy surface after having passed across the nonadiabatic region, which is totally unphysical. Unfortunately, since SET seems intuitively correct, a naive and conventional derivation of this theory obscures how this critical difficulty arises. 

These simple conservation laws are nevertheless not assmed, in fact they are sometimes even severely violated, in many of existing calculation methods including mean-field methods like the semiclassical Ehrenfest theory (SET). The surface hopping method circumvents the problem by adjusting the linear momenta of nuclei so as to satisfy the asymptotic energy conservation law. 

Mean-field path representation Semiclassical Ehrenfest theory... 

We next turn to much simpler and easily calculable methods. The semiclassical Ehrenfest theory (SET) is based on rather an intuitive combination of the electronic dynamics on time-dependent nuclear configuration and the Newtonian dynamics of point-like nuclei using the wavepacket average of the Hellmann-Feynman force. 

The basic idea of the algorithm is to allow a path to jump with the prob-abihty estimated internally (in a self-contained maimer) with the electronic-state mixing as in the semiclassical Ehrenfest theory but the number of transitions (hops) should be minimized. Suppose we have an electronic wavepacket (t)) = X]/C /(t) >/) as in the SET, the dynamics of the corresponding density matrix, pij t) = Cj t)Cj t), is written as... 

Thus the theoretical basis of the semiclassical Ehrenfest theory is found on a purely theoretical ground. 

This section follows the above basic theory and method of path-branching representation of nonadiabatic dynamics with numerical examples. We also show the performance of the semiclassical Ehrenfest theory to compare with. What we present below is illustrative numerical realization rather than optimized practice for actual applications. 

Practices in the semiclassical Ehrenfest theory and the full quantum dynamics of nuclear wavepacket dynamics... 

All the initial electronic state populations are set to be localized at the energetically highest adiabatic state to be compatible with the calculation for PSANB and the semiclassical Ehrenfest theory. The momentum of the wavepacket is varied by changing the initial wave number as k =16.05, 32.11, 48.16, 64.22, 80.28, 96.33, 112.39, 128.44, 144.50 and 160.56 a.u., which specifies the initial Gaussian wavepacket of minimum uncertainty... 

As observed above, both the semiclassical Ehrenfest and PSANB give highly accurate electronic transition probabilities, which are quite close to the full quantum counterparts. In particular, the semiclassical Ehrenfest offers a simpler and convenient computational scheme as far as the electronic transition probability is concerned. Indeed, we have been studying electron current dynamics in chemical reactions with this method. [307] On the other hand, there are situations in which the semiclassical Ehrenfest theory totally fails to reproduce correct dynamics. Even in such a case, PSANB provides not only accurate transition probability but also physically correct view of nuclear motion. We next show such an illustrative example. 

The first escape probabilities thus calculated are tabulated in the Table. The semiclassical Ehrenfest fails completely in this tough situation. On the other hand, PSANB reproduces the probability remarkably well. No further comment is necessary for the quality of the transition probability given by PSANB, since it has been already described in a great detail. Thus, it turns out that PSANB scheme may be used as a promising alternative to the semiclassical Ehrenfest theory. 

Fig. 6.23 PSANB against ID-PSANB in the trajectory branching in LiH+H+ collision dynamics. CISD/6-31G. Initial condition is the second state and the same as the example for 0=90° used in the Advances in Chemical Physics review article. Dotted lines correspond to semiclassical Ehrenfest theory. Branching paths construction in the multidimensional PSANB is totally the same as in previous articles involved with the conical intersection. Employed inverse energy thresholds for coupling entrance and branching were 1/0.010 and 1/0.015 Hartree, respectively.

With increasing system size, the implementation of ab initio electron wavepacket dynamics, such as the semiclassical Ehrenfest theory, using nuclear derivative coupling tends to be computationally more demanding because of the necessity of solving coupled perturbed equations. We therefore propose a useful treatment of nonadiabatic coupling, in which one can avoid the tedious coupled perturbed equations for the nuclear derivative of molecular orbitals and CSFs. 

The semiclassical Ehrenfest theory coupled with this representation was applied in an electron flux analysis in chemical reactions where large charge transfer occurs caused by significant nonadiabatic transition. The chemical systems treated in the summaries below are Na - - Cl and formic acid dimer (FAD). The time shift flux operator stated in the previous subsection was utilized in an analysis of the microscopic electron d3mamics in this chemically representative case. 

The electronic wavepackets are determined in an expansion with the configuration state functions (CSF) of single and double excitations (CISD) with the STO-6G basis set. This basis set is obviously small, but the main concern in this work is not the accm-acy but qualitative insights about the dynamical electrons in the course of chemical reactions. The program codes for the semiclassical Ehrenfest theory (SET) have been implemented in the GAMESS package [357]. 

Among the sampled paths chosen as above for the semiclassical Ehrenfest theory, we pick a couple of examples as a generic case-study of proton transfer . 

A very basic and simple molecular collision, LiH + H+ in a laser field, is investigated with the semiclassical Ehrenfest theory. The potential surfaces for (LiH2)+ system are drawn in Figs. 8.7 and 8.8. 

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