Mean-field trajectory method

The goal of this chapter is twofold. First we wish to critically compare—from both a conceptional and a practical point of view—various classical and mixed quantum-classical strategies to describe non-Born-Oppenheimer dynamics. To this end. Section II introduces five multidimensional model problems, each representing a specific challenge for a classical description. Allowing for exact quantum-mechanical reference calculations, aU models have been used as benchmark problems to study approximate descriptions. In what follows, Section III describes in some detail the mean-field trajectory method and also discusses its connection to time-dependent self-consistent-field schemes. The surface-hopping method is considered in Section IV, which discusses various motivations of the ansatz as well as several variants of the implementation. Section V gives a brief account on the quantum-classical Liouville description and considers the possibility of an exact stochastic realization of its equation of motion. 

Figure 3. Time-dependent simulations of the nonadiabatic photoisomerization dynamics exhibited by Model III, comparing results of the mean-field-trajectory method (dashed lines), the surface-hopping approach (thin lines), and exact quanmm calculations (full lines). Shown are the population probabilities of the initially prepared (a) adiabatic and (b) diabatic electronic state, respectively, as well as (c) the probability Pdsit) that the sytem remains in the initially prepared cis conformation.

III. MEAN-FIELD TRAJECTORY METHOD A. Classical-Path Approximation... 

In the MQC mean-field trajectory scheme introduced above, all nuclear DoF are treated classically while a quantum mechanical description is retained only for the electronic DoF. This separation is used in most implementations of the mean-field trajectory method for electronically nonadiabatic dynamics. Another possibility to separate classical and quantum DoF is to include (in addition to the electronic DoF) some of the nuclear degrees of freedom (e.g., high frequency modes) into the quantum part of the calculation. This way, typically, an improved approximation of the overall dynamics can be obtained—albeit at a higher numerical cost. This idea is the basis of the recently proposed self-consistent hybrid method [201, 202], where the separation between classical and quantum DoF is systematically varied to improve the result for the overall quantum dynamics. For systems in the condensed phase with many nuclear DoF and a relatively smooth distribution of the electronic-vibrational coupling strength (e.g.. Model V), the separation between classical and quanmm can, in fact, be optimized to obtain numerically converged results for the overall quantum dynamics [202, 203]. 

The mapping approach outlined above has been designed to furnish a well-defined classical limit of nonadiabatic quantum dynamics. The formalism applies in the same way at the quantum-mechanical, semiclassical (see Section VIII), and quasiclassical level, respectively. Most important, no additional assumptions but the standard semiclassical and quasi-classical approximations are needed to get from one level to another. Most of the established mixed quantum-classical methods such as the mean-field-trajectory method or the surface-hopping approach do invoke additional assumptions. The comparison of the mapping approach to these formulations may therefore (i) provide insight into the nature of these additional approximation and (ii) indicate whether the conceptual virtues of the mapping approach may be expected to result in practical advantages. 

Let us first consider the relation to the mean-field trajectory method discussed in Section III. To make contact to the classical limit of the mapping formalism, we express the complex electronic variables imaginary parts, that is, mean-field Hamiltonian function which may be defined as... 

Figure 19. Time-dependent (a) diabatic and (b) adiabatic electronic excited-state populations and (c) vibrational mean positions as obtained for Model 1. Shown are results of the mean-field trajectory method (dotted lines), the quasi-classical mapping approach (thin full lines), and exact quantum calculations (thick full lines).

A value of y = 1 corresponds to the original mapping formulation, which takes into account the full amount of ZPE. If, on the other hand, all electronic ZPE is neglected (i.e., y = 0), the mapping approach becomes equivalent to the mean-field trajectory method. 

Let us briefly review what MQC methods have so far been applied to the dynamics at conical intersections. Here the surface-hopping method has been the most popular approach, " in particular in combination with an on-the-fly ab initio evaluation of the potential-energy. Furthermore, various self-consistent-field methods have been employed to describe internal-conversion dynamics associated with a conical intersection, including the mean-field trajectory method, the classical electron... 

Finally, we consider Model III, which describes an ultrafast photoin-duced isomerization process. Figure 13 shows quantum mechanical results as well as results of the ZPE-corrected mapping approach for three different observables the adiabatic and diabatic population of the excited state, and Pcis> the probability that the system remains in the initially prepared cis-conformation. A ZPE correction of 7 = 0.5 has been used in the mapping calculation, based on the criterion to reproduce the quantum mechanical long-time limit of the adiabatic population. It is seen that the ZPE-corrected mapping approach represents an improvement compared to the mean-field trajectory method (cf. Fig. 3), in particular for the adiabatic population. The influence of the ZPE correction on the dynamics of the observables is, however, not as large as in the two other models. 

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