In this book, we have discussed the problems concerning mixing of classical and quantum mechanics, and we have given several possible solutions to the problem and a number of suggestions for the setup of working computational schemes. In the present chapter, we give some recommendations as to which methods one should use for a given type of system and problem. As can be seen from the tables and what is apparent from the discussion in the previous chapters, the quantum-classical method has been and is used for solving many different molecular dynamics problems. Recommendations, as far as molecule surface or processes in solution are concerned, have not been incorporated, the reason being that the methods here are still to some extent under development. We have seen that the quantum-classical approach can be derived in two different fashions. In one method the classical limit ħ→ 0 is taken in some degrees of freedom. In the other approach the quantum mechanical equations are parameterized in such a fashion that classical equations of motions are either pulled out of or injected into the quantum mechanical. Thus the first method involves and introduces the classical picture in certain particular degrees of freedom—in the second method the classical picture is in principle not introduced—it is just a reformulation of quantum mechanics. This reformulation has the exact dynamics as the limit. However, if exact calculations are to be performed, the reformulation may not be advantageous from a computational point of view, and, hence, standard methods are often more conveniently applied. We prefer the second approach for introducing the quantum-classical scheme because, as mentioned, it automatically has the exact formulation as the limit. The approach is most conveniently implemented through the trajectory driven DVR, or the so-called TDGH-DVR method, which gives the systematic way of approaching the quantum mechanical limit from the classical one. Thus, the method interpolates continuously between the classical and the quantum limit—a property it shares with, for instance, the FMS method and the Bohm formulation.
QUANTUM SINGULARITIES IN STATIC SPACETIMES