Path integrals approach advantages

Other interesting treatments of the solid motion have been developed in which the motion of the solid s atoms is described by quantum mechanics [Billing and Cacciatore 1985, 1986]. This has been done for a harmonic solid in the context of treatment of the motion of the molecule by classical mechanics and use of a TDSCF formalism to couple the quantum and classical subsystems. The impetus for this approach is the fact that, if the entire solid is treated as a set of coupled harmonic oscillators, the quantum solution can be evaluated directly in an operator formalism. Then, the effect of solid atom motion can be incorporated as an added force on the gas molecule. Another advantage is the ability to treat the harmonic degrees of freedom of the solid and the harmonic electron -hole pair excitations on the same footing. The simplicity of such harmonic degrees of freedom can also be incorporated into the previously defined path-integral formalism in a simple manner to yield influence functionals (Feynman and Hibbs 1965). 

In the recent years, a new approaeh has been developed for eliminating this disadvantage [5]. This approach is called the quasi-sequential approach and takes the advantages of both the sequential and the simultaneous approaches sinee both the control and the state variables are diseretized, the path constraints for the state variables can be handled the DAE system is integrated only once, so the computation becomes more effieient. 

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