Frozen Gaussian approximation

The Hemian-Kluk method has been developed further [153-155], and used in a number of applications [156-159]. Despite the formal accuracy of the approach, it has difficulties, especially if chaotic regions of phase space are present. It also needs many trajectories to converge, and the initial integration is time consuming for large systems. Despite these problems, the frozen Gaussian approximation is the basis of the spawning method that has been applied to... 

As usual there is the question of the initial conditions. In general, more than one frozen Gaussian function will be required in the initial set. In keeping with the frozen Gaussian approximation, these basis functions can be chosen by selecting the Gaussian momenta and positions from a Wigner, or other appropriate phase space, distribution. The initial expansion coefficients are then defined by the equation... 

The previous subsections defined the AIMS method, the various approximations that one could employ, and the resulting different limits classical mechanics, Heller s frozen Gaussian approximation, and exact quantum mechanics. As emphasized throughout the derivation, the method can be computationally costly, and this is one of the reasons for developing and investigating the accuracy of various approximations. Alternatively, and often in addition, one could try to develop algorithms that reduce the computational cost of the method without compromising its accuracy. In this subsection we discuss two such extensions. Each of these developments has been extensively discussed in a publication, and interested readers should additionally consult the relevant papers (Refs. 125 and 41, respectively). We conclude this subsection with a discussion of the first steps... 

Free and Lombardi (FL) models, Renner-Teller effect, triatomic molecules, 618-621 Free electrons, electron nuclear dynamics (END), time-dependent variational principle (TDVP), 333-334 Frozen Gaussian approximation direct molecular dynamics ... 

One can (at one s peril) freeze the elements of the cross correlation matrix at their initial value (which is the frozen Gaussian approximation) or evaluate their time dependence from the closed set of equations of motion... 

On the other hand, to mimic the full Gaussian quantum wavefunction, we employed the so-called frozen Gaussian approximation to generate nuclear (basis) functions, that is. 

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