Path integral Monte Carlo representation

In Eq. (3.28), erf(2) is the error function of a complex variable [98]. The third, and perhaps most intriguing aspect of Eqs. (3.25), (3.27), and (3.28) is that they have been derived before in a completely different context by Makri [69]. In particular, she was seeking a well behaved (i.e. less oscillatory) representation of the short-time kernel for use in real time path integral Monte Carlo calculations [63-68, 70-76] The advantage gained from the matrix element in Eqs. (3.27) and (3.28) derives from the asymptotic behavior of the smoothing function [69, 81], namely... 

Implementation of the FBSD schemes requires knowledge of the initial density matrix in the coherent state representation. Usually, the initial density corresponds either to the ground vibrational state of a polyatomic molecule or a Boltzmann distribution. Below we describe ways of obtaining the coherent state matrix element through closed form expressions or in terms of an imaginary time path integral evaluated along the same Monte Carlo random walk which samples the trajectory initial conditions. 

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