**Forward – Backward Semiclassical Dynamics**

Forward-backward semiclassical dynamics (FBSD) provides a practical methodology for including quantum mechanical effects in classical trajectory simulations of polyatomic systems. The basic FBSD idea is to combine the evolution operator and its adjoint, both of which enter quantum mechanical expressions for time correlation functions or expectation values, into a single operator along a forward-backward time contour, which is evaluated using the semiclassical approximation in a coherent state representation. The phase in the FBSD expression is given by the net forward-backward classical action, which generally is small, leading to a natural smoothing of the integrand and alleviating the Monte Carlo sign problem. The forward-backward semiclassical approximation to influence functionals, in conjunction with an iterative decomposition of the path integral, leads to a rigorous quantum-semiclassical methodology for simulating the dynamics of a system interacting with a general anharmonic environment.

Our group has developed the “derivative-FBSD” formulation of time correlation functions, which is practical for application to large clusters and condensed phase processes. In this, FBSD expressions for time-dependent expectation values or correlation functions take the form of phase space integrals with respect to trajectory initial conditions, weighted by the coherent state transform of a corrected density operator. The discretized path integral representation of statistical mechanics allows an accurate evaluation of the relevant phase space density, thus ensuring a proper treatment of zero point effects and capturing important imaginary components that are absent from purely classical trajectory methods. Optimal sampling is achieved through Monte Carlo or molecular dynamics techniques. The methodology has been applied to investigate the dynamical properties of various quantum fluids.

*Related articles:*

- Semiclassical influence functionals for quantum systems in anharmonic environments.
- Influence functionals with semiclassical propagators in combined forward-backward time.
- Rigorous forward-backward semiclassical formulation of many-body dynamics.
- Forward-backward semiclassical dynamics without prefactors.
- Forward-backward semiclassical dynamics with linear scaling.
- Forward-backward semiclassical calculation of spectral lineshapes.
- Forward-backward semiclassical dynamics in the interaction representation.
- Finite temperature correlation functions via forward-backward semiclassical dynamics.
- Quasiclassical dynamics methods from semiclassical approximations.
- Coherent state semiclassical initial value representation for the Boltzmann operator in thermal correlation functions.
- Monte Carlo evaluation of forward-backward semiclassical correlation functions with a quantized coherent state density.
- Forward-backward semiclassical dynamics for condensed phase time correlation functions.
- Forward-backward semiclassical dynamics for quantum fluids using pair propagators: application to liquid
*para*-hydrogen.

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