The influence functional can be obtained exactly in some cases (e.g. separable or quadratic bath, and extended systems of a one-dimensional topology) and by approximate methods (e.g. semiclassical theory) in other cases. See the Modular Path Integral (MPI) decomposition and Quantum-Semiclassical Dynamics for more information.

As discussed in the context of the linear response approximation derived through a cumulant expansion of the influence functional, arguments similar to those employed in the i-QuAPI methodology lead to an iterative decomposition of the path integral even with anharmonic, fully coupled condensed phase environments.  In this case, the effective system propagator is obtained from the values of the influence functional for appropriate configurations of forward and backward system path segments that span the relevant memory length.

Related articles:

• G. Ilk and N. Makri, “Real time path integral methods for a system coupled to an anharmonic bath”, J. Chem. Phys. 101, 6708-6716 (1994).
• K. Forsythe and N. Makri, “Dissipative tunneling in a bath of two-level systems”, Phys. Rev. B 60, 972-978 (1999).
• N. Makri, “Iterative evaluation of the path integral for a system coupled to an anharmonic bath”, J. Chem. Phys. (Rapid Commun.) 111, 6164-6167 (1999).
• J. Shao and N. Makri, “Influence functional from a bath of coupled time-dependent harmonic oscillators”, Phys. Rev. E 59, 269-274 (1999).
• N. Makri and K. Thompson, “Semiclassical influence functionals for quantum systems in anharmonic environments”, Chem. Phys. Lett. 291, 101-109 (1998).
• K. Thompson and N. Makri, “Influence functionals with semiclassical propagators in combined forward-backward time”, J. Chem. Phys. 110, 1343-1353 (1999).