### Abstract

In the path integral representation of quantum theory, a few body quantum problem becomes a classical many body problem. To exploit this isomorphism, it becomes necessary to develop methods by which degrees of freedom can be explicitly removed from consideration. The interactions among the remaining relevant variables are described by effective interactions. In this paper, we present a general methodology to carry out the reduction in numbers of degrees of freedom. Certain path integral algorithms are shown to correspond to reference systems for the full isomorphic classical many body problem. The correspondence allows one to determine systematic corrections to the algorithms by low order perturbation approximations familiar in the theory of simple classical fluids. We show how to use discretized path integrals to compute rigorous upper and lower bounds to the free energy for nontrivial quantum systems, and we discuss how to optimize the upper bounds with variational theories. Several illustrative examples are provided including treatments of a tunneling system that is strongly coupled to a nonadiabatic heat bath. The examples demonstrate that even for strongly perturbed systems in which the quantum energy scales are as large as 40k_{B}T, the methods discussed herein provide a practical route for obtaining equilibrium properties with both high accuracy and analytical simplicity.

Original language | English (US) |
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Pages (from-to) | 1347-1364 |

Number of pages | 18 |

Journal | The Journal of Chemical Physics |

Volume | 75 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 1981 |

### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Physical and Theoretical Chemistry

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## Cite this

*The Journal of Chemical Physics*,

*75*(3), 1347-1364. https://doi.org/10.1063/1.442141