Propagators in Quantum Chemistry, 2nd Edition

Appendix E

Temperature-Dependent Propagators

E.1   Preliminaries

A diagram technique can be constructed for so-called “temperature Green’s functions”, which depend on a temperature T, or rather on , which varies from 0 to 1/kT, where k is Boltzmann’s constant.

One seeks a perturbation expansion of the partition function

with β = 1/kT, H the many-electron Hamiltonian, N₀ the number operator for electrons, and μ a parameter, which may be called the “chemical potential”. The trace is taken in Fock space; i.e., the summation is taken over all possible states of the system with a given number of electrons and over all numbers of electrons.

The Hamiltonian is partitioned into a reference or unperturbed part H₀ and a perturbation V as

with

in diagonal form. The number operator in the same basis is

A partition function for the unperturbed system is

and the trace can be taken over a direct product ...