# Converging finite-temperature many-body perturbation theory that conserves the average number of electrons

## TL;DR

This paper introduces a finite-temperature perturbation theory for the grand canonical ensemble that maintains charge neutrality by conserving the average number of electrons at each perturbation order, with analytical formulas verified against benchmark data.

## Contribution

It develops a new finite-temperature perturbation framework that conserves electron number and provides analytical formulas for first-order corrections, bridging zero-temperature theories.

## Key findings

- Formulas verified against benchmark data
- Reduces to Møller-Plesset theory for nondegenerate ground states
- Correct zero-temperature limit for degenerate ground states

## Abstract

A finite-temperature perturbation theory for the grand canonical ensemble is introduced that expands chemical potential in a perturbation series and conserves the average number of electrons, ensuring charge neutrality of the system at each perturbation order. Two classes of (sum-over-state and reduced) analytical formulas are obtained in a straightforward, algebraic, time-independent derivation for the first-order corrections to chemical potential, grand potential, and internal energy, with the aid of several identities of the Boltzmann sums also introduced in this study. These formulas are numerically verified against benchmark data from thermal full configuration interaction. In the zero-temperature limit, the finite-temperature perturbation theory reduces analytically to and is consistent with the M{\o}ller-Plesset perturbation theory, but only for a nondegenerate ground state. For a degenerate ground state, its correct zero-temperature limit is not the M{\o}ller-Plesset perturbation theory, but the Hirschfelder-Certain degenerate perturbation theory.