Numerical evidence invalidating finite-temperature many-body perturbation theory

Low-order perturbation corrections to the electronic grand potential, internal energy, chemical potential, and entropy of an ideal gas of noninteracting, identical molecules at a nonzero temperature are determined numerically as the λ-derivatives of the respective quantity calculated exactly (by thermal full configuration interaction) with a perturbation-scaled Hamiltonian, Hˆ₀+λVˆ. The data thus obtained from the core definition of any perturbation theory serve as a benchmark against which analytical formulas can be validated. The first- and second-order corrections from finite-temperature many-body perturbation theory discussed in many textbooks disagree with these benchmark data. This is because the theory neglects the variation of chemical potential with λ, thereby failing to converge at the exact, full-interaction (λ = 1) limit, unless the exact chemical potential is known in advance. The renormalized finite-temperature perturbation theory (Hirata and He, 2013) (15) is also found to be incorrect.