Converging finite-temperature many-body perturbation theory in the grand canonical ensemble that conserves the average number of electrons

Abstract

A finite-temperature perturbation theory for the grand canonical ensemble is introduced that expands the chemical potential in a perturbation series and conserves the average number of electrons, ensuring charge neutrality of the system at each perturbation order. Two (sum-over-state and reduced) classes of analytical formulas are obtained in a straightforward, algebraic, time-independent derivation for the first-order corrections to the 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. For a nondegenerate ground state, the finite-temperature perturbation theory reduces analytically to and is consistent with the Møller-Plesset perturbation theory as temperature (T ) tends to zero. For a degenerate ground state, it should instead reduce to the Hirschfelder-Certain degenerate perturbation theory as T → 0.

Finite-temperature many-body perturbation theory for electrons: Algebraic recursive definitions, second-quantized derivation, linked-diagram theorem, general-order algorithms, grand canonical and canonical ensembles

“…In this section, we derive the reduced analytical formulas of Ω (n) and µ (n) for the few lowest n's, 54,55 starting from the sumover-states analytical formulas obtained from the recursions.…”

“…For instance, the grand potentials and chemical potentials are treated differently 53 with no analytical formulas available for the lowest-order corrections to the chemical potential, internal energy, or entropy for a long time. 54,55 The time-dependent, diagrammatic formulation exploiting the isomorphism of the Schrödinger and Bloch equations is elegant at the start, [46][47][48][49][50][51][52] but it becomes quickly inscrutable with the exhaustiveness of diagram enumeration being uncertain. Convergence of the finite-temperature MBPT to the well-established zerotemperature MBPT has been suspect, 48,49,56 compelling Kohn and Luttinger 48 to conclude at one point that the theory "is in general not correct."…”

“…The relationship between the widely used finite-temperature Hartree-Fock (HF) theory 58 and finite-temperature MBPT is also unknown, and as a result, the physical meaning 59 of their orbital energies and thus (quasiparticle) energy bands has not been established. 60 We recently introduced 54,55 a new finite-temperature MBPT in the grand canonical ensemble, which expands in power series the grand potential, chemical potential, internal energy, and entropy on an equal footing. We obtained sum-overstates and sum-over-orbitals analytical formulas for the firstand second-order perturbation corrections to these thermodynamic functions in a transparent algebraic derivation involving only the combinatorial identities and the energy sum rules of the Hirschfelder-Certain degenerate perturbation theory (HCPT).…”

“…Fortunately, however, we can still derive analytical formulas without knowing the eigenvalues. 54,55 Since these energy corrections are thermally averaged [Eq. (12)] with the same Boltzmann weight within each degenerate subspace, we only need the sum of the eigenvalues (not the individual eigenvalues) in order to calculate the thermal average correctly.…”

“…Each matrix element (before diagonalization) can further be written in a closed form, lending the trace and thus thermal average to analytical expressions. In our previous studies, 54,55 we used the Slater-Condon rules 73 to evaluate these matrix elements and derived compact analytical formulas for the thermal averages with custommade combinatorial identities. Such an order-by-order approach, however, reaches an impasse at higher orders, and hence, in this study, we switch to the second-quantized and diagrammatic formulations expounded on in the subsequent sections.…”