Finite-temperature many-body perturbation theory for electrons

Correct benchmark perturbation corrections, up to third order, to the electronic internal energy, chemical potential, grand potential, and entropy of an ideal gas of noninteracting, identical molecules in a wide range of temperature were determined numerically as the λ-derivatives of the respective quantity calculated exactly by a thermal full configuration interaction method using a perturbation-scaled modified Hamiltonian, H^0 +λV^ . We refer to this numerical framework as the λ-variation method. Numerical results obtained from first- and second-order corrections from finite-temperature many-body perturbation theory described in textbooks were found to be not identical to the benchmark λ-variation results calling into question the validity of this theory. The results obtained from renormalized finite-temperature perturbation theory [S.Hirata and X. He, J. Chem. Phys., 138, 204112 (2013)] also did not match with the benchmark λ-derivative results and hence were found to be incorrect. Further analysis of this discrepancy between the textbook finite-temperature perturbation theory and the benchmark numerical λ-variation results was traced not so much to mathematical issues but to a misuse of the grand canonical ensemble for systems with a small number of electrons. This misuse of the definition of the grand canonical ensemble lead to the breakdown of the charge neutrality of the system when a perturbation is introduced. A new finite-temperature perturbation theory for the grand canonical ensemble was introduced that expanded the electronic grand potential, chemical potential, internal energy, and entropy while the average number of electrons in the grand canonical system was kept fixed. This new ansatz which conserved the average number of electrons, ensured charge neutrality of the system at each perturbation order. Two classes of analytical formulas, sum-over-states, and sum-over-orbitals, were obtained algebraically in a time-independent and nondiagrammatic derivation for the first and second-order corrections to chemical potential, grand potential, and internal energy. This derivation made use of several Boltzmann identities and sum rules from the Hirschfelder--Certain degenerate perturbation [J. O. Hirschfelder and P. R. Certain, J. Chem. Phys., 60, 1118 (1974)] energies in a degenerate subspace. These analytical formulas reproduced the benchmark λ-derivative results numerically exactly, and are, therefore, correct. Finite-temperature many-body perturbation theory was extended to the canonical ensemble. Benchmark perturbation corrections for the zeroth- through third-order many-body perturbation corrections to the electronic Helmholtz energy, internal energy, and entropy in the canonical ensemble in a wide range of temperature using the λ-variation method. Sum-over-states analytical formulas for up to the third-order correction to the thermodynamic quantities in the canonical ensemble were also derived as analytical λ-derivatives in terms of the Hirschfelder--Certain degenerate perturbation energies and are valid for both degenerate and nondegenerate reference states at all temperatures. These analytical formulas were in exact agreement with the benchmark numerical data obtained from the λ-variation method.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Chemistry

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2022

Author dc:creator

• Jha, Punit K.

Contributors dc:contributor

• Hirata, So
• Ceperley, David
• Makri, Nancy
• Schweizer, Kenneth

• finite-temperature
• perturbation theory
• many-body methods

Statement dc:rights

• Copyright 2021 by Punit K. Jha.

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/112981

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/112981