Hirata So
Department of Chemistry, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA.
J Chem Phys. 2021 Sep 7;155(9):094106. doi: 10.1063/5.0061384.
A comprehensive and detailed account is presented for the finite-temperature many-body perturbation theory for electrons that expands in power series all thermodynamic functions on an equal footing. Algebraic recursions in the style of the Rayleigh-Schrödinger perturbation theory are derived for the grand potential, chemical potential, internal energy, and entropy in the grand canonical ensemble and for the Helmholtz energy, internal energy, and entropy in the canonical ensemble, leading to their sum-over-states analytical formulas at any arbitrary order. For the grand canonical ensemble, these sum-over-states formulas are systematically transformed to sum-over-orbitals reduced analytical formulas by the quantum-field-theoretical techniques of normal-ordered second quantization and Feynman diagrams extended to finite temperature. It is found that the perturbation corrections to energies entering the recursions have to be treated as a nondiagonal matrix, whose off-diagonal elements are generally nonzero within a subspace spanned by degenerate Slater determinants. They give rise to a unique set of linked diagrams-renormalization diagrams-whose resolvent lines are displaced upward, which are distinct from the well-known anomalous diagrams of which one or more resolvent lines are erased. A linked-diagram theorem is introduced that proves the size-consistency of the finite-temperature many-body perturbation theory at any order. General-order algorithms implementing the recursions establish the convergence of the perturbation series toward the finite-temperature full-configuration-interaction limit unless the series diverges. The normal-ordered Hamiltonian at finite temperature sheds light on the relationship between the finite-temperature Hartree-Fock and first-order many-body perturbation theories.
本文给出了电子有限温度多体微扰理论的全面详细描述，该理论在同等基础上以幂级数展开所有热力学函数。针对巨正则系综中的巨势、化学势、内能和熵以及正则系综中的亥姆霍兹自由能、内能和熵，推导了瑞利 - 薛定谔微扰理论风格的代数递归关系，从而得到任意阶的态求和解析公式。对于巨正则系综，通过扩展到有限温度的正规序二次量子化和费曼图的量子场论技术，将这些态求和公式系统地转换为轨道求和简化解析公式。研究发现，进入递归关系的能量微扰修正必须作为非对角矩阵处理，其非对角元素在由简并斯莱特行列式所张成的子空间内通常非零。它们产生了一组独特的链接图——重整化图——其解析线向上移动，这与众所周知的异常图不同，异常图中一条或多条解析线被擦除。引入了一个链接图定理，证明了有限温度多体微扰理论在任意阶的大小一致性。实现递归关系的一般阶算法确定了微扰级数朝着有限温度全组态相互作用极限的收敛性，除非级数发散。有限温度下的正规序哈密顿量揭示了有限温度哈特里 - 福克理论与一阶多体微扰理论之间的关系。
