Notes on density matrix perturbation theory

TL;DR

This paper explores and compares three types of density matrix perturbation theory (DMPT), reformulating existing methods and extending them to new contexts, demonstrating improved computational performance over traditional sum-over-states approaches.

Contribution

The paper reformulates two existing DMPT methods using the Sylvester equation and extends a recursive DMPT to hole-particle canonical purification, enhancing computational efficiency.

Findings

Reformulated Kussmann and Ochsenfeld's DMPT via Sylvester equation.

Extended recursive DMPT to hole-particle canonical purification.

Demonstrated improved performance over sum-over-states methods.

Abstract

Density matrix perturbation theory (DMPT) is known as a promising alternative to the Rayleigh-Schr\"odinger perturbation theory, in which the sum-over-state (SOS) is replaced by algorithms with perturbed density matrices as the input variables. In this article, we formulate and discuss three types of DMPT, with two of them based only on density matrices: the approach of Kussmann and Ochsenfeld [J. Chem. Phys.127, 054103 (2007)] is reformulated via the Sylvester equation, and the recursive DMPT of A.M.N. Niklasson and M. Challacombe [Phys. Rev. Lett. 92, 193001 (2004)] is extended to the hole-particle canonical purification (HPCP) from [J. Chem. Phys. 144, 091102 (2016)]. Comparison of the computational performances shows that the aformentioned methods outperform the standard SOS. The HPCP-DMPT demonstrates stable convergence profiles but at a higher computational cost when compared to the original recursive polynomial method