A comprehensive, self‐contained derivation of the one‐body density matrices from single‐reference excited‐state calculation methods using the equation‐of‐motion formalism

Abstract

In this contribution, we review in a rigorous, yet comprehensive fashion the assessment of the one‐body reduced density matrices derived from the most used single‐reference excited‐state calculation methods in the framework of the equation‐of‐motion formalism. Those methods are separated into two types: those which involve the coupling of a deexcitation operator to a single‐excitation transition operator, and those which do not involve such a coupling. The case of many‐body auxiliary wave functions for excited states is also addressed. For each of these approaches we were interested in deriving the elements of the one‐body transition and difference density matrices, and to highlight their particular structure. This has been accomplished by applying a decomposition of integrals involving one‐determinant quantum electronic states on which two or three pairs of second quantization operators can act. Such a decomposition has been done according to a corollary to Wick's theorem, which is brought in a comprehensive and detailed manner. A comment is also given about the consequences of using the equation‐of‐motion formulation in this context, and the two types of excited‐state calculation methods (with and without coupling excitations to deexcitations) are finally compared from the point of view of the structure of their transition and difference density matrices.

Visualizing and characterizing excited states from time-dependent density functional theory

“…Often, eqn (2.2) is simplified by invoking the Tamm-Dancoff approximation (TDA), 3,75 in which the de-excitation amplitudes y ia are neglected. These amplitudes arise naturally in the equation-of-motion formalism for the one-particle density matrix, 4,74 yet in molecular TD-DFT calculations they are typically B100Â smaller than the largest x ia . (This may not always be the case for solids.…”

“…It arises from a unique feature of single-excitation theories, namely, a direct correspondence between CI coefficients and matrix elements of the TDM. 18,73,74,112 In the CIS case, for example, x ia = hC exc |a ˆ † a a ˆi|C 0 i.…”

“…These are subject to an unconventional normalization,consistent with the metric matrix in eqn (2.2). 4,72–74 For brevity, we omit the state index n in eqn (2.4) and subsequent expressions. Amplitudes { x ia } and { y ia } parameterize the transition density matrix (TDM) for the excitation in question.…”

“…(2.2). 4,[60][61][62] For brevity, we omit the state index n in Eq. (2.4) and subsequent expressions.…”

“…It arises from a unique feature of single-excitation theories, namely, a direct correspondence between CI coefficients and matrix elements of the TDM. 14,61,62,100 For example, x ia = Ψ exc |â † a âi |Ψ 0 in the CIS case. Considering the specific case of ∆P in Eq.…”