Is the TDDFRT representation of a molecular electronic transition unique?

Abstract

In this article a piece of the reference time-dependent density-functional response theory (TDDFRT) representation of molecular electronic transitions is studied through the one-body reduced difference density matrix. A first derivation route for that object has been reported in the literature based on the substitution of TDDFRT-related objects into an exact functional expression. We show in the text that the functional providing the exact object is not unique, and we question whether substituting the same TDDFRT-related objects into the alternative functional expressions leads to acceptable candidates for the approximate TDDFRT one-body reduced difference density matrix, which would question the unequivocality of the TDDFRT representation of molecular electronic transitions. We first directly address the problem using a novel diagrammatic language, inspired by Dyck languages, and specifically designed for dealing with this kind of issue. In the last part of the article we question whether it is possible or not to recast the problem in a more general framework - the equations-of-motion framework. Through all the text, the case of time-dependent Hartree-Fock is compared with that of TDDFRT to highlight what makes them so different when those questions are at stake.