Boundary values for the charge transferred during an electronic transition: insights from matrix analysis

TL;DR

This paper proves and generalizes a decades-old conjecture relating the integral of detachment/attachment densities to electronic charge transfer during molecular transitions, providing a simple computational method for boundary values.

Contribution

It introduces a new theoretical framework combining matrix theorems to bound charge transfer, simplifying calculations to basic matrix operations.

Findings

Established an upper bound for charge transfer during electronic transitions.

Demonstrated the bound can be computed using singular value decomposition.

Generalized a longstanding conjecture with new matrix analysis techniques.

Abstract

In this contribution we start by proving and generalizing a conjecture that has been established few decades ago, relating the value of the integral of the detachment/attachment density in two pictures - one accounting for transition-induced basis relaxation and one which does not account for such a relaxation. To this end, we show that it is possible to follow two ways: one combines Haynsworth and Courant-Fischer theorems with a corollary to Lidskii-Wielandt theorem, the other combines two twin theorems extending Cauchy's interlacing theorem, together with the abovementioned corollary to Lidskii-Wielandt theorem. These derivations allow us to provide an upper bound for the electronic charge that is effectively displaced during the molecular electronic transition from one electronic quantum state to another. This quantity can be regarded as the neat charge that has been transferred during the transition. Our derivations ultimately show that this boundary value can be determined from a simple singular value decomposition and at most two matrix trace-computing operations.