Boundary values for the charge transferred during an electronic transition: insights from matrix analysis
Enzo Monino, Jérémy Morere, Thibaud Etienne
TL;DR
This work addresses boundary values on the charge transferred during a molecular electronic transition by comparing relaxed and unrelaxed detachment/attachment density pictures. It develops two rigorous derivations—one via Haynsworth inertia additivity with Courant-Fischer arguments and a Lidskii-Wielandt corollary, and a second via Cauchy interlacing—to establish a lower bound $S_+(\bm{\gamma}^\Delta_{rlx}) \ge \vartheta$, where $\vartheta = S_+(\bm{\gamma}^\Delta)$. It further shows that the basis-relaxation contribution yields an upper bound $q^{CT}_{rlx} \le \vartheta + \vartheta^Z$, with $\vartheta^Z = S_+(\bm{\gamma}^Z)$ equal to the sum of the singular values of $\mathbf{Z}$, so the total boundary value is computable from at most two matrix-trace computations and an SVD. The results generalize a long-standing CIS conjecture to TDHF, TDDFT, and the Bethe-Salpeter framework, and reveal that excitation and relaxation contributions are not simply additive in the detachment/attachment description. These boundary values provide practical, quantitative limits on light-induced electronic reorganization and can be used as sharp checks in excited-state analyses.
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.