Excitation/Relaxation Analysis of Electronic Transitions Using Difference Density Natural Orbitals

Characterizing an electronic excitation in terms of its underlying orbital reorganization is central to understanding photochemical and photophysical processes. Here, we introduce an excitation/relaxation framework that separates electron promotion from orbital relaxation contributions within Δ-self-consistent-field (ΔSCF) treatments of electronic transitions. The framework defines the excitation number and the relaxation number, which quantify electron promotion and electron relaxation. Formulated in terms of difference density natural orbitals (DDNOs), the approach generalizes earlier attachment/detachment and natural ionization orbital models. A compact set of modified Slater-Condon rules derived in the DDNO basis enables direct evaluation of transition dipole moments and oscillator strengths. Application to a test set of 5 molecules and 19 ΔSCF excitations demonstrates that the model yields integer excitation numbers and interpretable relaxation numbers, and that the corresponding DDNOs can be visualized to display particle/hole and relaxation pairs.