An Energy Decomposition Method for Electronic Excitations in Two-Determinant Restricted Open-Shell Kohn-Sham Theory

Restricted open-shell Kohn-Sham (ROKS) theory is a widely used approach for accurately describing single-electron excitations. However, a proper treatment of ROKS excited states requires two determinants, which poses challenges for conventional single-determinant energy decomposition analysis (EDA). In this work, we develop a new EDA framework for ROKS states by mapping the problem onto an effective single-determinant form and applying the theories of natural orbitals for chemical valence (NOCV) and occupied-virtual orbitals for chemical valence (OVOCV) to analyze electronic excitations. The goal is to understand an electronic excitation as the union of a primary orbital excitation to convert a closed-shell ground state into an open-shell excited state (defined by an intermediate frozen state that is free of polarization effects), with the secondary polarization or relaxation of other electron pairs (defined by the change from the frozen ROKS state to the final ROKS state). Our theory attaches energy changes and electron promotion numbers to each component and also separates the relaxation process into separate occupied-to-virtual contributions ranked by significance using the OVOCV approach. In this way, our ROKS excitation EDA can yield chemical insights from the energetic contributions of relaxation processes and the accompanying charge redistribution that are not otherwise accessible. The method is demonstrated on several representative examples, including the n → π* and π → π* valence excitations of formaldehyde, the 1s_Cl → σ* core excitation of HCl, a long-range charge-transfer excitation in the NH₃-F₂ complex, and the lowest valence excitation of (dimethylamino)benzonitrile (DMABN) with a hydrogen-bonded water molecule, which exhibits notable intramolecular charge transfer (ICT) character in DMABN.