Electronic Excitation Energy Calculations with Configuration Interaction Based on Nonorthogonal Localized Molecular Orbitals

A method for electronic excited-state calculations with configuration interaction (CI) based on nonorthogonal localized molecular orbitals (NOLMOs) is developed in this work. The conventional CI is based on canonical Hartree-Fock molecular orbitals, and its total wave function is expanded as a linear combination of possible excitation states. In particular, CIS, configuration with single excitation states, is a widely used method for excited-state calculations. For noninteracting reference systems such as Hartree-Fock theory or Kohn-Sham density functional theory, NOLMOs had been developed for efficient representation and calculations of the determinant wave function instead of the canonical orbitals, which are the eigenstates of the reference one-electron Hamiltonian. In this work, we combine CI with NOLMOs and explore its application. With NOLMOs, the configurations beyond the ground state determinant are generated by the substitution of occupied NOLMOs with virtual NOLMOs, as in the conventional CI approach. However, unlike conventional canonical orbitals, virtual NOLMOs are not determined as the eigenstates of a reference one-electron Hamiltonian. Instead, they are determined from optimization as part of the total energy variational calculations. We first explore this combination at the single substitution level, denoted as NOLMO-CIS. Unlike the conventional CIS approach, where the number of single excitation states is equal to the number of occupied molecular orbitals multiplying to the number of virtual orbitals (<i>N</i>_occ<i>N</i>_vir), we show that the number of excited determinants of the single substitution excitation is independent of the number of basis functions but equals <i>N</i>, the number of electrons. Of course, the number of basis functions determines the NOLMOs. For ground-state calculation, unlike conventional CIS, NOLMO-CIS introduces a major portion of electron correlation but is not size consistent, similarly to truncated CI. For the excitation calculations, NOLMO-CIS shows very promising results. We use a state-averaged approach and minimize the sum of the ground and excited-state energies, maintaining the orthogonality of the many-electron ground- and excited-state wave functions. NOLMO-CIS significantly outperforms conventional CIS and time-dependent DFT in a test set of small molecules. Our work is a first step in the pathway for correlated electronic structure calculations with NOLMOs, which could lead to efficient computational methods for large systems.