The motivation for NOCISOosterbaan:2018, Oosterbaan:2019, Oosterbaan:2020 is the desire to improve on CIS while still maintaining a reasonably low computational scaling. It does so by including orbital relaxation, which CIS neglects altogether, and the non-orthogonal interaction between multiple core-hole references, such as the O orbitals in O2.
A brief overview of the NOCIS algorithm is as follows: after a ground-state orbital optimization, a Maximum Overlap Method (MOM)Gilbert:2008 is done for an ionization from each localized core orbital of interest. This introduces orbital relaxation, and also renders the excited states non-orthogonal to the ground state. The Hamiltonian, overlap, and total spin squared matrices are constructed using the Slater-Condon rules for matrix elements between determinants which share a common set of orbitals and NOCI for the remaining matrix elementsThom:2009b. Finally, the generalized eigenvalue problem is solved.
A key feature in open-shell NOCIS is a separate optimization of any open-shell references, which are states in which a core-electron is excited to a singly-occupied ground-state orbital. These separate optimizations render these states non-orthogonal to the other excited states.
NOCIS is spin-pure, size consistent, and maintains spatial symmetry. Like CIS, NOCIS produces excited states with the same as the reference but potentially with larger total spin. For example, performing NOCIS on a molecule with a singlet ground state will produce both singlet and triplet excited states.