The motivation for NOCIS^{Oosterbaan: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 $1s$ orbitals in O_{2}.

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 elements^{Thom: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 ${m}_{s}$ 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.