7.7.1 NOCIS

The motivation for NOCIS ⁹¹⁷ Oosterbaan K. J., White A. F., Head-Gordon M.
J. Chem. Phys.
(2018), 149, pp. 044116. Link ^{, 918} Oosterbaan K. J., White A. F., Head-Gordon M.
J. Chem. Theory Comput.
(2019), 15, pp. 2966. Link ^{, 916} Oosterbaan K. J. et al.
Phys. Chem. Chem. Phys.
(2020), 22, pp. 8182. Link 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) ³⁹² Gilbert A. T. B., Besley N. A., Gill P. M. W.
J. Phys. Chem. A
(2008), 112, pp. 13164. Link 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 ¹²²⁰ Thom A. J. W., Head-Gordon M.
J. Chem. Phys.
(2009), 131, pp. 124113. Link . 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.