7.7.1 Non-Orthogonal CIS (NOCIS)

The motivation for the NOCIS method ⁹⁵⁵ Oosterbaan K. J., White A. F., Head-Gordon M.
J. Chem. Phys.
(2018), 149, pp. 044116. Link ^{, 956} Oosterbaan K. J., White A. F., Head-Gordon M.
J. Chem. Theory Comput.
(2019), 15, pp. 2966. Link ^{, 954} 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: following a ground-state orbital optimization, an SCF calculation is performed using the maximum overlap method (MOM, see Section 4.5.13) to compute 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. ¹²⁶⁵ 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.