# 7.8.4 SOS-CIS(D${}_{0}$) Model

CIS(D) and its cousins explained in the above are all based on a second-order non-degenerate perturbative correction scheme on the CIS energy (“diagonalize-and-then-perturb” scheme). Therefore, they may fail when multiple excited states come close in terms of their energies. In this case, the system can be handled by applying quasi-degenerate perturbative correction scheme (“perturb-and-then-diagonalize” scheme). The working expression can be obtained by slightly modifying CIS(D) expression shown in Section 7.8.1.Head-Gordon:1999a

First, starting from Eq. (7.35), one can be explicitly write the CIS(D) energy asCasanova:2008a, Head-Gordon:1999a

 $\omega^{\rm CIS}+\omega^{(2)}=\bf b^{(0)^{\mathbf{t}}}{\rm{\bf A}_{SS}^{(0)}}% \bf b^{(0)}+\bf b^{(0)^{\mathbf{t}}}{\rm{\bf A}_{SS}^{(2)}}\bf b^{(0)}-\bf b^{% (0)^{\mathbf{t}}}{\rm{\bf A}_{SD}^{(1)}\left({\bf D}_{DD}^{(0)}-\omega^{CIS}% \right)^{-1}{\bf A}_{DS}^{(1)}}\bf b^{(0)}$ (7.41)

To avoid the failures of the perturbation theory near degeneracies, the entire single and double blocks of the response matrix should be diagonalized. Because such a diagonalization is a non-trivial non-linear problem, an additional approximation from the binomial expansion of the $\rm\left({\bf D}_{DD}^{(0)}-\omega^{CIS}\right)^{-1}$ is further applied:Head-Gordon:1999a

 $\rm\left({\bf D}_{DD}^{(0)}-\omega^{CIS}\right)^{-1}=\left({\bf D}_{DD}^{(0)}% \right)^{-1}\left(1+\omega\left({\bf D}_{DD}^{(0)}\right)^{-1}+\omega^{2}\left% ({\bf D}_{DD}^{(0)}\right)^{-2}+...\right)$ (7.42)

The CIS(D${}_{0}$) energy $\omega$ is defined as the eigen-solution of the response matrix with the zero-th order expansion of this equation. Namely,

 $\rm\left({\bf A}_{SS}^{(0)}+{\bf A}_{SS}^{(2)}-{\bf A}_{SD}^{(1)}({\bf D}_{DD}% ^{(0)})^{-1}{\bf A}_{DS}^{(1)}\right)\bf b=\omega\bf b$ (7.43)

Similar to SOS-CIS(D), SOS-CIS(D${}_{0}$) theory is defined by taking the opposite-spin portions of this equation and then scaling them with two semi-empirical parameters:Casanova:2008a

 $\rm\left({\bf A}_{SS}^{(0)}+{\it c_{T}}{\bf A}_{SS}^{OS(2)}-{\it c_{U}}{\bf A}% _{SD}^{OS(1)}({\bf D}_{DD}^{(0)})^{-1}{\bf A}_{DS}^{OS(1)}\right)\bf b=\omega\bf b$ (7.44)

Using the Laplace transform and the auxiliary basis expansion techniques, this can also be handled with a 4th-order scaling computational effort. In Q-Chem, an efficient 4th-order scaling analytical gradient of SOS-CIS(D${}_{0}$) is also available. This can be used to perform excited state geometry optimizations on the electronically excited state surfaces.