7.9.5 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 7.9.2.⁴³⁷

First, starting from Eq. (7.46), one can be explicitly write the CIS(D) energy as^{166, 437}

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 ${\left({𝐃}_{\mathrm{DD}}^{(0)}-{\omega }^{\mathrm{CIS}}\right)}^{-1}$ is further applied:⁴³⁷

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,

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:¹⁶⁶

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.