7.9.4 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 7.9.1.³⁵⁷

First, starting from Eq. (7.35), one can be explicitly write the CIS(D) energy as^{131, 357}

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

(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 ${\left({𝐃}_{\mathrm{DD}}^{(0)}-{\omega }^{\mathrm{CIS}}\right)}^{-1}$ is further applied:³⁵⁷

$${\left({𝐃}_{\mathrm{DD}}^{(0)}-{\omega }^{\mathrm{CIS}}\right)}^{-1}={\left({𝐃}_{\mathrm{DD}}^{(0)}\right)}^{-1}\left(1+\omega {\left({𝐃}_{\mathrm{DD}}^{(0)}\right)}^{-1}+{\omega }^2{\left({𝐃}_{\mathrm{DD}}^{(0)}\right)}^{-2}+\mathrm{\dots }\right)$$

(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,

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

(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:¹³¹

$$\left({𝐀}_{\mathrm{SS}}^{(0)}+c_T{𝐀}_{\mathrm{SS}}^{\mathrm{OS}(2)}-c_U{𝐀}_{\mathrm{SD}}^{\mathrm{OS}(1)}{({𝐃}_{\mathrm{DD}}^{(0)})}^{-1}{𝐀}_{\mathrm{DS}}^{\mathrm{OS}(1)}\right)𝐛=\omega 𝐛$$

(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.