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 22.214.171.1247
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 is further applied:357
The CIS(D) energy 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) theory is defined by taking the opposite-spin portions of this equation and then scaling them with two semi-empirical parameters:131
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) is also available. This can be used to perform excited state geometry optimizations on the electronically excited state surfaces.