7.8 Correlated Excited State Methods: The CIS(D) Family

7.8.4 SOS-CIS(D0) 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

ωCIS+ω(2)=𝐛(𝟎)𝐭𝐀SS(0)𝐛(𝟎)+𝐛(𝟎)𝐭𝐀SS(2)𝐛(𝟎)-𝐛(𝟎)𝐭𝐀SD(1)(𝐃DD(0)-ωCIS)-1𝐀DS(1)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 (𝐃DD(0)-ωCIS)-1 is further applied:Head-Gordon:1999a

(𝐃DD(0)-ωCIS)-1=(𝐃DD(0))-1(1+ω(𝐃DD(0))-1+ω2(𝐃DD(0))-2+) (7.42)

The CIS(D0) energy ω is defined as the eigen-solution of the response matrix with the zero-th order expansion of this equation. Namely,

(𝐀SS(0)+𝐀SS(2)-𝐀SD(1)(𝐃DD(0))-1𝐀DS(1))𝐛=ω𝐛 (7.43)

Similar to SOS-CIS(D), SOS-CIS(D0) theory is defined by taking the opposite-spin portions of this equation and then scaling them with two semi-empirical parameters:Casanova:2008a

(𝐀SS(0)+cT𝐀SSOS(2)-cU𝐀SDOS(1)(𝐃DD(0))-1𝐀DSOS(1))𝐛=ω𝐛 (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(D0) is also available. This can be used to perform excited state geometry optimizations on the electronically excited state surfaces.