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

(November 19, 2024)

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. ⁵⁰² Head-Gordon M., Oumi M., Maurice D.
Mol. Phys.
(1999), 96, pp. 593. Link

First, starting from Eq. (7.67), one can be explicitly write the CIS(D) energy as ¹⁸⁹ Casanova D., Rhee Y. M., Head-Gordon M.
J. Chem. Phys.
(2008), 128, pp. 164106. Link ^, ⁵⁰² Head-Gordon M., Oumi M., Maurice D.
Mol. Phys.
(1999), 96, pp. 593. Link

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

(7.73)

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: ⁵⁰² Head-Gordon M., Oumi M., Maurice D.
Mol. Phys.
(1999), 96, pp. 593. Link

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

(7.74)

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,

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

(7.75)

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: ¹⁸⁹ Casanova D., Rhee Y. M., Head-Gordon M.
J. Chem. Phys.
(2008), 128, pp. 164106. Link

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

(7.76)

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.

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7.9.5 SOS-CIS(D0<math id="m1" class="ltx_Math" alttext="{}_{0}" display="inline"><msub><mi></mi><mn>0</mn></msub></math>) Model

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