7.9.1 CIS(D) Theory

The CIS(D) excited state procedure is a second-order perturbative approximation to the computationally expensive CCSD, based on a single excitation configuration interaction (CIS) reference. The coupled-cluster wave function, truncated at single and double excitations, is the exponential of the single and double substitution operators acting on the Hartree-Fock determinant:

$$|\mathrm{\Psi }⟩=\exp \left(T_1+T_2\right)|{\mathrm{\Psi }}_0⟩$$

(7.30)

Determination of the singles and doubles amplitudes requires solving the two equations

$$⟨{\mathrm{\Psi }}_i^a\left|H-E\right|\left(1+T_1+T_2+\frac{1}{2}{T}_1^2+T_1{T}_2+\frac{1}{3!}{T}_1^3\right){\mathrm{\Psi }}_0⟩=0$$

(7.31)

and

$$⟨{\mathrm{\Psi }}_{ij}^{ab}\left|H-E\right|\left(1+T_1+T_2+\frac{1}{2}{T}_1^2+T_1{T}_2+\frac{1}{3!}{T}_1^3+\frac{1}{2}{T}_2^2+\frac{1}{2}{T}_1^2{T}_2+\frac{1}{4!}{T}_1^4\right){\mathrm{\Psi }}_0⟩=0$$

(7.32)

which lead to the CCSD excited state equations. These can be written

$$⟨{\mathrm{\Psi }}_i^a\left|H-E\right|\left(U_1+U_2+T_1{U}_1+T_1{U}_2+U_1{T}_2+\frac{1}{2}{T}_1^2{U}_1\right){\mathrm{\Psi }}_0⟩=\omega b_i^a$$

(7.33)

and

$$\begin{array}{c}\hfill ⟨{\mathrm{\Psi }}_i^a|H-E|(U_1+U_2+T_1{U}_1+T_1{U}_2+U_1{T}_2+\frac{1}{2}{T}_1^2{U}_1+T_2{U}_2\\ \hfill +\frac{1}{2}{T}_1^2{U}_2+T_1{T}_2{U}_1+\frac{1}{3!}{T}_1^3{U}_1|{\mathrm{\Psi }}_0⟩=\omega b_{ij}^{ab}\end{array}$$

(7.34)

This is an eigenvalue equation $\mathrm{𝐀𝐛}=\omega 𝐛$ for the transition amplitudes ($𝐛$ vectors), which are also contained in the $U$ operators.

The second-order approximation to the CCSD eigenvalue equation yields a second-order contribution to the excitation energy which can be written in the form

$${\omega }^{(2)}={𝐛}^{{(0)}^{𝐭}}{𝐀}^{(1)}{𝐛}^{(1)}+{𝐛}^{{(0)}^{𝐭}}{𝐀}^{(2)}{𝐛}^{(0)}$$

(7.35)

or in the alternative form

$${\omega }^{(2)}={\omega }^{\mathrm{CIS}(\mathrm{D})}=E^{\mathrm{CIS}(\mathrm{D})}-E^{\mathrm{MP2}}$$

(7.36)

where

$$E^{\mathrm{CIS}(\mathrm{D})}=⟨{\mathrm{\Psi }}^{\mathrm{CIS}}\left|V\right|U_2{\mathrm{\Psi }}^{\mathrm{HF}}⟩+⟨{\mathrm{\Psi }}^{\mathrm{CIS}}\left|V\right|T_2{U}_1{\mathrm{\Psi }}^{\mathrm{HF}}⟩$$

(7.37)

and

$$E^{\mathrm{MP2}}=⟨{\mathrm{\Psi }}^{\mathrm{HF}}\left|V\right|T_2{\mathrm{\Psi }}^{\mathrm{HF}}⟩$$

(7.38)

The output of a CIS(D) calculation contains useful information beyond the CIS(D) corrected excitation energies themselves. The stability of the CIS(D) energies is tested by evaluating a diagnostic, termed the “theta diagnostic”.⁷¹⁹ The theta diagnostic calculates a mixing angle that measures the extent to which electron correlation causes each pair of calculated CIS states to couple. Clearly the most extreme case would be a mixing angle of ${45}^{\circ }$, which would indicate breakdown of the validity of the initial CIS states and any subsequent corrections. On the other hand, small mixing angles on the order of only a degree or so are an indication that the calculated results are reliable. The code can report the largest mixing angle for each state to all others that have been calculated.