7.9.2 CIS(D) Theory

(February 4, 2022)

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:

\left|\Psi\right\rangle=\exp\left({T_{1}+T_{2}}\right)\left|{\Psi_{0}}\right\rangle

(7.41)

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

\left\langle{\Psi_{i}^{a}}\right|H-E\left|{\left({1+T_{1}+T_{2}+\frac{1}{2}T_{% 1}^{2}+T_{1}T_{2}+\frac{1}{3!}T_{1}^{3}}\right)\Psi_{0}}\right\rangle=0

(7.42)

and

\left\langle{\Psi_{ij}^{ab}}\right|H-E\left|{\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)\Psi_{0}}\right\rangle=0

(7.43)

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

\left\langle{\Psi_{i}^{a}}\right|H-E\left|{\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)\Psi_{0}}\right\rangle=% \omega b_{i}^{a}

(7.44)

and

\begin{array}[]{r}\left\langle{\Psi_{i}^{a}}\right|H-E\left|{\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.}\right.+T% _{2}U_{2}\\ \left.{+\frac{1}{2}T_{1}^{2}U_{2}+T_{1}T_{2}U_{1}+\frac{1}{3!}T_{1}^{3}U_{1}}% \right|\left.{\Psi_{0}}\right\rangle=\omega b_{ij}^{ab}\\ \end{array}

(7.45)

This is an eigenvalue equation $\mathbf{A}\mathbf{b}=\omega\mathbf{b}$ for the transition amplitudes ( $\mathbf{b}$ 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)}=\mathbf{b}^{(0)^{\mathbf{t}}}{\rm{\bf A}}^{(1)}\mathbf{b}^{(1)}+% \mathbf{b}^{(0)^{\mathbf{t}}}{\rm{\bf A}}^{(2)}\mathbf{b}^{(0)}

(7.46)

or in the alternative form

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

(7.47)

where

E^{\mathrm{CIS(D)}}=\left\langle{\Psi^{\mathrm{CIS}}}\right|V\left|{U_{2}\Psi^% {\mathrm{HF}}}\right\rangle+\left\langle{\Psi^{\mathrm{CIS}}}\right|V\left|{T_% {2}U_{1}\Psi^{\mathrm{{HF}}}}\right\rangle

(7.48)

and

E^{\mathrm{MP2}}=\left\langle{\Psi^{\mathrm{HF}}}\right|V\left|{T_{2}\Psi^{% \mathrm{HF}}}\right\rangle

(7.49)

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”.⁸⁴⁹ 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.

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7.9.2 CIS(D) Theory