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:

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

(7.30)

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

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

(7.31)

and

$\left\langle{\Psi_{ij}^{ab}}\right|H-E\left|{\left({1+T_{1}+T_{2}+\frac{1}{2}T% _{1}^{2}+T_{1}T_{2}+\frac{1}{3!}T_{1}^{3}+\frac{1}{2}T_{2}^{2}+\frac{1}{2}T_{1% }^{2}T_{2}+\frac{1}{4!}T_{1}^{4}}\right)\Psi_{0}}\right\rangle=0$

(7.32)

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

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

(7.33)

and

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

(7.34)

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

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

(7.35)

or in the alternative form

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

(7.36)

where

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

(7.37)

and

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

(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.

Search Results

7.9.1 CIS(D) Theory