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# 6.10.4 Quadratic Configuration Interaction (QCISD)

(July 14, 2022)

Quadratic configuration interaction with singles and doubles (QCISD) 972 Pople J. A., Head-Gordon M., Raghavachari K.
J. Chem. Phys.
(1987), 87, pp. 5968.
is a widely used alternative to CCSD, that shares its main desirable properties of being size-consistent, exact for pairs of electrons, as well as being also non variational. Its computational cost also scales in the same way with molecule size and basis set as CCSD, although with slightly smaller constants. While originally proposed independently of CCSD based on correcting configuration interaction equations to be size-consistent, QCISD is probably best viewed as approximation to CCSD. The defining equations are given below (under the assumption of Hartree-Fock orbitals, which should always be used in QCISD). The QCISD equations can clearly be viewed as the CCSD equations with a large number of terms omitted, which are evidently not very numerically significant:

 $E_{QCISD}=\left\langle{\Phi_{0}\left|{\hat{H}}\right|\left({1+\hat{T}_{2}}% \right)\Phi_{0}}\right\rangle_{C}$ (6.42)
 $0=\left\langle{\Phi_{i}^{a}\left|{\hat{H}}\right|\left({\hat{T}_{1}+\hat{T}_{2% }+\hat{T}_{1}\hat{T}_{2}}\right)\Phi_{0}}\right\rangle_{C}$ (6.43)
 $0=\left\langle{\Phi_{ij}^{ab}\left|{\hat{H}}\right|\left({1+\hat{T}_{1}+\hat{T% }_{2}+\frac{1}{2}\hat{T}_{2}^{2}}\right)\Phi_{0}}\right\rangle_{C}$ (6.44)

QCISD energies are available in Q-Chem, and are requested with the QCISD keyword. As discussed in Section 6.11, the non iterative QCISD(T) correction to the QCISD solution is also available to approximately incorporate the effect of higher substitutions.