Critical comparison of various connected quadruple excitation approximations in the coupled-cluster treatment of bond breaking

To assess the limits of single-reference coupled-cluster (CC) methods for potential-energy surfaces, several methods have been considered for the inclusion of connected quadruple excitations. Most are based upon the factorized inclusion of the connected quadruple contribution $\left(Q_f\right)$ [J. Chem. Phys. 108, 9221 (1998)]. We compare the methods for the treatment of potential-energy curves for small molecules. These include CCSD$\left(T{Q}_f\right)$ , where the initial contributions of triple (T) and factorized quadruple excitations are added to coupled-cluster singles (S) and doubles (D), its generalization to $\mathrm{CCSD}\left(\tilde{T}{\tilde{Q}}_f\right)$ , where instead of measuring their first contribution from orders in H, it is measured from orders in $\tilde{H}=e^{-(T_1+T_2)}H{e}^{(T_1+T_2)}$ ; renormalized approximations of both, and CCSD(2) defined in [J. Chem. Phys. 115, 2014 (2001)]. We also consider CCSDT, $\mathrm{CCSDT}\left(Q_f\right)$ , CCSDTQ, and CCSDTQP for comparison, where T, Q, and P indicate full triple, quadruple, and pentuple excitations, respectively. Illustrations for ${\mathrm{F}}_2$ , the double bond breaking in water, and ${\mathrm{N}}_2$ are shown, including effects of quadruples on equilibrium geometries and vibrational frequencies. Despite the fact that no perturbative approximation, as opposed to an iterative approximation, should be able to separate a molecule correctly for a restricted-Hartree–Fock reference function, some of these higher-order approximations have a role to play in developing new, more robust procedures.