  Ab initio methods based on the second-order and higher connected moments, or
cumulants, of a reference function have seen limited use in the determination
of correlation energies of chemical systems throughout the years. Moment-based
methods have remained unattractive relative to more ubiquitous methods, such as
perturbation theory and coupled cluster theory, due in part to the intractable
cost of assembling moments of high-order and poor performance of low-order
expansions. Many of the traditional quantum chemical methodologies can be
recast as a selective summation of perturbative contributions to their energy;
using this familiar structure as a guide in selecting terms, we develop a
scheme to approximate connected moments limited to double excitations. The
tractable Double Connected Moments (DCM(N)) approximation is developed and
tested against a multitude of common single-reference methods to determine its
efficacy in the determination of the correlation energy of model systems and
small molecules. The DCM(N) sequence of energies exhibits smooth convergence,
with compute costs that scale as a non-iterative O(N^6) with molecule size, M.
Numerical tests on correlation energy recovery for 55 small molecules
comprising the G1 test set in the cc-pVDZ basis show that DCM(N) strongly
outperforms MP2 and even CCD with a Hartree-Fock reference. When using an
approximate Brueckner reference from orbital-optimized (oo) MP2, the resulting
oo:DCM(N) energies converge to values more accurate than CCSD for 49 of 55
molecules.
