Doubles Connected Moments Expansion: A Tractable Approximate Horn-Weinstein Approach for Quantum Chemistry

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 over 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 Doubles 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 toward limiting values in the range of N = 11-14, with compute costs that scale as a noniterative O(M⁶) 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. The qualitative success of the method in regions where strong correlation effects begin to dominate, even while maintaining spin purity, suggests this may be a good starting point in the development of methodologies for the description of strongly correlated or spin-contaminated systems while maintaining a tractable single-reference formalism.