Parametrization of the two-electron reduced density matrix for its direct calculation without the many-electron wave function: Generalizations and applications

An improved parametrization of the two-electron reduced density matrix (2-RDM) [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)] was recently shown to yield energies and properties that are markedly better than those calculated by traditional ab initio methods of similar computational scaling. In this paper a family of such energy functionals, generalizing the ones obtained previously, is derived through the use of (i) $p$-particle contraction relations based on the contraction of the cumulant expansions of $p$-particle RDMs and (ii) Cauchy-Schwarz relations that arise from an important set of $N$-representability constraints known as the two-positivity conditions. The 2-RDMs are explicitly parameterized in terms of the first-order part of the cumulant 2-RDM and, for the inclusion of single excitations, a second-order part of the 1-RDM. In contrast to earlier formulations based on the coefficients from configuration interaction with single and double excitations (CISD), the cumulant-based parametric 2-RDM methods, from the properties of cumulants, are rigorously size extensive. We also show that writing the energy functionals in terms of correlated 1-RDMs and cumulant 2-RDMs reduces the computational cost of the parametric 2-RDM methods to that of CISD. Applications are made to ground-state energies of several molecules, equilibrium bond distances, and frequencies of HF, ${\mathrm{F}}_2$, and CO, the relative energy of the cis and trans isomers of ${\mathrm{HO}}_3$${}^{-}$, and the HCN-HNC isomerization reaction. For bond breaking in hydrogen fluoride the improved and more efficient parametric 2-RDM methods yield energies with similar accuracies at both equilibrium and nonequilibrium geometries in 6-31G** and polarized valence quadruple-$\zeta$ basis sets. Computed 2-RDMs very nearly satisfy well-known $N$-representability conditions.

Figure 1

Potential energy curves for hydrogen fluoride in the cc-pVQZ basis set from the $K$, $M$, CCSD, CCSD(T), and CR-CC(2,3) methods. Energy results of the $M$ functional are nearly indistinguishable from those of the computationally more expensive CR-CC(2,3). The length of the H-F bond is given as angstroms (Å).