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,
F2, and CO, the relative energy of the cis and trans isomers of
HO3${}^{\ensuremath{-}}$, 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-$\ensuremath{\zeta}$ basis sets. Computed 2-RDMs very nearly satisfy well-known
N-representability conditions.
We consider, for systems of indistinguishable fermions, approximate reconstruction of the three- and four-particle reduced density matrices (RDMs) from the one- and two-particle RDMs, $\mathit{\ensuremath{\gamma}}$ and $\mathit{\ensuremath{\Gamma}}.$ Our ansatz for reconstructing the four-particle RDM is the linear combination $a(\mathit{\ensuremath{\Gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\Gamma}})+b(\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\Gamma}})+c(\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\gamma}}),$ where ``\ensuremath{\wedge}'' denotes the antisymmetrized (Grassmann) product. This is a generalization of reconstruction functionals employed recently to perform direct RDM calculations without wave functions via the contracted Schr\"odinger equation. Here we consider relationships between the parameters a, b, and c that are required in order for the reconstruction functionals to respect the hierarchy of contraction relations between RDMs. To this end we establish several general theorems concerning contractions of antisymmetrized tensor products of $\mathit{\ensuremath{\gamma}},$ $\mathit{\ensuremath{\Gamma}},$ and various products thereof. The accuracy of proposed reconstruction functionals is evaluated using accurate density matrices for the ground state of Be.
For many-electron systems, the second-order reduced density matrix (2-RDM) provides sufficient information for characterizing their properties of interests in physics and chemistry. In this paper, we present a set of nontrivial constraints on 2-RDM based on the basic geometric property of Hilbert space and the commutation relations of operators. Numerical examples are provided to demonstrate the pronounced violation of these constraints by the variational 2-RDMs. It is shown that, for a strongly correlated model system, the constraint violation may be responsible for a considerable portion of the variational error in ground-state energy. Our findings provide new insights into the structural subtlety of many-electron 2-RDMs.
Two-electron reduced density matrices (2-RDMs) of many-electron molecules are directly determined without calculation of their wave functions by solving the anti-Hermitian contracted Schrödinger equation. Approximation of the 3-RDM in the anti-Hermitian contracted Schrödinger equation by a corrected cumulant expansion [Mazziotti, Phys. Rev. A 60, 3618 (1999)] permits the direct calculation of the energy and 2-RDM with many high-order correlation effects included. The method is illustrated for the molecules BeH2, H2O, NH3, CH4, and CO as well as the dissociation of BH. Correlation energies are obtained within 95%-100% of full-configuration interaction, and 2-RDMs very nearly satisfy known N-representability conditions.
Several measures of electron correlation are compared based on two criteria: (i) the presence of a unique mapping between the reduced variables in the measure and the many-electron wave function and (ii) the linear scaling of the measure and its variables with system size. We propose the squared Frobenius norm of the cumulant part of the two-particle reduced density matrix (2-RDM) as a measure of electron correlation that satisfies these criteria. An advantage of this cumulant-based norm is its ability to measure the correlation from spin entanglement, which is not contained in the correlation energy. Alternative measures based on the 2-RDM, such as the von Neumann entropy, do not scale linearly with system size. Properties of the measures are demonstrated with Be, F(2), HF, N(2), and a hydrogen chain.