Parametric two-electron reduced-density-matrix (p-2RDM) methods have enjoyed much success in recent years; the methods have been shown to exhibit accuracies greater than coupled cluster with single and double substitutions (CCSD) for both closed- and open-shell ground-state energies, properties, geometric parameters, and harmonic frequencies. The class of methods is herein discussed within the context of the coupled electron pair approximation (CEPA), and several CEPA-like topological factors are presented for use within the p-2RDM framework. The resulting p-2RDM/n methods can be viewed as a density-based generalization of CEPA/n family that are numerically very similar to traditional CEPA methodologies. We cite the important distinction that the obtained energies represent stationary points, facilitating the efficient evaluation of properties and geometric derivatives. The p-2RDM/n formalism is generalized for an equal treatment of exclusion-principle-violating (EPV) diagrams that occur in the occupied and virtual spaces. One of these general topological factors is shown to be identical to that proposed by Kollmar [C. Kollmar, J. Chem. Phys. 125, 084108 (2006)], derived in an effort to approximately enforce the D, Q, and G conditions for N-representability in his size-extensive density matrix functional.
Parametric two-electron reduced-density-matrix (p-2RDM) methods have enjoyed much success in recent years; the methods have been shown to exhibit accuracies greater than coupled cluster with single and double substitutions (CCSD) for both closed- and open-shell ground-state energies, properties, geometric parameters, and harmonic frequencies. The class of methods is herein discussed within the context of the coupled electron pair approximation (CEPA), and several CEPA-like topological factors are presented for use within the p-2RDM framework. The resulting p-2RDM/n methods can be viewed as a density-based generalization of CEPA/n family that are numerically very similar to traditional CEPA methodologies. We cite the important distinction that the obtained energies represent stationary points, facilitating the efficient evaluation of properties and geometric derivatives. The p-2RDM/n formalism is generalized for an equal treatment of exclusion-principle-violating (EPV) diagrams that occur in the occupied and virtual spaces. One of these general topological factors is shown to be identical to that proposed by Kollmar [C. Kollmar, J. Chem. Phys. 125, 084108 (2006)], derived in an effort to approximately enforce the D, Q, and G conditions for N-representability in his size-extensive density matrix functional.
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.
The energy and properties of a many-electron atom or molecule may be directly computed from a variational optimization of a two-electron reduced density matrix (2RDM) that is constrained to represent many-electron quantum systems. In this paper we implement a variational 2RDM method with a representability constraint, known as the T2 condition. The optimization of the 2RDM is performed with a first-order algorithm for semidefinite programming [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)] which, because of its lower computational cost in comparison to second-order methods, allows the treatment of larger basis sets. We also derive and implement a spin- and symmetry-adapted formulation of the T2 condition that significantly decreases the size of the largest block in the T2 matrix. The T2 condition, originally derived by Erdahl [Int. J. Quantum Chem. 13, 697 (1978)], was recently applied via a second-order algorithm to atoms and molecules [Z. Zhao et al., J. Chem. Phys. 120, 2095 (2004)]. While these calculations were restricted to molecules at equilibrium geometries in minimal basis sets, we apply the 2RDM method with the T2 condition to compute the electronic energies of molecules in both minimal and nonminimal basis sets at equilibrium as well as nonequilibrium geometries. Accurate potential energies curves are produced for BH, HF, and N2. Results are compared with the 2RDM method without the T2 condition as well as several wave-function methods.
Variational two-electron reduced-density-matrix (2RDM) theory is applied to computing energy spectra and properties of few-electron quantum dots. The model system is a two-dimensional electron gas with a central confinement potential. For each orbital angular momentum J, the energy and 2RDM are computed by the variational 2RDM method in which the energy is minimized as a functional of the 2RDM. In the minimization, which is performed by semidefinite programming, the 2RDM is constrained to represent a N-electron wave function with angular momentum J by N- and J-representability conditions [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. Advantages of the variational 2RDM method include (i) lower bounds on the energies for all J values, (ii) calculation of approximate 2RDMs in polynomial time without many-electron wave functions, (iii) exploitation of angular symmetry in the sparse block-diagonal structure of the 2RDM, (iv) accurate description of multireference correlation (entanglement) effects, and (v) direct calculation of one- and two-electron properties from the 2RDM. With the 2RDM we directly compute pair-correlation functions, radial charge densities, and average radial electron displacements in the quantum dot. Energies and properties are compared to those from solving the N-electron Schr\"odinger equation by large-scale, exact diagonalization. It is found that the accuracy of the variational 2RDM approach is sensitive to the total orbital angular momentum and the symmetry of the final wave function. For quantum dots of high symmetry, the variational algorithm isolates a highly accurate solution that recovers the correlation energy within a few percent.
Parametrization of a molecular two-electron reduced density matrix (2-RDM) enables energies and properties to be directly computed at a highly efficient computational cost. In this Letter an improved 2-RDM parametrization yields energies and properties that are markedly better than those calculated with a similar computational scaling by traditional ab initio methods. Computed 2-RDMs very nearly satisfy well-known N-representability conditions. Applications are made to ground-state energies of several molecules and equilibrium bond distances and harmonic frequencies of HF, F2, and CO. The method for single-bond breaking in HF and CH4 yields similar accuracy at equilibrium and nonequilibrium geometries.
The direct variational optimization of the two-electron reduced density matrix (2RDM) can provide a reference-independent description of the electronic structure of many-electron systems that naturally capture strong or nondynamic correlation effects. Such variational 2RDM approaches can often provide a highly accurate description of strong electron correlation, provided that the 2RDMs satisfy at least partial three-particle N-representability conditions (e.g., the T2 condition). However, recent benchmark calculations on hydrogen clusters [N. H. Stair and F. A. Evangelista, J. Chem. Phys. 153, 104108 (2020)] suggest that even the T2 condition leads to unacceptably inaccurate results in the case of two- and three-dimensional clusters. We demonstrate that these failures persist under the application of full three-particle N-representability conditions (3POS). A variety of correlation metrics are explored in order to identify regimes under which 3POS calculations become unreliable, and we find that the relative squared magnitudes of the cumulant three- and two-particle reduced density matrices correlate reasonably well with the energy error in these systems. However, calculations on other molecular systems reveal that this metric is not a universal indicator for the reliability of the reduced-density-matrix theory with 3POS conditions.
Tensor hypercontraction is a method that allows the representation of a high-rank tensor as a product of lower-rank tensors. In this paper, we show how tensor hypercontraction can be applied to both the electron repulsion integral tensor and the two-particle excitation amplitudes used in the parametric 2-electron reduced density matrix (p2RDM) algorithm. Because only O(r) auxiliary functions are needed in both of these approximations, our overall algorithm can be shown to scale as O(r(4)), where r is the number of single-particle basis functions. We apply our algorithm to several small molecules, hydrogen chains, and alkanes to demonstrate its low formal scaling and practical utility. Provided we use enough auxiliary functions, we obtain accuracy similar to that of the standard p2RDM algorithm, somewhere between that of CCSD and CCSD(T).