The Kernel Energy Method (KEM) delivers accurate full molecule energies using less computational resources than standard ab-initio quantum chemical approaches. KEM achieves this efficiency by decomposing a system of atoms into disjoint subsets called kernels. The results of full ab-initio calculations on each individual single kernel and on each double kernel formed by the union of each pair of single kernels are combined in an equation of a form that is specific to KEM to provide an approximation to the full molecule energy. KEM has been demonstrated to give accurate molecular energies over a wide range of systems, chemical methods and basis sets. The efficiency of KEM makes calculations on large systems tractable. It has been shown to be accurate for calculations on large biomolecules including proteins and DNA. KEM has recently been generalized to provide the one-body density matrix for the full molecule. This generalization greatly expands the utility of the method since the density matrix enables simple calculation of the expectation value of any observable associated with a one-body operator. More importantly, the generalization enables results from KEM to be constrained to satisfy essential quantum mechanical conditions. KEM is generalized to density matrices by taking the density matrices obtained from the single and double kernels and adapting them to the full molecule basis. These augmented kernel density matrices are summed according to the standard KEM expansion. This initial matrix does not necessarily correspond to a valid quantum mechanical N-electron wavefunction which is normalized and obeys the Pauli principle. Such a density matrix is called N-representable. N-representability is imposed on the initial matrix by using the Clinton equations, which deliver a single-determinant N-representable one-body density matrix. Single-determinant N-representability is ensured by enforcing the density matrix to be a normalized projector. Such normalized projectors are factorizable into matrices which deliver full molecule molecular orbitals. Although energies have been demonstrated to be accurate in the original energy expansion form of KEM, there is no requirement that the energy corresponds to the expectation value of a valid quantum mechanical wavefunction. Because of this, there is no requirement that the energies satisfy the variational theorem. Imposing single-determinant N-representability on the KEM density matrix by using the Clinton equations recovers the variational bound on the energy obtained from KEM. Recovery of the variational bound by the N-representable KEM density matrix has been demonstrated by calculations on several water clusters. Energies from full molecule restricted Hartree-Fock calculations were compared to energies obtained from the KEM energy expansion and to energies obtained from the KEM density matrix expansion which had N-representability imposed. In each case where the energy expansion gave energies that violated the variational bound the N-representable density matrix gave energies that satisfied the bound and in fact were more accurate than the simple energy expansion results. The effect of imposing N-representability on the KEM density matrix has also been investigated in the context of Kohn-Sham Density Functional Theory (KS/DFT). Calculations on a simple noble gas system and a single water cluster were done. KS/DFT energies were compared to energies from the KEM energy expansion and energies associated with the KEM density matrix expansion on which N-representability was imposed. The energies from the KEM density matrix expansion made N-representable were more accurate than those obtained from straightforward KEM energy expansions in nearly all cases for the noble gas system. For the water cluster the accuracy of the energy obtained from the N-representable KEM density matrix was nearly four times closer to the energy calculated for the full system than the KEM energy expansion result. The Clinton-purified N-representable matrix obtained from the KEM density expansion can be used to calculate expectation values for any one-electron operator. In particular, a KEM density matrix approach can be used in the study of quantum crystallography.
Slater determinants have underpinned quantum chemistry for nearly a century, yet their full potential has remained challenging to exploit. In this work, we show that a variational wavefunction composed of a few hundred optimized non-orthogonal determinants can achieve energy accuracies comparable to the state of the art. This is obtained by introducing an optimization method that leverages the quadratic dependence of the variational energy on the orbitals of each determinant, enabling an exact iterative optimization, and uses an efficient tensor-contraction algorithm to evaluate the effective Hamiltonian with a computational cost that scales as the fourth power of the number of basis functions. We benchmark the accuracy of the proposed method with exact full-configuration interaction results where available, and we achieve lower variational energies than coupled cluster (CCSD(T)) for several molecules in the double-zeta basis. Accurate description of electronic wave functions typically requires large numbers of Slater determinants. Here, the authors develop an optimization method that achieves state-of-the-art accuracy for small molecules with only few hundred determinants.
Molecular two-electron reduced density matrices (2-RDMs) are computed variationally without the many-electron wave function by constraining the 2-RDM with a set of three-particle N-representability conditions known as three-positivity conditions. These constraints restrict four distinct three-particle probability distributions, which can be defined for any N-particle system, to be nonnegative. The variational calculation of the 2-RDM with full three-positivity conditions is implemented with the first-order semidefinite programming algorithm [D.A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. We also derive and implement a generalization of the T2 representability condition, which is a subset of the three-positivity conditions. Energies and 2-RDMs are computed for several molecules as well as the nitrogen molecule at both equilibrium and nonequilibrium geometries. The ground-state energies for nitrogen are within 0.0015a.u. of the values from full configuration interaction at all internuclear distances.
The variational two-electron reduced-density-matrix (2-RDM) method allows for the computation of accurate ground-state energies and 2-RDMs of atoms and molecules without the explicit construction of an N-electron wave function. While previous work on variational 2-RDM theory has focused on calculating full configuration-interaction energies, this work presents the first application toward approximating multiconfiguration self-consistent-field (MCSCF) energies via low-rank restrictions on the 1- and 2-RDMs. The 2-RDM method with two- or three-particle N-representability conditions reduces the exponential active-space scaling of MCSCF methods to a polynomial scaling. Because the first-order algorithm [Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)] represents each form of the 1- and 2-RDMs by a matrix factorization, the RDMs are readily defined to have a low rank rather than a full rank by setting the matrix factors to be rectangular rather than square. Results for the potential energy surfaces of hydrogen fluoride, water, and the nitrogen molecule show that the low-rank 2-RDM method yields accurate approximations to the MCSCF energies. We also compute the energies along the symmetric stretch of a 20-atom hydrogen chain where traditional MCSCF calculations, requiring more than 17x10(9) determinants in the active space, could not be performed.
The latest results of density matrix theory are summarized in terms of analysis of off-diagonal components. For a given reduced density matrix (RDM) of arbitrary order, the full density matrix is explicitly constructed, from which the RDM is derived by means of the corresponding reduction procedure. Both the matrices satisfy all the necessary conditions, such antisymmetry, normalization, hermiticity, except that the non-negative definiteness of the full DM is not guaranteed. If an RDM satisfies the necessary inequalities with the same number of indices as it has itself, then, as proved here, there is an additive correction to the full DM matrix that turns it into a non-negative definite matrix leading to the same RDM and satisfying the other requirements. This establishes a fundamental fact that these inequalities are sufficient as well, providing the N-representability of the given RDM. The construction is based on the classification, introduced in the article, of the multi-index elements of the RDM and full DM according to their degree of off-diagonality. RDM elements with a predetermined degree of off-diagonality are expressed through full DM elements with the same off-diagonality. This shows that the conventional variational approach to calculating the energy of the system by convex programming methods under the conditions of N-representability (before and even after reaching the minimum energy) generally does not lead to constructing a mixed quantum state composed of solutions to the Schrödinger equation. Therefore, except for the energy, it does not guarantee a correct description of the other properties so long as the conditions of correctness for each particular property are not expressed in terms of the RDM and not incorporated into the specific conditions of N-representability (regardless of the requirement of energy minimization).
Variational quantum Monte Carlo (VMC) method applied to interacting electron systems has played a pivotal role for the studies of electronic states of atoms, molecules and condensed matters. First, the many-body wave functions optimized by the VMC method provide good guiding functions in more accurate, diffusion quantum Monte Carlo (DMC) calculations for identical systems. Second, the VMC calculations themselves can give a good upper bound for the exact ground-state energy of many-electron systems when an appropriate form for the variational wave function is employed. In addition to a conventional form for the variational wave function that is expressed by the product of the Jastrow factor and the single Slater determinant, recent investigations have found that the expression in terms of many Slater determinants can provide a very accurate description of electronic states, especially for intrinsically multi-reference systems, which corresponds to the configuration interaction (CI) expansion method in conventional quantum chemistry. Although the CI method needs an extremely large number of expansion coefficients as variational parameters, the number of parameters can be reduced efficiently by using an idea of (molecular) orbital pair correlations. The purpose of the present paper is to examine the feasibility and usefulness of this idea within the framework of VMC method with the Jastrow–Slater wave function, where the ground state of a water molecule is investigated as a test case. In this study we rely on a variational wave function composed of a product of the Jastrow and Slater parts as 1⁄4 J S; ð1Þ