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The Journal of Chemical Physics

Generalized spin in the variational determination of two-electron reduced density matrices within the doubly occupied configuration interaction framework

Tomás R. Ayala, Elías Ríos, Ofelia B. Oña, Luis Laín, Alicia Torre, Gustavo E. Massaccesi, Diego R. Alcoba

Published Sep 22, 2025• 1 citation
Publisher

Abstract

This work extends the variational determination of two-electron reduced density matrices to the generalized spin formulation within the doubly occupied configuration interaction treatment. We describe electronic states of N-electron systems by means of variational evaluation of the corresponding two-electron reduced density matrix elements, subject to determined N-representability conditions, arising from the restricted, unrestricted, and generalized spin approaches. The results are compared with those predicted by the counterpart treatments involving wave functions. We report potential energy curves for a series of hydrogen atom clusters with different geometries in the presence of an external uniform magnetic field. These results allow us to analyze and discuss crossings between potential energy curves involving transitions between electronic states of different spin multiplicities. The predictions provided by these methods have been contrasted with the results arising from the full configuration interaction procedure, confirming changes of ⟨Ŝ2⟩ and ⟨Ŝz⟩ expectation values.

DOI
10.1063/5.0275659

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This work extends the variational determination of two-electron reduced density matrices to the generalized spin formulation within the doubly occupied configuration interaction treatment. We describe electronic states of N-electron systems by means of variational evaluation of the corresponding two-electron reduced density matrix elements, subject to determined N-representability conditions, arising from the restricted, unrestricted, and generalized spin approaches. The results are compared with those predicted by the counterpart treatments involving wave functions. We report potential energy curves for a series of hydrogen atom clusters with different geometries in the presence of an external uniform magnetic field. These results allow us to analyze and discuss crossings between potential energy curves involving transitions between electronic states of different spin multiplicities. The predictions provided by these methods have been contrasted with the results arising from the full configuration interaction procedure, confirming changes of ⟨Ŝ2⟩ and ⟨Ŝz⟩ expectation values.
In this work, we analyze the effectiveness of different sets of well-known necessary N-representability conditions that are imposed on the variational determination of two-electron reduced density matrices corresponding to the ground and excited states of N-electron Hamiltonians within doubly occupied configuration interaction subspaces. The two-electron reduced density matrix elements are determined so that they are suitably related to those of higher-order reduced density matrices and other associated matrices, up to a given order, constituting positive semidefinite matrices. These conditions up to two-, three-, and four-order have been sequentially imposed on the variational calculation of the expectation values of the dispersion operator corresponding to several Hamiltonian models, obtaining the electronic spectra. These results have been contrasted with those arising from the full configuration interaction method, illuminating the sufficiency/insufficiency of the variational N-representability conditions applied.
This work proposes the variational determination of two-electron reduced density matrices corresponding to the ground state of N-electron systems within the doubly occupied-configuration-interaction methodology. The P, Q, and G two-index N-representability conditions have been extended to the T1 and T2 (T2') three-index ones and the resulting optimization problem has been addressed using a standard semidefinite program. We report results obtained from the doubly occupied-configuration-interaction method, from the two-index constraint variational procedure and from the two- and three-index constraint variational treatment. The discussion of these results along with a study of the computational cost demanded shows the usefulness of our proposal.
This work extends to the unrestricted orbital approach the procedure described in our previous report [Alcoba et al., J. Chem. Phys. 148, 024105 (2018)] for determining variationally the two-electron reduced density matrix arising from doubly occupied-configuration-interaction wave functions by imposing two- and three-index N-representability conditions. An analysis of the numerical results obtained in selected systems, from both restricted and unrestricted treatments, allows one to assess the performance of these methodologies as well as to show the influence of the P, Q, G, T1, and T2 positivity conditions. We highlight the satisfactory results obtained within the unrestricted scheme.
This work implements a variational determination of the elements of two-electron reduced density matrices corresponding to the ground and excited states of N-electron interacting systems based on the dispersion operator technique. The procedure extends the previously reported proposal [Nakata et al., J. Chem. Phys. 125, 244109 (2006)] to two-particle interaction Hamiltonians and N-representability conditions for the two-, three-, and four-particle reduced density matrices in the doubly occupied configuration interaction space. The treatment has been applied to describe electronic spectra using two benchmark exactly solvable pairing models: reduced Bardeen-Cooper-Schrieffer and Richardson-Gaudin-Kitaev Hamiltonians. The dispersion operator combined with N-representability conditions up to the four-particle reduced density matrices provides excellent results.
This work deals with the variational determination of the two-particle reduced density matrix (2-RDM) and the energy corresponding to the ground state of N-particle systems within the doubly occupied configuration interaction (DOCI) space. Here, we impose for the first time up to four-particle N-representability constraint conditions in the variational determination of the 2-RDM matrix elements using the standard semidefinite programming algorithms. The energies and 2-RDMs obtained from this treatment and the corresponding computational costs are compared with those arisen from previously reported less restrictive variational methods [D. R. Alcoba et al., J. Chem. Phys. 149, 194105 (2018)] as well as with the exact DOCI values. We apply the different approximations to the one-dimensional XXZ model of quantum magnetism, which has a rich phase diagram with one critical phase and constitutes a stringent test for the method. The numerical results show the usefulness of our treatment to achieve a high degree of accuracy.
We perform a direct variational determination of the second-order (two-particle) density matrix corresponding to a many-electron system, under a restricted set of the two-index N-representability P-, Q-, and G-conditions. In addition, we impose a set of necessary constraints that the two-particle density matrix must be derivable from a doubly occupied many-electron wave function, i.e., a singlet wave function for which the Slater determinant decomposition only contains determinants in which spatial orbitals are doubly occupied. We rederive the two-index N-representability conditions first found by Weinhold and Wilson and apply them to various benchmark systems (linear hydrogen chains, He, N2, and CN(-)). This work is motivated by the fact that a doubly occupied many-electron wave function captures in many cases the bulk of the static correlation. Compared to the general case, the structure of doubly occupied two-particle density matrices causes the associate semidefinite program to have a very favorable scaling as L(3), where L is the number of spatial orbitals. Since the doubly occupied Hilbert space depends on the choice of the orbitals, variational calculation steps of the two-particle density matrix are interspersed with orbital-optimization steps (based on Jacobi rotations in the space of the spatial orbitals). We also point to the importance of symmetry breaking of the orbitals when performing calculations in a doubly occupied framework.