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The Journal of Chemical Physics

Variational determination of ground and excited-state two-electron reduced density matrices in the doubly occupied configuration space: A dispersion operator approach

Diego R. Alcoba, Ofelia B. Oña, Luis Laín, Alicia Torre, P. Capuzzi, Gustavo E. Massaccesi, Elías Ríos, Álvaro Rubio-García, J. Dukelsky

Published Jun 10, 2021• 12 citations
Publisher

Abstract

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.

DOI
10.1063/5.0051793

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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.
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 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.
Abstract This work deals with the variational determination of two-particle reduced density matrices corresponding to eigenstates of N -boson systems. Stringent N -representability conditions have been imposed in the variational treatment, in which the resulting optimization problem is addressed within a standard semidefinite programming framework. A unified variational treatment, based on the dispersion operator technique, is proposed. This treatment allows to determine ground- and excited-state energies and their corresponding reduced density matrices, which have been evaluated in systems formulated by means of simple Hamiltonians. We report results from different N -representability levels of treatment. These results are compared with those arising from the full configuration interaction procedure, what allows to point out the relevance of the imposed N -representability conditions. We also highlight the differences found in applying this methodology to fermion and boson systems.
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.
In this work, the Hellmann-Feynman theorem is extended within the doubly occupied configuration interaction space to enable practical calculations of reduced density matrices and expected values. This approach is straightforward, employing finite energy differences, yet remains reliable and accurate even with approximate energies from successive approximation methods. The method's validity is rigorously tested against the Richardson-Gaudin-Kitaev and reduced Bardeen-Cooper-Schrieffer models using approximate excitation energies procured from the Hermitian operator method within the same space, effectively proving the approach's reliability with median error rates for reduced density matrix calculations around 0.1%. These results highlight the procedure's potential as a practical tool for computing reduced density matrices and expected values, particularly valuable as an ad hoc method in scenarios where only system energies are easily available.