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arXiv:2605.16666

Reducing the Complexity of Density-Matrix Functionals in a Real-Space-Decomposed DF+RDMF Scheme with the Adaptive Cluster Approximation

Konstantin Tamoev, Robert Schade, Thomas D. Kühne

Published May 15, 2026• 0 citations
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Abstract

Reduced density-matrix functional theory (RDMFT) provides a variational route to electronic correlations beyond conventional density-functional approximations, but explicit evaluations of density-matrix functionals still scale exponentially with the number of active one-particle states. We formulate and assess a real-space-decomposed density-functional plus reduced-density-matrix-functional (DF+RDMF) scheme in which the Coulomb interaction is partitioned locally in real space and the RDMF correction is evaluated only for the strongly correlated part of the interaction. The resulting local density-matrix functionals are further compressed using the adaptive cluster approximation (ACA), which performs a unitary rotation of the bath subspace before truncation and therefore preserves the local interaction while reducing the number of explicitly correlated bath states. As a molecular test case, we consider the bending potential of carbon suboxide, C$_3$O$_2$. While semilocal PBE favors a linear molecule, the DF+RDMF/ACA correction stabilizes a bent configuration in qualitative agreement with the quasilinear behavior inferred from spectroscopy. The approach provides a systematic embedding hierarchy for combining density functionals with explicitly correlated density-matrix corrections in extended or spatially inhomogeneous systems.

DOI
10.48550/arxiv.2605.16666

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Strong electronic correlations are at the heart of many interesting phenomena. For the theoretical description of these materials, a proper treatment of the local atomic physics is required. We propose a novel approach combining functionals of the electron density and functionals of the one-particle reduced density matrix to improve this description. The proposed method has a solid foundation in reduced density-matrix functional theory and no double-counting problem arises. It employs a decomposition of the electron-electron interaction in real space. The interaction close to the correlated orbitals, for example, the partially filled 3d-orbitals of transition-metal ions, is described with a density-matrix functional and otherwise with a local or semi-local density functional. We propose to evaluate the density-matrix functional from Levy’s constrained search problem, i.e., via a constrained minimization over an ensemble of many-particle wave functions. In contrast to approximate parametrized density-matrix functionals, this evaluation allows us to systematically improve the functional towards the exact result. In situations where the one-particle basis is too large to evaluate the density-matrix functional from Levy’s constrained minimization problem, we apply a series of approximations that each can be converged to the exact result: the first approximation step is the local approximation of the density-matrix functional proposed by Blöchl, Walther and Pruschke. For the density-matrix functionals within the local interactions, we propose the adaptive cluster approximation (ACA) that systematically truncates non-interacting one-particle states and drastically reduces the computational effort. The resulting density-matrix functional for a local interaction and a small number of non-interacting one-particle states is then evaluated with the constrained minimization problem. We explore different parametrizations of many-particle wave functions in this constrained minimization problem. A parametrization based on a configuration-interaction-like ansatz is shown to converge rapidly if suitable selection criterion for the Slater determinants is chosen. An impurity-bath-separation ansatz is shown to be suitable for single-impurity Anderson models. It is shown that the constrained minimization can be solved for matrix product states with a DMRG-like iterative minimization. Furthermore, we show that Gutzwiller-Jastrow-correlated wave functions can be used with a quantum Monte Carlo procedure as many-particle wave functions. Finally, we formulate an algorithm for the evaluation of the density-matrix functional on near-term quantum computers. Results from the execution of the algorithm on an existing quantum computer with transmon qubits are presented. The proposed approach combining density functionals and density-matrix functionals is implemented in the CP-PAW code based on the projector augmented-wave formalism. We present results for the dissociation curve of the hydrogen molecule as the prototypical case of strong static correlation and show that static correlation is well described with the new approach. We show results for the nonmagnetic state of the transition-metaoxide NiO that is described qualitatively wrong with the DFT+U method. The proposed method properly describes the nonmagnetic state of NiO as an insulator and predicts a qualitatively correct spectral function.
Ab-initio calculations are dominated by density functional theory (DFT), which provides an efficient and accurate description of the electronic structure for most materials. For materials with strong correlations, however, many of the available density functionals yield poor results. Most well known is the case of transition metal oxides, for which most density functionals produce a qualitatively incorrect description. There is a quest to improve the description by borrowing from methods specifically designed for strongly correlated materials. One of the method which is widely used to study strongly correlated electron system in the thermodynamic limit is the dynamical mean-field theory (DMFT). For calculations pertaining to real materials, hybrid approaches like DFT+DMFT are commonly used. We consider reduced density-matrix functional theory (rDMFT) to be a useful framework for a rigorous formulation of such hybrid theories. The link from rDMFT to many-particle wave functions has been established by Levy's constrained-search algorithm on the one hand. The link to many-body perturbation theory and Green's function, on the other hand, has been provided via the Luttinger-Ward functional. The development in the field of rDMFT proceeded analogously to the development of DFT. In order to avoid the full complexity of an explicit many-body description, most density-matrix functionals are not extracted from the exact expressions i.e. via Levy's constrained-search or Green's function based methods . Rather, one proceeds analogously to the development of density functionals, namely by searching models for the density-matrix functional, that capture the most essential physical effects while having an algebraic dependence on the density matrix. Among such models, the most prominent ones are the Müller functional, Sharma functional and Marques-Lathiotakis functional. One of the major arguments in favor of density-matrix functionals is that one of the most simple functionals, the Müller functional, seems to provide a correct description of the bond-dissociation problem, for which common density functionals fail. Despite the successes, the common density-matrix functionals also reproduce a number of features in a qualitatively incorrect manner. Several such discrepancies of the standard state of the art density-matrix functionals are discussed in details in this thesis. The objective of this thesis is to study the performance of the class of commonly used model density-matrix functionals and to set the stage for the desire to develop the framework for an approximate scheme to evaluate new class of functionals from exact Green's function based method.
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Recently, an approximate theoretical framework was introduced, called local reduced density matrix functional theory (local-RDMFT), where functionals of the one-body reduced density matrix (1-RDM) are minimized under the additional condition that the optimal orbitals satisfy a single electron Schrödinger equation with a local potential. In the present work, we focus on the character of these optimal orbitals. In particular, we compare orbitals obtained by local-RDMFT with those obtained with the full minimization (without the extra condition) by contrasting them against the exact NOs and orbitals from a density functional calculation using the local density approximation (LDA). We find that the orbitals from local-RMDFT are very close to LDA orbitals, contrary to those of the full minimization that resemble the exact NOs. Since local RDMFT preserves the good quality of the description of strong static correlation, this finding opens the way to a mixed density/density matrix scheme, where Kohn-Sham orbitals obtain fractional occupations from a minimization of the occupation numbers using 1-RDM functionals. This will allow for a description of strong correlation at a cost only minimally higher than a density functional calculation.
Reduced density matrix functional theory (RDMFT) offers a route beyond Kohn-Sham density functional theory for strongly correlated systems, yet practical calculations for periodic solids are still out of reach. We formulate RDMFT for extended systems in a basis-independent way and present a planewave implementation using iterative minimization for periodic solids, evaluating nonlocal exchange-correlation functionals through the existing adaptive compressed exchange machinery. Natural occupations are optimized under N-representability constraints with a spectral projected gradient (SPG) method or an first-order explicit-by-implicit (EBI) map, while natural orbitals are updated by Riemannian optimization on the complex Stiefel manifolds. Benchmarks on typical systems of \ce{H2}, silicon, and sodium with the Hartree-Fock functional show that SPG reproduces converged hybrid references, whereas EBI can stall when occupations approach 000 or 111. With the power and Müller functionals, SPG yields lower energies and more stable convergence than EBI. Applications to fractionally charged \ce{LiH}, dissociating \ce{H2} and \ce{N2} molecules, and equation of state of silicon show that the algorithm presented in this implementation is reliable and robust.