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.
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.
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.
Strongly correlated electrons have been the subject of substantial theoretical interest for many years. Most work has focused on obtaining the energy in a low-cost fashion. Here, we show that even methods with good energies can yield significant "delocalization errors" that affect the orbitals and density, leading to large errors in predicting other important properties such as dipole moments. We illustrate this point by comparing existing state-of-art approaches with an accurate exchange correlation functional based on a generalised valence bond ansatz, in which orbitals and fractional occupations are treated as variational parameters via a common optimized effective potential (OEP). We show that the OEP exhibits step and peak features which, similar to the exact Kohn-Sham (KS) potential of DFT, are crucial to prevent charge delocalization. We further show that the step is missing in common approximations within reduced density matrix functional theory resulting in delocalization errors comparable to those found in DFT approximations. Finally, we explain the delocalisation error as coming from an artificial mixing of the ground state with a charge-transfer excited state which is avoided if occupation numbers exhibit discontinuities.
The one-particle reduced density-matrix (1-RDM) functional theory is a promising alternative to density-functional theory (DFT) that uses the 1-RDM rather than the electronic density as a basic variable. However, long-standing challenges such as the lack of the Kohn-Sham scheme and the complexity of the pure N-representability conditions are still impeding its wild utilization. Fortunately, ensemble N-representability conditions derived in the natural orbital basis are known and trivial such that almost every functional of the 1-RDM is actually natural orbital functional, which does not perform well for all the correlation regimes. In this work, we propose a variational minimization scheme in the ensemble N-representable domain that is not restricted to the natural orbital representation of the 1-RDM. We show that splitting the minimization into the diagonal and off-diagonal parts of the 1-RDM can open the way toward the development of functionals of the orbital occupations, which remains a challenge for the generalization of site-occupation functional theory in chemistry. Our approach is tested on the uniform Hubbard model using the Müller and the Töws-Pastor functionals, as well as on the dihydrogen molecule using the Müller functional.
From drug design to material science, computational chemistry has become a valuable tool for many fields of applications in the past decades. Quantum chemistry in particular, can provide accurate descriptions of molecules and solids by relying on principles of quantum mechanics with the Schrödinger equation at its core. Unfortunately this equation becomes prohibitively expensive to solve for systems made of more than a few electrons and one has to rely on other approaches. Among the most popular is Density Functional Theory, relatively inexpensive and exact in principle. One has, however, to introduce approximations in practice, deteriorating its accuracy for challenging, so-called strongly correlated, systems. On the contrary, other methods like CASSCF remain reliable for these systems, but although cheaper than the solution of the Schrödinger equation, remain too expensive for extended systems. There is therefore room for methods in between, that could handle large correlated systems with reasonable accuracy. One such method is the One Body Reduced Density Matrix (1-RDM) Functional Theory (RDMFT), based on the fact that an electronic system can, in principle exactly, be described by its 1-RDM only. Although its theory dates back to the 1970s, computational hindrances have prevented this method from shining so far. Indeed, computations in RDMFT rely on a constrained optimization of the eigenvalues and eigenvectors of the 1-RDM, that can be reformulated as an unconstrained optimization by parameterizing the latter but in both cases suffers from convergence problems. In this thesis, the role of this parametrization is analyzed and motivates a now second order optimization algorithm, which improves the convergence speed by two orders of magnitude w.r.t. the state of the art algorithm. This understanding of the problem also inspires an alternative approach with the 1-RDM as the optimization variable. This however requires the development of new approximations and a first investigation in that direction, based on Machine Learning is performed. Lastly, a generalization of the method to excited states is attempted, highlighting some key computational challenges.
In materials with strong electron correlation the proper treatment of local atomic physics described by orbital occupations is crucial. Reduced density-matrix functional theory is a natural extension of density functional theory for systems that are dominated by orbital physics. We review the current state of reduced density-matrix functional theory (RDMFT). For atomic structure relaxations or ab-initio molecular dynamics the combination of density functional theory (DFT) and dynamical mean-field theory (DMFT) possesses a number of disadvantages, like the cumbersome evaluation of forces. We therefore describe a method, DFT+RDMFT, that combines many-particle effects based on reduced density-matrix functional theory with a density functional-like framework. A recent development is the construction of density-matrix functionals directly from many-particle theory such as methods from quantum chemistry or many-particle Green’s functions. We present the underlying exact theorems and describe current progress towards quantitative functionals.
In this work, we explore the performance of approximations to electron correlation in reduced density-matrix functional theory (RDMFT) and of approximations to the observables calculated within this theory. Our analysis focuses on the calculation of total energies, occupation numbers, removal/addition energies, and spectral functions. We use the exactly solvable Hubbard dimer at 1/4 and 1/2 fillings as test systems. This allows us to analyze the underlying physics and to elucidate the origin of the observed trends. For comparison, we also report the results of the GW approximation, where the self-energy functional is approximated, but no further hypothesis is made concerning the approximations of the observables. In particular, we focus on the atomic limit, where the two sites of the dimer are pulled apart and electrons localize on either site with equal probability, unless a small perturbation is present: this is the regime of strong electron correlation. In this limit, using the Hubbard dimer at 1/2 filling with or without a spin-symmetry-broken ground state allows us to explore how degeneracies and spin-symmetry breaking are treated in RDMFT. We find that, within the used approximations, neither in RDMFT nor in GW, the signature of strong correlation is present, when looking at the removal/addition energies and spectral function from the spin-singlet ground state, whereas both give the exact result for the spin-symmetry broken case. Moreover, we show how the spectroscopic properties change from one spin structure to the other.