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arXiv:1912.12493 · Physical review. B./Physical review. B

Density functional theory plus dynamical mean-field theory with natural atomic orbital projectors

Jae-Hoon Sim, Myung Joon Han

Published Sep 25, 2019• 5 citations
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Abstract

We introduce natural atomic orbitals as the local projector to define the correlated subspace for $\mathrm{DFT}+\mathrm{DMFT}$ (density functional theory plus dynamical mean-field theory) calculation. The natural atomic orbitals are found to be stably constructed against the number and the radius of basis orbitals. It can also be self-consistently updated inside the $\mathrm{DFT}+\mathrm{DMFT}$ loop. The spatial localization, electron occupation and the degree of correlation are investigated and compared with other conventional techniques. As a ``natural'' choice to describe the electron numbers, adopting natural atomic orbitals has advantage in terms of electron number counting. We further explore the reduction of computation cost by separating correlated orbitals into two subgroups based on the orbital occupancy. Our new recipe can serve as a useful choice for $\mathrm{DFT}+\mathrm{DMFT}$ and related methods.

DOI
10.1103/physrevb.100.115151

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We introduce natural atomic orbitals as the local projector to define the correlated subspace for DFT+DMFT\mathrm{DFT}+\mathrm{DMFT}DFT+DMFT (density functional theory plus dynamical mean-field theory) calculation. The natural atomic orbitals are found to be stably constructed against the number and the radius of basis orbitals. It can also be self-consistently updated inside the DFT+DMFT\mathrm{DFT}+\mathrm{DMFT}DFT+DMFT loop. The spatial localization, electron occupation and the degree of correlation are investigated and compared with other conventional techniques. As a ``natural'' choice to describe the electron numbers, adopting natural atomic orbitals has advantage in terms of electron number counting. We further explore the reduction of computation cost by separating correlated orbitals into two subgroups based on the orbital occupancy. Our new recipe can serve as a useful choice for DFT+DMFT\mathrm{DFT}+\mathrm{DMFT}DFT+DMFT and related methods.
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.
Occupation numbers of natural orbitals capture the physics of strong electron correlations in momentum space. A natural orbital density functional theory based on the antisymmetrized geminal product provides these occupation numbers and the corresponding electron momentum density. A practical implementation of this theory approximates the natural orbitals by the Kohn–Sham orbitals and uses a mean-field approach to estimate pairing amplitudes leading to corrections for the independent particle model. The method is applied to weakly doped La2CuO4.
Abstract Density functional theory (DFT) is usually formulated in terms of the electron density as a function of position n(r). Here we discuss an alternative formulation of DFT in terms of the orbital occupation numbers {n α } associated with a local-orbital orthonormal basis set {ϕ α }. First, we discuss how the building blocks of DFT, namely the Hohenberg–Kohn theorems, the Levy–Lieb approach and the Kohn–Sham method, can be adapted for a description in terms of {n α }. In particular, the total energy is now a function of {n α }, E[{n α }], and a Kohn–Sham-like Hamiltonian is derived introducing the effects of the electron–electron interactions via effective potentials, { V α eff = ∂ E e e [ { n β } ] / ∂ n α } . In a second step we consider the Hartree and exchange energies and discuss how to describe them, in the spirit of a DFT approach, in terms of the orbital occupation numbers. In this contribution special attention is paid to the description of the (intra-atomic) correlation energy and corresponding correlation potentials {V corr,α }. For this purpose, a model system is analyzed in detail, whereby an atomic Hamiltonian interacts with the environment via a simplified model; the use of this model allows us to obtain the correlation energy and potentials (in terms of {n α }) for different cases corresponding to low, intermediate and high electron correlations.
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.
Natural orbital functional (NOF) theory provides a valuable framework for studying strongly correlated systems at an affordable computational cost, with an accuracy comparable to highly demanding wave-function-based methods. However, its widespread adoption in cases involving a large number of correlated electrons has been limited by the extensive iterations required for convergence. In this Letter, we present an approach that integrates the techniques used for optimization in deep learning into NOF calculations, enabling a substantial expansion in the scale of accessible systems. The proposed procedure employs an adaptive momentum-based technique for orbital optimization, alternated with the optimization of occupation numbers, significantly improving the computational feasibility of challenging calculations. We illustrate the capabilities of our approach through three challenging test cases: (i) the symmetric dissociation of a large hydrogen cluster with 1000 electrons, (ii) an analysis of occupancy distributions in fullerenes, and (iii) a study of the singlet-triplet energy gap in linear acenes. These examples demonstrate the method's applicability to large-scale systems and strongly correlated electron phenomena, extending the reach of NOF theory to increasingly complex systems.