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arXiv:1507.01754 · Journal of Chemical Theory and Computation

Spectrum for Nonmagnetic Mott Insulators from Power Functional within Reduced Density Matrix Functional Theory

Yasushi Shinohara, S. Sharma, Sam Shallcross, Nektarios N. Lathiotakis, Eberhard K. U. Gross

Published Sep 11, 2015• 21 citations
arXivHTMLPDF

Abstract

We demonstrate that reduced density matrix functional theory (RDMFT), in conjunction with the power functional, can successfully treat the nonmagnetic insulating state of the transition metal oxides NiO and MnO, finding for both a gapped single particle spectrum. While long-range spin order is thus not necessary for qualitative agreement with experiment, we find that it is required for good agreement with the X-ray photoemission spectroscopy and Bremsstrahlung isochromat spectroscopy data. We further examine the nature of the natural orbitals in the materials, finding that they display significant Hubbard localization and are, as a consequence, very far from the corresponding Kohn-Sham orbitals. This contrasts with the case of the band insulator Si, in which the Kohn-Sham orbitals are found to be very close to the RDMFT natural orbitals.

DOI
10.1021/acs.jctc.5b00661

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We demonstrate that reduced density matrix functional theory (RDMFT), in conjunction with the power functional, can successfully treat the nonmagnetic insulating state of the transition metal oxides NiO and MnO, finding for both a gapped single particle spectrum. While long-range spin order is thus not necessary for qualitative agreement with experiment, we find that it is required for good agreement with the X-ray photoemission spectroscopy and Bremsstrahlung isochromat spectroscopy data. We further examine the nature of the natural orbitals in the materials, finding that they display significant Hubbard localization and are, as a consequence, very far from the corresponding Kohn-Sham orbitals. This contrasts with the case of the band insulator Si, in which the Kohn-Sham orbitals are found to be very close to the RDMFT natural orbitals.
We present a method for calculating the spectrum of extended solids within reduced density matrix functional theory. An application of this method to the strongly correlated transition metal oxide series demonstrates that (i) an insulating state is found in the absence of magnetic order and, in addition, (ii) the interplay between the change transfer and Mott-Hubbard correlation is correctly described. In this respect we find that while NiO has a strong charge transfer character to the electronic gap, with substantial hybridization between t2gt_{2g}t2g​ and oxygen-ppp states in the lower Hubbard band, for MnO this is almost entirely absent. As a validation of our method we also calculate the spectra for a variety of weakly correlated materials, finding good agreement with experiment and other techniques.
Abstract Strongly correlated systems pose a challenge for theoretical methods based on an independent electron approximation. Such methods struggle to predict a nonzero gap in Mott insulators or to capture the correct physics of the insulator-to-metal phase transition in strongly correlated materials. In a recent paper by Shinohara et al (2015 New J. Phys. 17 093038 ) it is shown that strongly correlated materials and correct descriptions of their phase transitions are within the reach of reduced density matrix functional theory (RDMFT) approximations. For a doping-induced phase transition, not only is a satisfactory agreement with experimental spectra found for NiO but it is also shown that the physical picture of the observed Mott transition stays in line with more computationally demanding many-body theories. This is an important step toward providing an RDMFT–based computation tool for studying strongly correlated materials.
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.
Density-Functional Theory (DFT), although widely used and very successful in the calculation of several observables, fails to correctly describe strongly correlated materials. In the first part of this work we, therefore, introduce reduced-density-matrix-functional theory (RDMFT) which is one possible way to treat electron correlation beyond DFT. Within this theory the one-body reduced density matrix (1-RDM) is used as the basic variable. Our main interest is the calculation of the fundamental gap which proves very problematic within DFT. In order to calculate the fundamental gap we generalize RDMFT to fractional particle numbers M by describing the system as an ensemble of an N and an N+1 particle system (with N≤ M≤N+1). For each fixed particle number, M, the total energy is minimized with respect to the natural orbitals and their occupation numbers. This leads to the total energy as a function of M. The derivative of this function with respect to the particle number has a discontinuity at integer particle number which is identical to the gap. In addition, we investigate the necessary and sufficient conditions for the 1-RDM of a system with fractional particle number to be N-representable. Numerical results are presented for alkali atoms, small molecules, and periodic systems. Another problem within DFT is the description of non-relativistic many- electron systems in the presence of magnetic fields. It requires the paramagnetic current density and the spin magnetization to be used as basic variables besides the electron density. However, electron-gas-based functionals of current-spin-density-functional Theory (CSDFT) exhibit derivative discontinuities as a function of the magnetic field whenever a new Landau level is occupied, which makes them difficult to use in practice. Since the appearance of Landau levels is, intrinsically, an orbital effect it is appealing to use orbital-dependent functionals. We have developed a CSDFT version of the optimized effective potential (OEP) method which allows for the use of explicitly orbital-dependent functionals in the context of CSDFT. We present the derivation of corresponding equations and show results for a quantum dot in external magnetic fields in the second part of this thesis.
The insulating state of matter is characterized by the excitation spectrum, but also by qualitative features of the electronic ground state. The insulating ground wavefunction in fact: (i) displays vanishing dc conductivity; (ii) sustains macroscopic polarization; and (iii) is localized. The idea that the insulating state of matter is a consequence of electron localization was first proposed in 1964 by W. Kohn. I discuss here a novel definition of electron localization, rather different from Kohn's, and deeply rooted in the modern theory of polarization. In fact the present approach links the two features (ii) and (iii) above, by means of essentially the same formalism. In the special case of an uncorrelated crystalline solid, the localization of the many-body insulating wavefunction is measured - according to our definition - by the spread of the Wannier orbitals; this spread diverges in the metallic limit. In the correlated case, the novel approach to localization is demonstrated by means of a two-band Hubbard model in one dimension, undergoing a transition from band insulator to Mott insulator.