[Paper Review] Orbital densities functional

В. И. Анисимов, Anton Kozhevnikov+3 more|arXiv (Cornell University)|Feb 8, 2006

Advanced Physical and Chemical Molecular Interactions5 citations

TL;DR

This paper introduces the Orbital Densities Functional (ODF), a density functional theory extension that enforces the exact discontinuity of the exchange-correlation potential at integer electron counts by using Wannier-function-based orbital densities. By incorporating a projection-operator potential that increases band separation, ODF corrects the systematic underestimation of band gaps in LDA, successfully predicting a paramagnetic insulator in LaH₃₋ₓ for x ≤ 0.3, consistent with experiment.

ABSTRACT

Local density approximation (LDA) to the density functional theory (DFT) has continuous derivative of total energy as a number of electrons function and continuous exchange-correlation potential, while in exact DFT both should be discontinuous as number of electrons goes through an integer value. We propose orbital densities functional (ODF) (with orbitals defined as Wannier functions) that by construction obeys this discontinuity condition. By its variation one-electron equations are obtained with potential in the form of projection operator. The operator increases a separation between occupied and empty bands thus curing LDA deficiency of energy gap value systematic underestimation. Orbital densities functional minimization gives ground state orbital and total electron densities. The ODF expression for the energy of orbital densities fluctuations around the ground state values defines ODF fluctuation Hamiltonian that allows to treat correlation effects. Dynamical mean-field theory (DMFT) was used to solve this Hamiltonian with quantum Monte Carlo (QMC) method for effective impurity problem. We have applied ODF method to the problem of metal-insulator transition in lanthanum trihydride LaH_{3-x}. In LDA calculations ground state of this material is metallic for all values of hydrogen nonstoichiometry x while experimentally the system is insulating for x < 0.3. ODF method gave paramagnetic insulator solution for LaH_3 and LaH_{2.75} but metallic state for LaH_{2.5}.

Motivation & Objective

• Address the systematic underestimation of band gaps in LDA for insulators and Mott insulators.
• Reproduce the exact discontinuity of the exchange-correlation potential at integer electron numbers, a known deficiency of LDA.
• Develop a functional based on orbital densities (via Wannier functions) to enforce this discontinuity by construction.
• Enable treatment of electron correlation effects through a derived fluctuation Hamiltonian.
• Apply the method to predict the metal-insulator transition in lanthanum trihydride (LaH₃₋ₓ), resolving LDA's failure to predict insulating behavior.

Proposed method

• Define the Orbital Densities Functional (ODF) using Wannier functions as localized orbitals, with electron density expressed as a sum over orbital densities.
• Construct the ODF such that its variational derivative produces a discontinuous potential at integer electron counts, matching exact DFT requirements.
• Derive one-electron equations with a correction potential in the form of a projection operator onto occupied Wannier orbitals.
• The projection operator lowers energy for occupied states and raises it for unoccupied states, increasing the band gap beyond LDA.
• Formulate a fluctuation Hamiltonian from the ODF energy functional to describe electron correlation effects.
• Solve the effective impurity problem of the fluctuation Hamiltonian using dynamical mean-field theory (DMFT) combined with quantum Monte Carlo (QMC).

Experimental results

Research questions

• RQ1 Can a density functional theory extension be constructed that enforces the exact discontinuity of the exchange-correlation potential at integer electron numbers?
• RQ2 Does using orbital densities defined via Wannier functions enable a functional that corrects the LDA band gap underestimation?
• RQ3 Can the ODF method predict a metal-insulator transition in LaH₃₋ₓ that LDA fails to reproduce?
• RQ4 How do correlation effects, treated via the ODF fluctuation Hamiltonian, influence the electronic structure of strongly correlated systems?
• RQ5 What is the optimal localization of Wannier functions to minimize the energy of orbital density fluctuations in the ODF framework?

Key findings

• The ODF method successfully reproduces the exact discontinuity of the exchange-correlation potential at integer electron numbers by construction.
• The ODF correction potential, derived as a projection operator, increases the separation between occupied and unoccupied bands, correcting LDA's underestimation of the band gap.
• For LaH₃, the ODF method predicts a paramagnetic insulator ground state, consistent with experimental observations.
• For LaH₂.₇₅, the ODF method also yields a paramagnetic insulator solution, matching experimental insulating behavior for x ≤ 0.3.
• For LaH₂.₅, the ODF method predicts a metallic ground state, consistent with the absence of an insulating phase in experiments at higher hydrogen content.
• The energy of orbital density fluctuations is minimized when Wannier functions are as spatially localized as possible, indicating optimal orbital definition for the ODF framework.