DFT+U - Theory

The problem

Electron correlation

Transition metal oxides and rare-earth metal compounds have highly localised d and/or f electrons. This leads to strong on-site electron correlation, of which, the exchange-correlation functionals of density functional theory struggle to describe. This can lead DFT to give metallic descriptions of insulating materials:


DFT incorrectly describes FeO as a metal - we would expect, from experiment, insulating behaviour with a band gap of around $2.5$ eV.¹

Other functionals

Some functionals perform slightly better than others, and some even try to account for a portion of the problem. For example, due to its lower self-interaction error, (r-)SCAN can attain slightly better band gap results in some cases.

A solution

The Hubbard U model

The Hubbard model discusses the hopping of electrons between different sites in a lattice. The penalty for electrons to hop to another site is denoted by the parameter, $U$ . See a. In a crystal, this repulsion term introduces a gap in the density of states proportional to $U$ . See b.


The Hubbard model that inspires the DFT+U method.

This energetic penality for the delocalisation of electrons can be appended to the DFT potential, to give the DFT+U functional,

$E_{\text{DFT+U}} = E_{\text{DFT}} + \frac{U}{2}\sum_{i}q_{i}\left(1-q_{i}\right),$

where $q_{i}$ is the occupancy of orbital $i$ . This potential encourages completely empty ( $q\simeq0$ ) or fully filled ( $q\simeq1$ ) orbitals. This term would typically be applied to the d or f electrons of the transition metal(s) or rare-earth metal(s) in the compound of investigation.

Note

This formulation is derivable by imposing that the magnetic interactions are $0$ , or alternatively that some portion of their effects are described through an effective $U_\text{eff}$ of both Coulombic and magnetic potentials. In short, the Hubbard $J$ is set to $0$ here.

Koopmans' compliance

A fractional occupancy of electrons should be described by a statistical mixture of energies,

$E = \left(1-\omega\right) E_{N} + \omega E_{N+1},$

where $N$ is an integer and $0\lt\omega\lt1$ . We expect, therefore, the energy with respect to occupancy to be given by a series of straight-line segments. However, DFT alone gives an unphysical curvature and spurious minima of the energy with respect to the occupancy. DFT+U attempts to restore the piecewise linearity, which is known as Koopmans' compliance.


The total energy with respect to the number of electrons.¹ DFT+U provides a correction to map DFT to the exact energy.

Orbital basis

As there is no unique or rigorous way to define atomic orbitals in a multi-electron system, a choice must be made on how to map the unit cell wavefunction onto the localised functions. In CASTEP, we use the atomic valence states obtained through the construction of the pseudopotential. Any chosen value for the $U$ parameter is therefore dependent on this choice of basis.

Results

The DFT+U formalism correctly introduces a band gap in the bandstructure of FeO:


DFT comparison with DFT+U.¹

Physical Review B 71, 035105 (2005) ↩↩↩