Perspective: Treating electron over-delocalization with the DFT+U method

Abstract

Many people in the materials science and solid-state community are familiar with the acronym “DFT+U.” For those less familiar, this technique uses ideas from model Hamiltonians that permit the description of both metals and insulators to address problems of electron over-delocalization in practical implementations of density functional theory (DFT). Exchange-correlation functionals in DFT are often described as belonging to a hierarchical “Jacob’s ladder” of increasing accuracy in moving from local to non-local descriptions of exchange and correlation. DFT+U is not on this “ladder” but rather acts as an “elevator” because it systematically tunes relative energetics, typically on a localized subshell (e.g., d or f electrons), regardless of the underlying functional employed. However, this tuning is based on a metric of the local electron density of the subshells being addressed, thus necessitating physical or chemical or intuition about the system of interest. I will provide a brief overview of the history of how DFT+U came to be starting from the origin of the Hubbard and Anderson model Hamiltonians. This history lesson is necessary because it permits us to make the connections between the “Hubbard U” and fundamental outstanding challenges in electronic structure theory, and it helps to explain why this method is so widely applied to transition-metal oxides and organometallic complexes alike.

Towards quantifying the role of exact exchange in predictions of transition metal complex properties

“…This linear result suggests that HF-derived localization of covalent, delocalized orbitals may not be suitable for understanding the effect of higher fractions of HF exchange. A comparison to self-interaction correction schemes 36,37 and the delocalizationpenalty +U approach 42 will likely be instructive in the future. Since ∆E HS−LS varies linearly with HF exchange, linearregression fits are very good approximations to the partial derivative of the energy with respect to HF exchange (a HF ),…”

“…Preliminary success has been made in identifying chemically motivated ways to tune functional parameters [60][61][62] outside of organometallic chemistry. The appropriate tuning for organometallic complexes or correlated materials is largely still approximated on local measures of the chemical potential of the subshell of interest, 42 and efforts to improve functionals on transition metal test sets have indicated no clear path to optimization.…”

“…34,35 One suggested reason for the failure of practical DFT in describing organometallics is that relatively localized 3d valence electrons suffer strongly from self-interaction error (SIE) present in pure density functionals, which is only approximately corrected in hybrid functionals. Strides have been made in systematic removal of SIE 36,37 and identification of paths to improve balance in spin-state ordering, 28,29,[38][39][40][41][42] but hybrid functionals remain a popular and straightforward approach to approximately correct for energetic errors driven by imbalances in SIE between spin states.…”

Density functional theory for modelling large molecular adsorbate–surface interactions: a mini-review and worked example

“…[108,109,110] Here, a Hubbard model [111] functional is added to the functional to assist in correcting electron overdelocalization for an nl subshell (e.g., 3d electrons) on atom I:…”

Applications of large-scale density functional theory in biology

“…The class of methods known as DFT+U attempts to use a corrective functional inspired by the Hubbard model to improve deficiencies of the standard DFT description [152,153,154]. In practice, DFT+U methods have the effect of penalising non-integer occupations of orbitals of a partially-filled correlated subspace such as the 3d electrons of a first-row transition metal.…”