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## Self-Interaction Correction: State of the Art and New Directions

#### Ramada Jarvis Hotel, Chester, UK

#### Organisers

**Registration:**

To register for this workshop, please click here.

**Motivation:**

The motivation for this workshop is to review different methodologies and different applications of self-interaction corrected local-density approximation (LDA). The self-interaction error, introduced by local and semi-local approximations to density functional theory (DFT), leads to some dramatic failures of DFT, ranging from wrong predictions in chemical reactions to the failure in describing the insulating state in many transition metal oxides, and qualitatively wrong pictures for lanthanide and actinide compounds. Nearly 30 years ago, Perdew and Zunger (PZ) suggested a remedy for this error, the so-called self-interaction corrected (SIC) local (spin) density (LSD) approximation [Phys. Rev. B23, 5048 (1981)].

During the years that have passed since the publication of this seminal paper, the method has led to a plethora of applications in different fields of physics and chemistry. The original paper also gave rise to a variety of implementations, generalizations, and extensions of the method. It has also become apparent that different branches of SIC have been developed nearly independently of each other, in particular in the field of quantum chemistry and solid state physics, with the experiences/advances gained in one field barely noticed in the other areas.

We think that the next year’s 30th anniversary of the original paper by Perdew and Zunger would be a good opportunity to bring together, for the first time, all the groups that have applied/worked on self-interaction correction, in order to discuss and assess the state of the art of all the different flavours of SIC, share the experiences and identify the most important and burning issues, unsolved problems, and perhaps find a common direction for the future development.

The semi-local approximations to DFT, including the local density approximation, the local spin-density approximation, but also generalized gradient approximations (GGA’s), have proven to be very successful in many areas of quantum chemistry and solid-state physics. However, there are important cases, in both fields, where these approximations fail in a qualitative way. Many of these shortcomings can be attributed to the spurious self-interaction of electrons, which is introduced by the semi-local approximations. Perdew and Zunger (PZ) [1] suggested removing this self-interaction explicitly for each occupied orbital. While applications to finite systems were more straightforward, this method had interesting implications for extended systems: Due to the orbital dependence of the effective potential in SIC-LSD, and the fact that the PZ self-interaction correction vanishes in the thermodynamic limit for extended, i.e. Bloch like, states, the SIC-Schroedinger equation can have multiple solutions, depending on whether a state is assumed localized or delocalized. By comparison of the total energy of these solutions, the ground-state character of an electron can be determined. This led to a clear definition of the valence of an atom in a solid which, in fact, is quite close to the concept of valence use in chemistry. Furthermore, it gave rise to a variety of studies of localization-delocalization transitions in solids (see e. g. [2]).

A somewhat different correction was proposed by Lundin and Eriksson who argued that one ought to subtract the orbital density from the total density first and then for this resulting quantity evaluate the energy functional [3].

Other applications focus more on spectroscopy. Here, the original PZ-SIC approach shifts the corrected states to too low energies with respect to the Fermi level. Attempts to improve on this over-correction include a scaling of SIC, which is motivated by an argument based on the electron removal energies [4]. The results of this approach show similar trends as the densities of state and spectra calculated with LDA+U, but do not depend on the parameter U, but are determined from first principles. Self-interaction corrections, together with relaxation corrections, have been incorporated into pseudo potentials and applied to a variety of materials [5].

Other modified versions of PZ-SIC, which assume that localized states are confined within a given atom, have been implemented for transport calculations [6], or in combination with the coherent potential approximation, which allows applications to disordered systems [7]. The latter does not only include chemical disorder, but can also be applied to describe magnetically disordered systems (above the ordering temperature), or systems showing valence disorder [8]. In particular, it was shown that this approach is able to describe the insulating but paramagnetic state of transition metal oxides above their critical temperature [9], which was previously believed to be a dynamical effect which could only be described using dynamical mean field theory (DMFT).

For finite systems, also a time-dependent version of SIC has been developed and applied to molecules and quantum dots [10].

In the field of quantum chemistry, much effort was put into the study of the self-interaction error and the quantitative effects of SIC [11]. In particular, it was shown that the self-interaction correction restores the derivative discontinuities of the exchange and correlation potential, which is lost in semi-local approximations, and hence corrects the failure to describe dissociated atoms and charge transfer. In this context, also implementations of SIC using the optimized effective potential formalism have been studied [12]. The findings of these investigations brought to light some shortcomings of the PZ implementations, which then lead to the development of so-called scaled-down versions of SIC, where the magnitude of the correction term is reduced in areas where the density has predominantly many-body character.

Recently, also other methods have been developed, which are addressing the self-interaction error, and related issues such as the derivative continuities. One promising candidate is the reduced density-matrix functional theory (RDMFT), which has been shown to give very good results for the fundamental gaps of a series of systems, including atoms, molecules, as well as semi-conductors and insulators [13].

Despite the many successes of SIC-DFT, there are still open questions, e.g. regarding the sphericalization, which is often performed in actual implementations, how important the orbital-dependent character of the effective potential is, or how to deal with the over-correction of PZ-SIC. Since some of these questions have already been addressed by some groups, while these questions at the same time are highly relevant in the development of a true ab initio theory for strongly correlated materials, it seems very timely to bring together these communities in order to share the experiences and define new directions for future developments.

## References

**Germany**

Martin Lueders (Max Planck Institute for the structure and dynamics of matter) - Organiser

**United Kingdom**

Leon Petit (STFC Daresbury Laboratory) - Organiser

Zdzislawa (Dzidka) Szotek (Daresbury Laboratory) - Organiser