TYC Masterclass: Mean field description of electronic structure: From Hartree-Fock to DFT and beyond

Prof. Hannes Jonsson, University of Iceland

TYC Masterclass: Mean field description of electronic structure: From Hartree-Fock to DFT and beyond Share on X

The simplest picture we have for describing systems of electrons is to assume that each electron is only subject to the average influence of the other electrons. This is the basis of ‘mean field’ approximations. A function describing a single electron in such a mean field is referred to as an ‘orbital’ and the probability distribution for the location of the electron is the ‘orbital density’. While Hartree-Fock (HF) theory appears to be the optimal mean field description it turns out not to be in part because of the infinite range of Fock exchange. Today, most calculations in chemistry and condensed matter physics are carried out using density functional theory (DFT) with some approximate functional of the Kohn-Sham (KS) form where the quantum mechanical aspects of the interaction between electrons is of finite range. This so-called ‘nearsightedness’ of the electrons is, for example, manifested in the chemical concept of functional groups. But, the goal of Kohn-Sham theory is to describe electronic systems with only the total electron density, thereby abandoning in principle the concept of orbitals. Orbitals are, however, introduced in KS-DFT only to obtain accurate enough approximation of the kinetic energy of the electrons, but are not used in the estimation of the classical Coulomb interaction, thereby introducing a self-interaction error (SIE). Even if the system consists of just one electron, the KS estimate of the Coulomb interaction gives a non-zero value. The SIE is the primary source of many of the shortcomings of practical implementations of KS-DFT, such as the tendency to overly delocalize electrons, and the incorrect long range form of the potential. By making use of the concept of orbitals and the associated orbital density, the self-interaction in the classical Coulomb interaction can be avoided, but this brings in additional complexity in the calculations since the functional is then orbital density dependent. Over 40 years ago, Perdew and Zunger proposed an orbital based self-interaction correction to KS functionals, but it has not become commonly used for several reasons, one being the added complexity of the numerical calculations. Several examples of such calculations will be given in the lecture, especially for systems where commonly used KS functionals give poor estimates or even fail to give qualitatively correct results. The application of a self-interaction correction to a KS functional is, however, just a small step in the direction of an optimal mean field description. The development of an optimal orbital density dependent and self-interaction free functional remains a future task.

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