Strongly Correlated Materials in Electronic Structure Theory - Summary

While in simple metals, semiconductors and many ionic materials first principles calculations now underpin our understanding of the electronic structure and bonding, attempts to apply first principles methods to `strongly correlated' systems such as NiO and the high-T_c cuprates have been fraught with difficulties. For example, when the LDA is used to describe the magnetically-ordered, insulating ground states of materials such as La₂CuO₄, a non-magnetic, metallic ground state is obtained (see e.g. Pickett 1989). For many years such results were thought to be a failure of the one-electron approximation per se and proof positive that these systems could not be described with mean-field methods. It is now apparent that this is not the case. The failure of the LDA (and the various GGAs) is essentially due to its approximate treatment of the exchange interaction.

The reasons for the failure of DFT approaches in calculations of strongly correlated materials are still not widely appreciated. Even today, it is easy to find textbooks which claim that `band theory' or, equivalently, the one-electron approximation is somehow inappropriate in such situations. The situation is not helped by confusing terminology. The description `strongly correlated' used by many-body physicists is often confused with 'correlation' in the quantum chemistry sense, that is, all interactions not included in the mean field (of HF theory). In fact the `strong correlation' refers to a strong on-site Coulomb interaction between localized electrons on the same ion which is part of the Hartree interaction, and therefore included in something as simple as HF theory.

A different approach to these systems allows us to understand the problem with many DFT approaches. The key to this approach is the fact that the magnetically-ordered insulating ground state may often be described by a single determinant wave function and may therefore be accurately represented using the Hartree-Fock approximation which, by definition, contains the exact exchange interaction. It has been demonstrated that HF calculations of strongly correlated magnetic insulators agree very well with experiment. It may seem surprising that single-determinant HF could be so successful, but this is an important characteristic of these ionic materials. The highly symmetric environment and long-range Coulomb forces tend to separate the orbitals into well-defined subsets with a significant gap between occupied and unoccupied states. Hence, the ground state of NiO is rather well described by a single determinant while one could easily imagine a covalently-bonded molecular complex (for example) for which this approximation would be poor. In this sense, a strongly correlated magnetic insulator is in many ways a `simpler system' than many molecules. The success of such calculations and also hybrid schemes using combinations of DFT and exact exchange has now been well documented in a variety of publications.

The failure of the LDA to describe the highly correlated oxides adequately, despite the fact that the ground state can be well-approximated by a single determinant wave function, is now understood. Within the LDA, the potential felt by each electron is computed from a functional of the total electron densities. For the density functionals in common use this leads to eigenvalues which are relatively weak functions of the particular occupancy. Ultimately this behaviour stems from the spurious inclusion of `self-interaction' effects in the exchange-correlation potential. In HF theory, the non-local exchange exactly cancels the self-interaction and introduces a strongly orbitally-dependent potential which splits the manifold of d states in precisely the manner expected from a simple empirical (`Hubbard model') estimate of the on-site interactions between electrons in different orbitals. Indeed a variety of new `DFT' schemes (e.g. LDA+U, SIC-LDA) which emulate important features of the Hartree-Fock Hamiltonian have now been developed which give better descriptions of the on-site interactions.

The mean-field approach to strongly correlated materials is therefore not as bad as commonly thought. The simple LDA and GGA approaches are indeed qualitatively incorrect, but for reasons which are not directly due to the one-electron approximation. However, for those calculations containing exact exchange or some appropriate approximation to it, a qualitatively reasonable ground state is obtained. This is all we need as input to a quantum Monte Carlo calculation. Work is proceeding in this direction.