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 the magnetic-insulating transition metal oxides and the high-T_c cuprates have been fraught with difficulties. For example, when the local density approximation of density functional theory is used to describe the magnetically-ordered, insulating ground states of materials such as La₂CuO₄, a non-magnetic, metallic ground state is obtained. 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.

For many years the reasons for the failure of DFT approaches in calculations of strongly correlated materials with the functionals then available were not widely appreciated. Even today, it is difficult to find textbooks which do not 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' of many-body physics refers to a strong on-site Coulomb interaction between localized electrons whose major component is part of the basic Hartree interaction, and therefore included in something as simple as HF theory.

A different approach to these systems was tried fairly recently. The key to this approach was the realization 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. I was able to show that unrestricted HF calculations of strongly correlated magnetic insulators agreed pretty 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 one-electron 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.

We have therefore begun a series of QMC calculations of important transition metal materials. We are concentrating in particular on those properties which have been extremely difficult to calculate accurately in any other way for solids. For example, with QMC one can calculate excitation energies that include the relaxation and screening effects that are missing from a consideration of rigid band structures. The calculation of accurate band gaps and excitation spectra in such materials has long been a goal of condensed matter physics. We also intend to perform many-body QMC calculations of magnetic interactions such as superexchange in magnetic insulators in order to evaluate the usefullness of this technique in studying magnetism. In particular it would be of considerable interest to determine in a quantitative way how magnetic interactions are influenced by charge and orbital ordering in for example, the perovskite manganites. These are the compounds in which the phenomenon of colossal magnetoresistance occurs. Accurate many-body calculations have never been performed on such materials, and would be of fundamental interest. We have begun these calculations by performing benchmark calculations of simple molecules adsorbed on transition metal oxide surfaces. Ultimately we would also want to perform calculations on materials where non-collinear spins are important.

We have already been approached by several groups to perform calculations on materials containing transition elements. In particular, Matt Segall and Mike Payne have asked us to perform accurate calculations of a porphyrin model of the iron-containing active site of the enzyme cytochrome P450 in connection with a problem for which density functional calculations have proved to be inconclusive. Nic Harrison of Imperial College has asked us to carry out calculations of pressure-induced charge-transfer excitations in FeTiO₃. These investigations are now in progress.