The accurate description of electronic systems by means of ab-initio computer simulation methods is one of the most important tasks in quantum chemistry and condensed matter physics. The principal problem, which manifests itself in different forms, is that of describing electron correlation, in either molecules or materials, beyond mean-field or pure density-functional theories, especially in their local form, e.g. the traditional local density approximation (LDA) and generalized gradient approximation (GGA).

Difficult chemical systems include strong correlation problems, often associated with partially filled and highly localized d- and f- shells, or else even simpler molecular dissociation problems. Similarly, in materials, the narrow band systems of the first row transition metals and rare-earth metals are very difficult to describe, as indeed are their oxides, and other compounds, where the effect of non-stoichiometry can profoundly affect the electronic structure, and properties such as magnetism, susceptibility, etc. Long-range electron correlation, as exemplified by the van-der-Waals interaction, is another manifestation which requires beyond-LDA/GGA treatment.

Of course, a huge variety of methodologies exist. In this proposal, we limit ourselves to the ab-initio type (i.e. those concerned with treating realistic systems, as opposed to lattice models, from a first-principles point of view), which we can broadly classify as follows:

(1) Quantum Monte Carlo (QMC) methods constitute the most recent advance in the simulation of electronic systems, which unlike the normal quantum chemical methods, are essentially stochastic in the way they seek to solve the electron correlation problem exactly.

Traditionally, they have been based on the variational MC method, or diffusion MC and Green's function MC. The latter two are projection approaches which dispense with quantum chemical basis sets, but they have to deal with the fermion sign problem, and the related fixed-node approximation. In general, all QMC methods have a good scaling with the number of electrons, enabling relatively large systems to be tackled, but with a computational cost much larger than traditional ab-initio methods based on DFT.

In this respect a remarkable progress was recently achieved by using appropriate

''energy consistent'' pseudopotentials which are free of the electron-ion cusp conditions and are appropriate for QMC methods, as they reduce the computational cost by a substantial factor, especially for heavy atoms [1].

On the other hand, the introduction of the fixed-node approximation leads to errors which do not in general cancel nicely. Progress in this regard is perhaps the hardest and most important for these methodologies.

Recent work by Alavi and his group [2] suggests that there may be fruitful interplay between QMC and quantum chemical approaches which do not involve fixed-node approximations. Moreover, a substantial improvement for the description of the chemical bond was obtained at the variational level by efficient optimization techniques, combined with new types of variational wave functions (correlated AGP [3], Pfaffians [4]), that describe quite accurately strongly correlated systems in a very efficient way. Determinantal auxiliary-field QMC methods should also be mentioned in this regard [5].

(2) In quantum chemistry, the treatment of ``static'' correlation is often centered around the (usually unattainable) full-Configuration interaction method. In practice, CASSCF and CASPT2 are among the most successful methodologies [6], where the complete active space (CAS) approach accounts reasonably well for strong correlation, with the multiconfigurational perturbation theory (PT2) adding on the necessary dynamical correlation. CASSCF has, however, exponential scaling. Probably the most challenging area is the development of numerical techniques which can extend the applicability of these methods. An interesting development is the application of Density-Matrix Renormalization Group (DMRG) formalism to the static correlation problem in quantum chemistry [7], with some new input coming from quantum information theory [8].

From a DFT perspective, the LDA+U has become popular as a means to treat strong correlation problems, even if that does introduce an element of semi-empiricism into the theory. For ``dynamical correlation" (or single reference) problems, the quantum chemists have developed Coupled Cluster methodologies such as CCSD(T), or perturbation theories such as MP2, etc. Here the challenge is computational cost and scaling of the methods, and much effort is being devoted to producing "linear" scaling methods, based on local orbitals and localization approximations (e.g. the local correlation method of Pulay and Saebo, local ansatz of Stollhof and Fulde, and method of increments of Stoll). Such methods can also find application to bulk systems, and the development of wave function based methods to bulk systems [9] is an active and developing one (Local MP2 as implemented in CRYSCOR). In addition, there is also interest in developing canonical MP and CCSD theories for bulk systems, based on Bloch orbitals. A complementary DFT-based approach to the non-local correlation problem is provided by the adiabatic-connection fluctuation dissipation theorem, which provides a route to construct non-local density functionals, e.g. as a means to tackle van der Waals interactions.