In this dissertation I have attempted to explain the motivation for performing computational quantum-mechanical simulations and to describe the major difficulties encountered when one attempts to do so. The progress made so far by the introduction of density-functional theory, incorporating a local density approximation for exchange and correlation and the use of pseudopotentials already allows these calculations to be performed on systems which are of interest to scientists working in a variety of fields today. However, the scope of these calculations is limited by the unfavourable scaling of computational effort and resources required. Methods which exhibit optimal scaling, that is scale in the same way as the complexity of the problem to be solved, offer the prospect of extending the range of accessible scales much further, and will also take full advantage of future improvements in computer technology.

The work in this dissertation is based upon the density-matrix formulation of density-functional theory, which avoids the necessity of dealing directly with the extended wave-functions (which resulted in the unfavourable cubic scaling) and leads naturally to a linear-scaling method.

The spherical-wave basis-set proposed in chapter 5 provides a solution to the problem of representing the density-matrix in real-space while maintaining the accuracy of the kinetic energy (which is naturally calculated in reciprocal-space) and also efficiently calculating the action of the non-local pseudopotential. The analytic results derived have been implemented within the scheme described in chapters 6 and 7.

Secondly, methods to impose the non-linear idempotency constraint by the use of penalty functionals have been described. The failure of the original proposals by Kohn in computational implementations are shown to be due to the functional form of the penalty functional which must be chosen to obtain a variational principle. An original scheme has been proposed in which penalty functionals are chosen to be compatible with efficient minimisation algorithms and to approximately impose the idempotency constraint. The resulting errors in the total energy are corrected by considering the functional form of the penalty functional, so that accurate estimates of the true ground-state energy can be made.

Thirdly the relationship between traditional plane-wave methods based upon the Kohn-Sham wave-functions and density-matrix based schemes are discussed. We show how it is possible to interchange information about the electronic structure of the system between these two methods, and in particular apply this to the problem of obtaining initial density-matrices for use in linear-scaling calculations.

The results in chapters 5 and 6 have all been implemented in a total energy code, which has been tested on bulk crystalline silicon. The convergence of the energy with respect to support region radius and density-kernel cut-off has been examined. Predictions of some physical properties of bulk silicon are then compared with experimental values and results from the CASTEP plane-wave code. Finally, the scaling of the method with system-size is confirmed to be linear, and the scaling with respect to support region radius and density-kernel cut-off discussed.