However, the numerical solution of Schrödinger's equation remains a difficult task. Exact solutions of the equation are, in general, only solvable in times scaling exponentially with system size. This scaling precludes exact calculations for all but the smallest and simplest of systems. Approximations may be introduced to reduce the equations to a form that can be solved in polynomial time, but at the penalty of losing some degree of accuracy and predictive power. The treatment of electron-electron interactions is the principle source of difficulty: the physical and chemical properties of a system depend principally on the interaction of the electrons with each other and with the atomic cores. These interactions cannot easily be separated out or treated without approximation.
The most successful electronic structure methods in current use, those of density functional theory and quantum chemistry, have been applied to a wide range of systems relevant to the real world. In practice, the density functional and quantum chemical approaches involve approximations for the electron-electron interactions, limiting the achievable accuracy.
In this thesis, stochastic methods for the solution of Schrödinger's equation are developed and applied to real systems. Quantum Monte Carlo (QMC) methods treat electron-electron interactions almost without approximation and with a computational cost scaling cubically with system size. Thieir accuracy enables an unprecedented degree of confidence to be placed in the results obtained. A number of technical developments are made in the earlier chapters of the thesis, prior to two significant applications: the computation of the one-body density matrix of silicon and several related quantities, and a study of the energetic stability of a series of carbon clusters. The use of Quantum Monte Carlo methods was essential in both of these applications, particularly in the study of carbon clusters where no other currently applicable computational method is sufficiently accurate.