Quantum Monte Carlo techniques provide a practical method for solving the many-body Schrödinger equation. They are closely related to the well established classical Monte Carlo methods that have been successfully applied to a wide range of problems involving stochastic behaviour ranging from scientific problems, engineering problems and modelling the financial markets. The common link between classical and quantum Monte Carlo techniques is the use of random numbers to evaluate multi-dimensional integrals.
In its simplest form, the variational Monte Carlo (VMC) technique is based on evaluating a high-dimensional integral by sampling the integrand using a set of randomly generated points. It can be shown that the integral converges faster using a Monte Carlo technique than more conventional techniques based on sampling the integrand on a regular grid for problems involving more than a few dimensions. Moreover, the statistical error in the estimate of the integral decreases as the square root of the number of points sampled, irrespective of the dimensionality of the problem.
In this chapter the QMC techniques are compared with more established methods of solving the many-body Schrödinger equation. The relative merits of the different techniques for calculating the electronic structure of atoms, molecules and solids are considered. The progress made over the past decade in developing QMC methods as a tool for tackling realistic continuum electronic structure problems is described. At the end of the chapter, a summary of the layout of the thesis is given.