The simplest Quantum Monte Carlo technique, variational Monte Carlo (VMC), is based on a direct application of Monte Carlo integration to the calculation of multi-dimensional integrals of expectation values such as the total energy. Monte Carlo methods are statistical and a key result is that the value of integrals computed using Monte Carlo converges faster than by using conventional methods of numerical quadrature, once the problem involves more than a few dimensions. Statistical methods therefore provide a practical means of solving the full many-body Schrödinger equation by direct integration, making only limited and well-controlled approximations.
Comparisons are made with several well-developed approaches for solving the many-body Schrödinger equation in this chapter. Numerous quantum chemical methods and density functional approaches have been developed for solving the ``electronic structure problem''. A review of several methods currently in general use is given along with an outline of their relative advantages and shortcomings. Finally, the advantages of Quantum Monte Carlo techniques are outlined prior to their full description in chapter 3.