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Quantum Monte Carlo methods are powerful probabilistic approaches for computing quantum averages of a N-body quantum system described by a Schrödinger equation. Many QMC variants known under various acronyms exist in the literature. We propose here to classify them as follows:

  • Zero-temperature (T=O) and Finite-temperature (T diff 0) QMC methods
  • QMC defined in continuous or discrete (lattice) configuration space
  • QMC for Boltzmanon, Fermion, or Boson particles.

In what follows a self-contained presentation of QMC is proposed. We shall concentrate on the T=0 QMC approaches defined in a continuous space corresponding to electronic structure theory conditions.

In order to make easier the presentation, we shall first consider the case of a one-dimensional system defined in a continuous space and no particular statistics. Second, we shall say a few words about the generalization to arbitrary number d of dimensions (actually, d=3N where N is the number of electrons). Remark that it is a nice aspect of the vast majority of Monte Carlo methods that such a generalization does not bring additional difficulties. Finally, we shall consider how to introduce into QMC the specific constraints due to the Pauli principle.

I. T=0 QMC for a continuous one-dimensional system with no specific symmetry constraints