where configurations R=[r1,r2,...,rN] are sampled from the distribution of the wavefunction squared using a Metropolis random walk. The functional E_VMC provides an upper bound to the exact ground state energy and will be a minimum when the trial function is equal to the exact ground state wavefunction.

One of the important features of the QMC methods is that arbitrary trial functions may be used, including ones with explicit dependence on interelectronic distances. Since in VMC all averages are evaluated with respect to the trial function, the choice of its form is very important. In general, there is a trade-off between trial functions that are accurate and those that are fast to evaluate, and an acceptable balance between the two is sought.

A common form for the trial function is the Slater-Jastrow type, which is a sum of Slater determinants of single electron spin-up and spin-down orbitals times a correlation factor:

where I corresponds to the ions, i, j to the electrons, and the r's are distances. The Slater determinants are typically built either from Hatree-Fock or Density Functional calculations, and the parameters in uare optimized by minimizing the variance of the energy.

This limit is found by solving the imaginary-time Schroedinger equation in integral form. The effeciency of the method is greatly increased by using the technique of importance sampling, in which the unknown wavefunction is multiplied by the known trial function, and the product of the two is sampled during the walk. This also allows one to use the fixed-node approximation, where the nodes of the ground state wavefunction are restricted to be identical with the nodes of the trial function. This guarantees that the product (which is what is sampled) is positive everywhere. The impact of the fixed-node approximation is usually around 5% of the correlation energy.