Green function fixed-node approximation

Dilfusion and Green s function QMC calculations are often done using a fixed-node approximation. Within this scheme, the nodal surfaces used define the state that is obtained as well as ensuring an antisymmetric wave function. 

For both bosonic systems and fermionic systems in the fixed-node approximation, G has only nonnegative elements. This is essential for the Monte Carlo methods discussed here. A problem specific to quantum mechanical systems is that G is known only asymptotically for short times, so that the finite-time Green function has to be constructed by the application of the generalized Trotter formula [6,7], G(r) = limm 00 G(z/m)m, where the position variables of G have been suppressed. 

As was shown above, the true fixed-node Green function vanishes outside the nodal pocket of the trial wavefunction. However, since we are using an approximate Green function, moves across the nodes will be proposed for any finite x. To satisfy the boundary conditions of the fixed-node approximation, these proposed moves are always rejected. 

The first description of QMC is attributed to E. Fermi in a classic paper by Metropolis and Ulam [2]. Some years later, Kalos [3] proposed Green s Function QMC which was applied to the calculation of the ground state of three- and four-body nuclei. In retrospect this paper was indicative of the versatility of QMC for studies of nuclear, condensed matter as well as atomic and molecular systems. It remained for Anderson [4] to make the initial significant foray into atomic and molecular systems in the mid-seventies. These classic papers introduced the fixed-node approximation and served as a bellwether for the DMC described below. 

---