The attraction of the DMC method is tempered by the nodal problem in many-fermion systems. The interpretation of the imaginary-time Schödinger equation as a diffusion equation rests upon the interpretation of the wavefunction as a probability density. The problem is that a probability density is positive by definition and corresponds to the concentration of the diffusing walkers.

The fact that the ground-state of the many-fermion system has nodes implies that some kind of external constraint must be assumed. We will now discuss such methods.

The first solution to the nodal problem was proposed by Anderson [35]. He took a wavefunction as a reference function then made the nodes of the reference function act as sinks for the walkers. This method is known as the fixed node approximation. The problem with the method is that if the nodal surfaces of the reference function do not coincide with the exact ones, a small bias is introduced in the diffusion process.

When we introduced the fixed-node approximation before we defined it slightly differently to above. This is because it has been shown [30] that the procedure of deleting walkers that cross a nodal surface introduces a bias proportional to the time step. Instead the procedure that we use, and indeed the one that is now most generally used, is to reject moves that cross a node; that is a node acts as an infinite potential barrier. This method still introduces a bias but it is of the same order, second order in the time step, as the bias due to the finite time step Green function.

One method of refining this technique, proposed by Ceperley *et
al*. [31] is to ``release'' the nodes. The
antisymmetric wavefunction is then obtained via the difference of two
populations of signed walkers, generated within the fixed node
approximation, in the different regions corresponding to the positive
and negative ones of a reference function. The problem with this is
the population of the two sets of the walkers grows exponentially.
This leads to large statistical noise and in general makes the
technique very difficult to apply.

An alternative method for improving upon the fixed-node approximation
was introduced for calculations on the two-dimensional electron gas by
Kwon et al. [36]. They replace the positions of the orbitals
in the Slater determinant by their *quasiparticle* coordinates,
, given by

where is a *backflow* correlation function
parametrised as

The idea of *backflow* was originally suggested by Feynman and
Cohen[37]. By changing the coordinates of the
particles for which the Slater determinant is being evaluated, one is
effectively changing the nodal structure of the determinant. Kwon et
al. found that the introduction of the backflow correction produced a
significant reduction in the total energy of the two-dimensional
electron gas at high densities. It is expected that the improvement
will not be so large in three-dimensional systems, although this is
yet to be fully tested.

Tue Nov 19 17:11:34 GMT 1996