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The Fixed-Node Approximation


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, tex2html_wrap_inline5885 , given by


where tex2html_wrap_inline6527 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.

next up previous contents
Next: DMC With Non-Local Pseudopotentials Up: Diffusion Quantum Monte Carlo Previous: A DMC Algorithm

Andrew Williamson
Tue Nov 19 17:11:34 GMT 1996