Node Methods

Negative psips arise due to the required fermion antisymmetry of the true colutions. Any pair of electrons with the same spin are indistinguishable, and may be exchanged without affecting the square of the wavefunction. However, because electrons are fermions, the wavefunction itself changes sign (bosons eg. photons and some nuclei do not change sign). There are three ways of dealing with these negative reflections.

The figure above shows two possible nodes (horizontal and verticle) separating a positive region from a negative region. Both node lines are consistent with the inversion symmetry, but a calculation of the ground state will indicate the the lowest energy solution with inversion symmetry has the verticle node.

Since one half of the graph is merely an inversion of the other half (with sign change), only half of the wavefunction need be calculated, as shown in the sample calculations below.

Fixed node

Destroy all negative particles. Only works if you know exactly which regions should be positive and which regions should be negative. You also have to take very short steps to avoid errors.

Released node

Don't do anything to the negative psips. This is correct, but difficult to implement. As the ground state is approached, the population of negative psips approaches the population of positive psips, causing any measured quantities to become uncertain. It becomes a race between accuracy and precision. The error bars diverge as the wavefunction reaches the ground state.

Cancellation

Have nearby positive and negative psips cancel each other. An exact method based on overlapping Green's functions is used to reduce the weights. This results in the destruction of negative psips in predominantly positive regions, and a distribution that resembles the fixed-node distribution above, without the errors.

Negative psips appear briefly along the node, and disappear in a frame or two.