    Next: Determinant derivatives Up: thesis Previous: Derivatives of the local   Contents

# Cusp conditions

When two Coulomb particles get close, the potential has singularity. We want modify the wave function in such a way to cancel this singularity. Let us consider the case of an electron close to a nucleus, the Schrödinger equation reduces to: (C.1)

where is the nuclear charge, notice that we used rescaled distances (see Eq. 4.23). Writing the first term in spherical coordinates, we get (C.2)

To cancel the singularity at small the term multiplying by must vanish. So we have (C.3)

If we must have . For the case of two electrons, when they are close each other the Schrödinger equation, using relative coordinates , reduces to (C.4)

Electrons with unlike spins have an extra factor of in the cusp condition compared with the electron-nucleus case. So we have . In the antisymmetric case, the electrons will be in a relative state, reducing the cusp condition by , so . Since the antisymmetry requirement keeps them apart anyway having the correct cusp for like spin electrons leads to a very little in the energy or the variance(see Ref. (39).    Next: Determinant derivatives Up: thesis Previous: Derivatives of the local   Contents
Claudio Attaccalite 2005-11-07