Cusp conditions

(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).