and the new distance is defined as

In doing so, we have only to compute gradients and Laplacian with the chain rule:

where

This transformation has been applied to all orbitals appearing in the wave-function and also to the one-body term and the two-body Jastrow.

We remark here that also the normalization constant of a given orbital has to be changed in a periodic system. Namely its integral over the simulation cell has to be equal to one.

(4.5) |

For instance a normal Gaussian in three-dimension:

becomes after the substitution 4.3:

(4.7) |

where is the modified Bessel function of the first kind and is the size of the simulation box and is the periodic distance 4.4.