Coulomb potential (-Z/r)

 

Our approach to funding the potential Vtilde from the nuclei for our periodic calculations is based on our standard approach to solving Poisson's equation,

phinuc=I(Linv((-4*pi)*O(J(rhonuc))));

and our the definition of Vtilde from HW#10,

Vtidle=Jdag(O(J(-phinuc)));

where the extra `-' in front of phinuc comes from the fact that the charge on the electron is (-e): the potential energy of an electron at is . Combining the above two results gives

Vtilde=Jdag(O(Linv((4*pi)*O(J(rhonuc)))));

where we have used the fact that J(I(x))=x.

Only difficulty with this approach is that rhonuc represents a point charge, whose density is zero everywhere but at a single point where the density is infinite. We can, however, find the expansion coefficients [J(rhonuc) mathematically. To do this, we note that

so that

where V is the volume of the cell. Therefore,

In our case, is a point charge centered at the location of the nucleus whose total integral is Z, the charge on the nucleus. If the nucleus is centered at point , then we have simply

• evalG()
• Nuc1(), Nuc2()
• Debugging