Electrostatics
There are three groups of electrostatic interactions which
we must consider: interactions of nuclei with nuclei, of
electrons with nuclei, and of electrons with electrons.
Coulomb's law states that the potential energy between two charges
and at separation is
where is Coulomb's constant^{1}.
The total potential energy arising from the interactions among all the
nuclei is just the sum of all pairwise interactions,

(1) 
where is the charge of the electron,
and index different nuclei, is the separation between
nuclei and , is the atomic number of nucleus , and
the factor of is the famous doublecounting correction to
ensure that we count each pairwise
interaction only once.
Similarly, the potential energy of a single electron at position
due to the nuclei is

(2) 
where is the distance from point to nucleus .
If
the volume density (number per unit volume) of electrons is , then the number of electrons in the volume element near
point is and the total potential energy of
the electrons interacting with the nuclei is

(3) 
where the integral is over all of space. (There is no
doublecounting correction here because the interaction between
electron #1 and nucleus #2 is not counted again when we do the
interaction between electron #2 and nucleus #1!)
Finally, the electrons interact not only with the nuclei, but also
with themselves. From Coulomb's law, the potential energy for a
single electron at point coming from the electrons at point
is
where
is the distance between points and . The total potential for a single electron at point is
then
Standard electrostatics tells us that doing this integral is
equivalent to solving Poisson's equation,

(4) 
Finally, once we have , the potential energy for the
electrons interacting with themselves follows the same logic as
(3) but with the doublecounting correction of
(1) because we are dealing with the total interaction
of a group of particles with itself,

(5) 
Tomas Arias
20040126