## 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 constant1.

The total potential energy arising from the interactions among all the nuclei is just the sum of all pair-wise 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 double-counting correction to ensure that we count each pair-wise 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 double-counting 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 double-counting correction of (1) because we are dealing with the total interaction of a group of particles with itself,
 (5)

Tomas Arias 2004-01-26