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 $q_1$ and $q_2$ at separation $r_{12}$ is

$$U=[k_c] \frac{q_1 q_2}{r_{12}},$$

where $[k_c]$ 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,

$$U_{nuc-nuc} = \frac{1}{2} [k_c] e^2 \sum_{I \ne J} \frac{Z_I Z_J}{R_{IJ}},$$ (1)

where $e$ is the charge of the electron, $I$ and $J$ index different nuclei, $R_{IJ}$ is the separation between nuclei $I$ and $J$, $Z_I$ is the atomic number of nucleus $I$, and the factor of $1/2$ 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 $\vec x$ due to the nuclei is

$$V_{nuc}(\vec x) = - [k_c] e^2 \sum_i \frac{Z_I}{R_I},$$ (2)

where $R_I$ is the distance from point $\vec x$ to nucleus $I$. If the volume density (number per unit volume) of electrons is $n(\vec x)$, then the number of electrons in the volume element $dV$ near point $\vec x$ is $n(\vec x)\,dV$ and the total potential energy of the electrons interacting with the nuclei is

$$U_{el-nuc} = \int V_{nuc}(\vec x) n(\vec x)\,dV,$$ (3)

where the integral is over all of space. (There is no $1/2$ 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 $\vec x$ coming from the electrons at point $\vec x'$ is $[k_c] e^2 \, n(\vec x') \, dV'/\vert\vec x-\vec x'\vert$ where $\vert\vec x-\vec x'\vert$ is the distance between points $\vec x$ and $\vec x'$. The total potential for a single electron at point $\vec x$ is then

$$\phi(\vec x) = [k_c] e^2 \int \frac{n(\vec x')\,dV'}{\vert\vec x - \vec x'\vert}.$$

Standard electrostatics tells us that doing this integral is equivalent to solving Poisson's equation,

$$\nabla^2 \phi(\vec x) = - 4 \pi [k_c] e^2 n(\vec x).$$ (4)

Finally, once we have $\phi(\vec x)$, 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,

$$U_{el-el} = \frac{1}{2} \int \phi(\vec x) n(\vec x)\,dV.$$ (5)