# Hamiltonian for a two electron system

What is the hamiltonian for an atom with two electrons and a nuclear charge $Z$?

The Hamiltonian always takes the general form: $\hat H=\hat T + \hat V$

The kinetic energy of each of the electrons needs to be taken into account so: $\hat T=-\frac{\hbar^2}{2\mu}(\nabla_1^2+\nabla_2^2)$

The potential operator is where I am finding trouble. There are two terms for the attraction towards the nucleus - one for each electron: $-\frac{Ze^2}{4\pi\epsilon_0r_1}-\frac{Ze^2}{4\pi\epsilon_0r_2}$. However, the potential energy involved in the electron electron repulsion needs to be taken into account. My question is: are there two terms for this? To clarify do I need to calculate the increase in potential of electron 1 caused by electron 2 AND the increase in potential energy of electron 2 caused by the presence of electron 1? Or would adding: $\frac{e^2}{4\pi\epsilon_0r_{12}}$ take into account all of the potential energy involved in the electron electron interaction?

• Potential is a characteristic of the system as a whole, and not of individual electrons. There are three charges, hence three pairwise interactions. Two of them, namely nucleus-electron1 and nucleus-electron2, you took into account once. Why in the world would you want to add third such value twice? Oct 8 '15 at 15:16

In general, an interaction potential that only depends on the distance of two particles can be written as $$U = \sum_{i<j} u(|\vec{r}_i - \vec{r}_j|),$$ where $u$ represents the interaction between only two particles. You see that for two electrons you will have only one term, since $i<j$. For three electrons you will have three terms (interaction between 1 and 2, between 1 and 3 and between 2 and 3) and so on.
Maybe your confusion comes from the fact that, sometimes, this interaction potential is written as $$U = \frac{1}{2}\sum_{i\neq j} u(|\vec{r}_i - \vec{r}_j|)$$ In this case, for two electrons, you will have two terms. However there is the factor $1/2$ which cancels out the double counting, since $$u(|\vec{r}_i - \vec{r}_j|)=u(|\vec{r}_j - \vec{r}_i|).$$