How do I simulate an atom? Let us assume I wish to simulate a Helium atom, since there does not exist a closed-form solution.
However, I presume I would need to simulate the time-dependent Schrodinger wave equation. I would like to know what the time-dependent hamiltonian looks like for this.
I think I would need the terms for electromagnetic, strong, and weak nuclear forces --- I do not think I need relativistic corrections, since I'm only interested in local information.
I wish to simulate an equation on simulation gives me the evolving probability distributions of the electrons. 
I understand that this might be computationally infeasible, but I still wish to know what the exact PDE is that I need to solve --- I'm not looking for approximations!
 A: The Hamiltonian for the He atom is:
$$H = -\frac{\hbar^2}{2m_e}(\nabla_1^2 + \nabla_2^2) - \frac{2e^2}{4\pi\epsilon_0 r_1}
 - \frac{2e^2}{4\pi\epsilon_0 r_2}
 + \frac{e^2}{4\pi\epsilon_0 r_{12}}$$
where the electrons are denoted 1 and 2, and $r_i$ is the distance to the nucleus at the origin and $r_{12}$ the distance between the electrons. Since the electrons have 3D position vectors $\mathbf{r_i}$ this corresponds to 6 degrees of freedom. 
One can reformulate the equation to only act in $r_1,r_2,r_{12}$ coordinates for the spherically symmetric $^1S$ state. With the right approximation methods one can get energy eigenvalues with absurd precision. But I suspect the aim of the question is rather to let loose an electrong wave packet and see how it sloshes around.
Simulating the nucleus requires quite a lot more elaboration. Now you have 4 particles, making a 12D state space (18 in total). The hamiltonian is about the same but with more interaction terms (see section 3.1), corresponding to nucleon-nucleon potentials that generally look somewhat messy.
