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!

• How much QM background have you got and why do you think you need to include any forces other than electromagnetic? That said, best of luck with it.
– user226006
Jan 23, 2020 at 23:03
• Are you trying to simulate the nucleus and the electrons or just find the energy levels of the atom? Jan 23, 2020 at 23:12
• ah that'll be a complicated hamiltonian then Jan 23, 2020 at 23:18
• Nobody does it like this, because the tiny nuclear effects are probably smaller than the numerical errors in solving for the electrons. Jan 23, 2020 at 23:44
• Brute force simulation of processes at widely different scales is usually not a productive approach to understanding physical systems. Jan 23, 2020 at 23:47

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.
• @G.Smith the OP wanted to avoid relativistic corrections, and magnetic term definitely has a $c^{-1}$ factor. Otherwise you'd need to also include all the other relativistic corrections. Jan 24, 2020 at 22:29