Quantum Model of an Atom with more than one electron

[da_steve]

Hi,

I just completed an assignment modelling an atom with one electron.
The model was very simple, assuming that only coulombs law applied, then solving for standing wave solutions to the Schrödinger equation to find ionization energies.

I was astounded how accurate it was. i thought it would be maybe +/- 5% but instead it was correct within 0.5% for the first 10 atoms. It gave me a bit of an ego boost which is probably why I am asking this.

I noticed that for more than one electron the model failed spectacularly. Which got me wondering what's the next step?

My guess is its the Pauli Exclusion Principle but how is it applied?

 

[jtbell]

For multi-electron atoms you need something like the Hartree-Fock method.

The basic problem is that you have to take into account not only the Coulumb potential energy between each electron and the nucleus, but also between each pair of electrons.

 

[da_steve]

Thanks :)
That was what i was looking for

 

[JeffKoch]

The next step is messy, and I'm not sure there's an end to the number of steps as you add more and more reality. In school one of my class assignments was to calculate the ground state energy of helium as accurately as possible, and most people including me approached this by treating the (unknown) eigenfunctions as sums of hydrogenic eigenfunctions while including the potential between the electrons in schroedinger's equation. As I recall this gets you within a few percent, and if you adjust the effective Z to minimize the energy you can get an upper bound that is within a fraction of a percent.

People spend careers and write thick books on calculating energy levels in complex ions, so it's as complicated as you want it to be.