# Natural Orbitals for Particles in a Box

1. Mar 14, 2012

### Morberticus

Natural Orbitals for "Particles in a Box"

Hi,

Are Sine waves the natural orbitals for particles in a box when electron-electron interactions are considered? Or is it only true for non-interacting electrons?

2. Mar 15, 2012

### Steger

Re: Natural Orbitals for "Particles in a Box"

I believe this is a rather complicated problem but I'm not sure if sine functions would be a good basis to use, the electrons would want to stay on opposite sides of the box but then you would have to also worry about their wavefunctions being antisymmetric. It's not a trivial problem.

3. Mar 17, 2012

### sam_bell

Re: Natural Orbitals for "Particles in a Box"

If by "natural" you mean the *eigenstates* of the interacting electron system, then no. But you could take Slater determinants of the sine wave solutions as a useful *basis* to work in for the interacting electron problem. If you really want to find a good ground state though, probably the best approach would be to use a test wave-function with a few free parameters and apply the variational principle.

4. Mar 18, 2012

### DrDu

Re: Natural Orbitals for "Particles in a Box"

Natural orbitals are defined technically as the orbitals which diagonalize the 1-density operator.

5. Mar 22, 2012

### Einstein Mcfly

Re: Natural Orbitals for "Particles in a Box"

If you're looking for "natural orbitals" in the Frank Weinhold sense, then do what sam bell suggested to get the eigenstates in that particular basis and then construct the density matrix as DrDu suggested and diagonalize it to get the linear combination of those eigenstates that gives you the natural orbitals.

6. Mar 22, 2012

### DrDu

Re: Natural Orbitals for "Particles in a Box"

I suppose the answer is no. To prove this it would be sufficient to show that inclusion of interaction in lowest order of perturbation theory destroys diagonality of the density matrix. Given that the energy levels aren't degenerate, I suppose this reduces to showing that the Hartree Fock orbtials cannot be chosen as sine waves.