Natural Orbitals for Particles in a Box

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Discussion Overview

The discussion revolves around the concept of natural orbitals for particles in a box, particularly in the context of electron-electron interactions. Participants explore whether sine waves serve as natural orbitals in both interacting and non-interacting scenarios, delving into the complexities of the problem.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants question whether sine waves can be considered natural orbitals when electron-electron interactions are included, suggesting that this may only hold true for non-interacting electrons.
  • One participant notes the complexity of the problem, mentioning that electrons would prefer to occupy opposite sides of the box and that their wavefunctions must be antisymmetric.
  • Another participant clarifies that if "natural" refers to the eigenstates of the interacting electron system, then sine waves would not qualify, but Slater determinants of sine wave solutions could serve as a useful basis for the interacting problem.
  • A similar point is reiterated regarding the use of Slater determinants and the variational principle to find a good ground state, emphasizing the need for a test wave-function with free parameters.
  • One participant suggests that to find natural orbitals in the Frank Weinhold sense, one should construct the density matrix from the eigenstates and diagonalize it.
  • Another participant posits that including interactions in perturbation theory likely disrupts the diagonal nature of the density matrix, implying that Hartree-Fock orbitals cannot be represented as sine waves.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of sine waves as natural orbitals, with no consensus reached on the matter. The discussion remains unresolved regarding the implications of electron interactions on the nature of these orbitals.

Contextual Notes

Limitations include the dependence on definitions of natural orbitals and the complexities introduced by electron interactions, which are not fully resolved in the discussion.

Morberticus
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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?
 
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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.
 


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.
 


sam_bell said:
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.

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


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.
 


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.
 

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