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Often, when solutions to the Hartree-Fock equations are sought, a self-consistent (SCF) method is employed, such as that outlined in the answer to this question.

My question is not about the convergence of the self consistent method for these equations (asked in this question). Instead, my question is about the solutions themselves (fixed points) in the dynamics of the SCF approach. Is there a general proof that all of the fixed points are attractors? What about periodic orbits: are they also attractors?

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  • $\begingroup$ Are you asking Why does iteratively solving the Hartree-Fock equations result in convergence? $\endgroup$
    – John Rennie
    Commented Jan 23, 2020 at 16:29
  • $\begingroup$ No. I'm asking if there is a proof that solutions to the HF equations are stable fixed points in the SCF dynamics.That does not necessarily mean that the SCF iterations converge, but only that if you are "close" to a solution, they will converge. $\endgroup$
    – Jyules
    Commented Jan 23, 2020 at 21:37

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I have observed limit cycle behavior in my own calculations with the relativistic HF equations. I have also heard that similar limit cycle behavior also occurs with Schrodinger equation based HF, so I infer from this that the answer to your question is no, not all HF solutions are stable fixed points.

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  • $\begingroup$ Doesn't the fact that you've observed limit cycle behavior suggest that the limit cycles you've seen are attractors? If they weren't stable orbits, wouldn't you be unlikely to "stumble" upon them in your calculations? $\endgroup$
    – Jyules
    Commented Jan 23, 2020 at 15:57
  • $\begingroup$ If the period doubling exhibits Feigenbaum unuversitality, then no, not all solutions are fixed points. $\endgroup$
    – Lewis Miller
    Commented Jan 23, 2020 at 16:01
  • $\begingroup$ All solutions to the HF equations are definitely fixed points, by definition. My question is about their stability. $\endgroup$
    – Jyules
    Commented Jan 23, 2020 at 21:40

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