It is well known that the Schrodinger equation rapidly becomes intractable as the complexity of the situation increases.

Solutions for a single quantum particle can often be obtained (depending on which potential function it is subject to). Famously, exact solutions can be found for the time-independent equation for the electron in a Hydrogen atom using separation of variables, provided that the nucleus is treated as a classical point particle.

I was wondering what the result would be if we tried to do the anslysis for a single elctron and multiple classical nuclei.

In the simplest case, we shall be concerned with of the time-indepedent Schrodinger Equation with 3 nuclei, and the system being in the ground case. Then, we are only interested in finding the minimum energy configurations of the nuclei, and the corresponding energy levels of the electronic wave functions, so the intractability of the many-body problem should not neecssarily affect this.

I found this post which appears to claim that an analytical solution is not possible in the case of two nuclei ($\text{H}_{2}^{+}$), but this (admittedly old) preprint claims to find one.

Regardless of which claim is correct, the case of 3 or more nuclei (which changes the problem from a simple question of internuclear distance and the corresponding energy levels to a multi-dimensional conumdrum) should be much more complicated. For one, there is no longer any obvious symmetry which could be used to obtain seperability, since the presence or absence of any symmetry is now dependent on the appropriate nuclear configuration, which shall be influenced by the electronic condiguration of the corresponding solution.

In general, what I am interested in knowing is whether there are any known theoretical arguments or results which prevent or ensure the existence of an analytical solution (and in the latter case, whether any solutions are known). Or whether the issue is currently an open question and cannot be answered one way or the other.

Please note that this question is not a duplicate since I am not asking about the quantum solutions for the nuclei themselves.

  • $\begingroup$ As far as I can tell, the preprint you linked to was never published in a peer-reviewed journal. (I'd be happy to be proven wrong if you have a link). That may be valuable context to have in evaluating the claims of that article. $\endgroup$
    – Andrew
    Commented Oct 16, 2022 at 18:19
  • $\begingroup$ This is weird. How you would treat a "non-classical" nuclei? What does "quantum solutions to the nuclei themselves" mean? It is possible to treat analytically one electron with a proton that has finite size and a spherical charge distribution but it's a mess. In the case of $H_2^+$, the potential is not central so it would be rare to find analytical solutions in such as case, although there might be exceptions. $\endgroup$ Commented Oct 16, 2022 at 18:27
  • $\begingroup$ I'm dubious that a stable bound solution exists for an electron and three nuclei as the "natural" stable state would be either all the particles free or the electron bound to one or at most two nuclei and the other nuclei free. I'm not personally aware of a formal analysis for this. $\endgroup$ Commented Oct 16, 2022 at 21:03
  • $\begingroup$ Muffin-tin approximation $\endgroup$
    – mmesser314
    Commented Oct 17, 2022 at 4:06
  • $\begingroup$ Molecular orbital theory $\endgroup$
    – mmesser314
    Commented Oct 17, 2022 at 4:07


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