How can I tell how many solutions I will have for an electronic Schrödinger equation ? For example, solving it for the hydrogen atom we get infinitely many solutions \begin{equation} H_e(\mathbf{R})\Psi_i(\mathbf{R},\mathbf{r}) = E_i(\mathbf{R}) \Psi_i(\mathbf{R},\mathbf{r}), \qquad i = 1, 2, ..., \infty \end{equation} They all are bound in the potential.

But for a different potential, e.g. Morse potential it gives a finite number.Wikipedia claims that "This failure [to match the real anharmonicity] is due to the finite number of bound levels in the Morse potential".

I am looking at molecules and wondered if there would be an infinite number of molecular orbitals in general.

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    $\begingroup$ In general, the best way to find out the number of solutions to an equation is to solve it. Do you have any reason to believe there's another way? $\endgroup$ – ACuriousMind May 13 at 16:22
  • $\begingroup$ Martin, I think your question is how many bound solutions there are, correct? $\endgroup$ – zonksoft May 13 at 17:50
  • $\begingroup$ @ ACuriousMind and zonksoft : Well, that of course. But I have molecules in my mind, so solutions will only be found computationally and/or with approximations. And yes, I guess bound solutions, since if the potential does not go to infinity, there will be a continuous space of solutions (thinking of free electrons). $\endgroup$ – Martin May 14 at 9:51
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    $\begingroup$ This is generally a hard problem. Atomic anions, for example, are known to have only a finite number of bound states (see ref. 7 here and other references here and here), but this is not territory for the faint of heart. For neutral molecules, one would expect a Rydberg series that's increasingly hydrogenic as it approaches the ionization threshold -- but I'm unsure how much of this has been rigorously proved. $\endgroup$ – Emilio Pisanty May 20 at 16:40

Sticking my head out with guesswork, if the potential has infinite range, or is infinitely deep but with finite range, the number of bound states is infinite. For an atom or a molecule the number of bound states is definitely infinite. The energy separation of the bound states goes to zero when the onset of unbound states is approached.

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