# Number of bound solutions of electronic Schrödinger equation

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.

• 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? – ACuriousMind May 13 at 16:22
• Martin, I think your question is how many bound solutions there are, correct? – zonksoft May 13 at 17:50
• @ 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). – Martin May 14 at 9:51
• 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. – Emilio Pisanty May 20 at 16:40