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.