The hydrogen molecular ion (a.k.a. dihydrogen cation) $\mathrm H_2^+$ is the simplest possible molecular system, and as such you'd hope to be able to make some leeway in solving it, but it turns out that it's much harder than you'd hope. As it turns out, if you phrase it in spheroidal coordinates then the stationary Schrödinger equation for the electron (with stationary nuclei),
$$
\left[-\frac12\nabla^2-\frac{1}{\|\mathbf r-\mathbf R_1\|}-\frac{1}{\|\mathbf r-\mathbf R_2\|}\right]\psi(\mathbf r)=E\psi(\mathbf r)
\tag 1
$$
becomes separable, but – last I heard – the resulting equations do not admit exact analytical equations in anything you'd call either closed form or special-function-like.
(More specifically, the separation is not as clean as in the hydrogen atom, where you get an angular and then a radial eigenvalue problems, but instead you get a coupled 'bi-eigenvalue problem' that's harder to solve.)
On the other hand, Wikipedia lists the system in its List of quantum-mechanical systems with analytical solutions with a note that there are "Solutions in terms of generalized Lambert W function", so maybe I'm missing something.
Tracing the Wikipedia references leads to arXiv:physics/0607081, which seems to me to only (i) only work for the eigenvalue, not the eigenfunctions, (ii) work with generalizations of the Lambert $W$ function, and (iii) not be particularly closed-form either. However, I may be missing the end of some reference trail here.
So: are there known eigenfunctions of $(1)$ in exact analytical form, or even in terms of special functions (whose definition goes beyond "the solution of this given equation")?
If the answer to this is negative, then that's probably a very tall order to prove, since statements of the kind "there is no result of that type in the literature" are inherently hard to tie down. In that case, though, I will settle for a thorough exploration of the literature pointed at by the Wikipedia claim, and an explanation of what it does and does not provide.
Edit, given the large number (currently 8) of non-answers that this thread has received. Apparently some clarifications are in order.
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The question of whether a given solution does or does not qualify for the description of 'analytic', 'closed-form' or 'exact' is obviously a subjective call to a nontrivial extent. However, there are a lot of interesting shades of gray between 'the solution is an elementary function' and 'if you define the special function $f$ as the solution of the equation, then the equation is solvable in terms of special functions', and I want to know where this problem sits between those two extremes.
As such, I would like to set the bar at functions that include at least one nontrivial connection. Thus, I would argue that a direct Frobenius-method series solution is not really sufficient if it has no further analysis and no additional connections to other properties of the resulting functions. (In particular, if one wants to allow series solutions with no further connections, then it is worth considering carefully what other systems then become 'solvable' to the same degree.)
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It is well known that there are perfectly good approximate and numerical solutions to this problem, including several that are systematically convergent; moreover, even if an analytical solution exists, those numerical and approximate solutions are probably more useful and quite possibly more accurate than the 'exact' solution. That is irrelevant to the question at hand, which is simply about how far (or lack thereof) one can take 'exact' analytical methods in quantum mechanics.
- Obviously the Schrödinger equation at stake here is an approximation (as it ignores e.g. nuclear motion and relativistic effects such as spin-orbit coupling and other fine-structure effects), but that is irrelevant to the question of whether this specific problem has exact solutions or not.
Best Answer
To avoid re-treading old ground, this answer contains some previous literature that has been mentioned on this thread, as well as the surface layer obtainable via naive google searches:
From this list, the papers by Wilson, Teller, Jaffé, Hylleraas, Burrau and Bates contain derivations of the separation of variables as well a series solution for the resulting coupled equations, in which the quantization condition usually appears, if I understand correctly, as the requirement that the separation constant $\mu$ be a zero of a function defined by a continued fraction, as $$ f(\mu) = \mu+ \frac{ \frac{1\cdot2\lambda^2}{2\cdot3}}{1-\frac{\mu}{2\cdot 3} - \frac{ \frac{3\cdot4\lambda^2}{2\cdot3\cdot4\cdot5}}{1-\frac{\mu}{4\cdot 5} - \frac{ \frac{5\cdot6\lambda^2}{2\cdot3\cdot4\cdot5\cdot6\cdot7}}{1-\frac{\mu}{6\cdot 7} - \cdots } } } =0, $$ where $\lambda$ is essentially the energy eigenvalue.
I am extremely reluctant to call these series solutions as 'exact' or 'analytical', though of course this involves a personal judgement call. (As a contrast, I'm not that reluctant to call the Braak solution of the Rabi model an analytical solution, even though it shares many features with the ones in this reference list. To some extent, that's because it's more recent, so there hasn't been enough time to tell whether there's more connections to be made with those solutions, but intuitively they feel like they have more 'structure' around them.) However, maybe someone can come along with a review and simplified exposition of the series solutions, and make the case that the functions they define are as 'closed-form' as, say, the Bessel eigenfunctions of a cylindrical well?