According to Wikipedia, there are not many problems for which the Schrödinger equation can be solved analytically. [1] The $\ce{H2+}$ ion is probably the most complex molecule that can be treated this way.
As far as I know, every diatomic homonuclear system like this can be solved, e.g., $\ce{U2+}$.
Up to this point the molecules always consisted of two equal atoms which makes me curious about the following:
Is the Schrödinger equation not only solvable analytically for diatomic homonuclear systems $\ce{A2+}$ but also for diatomic heteronuclear systems like $\ce{AB+}$?
Answer
First, one has to be really careful with the words: when it is said that the Schrödinger equation can be solved the dihydrogen cation completely analytically, several remarks has to be made.
There are few different Schrödinger equations out there, so it is better to explicitly mention that one which can be solved analytically for $\ce{H2+}$ is the electronic Schrödinger equation. So, first of all, we are talking about the time-independent Schrödinger equation here, and secondly, we employ the Born-Oppenheimer approximation and solve only the electronic problem. No claim is maid with respect to analytical solutions of the whole time-independent Schrödinger equation without any approximation introduced, as well as regarding analytical solutions of its time-dependent counterpart.
The second remark is about the meaning of the word "analytical", or "analytic". The definitions taken from the Wikipedia are as follows:
An equation is said to have an analytic solution if and only if at least one solution can be expressed as an analytic expression.
An analytic expression is a mathematical expression constructed using well-known operations that lend themselves readily to calculation.
These definition seems to be rather vague for me, especially taking into account that,
[..] the set of well-known functions allowed can vary according to context [...]
so we have to be careful relying on some intuitive notions of "analytic expression" and "analytic solution".
Having said so, and in hope that my intuitive notions of "analytic expression" and "analytic solution" are not miles away from the truth, it is important to note that $\ce{H2+}$, for which analytic solutions are known, is (a special case of) a three-body system. And since there is no claim about analytic solutions for a four-body system, a five-body one, etc., from the beginning we have to restrict ourselves for three-body systems.
From that viewpoint the notation $\ce{AB+}$ used by OP is misleading, because the only three-body chemical system of $\ce{AB+}$ kind is $\ce{H2+}$. Already for the system composed of $\ce{H}$ and $\ce{He}$ the charge has to be +2 since we have to remove 2 electrons so that we end up with just one of them which together with 2 nuclei constitutes a three-body system, and not a four-body one.
Now, what about analytical solutions for $\ce{AB^{n+}}$ systems where $n$ is chosen appropriately so that the system is a three-body one? Well, I think, as for $\ce{H2+}$, there are analytic solutions for electronic Schrödinger equation in the Born-Oppenheimer approximation. The reason is pretty simple in fact: the clamped nuclei model reduces the three-body problem to a single-body problem of the motion of a single electron within the fixed nuclear potential.
Why there are no related references (at least I could not find one)? It is very likely because chemist has little or no interest in three-body chemical systems of $\ce{AB^{n+}}$ type in general, and specifically, little or no interest in analytic solutions for these systems. Except for the simplest $\ce{H2+}$ system which serves as a primary model for the concept of the one-electron bond.
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