Analytical solution of the Schrödinger Equation for AB^+ systems

According to Wikipedia, there are not many problems for which the Schrödinger equation can be solved analytically. [[1](https://en.wikipedia.org/wiki/List_of_quantum-mechanical_systems_with_analytical_solutions)] 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 $\ce{AB+}$?




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](https://en.wikipedia.org/wiki/Closed-form_expression) 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".




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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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