Combining two overlapping wavefunctions

Let's assume that we solved the many-body Schrödinger equation for two molecules and we can now write their wave functions as
$$
|\Psi\_1\rangle = O\_1|\operatorname{vac}\rangle\,,\\
|\Psi\_2\rangle = O\_2|\operatorname{vac}\rangle\,.
$$
where $O\_{1,2}$, according to the definition above, are the operators creating states $|\Psi\_{1,2}\rangle$ from vacuum.


Moreover, let's assume that both equations are written in a basis wherein the two molecules are spatially separated yet their wave functions do have some noticeable overlap. Now we would like to construct a "composite state" of two molecules.


It seems natural to define such a composite state as
$$
|\Psi\_1,\Psi\_2\rangle = O\_1 O\_2|\operatorname{vac}\rangle\,.\\
$$


However, I think that this definition would only make sense if the two operators commuted or anti-commuted. (Otherwise, should we use $O\_1 O\_2|\operatorname{vac}\rangle$ or $O\_2 O\_1|\operatorname{vac}\rangle$?..) Note that for non-overlapping sets of orbitals the question does not arise since the two operators clearly commute with each other.


Now, consider two ways of defining such operators.


1. In the coupled cluster ($U=e^{T}$) or unitary coupled cluster ($U=e^{T-T^\dagger}$) approaches, operator $O$ can be expressed in terms of an exponential of a polynomial in creation and annihilation operators applied to an operator creating some reference state from vacuum:
$$
|\Psi\_{1,2}^{\vphantom{\dagger}}\rangle = U\_{1,2}^{\vphantom{\dagger}} |\rm ref\rangle = U\_{1,2}^{\vphantom{\dagger}} a^\dagger\_{\rm ref} |vac\rangle\,,
$$
Therefore one option is to postulate $O\_{1,2}^{\vphantom{\dagger}}=U\_{1,2}^{\vphantom{\dagger}} a^\dagger\_{\rm ref}$ for CC or $O\_{1,2}^{\vphantom{\dagger}}=U\_{1,2}^{\vphantom{\dagger}} a^\dagger\_{\rm ref} U\_{1,2}^\dagger$ for UCC. However, it is not clear to me how two such operators $O\_1$ and $O\_2$ commute with each other. Moreover, operators $U\_{1,2}$ are typically constructed in a way that their action is only sensibly defined when they are applied to the vacuum state. The latter, however, could be fixed if $U$ was defined adiabatically or through similarity transformation (basically, $U$ should be the *wave operator*[1]), in which case its action would be sensible on all possible states.
2. Once the states $|\Psi\_{1,2}\rangle$ are found, one can write them down as polynomials $P\_{1,2}$ of creation operators only acting on the vacuum state, and postulate $O\_{1,2}=P\_{1,2}$. In this case, it is clear that two such operators commute.


It is tempting to choose the second approach to defining $O\_{1,2}$, but I still don't understand the physical motivation besides using this particular definition and not something else.


The motivation for my question stems from the fact that I would like to be able to construct $O\_{1,2}$ in some way resembling approach 1. (even in an approximate way)


**UPDATE**


I've just noticed that on Physics SE there [already exists](https://physics.stackexchange.com/questions/54412/combining-two-finite-number-fock-spaces-into-one) precisely the question I ask here, and nobody has got close to answering it in 10 years.


### References:


1. Kato, T. Wave operators and similarity for some non-selfadjoint operators. *Math. Ann.* **1966,** *162* (2), 258–279. [DOI: 10.1007/BF01360915](https://doi.org/10.1007/BF01360915).


