Nugatory said:
No, but they're also not bosons. They are quantum systems consisting of two interacting fermions and that can be approximated as a single bosonic particle as long as we avoid interactions that probe too deeply.
Thank you for that tactfully administered dose of reality.
Now I see where you and
@vanhees71 are coming from (after all, H3 is an unstable molecule), and that my earlier contribution to this thread could be more than just a little bit misleading.
There is still a (admittedly somewhat trivial or contrived) sense in which hydrogen atoms can be thought of as bosons, however, which might be easier to see with second quantization. Consider a many body wave function with an equal number of electrons and protons. This can be expanded (for example, by rearranging operators in lexicographical order of their coordinate/spin values and relabeling ##j##-indices accordingly) as
\begin{align*}
\int_{(\mathbb R_e^{3}\times\{\pm\}_e)^N/S_N}\int_{(\mathbb R_p^{3}\times\{\pm\}_p)^N/S_N} d^{3N}x_ed^{3N}x_p \Psi_N(\vec x_e, \vec x_p, \vec \sigma_e, \vec\sigma_p)\prod_{j=1}^N \psi^\dagger_e(x_{e,j},\sigma_{e,j})\psi^\dagger_p(x_{p,j},\sigma_{p,j})|0\rangle
\end{align*}
(where the integral over the 'quotient space' omits points where coordinates coincide.) This is pretty standard, and explains the correspondence between 2nd quantization and 1st quantization in many-body theory for Fermions (EDIT: see eq. 21.58 from [1]; here a quotient space is used for the integral instead of the usual normalization factor of ##\frac{1}{\sqrt{N!}}##, while the wave function ##\Psi_N## can still be defined outside of the integration bounds through antisymmetry.)
On a configuration by configuration basis, we associate electron coordinates with the nearest unpaired proton, starting with the electron closest to a proton. There will be some configurations where the closest unpaired electron-proton isn't well-defined, but those configurations are on a set of measure zero (e.g. an infinitely thin hypersurface) and can be ignored.
After pairing the electrons with protons in this way, reorder the product ##\Pi_j\psi_{e,j}^\dagger\psi_{p,j}^\dagger## so that electrons are paired with their corresponding proton, and relabel coordinates (and change the bounds of integration) accordingly.
When this has been done, each pair ##\psi^\dagger_e(x_{e,j},\sigma_{e,j})\psi^\dagger_p(x_{p_j},\sigma_{p,j})## can (again in a somewhat trivial or contrived way) be thought of as a joint "H-atom" creation operator ##\phi_H^\dagger(x_{e,j},x_{p,j},\sigma_{e,j},\sigma_{p,j})## over a 6-dimensional + 2##\times## spin-1/2 d.o.f. configuration space. Claim: pairs of spin-1/2 fermions over a 3-dimensional space can be recast as (individual) spin-1 or spin-0 bosons over a 6-dimensional space.
These are very
special bosons, in that they satisfy additional constraints beyond those of "normal" garden-variety 6D+2##\times##spin-1/2 bosons, but I'm at least 90% sure they are still bosons, even if they don't look anything like a real-world, well-adjusted, healthy-boundary-setting hydrogen atom, and can be constructed from arbitrary fermionic states (with equal numbers of electrons and protons.)
EDIT:
References:
[1]
Merzbacher, E., Quantum Mechanics, 3rd Ed.
