# The Born-Oppenheimer approximation and muonic molecules

Does the Born-Oppenheimer approximation fail for muonic molecules (i.e. molecules where one or more electrons are replaced with muons)?

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Excellent question :-) –  David Z Oct 8 '11 at 17:43

Deepends on your definition of "fails". The accuracy of Born-Oppenheimer approximation is determined by the smallness of the electron/nucleus mass ratio. For hydrogen this ratio is $\approx 1/1800$. Replace the electron mass by the muon mass, which is 200 times larger, and you get $\approx 1/9$ which gives you a rough estimate for the relative accuracy in case of muonic molecules.
@Georg: Well, most of computational chemistry never goes beyond the 0th order of Born-Openheimer; So $1/9$ is really 200 worse news than $1/1800$. OF course, it all depends on the particular problem. –  Slaviks Oct 8 '11 at 12:34
Is it a power expansion or something exponential like $1+e^{-\frac{a}{\epsilon}}+...$, $\epsilon = m_e/m_{nucl}$ (why is it called an "adiabatic" approximation)? –  Vladimir Kalitvianski Oct 9 '11 at 13:15
I did not find the answer. Is it a series like $1+a\epsilon+...$ or not? Where does that $1/9$ stand? –  Vladimir Kalitvianski Oct 9 '11 at 17:55