# 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.

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Which means, the approximation still works, but convergence is much slower, You need more calculation loops. – Georg Oct 8 '11 at 12:21
@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
@VladimirKalitvianski: BO treats nucleai as static and allows separation of (fast) electron dynamics and (slow) nucleus dynamics. Going beyond BO requires taking into account nucleus-electron entanglement and thus greatly expands the Hilbert space that need to be approximated. See more on wikipedia‌​. – Slaviks Oct 9 '11 at 16:05
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