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Accurate experimental values for the vibrational ground tone or fundamentalvibrational energy splitting of H$_2$, HD, and D$_2$ are presented. Absoluteaccuracies of $2\times10^{-4}$ cm$^{-1}$ are obtained from Doppler-free laserspectroscopy applied in a collisionless environment. The vibrational splittingfrequencies are derived from the combination difference between separateelectronic excitations from the $X^{1}\Sigma_{g}^{+}, v=0, J$ and $v=1, J$vibrational states to a common $EF^{1}\Sigma^{+}_{g}, v=0, J$ state. Thepresent work on rotational quantum states $J=1,2$ extends the results reportedby Dickenson et al. on $J=0$ [Phys. Rev. Lett. 110 (2013) 193601]. Theexperimental procedures leading to this high accuracy are discussed in detail.A comparison is made with full \emph{ab initio} calculations encompassingBorn-Oppenheimer energies, adiabatic and non-adiabatic corrections, as well asrelativistic corrections and QED-contributions. The present agreement betweenthe experimental results and the calculations provides a stringent test on theapplication of quantum electrodynamics in molecules. Furthermore, the combinedexperimental-theoretical uncertainty can be interpreted to provide bounds tonew interactions beyond the Standard Model of Physics or fifth forces betweenhadrons.

Precision spectroscopy of the $X\,^{1}\Sigma_{g}^{+}, v=0\rightarrow 1$ ($J=0-2$) rovibrational splittings in H$_{2}$, HD and D$_{2}$