Relativistic corrections of $m{\alpha }^6$ order to the rovibrational spectrum of $\text{H}_2{}^{+}$ and ${\text{HD}}^{+}$ molecular ions

The major goal of high-precision studies of rovibrational states in the hydrogen molecular ions is to provide an alternative way for improving the electron-to-proton mass ratio, or the atomic mass of electron. By now the complete set of relativistic and radiative corrections have been obtained for a wide range of rovibrational states of $\text{H}_2{}^{+}$ and ${\text{HD}}^{+}$ up to order $R_{\infty }{\alpha }^4$. In this work we complete calculations of various contributions to the $R_{\infty }{\alpha }^4$ order by computing the relativistic corrections to the binding energy of electron.

Figure 1

Adiabatic “effective” potentials for the relativistic $m{\alpha }^6$ order correction for $\text{H}_2{}^{+}$ molecular ion $\left(Z_1=Z_2=1\right)$. Energies are in $\left(\text{atomic}\text{units}\right)\times {\alpha }^4$.