Canceling spin-dependent contributions and systematic shifts in precision spectroscopy of the molecular hydrogen ions

We consider the application of a basic principle of quantum theory, the tracelessness of a certain class of hamiltonians, to the precision spectroscopy of the molecular hydrogen ions. We show that it is possible to obtain the spin-averaged transition frequencies between states from a simple weighted sum of experimentally accessible spin-dependent transition frequencies. We discuss the cases ${\rm H}_{2}^{+}$ and ${\rm HD}^{+}$, which are distinct in the multipole character of their rovibrational transitions. Inclusion of additional frequencies permits canceling also the electric quadrupole shift, the Zeeman shift and partially the Stark shift. In this context, we find that measuring electric quadrupole transitions in ${\rm HD}^{+}$ is advantageous. The required experimental effort appears reasonable.