  We present improved theoretical calculations of transition frequencies for
the fundamental transitions $(L\!=\!0,v\!=\!1)\to(L'\!=\!0,v'\!=\!0)$ in the
hydrogen molecular ions H$_2^+$ and HD$^+$ with a relative uncertainty
$4\cdot10^{-11}$ and for the two-photon transitions in the antiprotonic helium
atom with a relative uncertainty $10^{-10}$. To do that, the one-loop
self-energy correction of order $\alpha(Z\alpha)^6$ is derived in the two
Coulomb center approximation, and numerically evaluated in the case of the
aforementioned transitions. The final results also include a complete set of
other spin-independent corrections of order $m\alpha^7$. The leading order
corrections of $\alpha^2\ln^3(Z\alpha)^{-2}(Z\alpha)^6$ are also considered
that allows to estimate a magnitude of yet uncalculated contributions.
