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

Theoretical transition frequencies beyond 0.1 ppb accuracy in H$_2^+$, HD$^+$, and antiprotonic helium