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Non-relativisitic second-order corrections to the wave function at origin inmuonic and exotic atoms are considered. The corrections are due to theelectronic vacuum polarization. Such corrections are of interest due to variouseffective approaches, which take into account QED and hadronic effects. Thewave function at origin plays a key role in the calculation of the pioniumlifetime, various finite nuclear size effects and the hyperfine splitting. Theresults are obtained for the $1s$ and $2s$ states in pionic and muonic hydrogenand deuterium and in pionium, a bound system of $\pi^+$ and $\pi^-$.Applications to the hyperfine structure and the Lamb shift in muonic hydrogenare also considered.

Second-order corrections to the wave function at origin in muonic hydrogen and pionium