Exchange-assisted tunneling and positron annihilation on inner atomic shells

It has long been known that the long-range asymptotic behavior of Hartree-Fock orbitals is different from that of the orbitals in the local potential. However, there is no consensus about observable physical effects associated with this asymptotics. Here we argue that a weaker decrease of the Hartree-Fock orbitals at large distances is responsible for the positron annihilation on the inner shell electrons, which is observed experimentally.