Exchange assisted tunneling and positron annihilation on inner atomic shells

It is known for a long time that the long range asymptotic behavior of the 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 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.