Zero potential energy criterion applied to Hartree–Fock wave functions

The zero potential energy criterion, which is a necessary condition for the exact wave function, is applied to Hartree–Fock wave functions for a closed‐shell system with doubly occupied spatial orbitals. With the help of the known long‐range asymptotic behavior of Hartree–Fock orbitals, we first derive a single‐electron zero potential energy criterion to be satisfied by Hartree–Fock orbitals. The Hartree–Fock wave function is then shown to never satisfy the zero potential energy criterion, which implies that the Hartree–Fock approximation cannot describe the correct long‐range asymptotic behavior of many‐electron wave functions. Some numerical illustrations are given