Many-body atomic potentials in Thomas-Fermi theory

Many-body atomic potentials, ε{lunate}, are functions of the nuclear coordinates, and are defined by differences of ground state energies, E, e.g., ε{lunate}(1, 2) ≡ E(1, 2) - E(1) - E(2). We prove that in Thomas-Fermi theory the n-body potential always has the sign (-1)ⁿ for all coordinates. We also prove that the remainder in the expansion of the total energy E in terms of the ε{lunate}'s, when truncated at the n-body terms, has the sign (-1)ⁿ⁺¹.