A study of the Bonham series representation of the Born–Oppenheimer exchange amplitude and the derivation of a local exchange potential

The convergence and range of applicability of the Bonham series representation of the Born–Oppenheimer exchange amplitude is investigated. Numerical calculations on 1 1S→2 3S and 1 1S→2 3P of He by electron impact demonstrate that the first three terms of the Bonham series can provide an adequate representation of the Born–Oppenheimer amplitude from high energies down to near threshold. The three-term Bonham series is then used to represent the Hartree–Fock exchange kernel in momentum space, which has been shown by Lassettre to be proportional to the Born–Oppenheimer amplitude. Inverse Fourier transform, plus an additional approximation of replacing the momentum which appears as an expansion parameter in the Bonham series by its averge value, gives us a local exchange potential. If the averge momentum in the Thomas–Fermi model is used, the first term of the local exchange potential is just the electron gas exchange potential. A second term corresponding to a correction for the inhomogeneity in the electron gas density, is also obtained. No adjustable parameter is involved. Exchange energies of He, Be, and Ne calculated using the local exchange potential agree much better with the Hartree–Fock results than the electron gas model.