Wigner - Kirkwood ℎ - expansion of the linear response function

For the treatment of inhomogeneous Fermi systems we derive a semi-classical expansion of the linear response function in powers of ℎ (up to Square ℎ) for the interacting (RPA) and non-interacting (Lindhardt function) case. We find that the series is rapidly converging and that the semi-classical result compares very well with exact model calculations for cases where the response does not exhibit narrow resonances (e.g. the quasielastic peak region) otherwise it still represents the average value