for its cation. The procedure followed by Hirao et al. is to search for combinations of F (q) = -ε i (q) and F (q ) = -ε i (q ), with different fractional occupancies q and q , in order to cancel leading-order errors in the Taylor series that defines ∆E in Eq. ( ). The separate quantities F (q) and F (q ) require separate fractionaloccupancy SCF calculations. We will formulate this process in a somewhat different way, by rewriting Eq. ( ) as a polynomial expansion in q = 1 -1/n where n is an integer and q represents the fractional charge that is removed from the MO to be ionized. This will define a sequence of approximations F[n] for n = 2, 3, . . .: