  Buckingham expansion is important for understanding molecular multipoles and
(hyper)polarizabilities. In this study, we give a complete derivation of
Buckingham expansion in the traced form using successive Taylor series. Based
on such derivation, a general Buckingham expansion in the traced form is
proposed, and from which numerical calculations with finite field method of
high accuracy can be achieved. The transformations from the traced multipoles
and multipole-multipole polarizabilities to the corresponding traceless
counterparts are realized with an auxiliary traced electric field gradient. The
applications of the finite field method in this study show good agreements with
previous theoretical calculations and experimental measurements.
