  The influence of oscillating quadrupole fields on atomic energy levels is
examined theoretically and general expressions for the quadrupole matrix
elements are given. The results are relevant to any ion-based clock in which
one of the clock states supports a quadrupole moment. Clock shifts are
estimated for $^{176}$Lu$^+$ and indicate that coupling to the quadrupole field
would not be a limitation to clock accuracy at the $\lesssim10^{-19}$ level.
Nevertheless, a method is suggested that would allow this shift to be
calibrated. This method utilises a resonant quadrupole coupling that enables
the quadrupole moment of the atom to be measured. A proof-of-principle
demonstration is given using $^{138}$Ba$^+$, in which the quadrupole moment of
the $D_{5/2}$ state is estimated to be $\Theta=3.229(89) e a_0^2$.
