Do the relations between E/B and D/H contain higher order multipole terms?

Question

Jackson writes in section 1.4 (third edition) that

\begin{align*} D_\alpha &= \epsilon_0 E_\alpha + \left(P_\alpha - \sum_\beta \frac{\partial Q'_{\alpha\beta}}{\partial x_\beta} + \ldots \right) \\ H_\alpha &= \frac{1}{\mu_0} B_\alpha - (M_\alpha + \ldots ) \end{align*}

and adds,

The quantities $\mathbf{P}$, $\mathbf{M}$, $Q'_{\alpha\beta}$, and similar higher order objects represent the macroscopically averaged electric dipole, magnetic dipole, and electric quadrupole, and higher moment densities of the material medium in the presence of applied fields.

In section 6.6 he derives these relations and states for D,

The third and higher terms are present in principle, but are almost invariably negligible.

and makes a similar statement about H.

So, is it true that the precise definitions of D and H really contain these higher-order terms, but that they're almost invariably dropped (e.g., by Griffiths, and all the lecture notes I could find online) because they're almost always negligible?

Or do we instead say that $\mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P}$ exactly and that $\mathbf{H} = \frac{1}{\mu_0} \mathbf{B} - \mathbf{M}$ exactly and that the macroscopic Maxwell's equations break down when the quadrupole and higher moment densities become significant?

Answer 1

We split $\rho=\rho_{\it ex}+\tilde\rho$ and $\jmath=\jmath_{\it ex}+\tilde\jmath$ into "explicit" and "medium" contributions. We can then define $P$ and $M$ by $\tilde\rho=-\nabla P$ and $\tilde\jmath=\nabla\times M$, and macroscopic fields by $D=E+P$ (I'm sloppy about units, there is a $4\pi$ in Gauss units) and $B=H+M$. The macroscopic equations are then exact, independent of the question where $\tilde\rho$ and $\tilde\jmath$ come from.

If moments beyond the dipole are important then what is lost is the realtion between $P$ and $M$ and the density of dipoles.