inorganic chemistry - Where does the label ‘Dq’ to denote the field split in coordination compounds come from?

As established in a previous question, coordination compounds typically have a field split between the $\mathrm{t_{2g}}$ and the $\mathrm{e_g}$ d-orbitals.^[1] This energy difference can be explained by the crystal field theory which assumes negative point-charges approaching a complex and destabilising orbitals pointing towards said point charges.

Commonly, the energy difference between the lower and higher orbitals is referred to by the relative value of $\Delta$ or the equally relative value of $10~\mathrm{Dq}$.

The latter is of interest to me. It is clearly used as a (relative) unit, because ligand field stabilisation enthalpies (LFSE’s) are typically given in values of $\mathrm{Dq}$ — e.g. $\ce{[Cr(H2O)6]^3+}$ (hexaaquachromium(III)) has an LFSE of $-12~\mathrm{Dq}$. But what does $\mathrm{Dq}$ mean and what does it derive from?

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Note:

[1]: This is explicitly considering the octahedral case. Tetrahedral compounds split the d-orbitals into $\mathrm{t_2}$ and $\mathrm{e}$. This distinction is, however, not relevant to the question.

Answer

I currently happen to have the book by Figgis that Max mentioned. Chapter 2 is devoted to a mathematical formulation of crystal field theory, which I did not bother reading in detail because I do not understand any of it. (The corollary is: If you have a better answer, please post it!) It seems that both $D$ and $q$ are collections of constants defined in such a way so as to make the octahedral splitting equal to $10Dq$. In particular it is defined that

$$D = \frac{35ze}{4a^5}$$

and

$$q = \frac{2e\langle r^4\rangle}{105}$$

where $ze$ is the charge on the ligands (in crystal field theory, the ligands are treated as point charges), $a$ is the distance of the ligands from the central ion, and $\langle r^4 \rangle$ is the mean fourth power radius of the d electrons in the central ion.

These quantities can be derived from first principles (that is precisely what crystal field theory is about) and lead to the expression of the crystal field electrostatic potential in Cartesian coordinates:

$$\mathbf{V}_{\mathrm{oct}} = D\left(x^4 + y^4 + z^4 - \frac{3r^4}{5}\right)$$

I followed some references given in the book which ultimately ended up in one of the papers in porphyrin's comment: Phys. Rev. 1932, 31, 194. There's more discussion there.

In particular, as far as I can tell, it seems that $D$ is called that simply because it is a quartic term in the potential (the first three terms having coefficients of $A,B,C$), and $q$ is called that because it is related to a series of constants $p_J$ which depend on the total spin quantum number $J$ (consistent with $p$ and $q$ being commonly used letters for mathematical constants).

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Just to clear up any potential confusion with porphyrin's comment which at first glance seems to be missing one $r^4$ term in the electrostatic potential, the above article writes:

It is sufficient to write the potential $V = D(x^4+y^4+z^4)$ since this can be made to satisfy Laplace's equation by adding a function of $r^2$ (viz. $-3Dr^4$). [The addition of this term] corresponds to superposing a spherically symmetrical field, and merely shifts all levels equally.

There's still a factor of $1/5$ missing compared to the expression above, but I'm not particularly inclined to figure out how it arises, and I don't think it is of much importance anyway since the only difference is effectively to shift the zero of energy.