# Crystal-Field Effects not yet Fully Incorporated

• C. A. Morrison
Part of the Lecture Notes in Chemistry book series (LNC, volume 47)

## Abstract

In section 12 we discussed the point-charge contribution to the multipolar field components Anm. It was early recognized by Hutchings and Ray (1963) that the multipolar components of the constituent ions contribute to the Anm at the site occupied by the unfilled shell ndN. For a point charge eZi located at (Math). from the origin ion site, we have the electric potential
$$\phi(\vec r)=\frac{{e{Z_i}}}{{|{{\vec R}_i}-\vec r|}}$$
(14.1)
. The potential energy of one of the ℓN electrons at r is
$$\begin{array}{*{20}{c}}{U(\vec r)=-ej(r)}\\{U(\vec r)=-{e^2}{Z_i}\sum\limits_{nm}{\frac{{{r^n}}}{{R_i^{n+1}}}{C_{nm}}(\hat r){C_{nm}}({{\hat R}_i})}}\\\end{array}$$
(14.2)
where we have expanded the denominator of equation (14.1) in the spherical tensors discussed in chapter 1. If we write equation (14.2) as
$$F(abk)=-\left({\alpha\frac{{{e^2}}}{2}}\right)<a(0)b(0)|k(q)>[(a+b+2)-k(k+1)]$$
(14.3)
then
$$A_{nm}^{(0)}=-{e^2}\sum\limits_i{\frac{{{Z_i}{C_{nm}}({{\hat R}_i})}}{{R_i^{n+1}}}}$$
(14.4)
where the sum on i covers all the ions of charge eZi in the solid. This result we derived in section 11, expressed in slightly different form. It seems natural to extend equation (14.3) to the form
$$U(\vec r)=\sum\limits_{n,m,k}{A_{nm}^{(k)*}{r^n}{C_{nm}}}(\hat r)$$
(14.5)
and relate the A nm (k) to the various k-pole moments of ligands at $${\vec R_i}$$.

## Keywords

Dipole Moment Multipole Moment Crystal Field Parameter Spherical Tensor Scheelite Structure
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Annotated Bibliography and References

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