7.2 Application: Crystal-Field Potentials
169
Fig. 7.2 The octahedral
crystal-field potential
corresponding to the |4A 1 |
hexadecapole. Grey and black
refer to positive and negative
values, respectively
We can use this result to write the functional form of the hexadecapolar invariant by
combining the squares of the d-functions:
|4A 1g |=
1
√
5
√
3
√
2
θ
2 + ǫ
2
−
√
2
√
3
ξ
2 + η
2 + ζ
2
∼
z
4 + x
4 + y
4 − 3
x
2 y
2 + x
2 z
2 + y
2 z
2
(7.19)
This function corresponds precisely to the crystal-field operator of Eq. (7.14). Figure 7.2 shows this invariant. It is clearly a function that mimics the octahedral symmetry. Moreover, it reflects the multipole character of the crystal-field potential. The
potential is repulsive along the coordinate axes where the ligands are and of opposite sign along the threefold directions, in between the ligands, corresponding to the
vertices of the cube. In fact, Eq. (7.19) provides a direct route to the crystal-field
splitting. The coefficient in front of θ 2 and ǫ 2 in this equation is proportional to the
interaction of these e g -orbitals with the crystal-field operator, and similarly for the
coefficient in front of the t 2g orbitals. The ratio of these coefficients can be reduced
to
√
3
√
2
−
√
2
√
3
=
3
−2
(7.20)
which reflects the crystal-field splitting of the d-shell, where the e g level is raised
by 6Dq, while the t 2g level is lowered by 4Dq, so that the barycentre is conserved
and that the total splitting remains 10Dq.