PCEM expressions

General PCEM expressions for the Appendix 3. Crystal-field potentials 242... 

PCEM expressions for the 5 crystal-field parameters can be written down for the different coordination polyhedra. By filling in the angular coordinates calculated for the HSM or the MFP, one can predict the sign of the parameters and the B Bq ratios. 

To derive general PCEM expressions it is instructive to consider a coordination polyhedron as consisting of a prismatic body and a variable number of equatorial and axial ligands. The only polyhedra which are difficult to describe in this simple scheme are the dodecahedron and the icosahedron (along the threefold axis). It will be shown that a dodecahedron can be considered as two interpenetrating tetrahedra. An icosahedron described along the fivefold axis gives no problems. 

The simplest case is one ligand on the positive z-axis (CN= 1). The angular position of the ligand is 0=0 . The point group is Cqov- Three parameters are required to describe the crystal field B, Bq and B. The PCEM expressions for these parameters are ... 

It is clear that the crystal-field parameters do not have the same sign. B and B are positive, Bq is negative. It is possible to generalize the PCEM expressions of the Bg parameters for planar coordination polyhedra with a coordination number larger than 3. The number of B parameters with will of course be symmetry-dependent. All these planar polyhedra have a D h symmetry. The B parameters are ... 

By combining the PCEM expressions for axial and equatorial ligands, one can build PCEM expressions for pyramids (C v) and bipyramids (D h). In this model the lanthanide... 

A next step is the study of the prismatic (D h) and antiprismatic (D, ) coordination polyhedra. The simplest prismatic polyhedron is the trigonal prism (Djh, CN=6) and the simplest antiprismatic polyhedron is the tetragonal disphenoid (D2d, CN=4). The tetrahedron is a special case of this tetragonal disphenoid. The PCEM expression for the B, B q and Bq parameters are ... 

By combining the expressions for axial, equatorial and prismatic ligands, one can derive the general PCEM expressions for the B, Bq and Bg crystal-field parameters ... 

For the parameters with 0 the PCEM expressions have to be derived by filling in the ligand coordinates and summing over all ligands. These parameters depend on the choice of the x- and y-axes, as already mentioned. Notice that axial ligands only have contributions to q = 0 parameters, because in all expressions for the 5 parameters with 0 a sin 0 dependence is found and sin0°=0. 

The sign is calculated by filling in the angular coordinates of the HSM in the PCEM expressions. Only one radial distance is considered. The coordination numbers and point symmetry groups can be found in table 9. A zero means that the parameter is not present in the symmetry in question. 

The tetrahedron (Tj) is considered here only theoretically, since CN=4 is too low for lanthanide systems. But it is interesting to compare both octahedron, cube and tetrahedron. The fourfold inversion axis or the threefold rotation axis can be chosen as z-axis. The angular coordinates are given in table 12. The PCEM expressions for the parameters with regard to the fourfold rotation axis are ... 

Because the octahedron and cube have the same symmetry (Oh), the crystal field generated inside a cube is described by the same crystal-field parameters as the octahedron (5g, B, BI and B ). The actual PCEM expressions for the parameters will be different, because of the difference in coordination number ... 

The PCEM expressions for the fourfold axis as the quantization axis are ... 

The cubic ratios are found for B Bl and The PCEM expression are simply the... 

With the aid of these relations and the expressions for the crystal-field parameters of the octahedron, it is easy to build the PCEM expressions for the tetrakishexadron along the threefold rotation axis. 

Other coordination polyhedra with D41, symmetry are the tetragonal bipyramid (CN = 6), which can be seen as a distortion of the octahedron, and the bicapped square prism (CN=10). The tetragonal bipyramid is not discussed here, because an angular distortion cannot lower the symmetry to D4. The distortion scheme of a bicapped square prism (BSP, D4h) to a bicapped square antiprism (BSAP, D4d) is analogue to the one described above. The PCEM expressions for the B and 5 parameters will be exactly the same as for the SP-SAP distortion scheme, since axial ligands do not have an influence on... 

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