The difference for the Mn compound is attributed to the Jahn-Teller effect. i am very not confident that I can explain this correctly, but in my coursework we never actually addressed trigonal prismatic configurations in inorganic chemistry. A trigonal prism is a good choice for a coordination geometry that leads to a high negative value of D and possible SMM behaviorind7 ions:trigonalprismaticcobalt(II)complexeshave been recently found to demonstrate slow relaxation in a small20 or even zero external static dc ﬁelds, with the largest barrier obtained25 being 76 cm−1. 's results in their figure 3 (shown below). Hence the orbitals that point "between" nearby ligands have the lowest energy. The average energy of the five d orbitals is the same as for a spherical distribution of a −6 charge, however. This splitting gives rise eventually to the energy gap in the trigonal prismatic case for d2, causing the crossover of the stabilities of the octahedral and trigonal-prismatic structures between d2 and d3. The tetragonality is along the z-axis, so 101 and 011 are equivalent and therefore not split. When two or more types of ligands are coordinated to an octahedral metal centre, the complex can exist as isomers. It is found that d-covalency provides a stabilizing factor for trigonal-prismatic coordination, as compared with octahedral coordination, for atoms with d 0, d 1 or d 2 configuration. Is the problem that the Wikipedia page Susi gave has things like "trigonal bipyramidal" and "square planar" but not explicitly "trigonal prismatic" (the term you used in your title)? How to calculate the orthogonal representation of a HCP cell? I think this depends on the lattice details. The d-z^2 orbital has its density mostly out of the x-y plane, but there is a ring of electron density in the x-y plane, so this orbital will have a somewhat higher energy than the d-xz and d-yz orbitals. Maekawa et al.
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