Tetrahedron angular coordinates

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 ... 

Angular coordinates (d cp,) for the ligands in a tetrahedron (Tj) with respect to the fourfold inversion axis... 

Figure 15. Left-. Geometry of the surface 6 = 0 in Eq. (46) with fixed total angular momenta S and N. Properties of the special points A, B, C, and D are listed in Table I. All other permissible classical phase points lie on or inside the surface of the rounded tetrahedron. Right. Critical section at J. Continuous lines are energy contours for y = 0.5 and N/S = 4. Dashed lines are tangents to the section at D. Axes correspond to normalized coordinates, /NS and K JiN + S). Taken from Ref. [2] with permission of Elsevier.

The constituent atomic orbitals are orthogonal and if they are also normalized, these hybrid orbitals can be seen to be normal and orthogonal by forming their squares and products. Since the s function is spherically symmetrical, and since the angular dependence of px, py, and pz is proportional to x/r, y/r, and z/r, respectively, the four sp3 hybrids can be seen to have their maxima directed respectively toward the four corners of a tetrahedron centred at the origin of coordinates. 

The fixation of parameters d (those of the shape of the hybridization tetrahedron) can be considered as FO fixed orbitals) picture [93]. In this case only the resonance energy is the orientation dependent contribution. The angular dependence of the energy (bending) can be described by introducing small rotation vectors S(prm which after applying them to vectors CRmLm (distorted) coordination tetrahedron. The final result of... 

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