Local equilibrium conditions for hybridization tetrahedra and quasitorques

Local equilibrium conditions for hybridization tetrahedra and quasitorques 

In the FAFO picture, when the form of the HOs is fixed, the equilibrium condition for the hybridization tetrahedron can be written as the equilibrium condition for the orientation of the latter. Due to the angular character of the variables involved, the corresponding set of the energy derivatives with respect to the components can be thought to be a (quasi)torque (here the prefix quasi as previously refers to the fact that no rotation of any physical body is involved in its definition rather that of a fictitious hybridization tetrahedron). As one can check, each (m-th) bond, incident to the given atom A, contributes to the quasitorque the following increment  

Assuming to simplify the notations that for all the incident bonds the atom A is the right-end atom A = Rm) we obtain the overall quasitorque acting upon the hybridization tetrahedron centered on the atom A and the corresponding energy minimum conditions with respect to orientations of all hybridization tetrahedra in the molecule  

These equilibrium conditions are completely analogous to the equilibrium conditions for a system of rigid bodies [42] which requires evanescence of all (quasi)torques. 

Though the equilibrium conditions eq. (3.87) require that a sum of the contributions eq. (3.86) vanishes, it is of interest to consider archetypal situations when some of these contributions vanish separately. These situations are twofold as two vector terms eq. (3.86) sum up to give a quasitorque contribution. The first one, proportional to x v m, vanishes if the HO on the right-end atom and the bond vector are collinear. If the same holds also for the left-end atom, one can see that the vector parts of both HOs ascribed to the bond under consideration are collinear so that the second vector term proportional to v m x also vanishes. This clearly corresponds to the equilibrium condition for two singly a-bonded hybridization tetrahedra. A quasitorque appears if an HO ascribed to the bond under consideration is not collinear with the bond axis it is ascribed to and the quasitorque tends to align them. 

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