Quasitorque

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 ... 

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. 

An alternative equilibrium condition is possible only for a pair of HOs with vanishing s-contributions. For two pure p-orbitals residing on the right- and left-end atoms of the bond, the numerical coefficients at the first vector terms vanish if the HOs are perpendicular to the bond axis. In this case the second vector term vanishes if two vectors representing the pure p-orbitals are parallel. This clearly corresponds to the picture of a 7r-bond between the hybridization tetrahedra. A quasitorque then appears, tending to orient two hybridization tetrahedra in such a way that the two heights of the unit length of two hybridization tetrahedra are parallel. 

Then the contributions to the quasitorques can be found. The simplest one is the contribution coming as an effect of the orientation of the dipole moment jlA. As the energy contribution of all terms involving the dipole can be written as... 

The one-center energy components have no clear correspondence in the standard MM setting. In our approach the one-center contributions E- arise due to deviations of the geminal amplitude related ES Vs (7>P and 41 ) from their transferable values. These deviations interfere with hybridization. The derivatives of E f s with respect to the angles Land uji, taken at the values characteristic for the stable hybridization tetrahedra shapes which appear in the FATO model, yield quasi- and pseudotorques acting upon the hybridization tetrahedron. In evaluating these quantities we notice that all the hybridization dependence which appears in the one-center terms is that of the matrix elements of eq. (2.71). In the latter, the only source of the hybridization dependence is that of the second and fourth powers of the coefficients of the s-orbital in the HOs. Since they do not depend on the orientation of the hybridization tetrahedra, we immediately arrive at the conclusion that no quasitorques caused by the variation of electron densities appear in the TATO setting ... 

The variations of the one-electron densities 6Pria with a = a, ,v,( and the polarity (/ — P[l) of the bond with m = 1 deserve some discussion. As it is seen from eqs. (3.86), (3.105) each bond incident to an atom contributes an increment to the quasitorque and to the pseudotorque acting upon its hybridization tetrahedron. In the equilibrium these increments separately sum up to zero. We can think that the equilibrium shape and orientation of the hybridization tetrahedron is obtained within a TATO DMM model applied to the entire system. Then, within such a model, there exists an atom corresponding to the left end of the bond with m = 1 having number Li according to our previous notation. The HOs obtained in this approximation provide an initial guess for HOs in the system including those of the atom Pi, which... 

These additional pseudo- and quasitorques produce the pseudo- and quasirotations of the hybridization tetrahedron of the boundary atom R. In the linear response approximation, it corresponds to the treatment of the corresponding pseudo- and quasitorques by the fV7/0 1 matrix which is simple (diagonal in the basis of the and SAi variables) in the case of symmetric hydride ... 

The quasitorque induced by the small variations of the one-center ES Vs is vanishing, thus resulting in no quasirotation of the hybridization tetrahedron. At the same time the pseudotorque appears due to the involvement of the frontier atom in the density redistribution within the QM part of the complex system. This contribution to the QM induced pseudotorque is collinear to the QM residing HO (m = 4). 

The total pseudo- and quasitorques which appear due to quantum behavior of electrons in the QM region then become ... 

The equilibrium conditions for the form and orientation of bridization tetrahedron can be obtained by zeroing the pseudo-torque N and quasitorque K which are coefficients at ScSif and Scot in the resonance energy expansion ... 

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