Hybridization tetrahedron

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A very elegant statement concerning the properties of hybridization tetrahedra belongs to Kennedy and Schaffer [30] in any hybridization tetrahedron two planes formed by any two pairs of HOs are orthogonal. It can be easily proven using the quaternion representation for the scalar product of the vectors normal to two said planes, the following chain of equalities holds (numeration is obviously arbitrary) ... 

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 ... 

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 ... 

In the linear response approximation, the above pseudotorques give the following pseudorotations of the hybridization tetrahedron on the atom under consideration ... 

Inserting this into the energy, yields the following contribution to the pseudotorque acting on the hybridization tetrahedron of the atom A ... 

First we consider the geometry issues. As mentioned, only the hybridization compatible deformations of geometry affect the shape of the hybridization tetrahedron. On the other hand one can easily see that variation of the valence angle Xmm with m < m reduces to rotations of the involved bond vectors enmLm and around the axis orthogonal to the both coordination tetrahedron vectors ... 

The reaction of hybridization tetrahedron on the changes of local geometry can be considered in the linear response approximation eq. (3.133). It is clear that any variation of the valence angle is a sum of equal amounts of hybridization- compatible and hybridization- incompatible deformations. The denominator in the linear response relation eq. (3.133) is the same as for eq. (3.136) while the relevant block of the yxyqE matrix (with q taken as a difference of two opposite valence angles) is proportional to For the methane molecule with the carbon atom put in the origin of the coordinate frame, substitution of matrices of second derivatives for the energy... 

The only reason why this term appears is the deformation of the carbon hybridization tetrahedron eq. (3.136) effectively coupling stretchings of two C-H bonds. Its... 

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 ... 

Analogously, torques acting upon the atoms Lm (the left-end atom for the m-th bond) with to > 2, which are neighboring to the boundary from the MM side, arise due to variation of the shape of the hybridization tetrahedron ... 

The derivatives of the energy correction (of the terms proportional to dP4rr and c)T4r) with respect to the angles J/ J/ yield additional quasi- and pseudotorques (K 4 and (V4, respectively) acting upon the hybridization tetrahedron of the frontier nitrogen atom ... 

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). 

As it is seen from the perturbative estimates, the density ESVs are not transferable and are rather sophisticated functions of those ESVs which define the shape and orientation of the hybridization tetrahedron on the donor atom. At this stage, it is possible to develop a mechanistic description which retains the variable x to keep track of details of the electronic structure. This picture is also suitable for constructing the QM/MM junctions. According to the perturbative estimates for the solutions of eq. (4.9) the equilibrium value of the y variable is always by one order of magnitude in (3Da smaller than x. Since only the combinations xy and y2 enter in the expression of the projection operator and thus in that for the energy, we may set y 0 without causing... 

To conclude this section, we notice that the described discrepancy between the molecular geometry at the oxygen donor atom and the shape of its hybridization tetrahedron is characteristic only for the ionic limit of the dative bond. If the additional... 

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