Quaternion form of the hybrid orbitals and hybridization tetrahedra

Quaternion form of the hybrid orbitals and hybridization tetrahedra 

In the previous section we used quaternions to construct a convenient parameterization of the hybridization manifold, using the fact that it can be supplied by the 50(4) group structure. However, the strictly local HOs allow for the quaternion representation for themselves. Indeed, the quaternion was previously characterized as an entity comprising a scalar and a 3-vector part h = (h0, h) = (s, v). This notation reflects the symmetry properties of the quaternion under spatial rotation its first component ho = s does not change under spatial rotation i.e. is a scalar, whereas the vector part h — v — (hx,hy,hz) expectedly transforms as a 3-vector. These are precisely the features which can be easily found by the strictly local HOs the coefficient of the s-orbital in the HO s expansion over AOs does not change under the spatial rotation of the molecule, whereas the coefficients at the p-functions transform as if they were the components of a 3-dimensional vector. Thus each of the HOs located at a heavy atom and assigned to the m-th bond can be presented as a quaternion  

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)  

General linear relations between the elements of the HOs residing on a heavy atom as taken in the quaternion form represent some interest. The orthonormality condition for the HOs written in the quaternion form allows us to establish the shape of the hybridization tetrahedra through eq. (3.61). On the other hand, the 4 x 4 matrix formed by HOs expansion coefficients is orthogonal not only with respect to rows, each representing one HO, but also with respect to columns, so that  

From this general relation one easily derives the linear dependence condition for the vectors vm forming hybridization tetrahedra  

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