Linear response relations for hybridization ESVs

The main use of eq. (3.133) is for exclusion of the angular variables describing the hybridization tetrahedra from the DMM mechanistic picture and for going by this to a more standard classical MM-like description of the PES. However, before doing that, we have to estimate the precision of the linear response relations eq. (3.133) between geometry and hybridization variations themselves by numerical study. This has been done in [26] on the example of elongation of C-H bonds and deformations of valence angles in the methane molecule. 

In the tetrahedral methane molecule (its parameters then correspond to subscript 0 in eqs. (3.132), (3.133)), we notice that the 57X57XE matrix further simplifies as s nm = 1 and, therefore, simple analytical expressions become possible. Also, we notice that the FA approximation is adequate here as, for example, even very large elongation of one C-H bond by 0.1 A leads to changes of the bond geminal amplitudes u, v, and w not exceeding 0.003. The same applies to the expectation values of the pseudospin (f) operators representing the one- and two-electron density matrix elements. 

Linear response of hybridization to bond elongation. First the relation between hybridization and elongation of the C-H bond is considered. For this we need the mixed second order derivatives coupling the bond stretching with the hybridization ESVs. For every C-H bond in methane we can introduce diatomic coordinate frame with the t-axis directed along the bond and express the resonance integral related to this bond as  

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 

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