FATO molecular mechanics of sp3 carbons

As previously, here, for the sake of simplicity of notation, we assume that for all bonds incident to atom A, this atom is a right-end atom of the bond. Though the equilibrium conditions are rather cumbersome, for symmetric cases they can be solved yielding obvious answers the carbon atom in symmetric tetrahedral environment acquires the sp3 hybridization with the HOs collinear to the bonds, etc. 

Second order corrections to the energy of sp3 carbon atom. In order to construct the required mechanistic picture, the estimate of the restoring force which opposes both the quasi- and pseudorotation (deformation) of the hybridization tetrahedra is necessary. That can be obtained by a linear response procedure. For the sp3 carbon atom in the symmetric tetrahedral environment, the related resonance energy is a diagonal quadratic form with respect to small quasi- and pseudorotations together with triply degenerate eigenvalues [44,45]  

Further terms are necessary to describe the interaction between the shape and orientation modes of two bonded tetrahedra, which appears when either of them is quasi-or pseudorotated in the vicinity of the equilibrium. These formulae can be obtained by considering those cross terms in the resonance integral expansion which are bilinear in SujRm and 6ujLm, respectively. This result can be represented as a matrix element  

Here V.ft stands for the 3 x 3 matrix representing the vector multiplication by vRm VRmx = vRm x x and the gRmLm subblock is defined by eq. (3.101). These subblocks couple small pseudo- and quasirotations of the hybridization tetrahedra corresponding to the right- and left-end atoms of the bond (in the specified order) in a bilinear fashion. Their form particularly simplifies for the sp3 carbon atom in a symmetric environment for which we have  

For the diagonal restoring force constants related to the deformations of the hybridization tetrahedra the following estimates can be obtained 30.58 eV/rad2 in CH4 34.49 eV/rad2 in neopentane. This estimate indirectly explains the observation extracted from the numerical experiments the shapes of the hybridization tetrahedra are more stable than their orientations as the diagonal force constant for deformations is 1.5 times larger than that for rotations of the hybridization tetrahedra. 

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