Rotation axes multiple

The square matrix A x transforms the vector x into a vector y by the product y=Ax. Multiplication by the matrix A associates two vectors from the Euclidian space fR" and therefore corresponds to a geometric transformation in this space. A is a geometric operator. Non-square matrices would associate vectors from Euclidian spaces with different dimensions. The ordered combination of geometric transformations, such as multiple rotations and projections, can be carried out by multiplying in the right order the vector produced at each stage by the matrix associated with the next transformation. 

Torsional interactions. In addition to all the terms described above, we often introduce to the force field a torsional tetm Ax-y-z-w (l cos nm) for each torsional angle co showing how V changes when a rotation to about the chemical bond YZ, in the sequence X-Y-Z-W of chemieai bonds, takes place (n is the multiplicity of the energy barriers per single turn ). Some rotational barriers already result from the van der Waals interaction of the X and W atoms, but in practice, the barrier heights have to be corrected by the torsional potentials to reproduce experimental values. 

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