Tensor algebra techniques

Calculation of the angular part of the matrix elements thus remains, which can be performed exactly using tensor algebra techniques based on group theory. Since the calculation of the matrix elements is not straightforward, we provide here some details on it for the interested reader. The treatment follows the procedure described in Ref. [17]. 

The angular parts (containing the coordinates of the electron) of the matrix element in eq. (36) can be calculated exactly using standard techniques of tensor algebra. We will illustrate this. Consider the matrix element... 

It is the nature of the subject that makes its presentation rather formal and requires some basic, mainly conceptual knowledge in mathematics and physics. However, only standard mathematical techniques (such as differential and integral calculus, matrix algebra) are required. More advanced subjects such as complex analysis and tensor calculus are occasionally also used. Furthermore, also basic knowledge of classical Newtonian mechanics and electrodynamics will be helpful to more quickly understand the concise but short review of these matters in the second chapter of this book. 

I now give, for completeness, a brief account of the derivation of the spherical tensor formulation of the multipole expansion. This does require an understanding of spherical tensor methods and of Racah algebra, but it may be omitted by readers who are unfamiliar with these techniques, who should skip to the beginning of the next section. 

Flory reviewed in 1969 the development and applications of the rotational isomeric state scheme calculations, which allow, by matrix algebra, the statistical mechanical averaging over the rotational states of chain properties which may be expressed as a vector or tensor quantity associated with the chain bonds, and estimations of the probabilities of chosen conformational sequences. The methods were generalized and schemes for reducing the dimensions of certain generator matrices were presented in 1974, when comparisons were also made with an alternative Fourier expansion method, currently in use for atactic polypropylene. These techniques have greatly contributed to an understanding... 

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