Tensor formulation

Graphical interpretations of the results are facilitated by the tensor formulation. 

S. Blenk, H. Ehrentraut, W. Muschik. Statistical foundation of macroscopic balances for liquid crystals in alignment tensor formulation. Physica A 77 119-138, 1991. 

In the tensor formulation the difference between electric and magnetic fields disappears. What one observer interprets as an electric process another may regard as magnetic, although the actual particle motions that they predict will be identical12 [37]. 

According to Eq. (7.2.1) we require the polarizability components azz t) and ayz t) in the laboratory-fixed coordinate system. For compactness, we use the spherical tensor formulation outlined in Appendix 7.C. Accordingly, the nine spherical components of the polarizability tensor can be expressed in terms of the nine Cartesian components according to... 

In spite of the disadvantages of the Cartesian formulation, it is preferred by many workers because the alternative, the spherical tensor formulation, is perceived as mathematically difficult. There is undoubtedly some truth in this view. Moreover the spherical-tensor formulation deals in complex quantities which are more difficult to comprehend than the cartesian-tensor components. However the power and versatility of the spherical tensor approach should not be abandoned lightly, and the main purpose of the present paper is to show that it is possible to combine the best features of the cartesian and spherical-tensor methods. We will show that this hybrid approach leads to very compact expressions for the electrostatic energy and related quantities such as the induction and dispersion energies, and that these can be expressed entirely in terms of real multipole moments referred to molecule-fixed coordinate systems. The transformation between molecule-fixed and space-fixed coordinates can be carried out once and for all, and the analogues in this method of the interaction tensors contain the necessary orientational information. 

A more practical objection to the spherical tensor formulation has been that it is inefficient to use in computationally-intensive applications such as molecular dynamics, because of the supposed need to describe the molecular orientations in terms of Euler angles and to evaluate trigonometrical functions of these angles. This objection is completely unfounded. The Euler angles have traditionally been used to describe orientation, but they are by no means necessary, and the formulae to be presented below will not mention them at all. It is possible, within the spherical tensor formalism, to describe orientation in terms of the same direction cosines /. ... 

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. 

The mechanics of a deformable body treated here is based on Newton s laws of motion and the laws of thermodynamics. In this Chapter we present the fundamental concepts of continuum mechanics, and, for conciseness, the material is presented in Cartesian tensor formulation with the implicit assumption of Einstein s summation convention. Where this convention is exempted we shall denote the index thus ( a). 

In our presentation of the atomic polar tensor formulation we shall follow the notation introduced by Person and Newton [33] since it is now generally accepted. The dipole moment changes induced by vibrational distortions are represented as functions of individual atom displacements... 

In general terms, there is a considerable similarity between the APT and BCT (bond charge tensor) formulations of irffiared intensities. Formulas cormecting the elements of Px and D matrices have been derived [131]. The audiors have shown that if the coordinate system and numbering of atoms are conveniently chosen the following relation holds [131]... 

Among the different models for interpretation of vibrational absorption intensities the atomic polar tensor formulation is by far the simplest to ply in transforming die experimental dp/dQi dipole derivatives into quantities associated with molecular subunits, atoms in molecules in the particular case. Besides, the transformation does not involve urmecessary approximations and assumptions. The APT formulation provides also the possibility to directly compare experimental data and theoretical ab initio results. The physical interpretation of atomic polar tensors is, however, hampered by die redundancies between the elements of atomic polar tensors as expressed by Eqs. (4.18) and (4.19). Rotational atomic polar tensors associated with the permanent dipole moment value can make, in the general case, substantial contributions to APT elements. 

Steps for Density Matrix Based Energy Minimization Tensor Formulation and Toy Applications. 

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