Reducible representations simplification

While we have chosen to proceed here by reducing representations for the full group D3h, it would have been simpler to take advantage of the fact that D3h is the direct product of C3u and C where the plane in the latter is perpendicular to the principal axis of the former. The behaviour of any atomic basis functions with respect to the C3 subgroup is trivial to determine, and there are only two classes of non-trivial operations in C3v. In more general cases, it is often worthwhile to look for such simplifications. It is seldom useful, for instance, to employ the full character table for a group that contains the inversion, or a unique horizontal plane, since the symmetry with respect to these operations can be determined by inspection. With these observations and the transformation properties of spherical harmonics given in the Supplementary Notes, it should be possible to determine the symmetries spanned by sets of atomic basis functions for any molecular system. Finally, with access to the appropriate literature the labour can be eliminated entirely for some cases, since... 

Another solution involves reduced representations of the macromolecules. For this purpose, many simplified models of proteins were developed first for folding processes [19-22] and subsequently for docking problems [17, 23]. The various proposed models differ in their degree of simplification. A residue may be described by only one point per side chain [21, 24, 25] or more [20, 22, 26]. Other researchers prefer to use discrete positions with a grid representation of the molecules [4, 27]. A small interval enables obtaining a nearly atomic representation [18] a larger one yields a sharply reduced model [17, 23]. Many other techniques have been proposed to simplify the systems, e.g. spherical harmonics [28, 29], Connolly surfaces [8, 30, 31] etc. 

Any collection of basis vectors that complies with the molecular symmetry can generate a character representation of the group, but in most cases it will be a reducible one and so can be simplified. In this section we will show that the simplification of a reducible representation r can be made using the data for the set of irreducible representations available in the standard character tables. 

A further simplification of the semiclassical mapping approach can be obtained by introducing electronic action-angle variables and performing the integration over the initial conditions of the electronic DoF within the stationary-phase approximation [120]. Thereby the number of trajectories required to obtain convergence is reduced significantly [120]. A related approach is discussed below within the spin-coherent state representation. 

In practice, a number of simplifications assist in this procedure. To determine the representation spanned by a set of basis functions, we must determine the behaviour of the functions under all operators in the group. However, since we are only interested at this stage in reducing the resulting representation, we can use characters and classes rather than full matrix representations. 

We could use this as a starting point for finite elasticity. It would be desirable to reduce the number of elastic constants a, b, and so on, by considerations such as material symmetry. Rather than developing a general theory of finite elasticity, however, we will introduce appropriate restrictions at an early stage, as appropriate for a representation of the behaviour of rubbers. The principal restrictions are driven by the simplifications that ... 

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