Other representations

Even more complex quaternary SOWA diagrams with three independent composition variables at constant formulation and T/p conditions have been proposed to be represented in an equilateral tetrahedron. Such diagrams maybe useful for some peculiar cases, but they are generally not amenable to simple interpretations and in most cases they are described by a series of bidimensional cuts, i.e. cuts through the tetrahedron, which are not amenable to Winsor s types as seen in Fig. 3.4, because the four types are arranged in a different way [17, 18]. 

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This makes it desirable to define other representations in addition to the electronically adiabatic one [Eqs. (9)-(12)], in which the adiabatic electronic wave function basis set used in the Bom-Huang expansion (12) is replaced by another basis set of functions of the electronic coordinates. Such a different electronic basis set can be chosen so as to minimize the above mentioned gradient term. This term can initially be neglected in the solution of the / -electionic-state nuclear motion Schrodinger equation and reintroduced later using perturbative or other methods, if desired. This new basis set of electronic wave functions can also be made to depend parametrically, like their adiabatic counterparts, on the internal nuclear coordinates q that were defined after Eq. (8). This new electronic basis set is henceforth refened to as diabatic and, as is obvious, leads to an electronically diabatic representation that is not unique unlike the adiabatic one, which is unique by definition. 

Fingerprints structural keys identify a molecule - code is highly compact represented in bits ambiguous not convertible to other representations dependent on the fragment library... 

Where helical secondaiy structures are represented by the cylinder model, the /i-strand. structures are visualized by the ribbon model (see the ribbons in Figure 2-124c). The broader side of these ribbons is oriented parallel to the peptide bond. Other representations replace the flat ribbons with flat arrows to visualize the sequence of the primary structure. 

The wave function is an irreducible entity completely defined by the Schrbdinger equation and this should be the cote of the message conveyed to students. It is not useful to introduce any hidden variables, not even Feynman paths. The wave function is an element of a well defined state space, which is neither a classical particle, nor a classical field. Its nature is fully and accurately defined by studying how it evolves and interacts and this is the only way that it can be completely and correctly understood. The evolution and interaction is accurately described by the Schrbdinger equation or the Heisenberg equation or the Feynman propagator or any other representation of the dynamical equation. 

The matrices (27) provide one representation of SU(2). Other representations can be constructed by taking symmetric product representations with itself. The transformations of the symmetric products u2,uv,v2(= x, x2,x2) according to (27) are... 

It is evident that methods analogous to the ones developed here could be applied to molecular properties which, instead of being pseudoscalar, belong to some other representation of the skeleton point group (vector, tensor, etc. properties). To treat such properties, one needs only to induce from a different representation of than the chiral one. 

Other representations may be obtained by subdividing the unit cell into a number of similar subcells. In the cubic system the subdivision is made along the three axes by the same factor which is used as a subscript in the new lattice complex... 

The qualitative study of electronic structure through the electron (number) density p(r) relies heavily on linecut diagrams, contour plots, perspective plots, and other representations of the density and density differences. There is a review article by Smith and coworkers [302] devoted entirely to classifying and explaining the different techniques available for the pictorial representation of electron densities. Beautiful examples of this type of analysis can be seen in the work of Bader, Coppens, and others [303,304]. 

It is reasonable to hope to assemble a complete set of representations to provide a full and non-redundant description of the symmetry species compatible with a point group The problem is that there are far too many representations of any group. On the one hand, matrices in representations derived from expressing symmetry operations in terms of coordinates - as in problem 5-18 - depend on the coordinate system. Thus there are an infinite number of matrix representations of C2v equivalent to example 7, derivable in different coordinate systems. These add no new information, but it is not necessarily easy to recognize that they are related. Even in the cases of representations not derived from geometric models via coordinate systems, an infinite number of other representations are derivable by similarity transformations. 

Apart of historical reasons, there are several features of the Dirac-Pauli representation which make its choice rather natural. In particular, it is the only representation in which, in a spherically-symmetric case, large and small components of the wavefunction are eigenfunctions of the orbital angular momentum operator. However, this advantage of the Dirac-Pauli representation is irrelevant if we study non-spherical systems. It appears that the representation of Weyl has several very interesting properties which make attractive its use in variational calculations. Also several other representations seem to be worth of attention. Usefulness of these ideas is illustrated by an example. 

The solutions of the Schrodinger equation show how j/ is distributed in the space around the nucleus of the hydrogen atom. The solutions for v / are characterized by the values of three quantum numbers and every allowed set of values for the quantum numbers, together with the associated wave function, strictly defines that space which is termed an atomic orbital. Other representations are used for atomic orbitals, such as the boundary surface and orbital envelopes described later in the chapter. 

In this chapter we discuss several natural ways to construct representations from other representations. 

Irreducible representations are the building blocks of all other representations. Just as each molecule is made up of particular atoms, each representation is made up of particular irreducible representations. Unlike a molecule, whose properties are determined not only by which atoms it is made of, but also by their configuration, a representation is merely the sum of its irreducible parts. Mathematically, irreducible representations are useful because one can often reduce an idea or a calculation involving representations to an easier one involving only irreducible representations. Physically, irreducible representations correspond to fundamental physical entities. 

Aside from the infinite number of representations equivalent to (9.25), there are other representations of C3 >. The matrices (9.25) are each in block-diagonal form. If we partition them into submatrices of the form (2.30), then the submatrices S12 and S2i will be column and row matrices whose elements are all zero. The product of any two of these partitioned... 

Thus, if we want to know the characters x(R) of a representation that is the direct product of two other representations with characters Xi(R) and X2W, these are given by... 

Programs now exist to convert Wiswesser Line Notation [29], Hayward [30], and IUPAC [18] linear notations to connection tables. Because fragment codes alone do not provide the complete description of all structural detail, conversion to other representations is typically not possible. 

However, you will find other representations of planar carbons with rather grossly distorted bond angles. For example, methanoic acid is planar with nearly 120° bond angles, but often is drawn with H—C—O angles of 90° and 180° ... 

Representations Are Linked to Portray Dynamic Phenomena Many graphs and other representations are paired with photographs or drawings of chemical systems, starting with the molecular model superimposed on each chapter s opening photograph. The combination can help students make sense of what they see. 

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