Representations one-dimensional

This means that these matriees are really a eombination of two separate group representations (mathematieally, it is ealled a direet sum representation). We say that Dl ) is redueible into a one-dimensional representation DlO and a three-dimensional representation formed by the 3x3 submatriees whieh we will eall Dl ). 

Another one-dimensional representation of the group ean be obtained by taking rotation about the Z-axis (the C3 axis) as the objeet on whieh the symmetry operations aet ... 

These one-dimensional matriees ean be shown to multiply together just like the symmetry operations of the C3V group. They form an irredueible representation of the group (beeause it is one-dimensional, it ean not be further redueed). Note that this one-dimensional representation is not identieal to that found above for the Is N-atom orbital, or the Ti funetion. 

FIGURE 21.1 A one-dimensional representation of the energy of all possible con-fonners of a simple molecule. 

The one-dimensional representation described above is too simple to describe the behavior of a detonation in response to boundary conditions. Denisov et al. 

Figure 16-14. One-dimensional representation of the ground- and cxcitcd-slalc potential surfaces for irons- and m-stilbene [721.

Fran the one dimensional representation of the potential energy diagram one cannot decide vhether the dissociative adsorption takes place throu intermediate tr plng in the molecular state or throu ... 

One-dimensional representation of the propagation of a concentration step-disturbance in a packed bed reactor (5). 

Wallis, G. B. Third Congress of the European Federation of Chemical Engineering (1962). Symposium on the Interaction between Fluids and Particles, 9. A simplified one-dimensional representation of two-component vertical flow and its application to batch sedimentation. 

Fig. 2.3. Schematic view of a porous nanocrystaUine sensing layer with a one-dimensional representation of the energetic conduction band. A inter-grain band bending, eVs, occms as a consequence of smTace phenomena, and a band bending, eVc, occurs at the grain-electrode contact. Eb denotes the minimmn conduction band energy in the bulk tin oxide, and Ep is the Fermi-energy in the electrode metal...

Thus this corresponds to a one-dimensional representation. The totally symmetric representation is created in a similar fashion, but with all of the boxes arranged vertically ... 

Figure 10.5 Increase of the signal to noise ratio in non-crystallographic symmetry averaging. In (a) is shown a one-dimensional representation of the electron density of a macromolecule. In (b), a graph of the noise that results from the sources of errors in the crystallographic process, including experimental phasing and measurement errors. In (c), the observed density composed of the true electron density with the noise component. In (d), the effect of non-crystallographic symmetry improves the signal from the macromolecule while decreasing the noise level, the dotted lines shows the level of bias.

For C2v, the representation Ai is symmetric with respect to every operation. Such a one-dimensional representation, in which every element is assigned the number 1, occurs for every group, and is called the totally symmetric representation. Two representations are symmetric with respect to C2 each is denoted A. Two are symmetric with respect to symmetry species a unique symbol, and " need not be used in this case. 

In this table all one-dimensional representations are symmetrical with respect to C3, the principal axis, so all are designated A. Subscripts 1 and 2 refer to the operations C 2, and and " refer to Oh-... 

Since the Fourier transformation of equation (2.2) yields only a radial distribution function about the absorber, we note that information obtained from EXAFS is limited to an average, one-dimensional representation of structure. Furthermore, in order that the transform be comparatively free of ripples, the data should extend to at least... 

Note, however, that this one-dimensional representation can be somewhat misleading if it is taken to be a computational protocol. The trouble is that the one-dimensional slice of... 

Exercise 6.1 Show that any one-dimensional representation is irreducible. 

One-dimensional irreducible representations are labeled either A or B according to whether the character of a 2irjn (proper or improper) rotation about the symmetry axis of highest order n is +1 or —1, respectively. For the point groups Wl9 and which have no symmetry axis, all one-dimensional representations are labeled A. For... 

If a one-dimensional representation has complex characters a, 6, c,. .. then there must be another equally acceptable representation with the characters a, b, c, . .. since for a one-dimensional representation the character of an operation equals the single matrix element representing the operation. These pairs are usually bracketed together and labeled Et In fact quite often the reduction which produces the pair of irreducible representations is not carried out, since no useful information is gained by it and anyway the two always occur together. 

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