Symmetrically equivalent atoms, sets

In contrast to the r -methcxl, where unsubstituted atomic positions are indeterminable, r0-derived methods can locate also unsubstituted atoms, although with less accuracy than for substituted positions. In practical cases, this capacity is limited to one or two unsubstituted atoms and yields useful results only when the SDS is otherwise large enough. Coordinates that are (for whatever reason) not included in a r0-type fit must be assigned estimated values, which are kept fixed in the fit but affect the result. In these cases, one should not do so without also setting error limits to these coordinates in order to find out, by Eq. 30, which additional errors of the positions are due to the error afflicted fixed positions. If symmetrically equivalent atomic positions are among those kept fixed, the relations established between the independent and the expanded set of variables (Eq. 56b) now denoted as B(A) and B, respectively, are applicable in the form ... 

In fact the relative coefficients within a set of symmetrically equivalent atoms, such as Cl, C3, C5, and Cg in para-benzosemiquinone, can be determined by group theory alone. The appropriate set of symmetrized orbitals, also listed in Table 1, can be obtained by use of character tables and procedures described in Refs. 3 to 6. In matrix notation, the symmetrized combinations are <1> = Up, where <1) and p are column vectors and U is the transformation matrix giving the relations shown in part (c) of Table 1. The transformation U and its transpose U can be used to simplify the solution of the secular matrix X since the matrix multiplication UXU gives the block diagonal form shown in part (6) of Table 1. 

Chutjian [16] derived expressions for molecules in which a whole set of symmetrically equivalent atoms are substituted. These expressions were reformulated in terms of the planar moments by Nygaard [17]. For double substitution of a pair of atoms related by the symmetry plane S , x and y] are obtained from equation (22) with = fil. defined by... 

We keep the 2 px orbital unchanged. This set is a variant of the canonical sp hybrid set in which, however, all three orbitals are not symmetrically equivalent. If is the H-C-H angle, ha and hb point directly at the H atoms. Because of the invariance... 

A positron is defined as a set of symmetrically equivalent coordinate points. Within the unit cell, atoms or molecules may be located at general positions that do not lie on any symmetry element or at special positions. If they do lie either on a symmetry... 

An ED study requires a complete set of intensity data, measured to a much higher Bragg angle (sin 9IX > 1.1 A , sometimes up to 1.77A ) than is required for an lAM study, and with much more accuracy and precision. It is necessary to measure all symmetrical equivalents of each reflection, carefully address all possible sources of systematic errors, especially extinction and TDS. With a point detector, the data collection required many weeks or even months, but with an area detector (especially using SR) this time has been reduced to a few days, without any loss in precision. There is an intrinsic difficulty to distinguish between the ED distribution of a stationary atom and an apparent smearing of its ED from thermal motion, therefore the data must be collected at as low a temperature as possible. ... 

Finally, 9 and 10 form a set of equivalent atoms on their own. These can only go together symmetrically as... 

Several examples will now be given to illustrate the subgroup associated with each symmetrically equivalent set of atoms in a molecule. For ethane (the staggered configuration), 9 = 3 the subgroup associated with the carbon atoms is 3C = Qu since the following operations of... 

In group theoretical language, the collections of operations Ei3C of (6), for i > 1, are called the cosets of g determined by the subgroup 3C = 7fi3C. In summary of this section, it is to be noted that the atoms of any molecule may be classified in symmetrically equivalent sets, each of which is associated with some subgroup 3C (which may in extreme cases be either g itself or consist merely of the identity element) which is determined by those symmetry operations which send an arbitrarily chosen atom of the given set into itself. ... 

The PRDDO method employs three mathematically equivalent basis sets. The first (p) consists of pure Slater orbitals. The second (( >) is a Schmidt orthogonalized, principal axes basis (see below). The third basis (X) consists of orthogonal atomic orbitals (OAOs), which are obtained from the 4> basis via a LOwdin symmetric orthogonalization ... 

On examining the wave function, we find that the eight SC valence orbitals form four sets of two orbitals each. The orbital pairs are symmetrically equivalent and are permuted amongst themselves by operations of the tetrahedral group, Tj. Within each pair, one of the orbitals is very largely localized on one of the H atoms and clearly resembles a H(ls) orbital with small deformations, almost exclusively onto the C atom. The second orbital of the pair is localized mainly on the C atom, but with some small degree of delocalization, almost exclusively onto the H atom of the pair. This is shown in Figure 5. 

The first row represents an alternative set of equivalent localized orbitals which is as strongly localized as the one just discussed. 56) They extend essentially over three atoms. Whereas the Kekuld type localized orbitals are symmetric with respect to the plane bisecting a bond, the localized orbitals in the first row are symmetric with respect to a plane containing two opposite atoms. The negative lobe extends only over one atom, and the... 

It is further possible to describe a given constitution by mapping the indexed set of atoms onto the independently indexed graph. The indexed set of atoms must be partially symmetrized with regard to the element equivalence classes of the atoms and the indices of the graph with respect to its symmetry. That constitution is defined as the reference constitution in which the atomic indices a and the graph indices g match. 

There is a very important feature of this situation, which we can turn to advantage. It will be observed that the set of four methylene carbon atoms, numbers 1, 4, 5, and 8, possess D4h symmetry by themselves and that the set of four carbon atoms in the ring, numbers 2, 3, 6, and 7, also by themselves constitute a set having D4ll symmetry. Furthermore, the atoms in one set are not equivalent symmetrically to any of those in the other. None of the outer atoms is ever interchanged with any of the inner atoms by any symmetry operation. Thus each of these sets can be used separately as the basis for a representation of the group, and if this is done we obtain from each set a representation, r x, which reduces as follows ... 

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