Symmetry equivalence

Small spherical viruses have a protein shell around their nucleic acid that is constructed according to icosahedral symmetry. Objects with icosahedral symmetry have 60 identical units related by fivefold, threefold, and twofold symmetry axes. Each such unit can accommodate one or severed polypeptide chains. Hence, virus shells are built up from multiples of 60 polypeptide chains. To preserve quasi-equivalent symmetry when packing subunits into the shell, only certain multiples (T = 1, 3, 4, 7...) are allowed. 

Again, the Pi are the equivalent densities obtained from symmetry breaking. Let us clarify this concept by using the examples given above. In the B2 case, the two equivalent symmetry broken Kohn-Sham Slater determinants are... 

Figure 2 Fragments of a randomly disordered and an ordered arrangement of one-dimensional Ising spins. The long arrows identify equivalent symmetry...

Digressing from reductive desulfurization into stereochemistry, we may use this experimental proof of the equivalent symmetry of D-mannitol as a basis for an independent proof of the configurations of D-mannitol and D-arabitol. The reduction of D-arabinose yields the optically active pentitol, D-arabitol application of the Sowden-Fischer synthesis to D-arabinose yields D-mannose86 which upon reduction gives D-mannitol. 

In the formula for D-mannitol (XIV), by Emil Fischer s second convention,84 the hydroxyl on carbon atom 5 is placed on the right. Since D-arabitol (XV) is optically active, the hydroxyl on carbon atom 3 must then be on the left, for regardless of the configuration at carbon atom 4, arabitol would otherwise be an optically inactive meso form. Finally, by reason of the equivalent symmetry of D-mannitol, the... 

Another proof of the configuration of D-mannitol and also of D-manno-n-manno-octitol (XVI), which is likewise dependent on the experimental proof of the equivalent symmetry of D-mannitol is the following. D-Mannose has been converted, by successive cyanohydrin syntheses, first to a mannoheptose and then to a mannooctose which on reduction yielded a mannooctitol whose octaacetyl derivative is optically inactive. (It was not possible to examine the octitol itself because of its very low solubility in water.)87 The meso character of the octaacetate shows that the mannooctitol must possess a meso configuration, with a plane of symmetry between carbon atoms 4 and 5. To write its formula, the hydroxyl at carbon atom 7 is placed on the... 

The equivalent symmetry element in the Schoenflies notation is the improper axis of symmetry, S which is a combination of rotation and reflection. The symmetry element consists of a rotation by n of a revolution about the axis, followed by reflection through a plane at right angles to the axis. Figure 1.14 thus presents an S4 axis, where the Fi rotates to the dotted position and then reflects to F2. The equivalent inversion axes and improper symmetry axes for the two systems are shown in Table 1.1. 

TABLE 1.1 Equivalent symmetry elements in the Schoenflies and Hermann-Mauguin Systems... 

Fig. 14.1. Lewis structures of the carboxylate ion. (a) and (b) Two equivalent symmetry breaking structures, (c) An alternative Lewis structure, (d) The average of (a) and (b) which preserves the inherent S3nnmetry of the ion.

If a symmetry element A is carried into the element B by an operation generated by a third element X, then of course B can be carried back into A by the application of X x. The two elements A and B are said to be equivalent. If A can be carried into still a third element C, then there will also be a way of carrying B into C, and the three.elements, A, B, and C, form an equivalent set. In general, any set of symmetry elements chosen so that any member can be transformed into each and every other member of the set by application of some symmetry operation is said to be a set of equivalent symmetry elements. 

Class 3 is obtained by introducing a twofold axis of rotation, symbolized by below the motif on the line of translation. The important thing to note here is that in addition to the C2 operation explicitly introduced (and all those just like it obtained by unit translation) a second set of C2 operations, with axes halfway between those in the first set is created. In space symmetry (even in ID space) the introduction of one set of (equivalent) symmetry elements commonly creates another set, which are not equivalent to those in the first set. It should also be noted that had we chosen to introduce explicitly the... 

S C2 S4 E. The identity operator E is always present (whether there is an axis of symmetry or not) and it must always be included once in any list of symmetry operators. The following convention is used in drawing up a list of symmetry operators where the same configuration may be generated by equivalent symmetry operators we list only the simplest form, that is the one of lowest n, with n < o < rc, avoiding redundancies. Thus C2 and not S, S4 and not Sj, E and not S%. The first part of this convention implies that whenever n/k in the operator C=k (or Sn k) is an integer p, then there is a Cp (or Sp), axis coincident with C (or S ), and this should be included in the list of symmetry elements. Thus, for example, a C6 axis implies coincident C3 and C2 axes, and the list of operators associated with C6 is therefore Cg" C fi O, O, C2 E. ... 

Consider the set of point symmetry operators associated with a pyramid based on an equilateral triangle. Choose z along the C3 axis. The set of distinct (non-equivalent) symmetry operators is G li O, O, [Pg.32]

The trace, i.e., the sum of the diagonal elements of the matrix representing a symmetry operator is called the character of this operator and is symbolized by x- Equivalent symmetry operators (conjugate operators) form a class. Symmetry operators of the same class have the same character, since the equivalent representation matrices have the same trace. 

The mirror plane (two-fold inversion axis) reflects a clear pyramid in a plane to yield the shaded pyramid and vice versa, as shown in Figure 1.12 on the right. The equivalent symmetry element, i.e. the two-fold inversion axis, rotates an object by 180" as shown by the dotted image of a pyramid with its apex down in Figure 1.12, right, but the simultaneous inversion through the point from this intermediate position results in the shaded pyramid. The mirror plane is used to describe this operation rather than the two-fold inversion axis because of its simplicity and a better graphical representation of the reflection operation versus the roto-inversion. The mirror plane also results in two symmetrically equivalent objects. 

Symmetries intrinsic to the A -body system lead to a partitioning of the configuration space into equivalent symmetry sectors. If y is the symmetry number for each particle, the number of sectors is... 

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