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Theoretical Background

The General Principle

1.The Concept
2.A little mathematics
3.Computational methods

1. The Concept
The design of the measurement tool is based on a most minimalistic and least demanding approach. Our answer to the question "How much of a given symmetry is there in a given structure?" is as follows:

Find the minimal distances that the vertices of a structure have to undergo, in order for it to attain the desired symmetry.

The symmetry measure is then a function of these distances as described below. Using this measure it is possible to evaluate quantitatively how much symmetry exists in a non-symmetric configuration; what is the nearest symmetry of any given configuration; and what is the actual shape of the nearest symmetric structure.

2. A little mathematics
For a structure composed of N vertices the coordinates of which are
{Qk, k=1,2...N} we wish to calculate how much of G-symmetry it has, where G is a symmetry point group. Let the coordinates of the searched, nearset perfectly G-symmetric object be at {Pk, k=1,2...N}. The symmetry measure is defined as:

where Q0 is the coordinates vector of the center of mass of the investigated structure.

The denominator in Eq.(1) is a mean square size normalization factor,
summing over all N distances from Q0 to the vertices of the original structure. The normalization factor is introduced to avoid size effects. The CSM defined in Eq.(1) is a normalized root-mean-square deviation from the closest structure with the desired symmetry and is independent of the position, orientation, and size of the original structure. The bounds of S(G) are between zero and 100. If a structure has the desired G-symmetry, then S(G) = 0 and the symmetry measure increases as it departs from G-symmetry, reaching a maximal value (not necessarily 100). The maximal value of 100 is obtained when finding, for example, the degree of pentagonality of a hexagon: The nearest pentagon to a hexagon is the collapsed pentagon into a single center point, the distance of which is, by definition, 100. The process of finding the S value for a distorted triangle is summarized in Fig. 6.

Figure 6

All S(G) values, regardless of G or of the structure are on the same scale and therefore comparable. The degree of, say, perfect tetrahedricity (Td-ness) and square planarity (the degree of D4h-ness) of various distorted four-ligands molecules can be compared; as can the D4h-ness of molecules with different number of ligands; or the different symmetry contents of different molecules.

3. Computational methods
In most standard distance functions a preset reference structure is provided. S(G) is a special distance function in that the object the distance to which is searched, is, in most cases unknown a-priori. For instance, suppose we would like to determine the deviation of a distorted AB4 structure from having a C3 rotational axis. Of the infinite number of AB4 structures that are C3-symmetric, only one structure fulfils the requirement of being the closest to the original structure, and this one has to be searched (see Fig. 7). There are simpler situations. If one asks what is the degree of tetrahedricity, S(Td), of that AB4 structure, then it is known a-priori that the nearest symmetric structure has the shape of an exact tetrahedron.

Figure 7

The main computational problem has been to find the nearest structure that has the desired symmetry, namely how to minimize Eq.(1) in order to get {Pk, k=1,2...N}. Several methods, both general and problem-specific, have been constructed towards this goal, and are described in the literature ( Zabrodsky & Avnir, 1995; Salomon & Avnir, 1999; Pinsky & Avnir, 1998 ).

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