The Symmetry Number Method

According to the symmetry number method, the statistical factor, K, is given by the ratio of symmetry numbers for reactant and product species in equilibrium  

The internal symmetry number is defined as the number of different but indistinguishable atomic arrangements that can be obtained by internal rotations around single bonds, or other intramolecular processes such as pyramidal inversion, Berry pseudorotation and ring circumrotation (in the case of catenanes). It is implied that the processes giving rise to the internal symmetry number are fast with respect to the time scale in which the equilibrium in Eq. [34] is attained and measured. For example, staggered 

In the case of a chiral molecule present at equilibrium as a racemic mixture, its symmetry number must be divided by two to account for the entropy of mixing of the two enantiomers. For example, chiral tartaric acid belongs to the C2 point group and contains a C2 rotation axis, thus for a pure enantiomer, CT = CText = 2, but for a mixture of ( )-tartaric acid one must consider CT — 7ex(/2 — 1. 

It is interesting to apply the symmetry number method to the equilibrium defining the EM (Eq. [6]). Since in Eq. [6] the chains M,+y and Mj have the same symmetry number, = 1/aQ, where a a is the symmetry number of the ring ohgomer C,-, equal to i in the case of rings of the type f-(A—B)/ C h symmetry), or 2i in the case of rings of the type c-(A—A), D h symmetry). As a result, the microscopic effective molarity, defined as the microscopic equihbrium constant of the reaction in Eq. [6], is given by EM = Go EM,. 

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Both techniques converge to the same result, but the symmetry number method, though less intuitive, is easier to handle and we will limit our discussion to this technique. According to Benson (1976), the symmetry number cr of a molecule affects its rotational entropy by a factor —i ln(cr). Consequently, the statistical contribution to the stability constant of equilibrium (79) is given by the ratio of the symmetry numbers of the reactants and products (Eq. (82)). 

The symmetry number method must be used with some care when dealing with optically active reactants or transition states (Poliak and Pechukas, 1978). Suppose that the transition state is optically active. The reaction can then be written as... 

An accurate and consistent evaluation of statistical factors in self-assembly processes is of crucial importance to predict the expected stability constant in the absence of cooperative effects and, therefore, to spodight the emergence of either positive or negative cooperativity as a marked deviation firom statistical behaviour. However, the evaluation of statistical factors can be controversial and doubtful sometimes. A critical re-examination of the methods to assess statistical factors in self-assembly processes has been published in 2007. " Two methods appear the most useful, namely, the symmetry number method and the direct counting method. The two methods if properly appHed give the same results however, the symmetry... 

From the latter example, it appears that the direct counting method is more suitable than the symmetry number method to obtain statistical factors... 

How does one determine the symmetry number As illustrated in the section above it is equal to the number of rotations that take the molecule into itself. Another and very attractive method is based on the use of group theory. Students who have taken a course in inorganic chemistry have been introduced to group theory. If the reader is uncomfortable with this topic the next few paragraphs can be skipped, especially since this method of finding molecular symmetry numbers need not to be used for finding the ratios of symmetry numbers, Si/s2, required to understand isotopomer fractionation. 

To calculate the free enthalpy of an ion, we need to know its symmetry number and its intrinsic free enthalpy. The symmetry number can be calculated methodically and algorithmically. The intrinsic free enthalpy is obtained from the assumptions of the single-events theory. For an ion, the intrinsic free enthalpy calculations involve two parameters ... 

Surprisingly, one of the major difficulties encountered when applying this model in coordination chemistry is cormected with the calculation of reliable statistical factors. Two parallel methods have been developed by using either the symmetry numbers of the molecules or the direct count of the microspecies formed in the reactants and products (Ercolani et al.,... 

FIGURE 78 Calculation of statistical factors by using the method of the symmetry numbers for equilibrium (83) and (84). (A) The metal ions are considered as nonsolvated species and (B) the metal ions are considered as nine-coordinate tricapped-trigonal prismatic solvates. 

Owing to the complexity and variety of multicyclic structures, a systematic method is needed for the evaluation of the statistical factors and ap. A practical way to evaluate statistical factors in difficult cases stems from the consideration that they coincide with the product of the symmetry numbers of reactants or products each raised to the corresponding stoichiometric coefficient [95]. The symmetry number of a molecule is defined as the total number of independent permutations of identical atoms or groups in a molecule that can be arrived at by simple rotations of the entire molecule, or by rotations about freely rotating single bonds within the molecule [96]. In practice. 

Note that in equation system (2.64) the coefficients matrix is symmetric, sparse (i.e. a significant number of its members are zero) and banded. The symmetry of the coefficients matrix in the global finite element equations is not guaranteed for all applications (in particular, in most fluid flow problems this matrix will not be symmetric). However, the finite element method always yields sparse and banded sets of equations. This property should be utilized to minimize computing costs in complex problems. 

Symmetry is one of the most important issues in the field of second-order nonlinear optics. As an example, we will briefly demonstrate a method to determine the number of independent tensor components of a centrosymmetric medium. One of the symmetry elements present in such a system is a center of inversion that transforms the coordinates xyz as ... 

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