Maximum symmetry principle

Then, choosing the main rotation axis, is based on the maximum symmetry principle as the S5mimetry is more elevated, the order of rotation... 

First, however, it is appropriate to introduce the Principle of maximum symmetry, an important heuristic that underlies the bond valence model and... 

Rule 3.1 (Principle of maximum symmetry.) As far as allowed by the chemical and geometric constraints, all atoms and all bonds in a compound will be chemically and geometrically indistinguishable. 

In NaCl (18189), this principle would require all atoms to be identical. Clearly this symmetry is already broken by the constraint imposed by the chemical formula which requires half the atoms to be Na" " and half CP. However, all the Na" " ions are indistinguishable from each other, and the same is true for the CP ions. The bonds likewise, six for each formula unit, are also equivalent in the bond graph (Fig. 2.4). The crystal structure (Fig. 1.1) is then determined by applying the principle of maximum symmetry to the constraints imposed by three-dimensional space as described in Section 11.2.2.4. The crystal structure is thus uniquely determined by the principle of maximum symmetry and the chemical and spatial constraints. 

The principle of maximum symmetry can be justified by recognizing that the free energy of any symmetric system must necessarily be either a maximum or a minimum with respect to small shifts that break that symmetry, since shifts in opposite directions will produce identical changes in the energy. Thus equilibrium structures will tend to adopt the most symmetric configuration that corresponds to a minimum in the free energy. 

While the principle of maximum symmetry is a heuristic with wide scientific application, Rules 3.3 to 3.5 define the bond valence model. They have each been discussed before but are brought together here for convenience. 

The valence sum rule is not, in general, sufficient to determine the distribution of the valence among the various bonds, but the principle of maximum symmetry suggests that the distribution will be the most symmetric one that is consistent with the valence sum rule. The condition that makes the bond valences most nearly equal is the loop, or equal valence rule. 

Fig. 7.5. Bond valences and lengths expected for KH2PO4 (68696, 201373) (a) assuming the principle of maximum symmetry, (b) observed structure, (c) assuming normal hydrogen bonds. Note that in addition to the bonds shown, each ion receives a valence of 0.25 vu from K+.

The principle of maximum symmetry requires that the crystal structure adopted by a given compound be the most symmetric that can satisfy the chemical constraints. We therefore expect to find high-symmetry environments around atoms wherever possible, but such environments are subject to constraints such as the relationship between site symmetry and multiplicity (eqn (10.2)) and the constraint that each atom will inherit certain symmetries from its bonded neighbours. The problems that arise when we try to match the symmetry that is inherent in the bond graph with the symmetry allowed by the different space groups are discussed in Section 11.2.2.4. 

AI2O3 is six coordinate, the O ions will be four coordinate but if Ar is four coordinate, the ions will be only 2.67 coordinate on average. The principle of maximum symmetry favours the first choice as the second requires at least two different types of ion. 

The third principle is the Principle of maximum symmetry that plays a major role in deciding between different possible structures as the example given under Rule 11.2 shows. This principle has been previously given as Rule 3.1 ... 

The first step in any chemical approach to crystalline structure is to determine the short-range order, i.e. which atoms are bonded. The most convenient way of doing this is by means of the bond graph described in Section 2.5. In many cases all or most of the bond graph can be determined from first principles, since, except for the weakest bonds created in the post-crystallization stage, the bond graph is determined by the rules of chemistry, particularly the hierarchical principle (Rule 11.5), the valence matching principle (Rule 4.2), and the principle of maximum symmetry (Rule 3.1). 

It may not always be possible to assign the weak bonds formed, e.g. by alkali metals both because they are numerous and because their connections are determined more by the constraints of three-dimensional space than by the principle of maximum symmetry. 

In cases where there are no electronically driven distortions, the orbital description provides no better account of the chemistry than the bond valence model. Rather it tends to make an essentially simple situation more complex. For example, consider the phosphate and nitrate anions, and NOJ. In orbital models the P atom is described as sp hybridized and the N atom as sp hybridized, but these descriptions are just representations of the spherical and cylindrical harmonics appropriate to the observed geometries. They provide no explanation for why P is four but not three coordinate, or why N is three but not four coordinate. The bond valence account given in Chapter 6 is simpler, more physical, and more predictive. The orbital description is merely a rather complicated way of saying that the ions obey the principle of maximum symmetry but implying that the constraints are related in some unspecified way to the properties of one-electron orbitals rather than to the ionic sizes. 

There are simple underlying principles that govern structural chemistry, of which the principle of maximum symmetry is one. Another is the notion that the chemical properties of an atom are determined by the potential at the surface of its electron core, the potential that the valence electrons experience. This potential is proportional to the ratio of the atom s charge to size, and it is no surprise that this ratio determines such varied atomic quantities as bonding strength, acid and base strength, and electronegativity. 

A program to solve these equations has been described by Orlov et al. (1998). O Keeffe (1989) has described an alternative method that is suitable for performing the calculation by hand. Rutherford (1990) has presented a way of inverting the matrix that retains the symmetry of the equations by including all Aa of the equations of type 3.3. Brown (1977) has described a robust iterative technique for solving the equations based on recognizing that eqn (3.4) is an expression of the principle of maximum symmetry (Rule 3.1). In this procedure... 

We have emphasized that the Diels-Alder reaction generally takes place rapidly and conveniently. In sharp contrast, the apparently similar dimerization of olefins to cyclobutanes (5-49) gives very poor results in most cases, except when photochemically induced. Fukui, Woodward, and Hoffmann have shown that these contrasting results can be explained by the principle of conservation of orbital symmetry,895 which predicts that certain reactions are allowed and others forbidden. The orbital-symmetry rules (also called the Woodward-Hoffmann rules) apply only to concerted reactions, e.g., mechanism a, and are based on the principle that reactions take place in such a way as to maintain maximum bonding throughout the course of the reaction. There are several ways of applying the orbital-symmetry principle to cycloaddition reactions, three of which are used more frequently than others.896 Of these three we will discuss two the frontier-orbital method and the Mobius-Huckel method. The third, called the correlation diagram method,897 is less convenient to apply than the other two. 

The determining principle in the formation of a structure is the principle of energy minimization which, for nondirectional forces, is expressed geometrically as the maximum filling principle. If the minimization of energy is attained when the structural units occupy low-symmetry positions, their proper symmetry does not completely coincide with the symmetry of the crystallographic positions. 

A guiding principle in the following analysis is the principle of maximum symmetry which states that ... 

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