Infinite symmetry operations

All examples considered above illustrate symmetry elements that traverse the origin of coordinates and do not have translations. When symmetry elements do not intersect the origin (0,0,0) or have translations (e.g. glide planes and screw axes), their symbolic description includes fractions of full translations along the corresponding crystallographic axes. For example  

This description formalizes symmetry operations by using the coordinates of the resulting points and, therefore, it is broadly used to represent both symmetry operations and equivalent positions in the International Tables for Crystallography (see Table 1.18). The symbolic description of symmetry operations, however, is not formal enough to enable easy manipulations involving crystallographic symmetry operations. 

---

At this point the reader may feel that we have done little in the way of explaining molecular synnnetry. All we have done is to state basic results, nonnally treated in introductory courses on quantum mechanics, connected with the fact that it is possible to find a complete set of simultaneous eigenfiinctions for two or more commuting operators. However, as we shall see in section Al.4.3.2. the fact that the molecular Hamiltonian //coimmites with and F is intimately coimected to the fact that //commutes with (or, equivalently, is invariant to) any rotation of the molecule about a space-fixed axis passing tlirough the centre of mass of the molecule. As stated above, an operation that leaves the Hamiltonian invariant is a symmetry operation of the Hamiltonian. The infinite set of all possible rotations of the... 

A symmetry operation can be repeated infinitely many times. The symmetry element is a point, a straight line or a plane that preserves its position during execution of the symmetry operation. The symmetry operations are the following ... 

In the examples presented in the previous section, the vectors % of displace meat coordinates [Eqs. (12) and (19)] were used as a basis. It should not be surprising that the matrices employed to represent the symmetry operations have different forms depending on the basis coordinates. In effect, there is an infinite number of matrices that can serve as representations of a given symmetry operation. Nevertheless, there is one quantity that is characteristic of the operation - the trace of the matrix - as it is invariant under a change of basis coordinates. In group theory it is known as the character. 

It is reasonable to hope to assemble a complete set of representations to provide a full and non-redundant description of the symmetry species compatible with a point group The problem is that there are far too many representations of any group. On the one hand, matrices in representations derived from expressing symmetry operations in terms of coordinates - as in problem 5-18 - depend on the coordinate system. Thus there are an infinite number of matrix representations of C2v equivalent to example 7, derivable in different coordinate systems. These add no new information, but it is not necessarily easy to recognize that they are related. Even in the cases of representations not derived from geometric models via coordinate systems, an infinite number of other representations are derivable by similarity transformations. 

The symmetry operations, G, of the space group acting on an atom placed at an arbitrary point in space will generate a set of mo equivalent atoms in the unit cell. Operation of the lattice translations, R, acting on this set generates an infinite array of such atoms, with the finite set of ma atoms being repeated at each point on the lattice. This is illustrated in Fig. 10.1 in which nia = 4 and each of the rectangles defined by the horizontal and vertical lines represents a unit cell that is identical with the one outlined with heavy lines. 

Because the group is of infinite order, (9.67) is cumbersome to use, and it is simplest to find the symmetry-adapted functions by inspection. Each symmetry operation either leaves lja and sb alone or transforms them into each other, and the following functions are easily seen to transform... 

First, it is necessary to define the structure. The structure of a planar zig-zag polyethylene chain is shown in Fig. 2, together with its symmetry elements. These are C2 — a two-fold rotation axis, C — a two-fold screw axis, i — a center of inversion, a — a mirror plane, and og — a glide plane. Not shown are the indentity operation, E, and the infinite number of translations by multiples of the repeat (or unit cell) distance along the chain axis. All of these symmetry operations, but no others, leave the configuration of the molecule unchanged. 

There are hypersymmetry phenomena in some crystal structures that are characterized by extra symmetry operations applicable to infinite chains of molecules. This kind of hypersymmetry has proved to be more easily detectable and has been reported often in the literature [104],... 

A predominant motif in the hydrogen-bonding structure in a-cyclodex-trin-7.57H20 is a ribbon of fused four- and six-membered antidromic cycles, shown in Figs. 18.5 c and 18.7 c, d. This is generated by the symmetry operation of a screw axis in the b direction. It contains an infinite chain of water molecules,... 

The number of elements in a group is called its order g. A group may have a finite or infinite order (finite or infinite group). All symmetry operators of a point or a space group fulfill the conditions 1 - 4. 

Symmetry operations which involve shifts, can apply only to regularly repeating infinite patterns, like crystal structmes. A repeated application of such an operation brings the structme not to the original position, but to a different one, separated from the original by an integer number of lattice translations. There are two types of such ( translational ) symmetry elements (see Table 2), besides primitive lattice translations a, b, c. 

The symmetry operations must be compatible with infinite translational repeats in a crystal lattice. 

Another classification is based on the presence or absence of translation in a symmetry element or operation. Symmetry elements containing a translational component, such as a simple translation, screw axis or glide plane, produce infinite numbers of symmetrically equivalent objects, and therefore, these are called infinite symmetry elements. For example, the lattice is infinite because of the presence of translations. All other symmetry elements that do not contain translations always produce a finite number of objects and they are called finite symmetry elements. Center of inversion, mirror plane, rotation and roto-inversion axes are all finite symmetry elements. Finite symmetry elements and operations are used to describe the symmetry of finite objects, e.g. molecules, clusters, polyhedra, crystal forms, unit cell shape, and any non-crystallographic finite objects, for example, the human body. Both finite and infinite symmetry elements are necessary to describe the symmetry of infinite or continuous structures, such as a crystal structure, two-dimensional wall patterns, and others. We will begin the detailed analysis of crystallographic symmetry from simpler finite symmetry elements, followed by the consideration of more complex infinite symmetry elements. 

It is practically obvious that simultaneously or separately acting rotations (either proper or improper) and translations, which portray all finite and infinite symmetry elements, i.e. rotation, roto-inversion and screw axes, glide planes or simple translations can be described using the combined transformations of vectors as defined by Eqs. 1.38 and 1.39. When finite symmetry elements intersect with the origin of coordinates the respective translational part in Eqs. 1.38 and 1.39 is 0, 0, 0 and when the symmetry operation is a simple translation, the corresponding rotational part becomes unity, E, where... 

As a result of symmetry transformation (Eqs. 2.111 and 2.112), both the magnitude of the structure amplitude and its phase may change. Finite symmetry operations (t, tj and tj are all 0) usually affect the phase angle, while infinite operations, i.e. those which have a non-zero translational component, affect both the magnitude and the phase. 

Consider an ideally infinite helical chain whose crystallographic repeat contains N chemical repeat units, each with P atoms. A screw symmetry operation transforms one chemical unit into the next, with a being the rotation about the helix axis and d the translation along the axis. Let r" denote the ith internal displacement coordinate associated with the nth chemical repeat unit. The potential energy, by analogy with Eq. (3), is given by... 

---