Symmetry operators rotation

It is worthwhile now to introduce a mechanism devoted to representing the symmetry operations of our ABe center. The symmetry operation (rotation) of Figure 7.1 transforms the coordinates (x, y, z)into(y, -x, z). This transformation can be written as a matrix equation ... 

But this does nob end the tale of possible arrangements. Hitherto we have considered only those symmetry operations which carry us from one atom in the crystal to another associated with the same lattice point—the symmetry operations (rotation, reflection, or inversion through a point) which by continued repetition always bring us back to the atom from which we started. These are the point-group symmetries which were already familiar to us in crystal shapes. Now in "many space-patterns two additional types of symmetry operations can be discerned--types which involve translation and therefore do not occur in point-groups or crystal shapes. 

Any symmetry operation (rotation as well as reflection) can be associated with a certain permutation. In other words, the symmetry operations of the molecular point group G generate a permutation group of skeletal position indices (Gg). Note that Gg is also as a rule nonisomorphic to the group (see above). 

FIGURE 21.2 Three types of symmetry operation rotation about an n-fold axis, reflection in a plane, and inversion through a point. 

Four simple symmetry operations - rotation, inversion, reflection and translation - are visualized in Figure 1.7. Their association with the corresponding geometrical objects and symmetry elements is summarized in Table 1.2. 

Mathematics (Hassel, 1830) has shown that there are only 32 combinations of symmetry operations (rotation, inversion, and reflection) that are consistent with a three-dimensional crystal lattice. These 32 point groups, or crystal classes, can be grouped into one of the seven crystal systems given in Table 2.1. There are four types of crystal lattices primitive (P), end-centered (C, B, and A), face-centered (/O, and body-centered (/). The primitive lattice contains a lattice point at each comer of the unit cell, the end-centered lattice has an additional lattice point on one of the lattice faces, the face-centered lattice has an extra lattice on each of the lattice faces, and the body-centered lattice has an extra lattice point at the center of the crystal lattice. By combining the seven crystal systems with the four lattice types (P, C, I, F), 14 unique crystal lattices, also known as Bravais lattices (Bravais, 1849), are produced. 

Point group - A group of symmetry operations (rotations, reflections, etc.) that leave a molecule invariant. Every molecular conformation can be assigned to a specific point group, which plays a major role in determining the spectrum of the molecule. 

Fig. C.l. Examples of an isometric operation, (a) Unitary operation rotation of a point ty angle a about the axis . The old position of the point is indicated by the vector r, the new position by r (of the same length), (b) Unitary operation rotation of the funetion f(r — rg) by the angle a about the axis As a result, we have the function f(r—Uro), which in general represents a funetion that differs from f(r — ro)- (c) The unitary operation that represents a symmetry operation rotation by the angle a = 120° of the function/(r) = exp - r-ryi j +exp - Ir-r pj +exp - rwhere the vectorsr,. rg.rc are of the same length and form the mercedes trademark (the angle 120°). The new functirai is identical to the old one. (d) Translational operator by the vector ri (ri) applied to the Gaussian function /(r) = exp - r-roPj gives (ri)/(r) = /(J r) =...

Ik) and (I k ) both lie on an element of symmetry. Then application of this symmetry operation (rotation axis or reflection plane) brings the pair of molecules into self-coincidence. This requires the ij(lk, I k ) to be identical before and after transformation, However, a change of sign may be required, and it can then be concluded that the particular force constant must vanish. For example, assume that the symmetry element is a twofold axis parallel to x. Then I k ) will not change sign hut... 

The structural characteristics of the polymer chain constructed as described above are that translational or roto-translational symmetry can be found along the chain in addition to the traditional point group symmetry operations (rotation axes, symmetry planes, inversion center, and identity) [18]. 

The representations of other point groups label the MOs of all nonlinear molecules. For linear molecule wavefunctions, we considered only four symmetry operations rotation about the z axis, inversion, reflection in the xy horizontal plane, and reflection in the vertical planes. Remember that when these operators act on the wavefunction, they may change if/ but not if/. The same principle remains true when we move on to polyatomic molecules, now with other possible symmetry elements. The symmetry properties of the orbital are denoted by the representation used to label the orbital. The Uj MOs of the Cjy molecule F2O, for example, have electronic wavefunctions that are antisymmetric with respect to reflection in either of the two vertical mirror planes (Fig. 6.10). 

We begin our discussion with the idea of a symmetry operation, which is a process that generates a configuration indistinguishable from the initial one. In total there are five different types of symmetry operation for a single object such as a molecule, namely rotation, inversion, reflection, rotation-reflection, and identity. While the symmetry operation describes the process, the symmetry element describes the property that the molecule must possess in order for that operation to be performed. For example, the symmetry operation rotation requires that the moleeule possesses the symmetry element an axis of rotation , the operation inversion requires the molecule to possess the element inversion center , and so on. There is a proposed convention that the symmetry operation should be written in an italic font and the element in an upright (Roman) font, so, for example, you can perform a C2 operation around a C2 axis, and so on. 

We will need a systematic way to deduce whether a molecular property is symmetric or antisymmetric with respect to the symmetry operations for that molecule s point group, as things will quickly get more complicated. To this end, we define a number, Xp( )> called a character, which expresses the behavior of our property p when operated on by the symmetry operation, R. A collection of characters, one for each symmetry operation present in a point group, forms a representation, Fp. The property is technically referred to as the basis vector of the representation, Fp. We can define aU sorts of basis vectors, some of which have very little apparent connection to our original molecule, such as the non-symmetry operation translate along the z axis , often given the symbol z, or the non-symmetry operation rotate by an arbitrary amount about the X axis, often referred to as R. Strictly speaking, the characters are the trace of the transformation matrix for each symmetry operation, applied to the property, p. This is described in more detail in the on-line supplementary section for Chapter 2 on derivation of characters. 

Atoms and molecules in solids arranged in a lattice can be related by four crystallographic symmetry operations - rotation, inversion, mirror, and translation - that give rise to symmetry elements. Symmetry elements include rotation axis, inversion center, mirror plane, translation vector, improper rotation axis, screw axis, and glide plane. The reader interested in symmetry and solving crystal stmctures from diffraction data is encouraged to refer to other sources (7-... 

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