Of symmetry operations

Here g is the order of the group (the number of symmetry operations in the group- 6 in this ease) and Xr(R) is the eharaeter for the partieular symmetry T whose eomponent in the direet produet is being ealeulated. 

The complete set of symmetry operations that can be performed on a molecule is called the symmetry group or point group of the molecule and the order of the point group is the number of symmetry operations it contains. Table A2.1 lists the various point groups, together with their elements of symmetry and with examples of each. 

It is important to distinguish between mmetiy properties of wave functions on one hand and those of density matrices and densities on the other. The symmetry properties of wave functions are derived from those of the Hamiltonian. The "normal" situation is that the Hamiltonian commutes with a set of symmetry operations which form a group. The eigenfunctions of that Hamiltonian must then transform according to the irreducible representations of the group. Approximate wave functions with the same symmetry properties can be constructed, and they make it possible to simplify the calculations. 

A geometric object can have several symmetry elements simultaneously. However, symmetry elements cannot be combined arbitrarily. For example, if there is only one reflection plane, it cannot be inclined to a symmetry axis (the axis has to be in the plane or perpendicular to it). Possible combinations of symmetry operations excluding translations are called point groups. This term expresses the fact that any allowed combination has one unique... 

The sum over symmetry operations in formula (16) can be rewritten by considering the effect of multiplying vector h7 by the rotation matrices The collection of distinct reciprocal vectors h7Rg is called the orbit of reflexion h7 [27] r7 is the set of symmetry operations in G whose rotation matrices are needed to generate the orbit ofh/ r, denotes the number of elements in the same orbit [50]. 

The tables of characters have the general form shown in Table 5. Each colipua represents a class of symmetry operation, while the rows designate the different irreducible representations. The entries in the table are simply the characters (traces) of the corresponding matrices. Two specific properties of the character tables will now be considered. 

The mathematical apparatus for treating combinations of symmetry operations lies in the branch of mathematics known as group theory. A mathematical group behaves according to the following set of rules. A group is a set of elements and the operations that obey these rules. 

First, it is apparent that reflection through the xz plane, indicated by transforms H into H". More precisely, we could say that H and H" are interchanged by reflection. Because the z-axis contains a C2 rotation axis, rotation about the z-axis of the molecule by 180° will take H into H" and H" into H, but with the "halves" of each interchanged with respect to the yz plane. The same result would follow from reflection through the xz plane followed by reflection through the yz plane. Therefore, we can represent this series of symmetry operations in the following way ... 

To provide further illustrations of the use of symmetry elements and operations, the ammonia molecule, NH3, will be considered (Figure 5.6). Figure 5.6 shows that the NH3 molecule has a C3 axis through the nitrogen atom and three mirror planes containing that C3 axis. The identity operation, E, and the C32 operation complete the list of symmetry operations for the NH3 molecule. It should be apparent that... 

This process could be continued so that all the combinations of symmetry operations would be worked out. Table 5.3 shows the multiplication table for the C3 point group, which is the point group to which a pyramidal molecule such as NH3 belongs. 

Multiplication tables can be constructed for the combination of symmetry operations for other point groups. However, it is not the multiplication table as such which is of interest. The multiplication table for the C2v point group is shown in Table 5.2. If we replace E, C2, and cryz by +1, we find that the numbers still obey the multiplication table. For example,... 

Seven crystal systems as described in Table 3.2 occur in the 32 point groups that can be assigned to protein crystals. For crystals with symmetry higher than triclinic, particles within the cell are repeated as a consequence of symmetry operations. The number of asymmetric units within the unit cell is related but not necessarily equal to the number of molecules in a unit cell, depending on how the molecules are related by symmetry operations. From the symmetry in the X-ray diffraction pattern and the systematic absence of specific reflections in the pattern, it is possible to deduce the space group to which the crystal belongs. 

Crystal lattices can be depicted not only by the lattice translation defined in Eq. (7.2), but also by the performance of various point symmetry operations. A symmetry operation is defined as an operation that moves the system into a new configuration that is equivalent to and indistinguishable from the original one. A symmetry element is a point, line, or plane with respect to which a symmetry operation is performed. The complete ensemble of symmetry operations that define the spatial properties of a molecule or its crystal are referred to as its group. In addition to the fundamental symmetry operations associated with molecular species that define the point group of the molecule, there are additional symmetry operations necessary to define the space group of its crystal. These will only be briefly outlined here, but additional information on molecular symmetry [10] and solid-state symmetry [11] is available. 

Proceeding in the spirit above it seems reasonable to inquire why s is equal to the number of equivalent rotations, rather than to the total number of symmetry operations for the molecule of interest. Rotational partition functions of the diatomic molecule were discussed immediately above. It was pointed out that symmetry requirements mandate that homonuclear diatomics occupy rotational states with either even or odd values of the rotational quantum number J depending on the nuclear spin quantum number I. Heteronuclear diatomics populate both even and odd J states. Similar behaviors are expected for polyatomic molecules but the analysis of polyatomic rotational wave functions is far more complex than it is for diatomics. Moreover the spacing between polyatomic rotational energy levels is small compared to kT and classical analysis is appropriate. These factors appreciated there is little motivation to study the quantum rules applying to individual rotational states of polyatomic molecules. 

Although the number of atoms in a crystal is extremely high, we can imagine the crystal as generated by a spatial reproduction of the asymmetric unit by means of symmetry operations. The calculation can thus be restricted to a particular portion of space, defined as the Brillouin zone (Brillouin, 1953). 

Under the action of the electric field of the 0 environment, the state of the ( ) 3d configuration will split up into two states. The orbital degeneracy of a D state is 2 X 2 -f- 1 = 5. From the above discussion each of the resulting states must belong to one of the irreducible representations of Oh given in Table I. The state W corresponds to an irreducible representation of the group of symmetry operations of a sphere, i.e., the full rotation group R(3). 

Several types of symmetry operations can be distinguished in a crystalline substance. Purely translational operations, such as the translations defining the crystal lattice, are represented by I 1, n3, with nu n2, n3 being integers. 

Is it possible that consideration of other examples might expose a case in which (C2 followed by av) does not give the same result as additional tests, it will prove more effective to treat the geometric manipulations as independent entities, without reference to illustrative objects on which they operate. The development of numerical representations of symmetry operations will then provide straightforward arithmetic tests for equality. 

In many extremely important cases, the analogy between a group of symmetry operations and a group of real numbers is more than superficial. For example, consider the molecule a—chloronaphthalene ... 

A group consists of a set (of symmetry operations, numbers, etc.) together with a rule by which any two elements of the set may be combined - which will be called, generically, multiplication - with the following four properties ... 

Problem 4-3. In Figure 4.1c, what symmetry operation or combination of symmetry operations takes the x just above the horizontal line (just above the 3 o clock line) into (a) the o closest to the top of the circle (b) the x closest to the top of the circle, (c) the x next to the x in part (b) of this problem. 

Although every symmetry operation can be represented by a matrix, many matrices correspond to linear transformations that do not have the properties of symmetry operations. For example, every symmetry operation has the property that the distance between any two points and the angles between any two lines are not altered by the operation. Such a geometric transformation, that does not distort any object that it acts on, is called an orthogonal transformation. A matrix that corresponds to such a transformation is called an orthogonal matrix. 

In some circumstances the magnitudes of the translation vectors must be taken into account. Let us demonstrate this with the example of the trirutile structure. If we triplicate the unit cell of rutile in the c direction, we can occupy the metal atom positions with two kinds of metals in a ratio of 1 2, such as is shown in Fig. 3.10. This structure type is known for several oxides and fluorides, e.g. ZnSb20g. Both the rutile and tlie trirutile structure belong to the same space-group type PAjmnm. Due to the triplicated translation vector in the c direction, the density of the symmetry elements in trirutile is less than in rutile. The total number of symmetry operations (including the translations) is reduced to... 

A symmetry operation is one which leaves the framework of a molecule unchanged, such that an observer who has not watched the operation cannot tell that an operation has been carried out on the molecule (of course me presupposes the structure of the molecule from other experimental sources). The geometry of the molecule is governed by the geometry of the orbitals used by the constituent atoms to form the molecule. There are five kinds of symmetry operations which are necessary for classifying a point group. 

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. 

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