Symmetry operations kinds

Symmetry Operation Kind of Rotation 0 1 2C3 Proper 3a Improper 180°... 

In Section 4.3.f it was shown that there are 3N — 5 normal vibrations in a linear molecule and 3N — 6 in a non-linear molecule, where N is the number of atoms in the molecule. There is a set of fairly simple rules for determining the number of vibrations belonging to each of the symmetry species of the point group to which the molecule belongs. These rules involve the concept of sets of equivalent nuclei. Nuclei form a set if they can be transformed into one another by any of the symmetry operations of the point group. For example, in the C2 point group there can be, as illustrated in Figure 6.18, four kinds of set ... 

Symmetry axes can only have the multiplicities 1,2,3,4 or 6 when translational symmetry is present in three dimensions. If, for example, fivefold axes were present in one direction, the unit cell would have to be a pentagonal prism space cannot be filled, free of voids, with prisms of this kind. Due to the restriction to certain multiplicities, symmetry operations can only be combined in a finite number of ways in the presence of three-dimensional translational symmetry. The 230 possibilities are called space-group types (often, not quite correctly, called the 230 space groups). 

A derivative structure can be considered as being obtained from a reference structure by ordered atomic substitution, subtraction or addition processes or by unit cell distortions (or both). The opposite kinds of transformation correspond to the so-called degeneration processes. A derivative structure has fewer symmetry operations than the reference structure (a degenerate structure has more). A derivative structure has either a larger cell or a lower symmetry (or both) than the reference structure. 

For each symmetry element of the second kind (planes of reflection and improper axes of rotation) one counts according to Eq. (1) the pairs of distinguishable ligands at ligand sites which are superimposable by symmetry operations of the second kind. 

A mathematical group consists of a set of elements which are related to each other according to certain rules, outlined later in the chapter. The particular kind of elements which are relevant to the symmetries of molecules are symmetry elements. With each symmetry element there is an associated symmetry operation. The necessary rules are referred to where appropriate. 

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. 

We consider four kinds of symmetry elements. For an n fold proper rotation axis of symmetry Cn, rotation by 2n f n radians about the axis is a symmetry operation. For a plane of symmetry a, reflection through the plane is a symmetry operation. For a center of symmetry /, inversion through this center point is a symmetry operation. For an n-fold improper rotation axis Sn, rotation by lir/n radians about the axis followed by reflection in a plane perpendicular to the axis is a symmetry operation. To denote symmetry operations, we add a circumflex to the symbol for the corresponding symmetry element. Thus Cn is a rotation by lit/n radians. Note that since = o, a plane of symmetry is equivalent to an S, axis. It is easy to see that a 180° rotation about an axis followed by reflection in a plane perpendicular to the axis is equivalent to inversion hence S2 = i, and a center of symmetry is equivalent to an S2 axis. 

The two things, symmetry elements and symmetry operations, are inextricably related and therefore are easily confused by the beginner. They are, however, different kinds of things, and it is important to grasp and retain, from the outset, a clear understanding of the difference between them. 

We have shown that complete sets of symmetry operations do constitute groups. Now we shall systematically consider what kinds of groups will be obtained from various possible collections of symmetry operations. 

A practical consequence of collecting all operations in the same class when writing down the complete set, for example, at the head of a character table, is that the notation used is a little different from what we have beep using thus far. This new and final form of notation will now be explained and illustrated for the four kinds of symmetry operations. 

This list is reproduced exactly as it appears in the International Tables. It tells us all the different kinds of locations that exist within one unit cell. In each instance we are given the multiplicity of the type of point, namely, how many of them there are that are equivalent and obtainable from each other by application of symmetry operations. There is also an italic letter, called the Wyckoff letter. This is simply an arbitrary code letter that some crystallographers sometimes find useful these letters need not concern us further. Next there is the symbol for the point symmetry that prevails at the site. Finally, there is a list of the fractional coordinates for each point in the set. 

The space group G of a crystal is the set of all symmetry operators that leave the appearance of the crystal pattern unchanged from what it was before the operation. The most general kind of space-group operator (called a Seitz operator) consists of a point operator R (that is, a proper or improper rotation that leaves at least one point invariant) followed by a translation v. For historical reasons the Seitz operator is usually written R v. However, we shall write it as (R ) to simplify the notation for sets of space-group operators. When a space-group operator acts on a position vector r, the vector is transformed into... 

There is more LC extended over two n.n. double-bonds within the (+) electron-hole symmetry class - and, of course, there is much more LC more extended - but we believe the two selected ones are the more important [37]. For instance, we saw that the S-LC is in the (-)class of symmetry, however, with the simple product form of the electron-hole symmetry operator, a LC with two S-LC - or more generally with an even number of S-LC - is in the (+) class of symmetry and should be considered here [37], However, this kind ofLC are sufficiently high in energy to be reasonably neglected. 

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