Symmetry operators notation

A rotoreflection is a coupled symmetry operation of a rotation and a reflection at a plane perpendicular to the axis. Rotoreflection axes are identical with inversion axes, but the multiplicities do not coincide if they are not divisible by 4 (Fig. 3.3). In the Hermann-Mauguin notation only inversion axes are used, and in the Schoenflies notation only rotoreflection axes are used, the symbol for the latter being SN. 

A symmetry operation can have both rotational and translational components, and is described in the Seitz notation as R s. The terms R and s are the rotational and translational parts of the 3d symmetry element, respectively, such that... 

Table A.l shows the symmetry operations of the point groups more commonly used in the book. The notation for the symmetry operations is the following ...

There are seven elements of symmetry which are commonly possessed by molecular systems. These elements of symmetry, their notations and their related symmetry operations are given in Table 2.1. 

If two symmetry operations combine together to give the identical operation E, e.g. eqn (2-4.2), then they are said to be the inverse of each other and the inverse of an operation P is written as P l (we have, in fact, already been unknowingly using this notation for rotational operations, cf Ct and C 1), the general situation is written as ... 

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. 

Symmetry Notation.—A state is described in terms of the behavior of the electronic wave function under the symmetry operations of the point group to which the molecule belongs. The characters of the one-electron orbitals are determined by inspection of the character table the product of the characters of the singly occupied orbitals gives the character of the molecular wave function. A superscript is added on the left side of the principal symbol to show the multiplicity of the state. Where appropriate, the subscript letters g (gerade) and u (ungerade) are added to the symbol to show whether or not the molecular wave function is symmetric with respect to inversion through a center of symmetry. 

A symmetry element (which is not to be confused with a group element) is a point, line, or plane with respect to which a point symmetry operation is carried out. The symmetry elements, the notation used for them, the corresponding operation, and the notation used for the symmetry operators are summarized in Table 2.1. It is not necessary to use both n and n since all configurations generated by h can be produced by n. ... 

Symmetry element Notation for symmetry element Symmetry operation Symmetry operator... 

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... 

Table 1-4 lists the point symmetry elements and the corresponding symmetry operations. The notation used by spectroscopists and chemists, and used here, is the so-called Schoenflies system, which deals only with point groups. Crystallographers generally use the Hermann-Mauguin system, which applies to both point and space groups. 

Now, any type of motion or symmetry operation which leaves a lattice invariant may be written in matrix notation. For example, if a lattice point is moved from point Q in Euclidean space with coordinates (xj, yi, Zi) to point P with coordinates (x2, ya, Z2), this can be written as ... 

Further, symmetry elements are defined, these are the geometrical loci of all points which remain invariant when a symmetry operation is carried out. The names of the symmetry elements introduced by Schoenflies (1891) are given below, followed by the international notation, introduced by Hermann (1928) and Mauguin (1931) ... 

Figure 2.7-6 A Assignment of the Cartesian coordinate axes and the symmetry operations of a planar molecule of point group C2,.. B Character table, 1 symbol of the point group after Schoen-flies 2 international notation of the point group 3 symmetry species (irreducible representations) 4 symmetry operations 5 characters of the symmetry operations in the symmetry species +1 means symmetric, -1 antisymmetric 6 x, y, z assignment of the normal coordinates of the translations, direction of the change of the dipole moment by the infrared active vibrations, R, Ry, and R stand for rotations about the axes specified in the subscript 7 x, xy,. .. assign the Raman active species by the change of the components of the tensor of polarizability, aw, (Xxy,. ...

Table B.4. Double group characters table for the Ta point group. The numbers before the symmetry operations correspond to the number of geometrically different axes or symmetry planes. Some of the operations of the double group belong to the same class as those of the original group. When more than one IR is indicated, the first one corresponds to the notation of Mulliken [11], the second one to Koster et al. [9] and the one in parentheses to [3]...

A word about notation. For the points groups we shall generally use the Schonflies notation. In this notation basically the same symbols are used for the point groups as for the symmetry operations. Whenever there is any doubt as to the meaning we shall use a caret (example Cn) to indicate a symmetry operation. 

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