Symmetry operation geometrical

Symmetry operation Geometrical representation Symmetry element... 

These elements of symmetry are best recognized by performing various symmetry operations, which are geometrically defined ways of exchanging equivalent parts of a molecule. The 5 symmetry operations are ... 

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... 

Although all molecules are in constant thermal motion, when all of their atoms are at their equilibrium positions, a specific geometrical structure can usually be assigned to a given molecule. In this sense these molecules are said to be rigid. The first step in the analysis of the structure of a molecule is the determination of the group of operations that characterizes its symmetry. Each symmetry operation (aside from the trivial one, E) is associated with an element of symmetry. Thus for example, certain molecules are said to be planar. Well known examples are water, boron trifluoride and benzene, whose structures can be drawn on paper in the forms shown in Fig. 1. 

Potential fluid dynamics, molecular systems, modulus-phase formalism, quantum mechanics and, 265—266 Pragmatic models, Renner-Teller effect, triatomic molecules, 618-621 Probability densities, permutational symmetry, dynamic Jahn-Teller and geometric phase effects, 705-711 Projection operators, geometric phase theory, eigenvector evolution, 16-17 Projective Hilbert space, Berry s phase, 209-210... 

The special class of transformation, known as symmetry (or unitary) transformation, preserves the shape of geometrical objects, and in particular the norm (length) of individual vectors. For this class of transformation the symmetry operation becomes equivalent to a transformation of the coordinate system. Rotation, translation, reflection and inversion are obvious examples of such transformations. If the discussion is restricted to real vector space the transformations are called orthogonal. 

The hfs (or quadrupole) tensors of geometrically (chemically) equivalent nuclei can be transformed into each other by symmetry operations of the point group of the paramagnetic metal complex. For an arbitrary orientation of B0 these nuclei may be considered as nonequivalent and the ENDOR spectra are described by the simple expressions in (B 4). If B0 is oriented in such a way that the corresponding symmetry group of the spin Hamiltonian is not the trivial one (Q symmetry), symmetry adapted base functions have to be used in the second order treatment for an accurate description of ENDOR spectra. We discuss the C2v and D4h covering symmetry in more detail. 

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. 

Each of the symmetry operations we have defined geometrically can be represented by a matrix. The elements of the matrices depend on the choice of coordinate system. Consider a water molecule and a coordinate system so oriented that the three atoms lie in the x-z plane, with the z—axis passing through the oxygen atom and bisecting the H-O-H angle, as shown in Figure 5.1. 

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. 

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 theory of optical activity would be understood in terms of symmetry considerations at the first stage. The elements of symmetry are the geometric elements in relation to which the symmetry operations are carried out, and are classified in the following ... 

A symmetry element is a geometrical entity such as a line, a plane, or a point, with respect to which one or more symmetry operations may be carried out. 

What we have just done is to substitute the algebraic process of multiplying matrices for the geometric process of successively applying symmetry operations. The matrices multiply together in the same pattern as do the symmetry operations it is clear that they must, since they were constructed to do just that. It will be seen in the next section that this sort of relationship between a set of matrices and a group of symmetry operations has great importance and utility. 

The sum of the diagonal elements of the MR r( ) of the symmetry operator R(o z) forms a geometric series which we have summed before in Section 7.2 with j replaced by /, an integer. As before,... 

A symmetry operation is an atom-exchange operation (or more precisely, a coordinate transformation) performed on a molecule such that, after the interchange, the equivalent molecular configuration is attained in other words, the shape and orientation of the molecule are not altered, although the position of some or all of the atoms may be moved to their equivalent sites. On the other hand, a symmetry element is a geometrical entity such as a point, an axis, or a plane, with respect to which the symmetry operations can be carried out. We shall now discuss symmetry elements and symmetry operations of each type in more detail. 

Matter is composed of spherical-like atoms. No two atomic cores—the nuclei plus inner shell electrons—can occupy the same volume of space, and it is impossible for spheres to fill all space completely. Consequently, spherical atoms coalesce into a solid with void spaces called interstices. A mathematical construct known as a space lattice may be envisioned, which is comprised of equidistant lattice points representing the geometric centers of structural motifs. The lattice points are equidistant since a lattice possesses translational invariance. A motif may be a single atom, a collection of atoms, an entire molecule, some fraction of a molecule, or an assembly of molecules. The motif is also referred to as the basis or, sometimes, the asymmetric unit, since it has no symmetry of its own. For example, in rock salt a sodium and chloride ion pair constitutes the asymmetric unit. This ion pair is repeated systematically, using point symmetry and translational symmetry operations, to form the space lattice of the crystal. 

Terms such as symmetrical , dissymmetric , asymmetric , are frequently encountered in descriptions of molecular structures. At the intuitive level of comprehension, there appears to exist some form of relationship between the degree of order and of symmetry displayed by a molecule, namely that the more ordered molecular structures are the more symmetrical. At the mathematical level, symmetry elements and symmetry operations have been devised which allow to describe rigorously a number of geometrical properties displayed by molecular entities, or for that matter by any object. A short description of symmetry as a mathematical tool will be given in this section, and the interested reader is referred to a number of valuable monographs [8-14] for more extensive treatments. 

The application of a symmetry element is a symmetry operation and the symmetry elements are the symmetry operators. The consequence of a symmetry operation is a symmetry transformation. Strict definitions refer to geometrical symmetry, and will serve us as guidelines only. They will be followed qualitatively in our discussion of primarily non-geometric symmetries, according to the ideas of the Introduction. 

The simpliest and most important molecule with a low barrier to inversion is ammonia, NH3. In its ground electronic state, NH3 has a pyramidal equilibrium configuration with the geometrical symmetry described by the point group C3V (Fig. 1). Configuration B which is obtained from A by the symmetry operation E is separated from A by an inversion barrier of about 2000 cm . A large amplitude... 

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