Symmetry operation, definition

An element of an electrostatic moment tensor can only be nonzero if the distribution has a component of the same symmetry as the corresponding operator. In other words, the integrand in Eq. (7.1) must have a component that is invariant under the symmetry operations of the distribution, namely, it is totally symmetric with respect to the operations of the point group of the distribution. As an example, for the x component of the dipole moment to be nonzero, p(r)x must have a totally symmetric component, which will be the case if p(r) has a component with the symmetry of x. The symmetry restrictions of the spherical electrostatic moments are those of the spherical harmonics given in appendix section D.4. Restrictions for the other definitions follow directly from those listed in this appendix. 

In Fig. 2-3.1 we illustrate these definitions for a square-based pyramid by labelling the four comers of the base. This labelling is merely to enable us to see that an operation has taken place and it has no physical significance the whole point of the symmetry operation is that the final orientation is indistinguishable from the original one. Of the operations shown in Fig. 2-3.1, only three, excluding E, are distinct Cit Cti and Cj. It is conventional when choosing the symbol for a rotational operation to do so in such a way that is as small as possible, e.g. Ct is used in preference to Cj. Finally, it is apparent that quite often symmetry elements will coincide and in such cases we will link the symmetry elements e.g. the C Cti and C axes in Fig. 2-3.1 will be written as CV-Cj-Cj. 

Our definition of 0M applied to functions of the coordinates xx, x and x% of a point in physical space, but it can be generalized to apply to functions of any number of variables, as long as we know how those variables change under the symmetry operations. For example, if we let X stand for a complete specification of the coordinates of all the electrons (or all the nuclei) of some molecule, i.e. 

We conclude this section by considering the operator 0R that corresponds to the symmetry operation R. From (1.271), the definition of 0R is... 

Let the functions F ...,FW form a basis for a representation of some point group. Since a symmetry operation R amounts to a rotation (and possibly a reflection) of coordinates, it cannot change the value of a definite integral over all space we have... 

A group is a collection of elements that are interrelated according to certain rules. We need not specify what the elements are or attribute any physical significance to them in order to discuss the group which they constitute. In this book, of course, we shall be concerned with the groups formed by the sets of symmetry operations that may be carried out on molecules or crystals, but the basic definitions and theorems of group theory are far more general. 

The reason for this definition can be seen in Fig. 1.2, where we see that the effect of Eq. 1.18 is to rotate the contours of / into those of Og f- the value of the function is preserved by the symmetry operation. 

Bent AH2 Molecules.—A bent AH molecule belongs to the symmetry class C2r. The definitions of the symbols appropriate to the non-localized orbitals of such a molecule are given below. The z axis bisects the HAH angle and lies in the molecular plane. The y axis also lies in the molecular plane and is parallel with the H H line. C2(z) means a rotation by 180° about the z axis. wave function does not or does change sign when one of the symmetry- operations C2(z) or av(y) is carried out. 

The parameter Z is used to denote the number of molecules or formula units in the asymmetric unit of a crystal structure, i.e. the number of molecules that cannot be related to one another by the symmetry operations defined by the crystal space group. Of course, Z is therefore crucially dependent on the somewhat subjective definition of what constitutes the formula unit . Strictly Z is defined as the number of formula units in the unit cell divided by the number of independent general positions. The... 

To determine the symmetry of a molecule, we first need to identify the symmetry elements it may possess and the symmetry operations generated by these elements. The twin concepts of symmetry operation and symmetry element are intricately connected and it is easy to confuse one with the other. In the following discussion, we first give definitions and then use examples to illustrate their distinction. 

The immediate question is are selected nuclei in a molecule chemical shift equivalent, or are they not If they are, they are placed in the same set. The answer can be framed as succinctly as the question Nuclei are chemical shift equivalent if they are interchangeable through any symmetry operation or by a rapid process. This broad definition assumes an achiral environment (solvent or reagent) in the NMR experiment the common solvents are achiral (Section 3.17). 

The definition of chemical shift equivalence given for protons also applies to carbon atoms interchangeability by a symmetry operation or by a rapid mechanism. The presence of equivalent carbon atoms (or coincidence of shift) in a molecule results in a discrepancy... 

In crystallography one is accustomed to the idea that a structure either has a particular symmetry element or it does not. The membership is thus either 1 or zero. In morphological analysis the symmetry has a value of zero through 1 depending upon how closely the profile approaches the symmetry being considered. The definitions of the symmetry operations are shown below ... 

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. 

For a symmetrical atomic or molecular system, these considerations place a severe restriction on the possible eigenfunctions of the system. All possible eigenfunctions must form bases for some irreducible representation of the group of symmetry operations. The form of the possible eigenfunctions is also determined to a large extent since they must transform in a quite definite way under the operations of the group. 

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