Elements of point group

A molecular point group is a set of symmetry elements. Each symmetry element describes an operation which when carried out on the molecular skeleton leaves the molecular skeleton unchanged. Elements of point groups may represent any of the following operations ... 

It is dear that the elements of point groups do not necessarily commute, that is the order in which one combines two symmetry operations can be important (see, for example, Fig. 2-4.1 and Table 3-4.1 where for the symmetric tripod Ctar j o C ). A group for which all the elements do commute is called an Abelian group. 

Table 14.2 shows that all three elements have remarkably low melting points and boiling points—an indication of the weak metallic bonding, especially notable in mercury. The low heat of atomisation of the latter element compensates to some extent its higher ionisation energies, so that, in practice, all the elements of this group can form cations in aqueous solution or in hydrated salts anhydrous mercuryfll) compounds are generally covalent. 

It is assumed that the reader has previously learned, in undergraduate inorganie or physieal ehemistry elasses, how symmetry arises in moleeular shapes and struetures and what symmetry elements are (e.g., planes, axes of rotation, eenters of inversion, ete.). For the reader who feels, after reading this appendix, that additional baekground is needed, the texts by Cotton and EWK, as well as most physieal ehemistry texts ean be eonsulted. We review and teaeh here only that material that is of direet applieation to symmetry analysis of moleeular orbitals and vibrations and rotations of moleeules. We use a speeifie example, the ammonia moleeule, to introduee and illustrate the important aspeets of point group symmetry. 

The Of, point group contains three C4 axes, four C3 axes, six C2 axes, three df, planes, six planes and a centre of inversion i. It also contains elements generated from these. 

List the symmetry elements and point groups of these molecules in both electronic states. 

It is instructive to add to these examples from the numerous instances of point group symmetry mentioned throughout the text. In this way a facility will gradually be acquired in discerning the various elements of symmetry present in a molecule. 

During the study of inorganic chemistry, the structures for a large number of molecules and ions will be encountered. Try to visualize the structures and think of them in terms of their symmetry. In that way, when you see that Pt2+ is found in the complex PtCl42 in an environment described as D4h, you will know immediately what the structure of the complex is. This "shorthand" nomenclature is used to convey precise structural information in an efficient manner. Table 5.1 shows many common structural types for molecules along with the symmetry elements and point groups of those structures. 

Consider a group whose elements can be put in one-to-one correspondence with the points of a subset of an n-dimensional real inner-product space Sn, as parameter space. Let x and x2 be any two elements of a group G with images P(x ) and P(x 2) in Sn. If it is possible to connect P x ) and P x2) by one or more paths lying entirely within the parameter space, the parameter space is said to be connected otherwise it is disconnected. 

The high melting points of the compounds of the noble metals may be noticed. With elements of the groups from Ag to Te, alkali metal compounds are formed... 

Table 1. Several examples of point groups and symmetry elements...

The elements of a group which are conjugate to each other are said to form a doss and the number of elements in the tth class is given the symbol g. For the symmetric tripod point group the elements fall into three classes E o, a" C C 1 with gx = 1, g% = 3 and g% — 2,... 

The trace of a matrix which represents an element of a group (or an operation of a point group) is called a character and is usually given the symbol X- X(R) is thus the character of the operation R in the representation which has matrices D(R)> i.e. 

System Crystallographic elements Essential symmetry Number of point groups... 

Draw the formulae of all the possible isomeric butenes and determine their symmetry elements and point groups. Use the flow chart in the appendix to assist you. 

How many isomers are there of diamminedichloridoplatinum(II), [PtCl2NH3)2] Determine the symmetry elements and point groups for all isomers and assign appropriate stereodescriptors. 

Draw formulae for all the isomers of dichloro cyclopropane and predict the relative signal intensities in their H NMR spectra. Compare your answer with results obtained from a determination of the symmetry elements and point groups. 

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