Symmetry, elements

These differences in crystal shape are described in terms of the symmetry characteristics or elements that are present in the crystal. We will now take a look at these symmetry elements. 

consider the block of wood shown below  

Certainly, this is not a very symmetric object. Does it have any symmetry  

The block has a plane of symmetry that is, an imaginary plane that cuts through the block as shown on page 43. The part of the block on the right side of the plane is exactly the same as that on the left side. In fact, we could cut 

Ionic Compounds, By Claude H. Yoder Copyright 2006 John Wiley Sons, Inc. 

To say that something is symmetric, or that it possesses symmetry, usually is to say that an internal motif is repeated in some fashion. In the context of chemistry, tliat motif is a spatial arrangement of atoms. We may classify the nature of an object s symmetry based on die fashion in which its repeated motifs are made manifest. In describing the symmetric positioning of atoms in a molecule, there are only a few different operations that aie relevant for chemical systems, and these operations are referred to as symmetry elements . When a particular symmetry element is present, the molecule is said to possess that symmetry element. The four symmetry elements that may be used to characterize a molecular structure are  

Proper rotation axis. If a molecule can be rotated about some axis so that the positions originally occupied by eveiy atom are subsequently occupied by identical atoms, the molecule is said to possess a proper rotation axis. The axis and the rotation operation perfonned about it are typically represented by the notation C , where n is the order of the rotation. The order is the largest value of n for which it is true that a rotation of In/n radians about the axis reproduces the original structure this is also referred to as a n-fold rotation axis. 

Essentials of Computational Chemistry, 2nd Edition Christopher J. Cramer 

Improper rotation axis. Rotation about an improper axis is analogous to rotation about a proper synunetry axis, except that upon completion of the rotation operation, the molecule is mirror reflected through a symmetry plane perpendicular to the improper rotation axis. These axes and their associated rotation/reflection operations are usually abbreviated X , where n is the order of the axis as defined above for proper rotational axes. Note that an axis is equivalent to a a plane of symmetry, since the initial rotation operation simply returns every atom to its original location. Note also that the presence of an X2 axis (or indeed any X axis of even order n) implies that for every atom at a position (x,y,z) that is not the origin, there will be an identical atom at position (—x,—y,—z) the origin in such a system is called a point of inversion , since one may regard every atom as having an identical 

Point of inversion. The action of a point of inversion is described above in the context of improper rotation axes. Note that planes of symmetry and points of inversion are somewhat redundant symmetry elements, since they are already implicit in improper rotation axes. However, they are somewhat more intuitive as separate phenomena than are S axes, and thus most texts treat them separately. 

Symmetry, in one or other of its aspects, is of interest in the arts, mathematics, and the sciences. The chemist is concerned with the symmetry of electron density distributions in atoms and molecules and hence with the symmetry of the molecules themselves. We shall be interested here in certain purely geometrical aspects of symmetry, namely, the symmetry of finite objects such as polyhedra and of repeating patterns. Inasmuch as these objects and patterns represent the arrangements of atoms in molecules or crystals they are an expression of the symmetries of the valence electron distributions of the component atoms. In the restricted sense in which we shall use the term, symmetry is concerned with the relations between the various parts of a body. If there is a particular relation between its parts the object is said to possess certain elements of symmetry. 

The simplest symmetry elements are the centre, plane, and axes of symmetry. A cube, for example, is symmetrical about its body-centre, that is, every point (xyz) on its surface is matched by a point (xyz). It is said to possess a centre of symmetry or to be centrosymmetrical a tetrahedron does not possess this type of symmetry. Reflection of one-half of an object across a plane of symmetry (regarded as a mirror, hence the alternative name mirror plane) reproduces the other half. It can easily be checked that a cube has no fewer than nine planes of symmetry. The presence of an -fold axis of symmetry implies that the appearance of an object is the same after rotation through 3607 l a cube has six 2-fold, four 3-fold, and three 4-fold axes of symmetry. We postpone further discussion of the symmetry of finite solid bodies because we shall adopt a more general approach to the symmetry of repeating patterns which will eventually bring us back to a consideration of the symmetry of finite groups of points. 

Rotation of the H around either of the lines shown gives the letter in the orientation shown. The third line is perpendicular to the page and passes through the center of the cross bar. 

If we consider the H20 molecule, which has the arrangement of atoms (electrons are not localized and therefore are not considered when determining symmetry) shown as 

The structure of the C1F3 molecule is based on there being 10 electrons around the central atom (seven valence electrons from Cl, one from each F atom). As we have seen earlier, unshared pairs of electrons are found in equatorial positions, so the structure can be shown as 

The formaldehyde molecule, ff2CO, has a structure that is shown as 

The subscripts on the hydrogen atoms are to identify their positions. Clockwise rotation by 120° around the C3 axis results in the molecule having the orientation 

1 FIXED POINT ROTATION AXIS. MIRROR PLANE 

A point which is transformed into itself by an affine transformation is called a fixed point, x = Rx +1 = x = Ex hence (R — E)x = — t. We distinguish four cases  

The matrix (R — E) has two eigenvalues equal to zero and one non-zero eigenvalue. This is the case for a reflection by a plane, Si = l2, which is represented by matrices like  

The matrix (R — E) has three eigenvalues equal to zero, thus R = E. The operation is a pure translation and clearly has no fixed point or preferred origin. 

The ensemble of fixed points (points, lines or planed) of a symmetry operation are called symmetry elements. To the fixed point of l or S , we must add the fixed line corresponding to the operation A or the plane which is perpendicular to it. Rotation axes correspond to the operations A , centers and rotoinversion axes to the operations l , and mirror planes and rotoreflection axes to the 

Reflection plane (mirror) n-fold rotation axis 

Reflection in a plane followed by a translation according to a vector parallel to the plane. 

Translation in the l/2(a + b) or face diagonal direction n. Translation in the l/2(a + b + c) or volume diagonal direction d. Vertical -fold axis, followed by a translation parallel to the axis Point group vnth an n-fold axis of rotary reflection. 

Tetrahedron Equilateral triangles a V2/12 a 3 faces 4 edges 6 vertices 4 

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Connecting the energy-ordered orbitals of reactants to those ofproducts according to symmetry elements that are preserved throughout the reaction produces an orbital correlation diagram. 

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. 

Symmetry mode Symmetry element E>egree of freedom Molecule Number of C—H M braiions Number of skeleton vibrations Activity of vibrations ... 

Asymmetric (Section 7 1) Lacking all significant symmetry elements an asymmetric object does not have a plane axis or center of symmetry... 

Many transition states of chemical reactions contain symmetry elements not present in the reactants and products. For example, in the umbrella inversion of ammonia, a plane of symmetry exists only in the transition state. 

Corresponding to every symmetry element is a symmetry operation which is given the same symbol as the element. For example, C also indicates the actual operation of rotation of the molecule by 2n/n radians about the axis. 

From the definition of, it follows that (7 = 51 i = 0082, since a and i are taken as separate symmetry elements the symbols 5i and 82 are never used. 

A molecule is chiral if it does not have any S symmetry element with any value of n. 

In Section 4.2.1 it will be pointed out that hydrogen peroxide (Figure 4.1 la) has only one symmetry element, a C2 axis, and is therefore a chiral molecule although the enantiomers have never been separated. The complex ion [Co(ethylenediamine)3], discussed in Section 4.2.4 and shown in Figure 4.11(f), is also chiral, having only a C3 axis and three C2 axes. 

The main symmetry elements in SFg can be shown, as in Figure 4.12(b), by considering the sulphur atom at the centre of a cube and a fluorine atom at the centre of each face. The three C4 axes are the three F-S-F directions, the four C3 axes are the body diagonals of the cube, the six C2 axes join the mid-points of diagonally opposite edges, the three df, planes are each halfway between opposite faces, and the six d planes join diagonally opposite edges of the cube. 

Figure 4.12 (a) Some C2 and elements in methane, (b) Some of the symmetry elements in... 

Molecules belonging to the 4 point group are very highly symmetrical, having 15 C2 axes, 10 C3 axes, 6 C5 axes, 15 n planes, 10 axes, 6 5io axes and a centre of inversion i. In addition to these symmetry elements are other elements which can be generated from them. 

In order to obtain the direct product of two species we multiply the characters under each symmetry element using the mles... 

Question. List the symmetry elements of each of the following molecules (a) 1,2,3-trifluorobenzene, (b) 1,2,4-trifluorobenzene, (c) 1,3,5-trifluorobenzene, (d) 1,2,4,5-tetrafluoro-benzene, (e) hexafluorobenzene, (f) l,4-dibromo-2,5-difluorobenzene. 

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

List all the symmetry elements of the following molecules, assign each to a point group, and state whether they form enantiomers (a) lactic acid, (b) trans-[Co(ethylenediamine)2Cl2], (c) c -[Co(ethylenediamine)2Cl2], (d) cyclopropane,... 

On no symmetry element On afxz) On On all symmetry elements Trans. Rot. vibrations... 

Condition 3 The AOs must have the same symmetry properties with respect to certain symmetry elements of the molecule. 

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