4 point group

A Cn point group contains a Cn axis of symmetry. By implication it does not contain a, i or Sn elements. However, it must contain C, C, . C 1. 

1 Point Groups The 32 point groups are built upon four symmetry operators. These operators, which are also called four elements, are as follows  

Rotation axis symbols 1 (identity), 2 (twofold rotation), 3 (threefold rotation), up to 6 (sixfold rotation). 

Rotation-inversion axes (combination of rotation and inversion) symbols 1 (a simple center of conversion), 2 (a 180° rotation followed by an inversion, 2 = m), 3 (=3-1), 6(=3-m). 

Using the general symbol X to denote a principal axis of any degree, we may have the following combinations  

Xm Rotation axis with a vertical plane of symmetry Im, 2m, 3m, 4m, 6m 

Generally, there is infinite number of point groups, but not all of them correspond to real physical objects such as molecules or crystals. For example, only 32 point groups are compatible with crystal lattices. Each of them is labeled by a certain symbol according to Schonflies or according to the International classifications. The Schonflies symbols are vivid and more often used in scientific literature. Here we present only those point groups we may encounter in the literature on liquid crystals. 

Since operation ct is equal to its own inverse, it is convenient to use it as X and analyze whether operations C . form a class or not. For example, consider NH3 molecule (group C3v) whose projection along the C3 axis is shown in Fig. 2.6. Let us take point P and, at first, make twice C3 operation to arrive at point P. Then, we start again fi om point P and, guided by arrows in the figure, make operation aC3CT We again arrive at point P. Therefore = C, and operations 

The set of symmetry elements and operations that characterize the symmetry of an individual molecule defines its point group. If only one rotational symmetry operation (besides E) is possible then the point group bears the same name (C2, S3, etc.) Otherwise, if there is just one symmetry element, the point group is called Cs if there is a mirror plane, a, and Q if there is an inversion center, i. Finally, if no other symmetry elements are present then the point group is Ci. 

The symmetry elements of the point groups (a) 3, (b) D2d and (c) Dgh- (a) A twisted form of ethane, neither perfectly staggered nor eclipsed, viewed along the C3 axis, (b) Allene, where the planes xz and yz are planes and the C 2 axes lie at 45° to the x and y axes in the xy plane, (c) Benzene. Note that the Cg axis includes C3, C2, Sg and S3 axes. Also shown are the three axes (one of which lies along x) and the three C axes (one of which lies along y). The xy plane is CTh and there are vertical mirror planes including each C 2 and C axis the center of the molecule is an inversion center, i. 

The degree of symmetry that the point group represents is given by the order, h, whieh is simply the sum of the number of symmetry elements that the point group possesses. For Cav /i = 4 for h = 7A and for Oh / = 48. The highest (non-infinite) symmetry group encountered in chemistry is the icosahedron, 4 (order h = 120), which describes a polyhedron sometimes encountered in cluster chemistry, e.g. and 

To describe the symmetry of a molecule in terms of one symmetry element (e.g. a rotation axis) provides information only about this property. Each of BF3 and NH3 possesses a 3-fold axis of symmetry, but their structures and overall symmetries are different BF3 is trigonal planar and NH3 is trigonal pyramidal. On the other hand, if we describe the symmetries of these molecules in terms of their respective point groups (1)311 3 ), we are providing information 

Before we look at some representative point groups, we emphasize that it is not essential to memorize the symmetry 

Molecules that appear to have no s)rmmetry at all, e.g. 4.9, must possess the symmetry element E and effectively possess at least one C axis of rotation. They therefore belong to the C] point group, although since C = E, the rotational s5Tnmetry operation is ignored when we list the symmetry elements of this point group. 

Point group Characteristic symmetry elements Comments 

The application of a systematic approach to the assignment of a point group is essential, otherwise there is the risk that 

Coo signifies the presence of an oo-fold axis of rotation, i.e. that possessed by a linear molecule (Fig. 3.7) for the molecular species to belong to the Cqov point group, it must also possess an infinite number of planes but no (Til plane or inversion centre. These criteria are met by asymmetrical diatomics such as HF, CO and [CN] (Fig. 3.7a), and linear polyatomics (throughout this book, polyatomic is used to mean a species containing three or more atoms) that do not possess a centre of symmetry, e.g. OCS and HCN. 

Symmetrical diatomics (e.g. H2, [02] ) and linear polyatomics that contain a centre of symmetry (e.g. [N3], CO2, HC=CH) possess a aj, plane in addition to a 

Examples of inversion axes. If they are considered to be rotoreflection axes, they have the multiplicities expressed by the Schoenflies symbols SN 

If N is an even number, the inversion axis automatically contains a rotation axis with half the multiplicity. If N is an odd number, automatically an inversion center is present. This is expressed by the graphical symbols. If N is even but not divisible by 4, automatically a reflection plane perpendicular to the axis is present. 

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. 

Screw rotation. The symmetry element is a screw axis. It can only occur if there is translational symmetry in the direction of the axis. The screw rotation results when a rotation of 360/1V degrees is coupled with a displacement parallel to the axis. The Hermann-Mauguin symbol is NM ( N sub M )-,N expresses the rotational component and the fraction M/N is the displacement component as a fraction of the translation vector. Some screw axes are right or left-handed. Screw axes that can occur in crystals are shown in Fig. 3.4. Single polymer molecules can also have non-crystallographic screw axes, e.g. 103 in polymeric sulfur. 

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 

A C poini group conlains a C axis of symmelry. By implication if does nol conlain a, i or elemenis. However, if musi conlain C, C,  

An S point group contains an S axis of symmetry. The group must contain also 

A C point group contains a C axis of symmetry and n a planes of symmetry, all of which contain the C axis. It also contains other elements which may be generated from these. 

For all remaining molecules, find the rotation axis with the highest n, the highest order C axis for the molecule. 

Does the molecule have a mirror plane (o/,) perpendicular to the C axis If so, it is classified as C h or D /j. If not, continue with Step 5. 

FIGURE 4-7 Diagram of the Point Group Assignment Method. 

Similar statements can be written to show the combined effects of successive operations. For example, in planar BCI3, the 5 3 improper axis of rotation corresponds to rotation about the C3 axis followed by reflection through the crj, plane. This can be written in the form of equation 3.4. 

[PtCLil is square planar to what rotational operation is C4 equivalent  

Draw a diagram to illustrate what the notation means with respect to rotational operations in benzene. 

To describe the symmetry of a molecule in terms of one symmetry element (e.g. a rotation axis) provides information only about this property. Each of BF3 and NH3 possesses a 

A group G =. .., t/i. is a set of elements related by an operation which we will call group multiply for convenience and which has the following properties  

The product of any two elements is in the set that is, the set is closed under group multiplication. 

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  

Reflections in planes containing the origin (center of mass)  

Improper rotations—a rotation about an axis through the origin followed by a reflection in a plane containing the origin and perpendicular to the axis of rotation  

Determine whether the molecule exhibits very low symmetry (Ci, Cj, C,) or high symmetry Oh, Ccov or If) as described in Tables 4.2 and 4.3. 

Does the molecule have any C2 axes perpendicular to the principal C axis If it does, there will be of such C2 axes, and the molecule is in the D set of groups. If not, it is in the C or 5 set. 

---

In general, a point group synnnetry operation is defined as a rotation or reflection of a macroscopic object such that, after the operation has been carried out, the object looks the same as it did originally. The macroscopic objects we consider here are models of molecules in their equilibrium configuration we could also consider idealized objects such as cubes, pyramids, spheres, cones, tetraliedra etc. in order to define the various possible point groups. 

As an example, we again consider the PH molecule. In its pyramidal equilibrium configuration PH has all tlnee P-H distances equal and all tlnee bond angles Z(HPH) equal. This object has the point group synnnetry where the operations of the group are... 

The complete multiplication table of the point group, worked out using arguments similar to those leading to (equation A1.4.23) and (equation A1.4.24), is given in table A1.4.2. It is left as an exercise for the reader to use this table to show that the elements of satisfy the group axioms given in section Al.4.2.1. 

Table Al.4.2 The multiplication table of the point group using the space-fixed axis convention (see text).

If we were to define the operations of the point group as also rotating and reflecting the (p.q.r) axis system (in which case the axes would be tied to the positions of the nuclei), we would obtain a different multiplication table. We could call this the nuclear-fixed axis convention. To implement this the protons in the o, O2 and planes in figure Al.4.2 would be numbered H, H2 and respectively. With this convention the operation would move the a plane to the position in space originally occupied by the 02 plane. If we follow such a C3 operation by the reflection (in the plane containing Ft ) we find that, in the nuclear-fixed axis convention ... 

The point groups T, and /j. consist of all rotation, reflection and rotation-reflection synnnetry operations of a regular tetrahedron, cube and icosahedron, respectively. 

Point groups are discussed briefly in sections 4.3 and 4.4 of [1] and very extensively in chapter 3 of Cotton... 

As an example we consider the group introduced in (equation Al.4,19) and the point group given in (equation Al.4.22). Inspection shows that the multiplication table of in table Al.4,2 can be obtained from the multiplication table of the group (table Al.4,1) by the following mapping ... 

Nuclear pemuitations in the N-convention (which convention we always use for nuclear pemuitations) and rotation operations relative to a nuclear-fixed or molecule-fixed reference frame, are defined to transfomi wavefunctions according to (equation Al.4.56). These synnnetry operations involve a moving reference frame. Nuclear pemuitations in the S-convention, point group operations in the space-fixed axis convention (which is the convention that is always used for point group operations see section Al.4.2,2 and rotation operations relative to a space-fixed frame are defined to transfomi wavefiinctions according to (equation Al.4.57). These operations involve a fixed reference frame. 

Rigid linear molecules are a special case in which an extended MS group, rather than the MS group, is isomorphic to the point group of the equilibrium structure see chapter 17 of [1]. 

A detailed discussion of the relation between MS group operations and point group operations is given in section... 

We now turn to electronic selection rules for syimnetrical nonlinear molecules. The procedure here is to examme the structure of a molecule to detennine what synnnetry operations exist which will leave the molecular framework in an equivalent configuration. Then one looks at the various possible point groups to see what group would consist of those particular operations. The character table for that group will then pennit one to classify electronic states by symmetry and to work out the selection rules. Character tables for all relevant groups can be found in many books on spectroscopy or group theory. Ftere we will only pick one very sunple point group called 2 and look at some simple examples to illustrate the method. 

Each such nonual mode can be assigned a synuuetry in the point group of the molecule. The wavefrmctions for non-degenerate modes have the following simple synuuetry properties the wavefrmctions with an odd vibrational quantum number v. have the same synuuetry as their nonual mode 2the ones with an even v. are totally symmetric. The synuuetry of the total vibrational wavefrmction (Q) is tlien the direct product of the synuuetries of its constituent nonual coordinate frmctions (p, (2,). In particular, the lowest vibrational state. 

In addition to the most abundant fullerene, namely [60]fullerene, a number of higher fullerenes have also been isolated and characterized, including [70] (point group D . ), chiral [76] (point group Dj), the D and Cj isomers... 

Shlang J J ef a/1996 Symmetry of annealed wurtzite CdSe nanocrystals assignment to the C3v point group J. Phys. Chem. 100 13 886... 

This is the central Jahn-Teller [4,5] result. Three important riders should be noted. First, Fg = 0 for spin-degenerate systems, because F, x F = Fo. This is a particular example of the fact that Kramer s degeneracies, aiising from spin alone can only be broken by magnetic fields, in the presence of which H and T no longer commute. Second, a detailed study of the molecular point groups reveals that all degenerate nonlinear polyatomics, except those with Kramer s... 

The Couplitig-Coefficierits lJ ABC abc) for the Complex Form of a Doubly Degenerate Representation in the Octahedral Group, Following G. F. Koster et al.. Properties of tke Thirt i-Two Point Groups, MIT Press, MA, 1963, pp, 8, 52. 

If the symmetries of the two adiabatic functions are different at Rq, then only a nuclear coordinate of appropriate symmeti can couple the PES, according to the point group of the nuclear configuration. Thus if Q are, for example, normal coordinates, xt will only span the space of the totally symmetric nuclear coordinates, while X2 will have nonzero elements only for modes of the correct symmetry. 

---