Proper point group

Croups with Ih, Oh, Tt, Clt C and C, symmetries were assigned in the text by inspection Take the molecules given as illustrations of these symmetries (Figs. 3.10 and 3.11) and run them through the flowchart (Fig. 3.16) to assign their proper point groups. 

We first describe the proper point groups, P, that is the point groups that contain the identity and proper rotations only. 

This completes the list of proper point groups, P. A summary is given in the first column of Table 2.6. All the remaining axial point groups may be generated from the proper point groups P by one or other of two methods. 

Table 2.6. Derivation of commonly used finite point groups from proper point groups. 

In each column, the symbol for the point group is given in International notation on the left and in Schonflies notation on the right. When n = 2, the International symbol for D2h is mmm. When n is odd, the International symbol for C v is nm, and when n is even it is nmm. Note that n = n/2. In addition to these groups, which are either a proper point group P, or formed from P, there are the three cyclic groups 1 or Ci = E, 1 or Q = EI, and morCs = E a. ... 

The second method is applicable to proper point groups P that have an invariant subgroup Q of index 2, so that... 

Exercise 2.3-4 Use the second method to derive the point group P corresponding to the proper point group C4. Show that C4 and P are isomorphous and find the classes of both groups. 

For crystals, the point group must be compatible with translational symmetry, and this requirement limits n to 2,3,4, or 6. (This restriction applies to both proper and improper axes.) Thus the crystallographic point groups are restricted to ten proper point groups and a total of... 

In addition to the proper point groups P and the improper point groups that are either isomorpous with P or equal to P C there is the non-axial group 1 or Cj = E). 

Improper group P Proper point group P isomorphous to P... 

Thus for improper point groups that are formed by the DP of a proper point group with C the character is a class property which is zero for all irregular classes, namely those formed from rotations about proper or improper BB axes. All other improper point groups are isomorphous ( ) with a proper point group and have the same characters and representations as that proper point group. For example C2v D2 D2d C4V D4. 

Electronic Wavefunctions Must Also Possess Proper Symmetry. These Include Angular Momentum and Point Group Symmetries... 

Here /, are the three moments of inertia. The symmetry index a is the order of the rotational subgroup in the molecular point group (i.e. the number of proper symmetry operations), for H2O it is 2, for NH3 it is 3, for benzene it is 12 etc. The rotational partition function requires only information about the atomic masses and positions (eq. (12.14)), i.e. the molecular geometry. 

If you have arrived at this point, you have found no improper axis but more than one proper axis of order n Or, you have identified at least one other axis of order n in addition to that collinear with Sin- Now, look for n binary axes perpendicular to C . If you do not find any, the point group is one of the t -type. Otherwise, there are n binary axes and the group is one of type 0. 

Based on extensive studies of the symmetry in crystals, it is found that crystals possess one or more of the ten basic symmetry elements (five proper rotation axes 1,2,3, 4,6 and five inversion or improper axes, T = centre of inversion i, 2 = mirror plane m, I, and 5). A set of symmetry elements intersecting at a common point within a crystal is called the point group. The 10 basic symmetry elements along with their 22 possible combinations constitute the 32 crystal classes. There are two additional symmetry... 

Figure B.2 Flow chart for point-group assignment. A symmetry plane that is perpendicular to a proper axis of rotation is a <7 , plane, one that includes the unique proper axis of rotation is a cr plane, and one that includes the highest order proper axis of rotation and bisects the remaining two-fold axes of rotation is a a, plane...

One-dimensional irreducible representations are labeled either A or B according to whether the character of a 2irjn (proper or improper) rotation about the symmetry axis of highest order n is +1 or —1, respectively. For the point groups Wl9 and which have no symmetry axis, all one-dimensional representations are labeled A. For... 

As noted earlier, point groups with no threefold or higher proper or improper axis have only one-dimensional representations hence a necessary condition for a molecule to have degenerate vibrational modes is that it possess a Cn or an S axis with n> 3. Asymmetric tops have no degenerate vibrational modes. 

The only proper axis is a C3 axi there are no C2 axes at all. Hence, the point group must be C3, C3l., or C There are three vertical planes, one passing through each hydrogen atom. The group is thus C3l.. 

The most important and frequent use for projection operators is to determine the proper way to combine atomic wave functions on individual atoms in a molecule into MOs that correspond to the molecular symmetry. As pointed out in Chapter 5, it is essential that valid MOs form bases for irreducible representations of the molecular point group, we encounter the problem of writing SALCs when we deal with molecules having sets of symmetry-equiv-... 

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