Proper axis

Values of ty larger than 1 indicate that the octahedral axes do no represent a proper axis system for the EFG in the sense of (4.32). To achieve 0 < rj < 1, the axes may be just interchanged... 

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. 

In drawing stereographic projection diagrams, it is conventional to indicate improper axes, by an open polygon at the center, and a proper axis, C n, by a filled polygon (see Figure 4.1c). 

Proper axis cr Rotate the molecule by 360/n degrees around the axis... 

Figure 2.1 An example of a proper axis of symmetry, in this case of order 3, C3...

A proper axis of symmetry, denoted by Cn, is an axis around which a molecule is rotated by 360°ln to produce an equivalent configuration. The trigonally planar molecule BF3 may be set up so that the molecular plane is contained by the xy Cartesian plane (that containing the x and y axes) and so that the z Cartesian axis passes through the centre of the boron nucleus, as is shown in Figure 2.1. 

If the molecule is rotated around the z axis by 120° (360°/3), an equivalent configuration of the molecule is produced. The boron atom does not change its position, and the fluorine atoms exchange places depending upon the direction of the rotation. The rotation described is the symmetry operation associated with the C3 axis of symmetry, and the demonstration of its production of an equivalent configuration of the BF3 molecule is what is required to indicate that the C3 proper axis of symmetry is possessed by that molecule. 

A Each vertical plane contains a C2 proper axis of rotation. 

The symmetry elements of the water molecule are easily detected. There is only one proper axis of symmetry, which is the one that bisects the bond angle and contains the oxygen atom. It is a C2 axis and the associated operation of rotating the molecule about the axis by 180° results in the hydrogen atoms exchanging places with each other. The demonstration of the effectiveness of the operation is sufficient for the diagnosis of the presence of the element. 

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...

Prolate symmetric top, 199, 211 Propane, dipole moment of, 225 Proper axis of symmetry, 53 Proper rotation, 395-396 Proton, 178 Pseudovector, 434 Pulse laser, 137,139 Purcell, E. M., 328, 360 Purely electronic energy, 57 Pure-rotation spectra, 165... 

Before discussing proper axes and rotations in a general way, let us take a specific case. A line drawn perpendicular to the plane of an equilateral triangle and intersecting it at its geometric center is a proper axis of rotation for that... 

In discussing planes of symmetry and inversion centers, attention was directed to the fact that only one operation, reflection, is generated by a symmetry plane, and only one operation, inversion, by an inversion center. A proper axis of order / , however, generates n operations, namely Cn, C , Cl.C +I, C2( = ). 

Let us consider now the cases where the only symmetry element is a proper axis, C . This generates a set of operations C , CJ, Cj,.. . , Cna = E. Hence a molecule with C as its only symmetry element would belong to a group of order , which is designated C . It may be noted that a C group is a cyclic group (see Section 2.2) and hence also Abelian. 

We have already seen that, if a molecule possesses a proper axis, C , and also a twofold axis perpendicular to it, there must then necessarily be n such twofold axes. The n operations, E, C , CJ,. . . , . C" l, plus the n twofold rotations constitute a complete set of symmetry operations, as may be verified... 

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.. 

As noted, there is a C2 axis lying) along the C=C=C axis. There is no higher-order proper axis. There are two more C2 axes perpendicular to this one, as shown in Figure 3.11. Thus, tlhe group must be a D type, and we proceed to step 5. 

As indicated in Figure 3.12, there is a C2 axis and no other proper axis. There are no planes of symmetry. The group is therefore C2. Note that the C2 symmetry is in no way related to the value of the angle 0 except when 0 equals 0° or 90°, in which case the symmetry is higher. We shall next examine these two nonequilibrium configurations of the molecule. 

Coincident with the S4 axis there is (by necessity) a C2 axis. No proper axis of higher order can be found, but there are two more, equivalent C2 axes in a plane perpendicular to the S4-C2 axis. Thus we are dealing with a D2 type of group. 

The unique, high-order, proper axis is a C5 axis, as shown. Perpendicular to this there are five C2 axes. 

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