Reflection operators

For the nanotubes, then, the appropriate symmetries for an allowed band crossing are only present for the serpentine ([ , ]) and the sawtooth ([ ,0]) conformations, which will both have C point group symmetries that will allow band crossings, and with rotation groups generated by the operations equivalent by conformal mapping to the lattice translations Rj -t- R2 and Ri, respectively. However, examination of the graphene model shows that only the serpentine nanotubes will have states of the correct symmetry (i.e., different parities under the reflection operation) at the K point where the bands can cross. Consider the K point at (K — K2)/3. The serpentine case always sat-... 

The remaining fifty-eight magnetic point groups include the time reversal operator only in combination with rotation and rotation-reflection operators. The representations of these groups may be obtained from Eq. (12-27). 

The molecules shown in Fig. 1 are planar thus, the paper on which they are drawn is an element of symmetry and the reflection of all points through the plane yields an equivalent (congruent) structure. The process of carrying out the reflection is referred to as the symmetry operation a. However, as the atoms of these molecules are essentially point masses, the reflection operations are in each case simply the inversion of the coordinate perpendicular to the plane of symmetry. Following certain conventions, the reflection operation corresponds to z + z for BF3 and benzene, as it is the z axis that is chq ep perpendicular to die plane, while it is jc —> —x for water. It should be evident that the symmetry operation has an effect on the chosen coordinate systems, but not on the molecule itself. 

Reflection operators with respect to these planes are... 

Figure 7.2 The different symmetry elements of the center ABg. (a) A trigonal axis, C3 (b) A binary axis, C2. (c) A symmetry axis belonging to both the 6C4 and SCj classes, (d) A symmetry plane, au- (e, f) Two of the six aj, reflection planes, (g) A view down the C3 axis in (a) to show a roto-reflection operation, S. ...

Figure 7.4 The effect of different symmetry operations over the three p orbitals (a) the initial positions (b) after an inversion symmetry operation (c) after a reflection operation through the x-y plane (d) after a rotation Cj about the z-axis.

By setting the origin of the coordinate system at the intersection of the two mirror reflection lines, it is easy to see that only Eq. (E.3) of the four corrugation functions is invariant under the mirror reflection operation. The fourfold rotational symmetry further requires n = m, and a = To the lowest nontrivial corrugation component, the general form of the corrugation function is... 

For example, reflection of an N-orbital product through the ov plane in NH3 applies the reflection operation to all N electrons. 

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... 

Two reflection operations o and o will belong to the same class provided there is a symmetry operation in the point group which moves all the points on the a symmetry plane into corresponding positions on the a symmetry plane. A similar rule holds true for two rotational operations CJ and Cj (or SJ and S ) about different rotational axes, i.e. the two operations belong to the same class provided there is a symmetry operation in the point group which moves all the points on the C (or ) axis to corresponding positions on the C (or S ) axis. 

To prove Rule (3) first consider the reflection operation Q xoQ where Q is a symmetry operation which moves all the points on some symmetry plane a to corresponding positions on the symmetry plane a. Taking the components of Q-1oQ one by one Q will move the... 

It should be explicitly noted that the existence of one symmetry plane gives rise to, requires, or, as usually stated, generates one symmetry operation. We may also note here, for future use, that the effect of applying the same reflection operation twice is to bring all atoms back to their original positions. Thus, while the operation or produces a configuration equivalent to the original,... 

Although we have followed conventional practice—and for general purposes will continue to do so—in setting out four kinds of symmetry elements and operations, a, /, C , and Sny we should note that the list can in principle be reduced to only two C and S . A reflection operation can be regarded as an... 

If the molecule has an S axis, we may place the plane so it coincides with the plane through which the reflectional part of an Sn operation takes place. If the S axis is of odd order, the pure reflection operation (5, or S") will actually exist as a symmetry operation. The molecule is then obviously superimposable on its mirror image. 

If we introduce a longitudinal reflection operation (i.e., the line of translation is also a mirror line) we get class 2. 

Why must a glide-reflection operation entail a translation of i the repeat distance and no other fraction of it ... 

---