Rotation—reflection

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

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). 

In chemistry, perhaps because of the significance in visualizing molecular strac-ture, there has been a focus on how students perceive three-dimensional objects from a two-dimensional representation and how students mentally manipulate rotated, reflected and inverted objects (Stieff, 2007 Tuckey Selvaratnam, 1993). Although these visualization skills are very important in chemistry, it is evident that they are not the only ones needed in school chemistry (Mathewson, 1999). For example, conceptual understanding of nature of different types of chemical bonding, atomic theory in terms of the Democritus particle model and the Bohr model, and... 

The traditional eharaeterisation of an electron density in a crystal amounts to a statement that the density is invariant under all operations of the space group of the crystal. The standard notation for sueh an operation is (Rim), where R stands for the point group part (rotations, reflections, inversion and combinations of these) and the direct lattice vector m denotes the translational part. When such an operation works on a vector r we get... 

The major problem is to find the rotation/reflection which gives the best match between the two centered configurations. Mathematically, rotations and reflections are both described by orthogonal transformations (see Section 29.8). These are linear transformations with an orthonormal matrix (see Section 29.4), i.e. a square matrix R satisfying = RR = I, or R = R" . When its determinant is positive R represents a pure rotation, when the determinant is negative R also involves a reflection. 

The first term on the right-hand side represents the total sum of squares of Y, that obviously does not depend on R. Likewise, the last term represents the total sum of squares of the transformed X-configuration, viz. XR. Since the rotation/reflection given by R does not affect the distance of an object from the origin, the total sum of squares is invariant under the orthogonal transformation R. (This also follows from tr(R" X XR) = tr(X rXRR T) = tr(X XI) = tr(X" X).) The only term then in eq. (35.2) that depends on R is tr(Y XR), which we must seek to maximize. 

Rotation-reflection. Has the symbol S , which is a rotation followed by a reflection in the plane perpendicular to the axis of rotation. 

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... 

But this does nob end the tale of possible arrangements. Hitherto we have considered only those symmetry operations which carry us from one atom in the crystal to another associated with the same lattice point—the symmetry operations (rotation, reflection, or inversion through a point) which by continued repetition always bring us back to the atom from which we started. These are the point-group symmetries which were already familiar to us in crystal shapes. Now in "many space-patterns two additional types of symmetry operations can be discerned--types which involve translation and therefore do not occur in point-groups or crystal shapes. 

The way in which we systematize our notion of symmetry is by introducing the concept of a symmetry operation, which is an action which moves the nuclear framework into a position indistinguishable from the original one. At first sight it would appear that there are very many such operations possible. We will see, however, that each falls into one of five clearly delineated types identity, rotation, reflection, rotation-reflection, and inversion. 

This is the operation of clockwise rotation by 2w/ about an axis followed by reflection in a plane perpendicular to that axis (or vice versa, the order is not important). If this brings the molecule into coincidence with itself, the molecule is said to have a n-fold alternating axis of symmetry (or improper axis, or rotation-reflection axis) as a symmetry element. It is the knight s move of symmetry. It is symbolized by Sn and illustrated for a tetrahedral molecule in Fig. 2-3.3.f... 

Rotation-reflection. Consider a rotation by 0 (— 2w/n) about the e, base veotor, followed by reflection in the cTu plane. The components of the point vector p (or, the coordinates of the point P) will be first transformed by the rotation, as in (1), and then these new components (coordinates) will be transformed by the reflection, as in (2). Using matrix notation, these two transformations can be combined into one step (see 4-3(3)) and we get... 

Let the Cartesian coordinate axes x y z have the same origin as the xyz axes. The x y z set is obtainable from the xyz set by rotation, reflection, or inversion, or some combination of these operations. (If the x y z set is left handed while the xyz set is right handed, we must perform a reflection or inversion as well as a rotation to generate the x y z axes from the xyz axes.) Let the vector r have coordinates (x,y,z) and (x, y, z ) in the two coordinate systems. If i is a vector of unit length along the x axis, then (1.55) gives r i —Let be the direction cosines of the x" axis... 

The simplest molecules are atoms, which belong to point group %h (often called the full rotation-reflection group). The character table (which we omit) contains irreducible representations of dimensions 1,3,5,... these representations correspond to energy levels with electronic orbital angular-momentum quantum number /=0,1,2,... we have the (2/+1)-fold degeneracy associated with different values of the quantum number... 

In the Schoenfiies system the improper axis is an axis of rotation-reflection (see page 52). In the Internationa) system the axis of rotatory inversion ( ) is ore of n-fold rotation followed by inversion (see Fig. 3.29). 

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