Operator rotation-reflection

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. 

Although there are two fundamental types of symmetry operations, such as rotation and reflection, but an examination of different molecules reveal that there are four operations, rotation, reflection, improper rotation and inversion which will now be considered. 

Point group - A group of symmetry operations (rotations, reflections, etc.) that leave a molecule invariant. Every molecular conformation can be assigned to a specific point group, which plays a major role in determining the spectrum of the molecule. 

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

Periodic boundary conditions can also be used to simulate solid state conditions although HyperChem has few specific tools to assist in setting up specific crystal symmetry space groups. The group operations Invert, Reflect, and Rotate can, however, be used to set up a unit cell manually, provided it is rectangular. 

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 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... 

It can be seen that there are seven other symmetry operations (roto-reflections) of this class, which is then denoted by SSe, the subscript 6 indicating a rotation through lit 16. In a similar way, we can analyze other symmetry operations of classes 65 4,15 2 (commonly called an inversion symmetry operation and denoted by /), and IE (the identity operation that leaves the octahedron unchanged). 

Symmetry operators leave the electronic Hamiltonian H invariant because the potential and kinetic energies are not changed if one applies such an operator R to the coordinates and momenta of all the electrons in the system. Because symmetry operations involve reflections through planes, rotations about axes, or inversions through points, the application of such an operation to a product such as H / gives the product of the operation applied to each term in the original product. Hence, one can write ... 

We shall need one more matrix, that for a particle obtained by the symmetry operations of reflection and rotation this follows readily from (13.13) and (13.18) ... 

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... 

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... 

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... 

---