Symmetry elements diagonal

The main symmetry elements in SFg can be shown, as in Figure 4.12(b), by considering the sulphur atom at the centre of a cube and a fluorine atom at the centre of each face. The three C4 axes are the three F-S-F directions, the four C3 axes are the body diagonals of the cube, the six C2 axes join the mid-points of diagonally opposite edges, the three df, planes are each halfway between opposite faces, and the six d planes join diagonally opposite edges of the cube. 

First, it is possible to simplify the secular equation (2) by means of symmetry. It can be shown by group theory (140) that, in general, the integrals Hi and Si are nonzero only if the orbitals < , and j have the same transformation properties under all the symmetry elements of the molecule. As a simple example, the interaction between an s and a pn orbital which have different properties with respect to the nodal plane of the pn orbital is clearly zero. Interaction above the symmetry plane is cancelled exactly by interaction below the plane (Fig. 13). It is thus possible to split the secular determinant into a set of diagonal blocks with all integrals outside these blocks identically zero. Expansion of the determinant is then simply the product of those lower-order determinants, and so the magnitude of the... 

The onl other symmetry element involving translation which remains to be meniiqued is the glide plane having a translation of one-quarter of a eell-faee diagonal this type of glide, symbolized d, is found only in a few space-groups. 

The q matrix is the negative of the electric-field gradient. Like the inertial tensor and the polarizability tensor, q is symmetric (since the order of partial differentiation is immaterial), and we can make an orthogonal transformation to a new set of axes a, ft, y such that q is diagonal, with diagonal elements qaa, q, q. Note, however, that the origin for q is at the nucleus in question and the axes for which q is diagonal need bear no relation to the principal axes of inertia (unless the nucleus happens to lie on a symmetry element). 

In low-symmetry molecules, diagonal and off-diagonal matrix elements of the electronic dipolar coupling tensor may contribute to ( h[)0 ) J ssl b ). Therefore, they are specified mostly in terms of their Cartesian components. If symmetry is C2V or higher, the off-diagonal matrix elements of the tensor operator in Eq. [163] vanish (i.e., the principal axes diagonalizing the SCC tensor coincide with the inertial axes). For triplet and higher multiplicity states, one then obtains... 

Pig. Symmetry elements and point groups of three type of molecules, <7, av ad denote verticalhorizontal and diagonal plane of symmetry. 

Characteristic symmetry elements of the cube are shown in Figure 2-50. Three different symmetry planes go through the center of the cube parallel to its faces. Furthermore, six symmetry planes connect the opposite edges and also diagonally bisect the faces. 

Figure 9-23. The diamond structure after Shubnikov and Koptsik [33], (a) A unit cell the edges of the cube are the a, b, and c axes (b) Two face-centered cubic sublattices displaced along the body diagonal of the cube (c) Projection of some symmetry elements of the Fd im space group onto a horizontal plane. The vertical screw axes 4 and 43 are marked by appropriate symbols. Used with permission.

Therefore, zero, one, two or all three coordinates change their signs, but this only holds for symmetry elements of the first and second order when they are aligned with one of the three major crystallographic axes. Symmetry operations describing both diagonal symmetry elements and symmetry elements with higher order (i.e. three-, four- and six-fold rotations) may cause permutations and more complex relationships between the coordinates. For example ... 

By symmetry, these diagonal elements do not depend on j. We see that in the submatrix of H associated with the same atomic level n defined on each atom, all diagonal elements are the same Non-diagonal elements of H result from the interatomic coupling, and if the atomic centers are not too close to each other these elements will be small relative to the spacings between different s, that is,... 

If we choose the linear combination of the degenerate orbitals such that one is symmetric and the other antisymmetric with respect to a symmetry element that is retained in the perturbed molecule, then the off-diagonal elements will vanish, hki = hik = 0, because integration of any function over the complete range of its variable x vanishes if the function is antisymmetric Jf(x)dx = 0 (Figure 4.11). A function is said to be antisymmetric (a) if f(—x) = —f(x). The product of two antisymmetric functions is symmetric (s) we write a a = s similarly, s s = s but a s = s a = a. 

---