2. Familiar Point Group Ideas2.1 Geometric operationsFigure 1 illustrates the four hydrogens of CH_{4} numbered [17] and placed at the corners of a cube whose faces are perpendicular to the Cartesian axes. At least six other schemes for associating numbered hydrogens with corners of the cube in Fig. 1 occur [5, 9, 18-21]. Examination of Fig. 1 shows it to be invariant (apart from atom numbering) under the following 24 point group operations: the identity; eight three-fold rotations; three two-fold rotations; six four-fold rotation-reflections; and six planar reflections. The point group consisting of these operations is called T_{d}. Its character table [2-4] is given in Table 1. |
Figure 1. The hydrogen atom numbering and methane orientation in the molecule-fixed axis system used in this article. |
E | 8C_{3} | 3C_{2} | 6S_{4} | 6σ_{d} | |||
---|---|---|---|---|---|---|---|
A_{1} | 1 | 1 | 1 | 1 | 1 | α_{xx} + α_{yy} + α_{zz} | |
A_{2} | 1 | 1 | 1 | -1 | -1 | ||
E | 2 | -1 | 2 | 0 | 0 | (2α_{zz} - α_{xx} - α_{yy} , α_{xx} - α_{yy}) | |
F_{1} | 3 | 0 | -1 | 1 | -1 | R | |
F_{2} | 3 | 0 | -1 | -1 | 1 | T | (α_{yz} , α_{zx} , α_{xy}) |
The individual operations occurring in this point group for a methane molecule oriented as shown in Fig. 1 will be denoted here by the symbols: E; C_{3}(1,1,1), C_{3}^{2}(1,1,1), C_{3}(-1,1,1), C_{3}^{2}(-1,1,1), C_{3}(-1,-1,1), C_{3}^{2}(-1,-1,1), C_{3}(1,-1,1), C_{3}^{2}(1,-1,1); C_{2}(x), C_{2}(y), C_{2}(z); S_{4}(x), S_{4}^{3}(x), S_{4}(y), S_{4}^{3}(y), S_{4}(z), S_{4}^{3}(z); and σ_{d}(0,1,1), σ_{d}(0,-1,1), σ_{d}(1,0,1), σ_{d}(1,0,-1), σ_{d}(1,1,0), σ_{d}(-1,1,0); respectively, where directions of three-fold rotation axes and of perpendiculars to the planes of reflection are specified by three indices representing coefficients of unit vectors along the Cartesian axes x,y,z. One of these geometrical symmetry operations in general will be denoted by P.