We shall demonstrate that the symmetrized wavefunctions in Table 2.9 indeed transform according to the irreducible representations given in that table. Hopefully the readers will be able then to verify the symmetry of the wavefunctions in Table 2.10. To do this we shall note that the functions: {sin x, sin y, sin z} transform like {x,y, z} under all the symmetry operations of the Td group. On the other hand the functions {cos x, cos y, cos z) are "even" under C2 rotations so they have the same transformation properties as {x2,y2,z2). Based on this observation we can see immediately that :
(a)the function sin(2px/a)sin(2py/a)sin(2pz/a)
transforms like {xyz} and therefore from Table 2.3 this function belongs
to the G 1 representation.
(b)similarly the function cos(2px/a)cos(2py/a)cos(2pz/a)
transforms like (xyz)2 or (G 1
)2 and therefore belongs to the G
1 representation also.
(c)by the same argument the functions : {sin(2px/a)sin(2py/a)cos(2pz/a),
sin(2px/a)cos(2py/a)sin(2pz/a),
cos(2px/a)sin(2py/a)sin(2pz/a)}
transform like {xyz2,xy2z,x2yz}=xyz{z,y,x}.
Since {xyz} transforms like G 1 and
{x,y,z} transform like G 4 the
three functions transform like G 4.
(d)the proof for the three functions {sin(2px/a)cos(2py/a)cos(2pz/a),
cos(2px/a)sin(2py/a)cos(2pz/a),
cos(2px/a)cos(2py/a)sin(2pz/a)}is
left for the readers to complete.