The change of basis function for the orthoexcitons from xz, yz in O_h symmetry to L_x, L_y in D_4h was allowable because these functions transform the same under all D_4h symmetry operations. The paraexciton basis function changes from (x² − y²)(y² − z²)(z² − x²) in O_h to just (x² − y²) in D_4h. Show that if you keep the full, original basis function for paraexcitons in the O_h group, it does not change the selection rules in D_4h symmetry for k(vector) along [100] or [110].

Note that when the symmetry is changed, matrix element integrals of the form (6.10.9) must have the integration over z changed to going from −L′ to L′, where L′ is different from L in the other two directions. Show that for the paraexciton state, if L′ = L, the quadrupole matrix elements to this state vanish, in agreement with the selection rule deduced for Oh symmetry.