Chapter Questions
Find the selection rules for clectric and magnetie dipole and electrie quadrupole transitions for the symmetry groups $\mathrm{C}_{\infty}$ and $D_{\text {en }}$.
Prove that a proper orthogonal transformation (real or complex) in an odd-dimensional space always possesses an axis (i.e., a line whose points are left unehanged.
Find the matrix of the transformation $R(\alpha, \beta, \gamma)$
Construet a table to show how levels of the full rotation-reflection. group split in a field with symmetry $D_{3 d}$.
Solve the seeular equation and find the zero-order funetions for the $E$-levels formed for $l=3$.
Find the zero-order erystal wave funetions for $l=3$ under cubie symmetry.
Describe the eharge density for the eubie erystal states resulting from an $F$-state of the free atom.
Find the double-valued representations of the group $\mathrm{C}_{3}$.
Find the characters of the two-valued representations of $T^{\prime \prime}$ by using Eq. (3-173).
Carry out the resolution of the two-valued representations of the rotation group for the erystal symmetries $D_{3}$ and $T$.