Q$1)$ A carbon disulfide molecule belongs to the point group $D_{}.$ The nine displacements of the three atoms span $A_{1g}+A_{1u}+A_{2g}+2E_{1u}+E_{1g}.$ What are the symmetry species of the normal modes of vibration?

Q$2)$ Which of the normal modes of $($i$)H_{2}O,($ii$)H_{2}CO$ are infrared active and which are Raman active?

Q$3)$ The configuration $...4p^{6}5d^1$ of rubidium has two levels at $25700\mathrm{.}56c{m}^{-1}$ and $25703\mathrm{.}52c{m}^{-1}$ above the ground state. What is the spin$-$orbit coupling constant in this excited state?