The CO molecule presents, in Raman Stokes diffusion, a rotation$-$vibration spectrum with the $3$ Q$,$S

and $O$ branches. The $Q$ branch is a quite large and intense band, centered on $\frac{?}{\text{b}\text{a}\text{r}}(v{)}_0=2143cm-1.$ The $S$

and $O$ branches are composed of numerous rays which positions are displayed in the table below and

related to the first transitions :

$C=12\mathrm{.}011$;$O=15\mathrm{.}995.$

Knowing that :

Demonstrate the following expressions :

$\frac{?}{\text{b}\text{a}\text{r}}(v{)}_S(J-2)\frac{-}{b}ar(v{)}_0(J+2)=4B_0(2J+1)\frac{?}{b}$

$ar(v{)}_S(J)\frac{-}{b}ar(v{)}_0(J)=4B_1(2J+1)$

Knowing that :

$\frac{?}{\text{b}\text{a}\text{r}}(v{)}_S\frac{?}{b}=ar(v{)}_0+6B_1+(5B_1-B_0)J+(B_1-B_0)J^2\frac{?}{b}$

$ar(v{)}_O\frac{?}{b}=ar(v{)}_0+2B_1-(3B_1+B_0)J+(B_1-B_0)J^2$

Determinate the rotational constants $B_1$ and $B_0,$ the inertia moments $I_1$ and $I_0,$ and the bond

lengths $r_1$ and $r_2.$

After calculating $B_e$ and ${}_e$ of the molecule at equilibrium, deduce the inertia moment I and

the interatomic distance of $CO.$Exercice $10$:

The $N_2{O}_2($cis$)$ molecule presents $6$ vibrational normal modes, displayed below :

Can degeneracy exists $?$ Explain why, focusing on the point group characters table of this

molecule.

For each normal mode, determine the symmetry characters for each operator type, thus the

symmetry species the mode belongs to$.$

What are the normal modes leading to an active fundamental transition band in Infrared $?$

Among the actie fundamental transitions in Raman, what are the polarized one$($s$)?$

Exercice $11$:

$C{O}_3^{2-}$ and $S{O}_3^{2-}$ structures have been investigated with vibrational spectroscopy, both Infrared and

Raman. The wave numbers for each observed fundamentals bands have been reported below :

What point group may be attributed to $a{B}_3^{2-}$ ion $($list of possibility$)?$

Without calculation, refine your answer using character tables and assign the point group of

the two ions.

Deduce and associate each observed bands with symmetry species, for both ions. Do all the

fundamental transitions corresponding to vibrational normal modes are effectively displayed

for each ions $?$