Asymmetric Stretching Mode

In this mode of vibration one bond stretches while other is compressed, and vice versa, as shown in figure

Figure: The asymmetric stretching vibration of the carbon dioxide molecule showing fluctuation in the dipole moment

The mode has periodic alteration in the dipole moment. The vibration is thus 'infrared active'. The frequency of this mode is

$$\nu_3 = 7.05\times 10^{13} s^{-1}$$

$$= 2350 /cm$$

The associated quantum number is denoted by $v_3$.

Each of the vibrational levels would have superimposed upon it a set of rotational levels described as

$$E_{rot} = \frac{J(J+1)h^2}{2I}$$

$$= J(J+1)Bhc$$ (15)

where

$$B=\frac{h}{8 \pi^2 cI}$$

The value of B for CO$_2$ is B= 39 m$^{-1}$ and thus

$$E_{rot} = J(J+1)Bhc$$

$$= J(J+1) (39 m^{-1}) (4.14 \times 10^{-15} eV-s) (3 \times 10^8 m/s)$$

$$= J(J+1) (4.72 \times 10^{-5}) eV$$

Hence the rotational energy levels are spaced by energies that are more than three orders of magnitude smaller than the vibrational energies. A typical vibrational level would, thus, consist of a series of rotational levels spaced according to the above formula. On account of the Boltzmann distribution between the rotational levels J' = 19 rotational level of 00$^0$1 state happens to be the most heavily populated.

Transitions would occur according to the selection rules $\Delta J = \pm 1$. For $\Delta J = +1$, we would have P branch of the spectrum and $\Delta J = -1$, we would have the R-branch.

For triatomic molecules such as CO$_2$, the selection rule $\Delta v = \pm 1$ is not relevant, since the transitions can occur from one vibrational mode to another. Because the absorption of a photon requires the molecule to take up one unit angular momentum, vibrational transitions are accompanied by a change in rotational state, which is subject to the same selection rules as for the pure rotational spectrum. For a molecule in a $\Sigma$ state (with $\Lambda$ = 0), the transitions between two rovibronic levels (v,J) and $v',J'$), with vibrational quantum number $v$ and $v'=v+1$ fall into two sets according to whether $\Delta J = +1$ (R branch) or $\Delta J = -1$ (P-branch). Both branches make up vibrational-rotational band.

Appendix B