Selection Rules for Pure Rotational Spectra

The rules are applied to the rotational spectra of polar molecules when the transitional dipole moment of the molecule is in resonance with an external electromagnetic field. Polar molecules have a permanent dipole moment and a transitional dipole moment within a pure rotational spectrum is not equal to zero.

In contrast, no rotational spectra exists for homonuclear diatomics; the same is true for spherical tops. Nevertheless, certain states of a such molecules allow unexpected interactions with the electromagnetic field; i.e. some vibrations, that introduce a time-dependent dipole moment high rotational speeds that cause some distortion of an originally spherical symmetry. A (weak) dipole moment emerges.

Typical values of the rotational constant $B$ are within $0.1 \ldots 10\, cm^{-1}$ and the corresponding radiative transitions lie in the microwave spectral region where the spontaneous emission is very slow. Therefore, the transitions are usually detected by measuring the net absorption of the microwave radiation.

The conservation of the angular momentum is fundamental for the selection rules that allow or prohibit transitions of a linear molecule:

$$\Delta J = \pm 1 \:\:\:\:\:\: \Delta M_J = 0, \pm 1$$ (48)

The transition $\Delta J = 1$ corresponds to absorption and the transition $\Delta J = -1$ corresponds to emission. The transition $\Delta M_J =0$ corresponds to the case when the transition dipole moment is parallel to the quantization $Z$ axis, while the transitions $\Delta M_J = \pm 1$ correspond to the case when the transition dipole moment is perpendicular to this axis.

For a symmetric top, an existing dipole moment is always parallel to the molecular axis. Thus, with respect to this axis, no changes of the rotational state occur:

$$\Delta K = 0$$ (49)

For energy difference corresponding to the transitions $J + 1 \leftarrow J$ can be presented as:

$${\bf\nu}(J) = B\cdot(J+1)(J+2)-B\cdot J(J+1) = 2 B (J+1)\:\:\:\: \mbox{with } J=1,2,\ldots,$$ (50)

where the energy is measured in wave numbers, $cm^{-1}$.

It is easy to see that the frequency difference between two neighbour absorption lines is constant: $\Delta\nu= \nu(J) - \nu(J-1) = 2B$. Therefore, the constant $B$ as well as the bond's length $r$ can be directly determined from the absorption spectrum.

A more accurate expression for $\nu$ is

$${\bf\nu}(J)$$

$$= B(J+1)(J+2)-D(J+1)^2(J+2)^2-BJ(J+1)+DJ^2(J+1)^2$$

$$= 2B(J+1)-4 D(J+1)^3$$ (51)

Therefore the frequency difference between two neighbour absorption lines is

$$\Delta\nu= \nu(J) - \nu(J-1) = 2B-4D(3J^2+9J+7)$$ (52)

and decreases with J. Thus, the centrifugal constant $D$ for diatomic molecules is in connection with the wavenumber $\nu_S$ that corresponds with the molecule's vibration.

$$D = \frac{4B^3}{\nu_S^2}$$ (53)

Reversely, $D$ provides information on $\nu_s$. Of course, the intensity of an absorption is dependent on the transitional dipole moment and on the population of the initial and the final state.

The intensities of spectral lines first increase with increasing $J$ and pass through a maximum before tailing off as $J$ becomes large. The most important reason for the maximum in intensity is the existence of a maximum in the population of rotational levels. According to the Boltzmann distribution the population of a rotational level at temperature $T$ is given by

$$N_J = N g_J\, e^{-\frac{E_J}{k T}},$$ (54)

where $g_J$ is the degeneracy of the level $J$, $E_J$ is the level energy, and $N$ is a constant.

The distribution in eq. (54) applies that the population of each state decays exponentially with increasing $J$, but the pre-exponent factor increases linearly with $J$. Competition between these two tendencies gives a maximum in population at a certain $J$ value for each rotational state.

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