This section explains the origins of rotational and vibrational spectra; absorptions of radiation that cause molecules to rotate and vibrate more energetically
The quantum rule for a molecule to become rotationally excited by interacting [absorbing] suitable radiation is that it should have a permanent dipole moment. The molecule should be polar, e.g., HCl [hydrogen chloride] is a very polar molecule with the hydrogen end being partially positively charged and the chlorine end being equally and oppositely partially negatively charged. The HCl molecule does have a rotational spectrum. On the other hand, the nitrogen molecule, N_{2}, is non-polar and is forbidden by the quantum rule from acquiring rotational energy by absorbing radiation. That does not mean that the nitrogen molecule cannot rotate, it can acquire rotational energy by colliding with another molecule and it can also lose rotational energy by the same means.
The quantum rule for molecules to become vibrationally excited is that it should have at least one mode of vibration that causes it to have a change in its dipole moment [which may be zero as in the case of linear symmetric molecules like CO_{2}]. Then it can interact with suitable radiation of the correct energy to cause the molecule to absorb the energy of the radiation and to vibrate more energetically.
The precise condition for absorption of radiation is given by the equation:
hv = E_{2} - E_{1}
h is Planck's constant = 6.626 x 10^{-34} Joule-seconds, v is the frequency of the radiation and E_{1} and E_{2} are two vibrational energies corresponding to levels 1 and 2, with E_{2} having a higher energy than E_{1}.
A quantum of radiation, sometimes referred to as a photon, of frequency v may be absorbed by the molecule with the two energy levels 1 and 2 such that the rotational/vibrational energy of the molecule increases.
Vibrational spectra are complicated by the additional possibilities that in the absorption process the rotational energy of the molecule can change [either increase or decrease] as the vibrational energy increases.
This is a simulated [HITRAN database-generated] spectrum of the first vibrational transition of the CO_{2} bending mode. It is for a concentration of 380 ppmv and has a 1 metre path length. The data are plotted as the fraction of radiation absorbed versus its wavenumber. There are three sections to explain. One is the central 'Q branch' showing a very strong absorption and is a composite of many transitions in which the vibrational quantum number changes from 0 ® 1, all of them beginning and ending with the same rotational energy, i.e., no rotational changes are occurring in the Q branch. The P branch occurs when the same vibrational change takes place, but there are many rotational changes involving the loss of rotational energy. The opposite situation occurs for the generation of the R branch; the same vibrational change occurs with many different gains of rotational energy. The rotational changes are governed by another quantum rule in which the rotational quantum number is allowed to be either 0 [the Q branch], -1 [the P branch] or +1 [the R branch]. More explanation follows.
A diagram [not to scale] of some of the rotational energy levels associated with the zero vibrational state and that of the first vibrationally excited state. The arrows indicate the allowed transitions and at the bottom of the diagram is the simulated spectrum arising from the transitions, the Q branch transitions all coinciding at the centre of the spectrum.
This diagram seeks to explain the details of the main absorption band of CO_{2}. The ground state - the one with lowest energy - has a zero value for the vibrational quantum number, v. The excited state has v = 1. In the ground state the rotational quantum numbers [J] are restricted to even values only and the selection rules that apply are that the allowed changes in the value of J are DJ = 0, ±1 except that the transition J = 0 ® J = 0 is forbidden. If DJ = 0 the absorptions occur at much the same wavenumber and give rise to a very intense Q branch. If DJ = -1 the transitions produce the absorptions of the P branch with equally spaced wavenumbers. The R branch is produced by transitions in which DJ = +1 and give rise to another series of equally spaced wavenumbers. Each transition is broadened by collisional processes and by the participating molecules having a distribution of speeds and appear as bands rather than lines of definite wavenumber.
Although this single vibrational transition with its accompanying rotational transitions seems complicated there are other transitions of the CO_{2} molecule which contribute to its activity as a greenhouse gas. Some of these transitions are shown in the next diagram.
This diagram shows the somewhat complex situation in the region of the spectrum where CO_{2} absorbs. The various transitions are indicated by the arrows [colours have no significance, they are to make the diagram clearer] and the wavenumbers [in cm^{-1}] of the centres of the bands are given. Thus, in the range from 544.3 to 1063.7 cm^{-1} there are 13 transitions, all relating to the fundamental bending mode of the molecule. The most important and most intense transition is that between the ground state of the molecule in which the vibrational quantum number changes from zero to 1. This may be described by the symbolism:
n_{2}^{0}(0) ® n_{2}^{1}(1)
The three fundamental vibrational modes of the molecule are labeled n_{1}, n_{2} and n_{3} referring respectively to the symmetric stretch, the bend and the anti-symmetric stretch.
The superscripts 0 and 1 in the transition above refer to the change in the rotational energy of the molecule with respect to the molecular axis. The values of the vibrational quantum numbers are given in brackets.
The next three transitions centered at 618, 667.8 and 720.8 cm^{-1} all originate from the 1^{st} vibrationally excited state, n_{2}^{1}(1). In the 618.0 cm^{-1} transition the vibrational quantum number increases from 1 to 2, but the excited state has lost the axial rotational energy. The 667.8 cm^{-1} transition is an 'overtone' of the n_{2}^{0}(0) ® n_{2}^{1}(1) transition; there is an increase from 1 to 2 in the vibrational quantum number and an extra quantum of axial rotation is acquired.
The 720.8 cm^{-1} transition is from the first excited bending vibrational state to the first excited state of the symmetric stretch; n_{2}^{1}(1) ® n_{1}(1).
The remaining transitions are based on previously produced excited states and are summarized in the Table.
Wavenumber/cm^{-1} |
Lower state |
Higher state |
Comment |
544.3 |
n_{1}(1) |
n_{2}^{1}(3) |
Symmetric stretch to 3^{rd} bending vibrational level, plus axial rotation |
597.3 |
n_{2}^{2}(2) |
n_{2}^{1}(3) |
Increased bending vibration, loss of some axial rotation |
647.1 |
n_{2}^{0}(2) |
n_{2}^{1}(3) |
Increased bending vibration, extra axial rotation |
668.1 |
n_{2}^{2}(2) |
n_{2}^{3}(3) |
Increased bending vibration, extra axial rotation |
688.7 |
n_{1}(1) |
n_{1}(1) + n_{2}^{1}(1) |
Extra bending vibration and axial rotation, retaining sym. stretch |
741.7 |
n_{2}^{2}(2) |
n_{1}(1) + n_{2}^{1}(1) |
Loss of bending vibration and some axial rotation with excitation of sym. stretch |
791.4 |
n_{2}^{0}(2) |
n_{1}(1) + n_{2}^{1}(1) |
Loss of bending vibration, gain of axial rotation and excitation of sym. stretch |
961.0 |
n_{1}(1) |
n_{3}(1) |
Symmetrical stretch to anti-symmetric stretch |
1063.7 |
n_{2}^{0}(2) |
n_{3}(1) |
2^{nd} bending level to anti-symmetric stretch |
All but the last two transitions participate in the construction of the 'wells' in the graphs of emission spectra given on previous pages. The 961.0 and 1063.7 cm^{-1} transitions overlap with the symmetric stretching vibration of ozone that appears centered at 1043.4 cm^{-1}, but which does not have a Q branch. The very minor contributions to absorption and emission by the two CO_{2} transitions are blotted out by those from ozone.
The population of any upper state relates to that of the lower state by the value of the Boltzmann factor: exp(-E/RT) or exp(-e/kT), where E [and e] represent the transition energy and k is the Boltzmann constant, R = Nk, where N is the Avogadro number.
The Table below shows the values of the Boltzmann factors for the CO_{2} transitions with respect to the population of the ground state at 288 K.
Wavenumber/cm^{-1} |
Lower state |
Higher state |
Boltzmann factor |
667.4 |
n_{2}^{0}(0) |
n_{2}^{1}(1) |
0.036 |
618.0 |
n_{2}^{1}(1) |
n_{2}^{0}(2) |
1.63 ´ 10^{-3} |
667.8 |
n_{2}^{1}(1) |
n_{2}^{2}(2) |
1.27 ´ 10^{-3} |
720.8 |
n_{2}^{1}(1) |
n_{1}(1) |
9.75 ´ 10^{-4} |
544.3 |
n_{1}(1) |
n_{2}^{1}(3) |
6.43 ´ 10^{-5} |
597.3 |
n_{2}^{2}(2) |
n_{2}^{1}(3) |
6.43 ´ 10^{-5} |
647.1 |
n_{2}^{0}(2) |
n_{2}^{1}(3) |
6.43 ´ 10^{-5} |
668.1 |
n_{2}^{2}(2) |
n_{2}^{3}(3) |
4.51 ´ 10^{-5} |
688.7 |
n_{1}(1) |
n_{1}(1) + n_{2}^{1}(1) |
3.13 ´ 10^{-5} |
741.7 |
n_{2}^{2}(2) |
n_{1}(1) + n_{2}^{1}(1) |
3.13 ´ 10^{-5} |
791.4 |
n_{2}^{0}(2) |
n_{1}(1) + n_{2}^{1}(1) |
3.13 ´ 10^{-5} |
961.0 |
n_{1}(1) |
n_{3}(1) |
8.02 ´ 10^{-6} |
1063.7 |
n_{2}^{0}(2) |
n_{3}(1) |
8.02 ´ 10^{-6} |
The transitions have intensities that are related to the values of the Boltzmann factors of their initial states, so that, for instance, the 1063.7 cm^{-1} transition would be expected to be of the order of a factor of 8 ´ 10^{-6}/0.036 = ~2.2 ´ 10^{-4} times weaker that the main 667.4 cm^{-1} band. In the real atmosphere none of these other transitions are anywhere near being saturated so that an increase of CO_{2} concentration would be expected to cause an increase in the forcing of the climate system.