This diagram shows the spectra of the main molecules that participate in the greenhouse effect and the emission spectra of the Sun and the Earth. An idealized solar spectrum is included and there is also a description of the details of the carbon dioxide spectrum and a spectrum of water vapour that shows the region described as the infrared 'window'.

The red area of the Sun's spectrum is absorbed by the atmosphere and the Earth's surface. The warmed surface emits infrared radiation as indicated by the white areas on the individual molecule's spectrum. The grey bits are the parts of the spectra that are absorbed by the atmosphere. The blue area on the Earth's emission spectrum is known as the **infrared window** through which most of the Earth's radiation passes to space unhindered by being absorbed by any of the greenhouse gases.

The last row of the spectra shows the extent of what is known as Rayleigh scattering. This is what happens to high energy quanta and applies to the UV/blue end of the solar radiation coming into the atmosphere. It is this scattering of 'blue' photons by the molecules of the atmosphere which causes the clear sky to be blue. To be precise about this, the sky should really be violet as 'violet' photons are scattered even more than blue ones. Because the human eye is very much less sensitive to violet light than it is to blue light we percieve the sky to be blue.

**The solar spectrum**

This spectrum is divided into the main components of the electromagnetic spectrum and shows that 51% of solar radiation hitting the top of the atmosphere is in the infrared region; that known as the near IR region, i.e, near the visible region. On the wavenumber scale used the higher limit of terrestrial radiation, i.e., that emitted by the Earth's surface is 1600 cm^{-1} . Most of the ultraviolet radiation is absorbed in the stratosphere leading to ozone formation and an even greater protection of the Earth's surface. Nevertheless some UV is transmitted by the atmosphere and cause skin damage to the unwary. The solar spectrum above is for an idealized Stefan-Boltzmann emitter at the temperature of the Sun's surface. The 12% of UV represents the radiation hitting the top of the atmosphere. Much of it is reflected back to space.

The spectrum below is for the real Sun's output and shows the various absorbers.

Solar spectrum (A) above the atmosphere, (B) near the Earth's surface and (C) the spectrum detectable by the eye, The x-axis is wavelength in microns. The difference between A and B represents the albedo; the fraction of the incoming solar radiation that is reflected to space. No spectrum of that seems to be available, but I'm still trying to find one. According to the K/T diagram the radiation absorbed by the atmosphere amounts to 67 W/m^{2} with only 10 W/m^{2} of that absorbed by the stratosphere.

**Carbon Dioxide Infrared Spectrum**

There are several transitions of the CO_{2} molecule that contribute to its IR spectrum and which are relevant to the understanding of its role in global warming. For an apparently simple triatomic molecule its IR spectrum is quite complex. The relevant transitions are shown in Figure 1.

**Figure 1** Energy levels of the CO_{2} molecule and the various allowed transitions

The transition wavenumbers are given in reciprocal centimetres. The nomenclature of the energy levels requires some explanation. They have the form **ν _{a}^{b}(n**).

**a**has the value 1, 2 or 3 and represents the symmetric stretch, the bend, and the anti-symmetric stretch respectively.

**b**has the values 0, 1, 2… and represents the quantum number relevant to the angular momentum around the O=C=O axis. (

**n**) refers to the ground state (0) and excited states 1, 2, 3… of any of the three modes of motion. The fundamental vibrations have central wavenumbers of

**ν**= 1388 cm

_{1}^{-1},

**ν**= 667.4 cm

_{2}^{-1}and

**ν**= 2349 cm

_{3}^{-1}.

The ground state of the molecule's bending vibration is **ν _{2}^{0}(0)**; it is in its vibrational ground state and has zero angular momentum around the molecular axis. It is doubly degenerate in that the bending from the molecular axis, taken as the Cartesian z axis, occurs in the xz and yz planes at right angles to each other.

The fundamental bending vibration has a central wavenumber of 667.4 cm^{-1} and it is a transition **ν _{2}^{0}(0) → ν_{2}^{1}(1);** the energy change includes a quantum of angular momentum around the molecular axis and a quantum of vibrational energy. There are complications with regard to changes in rotational energy which are dealt with below.

The transition is the most intense of the ones shown in the diagram of Figure 1 because it originates in the ground state of the molecule which is the most populated energy level. The population of the excited level is dependent upon the value of the Boltzmann factor, exp(-ΔE/*RT*) where ΔE is the transition energy - the difference in energy of the two participating states, *R* is the gas constant and **T** is the absolute temperature.

The wavenumber of any transition is related to its corresponding energy by the equation:

1 cm^{-1} = 11.9624 J mol^{-1}

667.4 cm^{-1} = 667.4 × 11.9624/1000 = 7.98 kJ mol^{-1}

The Boltzmann factor at 288 K has the value exp(-7980/288R) = 0.036 which means that only 3.6% of the CO_{2} molecules are in this particular excited state. These are the molecules that form the lower energy state for the next higher transitions which have an even lower population.

There are three transitions emanating from the first excited state of the bending mode with wavenumbers of 618, 667.8 and 720.8 cm^{-1}. The second of these transitions is to the higher state, **ν _{2}^{2}(2)** with the molecule picking up an extra quantum of angular momentum and an extra quantum of vibrational energy; sometimes referred to as the first harmonic of the fundamental vibration. There is another transition in the same series in which the molecule acquires an extra quantum of angular momentum and another quantum of vibrational energy; the second harmonic of the fundamental vibration.

The transition at 618 cm^{-1} is **ν _{2}^{1}(1) → ν_{2}^{0}(2),** i.e., there is a loss of the one quantum of angular momentum around the molecular axis and a gain of one quantum of vibrational energy. The transition at 720.8 cm

^{-1}is

**ν**the angular momentum quantum is lost, but the excited molecule has now acquired one quantum of the symmetrical stretching vibration and has lost the quantum of the bending vibration. In terms of fundamental vibrations the transition energy is

_{2}^{1}(1) → ν_{1}^{0}(1),**ν**= 1388 - 667 = 721 cm

_{1}- ν_{2}^{-1}.

The three transitions at 618, 667.4 and 720.8 cm^{-1} are necessarily lower in intensity than the fundamental bending transition since the population of the first excited bend is only 3.6% of the gas.

The next higher transitions are suitably labelled and may be understood from the nomenclature. They are necessarily even weaker than the transitions from the first excited bend, but they are ready and waiting to contribute to further warming of the troposphere if the atmospheric concentration of the gas increases.

The highest energy level in the diagram of Figure 1 is **ν _{3}(1),** the first excited state of the fundamental anti-symmetric stretching vibration with a wavenumber of 2349.3 cm

^{-1}and is populated via two routes in which three transitions participate.

One is **ν _{2}^{0}(0) → ν_{2}^{1}(1)** [667.4 cm

^{-1}] plus

**ν**[618 cm

_{2}^{1}(1) → ν_{2}^{0}(2)^{-1}] plus

**ν**[1063.7 cm

_{2}^{0}(2) → ν_{3}(1)^{-1}] with a summed wavenumber of 2349.1 cm

^{-1}.

The second route is **ν _{2}^{0}(0) → ν_{2}^{1}(1)** [667.4 cm

^{-1}] plus

**ν**[720.8 cm

_{2}^{1}(1) → ν_{1}(1)^{-1}] plus

**ν**[961 cm

_{1}(1) → ν_{3}(1)^{-1}], with a summed wavenumber of 2349.2 cm

^{-1}.

The existence of the **ν _{3}(1)** level is highly important with regard to its collisional excitation by nitrogen molecules in their first excited vibrational mode that has a transition wavenumber of 2359.6 cm

^{-1}, almost a perfect match for the excitation of the anti-symmetric stretching vibration of CO

_{2}:

**N _{2} [ν_{1}(1)] + CO_{2} [ν_{2}^{0}(0)] → N_{2} [ν_{1}(0)] + CO_{2} [ν_{3}(1)] **

The CO_{2} **[ν _{3}(1)]** state can then emit radiation or be degraded to the lower energy levels from which radiation may be emitted.

**Details of the fundamental bending transition ν _{2}^{0}(0) → ν_{2}^{1}(1)**

The ground state of the CO_{2} molecule follows Bose-Einstein statistics rather than Fermi-Dirac statistics and this is the key to understanding the rotational fine structure of the fundamental transition of the bending mode. Herzberg (*Infrared and Raman Spectra*, page 16) states that for molecule with D_{¥h} symmetry (such as CO_{2} which is linear and centrosymmetric) in its ground state alternate rotational levels have different statistical weights and if the spins of the nuclei are all zero (as is the case with CO_{2}) the odd numbered rotational levels are missing entirely. The consequences of this are understood by studying the energy level diagram given in Figure 2. The differences in wavenumbers between adjacent P and R branch bands are twice that expected for most rotors.

**Figure 2** Energy levels of CO_{2} and how they interact to give the spectrum of the fundamental of the bending vibration

The ground state is that with a vibrational quantum number (v) of zero and a rotational quantum number (J) of zero. The vibrational quantum number of zero is associated with even J values. In the vibrationally excited state with v = 1, all the rotational quantum numbers are allowed except for J = 0. This is because the J = 0 → J = 0 transition is forbidden.

The energy of a rotational level is given by the equation:

E_{J} = J(J + 1)h/(8π^{2}Ic)

J is the rotational quantum number, h is Planck's constant, I is the moment of inertia of the molecule and c is the speed of light.

The equation is usually given as:

E = BJ(J + 1)

B = h/(8π^{2}Ic)

The lowest value of J is zero, so the lowest energy for the molecule is zero.

The selection rule for rotational transitions is **ΔJ = 0 or ±1**, except that J = 0 → J= 0 is forbidden.

In the rotation-vibration transition in which the vibrational quantum number change is **v = 0 → v = 1**, as in Figure 2, the permitted lower rotational energy levels are shown (not to scale).

Three transitions are shown that form part of the Q branch. They are transitions in which ΔJ = 0; there is no change in rotational energy. The lowest Q branch transition is from J = 2 → J = 2, since the 0 → 0 transition is forbidden. The other two Q branch transitions shown are J = 4 → J = 4 and J = 6 → J = 6. All the transitions of the Q branch have the same energy; that of the change in vibrational energy centred at 667.4 cm^{-1}.

The energies of the rotational levels in the diagram of Figure 2 are:

E_{0 }= 0, E_{1} = 2B, E_{2} = 6B, E_{3} = 12B, E_{4} = 20B, E_{5} = 30B, E_{6} =42B, E_{7} = 56B, E_{8} =72B

The R branch of the spectrum is composed of transitions of the vibrational quantum number from 0 → 1 with rotational transitions superimposed according to the selection rules and which increase the transition energy.

The R branch transitions in the diagram of Figure 2 are:

v = 0 → v =1 + E_{0} = 0 → E_{1} = 2B, i.e. an energy change of ΔE_{v} + 2B

v = 0 → v =1 + E_{2} = 6B → E_{3} = 12B, i.e. an energy change of ΔE_{v} + 6B

v = 0 → v =1 + E_{4} = 20B → E_{5} = 30B, i.e. an energy change of ΔE_{v} + 10B

v = 0 → v =1 + E_{6} = 42B → E_{7} = 56B, i.e. an energy change of ΔE_{v} + 14B

The transition energy increases by 4B for each successive rotation-vibration change as the lower J value increases.

The P branch transitions are those in which the same vibrational change occurs, but the rotational quantum number decreases by one unit. The P branch transitions in the diagram of Figure 2 are:

v = 0 → v =1 + E_{2} = 6B → E_{1} = 2B, i.e. an energy change of ΔE_{v} − 4B

v = 0 → v =1 + E_{4} = 20B → E_{3} = 12B, i.e. an energy change of ΔE_{v} − 8B

v = 0 → v =1 + E_{6} = 42B → E_{5} = 30B, i.e. an energy change of ΔE_{v} − 12B

v = 0 → v =1 + E_{8} = 72B → E_{7} = 56B, i.e. an energy change of ΔE_{v} − 16B

The transition energy decreases by 4B for each successive rotation-vibration change as the lower J value increases.

Figure 3 shows a spectrum of CO_{2} which exhibits the P, Q, and R branches described above.

**Figure 3** An absorption spectrum of CO_{2}

The value of 4B may be estimated from the spectrum, e.g., there are 13 spaces between 660 and 640 cm^{-1}, so 13 × 4B = 20 cm^{-1} and B = 0.385 cm^{-1}.

In terms of reciprocal centimetres B = h/(8π^{2}Ic)

The moment of inertia of the CO_{2} molecule is 2mr^{2} where m is the mass of the oxygen atom and r is the C=O bond length.

Rearranging the equation for B gives:

r = (hN_{A}/16π^{2}cmB)^{1/2}

N_{A} is the Avogadro constant.

The equation gives the C=O bond length as:

r(C=O) = (hN_{A}/(16π^{2}c × 0.016 × 38.5))^{1/2} = 1.17 × 10^{-10} = 117 pm [1 picometre = 10^{-12} m]

The length of the molecule between the oxygen nuclei is thus 117 × 2 = 234 pm, the accepted length from electron diffraction measurements is 232.6 pm.

*Details of the Q branch*

According to the theory given above the various Q branch absorptions should all have the same wavenumber since they all refer to the pure vibrational transition v = 0 → v = 1. The lowest energy pure vibrational transition occurs at the accepted centre of the whole band of absorptions of which the CO_{2} spectrum consists; 667.4 cm^{-1}. That transition arises when the rotational quantum number is zero and the rotational energy of the molecule is zero. The second Q branch transition arises from the same vibrational change, but the molecule has 1 unit of rotational energy and as the Q branch progression continues the molecules have greater and greater amounts of rotational energy. There is what is known as a Coriolis Effect with longer molecules being produced by the increasing rotational energy and thus the vibrational energy difference increases slightly for each successive Q branch member.

In terms of the distribution of energy in the whole band, the Q branch accounts for 50%, with the other 50% divided between the P and R branches. The Q branch is much more likely to be saturated in the atmosphere than any of the P and R branch members.

**The spectrum of water vapour**

The spectrum of water vapour is shown below. It is for the normal concentration of water vapour and with a path length of 100 metres.

This complex spectrum is composed mainly of the many rotational lines and beyond 1350 cm^{-1 } there are some of the components of the P branch of the fundamental bending mode centred at 1595 cm^{-1} and it shows the infrared 'window' through which none of the greenhouse gases absorb with the except ozone. The window is rather cloudy, but water vapour absorbs very little in that region: 750-1250 cm^{-1} . In the upper troposphere where the actual concentration is very small the window is much clearer.

**Line broadening**

The HITRAN database contains details of thousands of spectral lines representing the various transitions that occur when molecules absorb or emit radiation. They are all experimental values that have been produced with the most up-to-date spectrophotometers. The quoted values are the line centres given to four decimal places and don't imply that the transitions exhibit themselves in spectra as lines in the sense that they have no width. The lines have associated line strengths.

To extract a spectrum from HITRAN requires the hitranpc programme which has the facilities to provide a spectrum of between one and 40 gases at given concentrations and total pressures.

As an example, the hitranpc-generated spectrum of 400 ppmv CO_{2} with a path length of one metre and with a total atmospheric pressure of one atmosphere is shown below. The 'lines' have been broadened according to the Lorenzian theory.

It is obvious that there are no 'lines' as such and that the three absorptions have particular widths. The three transitions responsible for the absorptions are the second, third and fourth members of the P branch of the fundamental bending mode and have centres at 664.2679, 662.7143 and 661.1607 cm^{-1} respectively. The difference between adjacent centres is 1.554 cm^{-1} and is equivalent to 4B, making B = 0.388 cm^{-1}.

The next spectrum is for the same number density concentration of CO_{2} and with the same path length, but with only one tenth of the atmospheric pressure.

The three transitions that are members of the P branch of the fundamental bending mode show much lower transmission values, but are significantly narrower than in the 1 atmosphere pressure case. The other transitions observable are from an overlapping transition and in the higher pressure spectrum blend in so that they are not noticeable.

In theory, the transition probability for any transition is determined by the area under the absorption curve. This means that any broadening of a transition by increased pressure does not affect the total absorption. The bands become broader, but the maximum absorption decreases. In the atmosphere the lessening pressure broadening contributes to the 'lines' having higher emission levels in their centres, but lower emission levels 'off peak'.

This description of pressure broadening refers only to the effect of varying the total pressure rather than the partial pressure of the CO_{2}. The self-broadening which does occur in the Venusian atmosphere is not relevant to the Earth's atmosphere since CO_{2} - CO_{2} collisions have very low probability of occurrence at the partial pressure in the atmosphere.

**'Lines' overlap**

Pressure broadened lines or bands overlap so that over the spectral range there are no instances of 100% transmission. This is illustrated by spectra of CO_{2} with 100 m and 200 m path lengths respectively that show that over the spectral range there is some absorption at all wavenumbers. Please note that the spectra have been replaced with their correct titles. The previous titles were for 100 m and 200 m path lengths in error, pointed out by Brenden O'Connor.

This spectrum is for 380 ppmv CO_{2} with a path length of 100 m and shows that there is absorption at all wavenumbers, although for the extremes of the P and R branches this claim is not clear. The next spectrum is for the same concentration of CO_{2}, but with twice the path length.

This spectrum clearly shows that there is an underlying 'blanket' absorption over the wavenumber range. This is because the individual pressure broadened bands overlap.

Both spectra are presented for the wavenumber range of 630 cm^{-1} to 715 cm^{-1} which is why they are truncated.