https://www.hindawi.com/journals/ijas/2017/9251034/

3.4. RT Calculations with MRT Lineshape. While the preceding calculations are based on the standard molecular collision theory, considering collisional broadening of spectral transitions, which are characterized by a Lorentzian lineshape or at higher altitudes also by a Voigt profile, we have also performed extensive calculations using a more sophisticated lineshape as given by the molecular response theory (MRT) (Harde et al. [40–44]). This theory represents a generalization and unification of the classical collision theories of Lorentz and on the other hand of van Vleck and Weisskopf, considering thermalization of molecules during a collision, which in its limit is determined by the reciprocal of the molecular transition frequency and controlled by Heisen- berg’s uncertainty principle (Harde & Grischkowsky [44]). The specialty applying MRT is that it not only describes the near resonance absorption and emission behavior as already adequately characterized by a Lorentzian, but also reflects the far wing response, which is mainly caused by the ultrafast time response of molecules to an external electric field during collisions.

The results of such a calculation at a mean cloud cover of 66% and at otherwise same conditions as described in Section 3.3 is shown in Table 4. To compare the alterations caused by doubling the CO2 concentration it is again suffi- cient only to display the calculations for 380 and 760 ppm CO2. It is interesting to see that the more sophisticated lineshape and the continuum background absorption almost have no influence on Δ𝐹2×CO2 , the RF at doubled CO2, which

agrees within better than 1% with the simpler calculations shown in Table 3, neglecting the continuum absorption and using a Lorentzian lineshape. Also changes in the other fluxes at doubled CO2 concentration are consistent within 0.1 W/m2 ; only the back-radiation changes to Δ𝐼down = 22𝐴 total −3.18 W/m and is about 0.3 W/m smaller; thus, the power left in the atmosphere increases by this amount.

So, calculations based on the classical collision theory obviously reproduce quite reliable data of any radiation changes in the atmosphere. Since the far wings of the CO2 lines are found to decay more rapidly than a Lorentzian and thus should contribute less to the total absorption (see, e.g., Edwards and Strow [47]), such calculations even simulate slightly worse conditions and result in a more conservative assessment of global warming by CO2. For actual CO2 line- shape studies see also Happer [48], and for RT calculations including CO2 line mixing see Mlynczak et al. [49], and Ozak et al. [50].

]]>I checked literature and that confirmed my speculation. The temperature dependence can, indeed, be determined through quantum mechanical calculation of the transition probability. Here is one publication that discusses that point

In the previous comment I discussed the time between collisions and likelihood of a transition in a collision. That kind of division cannot really be made as telling whether a collision has occurred when that has not led to a transitions cannot be done unambiguously. There’s no precise dividing line between the cases where two molecules pass each other so far that their interaction cannot be considered a collision and a clear collision. The idea should, however, be clear. In some cases the is a sharp change from very weak interaction to a collision that leads to a transition with almost 100% certainty while in other cases the change goes more smoothly and leads to stronger temperature dependence.

One observation from that paper is that the power law dependence of the linewidth on temperature is not exact. For CO2 it’s a good approximation while it’s worse for water vapor.

]]>Eli,

I don’t fully agree. Whenever we see the Lorentzian lineshape, we have a situation where the lifetime of the excitation determines fully the lineshape. That means that the only properties of collisions that matter are their frequency and the likelihood that a collision leads to a state transition. Other details of the collision affect the form of far tail, i.e. its deviation from pure Lorentzian.

Temperature affects the linewidth trough its influence on density and average speed of the molecules. At constant pressure the density is inversely proportional to density. The speed is proportional to the square root of the temperature. The time between collisions is inversely proportional to both density and speed. These factors lead to the proportionality of the linewidth to *pressure/temperature^0.5*. The exponent of the temperature is not exactly 0.5. I haven’t checked what literature tells about that. Perhaps it’s due to the influence of collision energy on the probability of a transition in a collision, which is often close to one but may be less.

Increasing the number of collisions (pressure) and the average energy of a collision (temperature) affects the effective broadening. Energy transfer, obviously occurs on the collisional potential energy surface which is what determines the collisional lifetime.

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