Introduction to the problem
I computed the absorption by carbon dioxide in earth atmosphere at $14,7$ µm wavelength (absorption band of CO2 where the earth emission is the strongest by far), taking into account the carbon dioxide density decreasing with altitude, the exact solution with hyper geometric series and I found something which is very close to unity, which is completely wrong obviously, so this is a big issue ! That means an absorption of closely $1$ for $400$ ppm of CO2 without taking into account the other atmospheric phenomena ! Let's look into the details, and I think the major issue is the semi-classical expression of polarizability, which is not pressure-dependent.
Maxwell equations
First to compute absorption, we need to consider, to simply one polarization (we need to compute also the other one and do an average for the transmission but this is not a big deal here). For a plane wave who propagates in $z$ direction, we assume an electric field $E_x(z)$ who obeys to:
$$ \left[ \partial_z^2 + k^2 (1 + f(z)) \right] E_x(z) = 0 $$
with $1 + f(z) = \epsilon(z)$, with $\epsilon$ the relative permittivity of CO2 and $k = 2 \pi / \lambda$. It is obvious that $f(z) << 1$. We can rely permittivity to molecular polarizability by Debye formula:
$$ \epsilon - 1 \simeq N(z) \alpha = f(z) $$
with $\alpha$ the polarizability of one molecule of CO2 and $N(z)$ the number of CO2 particles by unit of volume. We assume the form of $N$ with exponential approximation:
$$ N(z) = N_0 exp(-z/H) $$
With $H \simeq 6$ km. The solution is exact in terms of hypergeometric serie, but we can have a more comprehensive approximation by posing into the propagation equation:
$$ E = A e^{i k z + w(z)} $$
with $e^{w(z)}$ non oscillatory at infinity, $E(0) = E_0$,and assuming (easy to verify) $\partial_z << k$:
$$ w'(z) = i f(z) k/2 $$
We find transmission coefficient:
$$T = \frac{ \vert E(+ \infty) \vert^2 }{E_0^2} = e^{- N_0 \text{imag}(\alpha) k H} $$
We need now the expressions of $\alpha$ and $N_0$.
Value of number of particles of CO2 per unit volume at earth surface, $N_0$
To determine $N_0$, we can use what we know as there is a ratio of $\beta = 400$ ppm of CO2 molecules in all the troposphere. So we can write:
$$ N_0 = \beta N_A \mu_{a} / M_a $$
with $N_A$, $\mu_a$ and $M_a$: Avogadro's number, $\mu_a$ mass density of air at surface and $M_a$ air molar mass.
Now we have to determine $\alpha$, the sensitive point.
Value of CO2 gas polarizability
For this paragraph I used this publication. According to a Drude model we have the following expression:
$$ \alpha = \sum_{i=1}^3 \frac{f_i e^2/m}{\omega_i^2 - \omega^2 - i\gamma_i \omega} $$
Since this expression must hold for a single molecule of CO2, I took the value from the curve fitting in the paper. This as been measure with 1 bar of CO2 pressure at 0°C.
Expression of transmission (or absorption) and problem exposed
Using the polarizability coefficients provided in the paper, I found $T \simeq exp(-51) \simeq 10^{-22}$ at $14.7$ µm for the transmission coefficient, which is way to small ! And that gives an absorption, almost equal to unity, by the way superior to the emmissivity of the whole atmosphere, which is just crazy.
Question, discussion
Is the value of the polarizability right ? Or did I make a basic mistake anywhere else ?
Don't you think the relaxation time $1 / \gamma $ or the oscillator strength are strongly dependent on inter-molecular force (and so on the pressure or density)? I don't have enough info on this. Maybe quantum/statistical physicists may help me to find the right expression of polarizability.
Thanks for your interest in the topic.
