Line-by-line Absorption

This section describes the physical process of absorption lines of different molecules absorbing and emitting spectral radiance in the atmosphere.

These are the types of line-by-line absorption considered here:

• Line-by-line Absorption Overview, where each absorption line is considered separately.

  • Without Zeeman effect

  • With Zeeman effect

• Line-mixing using Error-corrected Sudden, where the absorption lines of similar energies of a molecule are mixed together.

Line-by-line Absorption Overview

The absorption in plain line-by-line absorption is simply the sum of all absorption by each absorption line. The absorption of a single absorption line is described by the following equations:

\[\alpha = S(\cdots) F(\cdots),\]

where \(\alpha\) is the absorption coefficient, \(S\) is the Line Strength operator, and \(F\) is the Line Shapes operator.

Both \(S\) and \(F\) change slightly if Zeeman effect is considered. The main way that Zeeman effect changes the calculations is via the polarization it introduces to the propagation matrix summation.

Without Zeeman effect

The contribution to propagation matrix from all non Zeeman-split absorption lines is simply

\[K_{A, lbl} = \mathrm{Re} \sum_i \alpha_{i},\]

and from this the full matrix is

\[\begin{split}\mathbf{K}_{lbl} = \left[ \begin{array}{llll} K_{A, lbl}&0&0&0\\ 0&K_{A, lbl}&0&0\\0&0&K_{A, lbl}&0\\0&0&0&K_{A, lbl} \end{array} \right],\end{split}\]

where \(i\) is the pseudo-index of the absorption line and lbl is the pseudo-index of the plain line-by-line absorption for the sake of summing up absorption.

Note

Plain line-by-line absorption only contribute towards the diagonal of the propagation matrix.

With Zeeman effect

When Zeeman effect is considered, there are effectively 3 separate kinds of polarized absorption added to the propagation matrix

\[\begin{split}K_{\sigma_\pm, z} &= \sum_i \alpha_{i, \sigma_\pm} \\ K_{\pi, z} &= \sum_i \alpha_{i, \pi}\end{split}\]

and from this, the full matrix contribution is

\[\begin{split}\mathbf{K}_{z} =\sum_\pm\left(\\ \mathrm{Re} K_{\sigma_\pm,z} \left[\begin{array}{llll} 1 + \cos^2\theta_m & \sin^2\theta_m \cos 2 \eta_m & -\sin^2 \theta_m \sin 2 \eta_m & \mp 2 \cos \theta_m \\ \sin^2\theta_m \cos 2 \eta_m & 1 + \cos^2 \theta_m & 0 & 0 \\ -\sin^2\theta_m \sin 2 \eta_m & 0 & 1 + \cos^2 \theta_m & 0 \\ \mp 2 \cos\theta_m & 0 & 0 & 1 + \cos^2 \theta_m \end{array}\right] + \mathrm{Im} K_{\sigma_\pm,z} \left[\begin{array}{llll} 0 & 0 & 0 & 0 \\ 0 & 0 & \pm 2 \cos \theta_m & -\sin^2 \theta_m \sin 2 \eta_m \\ 0 & \mp 2 \cos \theta_m & 0 & -\sin^2 \theta_m \cos 2 \eta_m \\ 0 & \sin^2\theta_m \sin 2 \eta_m & \sin^2\theta_m \cos 2 \eta_m & 0 \end{array}\right] \right) +\\ \mathrm{Re} K_{\pi,z} \left[\begin{array}{llll} \sin^2\theta_m & -\sin^2\theta_m \cos 2 \eta_m & \sin^2 \theta_m \sin 2 \eta_m & 0 \\ -\sin^2\theta_m \cos 2 \eta_m & \sin^2\theta_m & 0 & 0 \\ \sin^2\theta_m \sin 2 \eta_m & 0 & \sin^2\theta_m & 0 \\ 0 & 0 & 0 & \sin^2\theta_m \end{array}\right] + \mathrm{Im} K_{\pi,z} \left[ \begin{array}{llll} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \sin^2 \theta_m \sin 2 \eta_m \\ 0 & 0 & 0 & \sin^2 \theta_m \cos 2 \eta_m \\ 0 & -\sin^2\theta_m \sin 2 \eta_m & -\sin^2\theta_m \cos 2 \eta_m & 0 \end{array}\right],\end{split}\]

where the somewhat weird \(\pm\)-sum is over the sigma components. Here the angles \(\theta_m\) and \(\eta_m\) are the angles with regards to the magnetic field.

Given a spherical coordinate observation system with zenith angle \(\theta_z\) and aziumuth angle \(\eta_a\) and a local magnetic field with upwards facing strength \(B_w\), eastward facing strength \(B_u\) and northward facing strength \(B_v\), these angles are given by

\[\begin{split}\theta_m = \arccos\left(\frac{B_v \cos\eta_a \sin\theta_z + B_u \sin\eta_a \sin\theta_z + B_w \cos\theta_z}{ \sqrt{B_w^2 + B_u^2 + B_v^2} } \right) \\ \eta_m = -\mathrm{atan2}\left(B_u \cos \eta_a-B_v \sin\eta_a,\; B_u \cos\theta_z \sin\eta_a + B_v \cos\theta_z\cos\eta_a - B_w\sin\theta_z \right)\end{split}\]

Line Shapes

Line shapes should distribute absorption as a function of frequency. By convention, the line shape is normalized to have an integral of 1.

Voigt Line Shape

\[F = \frac{1 + G_{lm} - iY_{lm}}{\sqrt{\pi}G_D} w(z),\]

where

\[z = \frac{\nu - \nu_0 - \Delta\nu_{lm} - \Delta_nu_Z - \Delta\nu_{P,0} + iG_{P,0}}{G_D},\]

where

Parameter	Description
\(\nu\)	The sampling frequency
\(\nu_0\)	The line center frequency
\(G_D\)	The scaled Doppler broadening half-width half-maximum
\(\Delta\nu_Z\)	The Zeeman shift
\(G_{P,0}\)	The pressure broadening - half width half maximum in the Lorentz profile
\(\Delta\nu_{P,0}\)	The pressure shift
\(Y_{lm}\)	The 1st order Line-mixing parameter
\(G_{lm}\)	The 2nd-order strength modifying line mixing parameter
\(\Delta\nu_{lm}\)	The 2nd-order line-mixing shift
\(w(z)\)	The Faddeeva function.

For more information about how \(G_{P,0}\), \(\Delta\nu_{P,0}\), \(Y_{lm}\), \(G_{lm}\), and \(\Delta\nu_{lm}\) are computed see Line Shape Parameters. The scaled Doppler broadening half width half maximum is given by

\[G_D = \sqrt{\frac{2000 R T}{mc^2}} \nu_0,\]

where

Parameter	Description
\(R\)	The ideal gas constant in Joules per mole per Kelvin,
\(T\)	The temperature in Kelvin,
\(m\)	The mass of the molecule in grams per mole, and
\(c\)	The speed of light in meters per second.

The factor 2000 is to convert to SI units.

The Faddeeva function is in ARTS computed using the MIT-licensed Faddeeva package, which is based in large parts on the work by Zaghloul and Ali [41].

The Zeeman line-shift is derived from the magnetic field strength and the magnetic quantum number of the transition. A linear Zeeman effect is assumed such that

\[\Delta\nu_Z = \frac{e} {4 \pi m_e} \left(M_l g_{l,z} - M_u g_{u,z} \right),\]

where \(e\) is the elementary charge, \(m_e\) is the mass of an electron, \(M_l\) and \(M_u\) are the projection of the lower and upper states, respectively, of the angular momentum quantum number on the magnetic field, and \(g_{l,z}\) and \(g_{u,z}\) are the lower and upper state Landé g-factors, respectively. The latter are generally computed ahead of time, e.g., as by Larsson and Lankhaar [14], Larsson et al. [15].

Note

It is important to not confuse the line-mixing parameters used here with full line mixing as described below. The line-mixing paramters here are still plain line-by-line absorption, but it is important that there are no cut lines and that the data for all line-paramters are derived toghether.

Line Shape Parameters

The line shape parameters supported by ARTS are

Parameter	Description	Pressure Dependency
\(G_{P,0}\)	Pressure broadening half width half maximum, collision-independent.	\(P\)
\(G_{P,2}\)	Pressure broadening half width half maximum, collision-dependent.	\(P\)
\(\Delta\nu_{P,0}\)	Pressure shift, collision-independent.	\(P\)
\(\Delta\nu_{P,2}\)	Pressure shift, collision-dependent.	\(P\)
\(\nu_{VC}\)	Velocity changing frequency.	\(P\)
\(\eta\)	Correlation parameter.	\(-\)
\(Y_{lm}\)	1st order Line-mixing parameter.	\(P\)
\(G_{lm}\)	2nd-order strength modifying line mixing parameter.	\(P^2\)
\(\Delta\nu_{lm}\)	2nd-order line-mixing shift.	\(P^2\)

These parameters are all computed species-by-species before being volume-mixing ratio weighted and summed up. In equation form:

\[L = \frac{\sum_i x_i L_i}{\sum_i x_i},\]

where \(L\) is a placeholder for any of the line shape parameters, and \(x\) is the volume-mixing ratio, and \(i\) is a species index. The normalization is there to allow fewer than all species to contribute to the line shape parameters.

The temperature dependencies of the individual \(L_i\) are computed based on avaiable data. There is no general form avaiable, so instead the temperature dependencies are computed based on the data avaiable for each species as:

Name	Equation	Description
T0	\(L_i(T) = X_0\)	Constant regardless of temperature
T1	\(L_i(T) = X_0 \left(\frac{T_0}{T}\right)^{X_1}\)	Simple power law
T2	\(L_i(T) = X_0 \left(\frac{T_0}{T}\right) ^ {X_1} \left[1 + X_2 \log\left(\frac{T_0}{T}\right)\right]\)	Power law with compensation.
T3	\(L_i(T) = X_0 + X_1 \left(T - T_0\right)\)	Linear in temperature
T4	\(L_i(T) = \left[X_0 + X_1 \left(\frac{T_0}{T} - 1\right)\right] \left(\frac{T_0}{T}\right)^{X_2}\)	Power law with compensation. Used for line mixing.
T5	\(L_i(T) = X_0 \left(\frac{T_0}{T}\right)^{\frac{1}{4} + \frac{3}{2}X_1}\)	Power law with offset.
AER	\(L_i(200) = X_0\), \(L_i(250) = X_1\), \(L_i(296) = X_2\), \(L_i(340) = X_3\). Linear interpolation inbetween.	Inspired by the way AER deals with linemixing.
DPL	\(L_i(T) = X_0 \left(\frac{T_0}{T}\right) ^ {X_1} + X_2 \left(\frac{T_0}{T}\right) ^ {X_3}\)	Double power law.
POLY	\(L_i(T) = X_0 + X_1 T + X_2 T ^ 2 + X_3 T ^ 3 + \cdots\)	Polynomial in temperature. Used internal in ARTS when training our own linemixing.

here, \(X_0\) … \(X_N\) are all model supplied constants whereas \(T\) is the temperature in Kelvin and \(T_0\) is the reference temperature of the model parameters.

Line Strength

Local thermodynamic equilibrium

For local thermodynamic equilibrium (LTE), the line strength is given by

\[S_{LTE} = \rho \frac{c^2\nu}{8\pi} \left[1 - \exp\left(-\frac{h\nu}{kT}\right)\right] \frac{g_u\exp\left(-\frac{E_l}{kT}\right)}{Q(T)} \frac{A_{lu}}{\nu_0^3},\]

where \(\rho\) is the number density of the absorbing species,

\[\rho = \mathrm{VMR}\frac{P}{kT},\]

where VMR is the volume-mixing ratio of the absorbing species.

Parameter	Description
\(\rho\)	Number density of the absorbing isotopologue, \(\mathrm{VMR} \cdot P / (kT)\)
\(\mathrm{VMR}\)	Volume-mixing ratio of the absorbing species
\(P\)	Atmospheric pressure
\(c\)	Speed of light
\(\nu\)	Sampling frequency
\(\nu_0\)	Line centre frequency
\(h\)	Planck constant
\(k\)	Boltzmann constant
\(T\)	Temperature
\(g_u\)	Degeneracy of the upper state
\(E_l\)	Energy of the lower state
\(Q(T)\)	Partition function at temperature \(T\)
\(A_{lu}\)	Einstein A coefficient for spontaneous emission

Non-local thermodynamic equilibrium

For non-LTE, the line strength is given by

\[S_{NLTE} = \rho \frac{c^2\nu}{8\pi} \left(r_l \frac{g_u}{g_l} - r_u\right) \frac{A_{lu}} {\nu_0^3},\]

and the added emissions are given by

\[K_{NLTE} = \rho \frac{c^2\nu}{8\pi} \left\{r_u\left[ 1 - \exp\left(\frac{h\nu_0}{kT}\right)\right] - \left(r_l \frac{g_u}{g_l} - r_u\right) \right\} \frac{ A_{lu}}{\nu_0^3},\]

where \(r_l\) and \(r_u\) are the ratios of the populations of the lower and upper states, respectively. Note that \(K_{LTE} = 0\), as it represents “additional” emission due to non-LTE conditions. Also note that \(K_{NLTE}\) may be negative.

To ensure ourselves that this can be turned into the expression for LTE, we can rewrite the above for the expression that \(r_l\) and \(r_u\) would have in LTE according to the Boltzmann distribution:

\[r_l = \frac{g_l\exp\left(-\frac{E_l}{kT}\right)}{Q(T)}\]

and

\[r_u = \frac{g_u\exp\left(-\frac{E_u}{kT}\right)}{Q(T)}\]

Putting this into the ratio-expression for \(S_{NLTE}\) with the following simplification steps:

Expansion:

\[\left(r_l \frac{g_u}{g_l} - r_u\right) = \frac{g_u}{Q(T)}\left[\exp\left(-\frac{E_l}{kT}\right) - \exp\left(-\frac{E_u}{kT}\right)\right].\]

Extract lower state energies:

\[\frac{g_u}{Q(T)}\left[\exp\left(-\frac{E_l}{kT}\right) - \exp\left(-\frac{E_u}{kT}\right)\right] \frac{\exp\left(-\frac{E_l}{kT}\right)}{\exp\left(-\frac{E_l}{kT}\right)} \rightarrow \left[1 - \exp\left(-\frac{h\nu_0}{kT}\right)\right]\frac{g_u\exp\left(-\frac{E_l}{kT}\right)}{Q(T)},\]

where this last step is possible because we estimate that \(E_u-E_l = h\nu_0\). Note how the expression for \(K_{NLTE}\) is 0 under LTE conditions. As it should be. This is seen by putting the above RHS and the expression for \(r_u\) into the expression for \(K_{NLTE}\):

\[K_{NLTE} = \rho \frac{c^2\nu}{8\pi} \left\{\frac{g_u\exp\left(-\frac{E_u}{kT}\right)}{Q(T)}\left[ 1 - \exp\left(\frac{h\nu_0}{kT}\right)\right] - \left[1 - \exp\left(-\frac{h\nu_0}{kT}\right)\right]\frac{g_u\exp\left(-\frac{E_l}{kT}\right)}{Q(T)} \right\} \frac{ A_{lu}}{\nu_0^3} = 0.\]

The ratio between LTE and non-LTE line strength remaining is:

\[\frac{S_{NLTE}}{S_{LTE}} = \frac{1 - \exp\left(-\frac{h\nu_0}{kT}\right)}{1 - \exp\left(-\frac{h\nu}{kT}\right)}.\]

It is clear that the non-LTE expression is the one that is incorrect here. The energy of the emitted photon is not \(h\nu_0\) but \(h\nu\), and as such the actual energy of the transition is \(E'_u-E'_l = h\nu\), but this should be relatively close in cases where we actually care about non-LTE (which is low density, low collision atmospheres).

Zeeman effect

If Zeeman effect is considered, the emission and absorption terms above are modified by quantum number state distribution. For O₂, for example, this introduces a factor of

\[\begin{split}S_z = f(\Delta M) \left( \begin{array}{ccc} J_l & 1 & J_u \\ M_l & \Delta M & - M_u \end{array} \right)^2,\end{split}\]

where \(\Delta M \in \left[-1,\;0,\;1\right]\) is the change in quantum number for angular rotational momentum projection along the magnetic field for \(\sigma_-\), \(\pi\), and \(\sigma_+\), respectively, \(f(\Delta M)\) is the 0.75 for \(\sigma_\pm\) and 1.5 for \(\pi\), and \(J_l\) and \(J_u\) are the lower and upper total angular rotational momentum quantum number. The \((:::)\) construct is the Wigner 3-j symbol. It can be computed using software such as that by Johansson and Forssén [12].

Line-mixing using Error-corrected Sudden

When the atmosphere is at sufficient pressure, collisions occur frequently enough that absorption lines of a vibrational-rotational band can no longer be treated independently. Molecules undergoing collisions may exchange rotational angular momentum, transferring population between rotational levels. At intermediate pressures this introduces off-diagonal couplings between lines in a spectral band, leading to the phenomenon of line mixing. At high pressures the lines collapse towards a pressure-broadened Q-branch.

The Error-corrected Sudden (ECS) approximation provides a rigorous, quantum-mechanical framework for computing this mixing. The “Sudden” part refers to the Infinite-Order-Sudden (IOS) approximation, in which the collision time is assumed short compared with the rotational period. The “Error-corrected” part refers to the subsequent rescaling of the relaxation matrix elements to satisfy the first-order optical sum rule exactly, removing a systematic bias that arises from the sudden approximation.

ECS Line Shape

For a band of \(n\) interacting absorption lines, the ECS complex absorption shape for a single broadening species (see Multiple Broadening Species for the full expression) is written in terms of the complex relaxation matrix \(\mathbf{W}\) as

\[\chi(\nu) \propto \mathrm{Im}\left[\mathbf{d}^T \left(\nu \mathbf{I} - \mathbf{W}\right)^{-1} \mathbf{p}\, \mathbf{d}\right],\]

where \(\mathbf{d}\) is the vector of reduced dipole matrix elements (one entry per line), \(\mathbf{p}\) is the diagonal matrix of lower-state thermal populations (\(p_j = g_j\exp(-E_l^{(j)}/kT)/Q(T)\), the Boltzmann fractional population of the lower state of line \(j\), following the Local thermodynamic equilibrium notation where \(g_j\) is the lower-state degeneracy and \(Q(T)\) the partition function), and \(\mathbf{W}\) is the full complex \(n \times n\) relaxation matrix whose diagonal and off-diagonal elements are described in Relaxation Matrix.

The relaxation matrix is diagonalised as \(\mathbf{W} = \mathbf{V} \tilde{\boldsymbol{\nu}} \mathbf{V}^{-1}\), where \(\tilde{\boldsymbol{\nu}}\) is the diagonal matrix of complex equivalent line positions. Each equivalent line \(k\) has a complex frequency \(\tilde{\nu}_k\) (real part: position, imaginary part: pressure broadening) and a complex equivalent strength \(\tilde{S}_k\).

Explicitly, the equivalent strength for line \(k\) is

\[\tilde{S}_k = \left(\sum_j d_j V_{jk}\right) \left(\sum_j p_j d_j V^{-1}_{kj}\right),\]

and the ECS line shape function is

\[F_{ECS}(\nu) = \frac{\sqrt{\ln 2}}{\sqrt{\pi}} \sum_k \tilde{S}_k \frac{w(z_k)}{G_{D,k}},\]

where \(w\) is the Faddeeva function and

\[z_k = \frac{\left(\tilde{\nu}_k - \nu\right)\sqrt{\ln 2}}{G_{D,k}}, \qquad G_{D,k} = G_D^{fac} \cdot \mathrm{Re}\!\left[\tilde{\nu}_k\right],\]

with the Doppler scale factor

\[G_D^{fac} = \sqrt{\frac{2000 R T}{m c^2}},\]

where \(R\) is the ideal gas constant in J mol^-1 K^-1, \(m\) the molar mass in g mol^-1, \(c\) the speed of light, and \(T\) the temperature (same symbols as in the plain LBL definition in Line Shapes).

Note that \(G_{D,k}\) is formed by multiplying \(G_D^{fac}\) by \(\mathrm{Re}[\tilde{\nu}_k]\) — the real part of the \(k\)-th eigenvalue — rather than by any original line centre \(\nu_{0,j}\). This is necessary because eigenvalue decomposition does not in general return eigenvalues in the same order as the input lines, so there is no well-defined mapping from equivalent line \(k\) to a single physical line \(j\). Using \(\mathrm{Re}[\tilde{\nu}_k]\) keeps the Doppler width self-consistent with the actual position of each equivalent line.

The contribution to the propagation matrix from the entire band is then

\[K_{A, ecs} = N \nu \left(1 - \exp\!\left(-\frac{h\nu}{kT}\right)\right) \mathrm{Re}\!\left[F_{ECS}(\nu)\right],\]

where \(N\) is the total number density of the absorbing species.

Note

Zeeman splitting within a band is not currently supported together with ECS line mixing.

Relaxation Matrix

The complex relaxation matrix \(\mathbf{W}\) has dimensions \(n \times n\) (lines in the band) and is constructed as follows.

The real part of the diagonal elements carries the (pressure-shifted) line centre frequencies:

\[\mathrm{Re}\, W_{ii} = \nu_{0,i} + \Delta\nu_{P,0,i},\]

where \(\nu_{0,i}\) is the vacuum line centre and \(\Delta\nu_{P,0,i}\) is the pressure shift of line \(i\).

The imaginary part of the diagonal elements carries the pressure broadening:

\[\mathrm{Im}\, W_{ii} = G_{P,0,i},\]

where \(G_{P,0,i}\) is the pressure-broadening half-width half-maximum.

The off-diagonal elements \(W_{ij}\) (\(i \neq j\)) encode the rate of transfer from line \(j\) to line \(i\) via collisions. Their construction from the ECS basis rates is described below.

ECS Basis Rates

The ECS approach introduces two species-dependent functions of the integer angular momentum transfer channel \(L\):

Basic rate \(Q(L)\):

This encodes the intrinsic probability of a collision transferring \(L\) units of angular momentum. Its temperature dependence is parameterised as

\[Q(L, T) = s(T) \cdot \frac{e^{-\beta(T)\, E_L / kT}}{[L(L+1)]^{\lambda(T)}},\]

where \(E_L\) is the rotational energy of level \(L\), and \(s(T)\), \(\beta(T)\), and \(\lambda(T)\) are temperature-dependent model parameters stored per broadening species (see Line Shape Parameters for the available temperature dependence forms).

Adiabatic factor \(\Omega(L)\):

The IOS approximation becomes inaccurate when the rotational period approaches the collision duration. The adiabatic factor corrects for this using the coupling model:

\[\Omega(L) = \frac{1}{\left[1 + \dfrac{\omega_{L,L-2}^2\, \tau_c^2}{24}\right]^2},\]

where \(\omega_{L,L-2} = (E_L - E_{L-2})/\hbar\) is the angular frequency of the \(L \to L-2\) rotational transition and

\[\tau_c = \frac{\sigma_c(T)}{\bar{v}}, \qquad \bar{v} = \sqrt{\frac{8kT}{\pi\mu}},\]

with \(\sigma_c(T)\) the temperature-dependent mean collisional diameter (a per-species model parameter, distinct from the reduced dipole \(d_j\)), \(\mu\) the reduced mass of the colliding pair, and \(\bar{v}\) the mean relative thermal speed.

Off-diagonal Elements

The off-diagonal elements of \(\mathbf{W}\) are computed species-by-species. All variants follow the formal IOS structure: they are written as a sum over even angular momentum transfer channels \(L\), weighted by the ratio \(Q(L)/\Omega(L)\) and by Wigner 3-j and 6-j coupling coefficients. After the IOS computation an error-correction (sum-rule rescaling) is applied (see Sum-rule Correction).

Detailed balance is enforced throughout: the rate of transfer from a lower strength line \(j\) to a higher strength line \(i\) is obtained from the downward rate \(W_{ij}\) via

\[W_{ji} = W_{ij} \exp\!\left(\frac{E_j - E_i}{kT}\right),\]

where \(E_i\) is the energy of the lower rotational state of line \(i\) (using the same \(E_l\) convention as in Local thermodynamic equilibrium).

The four variants implemented in ARTS, corresponding to the four line shape model types VP_ECS_HARTMANN, VP_ECS_MAKAROV, VP_ECS_STOTOP, and VP_ECS_SPHTOP, are described below.

Linear Molecules — Hartmann (CO₂)

For linear molecules (e.g. CO₂) the off-diagonal rate from line \(j\) (upper/lower rotational quantum numbers \(J'_i, J'_f\), vibrational angular momentum \(l\)) to line \(i\) (\(J_i, J_f\)) is [25]

\[\begin{split}W_{ij} = \Omega(J_i)\, (2J'_i+1)\sqrt{(2J_f+1)(2J'_f+1)} \sum_L (2L+1) \begin{pmatrix} J_i & J'_i & L \\ l & -l & 0 \end{pmatrix} \begin{pmatrix} J_f & J'_f & L \\ l & -l & 0 \end{pmatrix} \begin{Bmatrix} J_i & J_f & 1 \\ J'_f & J'_i & L \end{Bmatrix} \frac{Q(L)}{\Omega(L)},\end{split}\]

where the sum runs over even \(L \geq \max(|J_i-J'_i|, |J_f-J'_f|)\), \((\,\cdots)\) denotes a Wigner 3-j symbol, and \(\{\,\cdots\}\) a Wigner 6-j symbol.

The corresponding reduced dipole element used in the equivalent-strength calculation is

\[\begin{split}d(J_f, J_i) = (-1)^{J_f + l_f + 1} \sqrt{2J_f+1}\; \begin{pmatrix} J_f & 1 & J_i \\ l_i & l_f - l_i & -l_f \end{pmatrix}.\end{split}\]

The rotational energy entering \(Q\) and \(\Omega\) is the rigid-rotor expression \(E_J = B_0 J(J+1)\), where \(B_0\) is the effective ground-state rotational constant for the species. Energy levels provided by quantum-chemical calculations are used directly where available; the rigid-rotor expression serves to extrapolate to levels not covered by those calculations.

Symmetric Tops with Electron Spin — Makarov (O₂)

Molecular oxygen (O₂) has an unpaired electron spin \(S = 1\), so each rotational quantum number \(N\) gives rise to a triplet \(J = N-1, N, N+1\). The off-diagonal coupling between lines \((i: N_l J_l \to N_u J_u)\) and \((j: N'_l J'_l \to N'_u J'_u)\) is (using \(l\)/\(u\) for the lower/upper state of each transition, matching the convention of Local thermodynamic equilibrium) [19]

\[\begin{split}W_{ij} =& (-1)^{J'_l + J_l + 1}\, [N_l][N_u][N'_u][N'_l][J_u][J'_u][J_l][J'_l] \Omega(N_l) \\ & \begin{array}{llll} \sum_L (2L+1) & \begin{pmatrix} N'_l & N_l & L \\ 0 & 0 & 0 \end{pmatrix} & \begin{pmatrix} N'_u & N_u & L \\ 0 & 0 & 0 \end{pmatrix} \\ & \begin{Bmatrix} L & J_l & J'_l \\ S & N'_l & N_l \end{Bmatrix} & \begin{Bmatrix} L & J_u & J'_u \\ S & N'_u & N_u \end{Bmatrix} & \begin{Bmatrix} L & J_l & J'_l \\ 1 & J'_u & J_u \end{Bmatrix} \frac{Q(L)}{\Omega(L)}, \end{array}\end{split}\]

where \([X] \equiv \sqrt{2X+1}\), \((\cdots)\) denotes a Wigner 3-j symbol, and \(\{\cdots\}\) a Wigner 6-j symbol.

The reduced dipole is

\[\begin{split}d(J_u, J_l, N) = (-1)^{J_l + N} \sqrt{6(2J_l+1)(2J_u+1)} \begin{Bmatrix} 1 & 1 & 1 \\ J_l & J_u & N \end{Bmatrix}.\end{split}\]

The rotational energy for the O₂ microwave band is computed from the full ground-state Hamiltonian including spin–rotation coupling and magnetic interactions.

Symmetric Tops (NH₃, PH₃)

This part is mostly untested and may be incorrect. It has been generated by AI and is available in ARTS only for experimentation to see if it produces reasonable results.

For symmetric top molecules (e.g. NH₃, PH₃) with \(\Delta K = 0\) collisions, lines within the same \(K\) sub-band are coupled identically to the Hartmann linear-molecule formula with the vibrational angular momentum \(l\) replaced by \(K\): [10]

\[\begin{split}W_{ij} = \Omega(J_i)\, (2J'_i+1)\sqrt{(2J_f+1)(2J'_f+1)} \sum_L (2L+1) \begin{pmatrix} J_i & J'_i & L \\ K & -K & 0 \end{pmatrix} \begin{pmatrix} J_f & J'_f & L \\ K & -K & 0 \end{pmatrix} \begin{Bmatrix} J_i & J_f & 1 \\ J'_f & J'_i & L \end{Bmatrix} \frac{Q(L)}{\Omega(L)}.\end{split}\]

Lines with different \(K\) are not coupled. The reduced dipole is

\[\begin{split}d(J_f, J_i, K) = (-1)^{J_f + K + 1}\sqrt{2J_f+1}\; \begin{pmatrix} J_f & 1 & J_i \\ K & 0 & -K \end{pmatrix}.\end{split}\]

Rotational energy levels provided by quantum-chemical calculations are used directly where available; levels beyond those are extrapolated using the rigid-rotor expression \(E_J = B_0 J(J+1)\) with the species-specific ground-state rotational constant \(B_0\).

Spherical Tops (CH₄)

This part is mostly untested and may be incorrect. It has been generated by AI and is available in ARTS only for experimentation to see if it produces reasonable results.

For spherical top molecules (e.g. CH₄) the coupling reduces to the \(l = 0\) limit of the Hartmann formula: [27]

\[\begin{split}W_{ij} = \Omega(J_i)\, (2J'_i+1)\sqrt{(2J_f+1)(2J'_f+1)} \sum_L (2L+1) \begin{pmatrix} J_i & J'_i & L \\ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} J_f & J'_f & L \\ 0 & 0 & 0 \end{pmatrix} \begin{Bmatrix} J_i & J_f & 1 \\ J'_f & J'_i & L \end{Bmatrix} \frac{Q(L)}{\Omega(L)},\end{split}\]

and the reduced dipole is

\[\begin{split}d(J_f, J_i) = (-1)^{J_f+1}\sqrt{2J_f+1}\; \begin{pmatrix} J_f & 1 & J_i \\ 0 & 0 & 0 \end{pmatrix}.\end{split}\]

Rotational energy levels provided by quantum-chemical calculations are used directly where available; levels beyond those are extrapolated using the rigid-rotor expression \(E_J = B_0 J(J+1)\) with the species-specific ground-state rotational constant \(B_0\).

Sum-rule Correction

The pure IOS matrix elements computed above do not, in general, satisfy the first-order optical sum rule exactly due to the finite range of the \(L\) sum and the approximate nature of the adiabatic factor. The error-correction step rescales each column of off-diagonal elements to enforce

\[\sum_j d_j\, W_{ji} = 0 \quad \forall\, i.\]

This is done as follows. For each line \(i\), partition the off-diagonal elements into those coupling to lines with lower intensity-weighted frequency (summed into \(s_\downarrow\)) and those coupling to higher-frequency lines (\(s_\uparrow\)):

\[s_\downarrow = \sum_{j > i} d_j\, W_{ji}, \qquad s_\uparrow = \sum_{j < i} d_j\, W_{ji}.\]

All downward-coupling elements are then rescaled by \(-s_\uparrow / s_\downarrow\), and the corresponding upward-coupling elements are updated by detailed balance.

This rescaling constitutes the “error-corrected” part of the ECS method and ensures the resulting relaxation matrix produces physically consistent absorption profiles that recover the correct integrated line intensity at all pressures.

Multiple Broadening Species

When multiple broadening species are present, the ARTS implementation offers two modes:

In the single-W mode (used by default when calling calculate), the per-species relaxation matrices are first volume-mixing-ratio weighted and summed into a single effective \(\mathbf{W}\):

\[\mathbf{W}_{eff} = \sum_s x_s\, \mathbf{W}^{(s)},\]

and a single diagonalisation is performed.

In the multi-W mode (used by equivalent_values for pre-computing equivalent lines at multiple temperatures), the diagonalisation is performed separately per species and the resulting absorption contributions are VMR-weighted and summed.

Rosenkranz Approximation

The full ECS calculation requires the diagonalisation of an \(n \times n\) complex matrix at every temperature and pressure of interest, together with a VMR-weighted sum over broadening species. For many practical applications a simpler representation is desirable: the Rosenkranz approximation retains the ordinary Voigt line shape of each line but adds pressure-dependent first- and second-order correction terms that encode the effect of line mixing to a given order in pressure.

The corrected Voigt line shape for line \(i\) is exactly the Voigt profile already described,

\[F_i = \frac{1 + G_{lm,i} - iY_{lm,i}}{\sqrt{\pi}\,G_D}\,w(z_i),\]

where \(z_i\) contains \(\Delta\nu_{lm,i}\) as an additional shift, and the three correction parameters are:

Parameter	Physical meaning	Pressure scaling
\(Y_{lm,i}\)	First-order line-mixing: asymmetric intensity redistribution between nearby lines.	\(P\)
\(G_{lm,i}\)	Second-order strength correction: quadratic-in-pressure modification to the integrated area.	\(P^2\)
\(\Delta\nu_{lm,i}\)	Second-order frequency shift: quadratic-in-pressure displacement of the line centre due to the mixing.	\(P^2\)

The Rosenkranz parameters are not fitted to measured spectra directly; instead they are derived from the ECS equivalent lines or, equivalently, from the relaxation matrix itself via perturbation theory — both approaches are described below.

Perturbation Theory from the Relaxation Matrix

When the off-diagonal elements of \(\mathbf{W}\) are small compared with the spacings between line centres (the “weak coupling” limit, valid for resolved lines or moderate pressures), the Rosenkranz parameters can be obtained analytically by expanding the resolvent \((\nu\mathbf{I} - \mathbf{W})^{-1}\) in powers of the off-diagonal part. This is the original approach of Rosenkranz [32].

Write \(\mathbf{W} = \mathbf{D} + \mathbf{V}\), where \(\mathbf{D}\) is the diagonal part (line centres plus pressure broadening) and \(\mathbf{V}_{ij} = W_{ij}\) for \(i \neq j\) (the off-diagonal relaxation rates, purely imaginary in the ARTS convention: \(V_{ij} = i R_{ij}\) with \(R_{ij}\) real and proportional to \(P\)). Let \(g_i(\nu) = [\nu - W_{ii}]^{-1}\) be the unperturbed resolvent for line \(i\). The Neumann expansion then gives

\[(\nu\mathbf{I} - \mathbf{W})^{-1} = \mathbf{G}_0 + \mathbf{G}_0 \mathbf{V} \mathbf{G}_0 + \mathbf{G}_0 \mathbf{V} \mathbf{G}_0 \mathbf{V} \mathbf{G}_0 + \cdots,\]

where \(\mathbf{G}_0 = \mathrm{diag}(g_i(\nu))\).

Collecting all contributions to the absorption of line \(i\) through first and second order, and evaluating the slowly varying factors involving other lines \(j\) at \(\nu = \nu_i\), yields the three Rosenkranz parameters for line \(i\):

First-order mixing parameter (\(Y_i \sim P\)):

\[Y_i = \frac{2}{S_i} \sum_{j \neq i} S_j \frac{R_{ij}}{\nu_i - \nu_j},\]

where \(S_i = p_i d_i^2\) is proportional to the LBL line strength of line \(i\) (see ECS Line Shape for the definition of \(p_i\)), and \(R_{ij} = \mathrm{Im}[W_{ij}] / P\) is the pressure-normalised off-diagonal relaxation rate (transfer from line \(j\) to line \(i\); note \(\mathrm{Re}[W_{ij}] = 0\) for \(i \neq j\)), and the sum is over all other lines \(j\) in the band.

Second-order strength correction (\(G_i \sim P^2\)):

From the squared first-order cross terms, the fractional modification to the integrated area of line \(i\) is

\[G_i = -\frac{1}{S_i} \sum_{j \neq i} S_j \left(\frac{R_{ij}}{\nu_i - \nu_j}\right)^2.\]

Second-order line-centre shift (\(\Delta\nu_i \sim P^2\)):

The diagonal self-energy correction (virtual transition \(i \to j \to i\)) gives

\[\Delta\nu_i = -\frac{1}{S_i} \sum_{j \neq i} S_j \frac{R_{ij}^2}{\nu_i - \nu_j},\]

where detailed balance (\(S_i R_{ji} = S_j R_{ij}\)) has been used to express everything in terms of the downward rate \(R_{ij}\).

Note

Note that \(\Delta\nu_i = G_i (\nu_i - \nu_j)\) only for a single interfering line. In general they have different frequency denominators (\(\nu_i - \nu_j\) vs \((\nu_i - \nu_j)^2\)) and thus differ quantitatively when multiple lines contribute. The two parameters are both needed to correctly reproduce the second-order pressure dependence of the band profile.

The perturbation theory expressions above assume the off-diagonal elements are small relative to the line spacing. They break down for overlapping lines (e.g., at very high pressures or for lines very close in frequency). In that regime the full ECS calculation should be used instead.

Fitting from Equivalent Lines

The perturbation theory expressions above are analytically exact in the weak-coupling limit, but in practice it is often more accurate to extract the Rosenkranz parameters numerically from the ECS equivalent lines, because the equivalent-line calculation already includes the full resummation of the relaxation matrix (not just the first few terms of the Neumann series). The two approaches agree at low pressure but the equivalent-line fit is preferred at higher pressures where the perturbation series converges slowly.

Given the complex equivalent lines \((\tilde{S}_{k,s}, \tilde{\nu}_{k,s})\) (indexed by \(k\) in eigenvalue-decomposition order, which carries no physical meaning) computed by ECS for broadening species \(s\) at pressure \(P_0\) and a grid of temperatures \(T_1, \ldots, T_M\), the Rosenkranz coefficients are extracted by abs_bandsLineMixingAdaptation as follows. Here, \(i\) will denote the index of the physical LBL lines (the rows/columns of \(\mathbf{W}\)) and \(k\) the index of the equivalent lines.

Step 1 — Sort and match. The \(n\) equivalent lines are sorted by \(\mathrm{Re}[\tilde{\nu}_{k,s}]\) and the \(n\) physical LBL lines are sorted by \(\nu_{0,i}\). Equivalent line at sorted position \(n\) is then identified with the physical line at sorted position \(n\), giving a bijection \(k \leftrightarrow i\) between the two index sets. This is a heuristic matching: because eigenvalue decomposition does not guarantee any particular ordering of eigenvalues, sorting is the only way to establish a correspondence. The identification is reliable as long as the second-order frequency shifts and pressure shifts remain small compared with the separations between adjacent line centres; it can fail if either effect is comparable in magnitude to the line spacing.

Step 2 — Form normalised differences. For each matched pair \((k, i)\), the LTE line strength of physical line \(i\) at temperature \(T\) is \(S_{LTE,i}(T)\) as defined in Local thermodynamic equilibrium (without the number density factor \(\rho\); the equivalent per-molecule strength is \(s_i(T) = S_{LTE,i}(T)/\rho\)). The unperturbed complex frequency of physical line \(i\) is

\[\nu_i^{LBL}(T) = \nu_{0,i} + \Delta\nu_{P,0,i}(T,P_0) + i\,G_{P,0,i}(T,P_0).\]

The residual strength ratio and frequency residual are then formed:

\[\begin{split}r_{i,s}(T) &= \frac{\tilde{S}_{k,s}(T)}{S_{LTE,i}(T)}, \\ \delta\nu_{i,s}(T) &= \tilde{\nu}_{k,s}(T) - \nu_i^{LBL}(T),\end{split}\]

where \(k\) is the equivalent line matched to physical line \(i\) in Step 1.

Step 3 — Extract pressure-normalised Rosenkranz coefficients. The three coefficients for physical line \(i\) at the reference pressure \(P_0\) are read off as:

\[\begin{split}Y_{lm,i,s}(T) &= \frac{\mathrm{Im}\!\left[r_{i,s}(T)\right]}{P_0}, \\[4pt] G_{lm,i,s}(T) &= \frac{\mathrm{Re}\!\left[r_{i,s}(T)\right] - 1}{P_0^2}, \\[4pt] \Delta\nu_{lm,i,s}(T) &= \frac{\mathrm{Re}\!\left[\delta\nu_{i,s}(T)\right]}{P_0^2}.\end{split}\]

The imaginary part of \(\delta\nu_{i,s}(T)\) — which represents the correction to the pressure-broadening half-width — is divided by \(P_0^3\) but is not retained as a separate Rosenkranz coefficient (it is already captured by the diagonal of \(\mathbf{W}\) at first order in \(P\)).

Step 4 — Polynomial fit in temperature. Each of the three coefficients is fitted as a polynomial in temperature of configurable degree \(d\):

\[\begin{split}Y_{lm,i,s}(T) &\approx \sum_{n=0}^{d} a_n^{(Y)}\,T^n, \\ G_{lm,i,s}(T) &\approx \sum_{n=0}^{d} a_n^{(G)}\,T^n, \\ \Delta\nu_{lm,i,s}(T) &\approx \sum_{n=0}^{d} a_n^{(DV)}\,T^n.\end{split}\]

These polynomials are stored using the POLY temperature model (see Line Shape Parameters) for each broadening species separately and are then evaluated at runtime using the ordinary VMR-weighted sum of the line shape parameter framework, with the coefficients attached to physical line \(i\).

Note

Setting rosenkranz_fit_order = 1 retains only \(Y_{lm}\) and is appropriate for moderate pressures where the quadratic-in-pressure corrections are negligible. Setting it to 2 also fits \(G_{lm}\) and \(\Delta\nu_{lm}\), which is necessary at higher pressures or when second-order effects are important (e.g. near the Q-branch of O ₂ at tens of GHz).

The polynomial fits implicitly assume that the reference pressure \(P_0\) is fixed; the resulting coefficients must be used at the same pressure normalisation. This is handled automatically when the output of abs_bandsLineMixingAdaptation is fed back into the ordinary line-by-line calculation.