This short guide is written to provide users with the steps needed for the conversion of intensities between the HITRAN database [1] and the Jet Propulsion Laboratory (JPL) [2] or Cologne Database for Molecular Spectroscopy (CDMS) [3] spectral databases. The JPL and CDMS lists, which are identical in their intensity formalisms, provide base-10 logarithms of the integrated intensity at 300 K (in nm2MHz), while HITRAN gives the intensity at 296 K [in cm-1/(molecule/cm2)]. Apart from these differences, there are certain differences in the formalism of intensities and this guide provides steps for the most accurate conversion. If accuracy better than 2 % is not required, it is fairly safe to use an approximation given in Section 5 below. Some other intensity unit conversions are described in the textbook of Bernath [4].

### 1. Unit Conversion

The JPL [2] and CDMS [3] catalogs use nm2MHz as units of intensity. In order to conver to HITRAN [1] units, cm-1/(molecule/cm2), one has to divide the JPL intensity (not its logarithm) by a factor related to the speed of light, namely 2.99792458 × 1018. It should be recalled that the HITRAN units were constructed with application to atmospheric transmission calculations in mind, hence the emphasis on writing the units as wavenumber per column density and not simplifying it to the equivalent cm/molecule. $$I_\mathrm{JPL}(\mathrm{cm^{-1}/(molec\,cm^{-1})}) = \frac{I_\mathrm{JPL}(\mathrm{nm^2\,MHz})}{2.99792458\times 10^{18}} \label{eqn-jpl-units}$$

### 2. Isotopic Abundance

The JPL and CDMS catalogs assume 100 % abundance of every isotopologue, whereas the HITRAN database incorporates a terrestrial abundance scaling. Therefore one has to multiply the JPL (CDMS) intensity by the isotopologue abundance value ($I_\mathrm{a}$) assumed by HITRAN. The isotopic abundances used in HITRAN are given here and in, for example, Table 1 of Ref. [5]. $$S_\mathrm{JPL} = I_\mathrm{a}\frac{I_\mathrm{JPL}(\mathrm{nm^2\,MHz})}{2.99792458\times 10^{18}} \label{eqn-iso-scaling}$$

### 3. Scaling of the Partition Sum

After conversion to cm-1/(molecule/cm2) and scaling by isotopic abundance, one needs to consider the intensity defined in the JPL catalog, $S_\mathrm{JPL}$, which is given by: $$S_\mathrm{JPL}(T) = \frac{g'_\mathrm{JPL}}{Q_\mathrm{JPL}(T)} \frac{A}{8\pi c \nu_0^2} \exp\left( -\frac{hcE''}{k_\mathrm{B}T}\right) \left[ 1 - \exp\left( -\frac{hc\nu_0}{k_\mathrm{B}T}\right)\right], \label{eqn-jpl-intensity}$$ where $g'$ is the statistical weight of the upper level and $Q(T)$ is the total partition sum. The other factors in Eq. (\ref{eqn-jpl-intensity}) are defined here. The subscript label "JPL" refers to the fact that in some cases in the JPL and CDMS catalogs the common factors are factored out in $g'$ and $Q(T)$. This common factor is a state-independent statistical weight, $g_i$, which is not ignored in the HITRAN database. Nevertheless, $$\frac{g'_\mathrm{HIT}}{Q_\mathrm{HIT}(T)} \approx \frac{g'_\mathrm{JPL}}{Q_\mathrm{JPL}(T)} \label{eqn-jpl-hitran-approx-q}$$ The approximation here is due to the fact that partition sums are not calculated exactly the same way in JPL and HITRAN. Unlike HITRAN, the partition sums in the JPL catalog do not include the vibrational contribution in most cases. However this contribution may be significant for molecules possessing low vibrational modes. Therefore it is recommended that one scale the JPL intensities to HITRAN formalism in the following manner: $$S_\mathrm{HIT}(T) = S_\mathrm{JPL}(T) \frac{Q_\mathrm{JPL}(T)}{Q_\mathrm{HIT}(T)} \frac{g'_\mathrm{HIT}}{g'_\mathrm{JPL}} \label{eqn-jpl-hitran-intensity}$$ For 300 K, which is the reference temperature in the JPL and CDMS databases, $$S_\mathrm{HIT}(300\;\mathrm{K}) = I_\mathrm{a} \frac{I_\mathrm{JPL}(\mathrm{nm^2\,MHz})}{2.99792458\times 10^{18}} \frac{Q_\mathrm{JPL}(300\;\mathrm{K})}{Q_\mathrm{HIT}(300\;\mathrm{K})} \frac{g'_\mathrm{HIT}}{g'_\mathrm{JPL}} \label{eqn-jpl-hitran-refT-intensity}$$

It is not always immediately obvious whether or not $g'_\mathrm{JPL}= g'_\mathrm{HIT}$. Therefore it is always useful to obtain a ratio between partition sums in JPL and HITRAN at the same temperature and then round that ratio to an integer which will be the ratio between statistical weights in HITRAN and JPL. The HITRAN partition sums at 296 K are available here and are also listed in Table 1 of Ref. [5]. The partition sums (or their logarithms) for the JPL and CDMS catalogs are provided by JPL here and by CDMS here.

All intensities in the JPL and CDMS catalogs are calculated at 300 K, whereas HITRAN gives intensities at 296 K. By definition, $$S_\mathrm{HIT}(300\;\mathrm{K}) = \frac{g'_\mathrm{HIT}}{Q_\mathrm{HIT}(300\;\mathrm{K})} \frac{A}{8\pi c \nu_0^2} \exp\left( -\frac{hcE''}{300k_\mathrm{B}}\right)\left[ 1 - \exp\left(-\frac{hc\nu_0}{300k_\mathrm{B}}\right)\right],$$ and, $$S_\mathrm{HIT}(296\;\mathrm{K}) = \frac{g'_\mathrm{HIT}}{Q_\mathrm{HIT}(296\;\mathrm{K})} \frac{A}{8\pi c \nu_0^2} \exp\left( -\frac{hcE''}{296k_\mathrm{B}}\right)\left[ 1 - \exp\left(-\frac{hc\nu_0}{296k_\mathrm{B}}\right)\right].$$ Comparing these two equations, one obtains: \begin{align*} &S_\mathrm{HIT}(296\;\mathrm{K}) = S_\mathrm{HIT}(300\;\mathrm{K}) \exp\left[ -\frac{hcE''}{k_\mathrm{B}}\left(\frac{1}{300} - \frac{1}{296} \right)\right] \frac{\left[ 1 - \exp\left(-\frac{hc\nu_0}{296k_\mathrm{B}}\right)\right]}{\left[ 1 - \exp\left(-\frac{hc\nu_0}{300k_\mathrm{B}}\right)\right]} \frac{Q_\mathrm{HIT}(300\;\mathrm{K})}{Q_\mathrm{HIT}(296\;\mathrm{K})}\\ &= I_\mathrm{a} \frac{I_\mathrm{JPL}(\mathrm{nm^2\,MHz})}{2.99792458\times 10^{18}} \frac{Q_\mathrm{JPL}(300\;\mathrm{K})}{Q_\mathrm{HIT}(300\;\mathrm{K})} \frac{g'_\mathrm{HIT}}{g'_\mathrm{JPL}} \exp\left[ -\frac{hcE''}{k_\mathrm{B}}\left(\frac{1}{300} - \frac{1}{296} \right)\right] \frac{\left[ 1 - \exp\left(-\frac{hc\nu_0}{296k_\mathrm{B}}\right)\right]}{\left[ 1 - \exp\left(-\frac{hc\nu_0}{300k_\mathrm{B}}\right)\right]} \frac{Q_\mathrm{HIT}(300\;\mathrm{K})}{Q_\mathrm{HIT}(296\;\mathrm{K})}. \end{align*}

It should be noted that in the majority of cases in the literature, step 3 is omitted and the last equation in step 4 can be approximated by \begin{equation*} S_\mathrm{HIT}(296\;\mathrm{K}) \approx I_\mathrm{a} \frac{I_\mathrm{JPL}(\mathrm{nm^2\,MHz})}{2.99792458\times 10^{18}} \exp\left[ -\frac{hcE''}{k_\mathrm{B}}\left(\frac{1}{300} - \frac{1}{296} \right)\right] \frac{\left[ 1 - \exp\left(-\frac{hc\nu_0}{296k_\mathrm{B}}\right)\right]}{\left[ 1 - \exp\left(-\frac{hc\nu_0}{300k_\mathrm{B}}\right)\right]} \left(\frac{300}{296}\right)^n, \end{equation*} where $n=1$ for linear molecules and $n=3/2$ for nonlinear molecules. Sometimes an even coarser approximation is used: This approximation is best at small wavenumber, for which $hc\nu_0/k_\mathrm{B} \ll 300\;\mathrm{K}$ and hence $1 - e^{-hc\nu_0/300k_\mathrm{B}} \approx 300$, etc. \begin{equation*} S_\mathrm{HIT}(296\;\mathrm{K}) \approx I_\mathrm{a} \frac{I_\mathrm{JPL}(\mathrm{nm^2\,MHz})}{2.99792458\times 10^{18}} \exp\left[ -\frac{hcE''}{k_\mathrm{B}}\left(\frac{1}{300} - \frac{1}{296} \right)\right] \left(\frac{300}{296}\right)^{(n+1)}. \end{equation*}

#### References

[1] L. S. Rothman, et al., "The HITRAN 2012 Molecular Spectroscopic Database", J. Quant. Spectrosc. Radiat. Transfer 130, 4-50 (2013). [ADS]

[2] H.M. Pickett, R.L. Poynter, E.A. Cohen, M.L. Delitsky, J.C. Pearson, H.S.P. Müller, "Submillimeter, Millimeter, and Microwave Spectral Line Catalog", Journal of Quantitative Spectroscopy and Radiative Transfer 60, 883-890 (1998). [ADS]

[3] H.S.P. Müller, S. Thorwirth, D.A. Roth, G. Winnewisser, "The Cologne Database for Molecular Spectroscopy, CDMS", Astronomy and Astrophysics 370, L49-L52 (2001). [ADS]

[4] P. F. Bernath, Spectra of Atoms and Molecules, OUP (New York), 2nd ed. (2005)

[5] M. Šimečková, D. Jacquemart, L. S. Rothman, R. R. Gamache, A. Goldman, "Einstein A-coefficients and statistical weights for molecular absorption transitions in the HITRAN database", Journal Of Quantitative Spectroscopy and Radiative Transfer 98, 130-155 (2006). [ADS]