<caption>**Table A.1**</caption>

Definitions of parameters, integration variables and trajectory integrals for 12-m Lennard-Jones interactions (m = 6, 5, 4). Both initial and practically used for numerical integration expressions are given for B(a) and S(a).

<table><caption>**Table A.2**</caption><thead><tr><th>Potential form</th><th>α</th><th>x</th><th>B(α)</th><th>S(α)</th></tr></thead><tbody><tr><td rowspan="2">12-6</td><td><sup>7</sup>&frasl;<sub>2<sup>2/3</sup>3<sup>1/3</sup>π<sup>4/3</sup></sub><span style="text-decoration: overline;">V</span><sup>6/5</sup> <span style="text-decoration: overline;">|ΔC'<sub>12</sub>|</span><sup>11/8</sup></td><td>b<span style="text-decoration: overline;">(8<span style="text-decoration: overline;">π</span>3<sup>1/3</sup>ΔC'<sub>6</sub>)</span><sup>1/5</sup></td><td><span style="text-decoration: overline;">f</span><sub>0</sub><sup>∞</sup> sin<sup>2</sup><span style="text-decoration: overline;">(1/2)(ax<sup>-11</sup> - x<sup>-5</sup>)</span> xdx</td><td><span style="text-decoration: overline;">f</span><sub>0</sub><sup>∞</sup> sin(ax<sup>-11</sup> - x<sup>-5</sup>)xdx</td></tr><tr><td>≈ 0.5391<span style="text-decoration: overline;">V</span><sup>6/5</sup>ΔC'<sub>12</sub> <span style="text-decoration: overline;">|ΔC'<sub>6</sub>|</span><sup>-11/5</sup></td><td></td><td><sup>1</sup>&frasl;<sub>4</sub> <span style="text-decoration: overline;">f</span><sub>0</sub><sup>∞</sup> sin(ax<sup>-11</sup> - x<sup>-5</sup>)(11ax<sup>-10</sup> - 5x<sup>-4</sup>)dx</td><td><sup>1</sup>&frasl;<sub>2</sub> <span style="text-decoration: overline;">f</span><sub>0</sub><sup>∞</sup> cos(ax<sup>-11</sup> - x<sup>-5</sup>)(11ax<sup>-10</sup> - 5x<sup>-4</sup>)dx</td></tr><tr><td rowspan="2">12-5</td><td><sup>7x<sup>3</sup>3<sup>1/3</sup>π<sup>4/3</sup></sup>&frasl;<sub>2<sup>2/3</sup>2<sup>1/3</sup></sub><span style="text-decoration: overline;">V</span><sup>7/4</sup> <span style="text-decoration: overline;">|ΔC'<sub>4</sub>|</span><sup>11/8</sup></td><td>b<span style="text-decoration: overline;">(3<span style="text-decoration: overline;">π</span>4<sup>1/3</sup>ΔC'<sub>5</sub>)</span><sup>1/4</sup></td><td><span style="text-decoration: overline;">f</span><sub>0</sub><sup>∞</sup> sin<sup>2</sup><span style="text-decoration: overline;">(1/2)(ax<sup>-11</sup> - x<sup>-4</sup>)</span> xdx</td><td><span style="text-decoration: overline;">f</span><sub>0</sub><sup>∞</sup> sin(ax<sup>-11</sup> - x<sup>-4</sup>)xdx</td></tr><tr><td>≈ 0.3505<span style="text-decoration: overline;">V</span><sup>7/4</sup>ΔC'<sub>12</sub> <span style="text-decoration: overline;">|ΔC'<sub>5</sub>|</span><sup>-11/4</sup></td><td></td><td><sup>1</sup>&frasl;<sub>4</sub> <span style="text-decoration: overline;">f</span><sub>0</sub><sup>∞</sup> sin(ax<sup>-11</sup> - x<sup>-4</sup>)(11ax<sup>-10</sup> - 4x<sup>-3</sup>)dx</td><td><sup>1</sup>&frasl;<sub>2</sub> <span style="text-decoration: overline;">f</span><sub>0</sub><sup>∞</sup> cos(ax<sup>-11</sup> - x<sup>-4</sup>)(11ax<sup>-10</sup> - 4x<sup>-3</sup>)dx</td></tr><tr><td rowspan="2">12-4</td><td><sup>7.5<sup>3</sup>π<sup>4/3</sup></sup>&frasl;<sub>2<sup>3/3</sup>3<sup>1/3</sup>π<sup>4/3</sup></sub><span style="text-decoration: overline;">V</span><sup>8/3</sup> <span style="text-decoration: overline;">|ΔC'<sub>12</sub>|</span><sup>13/3</sup></td><td>b<span style="text-decoration: overline;">(2<span style="text-decoration: overline;">π</span>π<sup>4/3</sup>ΔC'<sub>4</sub>)</span><sup>1/3</sup></td><td><span style="text-decoration: overline;">f</span><sub>0</sub><sup>∞</sup> sin<sup>2</sup><span style="text-decoration: overline;">(1/2)(ax<sup>-11</sup> - x<sup>-3</sup>)</span> xdx</td><td><span style="text-decoration: overline;">f</span><sub>0</sub><sup>∞</sup> sin(ax<sup>-11</sup> - x<sup>-3</sup>)xdx</td></tr><tr><td>≈ 0.1476<span style="text-decoration: overline;">V</span><sup>8/3</sup>ΔC'<sub>12</sub> <span style="text-decoration: overline;">|ΔC'<sub>4</sub>|</span><sup>-11/3</sup></td><td></td><td><sup>1</sup>&frasl;<sub>4</sub> <span style="text-decoration: overline;">f</span><sub>0</sub><sup>∞</sup> sin(ax<sup>-11</sup> - x<sup>-3</sup>)(11ax<sup>-10</sup> - 3x<sup>-2</sup>)dx</td><td><sup>1</sup>&frasl;<sub>2</sub> <span style="text-decoration: overline;">f</span><sub>0</sub><sup>∞</sup> cos(ax<sup>-11</sup> - x<sup>-3</sup>)(11ax<sup>-10</sup> - 3x<sup>-2</sup>)dx</td></tr></tbody></table>

Linewidth and shift expressions for 12-m Lennard-Jones interactions (m = 6, 5, 4); see Table A.1 for the corresponding expressions of trajectory integrals.

<table><thead><tr><th>Potential form</th><th>γ/N</th><th>δ/N</th></tr></thead><tbody><tr><td>12-6</td><td>2(<sup>3π</sup>8)<sup>2/5</sup> |ΔC'<sub>6</sub>|<sup>2/5</sup> <span style="text-decoration: overline;">V</span><sup>3/5</sup> B(a)</td><td>(<sup>3π</sup>8)<sup>2/5</sup> |ΔC'<sub>6</sub>|<sup>2/5</sup> <span style="text-decoration: overline;">V</span><sup>3/5</sup> S(a)</td></tr><tr><td>12-5</td><td>2(<sup>4</sup>3)<sup>1/2</sup> |ΔC'<sub>5</sub>|<sup>1/2</sup> <span style="text-decoration: overline;">V</span><sup>1/2</sup> B(a)</td><td>(<sup>4</sup>3)<sup>1/2</sup> |ΔC'<sub>5</sub>|<sup>1/2</sup> <span style="text-decoration: overline;">V</span><sup>1/2</sup> S(a)</td></tr><tr><td>12-4</td><td>2(<sup>5</sup>2)<sup>3/3</sup> |ΔC'<sub>4</sub>|<sup>2/3</sup> <span style="text-decoration: overline;">V</span><sup>1/3</sup> B(a)</td><td>(<sup>5</sup>2)<sup>3/3</sup> |ΔC'<sub>4</sub>|<sup>2/3</sup> <span style="text-decoration: overline;">V</span><sup>1/3</sup> S(a)</td></tr></tbody></table>

tests of extrapolation to the region of very small r-values. Finally, the extrapolation type was found to be not important for linewidths (because of strong oscillations of the cosine function giving nearly zero contribution from this region), but the broadening coefficient values at both temperatures considered were much bigger than the measurements. For line shifting, although the correct negative signs were reproduced, underestimations of absolute values by several orders of magnitude were observed. Therefore, an “isolated” improvement of the potential in the frame of the traditional phase-shift theory with rectilinear trajectories was inefficient to get more realistic line-shape estimates.

As a next step, we tried (simultaneous with and separate from the potential description) improvements of the trajectory model, using for simplicity the mean-thermal-velocity approximation. Two types of trajectories curved by the force derived from the isotropic potential were considered: parabolic and exact ones, assuming that the two halves of each trajectory are driven, respectively, by the isotropic potentials in the initial and final radiator’s states. Applications made for the NO–Ar and NO–N₂ systems demonstrated quite disparate predictions, so we can also conclude that the trajectory improvements are still insufficient to match completely the measurements.

The results of our two improvement attempts (numerical potentials and curved trajectories) support the conclusion made by Szudy and Baylis [30] that going beyond the straight-line trajectories requires at least a first-order correction to the usual phase-shift integral and significant differences with respect to the standard approach can be expected. Developing such a formalism represents a serious independent work and will be addressed in a future study. Another subject of future investigations could be the rotational dependence of the line-shape parameters, which is generally neglected by the theory but still detectable experimentally for some molecular systems; considering this dependence will require an account of anisotropic interactions.

### CRediT authorship contribution statement

**Jeanna Buldyreva:** Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Writing – original draft, Writing – review & editing, Visualization, Supervision, Project administration. **Ryan P. Brady:** Conceptualization, Methodology,

Software, Validation, Formal analysis, Investigation, Writing – original draft, Writing – review & editing, Visualization. **Sergei N. Yurchenko:** Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Writing – original draft, Writing – review & editing, Visualization, Funding acquisition. **Jonathan Tennyson:** Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Writing – original draft, Writing – review & editing, Visualization, Supervision, Project administration, Funding acquisition.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Data availability

The data is provided as part of the publication.

### Acknowledgements

This work was supported by the European Research Council under Advanced Investigator Project 883830 and by UK STFC under grant ST/R000476/1. The authors acknowledge the use of the UCL Myriad High Performance Computing Facility and associated support services in the completion of this work. JB acknowledges the mobility grant provided by the Region Bourgogne-Franche-Comté for her visit to UCL.

### Appendix A

See Tables A.1 and A.2.

### Appendix B. Supplementary data

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.jqsrt.2023.108843.

### References

[1] Romps DM, Seeley JT, Edman JP. Why the forcing from carbon dioxide scales as the logarithm of its concentration. J Clim 2022;35:4027–47. http://dx.doi.org/10.1175/JCLI-D-21-0275.1.

[2] Hartmann J-M, Boulet C, Robert D. Collisional effects on molecular spectra. Laboratory experiments and models, consequences for applications. Amsterdam: Elsevier; 2008.

[3] Buldyreva J, Lavrentieva N, Starikov V. Collisional line broadening and shifting of atmospheric gases. A practical guide for line shape modelling by current semi-classical approaches. World Scientific, Imperial College Press; 2010.
<caption>**Table A.1**</caption>

Definitions of parameters, integration variables and trajectory integrals for 12-m Lennard-Jones interactions (m = 6, 5, 4). Both initial and practically used for numerical integration expressions are given for B(a) and S(a).

<table><caption>**Table A.2**</caption><thead><tr><th>Potential form</th><th>α</th><th>x</th><th>B(α)</th><th>S(α)</th></tr></thead><tbody><tr><td rowspan="2">12-6</td><td><sup>7</sup>&frasl;<sub>2<sup>3/3</sup>π<sup>5/3</sup></sub><span style="text-decoration: overline;">V</span><sup>6/5</sup> |<span style="text-decoration: overline;">ΔC'<sub>12</sub></span>|<sup>11/5</sup></td><td>b<span style="text-decoration: overline;">(8<span style="text-decoration: overline;">ε</span> / 3π)|ΔC'<sub>6</sub>|</span><sup>1/5</sup></td><td><span style="text-decoration: overline;">f</span><sub>0</sub><sup>∞</sup> sin<sup>2</sup>(<sup>1</sup>&frasl;<sub>2</sub>(ax<sup>-11</sup> - x<sup>-5</sup>))xdx</td><td><span style="text-decoration: overline;">f</span><sub>0</sub><sup>∞</sup> sin(ax<sup>-11</sup> - x<sup>-5</sup>)xdx</td></tr><tr><td>≈ 0.5391<span style="text-decoration: overline;">V</span><sup>6/5</sup> |<span style="text-decoration: overline;">ΔC'<sub>12</sub></span>|<span style="text-decoration: overline;">ΔC'<sub>6</sub></span>|<sup>-11/5</sup></td><td></td><td><sup>1</sup>&frasl;<sub>4</sub> <span style="text-decoration: overline;">f</span><sub>0</sub><sup>∞</sup> sin(ax<sup>-11</sup> - x<sup>-5</sup>)(11ax<sup>-10</sup> - 5x<sup>-4</sup>)dx</td><td><sup>1</sup>&frasl;<sub>2</sub> <span style="text-decoration: overline;">f</span><sub>0</sub><sup>∞</sup> cos(ax<sup>-11</sup> - x<sup>-5</sup>)(11ax<sup>-10</sup> - 5x<sup>-4</sup>)dx</td></tr><tr><td rowspan="2">12-5</td><td><sup>7x<sup>31/9</sup>π<sup>5/9</sup></sup>&frasl;<sub>2<sup>27/2</sup></sub><span style="text-decoration: overline;">V</span><sup>7/4</sup> |<span style="text-decoration: overline;">ΔC'<sub>12</sub></span>|<sup>11/2</sup></td><td>b<span style="text-decoration: overline;">(3<span style="text-decoration: overline;">ε</span> / 4)|ΔC'<sub>6</sub>|</span><sup>1/4</sup></td><td><span style="text-decoration: overline;">f</span><sub>0</sub><sup>∞</sup> sin<sup>2</sup>(<sup>1</sup>&frasl;<sub>2</sub>(ax<sup>-11</sup> - x<sup>-4</sup>))xdx</td><td><span style="text-decoration: overline;">f</span><sub>0</sub><sup>∞</sup> sin(ax<sup>-11</sup> - x<sup>-4</sup>)xdx</td></tr><tr><td>≈ 0.3505<span style="text-decoration: overline;">V</span><sup>7/4</sup> |<span style="text-decoration: overline;">ΔC'<sub>12</sub></span>|<span style="text-decoration: overline;">ΔC'<sub>6</sub></span>|<sup>-11/4</sup></td><td></td><td><sup>1</sup>&frasl;<sub>4</sub> <span style="text-decoration: overline;">f</span><sub>0</sub><sup>∞</sup> sin(ax<sup>-11</sup> - x<sup>-4</sup>)(11ax<sup>-10</sup> - 4x<sup>-3</sup>)dx</td><td><sup>1</sup>&frasl;<sub>2</sub> <span style="text-decoration: overline;">f</span><sub>0</sub><sup>∞</sup> cos(ax<sup>-11</sup> - x<sup>-4</sup>)(11ax<sup>-10</sup> - 4x<sup>-3</sup>)dx</td></tr><tr><td rowspan="2">12-4</td><td><sup>7.5<sup>2</sup>π<sup>5</sup>π<sup>7/3</sup></sup>&frasl;<sub>2<sup>13/3</sup>π<sup>8/3</sup></sub><span style="text-decoration: overline;">V</span><sup>8/3</sup> |<span style="text-decoration: overline;">ΔC'<sub>12</sub></span>|<sup>13/3</sup></td><td>b<span style="text-decoration: overline;">(2<span style="text-decoration: overline;">ε</span> / π)|ΔC'<sub>6</sub>|</span><sup>1/3</sup></td><td><span style="text-decoration: overline;">f</span><sub>0</sub><sup>∞</sup> sin<sup>2</sup>(<sup>1</sup>&frasl;<sub>2</sub>(ax<sup>-11</sup> - x<sup>-3</sup>))xdx</td><td><span style="text-decoration: overline;">f</span><sub>0</sub><sup>∞</sup> sin(ax<sup>-11</sup> - x<sup>-3</sup>)xdx</td></tr><tr><td>≈ 0.1476<span style="text-decoration: overline;">V</span><sup>8/3</sup> |<span style="text-decoration: overline;">ΔC'<sub>12</sub></span>|<span style="text-decoration: overline;">ΔC'<sub>6</sub></span>|<sup>-11/3</sup></td><td></td><td><sup>1</sup>&frasl;<sub>4</sub> <span style="text-decoration: overline;">f</span><sub>0</sub><sup>∞</sup> sin(ax<sup>-11</sup> - x<sup>-3</sup>)(11ax<sup>-10</sup> - 3x<sup>-2</sup>)dx</td><td><sup>1</sup>&frasl;<sub>2</sub> <span style="text-decoration: overline;">f</span><sub>0</sub><sup>∞</sup> cos(ax<sup>-11</sup> - x<sup>-3</sup>)(11ax<sup>-10</sup> - 3x<sup>-2</sup>)dx</td></tr></tbody></table>

Linewidth and shift expressions for 12-m Lennard-Jones interactions (m = 6, 5, 4); see Table A.1 for the corresponding expressions of trajectory integrals.

<table><thead><tr><th>Potential form</th><th>γ/N</th><th>δ/N</th></tr></thead><tbody><tr><td>12-6</td><td>2(<sup>3ε</sup>/8)<sup>2/5</sup> |ΔC'<sub>6</sub>|<sup>2/5</sup> <span style="text-decoration: overline;">V</span><sup>3/5</sup> B(α)</td><td>(<sup>3ε</sup>/8)<sup>2/5</sup> |ΔC'<sub>6</sub>|<sup>2/5</sup> <span style="text-decoration: overline;">V</span><sup>3/5</sup> S(α)</td></tr><tr><td>12-5</td><td>2(<sup>5</sup>/3)<sup>1/2</sup> |ΔC'<sub>5</sub>|<sup>1/2</sup> <span style="text-decoration: overline;">V</span><sup>1/2</sup> B(α)</td><td>(<sup>4</sup>/3)<sup>1/2</sup> |ΔC'<sub>5</sub>|<sup>1/2</sup> <span style="text-decoration: overline;">V</span><sup>1/2</sup> S(α)</td></tr><tr><td>12-4</td><td>2(<sup>ε</sup>/2)<sup>2/3</sup> |ΔC'<sub>4</sub>|<sup>2/3</sup> <span style="text-decoration: overline;">V</span><sup>1/3</sup> B(α)</td><td>(<sup>ε</sup>/2)<sup>2/3</sup> |ΔC'<sub>4</sub>|<sup>2/3</sup> <span style="text-decoration: overline;">V</span><sup>1/3</sup> S(α)</td></tr></tbody></table>

tests of extrapolation to the region of very small r-values. Finally, the extrapolation type was found to be not important for linewidths (because of strong oscillations of the cosine function giving nearly zero contribution from this region), but the broadening coefficient values at both temperatures considered were much bigger than the measurements. For line shifting, although the correct negative signs were reproduced, underestimations of absolute values by several orders of magnitude were observed. Therefore, an “isolated” improvement of the potential in the frame of the traditional phase-shift theory with rectilinear trajectories was inefficient to get more realistic line-shape estimates.

As a next step, we tried (simultaneous with and separate from the potential description) improvements of the trajectory model, using for simplicity the mean-thermal-velocity approximation. Two types of trajectories curved by the force derived from the isotropic potential were considered: parabolic and exact ones, assuming that the two halves of each trajectory are driven, respectively, by the isotropic potentials in the initial and final radiator's states. Applications made for the NO-Ar and NO-N₂ systems demonstrated quite disparate predictions, so we can also conclude that the trajectory improvements are still insufficient to match completely the measurements.

The results of our two improvement attempts (numerical potentials and curved trajectories) support the conclusion made by Szudy and Baylis [30] that going beyond the straight-line trajectories requires at least a first-order correction to the usual phase-shift integral and significant differences with respect to the standard approach can be expected. Developing such a formalism represents a serious independent work and will be addressed in a future study. Another subject of future investigations could be the rotational dependence of the line-shape parameters, which is generally neglected by the theory but still detectable experimentally for some molecular systems; considering this dependence will require an account of anisotropic interactions.

### CRediT authorship contribution statement

**Jeanna Buldyreva:** Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Writing – original draft, Writing – review & editing, Visualization, Supervision, Project administration. **Ryan P. Brady:** Conceptualization, Methodology,

Software, Validation, Formal analysis, Investigation, Writing – original draft, Writing – review & editing, Visualization. **Sergei N. Yurchenko:** Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Writing – original draft, Writing – review & editing, Visualization, Funding acquisition. **Jonathan Tennyson:** Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Writing – original draft, Writing – review & editing, Visualization, Supervision, Project administration, Funding acquisition.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Data availability

The data is provided as part of the publication.

### Acknowledgements

This work was supported by the European Research Council under Advanced Investigator Project 883830 and by UK STFC under grant ST/R000476/1. The authors acknowledge the use of the UCL Myriad High Performance Computing Facility and associated support services in the completion of this work. JB acknowledges the mobility grant provided by the Region Bourgogne-Franche-Comté for her visit to UCL.

### Appendix A

See Tables A.1 and A.2.

### Appendix B. Supplementary data

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.jqsrt.2023.108843.

### References

[1] Romps DM, Seeley JT, Edman JP. Why the forcing from carbon dioxide scales as the logarithm of its concentration. J Clim 2022;35:4027–47. http://dx.doi.org/10.1175/JCLI-D-21-0275.1.

[2] Hartmann J-M, Boulet C, Robert D. Collisional effects on molecular spectra. Laboratory experiments and models, consequences for applications. Amsterdam: Elsevier; 2008.

[3] Buldyreva J, Lavrentieva N, Starikov V. Collisional line broadening and shifting of atmospheric gases. A practical guide for line shape modelling by current semi-classical approaches. World Scientific, Imperial College Press; 2010.
[4] Hartmann J-M, Tran H, Armante R, Boulet C, Campargue A, Forget F, Gianfrani L, Gordon I, Guerlet S, Gustafsson M, Hodge JT, Kassi S, Lisak D, Thibault F, Toon GC. Recent advances in collisional effects on spectra of molecular gases and their practical consequences. J Quant Spectrosc Radiat Transfer 2018;213:178-227. http://dx.doi.org/10.1016/j.jqsrt.2018.03.016.

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[7] Pavanello M, Adamowicz L, Alijah A, Zobov NF, Mizus II, Polansky OL, Tennyson J, Szidarovszky T, Császár AG, Berg M, Petrignani A, Wolf A. Precision measurements and computations of transition energies in rotationally cold triatomic hydrogen ions up to the mid-visible spectral range. Phys Rev Lett 2012;108:023002. http://dx.doi.org/10.1103/PhysRevLett.108.023002.

[8] Anisman L, Chubb KL, Changeat Q, Edwards B, Yurchenko SN, Tennyson J, Tinetti G. Cross-sections for heavy atmospheres: H₂O self-broadening. J Quant Spectrosc Radiat Transfer 2022;283:108146. http://dx.doi.org/10.1016/j.jqsrt.2022.108146.

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