UV-Vis spectral simulation of polysulfur species using the nuclear ensemble approximation

Abstract

Polysulfurs (S_X) are thought to be an important part of the Venusian atmospheric sulfur cycle. Among these, S₃, S₄, and S₈ have been proposed to be contributors to the enigmatic near-UV absorption feature observed between 320 and 400 nm in the atmosphere of Venus. Furthermore, the individual UV-Vis spectral signatures of polysulfur isomers remain poorly characterized, hindering the accurate photochemical modeling of these species. In this work, we use the nuclear ensemble approach to simulate the UV-Vis spectra for a comprehensive set of S₂–S₈ structural isomers, generating wavelength-dependent absolute absorption cross sections suitable for use in photochemical models. Benchmark calculations using a set of molecules and radicals were used to validate the chosen spectral simulation approach. The results show that the smaller S₂–S₄ systems have open-chain geometric isomers as their global minima, while cyclic structures are the most stable for the larger S₅–S₈ species. We also identify two low-energy triplet S₄ isomers (cis-³S₄ and trans-³S₄), which may serve as important intermediates. The simulated spectra indicate that the global minimum S₃ structural isomer, linear-S₃, and a high-energy structural isomer of S₄, trigonal-S₄, absorb in the 320–400 nm window, while S₈ shows negligible absorption in the 320–400 nm range. This work advances our understanding of atmospheric sulfur species beyond Earth and supports the interpretation of observations of the atmosphere of Venus and other exoplanets.

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1 Introduction

The atmosphere of Earth's neighboring planet Venus consists mainly of carbon dioxide (CO₂) and nitrogen (N₂), though several different sulfur-containing compounds can also be found. The most abundant sulfur species in the gas phase is sulfur dioxide (SO₂), with a mixing ratio of 100–150 ppmv around 40 km, and 10–100 ppbv between 60–85 km.^1–4 Other observed sulfur-containing species in the Venusian atmosphere include carbonyl sulfide (OCS), carbon monosulfide (CS), and carbon disulfide (CS₂).⁵ Through oxidation of SO₂ and subsequent hydrolysis, sulfuric acid (H₂SO₄) is formed, which is understood to be the main component of the Venusian cloud and haze layer between 48–70 km.^1,2,4

Atmospheric sulfur species are not limited to Venus. Using the James Webb Space Telescope, spectral signals matching SO₂ have been recorded from the atmospheres of the hot Jupiter WASP-37b,⁶ the super-Neptune WASP-107b,⁷ the sub-Neptune GJ 3470 b,⁸ and the Super-Earth L 98-59 d.⁹ Recent infrared measurements from the atmosphere of the sub-Neptune K2-18b revealed features suggested to match dimethyl sulfide (DMS) and dimethyl disulfide (DMDS), both potential biosignatures.¹⁰ In the solar system, SO₂ was confirmed on Jupiter's moon Io using the infrared spectroscopy instrumentation onboard the Voyager 1 space probe.¹¹ Disulfur (S₂) was detected by the Hubble Space Telescope in the plume of Pele, a volcano on Io.¹²

In the atmosphere on Earth, sulfur species play a major role in several important atmospheric processes. The most abundant species are SO₂ and OCS, but others, such as DMS, hydrogen sulfide (H₂S), and methanesulfonic acid (MSA) are also important.^13,14 In the Venusian atmosphere SO₂ ultimately forms H₂SO₄ through oxidation and hydrolysis, and H₂SO₄ is a key contributor to acid rain¹⁵ and atmospheric nucleation.^16–18

Due to the relatively high mixing ratio of SO₂ in the Venusian atmosphere, it is the main absorber between 200–320 nm.¹⁹ In addition, another weaker absorption feature can be seen in the 320–400 nm range.²⁰ Unfortunately, even after half a century since its discovery, there is still no scientific consensus on the identity of this unknown absorber (UA). Numerous different species have been proposed, including Cl₂,¹⁹ SCl₂,²¹ FeCl₃ in H₂SO₄ aerosols,^21–25 the SSCO isomer of the [C, O, S, S] system,²⁶ the cis and trans isomers of OSSO,^27,28 and even complex organic species formed in H₂SO₄ aerosols.²⁹ The latest UA candidate suggestion are different structural isomers of S₃O.³⁰

One proposed set of candidate compounds as the UA are polysulfurs (S_X), specifically S₃, S₄, and S₈.^31,32 Using the data from Venera 14, Krasnopolsky³³ estimated the mixing ratio of S₃ to be (80 ± 30) ppt_V in the lower atmosphere of Venus between 5 and 25 km. By reanalyzing the spectra from Venera 11, Maiorov et al.³⁴ estimated the mixing ratio of S₃ to be 30–100 ppt_V between 3–19 km, and 0 ppt_V between 19–36 nm. Krasnopolsky³⁵ also reanalyzed the spectra from Venera 11, but included S₄ in the analysis. The mixing ratios of (11 ± 3) ppt_V at 3–10 km and (18 ± 3) ppt_V at 10–19 km were obtained for S₃, and (4 ± 4) ppt_V at 3–10 km and (6 ± 2) ppt_V at 10–19 km for S₄.³⁵ Polysulfurs are also hypothesized to be a part of the Venusian atmospheric sulfur cycle^36,37 and therefore included in several Venusian atmospheric models.^36–41 A possible polysulfur production pathway was reported by Francés-Monerris et al.⁴² using high-level photodynamics simulations.

For S₂, quite a few studies on UV-Vis absorption have been published.^43–47 To the best of our knowledge, the most recent experimental UV-Vis spectrum of S₂ was published by Stark et al.⁴⁸ For the larger S_X species, Meyer et al.⁴⁹ identified the absorption features of S₃ and S₄ using matrix isolation spectroscopy. Later, Billmers and Smith⁵⁰ published the experimental spectra of S₃ and S₄ from sulfur vapor at 673–903 K. Hassanzadeh and Andrews⁵¹ identified two S₄ isomers using matrix isolation spectroscopy. The absorption peak at 518 nm was assigned to the cis form of the linear open-chain S₄ isomer (see Fig. 1), and the peak at 560–660 nm to the cyclic-S₃ (S) isomer. Based on experimental Raman spectra of sulfur vapor, Boumedien and Picquenard⁵² also assigned the peak at 560–660 nm to the cyclic-S₃ (S) isomer, but the 518 nm peak was instead attributed to the trans form of the open-chain isomers. Finally, Steudel et al.⁵³ measured the spectra of homocycles S₆, S₇, S₈, S₉, S₁₀, S₁₂, S₁₅, and S₂₀ in solutions of methanol or methylcyclohexane.

Fig. 1 All S₂–S₈ structures considered in this work, ordered by increasing zero-point vibrational corrected electronic energy.

The tendency of small polysulfur species to react and form larger ones makes it difficult to experimentally study individual structures of, for example, the S₄ system. Therefore, for distinguishing absorption features for individual isomers, a computational approach is appropriate. Several studies have investigated the excited states of S₂ using ab initio approaches.^54–59 Recently, Sarka et al.⁶⁰ performed high-level calculations and spectral simulations using the R-matrix approach for S₂, with isotopic effects included. The approach used was then further refined in a subsequent publication by Sarka and Nanbu.⁶¹ Peterson et al.⁶² explored the low-energy excited states of S₃ using multireference configuration interaction methods. The S₄ system was explored by Wong and Steudel,⁶³ who calculated vertical excitations for the different S₄ isomers. Based on these calculations, the authors concluded that the linear open-chain cis form of S₄ was the species absorbing at 518 nm, while the trans form was the absorber at 560–660 nm. Furthermore, several studies have also calculated the potential energy surfaces (PES) for different polysulfur species.^64–73

An attractive approach for computationally simulating UV-Vis spectra for different species is the nuclear ensemble approximation (NEA). When using the NEA approach more natural absorption band widths are obtained with wavelength-dependent absolute absorption cross sections, as opposed to the stick spectra from similar vertical excitation calculations. Although the approach does not recover vibrational progression, it has been used to qualitatively reproduce the absorption spectra for a variety of species, such as sulfur-containing molecules relevant to the atmospheres of both Earth and Venus,^28,74 nucleobases,⁷⁵ isomers of Cl₂O₂,⁷⁶ the ClCO radical,⁷⁷ gaseous mercury compounds,⁷⁸ pyrrole,⁷⁹ and different volatile organic compounds.^80,81 The NEA approach was used to simulate the spectra for different UA candidates, such as the two OSSO isomers,²⁸ the SSCO molecule,²⁶ and the isomers of S₃O.³⁰ The approach is also used in AtmoSpec,⁸² a tool for calculating photo absorption cross sections for volatile organic compounds in the troposphere of Earth.

In this work, we use the NEA approach to simulate the UV-Vis spectra for the polysulfur species S₂–S₈ to produce wavelength-dependent absolute absorption cross sections for the different species to be included in photochemical models. To confirm the validity of the chosen spectral simulation approach we performed benchmark calculations for several molecules and radicals (CS₂, H₂S, HO₂, O₃, OCS, PH₃, SO₂, and SO₃), comparing the obtained results to experimental spectra.

2 Theoretical methods

2.1 Benchmark calculations

To test the validity of the nuclear ensemble approximation method for spectral simulations of the different polysulfur species, we performed benchmark calculations using a set of molecules that have experimental spectra cataloged in the MPI-Mainz UV/VIS Spectral Atlas⁸³ and are small enough for high-level calculations. The molecules chosen were CS₂, H₂S, HO₂, O₃, OCS, PH₃, SO₂ and SO₃. Geometry optimizations and harmonic frequency calculations for these molecules were performed at the CCSD/aug-cc-pV(T+d)Z level of theory using the Cfour 2.1 program suite^84,85 with default convergence criteria. An unrestricted Hartree–Fock (UHF) reference was used for the HO₂ radical; for all other benchmark species, a restricted Hartree–Fock (RHF) reference was used. The frozen core approximation was employed in all coupled cluster calculations done in this work.

We performed initial test using 11 different density functionals, combined with 13 basis sets [see the SI for details]. Based on these test, we selected the overall best performing levels of theory for the benchmark calculations. The wavefunction-based method chosen was the equation-of-motion coupled cluster singles and doubles (EOM-CCSD). Farahani et al.⁷⁴ also found that EOM-CCSD was a good approach for accurate spectral simulation. The density functionals chosen were the hybrid GGA functional BHandHLYP,⁸⁶ the range-separated hybrid GGA functionals CAM-B3LYP⁸⁷ and ωB97X-D3,⁸⁸ and the range-separated double hybrid functional ωB2PLYP⁸⁹ without the Tamm–Dancoff approximation (TDA). In all calculations the aug-cc-pV(T+d)Z basis set^90–93 was used. EOM-CCSD calculations were performed with the Gaussian 16 Rev C.02 program.⁹⁴ All other calculations were performed with the Orca 5.0.4 quantum chemistry code.⁹⁵ The basis set exchange⁹⁶ was used to import basis sets not implemented in the quantum chemistry packages.

The theory of the nuclear ensemble approximation for spectral simulation is described in detail in the literature.^97,98 Briefly, the absorption cross section σ is calculated using eqn (1):⁹⁷

where E is the photon energy, e the electron charge, m_e the electron mass, c the speed of light, ε₀ the vacuum permittivity, and n_r the refractive index. ΔE_0,n(R_l) is the vertical excitation energy between the ground state and the excited state n with nuclear configuration R_l. f_0,n(R_l) is the oscillator strength between the ground state and the excited state n with nuclear configuration R_l. g(E − ΔE(R_l), δ_n) is a normalized Gaussian function, with an empirical widening of δ_n. The first sum runs over the number of excited states included (N_s), while the second sum runs over the number of geometries included in the nuclear ensemble (N_p). The vertical excitation energies and oscillator strengths are calculated with an appropriate method for each geometry in the ensemble.

In this work, the Newton-X 2.6 (build 01) program^99,100 was used for all spectral simulations. A nuclear ensemble consisting of 2000 geometries was created based on a Wigner distribution using the calculated normal modes. The ensemble size was chosen as the best compromise between minimizing the error due to statistical sampling and the computational resources needed, as shown by Farahani et al.⁷⁴ The ensemble size chosen in this work is also on the larger side when compared to other works using the NEA for spectral simulation.^26,30,80,101 Calculated vertical excitations were convoluted with a 0.1 eV full width at half-maximum (FWHM) normalized Gaussian function.⁷⁴ The ten lowest-energy excited states were calculated for all species. All spectral simulations were done at a temperature of 0 K. The simulated spectra were visualized using Matlab.¹⁰²

2.2 Spectral simulations of polysulfur species

The search for stable structural isomers is trivial for S₂–S₄, since for these systems the atoms can only be arranged in a reduced number of ways. However, in the case of S₅–S₈ this becomes more difficult, especially for flexible structural isomers. Different polysulfur species previously reported in the literature were used as starting points.^64,67,69,71 For open chain structures, the Crest conformer-rotamer sampling tool^103,104 with the semi-empirical GFN2-xTB method¹⁰⁵ was used to find stable structures. The generated conformer ensemble obtained from the Crest search (those conformers within 6.0 kcal mol⁻¹ of the lowest energy one found) was then further optimized using the ωB97X-D3 functional with the 6-31+G* basis set¹⁰⁶ using the Orca 5.0.4 program to obtain the final minimum energy open chain structure. All isomers/conformers identified in this work are presented in Fig. 1.

Geometry optimizations and frequency calculations for all S₂–S₈ minima and transition states were performed at the ωB97X-D/aug-cc-pV(T+d)Z^107,108 level of theory, with the Int = UltraFine and Opt = tight keywords added. Calculated normal nodes were used to verify that the optimized geometries were true minima or transition states. Intrinsic reaction coordinate (IRC) calculations were used to verify that the transition states were connected to the correct minima. In addition, the S₂–S₄ isomers were also optimized at the CCSD/aug-cc-pV(T+d)Z level of theory. Single-point energy corrections were performed at the CCSD(T)/aug-cc-pV(T+d)Z level of theory for all optimized geometries. All density functional geometry optimizations were performed with the Gaussian 16 Rev C.02 program. Coupled cluster geometry optimizations and single point energy corrections were performed with the Cfour 2.1 program suite, with the exception of the cyclic-S₄ and trigonal-³S₄ structures. For these two, the Gaussian 16 Rev C.02 program was used for the CCSD/aug-cc-pV(T+d)Z geometry optimization.

Miliordos and Xantheas¹⁰⁹ showed that S₃ has partial biradical character. In addition, Frandsen et al.²⁷ noted that some structures of the S₂O₂ system, which is isovalent with S₄, also showed multiconfigurational character. Motivated by this, we performed additional calculations on the CCSD/aug-cc-pV(T+d)Z optimized S₂–S₄ geometries. Using the MRCC program,¹¹⁰ single-point energies were calculated at the CCSDT/aug-cc-pV(T+d)Z level of theory to gauge the effect of including canonical coupled cluster triples. Complete active space self-consistent field (CASSCF) calculations, with dynamical correlation corrections accounted for with n-electron valence state perturbation theory (NEVPT2), were performed with the aug-cc-pV(T+d)Z basis set using the Orca 6.0.1 program.¹¹¹ The active spaces included all valence orbitals for the sulfur atoms, resulting in 12 electrons in 8 orbitals [CASSCF(12,8)] for S₂, CASSCF(18,12) for S₃, and CASSCF(24,16) for S₄.

For S₂–S₄, the CCSD/aug-cc-pV(T+d)Z optimized geometries and normal modes were used to create a Wigner ensemble, while ωB97X-D/aug-cc-pV(T+d)Z geometries and normal modes were used for S₅–S₈. We used the same levels of theory for the spectral simulations of the polysulfur species as in the benchmark calculations, these being EOM-CCSD, BHandHLYP, CAM-B3LYP, ωB97X-D3, and ωB2PLYP, combined with the aug-cc-pV(T+d)Z basis set. All EOM-CCSD calculations were again performed with the Gaussian 16 Rev C.02 program, and TD-DFT calculations with the Orca 5.0.4 quantum chemistry code. Spectral simulations for the S₅–S₈ species were only performed using TD-DFT, as calculations using EOM-CCSD were too computationally demanding.

To concisely communicate the levels of theory calculations were performed at we will use Method2/Basis2//Method1/Basis1 notation, where Method1/Basis1 indicates the level of theory for the geometry optimization and frequency calculation, and Method2/Basis2 refers to a single-point or vertical excitation calculation performed on the optimized geometry. If the same basis set was used in both calculations (meaning Basis1 is the same as Basis2), this is shortened to Method2//Method1/Basis1.

3 Results and discussion

3.1 Benchmark calculations

The simulated spectra for the benchmark calculations are presented in Fig. 2, and a more detailed discussion can be found in the SI. Our simulated spectra capture well the overall shape of their experimental counterparts. The only noticeable exception is O₃, where a fall-off in the absorption cross section below 230 nm is seen in the simulated spectra. However, O₃ is a well-known multiconfigurational system,¹⁰⁹ and the spectral simulations in this work use single reference levels of theory, which explains the poor quality of the simulated spectra.

Fig. 2 The simulated spectra for the benchmark set compared to experimental spectra. Note the different scales for the axes. The references for the experimental spectra are: CS₂ – Grosch et al.¹¹² (206–350 nm), Sunanda et al.¹¹³ (185–206 nm), and Rabalais et al.¹¹⁴ (180–185 nm); H₂S – Wu and Chen¹¹⁵ (160–260 nm); HO₂ – JPL 2010 recommendation¹¹⁶ (160–260 nm); O₃ – JPL 2010 recommendation¹¹⁶ (120–350 nm); OCS – JPL 2010 recommendation¹¹⁶ (258–280 nm), Limão-Vieira et al.¹¹⁷ (140–258 nm); PH₃ – Chen et al.¹¹⁸ (130–230 nm); SO₂ – compilation by Manatt and Lane¹¹⁹ (130–340 nm); SO₃ – Burkholder and McKeen¹²⁰ (196–300 nm), Hintze et al.¹²¹ (140–194 nm).

Based on the benchmark calculations and the numerous successful applications of the NEA,^{26,28,30,74–76,78,80,82} we conclude that using the NEA approach to simulate the UV-Vis spectra for the different S_X species is appropriate.

3.2 Spectral simulations of polysulfur species

The relative energies for all S₂–S₈ structures considered in this work are presented in Table 1. For the spin triplet systems, where unrestricted Hartree–Fock (UHF) references were used in coupled cluster calculations, spin contamination might affect the results. To address this, we monitored the projected expectation value of the S² operator, 〈S²〉, in the UHF part of the coupled cluster calculations. For a triplet system, the ideal 〈S²〉 value is 2.00. The highest value was found for structure ³S₅ (J), with 〈S²〉 = 2.11. While this is somewhat high, ³S₅ (J) is a relatively high energy S₅ species. All other structures had a 〈S²〉 value lower than 2.09, which we still consider acceptable.

Table 1 Relative, zero-point vibrational energy corrected electronic energies (in kcal mol⁻¹) for the S₂–S₈ species considered in this work. All calculations use the aug-cc-pV(T+d)Z basis set, which has been omitted for brevity

	Structure	ωB97X-D^a	CCSD(T)//ωB97X-D^a	CCSD^a	CCSD(T)//CCSD^a	CCSDT//CCSD^a	NEVPT2//CCSD^a
a See the end of Section 2.2 for an explanation of the convention used to indicate the level of theory used. b Calculated using NEVPT2(12,8) on top of a CASSCF(12,8) reference. c Calculated using NEVPT2(18,12) on top of a CASSCF(18,12) reference. d Calculated using NEVPT2(24,16) on top of a CASSCF(24,16) reference. e Calculation did not converge.
S2	^3S2	0.0	0.0	0.0	0.0	0.0	0.0^b
	^1S2	22.3	17.7	19.2	17.4	16.5	13.3^b
S3	Linear-S3	0.0	0.0	0.0	0.0	0.0	0.0^c
	Cyclic-S3	2.6	5.0	1.3	5.4	5.5	4.3^c
	^3S3	17.2	20.6	18.6	21.0	20.4	21.5^c
S4	cis-S4	0.0	0.0	0.0	0.0	0.0	0.0^d
	trans-S4	8.3	8.5	8.3	8.4	7.8	8.2^d
	cis-^3S4	1.5	9.7	0.9	10.6	9.1	13.2^d
	trans-^3S4	1.8	10.9	0.9	11.3	9.8	13.7^d
	Cyclic-S3 (S)	7.6	12.7	7.2	13.1	—^e	14.3^d
	Cyclic-S4	8.2	14.5	5.1	11.3	—^e	17.2^d
	Trigonal-S4	18.5	23.5	18.9	23.7	23.2	25.2^d
	Trigonal-^3S4	38.7	46.5	39.9	47.7	45.2	49.2^d
S5	Cyclic-S5 (A)	0.0	0.0
	^3S5 (B)	18.2	23.0
	^3S5 (C)	18.2	23.1
	S5 (D)	34.3	26.9
	S5 (E)	30.9	28.8
	S5 (F)	31.3	30.8
	S5 (G)	31.5	31.3
	S5 (H)	47.4	43.7
	S5 (I)	57.5	56.9
	^3S5 (J)	78.0	97.9
S6	Chair-S6 (A)	0.0	0.0
	Boat-S6 (B)	14.3	13.2
	S6 (C)	29.3	23.5
	S6 (D)	27.0	24.2
	^3S6 (E)	27.8	29.1
	S6 (F)	32.6	29.3
	^3S6 (G)	28.7	32.2
	S6 (H)	48.2	43.8
	S6 (I)	54.2	51.3
	S6 (J)	56.5	53.5
S7	Chair-S7 (A)	0.0	0.0
	Boat-S7 (B)	5.8	4.2
	S7 (C)	21.2	19.1
	^3S7 (D)	29.0	32.5
	^3S7 (E)	28.9	33.1
	S7 (F)	45.0	35.5
	S7 (G)	42.2	36.0
S8	Crown-S8 (A)	0.0	0.0
	Twisted-S8 (B)	10.4	8.8
	exo–endo-S8 (C)	11.4	9.7
	Boat-S8 (D)	21.2	17.0
	S8 (E)	25.7	18.3
	S8 (F)	30.8	26.6
	^3S8 (G)	36.4	37.4
	S8 (H)	68.1	49.2

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Below we discuss the results for the different S_X systems in more detail.

3.2.1 S₂. Analogously to O₂, the ground state of S₂ has triplet multiplicity. The experimental energy difference between the singlet and triplet structures used in the work by Swope et al.⁵⁴ is 13.4 kcal mol⁻¹. However, Barnes et al.¹²² reported the energy difference as 16.4 kcal mol⁻¹. Calculated energy differences range between 12.4 and 15.8 kcal mol⁻¹.^54,55,57,123 As seen in Table 1, the NEVPT2(12,8)//CCSD/aug-cc-pV(T+d)Z energy is more in line with the experimental value used by Swope et al.,⁵⁴ while the CCSDT//CCSD/aug-cc-pV(T+d)Z, CCSD(T)//CCSD/aug-cc-pV(T+d)Z, and CCSD(T)//ωB97X-D/aug-cc-pV(T+d)Z levels of theory align better with the result by Barnes et al.¹²² Based on the configuration state function (CSF) weights of the single-point NEVPT2(12,8)//CCSD/aug-cc-pV(T+d)Z calculations, shown in Table A1 in the SI, static correlation plays a significant role, especially for the singlet state. Therefore, we consider the 13.3 kcal mol⁻¹ energy gap, calculated at the NEVPT2(12,8)//CCSD/aug-cc-pV(T+d)Z level of theory, to be the more accurate value.

A high-resolution UV-Vis spectrum of S₂ was recently published by Stark et al.⁴⁸ Therefore, we present our simulated spectra in Fig. A9 in the SI. The simulated spectra do not reproduce the vibrational fine structure seen in the experimental spectrum due to neglecting Franck–Condon factors.

3.2.2 S₃. There are two minima on the singlet S₃ potential energy surface (PES): linear-S₃ and cyclic-S₃. By analyzing the rotational spectrum of S₃, McCarthy et al.¹²⁴ and Thorwirth et al.¹²⁵ derived the experimental structural parameters for linear-S₃. Calculations by Raghavachari et al.⁶⁶ showed that linear-S₃ was the global minimum energy structural isomer. Peterson et al.⁶² reached the same conclusion, as they calculated cyclic-S₃ to be (4.4 ± 0.5) kcal mol⁻¹ higher in energy than linear-S₃. As seen in Table 1, we also find that linear-S₃ is the minimum energy isomer. Energies calculated in this work at the CCSD/aug-cc-pV(T+d)Z and ωB97X-D/aug-cc-pV(T+d)Z levels of theory somewhat underestimate the difference between linear-S₃ and cyclic-S₃, but the other levels of theory are in good agreement. As seen in Table A2 in the SI, based on the CSF weights, the S₃ structures show some multiconfigurational character. Due to this finding, using the NEVPT2(18,12)//CCSD/aug-cc-pV(T+d)Z relative energy is recommended. Also notable is the fact that the isomerization reaction connecting the two minima is orbital-symmetry-forbidden,¹²⁶ having a barrier of about 27.6–29.4 kcal mol⁻¹.^127,128

The simulated spectra for the S₃ isomers are presented in Fig. 3. For the cyclic-S₃ isomer, all simulated spectra are in good agreement with each other. However, with the linear-S₃ isomer, a blue shift in the EOM-CCSD/aug-cc-pV(T+d)Z simulated spectrum is observed when compared to the TD-DFT spectra. To investigate this discrepancy, we performed further single-point vertical excitation calculations on the CCSD/aug-cc-pV(T+d)Z optimized geometry. The vertical excitation energies and oscillator strengths for the three lowest energy excited states were calculated using multiconfigurational random phase approximation, MC-RPA(18,12)/aug-cc-pV(T+d)Z, on top of a CASSCF(18,12)/aug-cc-pV(T+d)Z reference using the Orca 6.0.1 program.¹¹¹ The MC-RPA approach was chosen based on its reasonable computational cost and relatively good performance compared to other multireference methods.^129,130 Furthermore, EOM-CC3/aug-cc-pV(T+d)Z vertical excitation calculations were performed using the e^T 1.9.13 program.¹³¹

Fig. 3 Simulated spectra for the two singlet S₃ isomers, compared with the experimental spectrum of S₃ by Billmers and Smith,⁵⁰ measured at temperatures in the range of 673–903 K. The experimental spectrum was digitized using the PlotDigitizer¹³⁴ tool. Note the different scales on the axes. The experimental spectrum has been scaled down by 80% for clarity in the lower part of the figure.

The results from these additional calculations are presented in Table A4 in the SI. When specifically looking at the third transition (S₃ ← S₀), the result from the EOM-CCSD/aug-cc-pV(T+d)Z calculation is slightly better in line with the MC-RPA(18,12)/aug-cc-pV(T+d)Z and EOM-CC3/aug-cc-pV(T+d)Z results. Using the MC-RPA(18,12)/aug-cc-pV(T+d)Z results, we scale the EOM-CCSD/aug-cc-pV(T+d)Z simulated spectrum and visualize this in Fig. 3, which we recommend as our highest quality linear-S₃ UV-Vis spectrum (the scaling procedure is outlined in Section A2.3 in the SI). Scaling based on the MR-RPA result was motivated by the non-negligible amount of multiconfigurational character present in linear-S₃. For cyclic-S₃ we recommend the EOM-CCSD/aug-cc-pV(T+d)Z spectrum.

All simulated spectra in Fig. 3 predict a lower absorption cross section than the experimental spectrum. This is especially relevant for linear-S₃, as this is the minimum energy isomer. However, it should be noted that the experimental spectrum was measured at very high temperatures, 673–903 K.⁵⁰ Therefore, we postulate that the primary difference between the experimental spectrum and the simulated spectra originates from the different temperatures used, such as by the inclusion of hot bands in the experimental spectrum. Because of the high temperatures used in the experimental work, we consider the simulated spectra to be representative of a lower temperature (0–300 K) absorption spectra for use in photochemical models.

S₃ was one of the polysulfur species suggested as the unknown absorber (UA).³² Indeed, based on our simulated spectra, linear-S₃ absorbs in the same wavelength region as the UA (320–400 nm) with a relatively large absorption cross section. But the spectral match is only one part, as there must also be an appreciable abundance of linear-S₃ in the Venusian atmosphere to have an impact. The model by Zhang et al.³⁷ predicts a S₃ mixing ratio of approximately 0.01 ppt_V at an altitude of 58 km. However, while accurate reaction rates have been calculated for the S + S₂ reaction,¹³² other reaction rates for polysyulfurs are still based on estimates.¹³³ Updated reaction rates are needed for a more accurate determination.

3.2.3 S₄. There are several minima on the S₄ PES. While most studies agree that the global minimum of S₄ is the cis form of the linear open chain structures (cis-S₄ in Fig. 1),^63,65,66,72 some report a rectangular structure to be the global minimum.^71,135 Our results agree with the former, as we predict cis-S₄ to be the lowest energy isomer, being approximately 8 kcal mol⁻¹ lower in energy than the linear open-chain trans structure. Thorwirth et al.¹²⁵ also identified cis-S₄ through rotational spectroscopy. Curiously, we found that the energy differences between the singlet and triplet linear open-chain cis and trans structures are quite small when using the ωB97X-D/aug-cc-pV(T+d)Z and CCSD/aug-cc-pV(T+d)Z levels of theory. However, when adding energy corrections using CCSD(T)/aug-cc-pV(T+d)Z, CCSDT/aug-cc-pV(T+d)Z, and NEVPT2(24,16)/aug-cc-pV(T+d)Z, the energy differences increase, yet remain below 15 kcal mol⁻¹, as seen in Table 1.

The singlet PES for the S₄ system is shown in Fig. 4. We find, like Wong and Steudel,⁶³ that the rectangular structure (labeled TS1 in Fig. 4) is a transition state. At the ωB97X-D/aug-cc-pV(T+d)Z level of theory this transition state corresponds to an interconversion of cis-S₄ to itself, which might be relevant for isotope scrambling. We also note that, despite our best efforts, we were unable to locate a transition state connecting cis-S₄ to trans-S₄ at the ωB97X-D/aug-cc-pV(T+d)Z level of theory. As can be seen in Table A3 in the SI, the CSF weights of cis-S₄ and trans-S₄ indicate that multiconfigurational approaches should be used. Therefore, this transition state might exist only on a multireference PES, or proceed via the spin triplet PES. However, based on the relatively high isomerization energies shown in Fig. 4, we conclude, as did Wong and Steudel,⁶³ that the different singlet structural isomers are well separated and should be experimentally accessible, especially in low-temperature conditions.

Fig. 4 The singlet potential energy surface (PES) for the S₄ system. Energies are relative zero-point vibrational energy corrected electronic energies (kcal mol⁻¹). The coloring corresponds to the level of theory used: black – CCSD(T)//CCSD/aug-cc-pV(T+d)Z; blue – CCSDT//CCSD/aug-cc-pV(T+d)Z; red – NEVPT2(24,16)//CCSD/aug-cc-pV(T+d)Z; green – CCSD(T)//ωB97X-D/aug-cc-pV(T+d)Z. Note that the calculations using CCSDT//CCSD/aug-cc-pV(T+d)Z did not converge for the cyclic-S₃ (S) and cyclic-S₄ structures.

The simulated spectra for the S₄ singlet species are presented in Fig. 5–7. The simulated spectra for cyclic-S₃ (S), cyclic-S₄, and trigonal-S₄ are similar. But for cis-S₄ and trans-S₄, the first peak in the EOM-CCSD/aug-cc-pV(T+d)Z spectra is blue-shifted compared to the spectra simulated using TD-DFT. Therefore, as with linear-S₃, we performed additional vertical excitation calculations on the CCSD/aug-cc-pV(TT+d)Z optimized geometries using EOM-CC3/aug-cc-pV(T+d)Z and MC-RPA(24,16)/aug-cc-pV(T+d)Z.

Fig. 5 Simulated spectra for cis-S₄ and trans-S₄, compared with the scaled experimental spectrum of S₄ by Billmers and Smith,⁵⁰ measured at temperatures in the range of 673–903 K. The experimental spectrum was digitized using the PlotDigitizer¹³⁴ tool. The experimental spectrum has been scaled up by 400% for clarity. Note the different scales on the axes.

Fig. 6 Simulated spectra for cyclic-S₃ (S) and cyclic-S₄, compared with the experimental spectrum of S₄ by Billmers and Smith,⁵⁰ measured at temperatures in the range of 673–903 K. The experimental spectrum was digitized using the PlotDigitizer¹³⁴ tool. Note the different scales on the axes. The experimental spectrum has been scaled down by 80% for clarity in the lower part of the figure.

Fig. 7 Simulated spectra for trigonal-S₄, compared with the experimental spectrum of S₄ by Billmers and Smith,⁵⁰ measured at temperatures in the range of 673–903 K. The experimental spectrum was digitized using the PlotDigitizer¹³⁴ tool.

The vertical excitation energies and oscillator strengths for the three lowest excited states of cis-S₄ and trans-S₄ are presented in the SI in Tables A5 and A6. It is clear that the vertical excitation energies calculated at the EOM-CCSD/aug-cc-pV(T+d)Z level of theory are in good agreement with those calculated using EOM-CC3/aug-cc-pV(T+d)Z and MC-RPA(24,16)/aug-cc-pV(T+d)Z. In contrast, TD-DFT energies differ by approximately 100 nm. We use the results from the MC-RPA(24,16)/aug-cc-pV(T+d)Z calculations and scale the EOM-CCSD/aug-cc-pV(T+d)Z spectra for both cis-S₄ and trans-S₄. These scaled spectra are also visualized in Fig. 5, and we recommend these as our highest quality UV-Vis spectra for cis-S₄ and trans-S₄ to be used in photochemical models. The EOM-CCSD/aug-cc-pV(T+d)Z spectra are recommended for the other singlet S₄ isomers.

In Fig. 8 we show the simulated spectra for the spin triplet S₄ species cis-³S₄ and trans-³S₄. In the case of cis-³S₄, simulated spectra are similar, except for the ωB2PLYP/aug-cc-pV(T+d)Z spectrum. The ωB2PLYP/aug-cc-pV(T+d)Z spectrum is more uniform, while the other simulated spectra show two different peaks. The ωB2PLYP/aug-cc-pV(T+d)Z spectrum also predicts a larger absorption cross section. For trans-³S₄, all simulated spectra are similar. Furthermore, as can be seen and Fig. A10 and A11 in the SI, the simulated spectra predict that the triplet S₄ species weakly absorb in the near-infrared region.

Fig. 8 Simulated spectra for cis-³S₄ and trans-³S₄. Note the different scales on the axes.

Of the S₄ isomers we have studied in this work, based on the simulated spectra, we find that trigonal-S₄ absorbs in the same wavelength window as the UA. Furthermore, the triplet structures cis-³S₄ and trans-³S₄ also show some absorption between 320 and 400 nm. However, cis-³S₄ and trans-³S₄ are very unlikely to contribute to the enigmatic absorption, but they might be a useful addition as intermediates in photochemical models. Zhang et al.³⁷ estimated a mixing ratio of approximately 0.1 ppt_V using their Venusian atmospheric model. However, the reaction rates are again based on estimates.¹³³ In addition, atmospheric models do not differentiate between the S₄ species.^36–41 Based on the PES in Fig. 4, the S₄ isomers are separated by quite high barriers, meaning that several isomers could form and persist. Therefore, individual reaction rates for the formation and consumption of the different isomers should be determined.

3.2.4 S₅–S₈. The number of possible minima for the S₅–S₈ systems are greater than for S₂–S₄, simply because of the increased number of atoms in the system. However, when looking at Table 1 a clear trend is seen; the cyclic structures are the lowest energy structural isomers. Still, the higher energy S₅–S₈ structures might be important intermediates in polysulfur reactions. We simulate the UV-Vis spectra for the lowest energy S₅–S₈ isomers and those isomers found in Table 1 within 5.0 kcal mol⁻¹ of the lowest energy one. Because these structures are closed-shell singlet species, we believe TD-DFT is an appropriate level of theory for the spectral simulations. The results of these spectral simulations are presented in Fig. 9–12.

Fig. 9 The simulated spectra for cyclic-S₅ (A).

Fig. 10 The simulated spectra for chair-S₆, compared to the experimental spectrum by Steudel et al.⁵³ measured in a methylcyclohexane (C₇H₁₄) solution. The experimental spectrum was digitized using the PlotDigitizer¹³⁴ tool.

Fig. 11 The simulated spectra for the two lowest energy S₇ conformers, chair-S₇ and boat-S₇, compared to the experimental spectrum by Steudel et al.⁵³ measured in a methylcyclohexane (C₇H₁₄) solution. The experimental spectrum was digitized using the PlotDigitizer¹³⁴ tool. Note the different scales on the axes.

Fig. 12 The simulated spectra for crown-S₈, compared to the experimental spectrum by Steudel et al.⁵³ measured in a methylcyclohexane (C₇H₁₄) solution. The experimental spectrum was digitized using the PlotDigitizer¹³⁴ tool.

The cyclic-S₅ structure is clearly the most stable S₅ structural isomer, being 20 kcal mol⁻¹ lower in energy than the next lowest energy S₅, which is ³S₅ (B). To the best of knowledge, no experimental UV-Vis spectrum for S₅ has been published. As seen in Fig. 9, three of the TD-DFT methods, BHandLHYP/aug-cc-pV(T+d)Z, CAM-B3LYP/aug-cc-pV(T+d)Z, and ωB97X-D3/aug-cc-pV(T+d)Z, predict similar absorption profiles for cyclic-S₅, with a maximum around 290 nm. The ωB2PLYP/aug-cc-pV(T+d)Z spectrum differs slightly from the others. Although there is a clear shoulder around 290 nm, the maximum is predicted to be at 265 nm.

Like cyclohexane, there are two different conformers of cyclic S₆. Of these two, the chair conformer is the minimum, as the boat conformer is calculated to be 13.2 kcal mol⁻¹ higher in energy at the CCSD(T)//ωB97X-D/aug-cc-pV(T+d)Z level of theory. This is in excellent agreement with the energy difference reported by Cioslowski et al.⁶⁸ Other studies also report chair-S₆ to be the minimum energy structure.^66,71 The experimental spectrum of S₆ by Steudel et al.⁵³ seen in Fig. 10 shows a steady increase in the absorption cross section when going below 330 nm. All simulated spectra predict a lower absorption cross section than the experiment, but the steady increase is reproduced.

For S₇, the general features of the experimental spectrum are similar to that of S₆; at wavelengths shorter than 340 nm, the absorption cross section steadily increases. Spectral simulations were performed for two structures, these being chair-S₇ and boat-S₇, as the energy difference is only 4.2 kcal mol⁻¹ at the CCSD(T)//ωB97X-D/aug-cc-pV(T+d)Z level of theory – an energy gap consistent with previous results.^66,68,71 The simulated spectra in Fig. 11 show a lower absorption cross section compared to the experimental counterpart, but once again the general shape is reproduced. The overall absorption cross section of the boat-S₇ conformer is predicted to be slightly higher than that of the chair-S₇ conformer, even surpassing the experimental value below 270 nm.

There are several unique cyclic S₈ structures. Our calculations show that the lowest energy structure is crown-S₈, which is consistent with previously reported results.^{66–68,70,71} Contrary to the findings by Wong et al.,⁷⁰ we find that the twisted-S₈ is slightly lower in energy compared to the exo–endo-S₈ conformer. Using simulated annealing, Jones and Hohl¹³⁵ reported a S₈ cluster (see Fig. 1b in ref. 135) that was only 9.9 kcal mol⁻¹ higher in energy than crown-S₈. This cluster was also reported by Wong et al.⁷⁰ and Jones and Ballone.⁷¹ However, using ωB97X-D/aug-cc-pV(T+d)Z, this cluster optimizes to the S₈ (E) structure, which is 18.3 kcal mol⁻¹ higher in energy than crown-S₈ at the CCSD(T)//ωB97X-D/aug-cc-pV(T+d)Z level of theory. The experimental spectrum for S₈ is presented in Fig. 12, together with the simulated spectra for crown-S₈. The simulated spectra predict a lower absorption cross section compared to the experimental spectrum between 320–270 nm, but reach agreement at 260 nm.

An important detail with respect to the experimental UV-Vis spectra published by Steudel et al. used for S₆–S₈ is that these are liquid phase spectra, measured in methylcyclohexane (C₇H₁₄). Different effects, such as clustering or solvent interactions, might affect the final spectrum. In contrast, our simulated spectra are in the gas phase. Therefore, the experimental spectra are not directly comparable to our spectra. We consider our simulated spectra to be a more appropriate representation of gas phase spectra. For the S₅–S₈ species, we recommend using the spectra simulated at the ωB97X-D3/aug-cc-pV(T+d)Z level of theory.

The experimental spectra S₆–S₈ reach down to approximately 220 nm. To construct the simulated spectra, we used the ten lowest excited states. This number of excited states limits us to about 250–260 nm. Increasing the number of excited states would extend the simulated spectra, but this was prohibitively expensive and thus not done in this work.

We estimated if spin–orbit coupling (SOC) would influence the simulated spectra in any significant way. We performed single-point vertical excitation calculations using the ωB97X-D3 functional with the aug-cc-pV(T+d)Z basis set on the ωB97X-D/aug-cc-pV(T+d)Z optimized geometries [specifically, linear-S₃, cyclic-S₃, cis-S₄, trans-S₄, cyclic-S₅ (A), chair-S₆ (A), chair-S₇ (A), boat-S₇ (B), and crown-S₈ (A)]. The calculations were performed with the Orca 6.0.1 program, where SOC is implemented for TD-DFT.¹³⁶ We saw shifts of less than 0.02 eV when comparing the calculated singlet transitions without SOC with those calculated including SOC. Furthermore, while including SOC allowed some triplet transitions to have nonzero oscillator strengths, the oscillator strengths were at least two orders of magnitude lower when compared to the singlet transitions. We therefore conclude that the effect of SOC is negligible.

Although S₈ was suggested as a possible UA candidate,³¹ based on our simulated gas phase spectra crown-S₈ has no appreciable absorption cross section in the UA wavelength range of 320–400 nm. We therefore refute gas phase S₈ as an UA candidate.

4 Conclusions

In this work, we used the nuclear ensemble approximation approach to simulate UV-Vis spectra for the polysulfur species S₂–S₈. Our results provide wavelength-dependent absolute absorption cross sections for individual structural isomers, which can be included into photochemical models. Successful benchmark calculations using a set of small molecules and radicals (CS₂, H₂S, HO₂, O₃, OCS, PH₃, SO₂, and SO₃) indicate that the nuclear ensemble approach is a valid approach to simulate UV-Vis spectra.

Our results show that polysulfurs display great structural diversity. The lowest energy S₂–S₄ structures tend to be linear, while cyclic structures are the lowest energy structures for the larger S₅–S₈ species. Furthermore, we identified two low-energy triplet S₄ species, cis-³S₄ and trans-³S₄, that could be important intermediates in polysulfur reactions.

We recommend using the MC-RPA scaled EOM-CCSD/aug-cc-pV(T+d)Z spectra for linear-S₃, cis-S₄, and trans-S₄ when adding these species to photochemical models. For the other S₃ and S₄ species, the EOM-CCSD/aug-cc-pV(T+d)Z spectra are recommended. The ωB97X-D3/aug-cc-pV(T+d)Z spectra are recommended for the S₅–S₈ species. Finally, for S₂ we recommend using previously published experimental gas phase spectra or high-level simulated spectra.^48,60,61

Based on the simulated spectra, we find two structures that absorb in the same wavelength region (320–400 nm) as the unknown absorber in the Venusian atmosphere, these being linear-S₃ and trigonal-S₄. However, the low concentrations predicted by Venusian atmospheric models makes it unlikely that polysulfurs are a major contributor to the enigmatic absorption. A conclusive assessment could be possible by investigating the accuracy of the polysulfur reaction rates used in Venusian atmospheric models, either using experimental methods or through high-level ab initio calculations.

For future work, the simulated spectra for the S₅–S₈ species should be extended down to 200 nm, as the high spectral resolution channel of the VenSpec-U UV-Vis spectrometer onboard the EnVision spacecraft reaches down to 205 nm.¹³⁷ Finally, with the upcoming mission to Venus by NASA (DAVINCI¹³⁸ and VERITAS¹³⁹) and ESA (EnVision¹⁴⁰), investigating sulfur chemistry in the context of Venus is vital for correctly interpreting the data these missions will produce in the coming decade.

Author contributions

R. S.: investigation; formal analysis; writing – original draft preparation; writing – review & editing; funding acquisition. B. N. F.: conceptualization; writing – review & editing; supervision; funding acquisition; project administration.

Conflicts of interest

There are no conflicts to declare.

Data availability

Supplementary information is available. See DOI: https://doi.org/10.1039/d5cp02991k.

Data supporting the findings in this article can be found in the SI. The simulated spectra for all species are included in the Zenodo archive DOI: https://doi.org/10.5281/zenodo.16361268.

Acknowledgements

R. S. acknowledges the support from the Doctoral Programme in Chemistry and Molecular Sciences (CHEMS-DP) at the University of Helsinki. R. S. thanks Dr Alyce Whipp and Mr Hugo Åström (University of Helsinki) for language edits, and Dr Nino Runeberg (CSC – IT Center for Science) and Prof. Sarai Dery Folkestad (Norwegian University of Science and Technology) for helping with the calculations. R. S. and B. N. F. thank Ms Nanna Falk Christensen for helpful discussions on spectral simulations. B. N. F. thanks for the fellowship support from the Carlsberg Foundation (CF22-0754) and the Magnus Ehrnrooth Foundation. The authors thank Prof. Theo Kurtén (University of Helsinki) for helpful discussions, and acknowledge Research Council of Finland Center of Excellence VILMA grant number 346369. Computational resources were generously provided by CSC – IT Center for Science (Espoo, Finland) project number 2008552.

References

1. D. V. Titov, H. Svedhem and F. W. Taylor, in The Atmosphere of Venus: Current Knowledge and Future Investigations, ed. P. Blondel and J. W. Mason, Springer-Verlag, Berlin, Heidelberg, 2006, pp. 87–110.
2. B. Fegley, Treatise on Geochemistry, Elsevier, Oxford, 2nd edn, 2014, pp. 127–148.
3. A. Vandaele, O. Korablev, D. Belyaev, S. Chamberlain, D. Evdokimova, T. Encrenaz, L. Esposito, K. Jessup, F. Lefèvre, S. Limaye, A. Mahieux, E. Marcq, F. Mills, F. Montmessin, C. Parkinson, S. Robert, T. Roman, B. Sandor, A. Stolzenbach, C. Wilson and V. Wilquet, Icarus, 2017, 295, 16–33.
4. C. Gillmann, G. N. Arney, G. Avice, M. Dyar, G. J. Golabek, A. J. Gülcher, N. M. Johnson, M. Lefèvre and T. Widemann, Treatise on Geochemistry, Elsevier, Oxford, 3rd edn, 2025, pp. 289–323.
5. A. Mahieux, S. Robert, F. Mills, K. Jessup, L. Trompet, S. Aoki, A. Piccialli, J. Peralta and A. Vandaele, Icarus, 2023, 399, 115556.
6. S.-M. Tsai, E. K. H. Lee, D. Powell, P. Gao, X. Zhang, J. Moses, E. Hébrard, O. Venot, V. Parmentier, S. Jordan, R. Hu, M. K. Alam, L. Alderson, N. M. Batalha, J. L. Bean, B. Benneke, C. J. Bierson, R. P. Brady, L. Carone, A. L. Carter, K. L. Chubb, J. Inglis, J. Leconte, M. Line, M. López-Morales, Y. Miguel, K. Molaverdikhani, Z. Rustamkulov, D. K. Sing, K. B. Stevenson, H. R. Wakeford, J. Yang, K. Aggarwal, R. Baeyens, S. Barat, M. De Val-Borro, T. Daylan, J. J. Fortney, K. France, J. M. Goyal, D. Grant, J. Kirk, L. Kreidberg, A. Louca, S. E. Moran, S. Mukherjee, E. Nasedkin, K. Ohno, B. V. Rackham, S. Redfield, J. Taylor, P. Tremblin, C. Visscher, N. L. Wallack, L. Welbanks, A. Youngblood, E.-M. Ahrer, N. E. Batalha, P. Behr, Z. K. Berta-Thompson, J. Blecic, S. L. Casewell, I. J. M. Crossfield, N. Crouzet, P. E. Cubillos, L. Decin, J.-M. Désert, A. D. Feinstein, N. P. Gibson, J. Harrington, K. Heng, T. Henning, E. M.-R. Kempton, J. Krick, P.-O. Lagage, M. Lendl, J. D. Lothringer, M. Mansfield, N. J. Mayne, T. Mikal-Evans, E. Palle, E. Schlawin, O. Shorttle, P. J. Wheatley and S. N. Yurchenko, Nature, 2023, 617, 483–487.
7. A. Dyrek, M. Min, L. Decin, J. Bouwman, N. Crouzet, P. Mollière, P.-O. Lagage, T. Konings, P. Tremblin, M. Güdel, J. Pye, R. Waters, T. Henning, B. Vandenbussche, F. Ardevol Martinez, I. Argyriou, E. Ducrot, L. Heinke, G. Van Looveren, O. Absil, D. Barrado, P. Baudoz, A. Boccaletti, C. Cossou, A. Coulais, B. Edwards, R. Gastaud, A. Glasse, A. Glauser, T. P. Greene, S. Kendrew, O. Krause, F. Lahuis, M. Mueller, G. Olofsson, P. Patapis, D. Rouan, P. Royer, S. Scheithauer, I. Waldmann, N. Whiteford, L. Colina, E. F. Van Dishoeck, G. Östlin, T. P. Ray and G. Wright, Nature, 2024, 625, 51–54.
8. T. G. Beatty, L. Welbanks, E. Schlawin, T. J. Bell, M. R. Line, M. Murphy, I. Edelman, T. P. Greene, J. J. Fortney, G. W. Henry, S. Mukherjee, K. Ohno, V. Parmentier, E. Rauscher, L. S. Wiser and K. E. Arnold, Astrophys. J., Lett., 2024, 970, L10.
9. A. Banerjee, J. K. Barstow, A. Gressier, N. Espinoza, D. K. Sing, N. H. Allen, S. M. Birkmann, R. C. Challener, N. Crouzet, C. A. Haswell, N. K. Lewis, S. R. Lewis and J. Yang, Astrophys. J., Lett., 2024, 975, L11.
10. N. Madhusudhan, S. Constantinou, M. Holmberg, S. Sarkar, A. A. A. Piette and J. I. Moses, Astrophys. J., Lett., 2025, 983, L40.
11. J. Pearl, R. Hanel, V. Kunde, W. Maguire, K. Fox, S. Gupta, C. Ponnamperuma and F. Raulin, Nature, 1979, 280, 755–758.
12. J. R. Spencer, K. L. Jessup, M. A. McGrath, G. E. Ballester and R. Yelle, Science, 2000, 288, 1208–1210.
13. S. Kremser, L. W. Thomason, M. Von Hobe, M. Hermann, T. Deshler, C. Timmreck, M. Toohey, A. Stenke, J. P. Schwarz, R. Weigel, S. Fueglistaler, F. J. Prata, J.-P. Vernier, H. Schlager, J. E. Barnes, J.-C. Antuña-Marrero, D. Fairlie, M. Palm, E. Mahieu, J. Notholt, M. Rex, C. Bingen, F. Vanhellemont, A. Bourassa, J. M. C. Plane, D. Klocke, S. A. Carn, L. Clarisse, T. Trickl, R. Neely, A. D. James, L. Rieger, J. C. Wilson and B. Meland, Rev. Geophys., 2016, 54, 278–335.
14. C. V. Brodowsky, T. Sukhodolov, G. Chiodo, V. Aquila, S. Bekki, S. S. Dhomse, M. Höpfner, A. Laakso, G. W. Mann, U. Niemeier, G. Pitari, I. Quaglia, E. Rozanov, A. Schmidt, T. Sekiya, S. Tilmes, C. Timmreck, S. Vattioni, D. Visioni, P. Yu, Y. Zhu and T. Peter, Atmos. Chem. Phys., 2024, 24, 5513–5548.
15. J. G. Calvert, A. Lazrus, G. L. Kok, B. G. Heikes, J. G. Walega, J. Lind and C. A. Cantrell, Nature, 1985, 317, 27–35.
16. R. J. Weber, G. Chen, D. D. Davis, R. L. Mauldin III, D. J. Tanner, F. L. Eisele, A. D. Clarke, D. C. Thornton and A. R. Bandy, J. Geophys. Res.:Atmos., 2001, 106, 24107–24117.
17. M. Sipilä, T. Berndt, T. Petäjä, D. Brus, J. Vanhanen, F. Stratmann, J. Patokoski, R. L. Mauldin, A.-P. Hyvärinen, H. Lihavainen and M. Kulmala, Science, 2010, 327, 1243–1246.
18. R. Zhang, A. Khalizov, L. Wang, M. Hu and W. Xu, Chem. Rev., 2012, 112, 1957–2011.
19. J. B. Pollack, O. B. Toon, R. C. Whitten, R. Boese, B. Ragent, M. Tomasko, L. Esposito, L. Travis and D. Wiedman, J. Geophys. Res.:Space Phys., 1980, 85, 8141–8150.
20. S. Pérez-Hoyos, A. Sánchez-Lavega, A. García-Muñoz, P. G. J. Irwin, J. Peralta, G. Holsclaw, W. M. McClintock and J. F. Sanz-Requena, J. Geophys. Res.:Planets, 2018, 123, 145–162.
21. V. A. Krasnopolsky, Photochemistry of the Atmospheres of Mars and Venus, Springer Berlin Heidelberg, Berlin, Heidelberg, 1986, vol. 13.
22. L. Zasova, V. Krasnopolsky and V. Moroz, Adv. Space Res., 1981, 1, 13–16.
23. V. A. Krasnopolsky, Icarus, 2017, 286, 134–137.
24. C. Z. Jiang, P. B. Rimmer, G. G. Lozano, N. J. Tosca, C. L. Kufner, D. D. Sasselov and S. J. Thompson, Sci. Adv., 2024, 10, eadg8826.
25. J. V. Egan, A. D. James and J. M. C. Plane, ACS Earth Space Chem., 2025, 9, 2127–2136.
26. T. Trabelsi and J. S. Francisco, Astrophys. J., 2024, 977, 92.
27. B. N. Frandsen, P. O. Wennberg and H. G. Kjaergaard, Geophys. Res. Lett., 2016, 43, 11146–11155.
28. B. N. Frandsen, S. Farahani, E. Vogt, J. R. Lane and H. G. Kjaergaard, J. Phys. Chem. A, 2020, 124, 7047–7059.
29. J. Spacek, P. Rimmer, G. E. Owens, S. R. Cady, D. Sharma and S. A. Benner, ACS Earth Space Chem., 2024, 8, 89–98.
30. T. Trabelsi, J. Phys. Chem. A, 2025, 129, 4870–4878.
31. B. Hapke and R. Nelson, J. Atmos. Sci., 1975, 32, 1212–1218.
32. O. B. Toon, R. P. Turco and J. B. Pollack, Icarus, 1982, 51, 358–373.
33. V. Krasnopolsky, Adv. Space Res., 1987, 7, 25–27.
34. B. S. Maiorov, N. I. Ignat’ev, V. I. Moroz, L. V. Zasova, B. E. Moshkin, I. V. Khatuntsev and A. P. Ekonomov, Sol. Syst. Res., 2005, 39, 267–282.
35. V. A. Krasnopolsky, Icarus, 2013, 225, 570–580.
36. F. P. Mills and M. Allen, Planet. Space Sci., 2007, 55, 1729–1740.
37. X. Zhang, M. C. Liang, F. P. Mills, D. A. Belyaev and Y. L. Yung, Icarus, 2012, 217, 714–739.
38. V. A. Krasnopolsky, Icarus, 2012, 218, 230–246.
39. A. Stolzenbach, F. Lefèvre, S. Lebonnois and A. Määttänen, Icarus, 2023, 395, 115447.
40. C. J. Bierson and X. Zhang, J. Geophys. Res.:Planets, 2020, 125, e2019JE006159.
41. P. B. Rimmer, S. Jordan, T. Constantinou, P. Woitke, O. Shorttle, R. Hobbs and A. Paschodimas, Planet. Sci. J., 2021, 2, 133.
42. A. Francés-Monerris, J. Carmona-García, T. Trabelsi, A. Saiz-Lopez, J. R. Lyons, J. S. Francisco and D. Roca-Sanjuán, Nat. Commun., 2022, 13, 4425.
43. J. I. Graham and W. N. Hartley, Proc. R. Soc. London, Ser. A, 1910, 84, 311–324.
44. G. Herzberg and L. G. Mundie, J. Chem. Phys., 1940, 8, 263–273.
45. L. Brewer, G. D. Brabson and B. Meyer, J. Chem. Phys., 1965, 42, 1385–1389.
46. A. Kumar, P. Chowdhury, K. Rama Rao and J. Mittal, Chem. Phys. Lett., 1992, 198, 406–412.
47. J. Lindner, R. Niemann and E. Tiemann, J. Mol. Spectrosc., 1994, 165, 358–367.
48. G. Stark, H. Herde, J. R. Lyons, A. N. Heays, N. de Oliveira, G. Nave, B. R. Lewis and S. T. Gibson, J. Chem. Phys., 2018, 148, 244302.
49. B. Meyer, T. Stroyer-Hansen and T. Oommen, J. Mol. Spectrosc., 1972, 42, 335–343.
50. R. I. Billmers and A. L. Smith, J. Phys. Chem., 1991, 95, 4242–4245.
51. P. Hassanzadeh and L. Andrews, J. Phys. Chem., 1992, 96, 6579–6585.
52. M. S. Boumedien, J. Corset and E. Picquenard, J. Raman Spectrosc., 1999, 30, 463–472.
53. R. Steudel, D. Jensen, P. Göbel and P. Hugo, Ber. Bunsen-Ges. Phys. Chem., 1988, 92, 118–122.
54. W. C. Swope, Y. Lee, I. Schaefer and F. Henry, J. Chem. Phys., 1979, 70, 947–953.
55. G. Theodorakopoulos, S. Peyerimhoff and R. J. Buenker, Chem. Phys. Lett., 1981, 81, 413–420.
56. B. A. Hess, R. J. Buenker, C. M. Marian and S. D. Peyerimhoff, Chem. Phys., 1982, 71, 79–85.
57. T. Kiljunen, J. Eloranta, H. Kunttu, L. Khriachtchev, M. Pettersson and M. Räsänen, J. Chem. Phys., 2000, 112, 7475–7483.
58. W. Xing, D. Shi, J. Sun, H. Liu and Z. Zhu, Mol. Phys., 2013, 111, 673–685.
59. W. Xing, D. Shi and J. Sun, J. Quant. Spectrosc. Radiat. Transfer, 2020, 242, 106805.
60. K. Sarka, S. O. Danielache, A. Kondorskiy and S. Nanbu, Chem. Phys., 2019, 516, 108–115.
61. K. Sarka and S. Nanbu, ACS Earth Space Chem., 2023, 7, 2374–2381.
62. K. A. Peterson, J. R. Lyons and J. S. Francisco, J. Chem. Phys., 2006, 125, 084314.
63. M. W. Wong and R. Steudel, Chem. Phys. Lett., 2003, 379, 162–169.
64. D. Hohl, R. O. Jones, R. Car and M. Parrinello, J. Chem. Phys., 1988, 89, 6823–6835.
65. G. E. Quelch, H. F. I. Schaefer and C. J. Marsden, J. Am. Chem. Soc., 1990, 112, 8719–8733.
66. K. Raghavachari, C. M. Rohlfing and J. S. Binkley, J. Chem. Phys., 1990, 93, 5862–5874.
67. M. Chen, M. Liu, H. Luo, Q. Zhang and C. Au, J. Mol. Struct. THEOCHEM, 2001, 548, 133–141.
68. J. Cioslowski, A. Szarecka and D. Moncrieff, J. Phys. Chem. A, 2001, 105, 501–505.
69. S. Millefiori and A. Alparone, J. Phys. Chem. A, 2001, 105, 9489–9497.
70. M. W. Wong, Y. Steudel and R. Steudel, Chem. Phys. Lett., 2002, 364, 387–392.
71. R. O. Jones and P. Ballone, J. Chem. Phys., 2003, 118, 9257–9265.
72. M. H. Matus, D. A. Dixon, K. A. Peterson, J. A. W. Harkless and J. S. Francisco, J. Chem. Phys., 2007, 127, 174305.
73. C. Tian, Z. Wang, S. Pan, W. Zhao, Q. Guo and M. Jin, J. Mol. Struct. THEOCHEM, 2009, 906, 1–5.
74. S. Farahani, B. N. Frandsen, H. G. Kjaergaard and J. R. Lane, J. Phys. Chem. A, 2019, 123, 6605–6617.
75. M. Barbatti, A. J. A. Aquino and H. Lischka, Phys. Chem. Chem. Phys., 2010, 12, 4959–4967.
76. O. Ferchichi, N. Derbel, T. Cours and A. Alijah, Phys. Chem. Chem. Phys., 2020, 22, 4059–4071.
77. W. Chao, R. Skog, B. N. Frandsen, G. H. Jones, K. T. Pham, M. Okumura, M. P. Sulbaek Andersen, C. J. Percival and F. A. F. Winiberg, Commun. Chem., 2025, 8, 163.
78. A. Saiz-Lopez, S. P. Sitkiewicz, D. Roca-Sanjuán, J. M. Oliva-Enrich, J. Z. Dávalos, R. Notario, M. Jiskra, Y. Xu, F. Wang, C. P. Thackray, E. M. Sunderland, D. J. Jacob, O. Travnikov, C. A. Cuevas, A. U. Acuña, D. Rivero, J. M. C. Plane, D. E. Kinnison and J. E. Sonke, Nat. Commun., 2018, 9, 4796.
79. M. Barbatti and K. Sen, Int. J. Quantum Chem., 2016, 116, 762–771.
80. A. Prlj, E. Marsili, L. Hutton, D. Hollas, D. Shchepanovska, D. R. Glowacki, P. Slavícek and B. F. E. Curchod, ACS Earth Space Chem., 2022, 6, 207–217.
81. A. Prlj, D. Hollas and B. F. E. Curchod, J. Phys. Chem. A, 2023, 127, 7400–7409.
82. D. Hollas and B. F. E. Curchod, J. Phys. Chem. A, 2024, 128, 8580–8590.
83. H. Keller-Rudek, G. K. Moortgat, R. Sander and R. Sörensen, Earth Syst. Sci. Data, 2013, 5, 365–373.
84. D. A. Matthews, L. Cheng, M. E. Harding, F. Lipparini, S. Stopkowicz, T.-C. Jagau, P. G. Szalay, J. Gauss and J. F. Stanton, J. Chem. Phys., 2020, 152, 214108.
85. J. F. Stanton, J. Gauss, L. Cheng, M. E. Harding, D. A. Matthews and P. G. Szalay, CFOUR, Coupled-Cluster techniques for Computational Chemistry, a quantum-chemical program package, with contributions from A. Asthana, A.A. Auer, R.J. Bartlett, U. Benedikt, C. Berger, D.E. Bernholdt, S. Blaschke, Y. J. Bomble, S. Burger, O. Christiansen, D. Datta, F. Engel, R. Faber, J. Greiner, M. Heckert, O. Heun, M. Hilgenberg, C. Huber, T.-C. Jagau, D. Jonsson, J. Jusélius, T. Kirsch, M.-P. Kitsaras, K. Klein, G.M. Kopper, W.J. Lauderdale, F. Lipparini, J. Liu, T. Metzroth, L. Monzel, L.A. Mück, D.P. O’Neill, T. Nottoli, J. Ostwald, D. R. Price, E. Prochnow, C. Puzzarini, K. Ruud, F. Schiffmann, W. Schwalbach, C. Simmons, S. Stopkowicz, A. Tajti, T. Uhlířová, J. Vázquez, F. Wang, J. D. Watts, P. Yergün. C. Zhang, X. Zheng, and the integral packages MOLECULE (J. Almlöf and P. R. Taylor), PROPS (P. R. Taylor), ABACUS (T. Helgaker, H. J. Aa. Jensen, P. Jørgensen, and J. Olsen), and ECP routines by A. V. Mitin and C. van Wüllen, For the current version, see https://www.cfour.de.
86. A. D. Becke, J. Chem. Phys., 1993, 98, 1372–1377.
87. T. Yanai, D. P. Tew and N. C. Handy, Chem. Phys. Lett., 2004, 393, 51–57.
88. Y.-S. Lin, G.-D. Li, S.-P. Mao and J.-D. Chai, J. Chem. Theory Comput., 2013, 9, 263–272.
89. M. Casanova-Páez, M. B. Dardis and L. Goerigk, J. Chem. Theory Comput., 2019, 15, 4735–4744.
90. J. Dunning and H. Thom, J. Chem. Phys., 1989, 90, 1007–1023.
91. J. Dunning, H. Thom, K. A. Peterson and A. K. Wilson, J. Chem. Phys., 2001, 114, 9244–9253.
92. R. A. Kendall, J. Dunning, H. Thom and R. J. Harrison, J. Chem. Phys., 1992, 96, 6796–6806.
93. D. E. Woon, J. Dunning and H. Thom, J. Chem. Phys., 1993, 98, 1358–1371.
94. M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Izmaylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. J. Bearpark, J. J. Heyd, E. N. Brothers, K. N. Kudin, V. N. Staroverov, T. A. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. P. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene, C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin, K. Morokuma, O. Farkas, J. B. Foresman and D. J. Fox, Gaussian 16 Revision C.02, Gaussian Inc., Wallingford CT, 2021.
95. F. Neese, Wiley Interdiscip. Rev.:Comput. Mol. Sci., 2022, 12, e1606.
96. B. P. Pritchard, D. Altarawy, B. Didier, T. D. Gibson and T. L. Windus, J. Chem. Inf. Model., 2019, 59, 4814–4820.
97. R. Crespo-Otero and M. Barbatti, Theor. Chem. Acc., 2012, 131, 1237.
98. R. Crespo-Otero and M. Barbatti, Chem. Rev., 2018, 118, 7026–7068.
99. M. Barbatti, M. Bondanza, R. Crespo-Otero, B. Demoulin, P. O. Dral, G. Granucci, F. Kossoski, H. Lischka, B. Mennucci, S. Mukherjee, M. Pederzoli, M. Persico, M. Pinheiro Jr, J. Pittner, F. Plasser, E. Sangiogo Gil and L. Stojanovic, J. Chem. Theory Comput., 2022, 18, 6851–6865.
100. M. Barbatti, G. Granucci, M. Ruckenbauer, F. Plasser, R. Crespo-Otero, J. Pittner, M. Persico and H. Lischka, NEWTON-X: a package for Newtonian dynamics close to the crossing seam, version 2.6 build 01, 2023, https://www.newtonx.org.
101. S. Mai, H. Gattuso, A. Monari and L. González, Front. Chem., 2018, 6, 495.
102. MATLAB, version: 24.1.0.2653294 (R2024a) Update 5, 2024, The MathWorks Inc., Natick, Massachusetts, https://www.mathworks.com.
103. P. Pracht, F. Bohle and S. Grimme, Phys. Chem. Chem. Phys., 2020, 22, 7169–7192.
104. P. Pracht, S. Grimme, C. Bannwarth, F. Bohle, S. Ehlert, G. Feldmann, J. Gorges, M. Müller, T. Neudecker, C. Plett, S. Spicher, P. Steinbach, P. A. Wesolowski and F. Zeller, J. Chem. Phys., 2024, 160, 114110.
105. C. Bannwarth, S. Ehlert and S. Grimme, J. Chem. Theory Comput., 2019, 15, 1652–1671.
106. M. S. Gordon, J. S. Binkley, J. A. Pople, W. J. Pietro and W. J. Hehre, J. Am. Chem. Soc., 1982, 104, 2797–2803.
107. J.-D. Chai and M. Head-Gordon, J. Chem. Phys., 2008, 128, 084106.
108. J.-D. Chai and M. Head-Gordon, Phys. Chem. Chem. Phys., 2008, 10, 6615–6620.
109. E. Miliordos and S. S. Xantheas, J. Am. Chem. Soc., 2014, 136, 2808–2817.
110. D. Mester, P. R. Nagy, J. Csóka, L. Gyevi-Nagy, P. B. Szabó, R. A. Horváth, K. Petrov, B. Hégely, B. Ladóczki, G. Samu, B. D. Lorincz and M. Kállay, J. Phys. Chem. A, 2025, 129, 2086–2107.
111. F. Neese, Wiley Interdiscip. Rev.:Comput. Mol. Sci., 2025, 15, e70019.
112. H. Grosch, A. Fateev and S. Clausen, J. Quant. Spectrosc. Radiat. Transfer, 2015, 154, 28–34.
113. K. Sunanda, A. Shastri, A. K. Das and B. Raja Sekhar, J. Quant. Spectrosc. Radiat. Transfer, 2015, 151, 76–87.
114. J. W. Rabalais, J. R. McDonald, V. M. Scherr and S. P. McGlynn, Chem. Rev., 1971, 71, 73–108.
115. C. Wu and F. Chen, J. Quant. Spectrosc. Radiat. Transfer, 1998, 60, 17–23.
116. S. P. Sander, J. Abbatt, J. R. Barker, J. B. Burkholder, R. R. Friedl, D. M. Golden, R. E. Huie, C. E. Kolb, M. J. Kurylo, G. K. Moortgat, V. L. Orkin and P. H. Wine, Chemical Kinetics and Photochemical Data for Use in Atmospheric Studies, Evaluation No. 17, JPL Publication 10-6, Jet Propulsion Laboratory, Pasadena, 2011, https://jpldataeval.jpl.nasa.gov.
117. P. Limão-Vieira, F. Ferreira da Silva, D. Almeida, M. Hoshino, H. Tanaka, D. Mogi, T. Tanioka, N. J. Mason, S. V. Hoffmann, M.-J. Hubin-Franskin and J. Delwiche, J. Chem. Phys., 2015, 142, 064303.
118. F. Chen, D. L. Judge, C. Y. Robert Wu, J. Caldwell, H. P. White and R. Wagener, J. Geophys. Res.:Planets, 1991, 96, 17519–17527.
119. S. L. Manatt and A. L. Lane, J. Quant. Spectrosc. Radiat. Transfer, 1993, 50, 267–276.
120. J. B. Burkholder and S. McKeen, Geophys. Res. Lett., 1997, 24, 3201–3204.
121. P. E. Hintze, H. G. Kjaergaard, V. Vaida and J. B. Burkholder, J. Phys. Chem. A, 2003, 107, 1112–1118.
122. I. Barnes, K. Becker and E. Fink, Chem. Phys. Lett., 1979, 67, 314–317.
123. L. Xiao, B. Yan and B. F. Minaev, Physchem, 2023, 3, 110–124.
124. M. C. McCarthy, S. Thorwirth, C. A. Gottlieb and P. Thaddeus, J. Am. Chem. Soc., 2004, 126, 4096–4097.
125. S. Thorwirth, M. C. McCarthy, C. A. Gottlieb, P. Thaddeus, H. Gupta and J. F. Stanton, J. Chem. Phys., 2005, 123, 054326.
126. B. Flemmig, P. T. Wolczanski and R. Hoffmann, J. Am. Chem. Soc., 2005, 127, 1278–1285.
127. J. Ivanic, G. J. Atchity and K. Ruedenberg, J. Chem. Phys., 1997, 107, 4307–4317.
128. J. D. Goddard, X. Chen and G. Orlova, J. Phys. Chem. A, 1999, 103, 4078–4084.
129. B. Helmich-Paris, J. Chem. Phys., 2019, 150, 174121.
130. A. D. Torres, M. A. S. Francisco, R. R. Oliveira and A. B. Rocha, J. Phys. Chem. A, 2023, 127, 3200–3209.
131. S. D. Folkestad, E. F. Kjønstad, R. H. Myhre, J. H. Andersen, A. Balbi, S. Coriani, T. Giovannini, L. Goletto, T. S. Haugland, A. Hutcheson, I.-M. Høyvik, T. Moitra, A. C. Paul, M. Scavino, A. S. Skeidsvoll, S. H. Tveten and H. Koch, J. Chem. Phys., 2020, 152, 184103.
132. S. Du, T. C. Germann, J. S. Francisco, K. A. Peterson, H.-G. Yu and J. R. Lyons, J. Chem. Phys., 2011, 134, 154508.
133. J. I. Moses, M. Y. Zolotov and B. Fegley, Icarus, 2002, 156, 76–106.
134. PlotDigitizer: Version 3.1.6, PORBITAL Technologies, 2025, https://plotdigitizer.com.
135. R. O. Jones and D. Hohl, Int. J. Quantum Chem., 1990, 38, 141–151.
136. B. de Souza, G. Farias, F. Neese and R. Izsák, J. Chem. Theory Comput., 2019, 15, 1896–1904.
137. E. Marcq, F. Montmessin, J. Lasue, B. Bézard, K. L. Jessup, Y. J. Lee, C. F. Wilson, B. Lustrement, N. Rouanet and G. Guignan, Adv. Space Res., 2021, 68, 275–291.
138. J. B. Garvin, S. A. Getty, G. N. Arney, N. M. Johnson, E. Kohler, K. O. Schwer, M. Sekerak, A. Bartels, R. S. Saylor, V. E. Elliott, C. S. Goodloe, M. B. Garrison, V. Cottini, N. Izenberg, R. Lorenz, C. A. Malespin, M. Ravine, C. R. Webster, D. H. Atkinson, S. Aslam, S. Atreya, B. J. Bos, W. B. Brinckerhoff, B. Campbell, D. Crisp, J. R. Filiberto, F. Forget, M. Gilmore, N. Gorius, D. Grinspoon, A. E. Hofmann, S. R. Kane, W. Kiefer, S. Lebonnois, P. R. Mahaffy, A. Pavlov, M. Trainer, K. J. Zahnle and M. Zolotov, Planet. Sci. J., 2022, 3, 117.
139. G. Cascioli, S. Hensley, F. De Marchi, D. Breuer, D. Durante, P. Racioppa, L. Iess, E. Mazarico and S. E. Smrekar, Planet. Sci. J., 2021, 2, 220.
140. R. Ghail, C. Wilson, T. Widemann, L. Bruzzone, C. Dumoulin, J. Helbert, R. Herrick, E. Marcq, P. Mason, P. Rosenblatt, A. C. Vandaele and L.-J. Burtz, EnVision: understanding why our most Earth-like neighbour is different, arXiv, 2017, preprint, arXiv:1703.09010 DOI:10.48550/arXiv.1703.09010.

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