Elemental sulfur aerosol-forming mechanism

Significance

The elemental sulfur aerosols are an important constituent in the atmospheres of Earth, Mars, and Venus. There is now evidence suggesting that these aerosols have also played a role in the evolution of early life on Earth. Traditionally, the photolysis of sulfur gases by UV light is thought to be the main mechanism for the formation of sulfur particles in these atmospheres. But, in the theoretical calculations reported here, we propose a nonphotochemical mechanism for the formation of elemental sulfur aerosols that takes advantage of the interaction between sulfur oxides and hydrogen sulfide under water or sulfuric acid catalysis. These results provide a chemical framework for understanding the formation mechanism of S⁰ aerosols in planetary atmospheres.

Abstract

Elemental sulfur aerosols are ubiquitous in the atmospheres of Venus, ancient Earth, and Mars. There is now an evolving body of evidence suggesting that these aerosols have also played a role in the evolution of early life on Earth. However, the exact details of their formation mechanism remain an open question. The present theoretical calculations suggest a chemical mechanism that takes advantage of the interaction between sulfur oxides, SO_n (n = 1, 2, 3) and hydrogen sulfide (nH₂S), resulting in the efficient formation of a S_n+1 particle. Interestingly, the SO_n + nH₂S → S_n+1 + nH₂O reactions occur via low-energy pathways under water or sulfuric acid catalysis. Once the S_n+1 particles are formed, they may further nucleate to form larger polysulfur aerosols, thus providing a chemical framework for understanding the formation mechanism of S⁰ aerosols in different environments.

Sulfur chemistry is a ubiquitous component in the atmospheres of Venus, early Earth, and Mars (1). The different forms of sulfur (e.g., S₂⁻, S⁻, S⁰, S₂O₃²⁻, SO₃²⁻, SO₄²⁻) provide energy for different types of sulfur metabolisms in different environments. The sulfur cycle in the Archean atmosphere is also believed to have played a role in the early evolution of life on Earth (2–7). The emerging photochemical picture suggests that reduced elemental sulfur (S⁰) and sulfate (SO₄) are the dominant sulfur species in the Archean (2–9). However, the latest isotope signatures of microscopic sulfides in marine sulfate deposits indicate that the ultimate source for this metabolic sulfur cycling was atmospherically derived S⁰ (10). One of the most important sources of sulfur into the atmosphere is from volcanoes, and the most abundant sulfur gases are SO₂ and H₂S. The photochemistry of these gases in the atmosphere yields elemental sulfur, sulfur particles, sulfuric acid, and oceanic sulfate. Scheme 1 illustrates the chemical processes suggested to be important in the photochemical oxidation of volcanic sulfur species in the early atmosphere of Earth.

Scheme 1.

Sulfur photochemistry in an anoxic early atmosphere. Sulfur is emitted to the atmosphere from volcanoes as sulfur dioxide and hydrogen sulfide, and is removed by rainout of soluble gases and by formation and deposition of sulfate and elemental sulfur particles.

The S⁰ aerosols are not only involved in the Archean life, but are also implicated in other environments (1, 11–19). For example, polysulfur (S_x = S_2→8) aerosols are thought to exist in clouds of Venus and their role as the unknown UV absorber in its lower atmosphere has been discussed in the literature (14). The S₈ particles are also observed in the marine troposphere (15). Finally, the role of S₈ aerosols in explaining the early climate of Mars atmosphere has also been debated (16).

Despite being of broad appeal, the formation mechanism of S⁰ aerosols remains an open question. The photolysis of SO₂ and SO by UV light with λ < 220 nm has generally been invoked to explain the mass-independent fractionation (MIF) of isotope effects in the sulfur cycle during the Archean (2–9). However, the contribution of other mass-independent chemical reactions to this geologic record remains unclear. To fully understand the sulfur cycle, it is necessary to identify all sources of sulfur compounds and account for all species which can occur in the atmosphere.

Results and Discussion

SO_n (n = 1, 2, 3) + nH₂S Potential Energy Surface.

As can be seen from Scheme 1, which summarizes the current state of sulfur chemistry in the atmosphere, there is a gap in our understanding of the connection between sulfur oxide chemistry and sulfur aerosol formation. Herein, we describe a nonphotochemical reaction mechanism that may possibly convert the SO_n + nH₂S (n = 1, 2, 3) chemistries into the S₈ aerosol in the gas phase (Scheme S1). It is the thermodynamics of these processes, and their catalysis by water and sulfuric acid, that we investigate here. This mechanism may not only help in better understanding the role of sulfur cycle involving SO_n, S₈, and H₂S as the potential S MIF carrier from the atmosphere to the ocean surface, but may also provide deeper insight into the formation mechanism of S⁰ aerosols in various other environments.

Scheme S1.

Proposed nonphotochemical mechanisms for the conversion of SO_n (n = 1, 2, 3) + nH₂S chemistries into the elemental sulfur aerosols, respectively. The green arrowed pathways are the most probable ones whereas the black pathways may also involve high-barrier steps.

We first explored the uncatalyzed gas-phase reactions of SO_n with nH₂S using quantum-chemical calculations at the coupled cluster single and double substitution method with a perturbative treatment of triple excitations [CCSD(T)]/aug-cc-pVTZ//M06-2X/aug-cc-pVTZ level of theory. We considered both singlet and triplet states for SO. Although the triplet ground state of SO is more stable than its singlet state, the calculations suggest that the ³SO + H₂S reaction leads to the formation of HS and HOS radicals, and is endothermic by 33.5 kcal/mol (Fig. S1). By contrast, the ¹SO + H₂S reaction is highly exothermic (Fig. S2). The relative energies of the computed transition-state structures and minima for the uncatalyzed ¹SO + H₂S reaction are shown in Fig. S2. The possible source of ¹SO is either the photolysis of SO₂ at λ < 220 nm or the partial oxidation of H₂S. There have also been reports that ¹SO could be ejected directly from the volcanic vent (20). However, the ¹SO + H₂S reaction would face competition from the ¹SO + O₂ → SO₂ + O in atmosphere, suggesting that the ¹SO + H₂S is more likely to happen locally where the concentration of sulfur gases is expected to be high.

Fig. S1.

Calculated reaction profiles for the uncatalyzed gas-phase reaction of H₂S with ¹SO. A shows a S₂-forming reaction pathway, whereas B and C show S₃-forming reaction pathways. Relative energies (kcal mol⁻¹) of minima and transition-state structures are calculated at the uCCSD(T)/aug-cc-pVTZ//uM06-2X/aug-cc-pVTZ level of theory.

Fig. S2.

Calculated potential energy surface for the uncatalyzed gas-phase reaction of H₂S with ³SO. Relative energies (kcal mol⁻¹) of minima and transition-state structures are calculated at the CCSD(T)/aug-cc-pVTZ//M06-2X/aug-cc-pVTZ level of theory.

The ¹SO + H₂S reaction results in the stepwise formation of H₂S₂O, which involves a barrier of 23.2 kcal/mol and has an exothermicity of 30.1 kcal/mol. The comparative analysis of the potential energy surfaces for the ¹SO and SO₂ (Fig. S3) reactions reveals that the ¹SO reaction is relatively more favorable. Although the uncatalyzed SO₂ + H₂S reaction has been previously calculated (13), we reexamined the reaction here in greater detail at the same level of theory to facilitate the comparison between the ¹SO and SO₂ reactions. The effective barrier for the ¹SO reaction is 7.3 kcal/mol lower than that for the SO₂ reaction. The exothermicity of the H₂S₂O formation is dramatically higher than that for the H₂S₂O₂ formation via the SO₂ + H₂S reaction. The higher reactivity of ¹SO may provide an alternate mechanistic explanation as to why SO is rarely observed in the troposphere.

Fig. S3.

Calculated potential energy surface for the uncatalyzed gas-phase reaction of H₂S with SO₂. Relative energies (kcal mol⁻¹) of minima and transition-state structures are calculated at the CCSD(T)/aug-cc-pVTZ//M06-2X/aug-cc-pVTZ level of theory.

Once H₂S₂O is formed, it can either dehydrate to S₂ or dehydrogenate to S₂O that can subsequently react with H₂S, resulting in the formation of S₃. Thus, the overall ¹SO + H₂S reaction leads to the formation of a S₂ or S₃ particle. Analogously, the SO₂ + 2H₂S and SO₃ + 3H₂S (Fig. S4) chemistries produce S₃ or S₄ and S₄ or S₅ particles, respectively. The SO₃ reaction is predicted to be more favorable than the SO₂ one. These mechanistic outcomes point toward a more generalized interaction between a sulfur oxide, SO_n and H₂S, which can be qualitatively summarized in the form of a reaction, SO_n + n/(n+1)H₂S → S_n+1 + S_n+2 + n/(n+1)H₂O. The formation of S_n+1 particle is predicted to be energetically more viable because it only involves low-barrier dehydration whereas the S_n+2 pathway must also go through high-barrier dehydrogenation in addition to low-barrier dehydration. Thus, it is reasonable to suggest that the SO_n + nH₂S → S_n+1 + nH₂O is the most efficient reaction except for n = 1, where the S_n+2 formation may also become feasible under certain conditions.

Fig. S4.

Calculated potential energy surface for the uncatalyzed gas-phase reaction of H₂S with SO₃. Relative energies (kcal mol⁻¹) of minima and transition-state structures are calculated at the CCSD(T)/aug-cc-pVTZ//M06-2X/aug-cc-pVTZ level of theory.

Reactivity Under Catalysis.

Although the formation of a S_n+1 particle via the SO_n + nH₂S reaction is more favorable, it still involves an appreciable thermal barrier that seems insurmountable under atmospheric conditions. However, recent studies (21–24) suggest that there are certain species in the atmospheres of Earth and Venus that may be able to catalyze these chemistries to such an extent that these processes become accessible. For example, H₂O is the most dominant species in the troposphere and has been shown to catalyze hydrogen atom transfer (HAT)-based addition reactions (21). Sulfuric acid (H₂SO₄) is an important constituent in the Venus atmosphere (25) and has been predicted to be one of the most efficient catalysts available for the HAT-based reactions (22–24). Building upon these recent developments, we next examined the SO_n + nH₂S reaction, which also involves an HAT reaction, in the presence of H₂O and H₂SO₄.

The formation of H₂S₂O from the ¹SO + H₂S reaction and its subsequent dehydration to S₂ becomes facile under H₂O or H₂SO₄ catalysis (Fig. 1). H₂SO₄ turns out to be a better catalyst than water because of its ability to stabilize reactants and products by forming sterically more favorable double hydrogen-bonding interactions. The alternate decomposition pathway for H₂S₂O, which leads to S₂O, is also significantly impacted under catalysis. However, the barriers for the S₂O-forming decomposition under catalysis are relatively higher than the dehydration one, suggesting that the probability of this decomposition pathway in water-rich surfaces or acidic environments may be quite low. H₂S₂O in the ¹SO + H₂S reaction is formed with an excess energy of 30.1 kcal/mol, which may play a role in making a S₂O-based S₃ channel accessible under catalytic conditions.

Fig. 1.

Calculated reaction profiles for the gas-phase reactions of SO_n (n = 1, 2, 3) with H₂S, (black), H₂S–H₂O (magenta) and H₂S–H₂SO₄ (green), respectively. Relative energies (kcal mol⁻¹) of minima and transition-state structures are calculated at the CCSD(T)/aug-cc-pVTZ//M06-2X/aug-cc-pVTZ level of theory. Note that the SO reactions have been calculated at the uCCSD(T)/aug-cc-pVTZ//uM06-2X/aug-cc-pVTZ level.

On the other hand, we only examined the SO₂ + 2H₂S → S₃ + 2H₂O-forming pathway in the presence of a single H₂O and H₂SO₄ molecule (Fig. 1). This is because the S₄-particle-forming pathways are mediated by very high-lying transition states (Fig. S3) and are not expected to become accessible even under H₂O or H₂SO₄ catalysis. The overall SO₂ + 2H₂S → S₃ + 2H₂O reaction is calculated to be 5.7 kcal/mol exothermic. The uncatalyzed H₂S₂O₂ formation involves an effective barrier of 30.5 kcal/mol. Under H₂O and H₂SO₄ catalysis, the reaction barrier is appreciably lowered to 14.6 and 14.7 kcal/mol, respectively. The subsequent dehydration of H₂S₂O₂, which produces S₂O, has a barrier of 28.4 kcal/mol and an exothermicity of 2.5 kcal/mol that are significantly impacted under catalysis. H₂SO₄ produces more catalytic effect than water in this case; the dehydration barriers for the H₂SO₄- and H₂O-catalyzed reactions are lowered to 12.1 and 15.8 kcal/mol, respectively. The reaction of S₂O with H₂S and the eventual decomposition of H₂S₃O into S₃ + 2H₂O are significantly influenced under catalysis. Although the S₃ formation via the SO₂ + 2H₂S reaction is 11.4 kcal/mol less exothermic than that in the ¹SO + 2H₂S reaction, it is still expected to be the favored pathway because it bypasses the high-barrier dehydrogenation step, which significantly lowers the energetics of the overall reaction. The effect of catalysis on the S₄-forming pathway from the SO₃ + 3H₂S reaction is also appreciable; all of the transition states are submerged below the separated reactants and the overall reaction occurs in a barrierless manner (Fig. 1). Moreover, the complexed S₄ particle, which is hydrogen-bonded with catalyst, H₂O or H₂SO₄, is at least 18.0 kcal/mol more stable than SO₃ and ∼7.0 kcal/mol more stable than free S₄ and catalyst. Among all of the sulfur gases considered, the sulfur particle formation via the SO₃ reactions is the most favorable. Fig. 2 summarizes the optimum path for the S_n+1 formation from the SO_n + nH₂S reaction under H₂SO₄ catalysis. This qualitative profile reveals an interesting reactivity pattern; for a given interaction between SO_n and nH₂S, there are n high-energy dehydrogenation channels that may or may not open up depending upon reaction conditions.

Fig. 2.

Reaction scheme network showing the optimum path for the sulfuric acid-assisted formation of elemental S_n+1 aerosols from the SO_n (n = 1, 2, 3) + nH₂S reaction. The blue and green connections represent the favorable pathways whereas the red connections represent high-energy less likely pathways.

Redefining the Formation Mechanism of S⁰ Aerosols.

The S⁰ aerosols in the anoxic atmosphere of Archean are suggested to be produced by the UV photolysis of SO₂ at λ< 220 nm (2–9). However, the present results suggest a nonphotochemical mechanism for the formation of these S⁰ aerosols, in which the S_n+1 particle is formed from the interaction of H₂O- or H₂SO₄-bound H₂S with SO_n (Fig. 3). This indicates that once S₂, S₃, or S₄ is formed, it could initiate a self-reaction that would build larger S⁰ particles. The mechanistic beauty of this aerosol-forming chemical process is that it does not require the conventional gas-phase three-body sulfur atom recombination, S + S, required for forming S₂ (5, 26, 27) or the UV photon-induced reaction between S and SH to form S₂ and H (28). It should be noted that several S_2–4 allotropes generally have very low vapor pressures in planetary atmospheres (7) However, the consideration of the kinetics of condensation may make these gas-phase sulfur self-reactions important under certain conditions.

Fig. 3.

Reaction scheme showing the optimum path for the formation of elemental S₈ aerosols from the SO_n (n = 1, 2, 3) + nH₂S reaction. The green connections represent the probable pathways whereas the red connections represent less likely pathways.

This mechanism has broad implications. For example, in Venus clouds, the exact source of polysulfur particle, which absorbs UV light, is unknown (14). The present calculations suggest that the S_n+1 formation is the most favorable under H₂SO₄ catalysis. The S_n+1 particle may then nucleate into the polysulfur particle. Sulfur species are abundantly available in the Venus atmosphere. The estimated concentration of SO₂ lies in the range of 180 ± 50 ppm (25) and 130 ± 35 ppm (29), whereas the H₂S concentration has been predicted to be 80 ± 40 ppm (30). H₂SO₄ and H₂O are present in ∼5- and ∼30-ppm amounts, respectively (31), which lends support to such a mechanism. In a recently studied kinetic model of Venus chemistry (1), the SO₃ mixing ratio in the 40–45-km altitude inversely correlates with the sulfur particle mixing ratio. The reaction of SO₃ with 3H₂S to form S₄ may explain this inverse correlation. The SO₃ + H₂S reaction involves a relatively smaller barrier than that of SO₂ + H₂S. In 35–45-km altitude, the H₂SO₄ mixing ratio is ∼5 ppm, implying that a significant fraction of H₂SO₄ may exist as a van der Waals complex with H₂S, and thus may catalyze the SO₃ + H₂S chemistry by invoking a bimolecular reaction between SO₃ and H₂SO₄••H₂S.

The CCSD(T) calculated binding energy of H₂SO₄••H₂S complex is 5.3 kcal/mol. The equilibrium constants for the H₂SO₄••H₂S complex calculated at various temperatures are collected in Table 1. The SO₃ + H₂SO₄••H₂S reaction occurs in a low-energy manner and is strongly exothermic. Note that the SO₂ mixing ratio in the middle atmosphere is the highest among the sulfur gases (17, 18) implying that the SO₂ + H₂SO₄••H₂S reaction would also make an important contribution toward the overall elemental sulfur production via SO_n + nH₂S reaction. These conclusions are consistent with the fact that the sulfur cycles in the middle atmosphere are fast. Although chlorosulfane chemistry has been previously proposed to explain the elemental sulfur aerosols (30), the present results suggest a direct connection between the chemistries of sulfur gases and elemental sulfur particles.

Table 1.

Calculated equilibrium constants for the complexes of hydrogen sulfide (H₂S) with water (H₂O), thiosulfurous acid (H₂S₂O₂), sulfuric acid (H₂SO₄), and carbonic acid (H₂CO₃) at various temperatures

	Equilibrium constant, Keq, cm3•molecule−1
Temperature, K	H2O••H2S (∆E = −1.36)	H2S2O2••H2S(∆E = −5.32)	H2CO3••H2S(∆E = −5.07)	H2SO4••H2S(∆E = −6.69)
200	4.35 × 10−22	1.73 × 10−20	6.21 × 10−20	2.79 × 10−18
210	3.71 × 10−22	8.99 × 10−21	3.42 × 10−20	1.26 × 10−18
220	3.23 × 10−22	4.97 × 10−21	1.99 × 10−20	6.17 × 10−19
230	2.85 × 10−22	2.90 × 10−21	1.23 × 10−20	3.22 × 10−19
240	2.56 × 10−22	1.78 × 10−21	7.87 × 10−21	1.78 × 10−19
250	2.32 × 10−22	1.14 × 10−21	5.25 × 10−21	1.03 × 10−19
260	2.13 × 10−22	7.54 × 10−22	3.63 × 10−21	6.28 × 10−20
270	1.98 × 10−22	5.17 × 10−22	2.58 × 10−21	3.97 × 10−20
280	1.85 × 10−22	3.65 × 10−22	1.89 × 10−21	2.60 × 10−20
290	1.74 × 10−22	2.64 × 10−22	1.42 × 10−21	1.76 × 10−20
298.15	1.67 × 10−22	2.07 × 10−22	1.14 × 10−21	1.31 × 10−20
300	1.65 × 10−22	1.96 × 10−22	1.09 × 10−21	1.23 × 10−20

CCSD(T)/aug-cc-pVTZ//M06-2X/aug-cc-pVTZ calculated binding energies of the H₂S complexes are given in parentheses.

In a recent Venus model by Zhang et al. (18), a strong anticorrelation between the sulfur gases (SO₃ and SO) and the elemental sulfur aerosol profiles below 65 km has also been observed. The thermal reactions of SO₃ and SO with H₂SO₄••H₂S clearly explain this anticorrelation. These thermal reactions not only provide useful mechanistic insights into an important sink of sulfur gases below 90 km in the Venus atmosphere, but may also help in understanding the mixing profiles of SO and SO₂ at higher altitudes (1, 17, 18, 32–36). At 96-km Venus atmosphere, the rates of the SO₃ hydration and the SO₃ photolysis are found to be comparable (17), which is suggestive of the fact that nearly half of the sulfur in H₂SO₄ goes into SO₃ and produces the inversion layers of SO₂ and SO. However, our calculations on the reactions of SO₂ and SO₃ with H₂X (X = O, and S) suggest that the bimolecular reactions of sulfur gases with H₂S involve smaller barriers than the analogous reactions involving H₂O (Fig. 4). This indicates that the SO₃ + H₂S reaction under H₂SO₄ catalysis may occur even at higher altitude, leading to the formation of elemental sulfur aerosol.

Fig. 4.

CCSD(T)/aug-cc-pVTZ//M06-2X/aug-cc-pVTZ calculated zero-point–corrected reaction profiles for the reactions of sulfur dioxide (Top) and sulfur trioxide (Bottom) with chalcogen hydrides, H₂X (X = O, and S). The energies are given in kcal/mol units.

In the Venus photochemistry model analyzed by Mills (37), the SO and S₂O sulfur gases, in addition to SO₂, are present in significant amounts in the middle atmosphere, which makes the low-barrier SO + H₂SO₄••H₂S and S₂O + H₂SO₄••H₂S chemistries important for the S₂- and S₃-formation mechanisms. A recently observed correlation between SO₂ depletion and enhancement in UV absorber also endorses this nonphotochemical mechanism (33). This mechanism also opens up a chemical channel for the formation of S₈ aerosols in the Mars atmosphere, which have been shown to play a role in Mars’ early climate (16).

Elemental sulfur deposits are also common at volcanic vents and fumaroles where high-temperature discharge gas is supersaturated in sulfur (38) as a result of equilibrium chemical reactions involving a variety of magmatic sulfur gases. The reaction between SO₂ and H₂S has been used to explain these sulfur sediments (39). It is important to note that in Kawah Ijen fumarole discharges (38), water is the most abundant gas species, followed by carbon dioxide (CO₂) and sulfur gases, SO₂ and H₂S. As a result, an appreciable fraction of H₂O may exist in a complexed form with H₂S. This may alter the energetics of the overall elemental sulfur formation by invoking the bimolecular reaction between SO₂ and H₂O••H₂S. Our results indicate that the barrier for the rate-determining step of the reaction between SO₂ and H₂O••H₂S is lowered by ∼40% compared with the uncatalyzed SO₂ + H₂S reaction. Alternatively, H₂O may add across either CO₂ to form carbonic acid (H₂CO₃) or SO₂ to form thiosulfurous acid (H₂S₂O₂), which may subsequently influence the SO₂ + H₂S reaction. Both H₂CO₃ (–C=O, OH) and H₂S₂O₂ (–S=O, OH) form strong complexes with H₂S (Table 1) and possess mandatory functionalities to allow the SO₂ + H₂S reaction to occur under acid catalysis. Considering that these proposed chemical processes are all H₂S-based, and no elemental sulfur formation has been seen in the absence of H₂S (40), the results could help in better understanding the geochemistry of the magmatic–hydrothermal systems.

In summary, we have used electronic structure calculations to suggest a nonphotochemical mechanism for the formation of elemental sulfur aerosols in planetary atmospheres. The mechanistic beauty of this proposal is that the reactions of sulfur oxides and hydrogen sulfide under water or sulfuric acid catalysis provide low-energy pathways for the formation of S₂–S₄ particles. Interestingly, the uncatalyzed reactions of sulfur oxides and hydrogen sulfide result in the intermediates that are functionally similar to sulfuric acid, which points to the fact that these sulfur chemistries could be autocatalyzed.

Methods

The SO_n (n = 1, 2, 3) + nH₂S reactions in the gas phase have been examined in the absence and presence of H₂O and H₂SO₄ catalysts. The uncatalyzed reactions have been briefly explored whereas the effect of catalysis on the most probable reactions has been examined in detail. In particular, the effect of H₂O or H₂SO₄ catalysis on the H₂S addition reactions and the subsequent H₂O elimination reactions have been explored. The effect of catalysis on the dehydrogenation reactions has not been examined because these reactions involve very high barriers and are less likely to be important in atmosphere. The impact of catalysis was quantified by calculating the reaction profiles for the bimolecular reactions between the H₂O- or H₂SO₄-bound H₂S and SO_n. All of the reactions have been calculated assuming the singlet ground state except for the SO reactions, which have been explored for both the singlet and triplet states. The singlet SO reactions have been examined because the spectroscopic signatures of the direct singlet SO ejection from the volcanic vent have been detected (20). All calculations were performed with Gaussian 09 (41). All geometries were optimized using the density-functional theory method, M06-2X (42), and the augmented correlation-consistent basis set, aug-cc-pVTZ (43). Because the open-shell ¹SO is more stable than the closed-shell one, the calculations involving ¹SO have been done using the uM06-2X/aug-cc-pVTZ level of theory. The energetics were further improved by performing single-point calculations at the CCSD(T) (44) and the aug-cc-pVTZ basis set. This level of theory has been found to provide an accurate description of hydrogen atom transfer-based chemistries in the recent past (23, 24). All stationary points were characterized by frequency calculations and reported energies include zero-point energy corrections (unscaled) from the method used for geometry optimization.