Characterization of the nucleation precursor (H₂SO₄–(CH₃)₂NH) complex: intra-cluster interactions and atmospheric relevance

Abstract

Amines have been proposed to participate in the nucleation process, but the electron density analysis and the determination of a temperature dependence of the clusters are still lacking. In this study, the clusters of (H₂SO₄)_m(CH₃NHCH₃)_n (m = 1–2, n = 1–3) are studied using the basin-hopping method coupled with density functional theory (DFT). Considering the high flexibility and complexity of a hydrogen bonding environment, the temperature dependence of the conformational population and the relative population fraction of the clusters are investigated. Moreover, the electron density is analyzed to identify the different types of intra-cluster interactions. The results indicate that the ratio between acid and base is very important for the cluster formation. The main interaction type changes from hydrogen bonding to a weak attraction as the number of bases increase. When the number of dimethylamine molecules is less than or equal to that of the sulfuric acid molecules as the most abundant clusters in the atmosphere, we tentatively suggest that the cluster contains less than two dimethylamine molecules because the critical clusters contain two or fewer sulfuric acid molecules. This means that the sulfuric acid–dimethylamine system can only form three main small clusters in the real atmosphere. Thus, other substances, such as water or organic acids, may be involved to promote the growth of clusters, and they may also affect the nucleation. This work predicts the possible forms of dimethylamine with sulfuric acid when participating in nucleation in a theoretical approach, and provides a reliable reference for the research on the nucleation mechanism containing dimethylamine in the atmosphere.

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1 Introduction

Aerosol particles are formed in the Earth's atmosphere via nucleation^1,2 and influence the Earth's climate by affecting cloud properties and precipitation.^3–7 They play an important role in global climate change^3,8 and are responsible for the detrimental impact on public health from airborne ultrafine particles, which can result in various cardiovascular diseases, lung cancer and enhanced mortality rates.^6,7,9 Nucleation produces a large fraction of atmospheric aerosols.^10,11 New particle formation (NPF) is commonly considered to be a two-step process:^12–14 an initial nucleation step to form the critical nucleus and a subsequent growth step in which the size of the critical clusters increases rapidly.^14,15 The initial nucleation step is critical, because its chemical composition and population would affect the NPF events and the nucleation rate.

Sulfuric acid (SA), for instance, is a major nucleating component in the atmosphere. Its presence in gaseous concentrations of 10⁶ to 10⁷ molecules cm⁻³ or more is a necessary condition for NPF.^4,14–19 In addition to SA, a number of other nucleating precursors, including atmospheric ions,^14,20,21 ammonia,^4,14,22–24 amines,^14,25–38 organic acids,^{10,11,39–41} and iodine oxides¹⁴ are proposed to be involved in the formation of the critical nucleus under different ambient environments. The abundance, volatility, and reactivity likely determine the potential of a chemical species to be a nucleation precursor, including such volatile organic compounds (VOCs)^39,42–48 as many saturated, unsaturated, or aromatic hydrocarbons. In addition, the highly elevated concentrations of gaseous aerosol precursors, most noticeably anthropogenic VOCs (aromatics), NO_x, and SO₂ emitted from transportation and industrial activities may be the main contributor to the haze problem in some developing countries, for example China.⁴⁹ In addition to ammonia, amines are one of the few basic compounds present in the atmosphere, which could be important to the nucleation process.

The size and chemical composition of atmospheric critical nuclei are still not clear due to the lack of existing analytical methods to directly probe the critical nucleus. Recently, dimethylamine (DMA) has been suggested to participate in the nucleation process in field experiments.^32,50–55 There has been theoretical research on SA and DMA,^26,56–58 but the theoretical study of the role of DMA molecule in the nucleation process is still lacking. There have been many studies analyzing the electron density of single molecules or dimers, such as for chemical and biological molecules.^59–67 However, to our knowledge before this study, there is little study on the analysis of the SA clusters via electron density, while this is essential for understanding these compounds' nucleation mechanism. As Hanson and Eisele^29,68 concluded that the critical clusters contain two or fewer H₂SO₄ molecules and as little as one ammonia molecule. Sipilä et al.¹⁶ suggested that freshly formed particles contain one or two sulfuric acid molecules, and Kürten et al.³⁸ also demonstrated that a cluster contains as few as two sulfuric acid and one or two dimethylamine molecules, and it was stable against evaporation during the experiments in the CLOUD chamber. Based on the above conclusions, the theoretical study of the number of dimethylamine molecule is an interesting and essential topic.

In this study, the structures, energetics, thermodynamics and vibrational frequencies of (H₂SO₄)_1–2[(CH₃)₂NH]_1–3 are attained with DFT method. Considering the high flexibility and complexity of hydrogen bonding environment, the temperature dependence of the conformational population and the relative population fraction of the clusters are investigated in detail. In addition to the thermodynamic analysis, we have also analyzed the clusters from the point of view of the electron density for the first time. We focus on the analysis of the non-covalent interactions (NCI) in all of the clusters, such as hydrogen bonds, van der Waals interactions, and steric clashes, which are useful to predict the attraction mechanism of the nucleation precursors and to understand the nucleation process. Based on our results, we tentatively conclude that the number of DMA molecules is less than or equal to the number of SA molecules. The critical cluster contains two or fewer H₂SO₄ molecules, and there may be one or two DMA molecules. Thus, the SA and DMA molecules can form a small cluster, such as (H₂SO₄)₁(CH₃NHCH₃)₁, (H₂SO₄)₂(CH₃NHCH₃)₁ or (H₂SO₄)₂(CH₃NHCH₃)₂, and the clusters may then grow larger by combining with other molecules, such as the water or organic acids, which would indirectly impact the Earth's climate and public health.

2 Methodology

Currently, DFT is the only option for studying large clusters due to the enormously large computational expenses associated with the application of ab initio MP2 and higher level methods which also preserve accuracy and speed.^{26,54,58,69–71}

The computation is conducted using three steps. In the first step, the initial geometries of the monomers and (H₂SO₄)_1–2[(CH₃)₂NH]_1–3 clusters in this study are searched using the Basin-Hopping (BH) algorithm^72–81 coupled with DFT, and the generalized gradient approximation in the Perdew–Burke–Ernzerhof (PBE) functional and the double numerical plus d-functions (DND) basis set which implemented in DMol³ software package⁸² are used in this step. The number of the BH search time is ten, and each search is performed with 400 Monte Carlo (MC) steps at different temperatures from 3000 K to 9000 K, with randomly generated initial structures. Then, the geometries of the clusters are optimized in two steps: an initial optimization step and a high level optimization step. The initial optimization is performed at the PW91PW91/6-311++G(d,p) level, and the PW91PW91/6-311++G(3df,3pd) level which implemented in the Gaussian 09 software package⁸³ is used for the second optimization and frequency calculation. The PW91PW91 method has shown fine performance on clusters containing common amines, and its predictions agree best with experiments compared to other density functionals.^58,71,84 To obtain more accurate structure parameters, we use the DF-LMP2-F12/aug-cc-pvdz-F12 method implemented in Molpro 2010.1.⁸⁵ for the higher level single point energy calculation in the third step.

Weak attractions play a crucial role in many chemical and biological systems,^59–67 especially in molecular clusters, that are primarily hydrogen-bonded complexes. To determine the weak interactions, Johnson et al.⁸⁶ constructed the NCI index that is based on the study of the reduced density gradient (RDG or s) as a function of electron density ρ(r).

By plotting the RDG with respect to the electron density or sign (λ₂)ρ, the NCI and the weak interaction regions can be identified when the RDG approaches zero. The spikes that appear in the plot are associated with the interaction of critical points (ICP).

Here, based on the global minima of (H₂SO₄)_1–2[(CH₃)₂NH]_1–3 calculated at the level of PW91PW91/6-311++G(3df,3pd), the plots of the electron density (ρ) and reduced density gradient (s = 1/(2(3π²)^1/3)|∇ρ|/ρ^4/3) are obtained. The DFT calculations are performed for a selected set of clusters using the multiwfn 3.3.3.⁸⁷ We have used a three-step method to analyze the weak attractions in the molecular clusters: firstly, we identify the NCI; secondly, we identify the interaction types; lastly, we perform the NCI analysis.

3 Results and discussion

3.1 Structures and nucleation mechanism

The representations of these structures are defined by using “xy” notation, “x” denotes the character of the BH search time, and the index “y” is used to distinguish among different structures in each BH search time. The structures of the studied clusters are shown in Fig. 1 and 2, the coordinates of (H₂SO₄)_m[(CH₃)₂NH]_n (m = 1–2, n = 1–3) isomers are displayed in the Date S1 (see ESI†), and the corresponding electronic energies, enthalpies and Gibbs free energies are presented in Table 1.

Fig. 1 The global minimum and the local minimum for (H₂SO₄)₁[(CH₃)₂NH]_n (n = 1–3) that were optimized at the PW91PW91/6-311++G(3df,3pd) level using DF-LMP2-F12/aug-cc-pV(T+d)Z for the single point energy calculation. The relative energies are given in kcal mol⁻¹ (red for oxygen, white for hydrogen, yellow for sulfur, gray for carbon and blue for nitrogen). The structures were drawn using Chemcraft 1.6 [http://www.chemcraftprog.com].

Fig. 2 The global minimum and the local minimum for (H₂SO₄)₂[(CH₃)₂NH]_n (n = 1–2) that were optimized at the PW91PW91/6-311++G(3df,3pd) level using DF-LMP2-F12/aug-cc-pV(T+d)Z for single the point energy calculation. The relative energies are given in kcal mol⁻¹ (red for oxygen, white for hydrogen, yellow for sulfur, gray for carbon and blue for nitrogen). The structures were drawn using Chemcraft 1.6 [http://www.chemcraftprog.com].

Table 1 Comparison of changes in the electron binding energies (ΔE, kcal mol⁻¹), enthalpies (ΔH, kcal mol⁻¹), and Gibbs free energies (ΔG, kcal mol⁻¹) that were associated with the formation of (H₂SO₄)_m[(CH₃)₂NH]_n (m = 1–2, n = 1–3) at a temperature of 298.15 K and at a pressure of 101.3 KPa

Reaction equation	ΔE	ΔH	ΔG
H2SO4 + (CH3)2NH → (H2SO4)[(CH3)2NH]	−23.54	−23.48	−13.92
(H2SO4)[(CH3)2NH] + (CH3)2NH → (H2SO4)[(CH3)2NH]2	−15.47	−15.16	−4.73
(H2SO4)[(CH3)2NH]2 + (CH3)2NH → (H2SO4)1[(CH3)2NH]3	−13.96	−14.24	−2.08
2H2SO4 + (CH3)2NH → (H2SO4)2[(CH3)2NH]1	−37.17	−37.26	−25.11
(H2SO4)2[(CH3)2NH]1 + (CH3)2NH → (H2SO4)2[(CH3)2NH]2	−27.88	−26.84	−19.85

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In the system of (H₂SO₄)[(CH₃)₂NH], there is a pair of mirror symmetric structures. One of the H protons in SA transfers to DMA to form the ions pair [(CH₃)₂NH₂⁺–HSO₄⁻], which associated with a N–H⋯O hydrogen bond, and the two N–H bonds in (CH₃)₂NH₂⁺ interact with the two O atoms in HSO₄⁻ to form a nearly symmetric hexagon. When add the second DMA molecule, the DMA molecule interact with the rest O atoms in HSO₄⁻ to form another hexagon. The two hexagonal planes are perpendicular to each other. The DMA molecules can also interact with the sulfuric acid in the same side to form an octagon, for example the configuration of A2. If there are three DMA molecules interact with the SA molecule, the global minima will have five hydrogen bonds, and the two DMA molecules combine with the SA in the same side to form a trihedron. The three hydrogen bonds are parallel in the trihedron. The others local minima are mainly composed with one or more polyhedral rings, such as C3, C7, and C8. The configurations of C7 and C8 can come from the local minima of (H₂SO₄)₁[(CH₃)₂NH]₂, the third DMA molecule interacts with a O atom in HSO₄⁻. Different from local minima, the three DMA molecules can also interact with SA molecule at the same side, such as the configuration of C4, four hydrogen bonds combine with the four molecules to form a decagon.

The configurations of (H₂SO₄)₂[(CH₃)₂NH]₁ and (H₂SO₄)₂[(CH₃)₂NH]₂ are shown in Fig. 2. In the cluster of (H₂SO₄)₂[(CH₃)₂NH]₁, two SA molecules interact with one DMA molecule. The proton hydrogen on one of the two SA molecules transfers to DMA molecule to form a N–H⋯O hydrogen bond. The other N–H bond in the DMA molecule combines with the other SA molecule to form another N–H⋯O hydrogen bond. In the configuration of A1, which is the global minima, the two H–O bonds in the SA with no proton transfer occurs interact with the two O atoms in HSO₄⁻ to form a symmetric octagon by two new hydrogen bonds, which achieve space stability. As for the isomers of (H₂SO₄)₂[(CH₃)₂NH]₂, the configuration of B1 is the global minima in this system. Each of the H protons in SA transfers to DMA to form two ions pair [HSO₄⁻–(CH₃)₂NH₂⁺–HSO₄⁻–(CH₃)₂NH₂⁺], which associated with four N–H⋯O hydrogen bonds. The other proton hydrogen interacts with the O atom in another HSO₄⁻ to form a hydrogen bond, just like a bridge connecting the two HSO₄⁻. In addition, there are other local minima may also stable, for example the configuration of B4, which with the lowest Gibbs free energy, the H–O in SA molecules are all in the outmost of the cluster of B4, forming a symmetric stable configuration.

In the (H₂SO₄)_m[(CH₃)₂NH]_n (m = 1–2, n = 1–3) system, the acid–base reaction is the mechanism of the nucleation. Proton transfer always occurs in the formation of the cluster, the DMA molecule obtains the proton from SA to form a symmetrical N–H bonding that is more stable for the clusters. However, the alkalinity of DMA is strong, and the SA is very easy to lose one hydrogen bond. The others, such as water, also form hydrogen bonds.

Moreover, as shown in Table 2, the energies of ΔG in this work are in qualitative agreement with previous studies.^{35,36,57,58,85} In our work, we have listed two ways to calculate the formation of Gibbs free energy, as the last two columns shown in the Table 2. In the previous column, the results without the single point energy correction by DF-LMP2-F12/aug-cc-pvdz-F12, and the values compare well with Nadykto et al.'s PW91/6-311++G(3df,3pd) results,⁵⁸ indicating the structures and the calculation method in our work are correct and reliable. The last column is the result of corrected with DF-LMP2-F12/aug-cc-pvdz-F12 for single point energy calculation, which is similar to the result of the B3RICC2 (B3LYP/CBSB7//RI-CC2/aug-cc-pv(T+d)z) method. As Nadykto et al. have demonstrated that the B3RICC2 method overestimate the stability of sulfuric acid–base clusters compared with conventional DFT and ab initio methods and experimental data, and fail to reproduce the base dependencies of nucleation and cluster formation rates. They have also shown that the thermochemistry predicted by the conventional PW91PW91 method is consistent with experiments and can be used for simulations of atmospheric nucleation rates.⁸⁴ If the view of Nadykto et al. is correct, then the DF-LMP2-F12/aug-cc-pvdz-F12 for energy correction has no effect on the accuracy of the calculation.

Table 2 Comparison of the Gibbs free energy of formation of global minima at T = 298.15 K in kcal mol⁻¹ with previous literature values for (H₂SO₄)_m[(CH₃)₂NH]_n, where m = 1–2, n = 1–3

Reaction equation	Kurtén et al.	Loukonen et al.	Ortega et al.	Nadykto et al.			This work
	RI-MP2^a	RI-MP2^b	B3RICC2^c	PW91^d	B3RICC2^e	B3LYP^f	PW91^g	PW91^h
a Ref. 36. RI-MP2/aug-ccpV(T+d)Z for geometry optimizations with RI-CC2/aug-ccpV(T+d)Z for energy correlation.b Ref. 35. BLYP/DZP for optimization and RI-MP2/aug-ccpV(T+d)Z for single-point energy calculation with anharmonicity corrections.c Ref. 57. B3LYP/CBSB7 for geometry optimizations and frequencies with RI-CC2/aug-cc-pV(T+d)Z single point energy calculations.d Ref. 58. PW91/6-311++G(3df,3pd) for geometry optimizations and frequencies without anharmonicity corrections.e Ref. 84. B3LYP/CBSB7 for geometry optimizations and frequencies with RI-CC2/aug-cc-pV(T+d)Z single point energy calculations.f Ref. 84. B3LYP/CBSB7 for geometry optimizations and frequency calculations.g PW91/6-311++G(3df,3pd) for geometry optimizations and frequencies without anharmonicity corrections.h PW91/6-311++G(3df,3pd) for geometry optimizations and frequencies with DF-LMP2-F12/VDZ-F12 single point energy calculations without anharmonicity corrections.
H2SO4 + (CH3)2NH	−13.66	−15.57	−15.40	−11.38	−15.40	−11.07	−11.88	−13.92
(H2SO4)[(CH3)2NH] + (CH3)2NH			−4.89	−3.92	−4.89	−2.76	−3.72	−4.73
(H2SO4)[(CH3)2NH]2 + (CH3)2NH			−4.79		−4.79	−0.76	−1.71	−2.08
2H2SO4 + (CH3)2NH			−26.99	−20.10	−26.98	−20.99	−20.66	−25.11
(H2SO4)2[(CH3)2NH]1 + (CH3)2NH			−21.29	−13.64	−19.38	−13.01	−13.35	−19.85

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3.2 Weak attraction analysis

3.2.1 Identifying the non-covalent interactions. To explore the features associated with the small reduced gradients, we examine the plots of RDG vs. ρ(r). These plots are generated by evaluating the PW91PW91 density and the reduced gradients on the cuboid grids with a 0.035 a.u. step size. The plotting of RDG vs. ρ(r), as in Fig. 3, reveals the basic pattern of intracluster interaction. The cluster of one SA and one DMA molecule is combined via the formation of a covalent bond. The mechanism of the cluster formation occurs via an acid–base reaction that is produced from the organic amine salt and the hydrogen sulfate root. However, in the other clusters, there are multiple weak attractions in the clusters.

Fig. 3 Plots of the reduced density gradient (s) vs. the electron density (ρ) for the global minimum of (H₂SO₄)_m[(CH₃)₂NH]_n, (m = 1–2, n = 1–3).

3.2.2 Identifying interaction types and non-covalent interactions analysis. Although localizing the low-density, low-gradient regions enables the identification of the weak interactions in a cluster system, we use the sign (λ₂)ρ to discern the different types of NCIs, as depicted in Fig. 4. Low density regions are obviously related to the weak interactions, such as van der Waals, whereas those with higher densities are related to stronger (either stabilizing or destabilizing) interactions, such as hydrogen bonds and steric clashes. In RDG gradient isosurface analysis, the gradient isosurfaces are colored according to the corresponding values of the sign (λ₂)ρ, which is a good indicator of the interaction strength. The gradient isosurfaces provide a rich visualization of the noncovalent interactions as broad regions of real space. The different colors represent different interaction types. Blue represents the highly attractive interactions (such as hydrogen bonds), green represents the weak interactions (such as dispersive-like van der Waals), and red represents the repulsive interactions (such as steric clashes).

Fig. 4 Plots of the reduced density gradient (s) vs. the electron density (ρ) multiplied by the sign of the second Hessian eigenvalue (λ₂) for the global minimum of (H₂SO₄)_m[(CH₃)₂NH]_n, (m = 1–2, n = 1–3).

As shown in Fig. 4, the hydrogen bonds and steric clashes interact evenly among the molecules, and the hydrogen-bond interaction plays a major role in the cluster formation. There is a pattern from the system of (H₂SO₄)₁[(CH₃)₂NH]₁ to (H₂SO₄)₁[(CH₃)₂NH]₃. As the number of bases increases, the number of hydrogen bonds increases, accounting for the high proportions. Meanwhile, the repulsive interaction and weak interaction will also be apparent. Thus, the types of interactions in the clusters are correlated with the ratio of the acid to the base.

From Fig. 4, there is a pattern among the five pictures. In the large negative values of sign (λ₂)ρ (−0.05, −0.06), there are two spikes that have similar hydrogen bond positions and similar interaction strengths. In addition to the pattern, there is another similar characteristic observed in the large and positive values of the sign (λ₂)ρ (0.045). The same features are observed in the gradient isosurface analysis shown in Fig. 5. The gradient isosurfaces provide a rich visualization of the noncovalent interactions as broad regions of real space. Two small spikes in the five systems correspond to the repulsive interaction. This result implies that the steric clashes are concurrent with the two hydrogen bonds in the clusters. By combining these results, we determine that the steric clashes exist between the two hydrogen bonds, and these co-exist and interact in a balanced manner within the clusters. When the size of the clusters increases, the steric clashes become strong, which occurs specifically for the clusters with one sulfuric acid and two dimethylamine molecules.

Fig. 5 Gradient isosurfaces (s = 0.5 a.u.) graph for the global minimum of (H₂SO₄)_m[(CH₃)₂NH]_n, (m = 1–2, n = 1–3).

3.3 Temperature dependence of conformational population

We have searched thousands of molecular geometries in each cluster system, and each configuration provides different contributions to the averaged free energy under different conditions. Additionally, the actual configurations measured in the experiment are not always the global minima. Thus, we have considered all of the possible local minimum structures and investigated the temperature dependence of the conformational population. The results are shown in Fig. 6.

Fig. 6 The Boltzmann averaging of free energies of the (H₂SO₄)_m[(CH₃)₂NH]_n, (m = 1–2, n = 1–3) cluster, taking into account the contributions of the local minima on the potential energy surface. The temperature (from 100 K to 1000 K) dependence of the conformational population for the isomers of (H₂SO₄)_m[(CH₃)₂NH]_n, (m = 1–2, n = 1–3) has been investigated. (a) (H₂SO₄)₁[(CH₃)₂NH]₁, (b) (H₂SO₄)₁[(CH₃)₂NH]₂, (c) (H₂SO₄)₁[(CH₃)₂NH]₃, (d) (H₂SO₄)₂[(CH₃)₂NH]₁, (e) (H₂SO₄)₂[(CH₃)₂NH]₂.

As shown in Fig. 6, with the temperature increases, the proportions of the different configurations in each system vary, and some cluster systems change significantly, such as Fig. 6(b)–(d). In Fig. 6(b), the structures of C1 and A2 account for a large proportion from 100 K to 900 K. For the structure of A1, the electron energy is the lowest, while the Gibbs free energy is higher. The entropy and the enthalpy (H_(T)–H₀) are all lower than the others from 100 K to 1000 K. Although it does not account for the high proportion, the structure will be the most stable. For the cluster of (H₂SO₄)[(CH₃)₂NH]₃ in Fig. 6(c), the proportions of the change in the structures of C2 and C1 overlap at the temperature of 200 K. However, the structure of A2 contributes more to the average free energy at the temperature from 800 K to 1000 K, and the A2 structure is more stable at high temperature. All of the proportions of the structures are less than 30%. Thus, perhaps several structures co-exist at the high temperatures.

In Fig. 6(d), the proportion varies significantly with a gap of only 2 K for the structures of A2 and A1 from 298.15 K to 300 K. The structure of A1 decreases by 10%, while the structure of A2 increases approximately 10%. This indicates that the entropy of structure A2 is large so that the proportion changes rapidly during very small temperature changes. At the following temperature, the proportions of the two structures decrease to 10% below, and for the other structures, such as A11, B2, A9, and B1, their proportions increase at high temperature. Thus, these structures will likely exist at high temperatures. Different conditions produce different temperatures on the Earth, so high temperatures can also exist on the Earth. Therefore, a discussion of the different configurations in each system is necessary.

In addition, there are some configurations which do not change with increase in temperature, such as the systems of (H₂SO₄)₁–(DMA)₁ and (H₂SO₄)₂–(DMA)₂. The stable configurations are not altered, and they are still in high proportion at different temperatures, especially for the system of (H₂SO₄)₂–(DMA)₂. This indicates that the structure most probably exists in the atmosphere under different conditions.

3.4 Atmospheric relevancy

The (H₂SO₄)_m[(CH₃)₂NH]_n system has strong hydrogen bonds. The strong primary hydrogen bond with the acid acts as the donor, and the amine acts as the receptor. The SA and DMA system is the most strong excitation in the atmosphere, because of the high concentration of the reactants and their reactivity. To estimate the actual concentrations of the SA and DMA clusters in the atmosphere, we use an approach that was described previously.^{15,24,88–91} The standard free energies, ΔG, give the equilibrium constants, K_f (at 298.15 K), for the formation of the clusters from the respective monomers:

ΔG = −RT ln(K_f) (2)

The equilibrium constant, K_f, is defined by the equation below:

where [H₂SO₄], [(CH₃)₂NH], and [(H₂SO₄)_m[(CH₃)₂NH]_n] are the vapor pressures of SA, DMA, and the cluster, respectively. The concentrations of SA and DMA can be obtained by an external field experiment.

A quantity referred as the relative population fraction (RPF) is defined by the following relationship:

where the total concentration of dimethylamine is approximately 1 ppt, which is the value corresponding to typical concentrations of these pollutant species, and the concentration of (H₂SO₄)_m[(CH₃)₂NH]_n, at a standard state of 1 atm and 298.15 K. In the system, the concentration of (H₂SO₄)_m[(CH₃)₂NH]_n (m = 1–2, n = 1–3) can be solved by the equilibrium equations, and eventually obtain the value of RPF. The concentration of SA in the atmospheric varies with time and geography. Berndt et al.²³ have shown that the concentration of SA in a flow tube via a mechanism needs to be above ∼10⁷ molecule cm⁻³ for nucleation to occur. We have thus used a SA concentration of 5 × 10⁷ molecule cm⁻³. Because the concentration of SA is much larger than that of the DMA, it is assumed to remain unchanged. The different atmosphere relations between the isomers in each system are considered and shown in Fig. 7.

Fig. 7 The atmospheric relevancy of the (H₂SO₄)_m[(CH₃)₂NH]_n, (m = 1–2, n = 1–3) cluster, taking into account the contributions of the local minimum on the potential energy surface at a temperature of 298.15 K and at a pressure of 101.3 KPa. (a) (H₂SO₄)₁[(CH₃)₂NH]₁, (b) (H₂SO₄)₁[(CH₃)₂NH]₂, (c) (H₂SO₄)₁[(CH₃)₂NH]₃, (d) (H₂SO₄)₂[(CH₃)₂NH]₁, (e) (H₂SO₄)₂[(CH₃)₂NH]₂.

In Fig. 7, for the standard condition of 298.15 K and 1 atm, the most abundant clusters are the global minimum, and they are part of the local minima. For the (H₂SO₄)₁[(CH₃)₂NH]₁ in Fig. 7(a), the main structures are the two mirror symmetry configurations, A1 and C1, and their energies are close to what occurs in the atmosphere. In Fig. 7(b), the main configurations in the atmosphere are the local minima structures for the system. However, for the other systems, the main configurations existed in the atmosphere are the lower-energy structures in the system (Fig. 7(c)–(e)). There are one or two stable configurations that have larger RPFs in the actual atmosphere.

For the (H₂SO₄)_m[(CH₃)₂NH]_n (m = 1–2, n = 1–3), the RPFs of the global minimum in each system are presented in Fig. 8. The cluster of the (H₂SO₄)₂[(CH₃)₂NH]₂ is the most abundant cluster in the atmosphere, and the (H₂SO₄)₁[(CH₃)₂NH]₁ and (H₂SO₄)₂[(CH₃)₂NH]₁ are less abundant in the atmosphere. The others have very small RPFs in the atmosphere. The abundance is related to the ratio of acid to base, and the number of DMA molecules is less than or equal to that of the sulfuric acid molecules which have high relative population fractions. Thus, these three clusters are the most likely to be formed during the nucleation process, and they contribute more to the atmospheric population. In addition to the temperature of the atmospheric layer (298.15 K), we have also considered the other states, at 273.15 K and 216.65 K; these are the average temperature of the troposphere and the temperature of the troposphere boundary layer, respectively. The results are shown in Fig. 9. Unlike the RPFs at 298.15 K, the RPFs at 273.15 K and 216.65 K change, and the RPF of (H₂SO₄)₁[(CH₃)₂NH]₁ is higher than the others, and all the values can be seen in the Table S1.† For example, for 216.65 K, the most stable cluster is (H₂SO₄)₁[(CH₃)₂NH]₁. As the temperature rises, all of the RPFs will increase, but the speeds of the clusters increase at different rates. The speed of the cluster (H₂SO₄)₂[(CH₃)₂NH]₂ is larger than the others; when the temperature reaches 298.15 K, the RPF of (H₂SO₄)₂[(CH₃)₂NH]₂ is the highest. The results can be interpreted by the changes in the Gibbs free energies, ΔG, in Table 3. At a temperature of 216.65 K and 273.15 K, the ΔG are positive values; and thus the reactions are not spontaneous processes, and the actual concentrations are so low that they may be non-existent under these atmospheric conditions.

Fig. 8 The atmospheric relevancy of the (H₂SO₄)_m[(CH₃)₂NH]_n, (m = 1–2, n = 1–3) cluster at a temperature of 298.15 K and a pressure of 101.3 KPa. The structure of every cluster is the global minima.

Fig. 9 The atmospheric relevancy of the (H₂SO₄)_m[(CH₃)₂NH]_n, (m = 1–2, n = 1–3) cluster at temperatures of 216.65 K and 273.15 K and at a pressure of 101.3 KPa. The structure of every cluster is the global minima.

Table 3 Comparison of changes in the Gibbs free energies (ΔG, kcal mol⁻¹) associated with the formation of (H₂SO₄)_m[(CH₃)₂NH]_n (m = 1–2, n = 1–3) at temperatures of 216.65 K, 273.15 K, and 298.15 K, and at a pressure of 101.3 KPa

Reaction equation	ΔG (216.65 K)	ΔG (273.15 K)	ΔG (298.15 K)
H2SO4 + (CH3)2NH → (H2SO4)[(CH3)2NH]	8.91	8.68	−13.92
H2SO4 + 2(CH3)2NH → (H2SO4)[(CH3)2NH]2	29.28	28.49	−18.65
H2SO4 + 3(CH3)2NH → (H2SO4)1[(CH3)2NH]3	51.75	50.40	−21.78
2H2SO4 + (CH3)2NH → (H2SO4)2[(CH3)2NH]1	19.47	18.94	−31.79
2H2SO4 + 2(CH3)2NH → (H2SO4)2[(CH3)2NH]2	18.37	17.26	−51.64

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From the analysis of the cluster (H₂SO₄)_m[(CH₃)₂NH]_n (m = 1–2, n = 1–3) and especially regarding the RPF and weak interaction analysis, we found that the cluster with the number of DMA molecules that is less or equal to the number of the SA molecules is the most stable and most abundant in the atmosphere. Hanson and Eisele^29,68 indicated that there are one or two SA molecules in the nucleation process, and one or two NH₃ molecules may therefore participate in the process. Kürten et al.³⁸ also demonstrated via the experiments from the CLOUD chamber that a cluster containing as few as two SA and one or two DMA molecules are already stable against evaporation. In view of these results, we tentatively conclude that the critical clusters contain two or fewer SA molecules and one or two DMA molecules. This system perhaps only forms small cluster in the atmosphere, and it requires other molecules for the subsequent growth.

4 Conclusions

In this study, the temperature dependence and atmospheric relevancy of the conformational population for the (H₂SO₄)_m[(CH₃)₂NH]_n (m = 1–2, n = 1–3) isomers are investigated, and the weak interactions in the cluster are also considered. Based on our calculations, the following conclusions are attained:

(a) Under different conditions, the populations of different conformations are different. There are several structures that co-exist in the actual atmosphere for some cluster sizes, and there is also only one configuration at the same size cluster.

(b) For the different cluster sizes, the (H₂SO₄)₂[(CH₃)₂NH]₂ exists in the highest proportion at 298.15 K and 1 atm. (H₂SO₄)₁[(CH₃)₂NH]₁ and (H₂SO₄)₂[(CH₃)₂NH]₁ are the less abundant clusters in the atmospheric layer. The DMA participates in the atmosphere nucleation, and the most probably cluster will be the (H₂SO₄)₂[(CH₃)₂NH]₂, which has the annular configuration and the high proportion of the hydrogen bond. At temperatures of 216.65 K and 273.15 K, the actual concentrations are so low that we believe they can co-exist in the troposphere and troposphere boundary layer.

(c) For the weak attraction analysis of the clusters using electron density method, the results indicate the following: as the number of DMA molecules increases, the types of attractions become complicated, such as hydrogen bonding, van der Waals interactions, steric clashes, etc. As the proportions of the weak attraction (such as van der Waals) increase, the stability of the clusters is relatively decreased.

For all of our analysis, there is one common conclusion: the clusters with the number of DMA molecules that is less than or equal to that of the SA are more stable in the atmosphere, and they account for a large proportion because the main type of interaction is much stronger (such as hydrogen bonding). When the number of sulfuric acid molecules is one or two in the nucleation process, one or two DMA molecules may also participate in the process. There are primarily three types of clusters in the environment of the SA and DMA, and they are small particles that are hard to settle and eliminate. The other elements will be involved in the cluster formation, such as water or organic acid, to grow a larger cluster, which is our next step for this research. The current work is fundamental and necessary for the study of nucleation clusters containing sulfuric acid, amines. This work predicts the possible forms of DMA with SA when participating in nucleation in a theoretical approach, and further theoretical and experimental studies are still needed to elucidate the nucleation mechanism.

Acknowledgements

The study was supported by grants from the National Natural Science Foundation of China (Grant No. 21403244, 21133008, 21573241 and 41527808), the National High Technology Research and Development Program of China (863 Program) (Grant No. 2014AA06A501), and the program of Formation Mechanism and Control Strategies of Haze in China (Grant No. XDB05000000). Acknowledgement is also made to the “Thousand Youth Talents Plan”, Scientific Research Equipment Development Program (YZ201422) and “Interdisciplinary and Cooperative Team” of CAS. The computation was performed in EMSL, a national scientific user facility sponsored by the department of Energy's Office of Biological and Environmental Research and located at Pacific Northwest National Laboratory (PNNL). PNNL is a multiprogram national laboratory operated for the DOE by Battelle. Part of the computation was performed at the Supercomputing Center of USTC.

References

1. R. Becker and W. Döring, Ann. Phys., 1935, 24, 719–752.
2. C. T. R. Wilson, Proc. R. Soc. London, 1897, 61, 240–242.
3. M. Kulmala, Science, 2003, 302, 1000–1001.
4. I. Napari, M. Noppel, H. Vehkamäki and M. Kulmala, J. Geophys. Res.: Atmos., 2002, 107, AAC6-1–AAC6-6.
5. C. D. O'Dowd, J. L. Jimenez, R. Bahreini, R. C. Flagan, J. H. Seinfeld, K. Hämeri, L. Pirjola, M. Kulmala, S. G. Jennings and T. Hoffmann, Nature, 2002, 417, 632–636.
6. P. Penttinen, K. Timonen, P. Tiittanen, A. Mirme, J. Ruuskanen and J. Pekkanen, Eur. Respir. J., 2001, 17, 428–435.
7. A. Saxon and D. Diaz-Sanchez, Nat. Immunol., 2005, 6, 223–226.
8. R. J. Charlson, J. H. Seinfeld, A. Nenes, M. Kulmala, A. Laaksonen and M. C. Facchini, Science, 2001, 292, 2025–2026.
9. G. Oberdörster and M. J. Utell, Environ. Health Perspect., 2002, 110, A440–A441.
10. W. Xu and R. Zhang, J. Phys. Chem. A, 2012, 116, 4539–4550.
11. R. Zhang, I. Suh, J. Zhao, D. Zhang, E. C. Fortner, X. Tie, L. T. Molina and M. J. Molina, Science, 2004, 304, 1487–1490.
12. L. Wang, A. F. Khalizov, J. Zheng, W. Xu, Y. Ma, V. Lal and R. Zhang, Nat. Geosci., 2010, 3, 238–242.
13. R. Zhang, Science, 2010, 328, 1366–1367.
14. R. Zhang, A. Khalizov, L. Wang, M. Hu and W. Xu, Chem. Rev., 2011, 112, 1957–2011.
15. B. Temelso, T. E. Morrell, R. M. Shields, M. A. Allodi, E. K. Wood, K. N. Kirschner, T. C. Castonguay, K. A. Archer and G. C. Shields, J. Phys. Chem. A, 2012, 116, 2209–2224.
16. M. Sipilä, T. Berndt, T. Petäjä, D. Brus, J. Vanhanen, F. Stratmann, J. Patokoski, R. L. Mauldin, A.-P. Hyvärinen and H. Lihavainen, Science, 2010, 327, 1243–1246.
17. A. Al Natsheh, A. B. Nadykto, K. V. Mikkelsen, F. Yu and J. Ruuskanen, J. Phys. Chem. A, 2004, 108, 8914–8929.
18. T. Kurtén, I. K. Ortega and H. Vehkamäki, J. Mol. Struct.: THEOCHEM, 2009, 901, 169–173.
19. T. Petäjä, M. Sipilä, P. Paasonen, T. Nieminen, T. Kurtén, I. K. Ortega, F. Stratmann, H. Vehkamäki, T. Berndt and M. Kulmala, Phys. Rev. Lett., 2011, 106, 228302.
20. I. Ortega, T. Kurtén, H. Vehkamäki and M. Kulmala, Atmos. Chem. Phys., 2008, 8, 2859–2867.
21. J. Herb, Y. Xu, F. Yu and A. Nadykto, J. Phys. Chem. A, 2012, 117, 133–152.
22. J. N. Smith, K. C. Barsanti, H. R. Friedli, M. Ehn, M. Kulmala, D. R. Collins, J. H. Scheckman, B. J. Williams and P. H. McMurry, Proc. Natl. Acad. Sci. U. S. A., 2010, 107, 6634–6639.
23. T. Berndt, F. Stratmann, M. Sipilä, J. Vanhanen, T. Petäjä, J. Mikkilä, A. Grüner, G. Spindler, L. Mauldin III and J. Curtius, Atmos. Chem. Phys., 2010, 10, 7101–7116.
24. K. H. Weber, Q. Liu and F.-M. Tao, J. Phys. Chem. A, 2014, 118, 1451–1468.
25. N. Bork, J. Elm, T. Olenius and H. Vehkamäki, Atmos. Chem. Phys., 2014, 14, 12023–12030.
26. A. B. Nadykto, J. Herb, F. Yu and Y. Xu, Chem. Phys. Lett., 2014, 609, 42–49.
27. J. W. DePalma, D. J. Doren and M. V. Johnston, J. Phys. Chem. A, 2014, 118, 5464–5473.
28. I. Ortega, T. Olenius, O. Kupiainen-Määttä, V. Loukonen, T. Kurtén and H. Vehkamäki, Atmos. Chem. Phys., 2014, 14, 7995–8007.
29. M. Chen, M. Titcombe, J. Jiang, C. Jen, C. Kuang, M. L. Fischer, F. L. Eisele, J. I. Siepmann, D. R. Hanson and J. Zhao, Proc. Natl. Acad. Sci. U. S. A., 2012, 109, 18713–18718.
30. H. Yu, R. McGraw and S. H. Lee, Geophys. Res. Lett., 2012, 39, L02807.
31. M. Erupe, A. Viggiano and S.-H. Lee, Atmos. Chem. Phys., 2011, 11, 4767–4775.
32. J. Zhao, J. Smith, F. Eisele, M. Chen, C. Kuang and P. McMurry, Atmos. Chem. Phys., 2011, 11, 10823–10836.
33. M. C. Facchini, S. Decesari, M. Rinaldi, C. Carbone, E. Finessi, M. Mircea, S. Fuzzi, F. Moretti, E. Tagliavini and D. Ceburnis, Environ. Sci. Technol., 2008, 42, 9116–9121.
34. S. Murphy, A. Sorooshian, J. Kroll, N. Ng, P. Chhabra, C. Tong, J. Surratt, E. Knipping, R. Flagan and J. Seinfeld, Atmos. Chem. Phys., 2007, 7, 2313–2337.
35. V. Loukonen, T. Kurtén, I. Ortega, H. Vehkamäki, A. A. Padua, K. Sellegri and M. Kulmala, Atmos. Chem. Phys., 2010, 10, 4961–4974.
36. T. Kurtén, V. Loukonen, H. Vehkamäki and M. Kulmala, Atmos. Chem. Phys., 2008, 8, 4095–4103.
37. B. Bzdek, D. Ridge and M. Johnston, Atmos. Chem. Phys., 2010, 10, 3495–3503.
38. A. Kürten, T. Jokinen, M. Simon, M. Sipilä, N. Sarnela, H. Junninen, A. Adamov, J. Almeida, A. Amorim and F. Bianchi, Proc. Natl. Acad. Sci. U. S. A., 2014, 111, 15019–15024.
39. J. Sun and P. A. Ariya, Atmos. Environ., 2006, 40, 795–820.
40. W. Xu and R. Zhang, J. Chem. Phys., 2013, 139, 064312.
41. J. Elm, T. Kurtén, M. Bilde and K. V. Mikkelsen, J. Phys. Chem. A, 2014, 118, 7892–7900.
42. R. Zhang, L. Wang, A. F. Khalizov, J. Zhao, J. Zheng, R. L. McGraw and L. T. Molina, Proc. Natl. Acad. Sci. U. S. A., 2009, 106, 17650–17654.
43. J. Smith, M. Dunn, T. VanReken, K. Iida, M. Stolzenburg, P. McMurry and L. Huey, Geophys. Res. Lett., 2008, 35, L04808-1–L04808-5.
44. J. Parshintsev, J. Nurmi, I. Kilpeläinen, K. Hartonen, M. Kulmala and M.-L. Riekkola, Anal. Biochem., 2008, 390, 913–919.
45. K. Sellegri, M. Hanke, B. Umann, F. Arnold and M. Kulmala, Atmos. Chem. Phys., 2005, 5, 373–384.
46. A. B. Nadykto and F. Yu, Chem. Phys. Lett., 2007, 435, 14–18.
47. A. Metzger, B. Verheggen, J. Dommen, J. Duplissy, A. S. Prevot, E. Weingartner, I. Riipinen, M. Kulmala, D. V. Spracklen and K. S. Carslaw, Proc. Natl. Acad. Sci. U. S. A., 2010, 107, 6646–6651.
48. P. Paasonen, T. Nieminen, E. Asmi, H. Manninen, T. Petäjä, C. Plass-Dülmer, H. Flentje, W. Birmili, A. Wiedensohler and U. Horrak, Atmos. Chem. Phys., 2010, 10, 11223–11242.
49. R. Zhang, G. Wang, S. Guo, M. Zamora, Q. Ying, Y. Lin, W. Wang, M. Hu and Y. Wang, Chem. Rev., 2015, 115, 3803–3855.
50. X. Ge, A. S. Wexler and S. L. Clegg, Atmos. Environ., 2011, 45, 524–546.
51. X. Ge, A. S. Wexler and S. L. Clegg, Atmos. Environ., 2011, 45, 561–577.
52. D. Hanson, P. McMurry, J. Jiang, D. Tanner and L. Huey, Environ. Sci. Technol., 2011, 45, 8881–8888.
53. H. Yu and S.-H. Lee, Environ. Chem., 2012, 9, 190–201.
54. T. Kurtén, T. Petäjä, J. Smith, I. Ortega, M. Sipilä, H. Junninen, M. Ehn, H. Vehkamäki, L. Mauldin and D. Worsnop, Atmos. Chem. Phys., 2011, 11, 3007–3019.
55. J. Almeida, S. Schobesberger, A. Kürten, I. K. Ortega, O. Kupiainen-Määttä, A. P. Praplan, A. Adamov, A. Amorim, F. Bianchi and M. Breitenlechner, Nature, 2013, 502, 359–363.
56. T. Kurtén, Entropy, 2011, 13, 915–923.
57. I. Ortega, O. Kupiainen, T. Kurtén, T. Olenius, O. Wilkman, M. McGrath, V. Loukonen and H. Vehkamäki, Atmos. Chem. Phys., 2012, 12, 225–235.
58. A. B. Nadykto, F. Yu, M. V. Jakovleva, J. Herb and Y. Xu, Entropy, 2011, 13, 554–569.
59. M. Masella and J. P. Flament, J. Chem. Phys., 1998, 108, 7141–7151.
60. W. Gao, J. Jiao, H. Feng, X. Xuan and L. Chen, J. Mol. Model., 2013, 19, 1273–1283.
61. C. Guo, H. Fang, R.-Y. Huang, H. Xu, G.-H. Wu and S.-Y. Ye, Chem. Phys. Lett., 2013, 588, 97–101.
62. Q. Gou, G. Feng, L. Evangelisti and W. Caminati, J. Phys. Chem. A, 2014, 118, 737–740.
63. N. Han, Y. Zeng, X. Li, S. Zheng and L. Meng, J. Phys. Chem. A, 2013, 117, 12959–12968.
64. H. Ding, Y. Lu, W. Wu and H. Liu, Chem. Phys., 2014, 441, 30–37.
65. J. Contreras-García, W. Yang and E. R. Johnson, J. Phys. Chem. A, 2011, 115, 12983–12990.
66. J. Contreras-García, E. R. Johnson, S. Keinan, R. Chaudret, J.-P. Piquemal, D. N. Beratan and W. Yang, J. Chem. Theory Comput., 2011, 7, 625–632.
67. R. Huang, R. Du, G. Liu, X. Zhao, S. Ye and G. Wu, J. Chem. Thermodyn., 2012, 55, 60–66.
68. D. Hanson and F. Eisele, J. Geophys. Res.: Atmos., 2002, 107, AAC10-11–AAC10-18.
69. T. Kurtén and H. Vehkamäki, Adv. Quantum Chem., 2008, 55, 407–427.
70. J. Elm, M. Bilde and K. V. Mikkelsen, J. Chem. Theory Comput., 2012, 8, 2071–2077.
71. J. Elm and K. V. Mikkelsen, Chem. Phys. Lett., 2014, 615, 26–29.
72. W. Huang, H.-J. Zhai and L.-S. Wang, J. Am. Chem. Soc., 2010, 132, 4344–4351.
73. W. Huang and L.-S. Wang, Phys. Rev. Lett., 2009, 102, 153401.
74. W. Huang, S. Bulusu, R. Pal, X. C. Zeng and L.-S. Wang, ACS Nano, 2009, 3, 1225–1230.
75. W. Huang, A. P. Sergeeva, H.-J. Zhai, B. B. Averkiev, L.-S. Wang and A. I. Boldyrev, Nat. Chem., 2010, 2, 202–206.
76. S. Jiang, Y. R. Liu, T. Huang, H. Wen, K. M. Xu, W. X. Zhao, W. J. Zhang and W. Huang, J. Comput. Chem., 2014, 35, 159–165.
77. H. Wen, Y.-R. Liu, K.-M. Xu, T. Huang, C.-J. Hu, W.-J. Zhang and W. Huang, RSC Adv., 2014, 4, 15066–15076.
78. K.-M. Xu, T. Huang, H. Wen, Y.-R. Liu, Y.-B. Gai, W.-J. Zhang and W. Huang, RSC Adv., 2013, 3, 24492–24502.
79. L.-L. Yan, Y.-R. Liu, T. Huang, S. Jiang, H. Wen, Y.-B. Gai, W.-J. Zhang and W. Huang, J. Chem. Phys., 2013, 139, 244312.
80. Y.-P. Zhu, Y.-R. Liu, T. Huang, S. Jiang, K.-M. Xu, H. Wen, W.-J. Zhang and W. Huang, J. Phys. Chem. A, 2014, 118, 7959–7974.
81. S. Jiang, T. Huang, Y.-R. Liu, K.-M. Xu, Y. Zhang, Y.-Z. Lv and W. Huang, Phys. Chem. Chem. Phys., 2014, 16, 19241–19249.
82. B. Delley, J. Chem. Phys., 1990, 92, 508–517.
83. M. Frisch, G. Trucks, H. B. Schlegel, G. Scuseria, M. Robb, J. Cheeseman, G. Scalmani, V. Barone, B. Mennucci and G. Petersson, Gaussian 09, Revision A. 02, Gaussian. Inc., Wallingford, CT, 2009, vol. 200.
84. A. B. Nadykto, J. Herb, F. Yu, E. S. Nazarenko and Y. Xu, Chem. Phys. Lett., 2015, 624, 111–118.
85. H.-J. Werner, P. J. Knowles, G. Knizia, F. R. Manby andM. Schütz, et al., MOLPRO, version 2010.1, A Package of Ab Initio Programs, 2010.
86. E. R. Johnson, S. Keinan, P. Mori-Sanchez, J. Contreras-Garcia, A. J. Cohen and W. Yang, J. Am. Chem. Soc., 2010, 132, 6498–6506.
87. T. Lu and F. Chen, J. Comput. Chem., 2012, 33, 580–592.
88. D. E. Husar, B. Temelso, A. L. Ashworth and G. C. Shields, J. Phys. Chem. A, 2012, 116, 5151–5163.
89. K. H. Weber, F. J. Morales and F.-M. Tao, J. Phys. Chem. A, 2012, 116, 11601–11617.
90. B. R. Bzdek, J. W. DePalma, D. P. Ridge, J. Laskin and M. V. Johnston, J. Am. Chem. Soc., 2013, 135, 3276–3285.
91. D. J. Bustos, B. Temelso and G. C. Shields, J. Phys. Chem. A, 2014, 118, 7430–7441.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c5ra22887e

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