The driving e ﬀ ects of common atmospheric molecules for formation of prenucleation clusters: the case of sulfuric acid, formic acid, nitric acid, ammonia, and dimethyl amine †

How secondary aerosols form is critical as aerosols' impact on Earth's climate is one of the main sources of uncertainty for understanding global warming. The beginning stages for formation of prenucleation complexes, that lead to larger aerosols, are di ﬃ cult to decipher experimentally. We present a computational chemistry study of the interactions between three di ﬀ erent acid molecules and two di ﬀ erent bases. By combining a comprehensive search routine covering many thousands of con ﬁ gurations at the semiempirical level with high level quantum chemical calculations of approximately 1000 clusters for every possible combination of clusters containing a sulfuric acid molecule, a formic acid molecule, a nitric acid molecule, an ammonia molecule, a dimethylamine molecule, and 0 – 5 water molecules, we have completed an exhaustive search of the DLPNO-CCSD(T)/CBS// u B97X-D/6-31++G ** Gibbs free energy surface for this system. We ﬁ nd that the detailed geometries of each minimum free energy cluster are often more important than traditional acid or base strength. Addition of a water molecule to a dry cluster can enhance stabilization, and we ﬁ nd that the (SA)(NA)(A)(DMA)(W) cluster has special stability. Equilibrium calculations of SA, FA, NA, A, DMA, and water using our quantum chemical D G (cid:1) values for cluster formation and realistic estimates of the concentrations of these monomers in the atmosphere reveals that nitric acid can drive early stages of particle formation just as e ﬃ ciently as sulfuric acid. Our results lead us to believe that particle formation in the atmosphere results from the combination of many di ﬀ erent molecules that are able to form highly stable complexes with acid molecules such as SA, NA, and FA.


Introduction
Aerosols affect the climate through light scattering, absorbing and emitting radiation, and by serving as cloud condensation nuclei (CCN). 1 They exert a substantial inuence on the microphysical properties of water and ice clouds, thereby affecting the processes that lead to the formation of various forms of precipitation. 2 Secondary aerosol particles form from atmospheric gases and are predicted to make up 50% or more of global CCN. 3,4 The precise way that gaseous vapor comes together to form prenucleation complexes, which eventually build to aerosols, and thus form CCN, is a very active area of research. 3, Much of this activity is driven by the uncertainty in how much aerosols and clouds will impact global warming. 75,76 While it is thought that in general more aerosols will lead to a cooling effect, the uncertainty in our knowledge exceeds the actual size of the predicted cooling. 75 Aerosols promote photochemical reactions at their surfaces and ampli-cation of light within aerosols accelerates photochemistry. [77][78][79][80][81] In addition, nanodroplets and microdroplets catalyze chemical reactions at the air-water interface. [82][83][84][85][86][87][88][89][90][91] Current instrumentation is capable of examining the various stages of aerosol formation, but no single instrument can concurrently analyze data from the sub-nanometer size regime to the 10 nm size regime, leading to gaps in our knowledge about initial aerosol formation. 67 Meanwhile accurate calculations of the Gibbs free energy for cluster formation (DG ) requires both a comprehensive exploration of the potential energy surfaces for each possible cluster along with high-level quantum chemical calculations on all the lowest free energy structures. 55,92 In this study we have used robust computational chemistry methods to determine the DG values for every possible cluster that can be formed in the sulfuric acid-nitric acid-formic acid-ammonia-dimethylamine-water system from monomers of each molecule along with 0-5 waters. This is the rst study to probe three different acid molecules and two different bases to date, and adds to the insights obtained from previous work.

Methodology
Congurational sampling of every possible combination of sulfuric acid (SA), formic acid (FA), nitric acid (NA), ammonia (A), dimethylamine (DMA), and 0-5 water (W) molecules was performed with a genetic-algorithm-based protocol using the OGO-LEM program 93,94 and semiempirical methods. This algorithm takes an initial pool of randomly generated congurations and changes them according to an evolutionary algorithm to produce a nal set of converged structures. We typically used a pool size of 1000 and set the number of global optimizations to 20 000. Two different semiempirical methods were used, PM7, 95 and GFN2, [96][97][98] to calculate approximately 1000 nal structures for each system. These geometries were then used for DFT geometry optimizations with the Gaussian 16 Rev. B01 program 99 using either the M08-HX 100 or uB97X-D 101,102 functionals. Clusters containing SA, FA, A, and W were previously optimized with the M08-HX functional and the MG3S basis set. 68,100 All of these M08-HX geometries were re-optimized with uB97X-D using the 6-31++G** basis set [103][104][105][106] to obtain a nal set of consistent geometries. The electronic energies of all structures within 8 kcal mol −1 of the uB97X-D global minimum for each system were recomputed using the domain-based local pair natural orbital coupled-cluster (DLPNO-CCSD(T)) method [107][108][109][110][111][112][113][114][115][116][117][118][119][120] with single, double, and semi-canonical perturbative triple excitations with three Dunning basis sets, cc-pVnZ (n ¼ D, T, Q), 121-124 using the ORCA 5.0.1 program. 125,126 The uB97X-D frequencies were scaled by 0.971 to partially correct for anharmonicity 127 and used to estimate the thermodynamic corrections for H , S , and G at a standard state of 1 atm pressure and temperatures of 217.65, 273.15, and 298.15 K using the THERMO.pl script 128 from the National Institute of Science and Technology. The three CCSD(T) electronic energies were used in a 4-5 inverse polynomial complete basis set (CBS) extrapolation 129 and combined with the scaled uB97X-D thermodynamic corrections to obtain the nal set of DG values for every system. For the tetramers of two acids and two bases and the pentamer systems, we include a comparison of the cc-pVnZ and aug-cc-pVnZ (n ¼ D, T, Q) for the dry clusters. Additionally, we show the difference between using the cc-pVnZ versus the aug-cc-pVnZ basis sets in the ESI (Table S1), † as well as a comparison of using scaled vs. unscaled frequencies for the calculation of DE and DG values (Table S2 †). The rate-limiting step in our computational methodology is the DFT geometry optimizations from the initial set of PM7 or GFN2 structures produced from the genetic algorithm. Semiempirical methods have long been known to do a better job at predicting structures for hydrogen-bonded clusters than energies. 130,131 Because of this the PM7 and GFN2 potential energy surfaces are quite at, such that we cannot rely on a cut-off in energy to select the lowest energy ensemble of structures, but must optimize every semiempirical structure. Switching to the uB97X-D functional and the smaller 6-31++G** basis set has sped up this ratelimiting step. 54,55,61 Further details on methodology are presented in our previous works [57][58][59]68 and a summary of the most effective methodology is presented in Fig. 1. This methodology allows for the determination of accurate DG values for every possible combination of our monomers SA, FA, NA, A, DMA, and up to ve waters. An example of how to calculate all possible stepwise energies is shown in the ESI. † Assuming a closed system, we can then calculate the equilibrium concentrations for all possible clusters from the DG values and initial concentrations of the monomers. 58,68 We note that two big caveats about our methodology are that (a) no search routine based on lower level methods can possibly nd every nal minimum energy cluster, and (b) that any combination of electronic energy theory such as CCSD(T) with DFT geometries like uB97X-D means that the electronic energy theory is calculated on the DFT PES. Both of these uncertainties will produce DG values that are more positive than the values one would obtain if (a) a lower energy conformer is obtained, or (b) the PES is improved to be closer to the electronic energy method used in the calculation.

Monomer and dimer systems
All of the monomer and dimer systems researched have been well examined prior to this study. Therefore, all of the structures and corresponding energies are presented in the ESI (Fig. S1-S6 and Tables S3-S8). † Our results are consistent with previous studies of these clusters, with some new structures designated as the minimum due to level of theory differences from past studies. 4,28,33,37,48,50,61,65,66,68,87,[127][128][129][130][131][132][133][134][135][136][137] In these systems, no minimum energy clusters undergo proton transfers. This is unlike two-acid dimer systems such as (SA) 2 (W) 0-5 , which have been found to undergo proton transfers in the 3-5 water range. 16,54,132 Additional studies of these smaller systems have been performed experimentally. For example, rapid growth of new atmospheric particles through condensation of NA and A has been observed in the CLOUD chamber at CERN to produce particles as small as a few nm in diameter. 53 This same system has been studied in a ow tube and NA-DMA particles were detected with diameters of 9-30 nm in dry and humid conditions. 48 The NA/ DMA ratio remained close to one, and modeling predicted that 1 : 1 clusters had smaller evaporation rates, resulting in the most favorable thermodynamic pathway for growth. 48

Trimers of a single acid and two bases
The structures and DG values for formation of clusters consisting of the sequential hydration of one acid (SA, NA, or FA alone) and both bases (A and DMA), are listed in the ESI ( Fig. S7 and Table S9) † as our results reveal they are less likely to be atmospherically relevant at any temperature (see Section 3.8). However, these systems do demonstrate the clear relationship that the strongest acid, sulfuric, which is a known driver of CCN formation, 133-138 forms more negative DG clusters with bases than either nitric or formic acid. The DG of formation for the dry formic acid cluster with the two bases is roughly 0 kcal mol −1 at 298 K. The dry nitric acid and sulfuric acid with two base clusters are more stable, with DG values of −5 and −15 kcal mol −1 respectively. This decrease in cluster formation Gibbs free energy likely results from the proton transfer from the acid to DMA. Even though nitric acid donates its proton and forms the same number of hydrogen bonds as sulfuric acid (Fig. S7 †), the (NA)(A)(DMA) cluster is still roughly 10 kcal mol −1 higher in energy, demonstrating the importance of sulfuric acid in prenucleation cluster formation. For all of the one acid-two base hydrated clusters, the acid always acts as a bridge between the two bases such that the bases are on opposite sides of the clusters. The (FA)(A)(DMA)(W) 0-2 clusters consist of neutral molecules, but in every other case one acid-two base cluster the acid always transfers its proton to the dimethylamine. In the unique case of (SA)(A)(DMA)(W) 5 , the sulfuric acid becomes a sulfate anion, with proton transfers forming both the protonated dimethyl amine and ammonium moieties. (This is the only instance where a sulfate anion was part of the lowest free energy cluster for any of the trimers and smaller systems in this work.) However, even with the additional proton transfer, the ve-water cluster is still uphill in free energy compared to (SA)(A)(DMA)(W) 4 at 298 K (Table S9 †).

Trimers of two acids and one base
In our gures we have denoted strong hydrogen bonds, which have hydrogen-bonded distances less than 2.2 A and hydrogen bond angles between 140 and 180 , with blue lines. We have used red lines to denote other van der Waals forces where the bond angle is less than 140 and/or the putative hydrogen bond distance is greater than 2.2 A. The clusters consisting of two acids paired with one base allow for an easy comparison of the strength of the driving force between the two bases. Prior studies have investigated the stabilities of sulfuric-acid based clusters with ammonia versus amines such as methylamine (MA), dimethylamine (DMA), and trimethylamine (TMA). 24,25,28,37,50,61,63,92,[139][140][141][142][143][144][145] Overall, the more substituted amines, specically DMA and TMA, tend to be better nucleators with SA, 145,146 with A forming the least stable complexes. However, the 2-3 orders of magnitude higher concentration of ammonia may compensate for the lower stabilities. When ammonia and amines are both present together, the effects are synergistic, with nucleation increasing by nearly a thousandfold. 50,138,139,143,[147][148][149][150][151][152] As seen from Tables 1 and 2, substituting DMA for A decreases the DG for formation of the dry clusters by roughly 6-8 kcal mol −1 , further supporting the idea that DMA is a better nucleator. As all of these clusters contain three strong hydrogen bonds ( Fig. 2 and 3), and all except (FA)(NA)(A) transfer a proton from the stronger acid to the base, the stronger base DMA forms more stable hydrogen bonds than ammonia. This view is consistent with past results. 92 While the dry cluster with DMA is a lot more stable than with A, adding waters to the ammonia clusters tends to yield lower DG values than adding to the dimethyl amine clusters (Tables 1  and 2). This is because ammonium is able to form more hydrogen bonds than the dimethyl ammonium ion. The ammonium ion remains near the middle of large clusters to donate three or four hydrogen bonds to the surrounding molecules, while protonated DMA can only donate two. This is in line with previous work where it was found that for small Table 1 DLPNO-CCSD(T)/CBS//uB97X-D/6-31++G** Gibbs free energy changes associated with the formation and sequential hydration of sulfuric acid-formic acid-ammonia, sulfuric acid-nitric acidammonia, and formic acid-nitric acid-ammonia trimers at atmospherically relevant temperatures and 1 atm pressure clusters, like the dry clusters reported here, the gas phase basicity determines gas phase base strength: DMA is a much stronger gas phase base than A and is better able to promote proton transfer. On the other hand, as a cluster is hydrated and grows in size, ammonia is better able to bind multiple molecules together in larger clusters. 92 The general trend is that ammonium will be located between waters, whereas DMA tends to remain on the outside of the cluster. One effect of this is that the DMA clusters tend to have spines of hydration, as can be seen for the ( Atoms are drawn in the following colors: hydrogenwhite, carbongray, nitrogenblue, oxygenred, sulfuryellow. Table 2 DLPNO-CCSD(T)/CBS//uB97X-D/6-31++G** Gibbs free energy changes associated with the formation and sequential hydration of sulfuric acid-formic acid-dimethyl amine, sulfuric acid-nitric acid-dimethyl amine, and formic acid-nitric acid-dimethyl amine trimers at atmospherically relevant temperatures and 1 atm pressure

Trimers of three acids
The system composed of all three acids (SA, FA, and NA) together with up to ve waters is shown in Fig. S8 with energies in Table S10. † Comparing this system's dry energies of formation with those of the other trimer systems, one will notice that even without a base present, this system is more stable than any other trimer without SA except for (FA)(NA)(DMA) (Tables 1, 2, S7 and S8 †). With multiple strong acids, a proton transfer might be expected with the rst available water molecule, as it can act as a base. However, proton transfer only occurs when there are three or more waters present. The proton transfer is always from the sulfuric acid to one of the waters, forming the bisulfate anion and a hydronium cation. The proton transfer occurring only from 3-5 waters is consistent with the results from Rasmussen et al., who studied (SA) 3 (W) n for n ¼ 0-5. 54 Addition of acetic acid to the (SA) 2 (W) n system results in proton transfer with two waters. 132 Our SA-NA-FA-(W) n system as well as the

Tetramers of three acids and one base
The energies of formation and sequential hydration of the tetramer systems containing SA, FA, NA and either A or DMA are listed in Table 3, with the corresponding minimum energy structures shown in Fig. 4. Forming the dry cluster with DMA is roughly 8.5 kcal mol −1 lower in free energy than forming the dry cluster with ammonia at all three temperatures. While part of Table 3 DLPNO-CCSD(T)/CBS//uB97X-D/6-31++G** Gibbs free energy changes associated with the formation and sequential hydration of sulfuric acid-formic acid-nitric acid-ammonia and sulfuric acid-formic acid-nitric acid-dimethyl amine tetramers at atmospherically relevant temperatures and 1 atm pressure Atoms are drawn in the following colors: hydrogenwhite, carbongray, nitrogenblue, oxygenred, sulfuryellow.
this is likely because of DMA's ability to form more stable clusters, it could also be due to the strength of the hydrogen bonds. Both clusters contain 5 strong hydrogen bonds; however, the cluster with ammonia has three of the type N-H/ O, whereas the cluster with DMA only has two of these. Since oxygen is more electronegative, this difference in the strength of the hydrogen bond could also play a role in the extra stability of the dry cluster with DMA. In all of these tetramer systems, a proton transfer occurs between the sulfuric acid and the base. Similar to smaller clusters, the energy of formation for the SA-FA-NA-base dry cluster has a more positive free energy for ammonia than for dimethylamine, but the sequential hydration energies decrease more rapidly for A than for DMA. This is because the ammonium cation usually donates three or four hydrogen bonds as the cluster is hydrated, whereas the dimethyl ammonium ion can only donate two in any system. The bisulfate anion is in the middle of these clusters which maximizes hydrogen bonding interactions, but this leaves NA on the outside where it can donate only one hydrogen bond and accept at most one hydrogen bond.

Tetramers of two acids and two bases
The interactions between the two acid and two base systems are shown in Fig. 5, with the energies corresponding to their formation listed in Table 4. Sulfuric acid has a strong driving effect, accounting for more than 10 kcal mol −1 of stabilization for the dry clusters with either acid and the two bases at all three temperatures. The two dry clusters SA-FA-A-DMA, and SA-NA-A-DMA are only 2-3 kcal mol −1 different in free energy ( Table  4). The extra stabilization for the NA cluster stems from the additional proton transfer from the nitric acid to one of the bases, forming an overall tetra-ionic cluster, resulting in a strong hydrogen bond between the two ions.
The addition of water molecules has some interesting effects on these systems. The (SA)(FA)(A)(DMA)(W) n system remains mostly di-ionic, only undergoing two proton transfers to form a tetra-ionic system when four or ve waters are added. In this case sulfuric acid loses both its protons and the bisulfate dianion is complexed with the protonated bases and a neutral formic acid (Fig. 5). In contrast, the (SA)(NA)(A)(DMA)(W) n system is tetra-ionic for n ¼ 0-5, with the sulfuric acid and nitric acid both losing one proton. The addition of one water molecule greatly stabilizes the dry tetra-ionic cluster, as both acids bridge the two bases together in (SA)(NA)(A)(DMA)(W) 1 , resulting in a stepwise DG value of −6.8 kcal mol −1 at 217 K and −4.0 kcal mol −1 at 298 K. This is the most negative drop in free energy for the addition of any water in all of the reported structures, such that the (SA)(NA)(A)(DMA)(W) 1 cluster has an outsized effect on the results of the simulation reported in Section 3.8. The water appears to greatly stabilize this tetra-ionic structure. The (FA)(NA)(A)(DMA)(W) n clusters became tetra-ionic with the addition of one water. Unlike the (SA)(FA)(A)(DMA)(W) [4][5] structures, in this case formic acid does lose its proton. However, this additional proton transfer does not account for a lot of stability, as the DG value at 298 K is still positive (0.50 kcal mol −1 ). These results are mostly consistent with our knowledge of acid strength. However, HSO 4 − is a weak acid, and yet it loses a proton to become SO 4 2− while formic acid remains neutral in the (SA)(FA)(A)(DMA)(W) [4][5] systems. This is probably because hydrogen bonding topology is more important than acid/base strength, 68,70 and the details of the hydrogen-bonded structure in the gas phase govern the overall free energy change. For all of the clusters that contain a proton transferred to ammonia, the ammonium cation donates at least three hydrogen bonds and resides near the middle of the cluster.

Pentamers of three acids and two bases
The system containing all ve molecules along with zero to ve waters is depicted in Fig. 6, and the respective energies for formation of the clusters are in Table 5. The system remains tetra-ionic at all stages of hydration, with the sulfuric acid and nitric acid donating their protons to the two bases. Without any water present, the pentamer system forms a tetra-ionic species with SA and NA donating protons to A and DMA. These four molecules make a ring with both A (1. 70 64 A, 175 ), making a triangular structure between those three molecules.
Atoms are drawn in the following colors: hydrogenwhite, carbongray, nitrogenblue, oxygenred, sulfuryellow. Table 4 DLPNO-CCSD(T)/CBS//uB97X-D/6-31++G** Gibbs free energy changes associated with the formation and sequential hydration of sulfuric acid-formic acid-ammonia-dimethyl amine, sulfuric acid-nitric acid-ammonia-dimethyl amine, and formic acid-nitric acidammonia-dimethyl amine tetramers at atmospherically relevant temperatures and 1 atm pressure. DLPNO-CCSD(T)/aug-CBS//uB97X-D/6-31++G** Gibbs free energy changes are shown in parenthesis for the formation of the dry tetramers Upon the addition of one water, FA and SA form the strong SA-FA dimer (1.68 A, 173 ; 1.67 A, 178 ). 68 The SA then receives an additional three strong hydrogen bonds: (1.85 A, 149 1.68 A, 173 ). This same bonding pattern is present in the two-water cluster. However, the DMA does not form a second strong hydrogen bond but instead forms two weak van der Waals interactions: one to the NA (2.05 A, 128 ) and one to the SA (2.18 A, 132 ). The SA uses its singular proton to donate a strong hydrogen bond to the NA (1.55 A, 172 ). Overall, this structure forms 10 strong hydrogen bonds with three weak van der Waals interactions. Interestingly, this threewater pentamer cluster is less stable than the two-water cluster by 4 kcal mol −1 at 298 K ( Table 5). The three-water system is also the only pentamer system in which another structure was identied within 1 kcal mol −1 of the DLPNO-CCSD(T)/CBS// uB97X-D/6-31++G**Gibbs free energy minimum. The next lowest three-water cluster is 0.42 kcal mol −1 higher in free energy than the minimum. In this cluster, the SA donates both of its protons to A and DMA, forming the sulfate anion, while NA and FA both remain neutral. In this cluster, A forms three strong hydrogen bonds with SA (1.  76 A, 169 ). The A connects these two dimers together by donating to one water molecule in each (1.69 A, 165 ; 1.68 A, 167 ). The A also donates to the NA (1.83 A, 160 ), which receives from the second of the waters in the second dimer (2.00 A, 144 ) and the DMA (1.73 A, 180 ). The DMA donates its second hydrogen bond to the SA (1.75 A, 173 ). SA donates one proton to the last water molecule (1.61 A, 178 ), which donates to the FA (1.77 A, 170 ). The FA donates to the same water that bonds to the NA (1.62 A, 171 ), resulting in that one water having four hydrogen bonds. Overall, the structure has 15 strong hydrogen bonds. Table 5 DLPNO-CCSD(T)/CBS//uB97X-D/6-31++G** Gibbs free energy changes associated with the formation and sequential hydration of sulfuric acid-formic acid-nitric acid-ammonia-dimethyl amine pentamer at atmospherically relevant temperatures and 1 atm pressure. DLPNO-CCSD(T)/aug-CBS//uB97X-D/6-31++G** Gibbs free energy changes are shown in parenthesis for the formation of the dry pentamer

Equilibrium concentrations and optimal pathways
To calculate the equilibrium concentrations of the clusters, equilibrium constants were calculated using the DG values at 217 and 298 K. For each temperature, a system of 186 equations was created according to Odbadrakh et al., 57 using the equilibrium constants, with 5 equations representing mass balances for the 5 monomers and 181 equations for the reactions. This system of equations solves for the thermodynamics and does not completely account for the kinetics. The ratio of each equilibrium constant is of course the ratio of the forward and reverse reaction of each elementary clustering reaction. However, the results would likely change when accounting for the collision and evaporation rates of these clusters in the atmosphere. The system of equations was solved simultaneously to predict the equilibrium cluster concentrations assuming initial starting concentrations of 5 Â 10 7 cm −3 for SA, 2 Â 10 11 cm −3 for FA, 9.8 Â 10 10 cm −3 for NA, 2 Â 10 11 cm −3 for A, and 2 Â 10 9 cm −3 for DMA at 298 K. These concentrations were picked as they are atmospherically relevant over continents, typically near urban areas. 16,133,[153][154][155][156] To account for the decrease in monomeric concentrations with an increase in altitude, the initial concentrations were decreased by three orders of magnitude at 217 K. We chose a water concentration of 7.7 Â 10 17 cm −3 at 298 K and 9.9 Â 10 14 cm −3 at 217 K, which corresponds to 100% humidity at the bottom and top of the troposphere. 133 The results are shown in Table 6. We only populated Table 6 with clusters that exceeded 1 cm −3 at either 217 or 298 K. While most monomers remain close to their starting concentrations, sulfuric acid is almost completely used up at 217 K. In urban areas where the DMA concentration is high, the rates of new particle formation are also quite high. 143,152,157 Interestingly, although ammonia has an initial concentration two orders of magnitude greater than DMA, clusters with DMA are still more common and typically are present in higher equilibrium concentrations, supporting the idea that DMA is a stronger driver for prenucleation than ammonia. 145,146 Using these concentrations, the optimal pathways for forming clusters can be mapped out by nding which smaller clusters have the highest equilibrium concentrations and sequentially adding monomers until the larger cluster is formed. From this, the three best pathways for forming the dry pentamer at both 217 and 298 K are shown in Table 7. While forming the (SA)(DMA) cluster rst is optimal at both temperatures, starting with NA instead of SA will also lead to good cluster growth. In ve of the six pathways, SA or NA rst forms a dimer with a base, and then a second acid is added to form a trimer, and then the third acid is added to form a tetramer.
The best pathway at 217 and 298 K forms the (SA)(FA)(NA)(DMA) tetramer. Of these six possible pathways for growing the dry pentamer, the only pathway that forms the (SA)(NA)(A)(DMA) tetramer is the second best at 298 K. Thus, the three acids tend to preferentially build up these clusters. For all of the hydrated and dehydrated tetramer and pentamer clusters, the only one that appears in an equilibrium concentration above 1 cm −3 is (SA)(NA)(A)(DMA)(W). This individual cluster, which was identied with exceptional stability in Section 3.6, demonstrates the synergistic effects between the two bases when there are also at least the same number of acids present. For this reason, the three best pathways to forming this cluster have been identied in Table 8. Interestingly, the second and third pathways are different at the two temperatures. Additionally, these clusters tend to add in SA later, preferring early growth with NA. In every pathway except the second best at 217 K, where the water adds second to form the trimer, the water adds rst to form the dimer, likely caused by the extreme amount of water present in the atmosphere. In four of these six pathways, the tetramer consists of SA, NA, DMA, and W. In the third best pathway at 217 K, A replaces DMA. The only pathway where one acid, both bases, and a water form a tetramer is for the second-best pathway at 298 K, where the (SA)(A)(SMA)(W) tetramer is predicted to form along this pathway, with NA adding last to form the nal (SA)(N-A)(A)(DMA)(W) cluster. Unlike the other molecules in our experiment, DMA is believed to have a very short lifetime in the atmosphere, of only 1-2 hours above coastal areas and 5-10 hours over other parts of populated continents. 158 Because of this, we have also performed the calculations where the concentration of DMA was decreased further. When the concentration of DMA remained above the concentration of SA, there were nearly no changes in the results. However, as soon as the concentration of DMA dropped below that of SA, the results changed drastically, with the DMA being depleted in place of the SA. As the DMA concentration was continually lowered, the system converged towards one more like that shown by Harold et al., where the clusters containing acids and ammonia had the highest concentrations (Tables S11-S13 †). 68

Conclusions
This is the rst study to probe three different acid molecules and two different bases to date, and adds to the insights obtained from previous work. Accurate DG values for the formation of every possible cluster that can be formed from a sulfuric acid molecule, a formic acid molecule, a nitric acid molecule, an ammonia molecule, a dimethylamine molecule, and 0-5 water molecules were determined from a comprehensive search of the PM7, GFN2, and uB97X-D potential energy surfaces combined with DLPNO-CCSD(T)/CBS electronic energy calculations on the DFT geometries. The detailed geometries of each minimum free Table 7 Optimal pathways for growing the dry pentamer at 1 atm pressure according to equilibrium concentrations calculated using initial energy cluster are more important than traditional acid or base strength in many cases. Addition of a water molecule to a dry cluster can enhance stabilization of that cluster. Dimethylamine forms stronger clusters than ammonia in the absence of water, but as waters are added, the ammonium cation is able to donate more hydrogen bonds than the dimethyl ammonium cation, increasing the stability of the ammonium-water clusters relative to DMA. Results presented here reveal that the (SA)(N-A)(A)(DMA)(W) cluster has special stability. Equilibrium concentrations were then calculated and used to map out the optimal pathways for forming the dry pentamer and this especially stable (SA)(NA)(A)(DMA)(W) cluster. The pathways indicate that NA could initiate new particle formation just as efficiently as SA, especially in the presence of water. This suggests that particle growth in the atmosphere results from the combination of many different molecules that are able to form highly stable complexes with nitric, formic, and sulfuric acids.
Pathways that lead to tetramers with all three acids and one base are predicted to be important in the growth of mixed acid/ base clusters.

Author contributions
The manuscript was written through contributions of all authors. All authors have given approval to the nal version of the manuscript.

Conflicts of interest
There are no conicts to declare.