(HSO4â)(NH3)(H2SO4) - ACS Publications - American Chemical Society

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Large Hydrogen-Bonded Pre-nucleation (HSO4−)(H2SO4)m(H2O)k and (HSO4−)(NH3)(H2SO4)m(H2O)k Clusters in the Earth’s Atmosphere Jason Herb,*,† Yisheng Xu,‡ Fangqun Yu,† and A. B. Nadykto*,†,§ †

Atmospheric Sciences Research Center, State University of New York at Albany, 251 Fuller Road, Albany, New York 12203, United States ‡ Atmospheric Chemistry and Aerosol Research Division, Chinese Research Academy of Environmental Science, Beijing 100012, China § Department of Applied Mathematics, Moscow State Unversity of Technology “Stankin”, Vadkovsky 1-3, Moscow, Russia S Supporting Information *

ABSTRACT: The importance of pre-nucleation cluster stability as the key parameter controlling nucleation of atmospheric airborne ions is well-established. In this Article, large ternary ionic (HSO4−)(H2SO4)m(NH3)(H2O)n clusters have been studied using Density Functional Theory (DFT) and composite ab initio methods. Twenty classes of clusters have been investigated, and thermochemical properties of common atmospheric (HSO4−)(H2SO4)m(NH3)0(H2O)k and (HSO4−)(H2SO4)m(NH3)1(H2O)n clusters (with m, k, and n up to 3) have been obtained. A large amount of new themochemical and structural data ready-to-use for constraining kinetic nucleation models has been reported. We have performed a comprehensive thermochemical analysis of the obtained data and have investigated the impacts of ammonia and negatively charged bisulfate ion on stability of binary clusters in some detail. The comparison of theoretical predictions and experiments shows that the PW91PW91/6-311++G(3df,3pd) results are in very good agreement with both experimental data and high level ab initio CCSD(T)/CBS values and suggest that the PW91PW91/6-311++G(3df,3pd) method is a viable alternative to higher level ab initio methods in studying large pre-nucleation clusters, for which the higher level computations are prohibitively expensive. The uncertainties in both theory and experiments have been investigated, and possible ways of their reduction have been proposed.

1. INTRODUCTION Secondary aerosols formed in the atmosphere via homogeneous or/and ion-mediated nucleation pathways are pervasive with direct implications to both the global climate changes in the Earth’s atmosphere and public health. The adverse public health impacts of nanoparticles and ultrafine particles (5 kcal/mol).52 The uncertainties in the van’t Hoff analysis and discrepancies between direct calorimetric and indirect van’t Hoff estimates of enthalpies are well-known; however, for now, these issues remain unresolved.52,60 These considerations lead us to the following logical conclusions: (i) high level ab initio and composite methods cannot be used as

Table 1. Comparison of the Theoretical and Experimental Standard Conditions (HSO4−)(H2SO4)0(H2O)n−1 + (H2O) ⇔ (HSO4−)(H2SO4)0(H2O)n Reaction Enthalpiesa n 1 2 3 4 5

CCSD(T)/CBS

PW91

exp.

♀

−14.13 −14.59 −12.21 −10.91 −13.07

−12.9 ± 0.6 −11.2 ± 0.7 −12.4 ± 0.4 −13.2 ± 0.6 −11.7 ± 1.5

−14.21 (−14.14 ) −12.91 −12.33 −11.39 −12.68

a Abbreviations CCSD(T)/CBS, PW91,♀ and exp. refer to ref 59, ref 60, G3MP2 (present study), and ref 54, respectively.

Table 2. Comparison of the Theoretical and Experimental Standard Conditions (HSO4−)(H2SO4)0(H2O)n−1 + (H2O) ⇔ (HSO4−)(H2SO4)0(H2O)n Reaction Gibbs Free Energiesa n 1 2 3 4 5

CCSD(T)/CBS ♀

−3.91 (−3.92 ) −3.32 −2.53 −1.44 −2.31

PW91

exp.

−4.58 −3.42 −2.99 −2.98 −1.38

−6.04 ± 1 −4.64 ± 1.1 −3.25 ± 0.6 −2.2 ± 0.9 −2.32 ± 2.4

a Abbreviations CCSD(T)/CBS, PW91,♀ and exp. refer to ref 59, ref 60, ref 53, G3MP2 (present study), and ref 54, respectively.

present the comparison of the best ab initio59 with our results obtained using the PW91PW91/6-311++G (3df, 3pd) method60 used in the present study and experimental data.54 As may be seen from Tables 1 and 2, the high level ab initio CCSD(T)/CBS study and DFT PW91PW91/6-311++G(3df,3pd) studies are in good agreement with each other and experimental data. However, it is important to note that the agreement of both quantum-chemical methods with the experimental Gibbs free energies is much better than that with the experimental enthalpy changes. This gives us another argument in favor of the importance of the above-mentioned “nonlinearity issue” leading to large variations of (ΔH, ΔS) derived from the same experimental data set for reaction constants using different forms of the van’t Hoff equation. The average difference from the experimental data in the case of the high level ab initio CCSD(T)/CBS and PW91PW91/6311++G(3df,3pd) in reaction enthalpies and free energies (ΔH; ΔG) is (1.3; 0.94) and (1.7; 0.9) kcal/mol, respectively. As it may be seen from the aforementioned comparison, the accuracy of both CCSD(T)/CBS and PW91PW91/6-311++G(3df,3pd) in predicting the Gibbs free energy changes, the key quantity controlling the thermochemical stability of the pre-nucleation clusters, is close to the chemical accuracy (∼1 kcal/mol). While CCSD(T)/CBS and PW91PW91/6-311++G(3df,3pd) methods are in very good agreement with each other, with the average deviation in reaction enthalpies and Gibbs free energies of ∼0.6 and ∼0.75 kcal/mol, respectively, and both agree well with the experimental data, the more affordable DFT 135

dx.doi.org/10.1021/jp3088435 | J. Phys. Chem. A 2013, 117, 133−152

The Journal of Physical Chemistry A

Article

Figure 1. Most stable (the lowest Gibbs free energy) isomers of (a−d) (HSO4−)(H2SO4)2; (e,f) (HSO4−)(H2SO4)2(H2O)1; (g,h) (HSO4−)(H2SO4)2(H2O)2; and (i−k) (HSO4−)(H2SO4)2(H2O)3.

have been obtained. Figures 1−5 present the structures of the lowest Gibbs free energy isomers (global minima and local minima located within 1 kcal mol−1). The complete information, including figures and Cartesian geometries, are given in the Supporting Information. (HSO4−)(H2SO4)2. Four stable conformers of (HSO4−)(H2SO4)2 have been identified. For the aforementioned conformers, average O−H bonds range from 1.577 to 1.656 Å, and O−O distances range from 4.01 to 4.51 Å. The most stable isomer has the average O−H length of 1.577 Å and O−O average distance of 4.01 Å. The global minimum is found to be fully bonded with all O−H bonds utilized. This configuration allows for this isomer to be more compact in comparison to the local minima, which are all located within 1 kcal mol−1 of the global minimum. This isomer is virtually free of O−O repulsion because the nonbonded oxygen atoms of each sulfuric acid are directed toward the outside edge of the cluster. This conformer includes four slightly overstretched SO−H bonds from the two H2SO4 molecules. The bisulfate shares two hydrogens from one sulfuric acid and one hydrogen from another H2SO4 molecule. The remaining hydrogen is shared by two H2SO4 molecules. The average O−O and O−H distances of the local minima were found to be larger than those of the global minimum by up to ∼8% and 5%, respectively.

PW91PW91/6-311++G(3df,3pd) slightly outperforms the high level ab initio CCSD(T)/CBS method in predicting the Gibbs free energy changes in nearly all of the cases studied here, showing no decline in its performance with the growing cluster size. These considerations led us to conclude that because the agreement of the PW91/6-311++G(3df,3pd) results with experimental data and ab initio results for both neutral and ionic unary, binary, and ternary systems containing the key atmospheric nucleation precursors goes well beyond satisfactory, the application of PW91/6-311++G(3df,3pd) method is fully justified. In addition to the DFT at PW91/6-311++G(3df,3pd) level, we also used the so-called high-accuracy composite ab initio-based G3MP2 method. Data for smaller clusters needed to compute some of the sulfuric acid affinities were adopted from refs 38,58,60. Computations have been carried out using the Gaussian 03 suite of programs.53

3. RESULTS AND DISCUSSION 3.1. Structure and Geometrical Properties of (HSO4−)(H2SO4)m(NH3)k(H2O)n. Twenty classes of clusters have been investigated, and, finally, over 200 equilibrium conformers obtained at the PW91PW91/6-311++G(3df,3pd) level of theory 136



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Figure 2. Most stable (the lowest Gibbs free energy) isomers of (a,b) (HSO4−)(H2SO4)3; (c,d) (HSO4−)(H2SO4)3(H2O)1; (e−g) (HSO4−)(H2SO4)3(H2O)2; and (h,−j) (HSO4−)(H2SO4)3(H2O)3.

(HSO4−)(H2SO4)2(H2O)1. Twelve equilibrium conformers have been found. Four out of the 12 isomers were obtained using the Cartesian coordinates from Froyd and Lovejoy (2003).54 The average O−H bonds range from 1.647 to 1.865 Å, and O−O distances range from 4.23 to 4.56 Å. The global minimum has O−H length of 1.646 Å and average O−O distance of 4.23 Å. The global minima isomer is similar to that found by Froyd and Lovejoy.54 One local minimum 1 kcal mol−1 from the global minimum and six out of the 12 equilibrium isomers are located within 2 kcal mol−1 from the global minimum. The global minimum is the only isomer to contain the water interacting with both two sulfuric acid molecules and bisulfate ion. The global minimum has the two sulfuric acid molecules and the bisulfate taking a crescent shape around the H2O molecule. With

this configuration, one sulfuric acid tends to transfer protons to the bisulfate ion, while the second sulfuric acid forms two O−H bonds with the bisulfate ion. The twisted acid chain around this single water adds rigidity to the cluster. The first local minimum includes, like the global minimum, the slightly overstretched SO−H bonds. However, the water molecule is located on the outside edge of the cluster between the bisulfate ion and a sulfuric acid. In this case, the water molecule forms O−H bonding with bisulfate ion and two O−H bonds with the sulfuric acid tending to transfer the proton to the water molecule. (HSO4−)(H2SO4)2(H2O)2. Twelve equilibrium conformers have been identified. Seven out of the 12 isomers are similar to those obtained by Froyd and Lovejoy.54 The molecular average O−H bonds range from 1.573 to 2.006 Å, and O−O distances range 137



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Figure 3. Most stable (the lowest Gibbs free energy) isomers of (a) (HSO4−)(H2SO4)(NH3); (b−d) (HSO4−)(H2SO4)(NH3)(H2O)1; (e−i) (HSO4−)(H2SO4)(NH3)(H2O)2; and (j−m) (HSO4−)(H2SO4)(NH3)(H2O)3-based.

of the global minimum and only one 1 kcal mol−1 of the global minimum. The global minimum with the structural formula of (HSO4−)−H−(SO42−)−H−(HSO4−)−H−(OH)−H−(H2O)

from 4.22 to 4.48 Å. The most stable isomer has the average O−H length of 1.645 Å and O−O distance of 4.48 Å. Out of the 12 equilibrium isomers, only four are located within 3 kcal mol−1 138



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Figure 4. Most stable (the lowest Gibbs free energy) isomers of (a) (HSO4−)(H2SO4)2(NH3); (b−e) (HSO4−)(H2SO4)2(NH3)(H2O)1; (f−i) (HSO4−)(H2SO4)2(NH3)(H2O)2; and (j) (HSO4−)(H2SO4)2(NH3)(H2O)3.

bisulfate ion and a sulfuric acid, while the second water molecule (W2) is bonded to the sulfuric acid (S1), bisulfate ion, and the first water molecule W1 sulfuric acid. (HSO4−)(H2SO4)2(H2O)3. Seven equilibrium isomers have been found. Three out of the seven isomers are similar to those from Froyd and Lovejoy.54 The average O−H bonds range from 1.628 to 1.838 Å, and O−O distances range from 4.39 to 4.79 Å. The global minimum has the average O−H length and O−O distance of 1.628 and 4.39 Å, respectively. All six local minima are located within 3 kcal mol−1 of the global minimum. None of the isomers obtained using cluster geometries of Froyd and Lovejoy54 appeared to be the global minimum. The smallest energy difference of those isomers from global minimum is 1.70 kcal mol−1.

contains two slightly overstretched SO−H bonds from the sulfuric acid (S1) to the bisulfate ion (B1), while the bisulfate ion has another slightly overstretched SO−H bond oriented toward to the other sulfuric acid (S2) in the cluster. In addition, the second sulfuric acid (S1) transfers a proton to a water (W1) molecule. The water molecule tends to transfer a proton to the second water (W2) molecule forming an overstretched SO−H bond. This second water then remains intact forming O−H bonds bridging the sulfuric acid (S1) and the bisulfate ion (B1). This isomer is the only one in which the water molecules interact with each other while also interacting with the bisulfate and sulfuric acid molecule. The first local minimum also contains the water molecules in a bridging position between the two H2SO4 molecules (S1−S2). However, one of the water molecules (W1) directly bridges the 139



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Figure 5. Most stable (the lowest Gibbs free energy) isomers of (a) (HSO4−)(H2SO4)3(NH3); (b−g) (HSO4−)(H2SO4)3(NH3)(H2O)1; (h−k) (HSO4−)(H2SO4)3(NH3)(H2O)2; and (l−m) (HSO4−)(H2SO4)3(NH3)(H2O)3.

bisulfate ion. The bisulfate ion tends to transfer the only hydrogen to the water on the cluster edge, while the second sulfuric acid transfers one proton to the water bridging between the bisulfate and water, resulting in the formation of the H3O+ ion.

The global minimum contains one transferred proton and a number of overstretched SO−H bonds. The sulfuric acid was located on the opposite side to that of the water molecule located on the edge of the cluster transferring both hydrogens to the 140



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The first local minimum also contains one transferred proton, has one hydrogen bond not utilized, and, as a result, the isomer structure is more elongated as compared to the global minimum. All of the water molecules take positions on the outside edges of the isomer. Two water molecules (W1,W2) bridge two sulfuric acids (S1,S2); however, the first water molecule forms two O−H bonds, one to the oxygen of the sulfuric acid (S2) and the second to the hydrogen of the other sulfuric acid (S1), while the second water molecules forms O−H bonds by directing one of its hydrogens toward both sulfuric acid molecules (S1,S2). This configuration results in an increase of the average O−O distance by ∼8% as compared to the global minimum and a relatively small energy difference of 0.29 kcal mol−1 from the most stable isomer. The second local minimum includes a number of overstretched SO−H bonds and no protons transferred. The water molecules in this isomer take position in the bends in the acid cluster chain. The second local minimum also contains a water molecule that serves as a bridge between two of the acid molecules (S1,S2) with both hydrogen bonds to the corresponding oxygen molecules of two acids, like the global minimum and the first local minimum. This bridging by the water (W2) results in the average distance in the O−O bonds increasing. As with the first local minimum, not all of the possible O−H bonds are utilized for this isomer. (HSO4−)(H2SO4)3. In this cluster class, eight stable isomers have been found. The intermolecular average O−H bonds range from 1.587 to 1.735 Å, and O−O distances range from 4.95 to 5.49 Å. The most stable isomer of (HSO4−)(H2SO4)3 has the average O−H length of 1.654 Å and average O−O distances of 5.49 Å. Only one isomer is located within 1.00 kcal mol−1 of the most stable isomer. Three out of the eight stable isomers were calculated using the geometries from Froyd and Lovejoy54 as initial guess geometries. However, none of them appeared to be the global minimum, and only one out of the three is located within 2 kcal mol−1 of the global minimum obtained in the present study. The most stable isomer has a twisted chain-like structure and includes several overstretched SO−H bonds. The structure of the global minimum includes no protons transferred. The first local minimum also contains no protons transferred, with the bisulfate having shared O−H bonding with each of the available oxygens from two H2SO4 molecules. The bisulfate ion is located relatively close to the center of the cluster. Two of the H2SO4 molecules share O−H bonding with oxygen from the bisulfate ion. The O−O repulsion in this isomer results in one sulfuric acid being twisted away from the other. (HSO4−)(H2SO4)3(H2O)1. Eleven equilibrium conformers have been identified. Four out of the 11 isomers of this class were obtained on the basis of the data of Froyd and Lovejoy,54 and these four isomers were also found to be the most stable isomers in this cluster class. The molecular average O−H bonds range from 1.541 to 1.823 Å, and O−O distance ranges from 4.61 to 5.02 Å. The most stable isomer of the cluster had an O−H length of 1.541 Å and an average O−O distance of 4.61 Å. The four most stable isomers were found to be within 3 kcal mol−1 of the global minimum and only one was within 1 kcal mol−1 of the energy of the most stable isomer. The global minimum is the only isomer in this class to contain several overstretched SO−H bonds along with the water molecule interacting with all three sulfuric acid molecules. The first local minimum was found to have a structure similar to the global minimum. The isomer geometry shows that the sulfuric

acids have a slight twist when interacting with the bisulfate ion. This subtle difference in geometry between the global minimum and first local minimum accounts for the ∼0.42 kcal mol−1 difference in the Gibbs free energy. (HSO4−)(H2SO4)3(H2O)2. Ten equilibrium conformers were identified for this cluster class. Out of the 10 conformers, only two appear to be similar to the isomers from Froyd and Lovejoy.54 Isomers obtained based on their results were found to be within 4 kcal mol−1 of the global minimum. The molecular average O−H bonds range from 1.559 to 1.748 Å, and O−O distances range from 5.04 to 5.42 Å. The global minimum has the average O−H length of 1.622 Å and average O−O distance of 5.04 Å. Two isomers are located within ∼1 kcal mol−1 of the global minimum, five isomers in this class are located within ∼3 kcal mol−1 of the global minimum, with the remaining five isomers having the free energy difference from the global minimum over 3 kcal mol−1. The geometries of the five most stable isomers are surprisingly different. The global minimum contains partly deprotonated sulfuric acid S2 yielding to the formation of the H3O+·HSO4− ion pair. The sulfuric acid molecules and the bisulfate ion take on a vshaped configuration around one of the H2O molecules, acting as a bridge between the sulfuric acid and the bisulfate that do not interact directly like the other two H2SO4 molecules. The second H2O molecule interacts with the H2SO4 molecules, creating the base of the “v” by interacting with the remaining oxygen without the formation of the O−H bond. The global minimum is the most compact as compared to the closest local minima. The first and second local minima were found to contain the H3O+·HSO4− pair along with the initial HSO4− ion. All four closest local minima take on chain-like structures with various placements of water molecules. The first and second local minima differ in energy from the global minimum by 0.14 and 0.63 kcal mol−1, respectively. These two isomers are close in geometry to the global minimum; however, the way water molecules interact in these two different isomers dictates the energy difference. The first local minimum has the first water (W1) molecule interacting with the sulfuric acid and the bisulfate ion as well as with the second H2O molecule (W2). Along with interacting with the first water molecule, the second water molecule is interacting with a sulfuric acid (S2) resulting in the formation of the H3O+·HSO4− ion pair. The second local minimum was also found to contain the H3O+·HSO4− ion pair. The H3O+ interacts with the second water in the same way as in the first local minimum; however; this water molecule interacts with HSO4− ion only. This difference leaves a sulfuric acid molecule at the end of the chain with two available oxygens, and leads to O−O repulsion between the sulfuric acid and bisulfate ion. These differences in the isomer structures between the first and second local minima account for the energy difference of 0.49 kcal mol−1. (HSO4−)(H2SO4)3(H2O)3. Seven equilibrium conformers were found in this cluster class. The global minimum has the average O−H length of 1.737 Å and average O−O distance of 5.12 Å. Two isomers are located within 1 kcal mol−1 of the global minimum, and six isomers are located within 2 kcal mol−1 of the global minimum. The difference between first two local minima, which were found to be 0.05 and 0.09 kcal mol−1 from the global minimum, is minor. All seven final isomers are similar and contain the H3O+·HSO4− ion pair along with the initial HSO4− bisulfate ion. The network formed by the H2O molecules in the global minimum is the largest as compared to other most stable isomers. 141



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ions. The first and second local minima have a dual H−O bond as a water competes to bond with an available oxygen from the HSO4−·NH4+ ion pair. Given the opposite ion charge, the NH4+ is clearly favored to bond with oxygen of (S1, W1) over the bisulfate ion (B1). The global minimum along with the first and second local minima contain the water and ammonia molecules between the sulfuric acid molecule and the bisulfate ion. (HSO4−)(NH3)(H2SO4)(H2O)3. Nine conformers with the average O−H bonds ranging from 1.776 to 1.981 Å and O−O distance ranging from 4.39 to 4.47 Å have been identified. As with the (HSO4−)(NH3)(H2SO4)(H2O)2, the most stable isomer does not have the shortest O−H bonds. The global minimum of the cluster has an average O−H length of 1.819 Å and average O−O distance of 4.47 Å, respectively. The ammonia in the global minimum is set in slightly toward the center of the cluster as compared to the other isomers, where the ammonia is forced to the outer edge of the cluster. Five local minima are located within 1 kcal mol−1 of the global minimum. All of the global minimum and closest local minima include the HSO4−·NH4+ ion pair in addition to the bisulfate ion and a number of overstretched SO−H bonds. The global minimum with the structural formula is not the most compact isomer and does not have the shortest O−H bonds. Two of the H2O molecules and ammonia are located between the sulfuric acid and the bisulfate ion. The third water takes position above and between the ammonia and the sulfuric acid molecule, bonding to both NH3 and H2SO4. The ammonia in this isomer interacts with two water molecules, which help to stabilize the cluster. W1 that the ammonia interacts with is located between the sulfuric acid and the bisulfate ion, yielding the proton transfer between the ammonia and water molecule. This configuration allows each of the hydrogens of the water to bond with the oxygen in each acid. The W2 molecule forms three O−H bonds between both H2SO4 molecule and bisulfate ion. The water has two bonds with the sulfuric acid and a third bond with the bisulfate ion. The configuration of the bonds is positioned in the following way: the hydrogen from the bisulfate ion bonds to the oxygen of the water, and a hydrogen bonds to the available oxygen of the bisulfate ion and the sulfuric acid. The global minimum was identified as the only convergent isomer to have the water molecules positioned between the sulfuric acid and the bisulfate ion as well as the third water, which is positioned on the outside edge between and bonds with NH4+ ion and the sulfuric acid (S1). Unlike the global minimum, two of the H2O molecules in the first local minimum interact together in which both compete for the O−H bond with an oxygen from the sulfuric acid molecule. The remaining water also competes with the ammonia for O−H bonding to the sulfuric acid. The second local minimum has a structure similar to that of the global minimum. However, the second water molecule that interacts with the ammonia in the global minimum is relegated to the edge of the cluster in the second local minimum, indicating lower hydration energy due to the water molecule not being able to effectively enter the cluster core. The remaining isomers have H2O molecules located on the outside edge of the cluster. (HSO4−)(NH3)(H2SO4)2. Ten stable conformers of (HSO4−)(NH3)(H2SO4)2 have been identified. Only two out of nine local minima are located within 3 kcal mol−1 of the global minimum, and none are located within 1 kcal mol−1 of the lowest energy conformer. The O−H bonds range from 1.605 to 1.810 Å, and O−O distances range from 4.14 to 4.36 Å. The most stable

The chain formed by the H2O molecules in the most stable isomer is perpendicular to that formed by the sulfuric acid and bisulfate molecules. This geometry results in additional stabilization as compared to other isomers. When the geometries of the seven isomers in this class are compared, the positions of the sulfuric acid and bisulfate appear to be similar. However, the location of the water molecules differs from one isomer to another. The local minima are found to contain a smaller number of overstretched SO−H bonds than the global minimum, while all of them have identical structural formula. Energies of the three most stable isomers are very close, and the locations of sulfuric acid molecules and bisulfate are essentially the same. In the case, the main cause of the minor energy difference is related to H2O molecules and their placement. The first and second local minima have the H2O molecules forming a chain, which is parallel to that formed by the H2SO4 and bisulfate molecules. (HSO4−)(NH3)(H2SO4). There was a single isomer found for this class (no local minima were found within 3 kcal/mol of the global minimum). The ammonia positions between the sulfuric acid and the bisulfate ion. This positioning results in the sulfuric acid and bisulfate ion to angle away from the ammonia, with a highly symmetric shape. (HSO4−)(NH3)(H2SO4)(H2O)1. There were four equilibrium isomers found in this cluster class. Two local minima are located within 1 kcal mol−1 of the global minimum. The molecular average O−H bonds range from 1.816 to 1.861 Å, and O−O distances range from 3.96 to 4.06 Å. The global minimum has the average O−H length of 1.816 Å and average O−O distance of 4.06 Å. The bonding angles between the sulfuric acid (S1) and the bisulfate (B1) are nearly identical with (O−H−O) 177.44° and 177.43° for the global minimum structure. In the first local minimum, the bonding angles between the sulfuric acid and bisulfate differ by 0.15° with angles of (O−H−O) 177.09° and 176.94°. The difference of 0.15° between the pair of (O−H−O) bonds for the first local minimum explains the 0.35 kcal mol−1 difference from the global minimum. The first local minimum is similar to the global minimum. However, the difference between the global minimum and first local minimum is in a slight twist in the positioning of the sulfuric acid molecule and bisulfate ion and location of the water molecule, which is located in the first local minimum on the opposite side to that of the global minimum. The water molecule bonding in the first local minima results in a dual hydrogen bond to one of the free oxygens of the sulfuric acid. The second local minimum differs from the first local minimum mainly in the placement of the water molecule. (HSO4−)(NH3)(H2SO4)(H2O)2. Four equilibrium isomers conformers were identified. The average O−H bonds range from 1.735 to 1.795 Å, and O−O distances range from 4.24 to 4.30 Å. The global minimum has an average O−H length of 1.740 Å and average O−O distance of 4.24 Å. Two minima are located within 1 kcal mol−1 of the global minimum. The geometry of all four isomers contains two slightly overstretched SO−H distances with the stretching of the SO−H bonds occurring between the ammonia and the sulfuric acid molecules. The most important factor affecting the Gibbs free energy appears to be the interaction of the H2O molecule and ammonia with the H2SO4 molecule. The global minimum includes the HSO4−·NH4+ ion pair. The ammonium ion interacts with both bisulfate ions while also interacting with one water, and the hydrogens from the water molecule act as a bridge between the available oxygens from the bisulfate ions, with the second water interacting only with the two bisulfate ions. Both water molecules, however, attempt to bond with the same oxygen of one of the bisulfate 142



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geometry similar to that of the most stable isomer from the (HSO4−)(NH3)(H2SO4)2(H2O)2 class. In the most stable isomer, each water molecule interacts with two to three molecules. The H2O molecules do not interact with each other in both global minimum and local minima. As for the (HSO 4 − )(NH 3 )(H 2 SO 4 ) 2 (H 2 O) 2 , and (HSO 4 − )(NH 3 )(H2SO4)2(H2O)3, the HSO4−·NH4+ pair interacts with the initial bisulfate ion. The ammonium ion is located in the center of the cluster. (HSO 4 − )(NH 3 )(H 2 SO 4 ) 3 . Eleven stable conformers of (HSO4−)(NH3)(H2SO4)2 have been identified. The molecular average O−H bonds range from 1.816 to 2.064 Å, and O−O distances range from 4.56 to 4.78 Å. The most stable isomer of the cluster had an O−H length of 1.870 Å and an average O−O distance of 4.68 Å. The difference in the thermodynamics of the four most stable isomers is driven by different types of interactions along with the number of molecules interacting with the ammonia, leading to the formation of ammonium ion. In the global minimum, the acid molecules are wrapped around the ammonia molecule, with two H2SO4 (S1, S3) molecules and bisulfate ion O−H bonding to the ammonia acids. The remaining H2SO4 (S2) molecule tends to bond with the bisulfate ion and the two H2SO4 molecules. In addition, the interaction of two sulfuric acids results in an O−O repulsion and slight twisting and alignment of the sulfuric acid bonds to the ammonia. In this isomer, there is a single vacant hydrogen located on the outside edge of this cluster. This leaves only two oxygens available on the outside edge of the cluster and reduces the energy of this isomer as compared to the first two local minima. The ammonia in the global minimum interacts with two sulfuric acid molecules (S1, S3) and the bisulfate (B1), while the remaining sulfuric acid (S2) is found to be relegated to the opposite side of the ammonia molecule. (HSO4−)(NH3)(H2SO4)3(H2O)1. The total number of equilibrium isomers identified for this cluster class is seven, with the average O−H bonds ranging from 1.624 to 1.889 Å and O−O distance ranging from 4.87 to 4.96 Å. The global minimum has an O−H length of 1.786 Å and average O−O distance of 4.87 Å. The Gibbs free energies of the five most stable isomers containing the HSO4−·NH4+ ion pair are close, with four isomers located within ∼0.75 kcal mol−1 of the global minimum. As with the (HSO4−)(NH3)(H2SO4)2(H2O)n clusters, there are several factors deciding which isomer is the global minimum: the number of H2SO4 molecule interacting with the ammonia, whether the water and ammonia molecules interact with each other, the formation of the ammonium ion, and the placement of the water molecule with respect to the cluster geometry. The number of H2SO4 molecules found to bond with the ammonia ranges from two to four. Of the total seven clusters, only two isomers were found to contain the ammonia interacting with the water molecule. In regards to the placement of the water molecule, only one isomer has the water molecule located on the outside most edge and interacting with the sulfuric acid. The global minimum has the following feature: the ammonium ion is interacting with three sulfuric acid molecules (S1, S2, S3) along with the water molecule. (HSO4−)(NH3)(H2SO4)3(H2O)2. Eleven isomers with average O−H bonds ranging from 1.707 to 1.774 Å and O−O distance ranging from 4.93 to 5.15 Å have been identified The global minimum has O−H length of 1.724 Å and average O−O distance of 4.94 Å. Three local minima have energies within ∼1 kcal mol−1 of the global minimum.

isomer has an O−H length of 1.642 Å and an average O−O distance of 4.14 Å. The most stable isomer was found to be the only isomer to contain the HSO4−·NH4+ ion pair in addition to the bisulfate ion. It also includes two sets of shared O−H bonds involving (H 2 SO 4 )−(HSO 4 − )−(H 2 SO 4 ) and (NH 3 )− (H2SO4)−(HSO4−) molecules. (HSO4−)(NH3)(H2SO4)2(H2O)1. Seven convergent isomers with the average O−H bonds ranging from 1.681 to 1.807 Å and O−O distances ranging from 4.29 to 4.43 Å have been identified. All of the global minimum and closest local minima include the HSO4−·NH4+ ion pair in addition to the bisulfate ion and a number of overstretched SO−H bonds. The global minimum has the average O−H length of 1.705 Å and average O−O distance of 4.29 Å. All isomers in this class were found to be within ∼1.40 kcal mol−1 of the global minimum, and four were within ∼1 kcal mol−1. The total number of slightly overstretched SO−H distance bonds ranges from two to four. The first local minimum contains three acids partially wrapped around the ammonia with each acid bonding directly with the ammonia and the water molecules positioned away from the cluster core. The water molecule bonds with a bisulfate ion with each hydrogen seeking the available oxygen. The water is only able to weakly bond to the oxygen in the sulfuric acid (S2), while the other oxygen is competing for O−H bonding from the (S1 and B1) molecules in the process of slightly overstretched SO−H distances to the bisulfate ion. The second local minimum is similar to the first local minimum; however, the water molecule is located on the outside edge of the cluster and forms two O−H bonds with the same sulfuric acid (S2). All four most stable local minima contain the water molecule on the outside edge of the cluster as well as interactions occurring with the sulfuric acid or bisulfate ion. The resulting geometries of the core sulfuric acid−bisulfate−ammonia structures of all five isomers are similar. However, the global minimum has the water molecule interacting with the ammonia and bisulfate ion, resulting in a more compact geometry as compared to the local minima. (HSO4−)(NH3)(H2SO4)2(H2O)2. Eight equilibrium isomers with average O−H bonds ranging from 1.739 to 1.881 Å and O−O distance ranging from 4.44 to 4.59 Å have been identified. The global minimum has an average O−H length of 1.739 Å and average O−O distance of 4.59 Å. Only three of the local minima have the Gibbs free energy within one kcal mol−1 of the global minimum. All of the global minimum and closest local minima include the HSO4−·NH4+ ion pair in addition to the bisulfate ion and a number of overstretched SO−H bonds. These four more stable isomers differ in numbers of overstretched SO−H bonds, and free hydrogen bonds range from one to three. Out of the four isomers, the most stable was the only conformer to contain only one free hydrogen from a water molecule. The global minimum contains the HSO4−·NH4+ pair bonding with four out of five remaining molecules. The centrally located ammonium ion allows for the sulfuric acid and the bisulfate ions to have strong bonds. (HSO4−)(NH3)(H2SO4)2(H2O)3. Eleven convergent isomers for this cluster class have been identified. The molecular average O−H bonds range from 1.708 to 1.888 Å, and O−O distances range from 4.55 to 4.80 Å. The global minimum of the cluster has an average O−H length of 1.888 Å and average O−O distance 4.80 Å. Out of the 11 isomers, only four are located within 2 kcal mol−1 of the global minimum and none within 1 kcal mol−1. The global minimum containing the HSO4−·NH4+ ion pair has a 143



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Table 3. Stepwise Hydration Reaction Enthalpies, Entropies, and Gibbs Free Energies of Hydration at Temperature of 298.15 K and Pressure of 101.3 kPa Obtained at PW91PW91/6-311++G(3df,3pd) Level of Theorya reaction (H2SO4)(HSO4−) + (H2O) ⇔ (H2SO4)(HSO4−)(H2O)

(H2SO4)(HSO4−)(H2O) + (H2O) ⇔ (H2SO4)(HSO4−)(H2O)2 (H2SO4)(HSO4−)(H2O)2 + (H2O) ⇔ (H2SO4)(HSO4−)(H2O)3 (H2SO4)2(HSO4−) + (H2O) ⇔ (H2SO4)2(HSO4−)(H2O)

(H2SO4)2(HSO4−)(H2O) + (H2O) ⇔ (H2SO4)2(HSO4−)(H2O)2 (H2SO4)2(HSO4−)(H2O)2 + (H2O) ⇔ (H2SO4)2(HSO4−)(H2O)3 (H2SO4)3(HSO4−) + (H2O) ⇔ (H2SO4)3(HSO4−)(H2O) (H2SO4)3(HSO4−)(H2O) + (H2O) ⇔ (H2SO4)3(HSO4−)(H2O)2 (H2SO4)3(HSO4−)(H2O)2 + (H2O) ⇔ (H2SO4)3(HSO4−)(H2O)3 (H2SO4)(HSO4−)(NH3) + (H2O) ⇔ (H2SO4)(NH3)(HSO4−)(H2O) (H2SO4)(HSO4−)(NH3)(H2O) + (H2O) ⇔ (H2SO4)(NH3)(HSO4−)(H2O)2 (H2SO4)(HSO4−)(NH3)(H2O)2 + (H2O) ⇔ (H2SO4)(NH3)(HSO4−)(H2O)3 (H2SO4)2(HSO4−)(NH3) + (H2O) ⇔ (H2SO4)2(NH3)(HSO4−)(H2O) (H2SO4)2(HSO4−)(NH3)(H2O) + (H2O) ⇔ (H2SO4)2(NH3)(HSO4−)(H2O)2 (H2SO4)2(HSO4−)(NH3)(H2O)2 + (H2O) ⇔ (H2SO4)2(NH3)(HSO4−)(H2O)3 (H2SO4)3(HSO4−)(NH3) + (H2O) ⇔ (H2SO4)3(NH3)(HSO4−)(H2O) (H2SO4)3(HSO4−)(NH3)(H2O) + (H2O) ⇔ (H2SO4)3(NH3)(HSO4−)(H2O)2 (H2SO4)3(HSO4)(NH3)(H2O)2 + (H2O) ⇔ (H2SO4)3(NH3)(HSO4−)(H2O)3 a

ΔH

ΔS

ΔG

−8.20 −9.35♀ −8.70 ± 0.70 −9.30 −9.50 ± 0.40 −18.00 −11.1 ± 2.00 −9.29 −9.40♀ −11.7 ± 0.90 −17.59 −13.60 ± 0.7 −11.11 −10.90 ± 1.3 −18.27 −15.10 ± 0.4 −9.46 −12.5 ± 0.4 −16.57 −12.90 ± 0.3 −8.82 −14.03 −11.03 −9.13 −9.56 −1.44 −13.33 −15.54 −9.98 −10.18

−25.50 −25.06♀ −21.10 ± 3.00 −27.50 −26.5 ± 1.5 −51.65 −33.5 ± 1.5 −26.37 −27.72♀ −33.5 ± 3.80 −44.54 −38.6 ± 2.7 −34.98 −33.20 ± 5.2 −49.17 −41.50 ± 1.5 −27.54 −36.8 ± 1.8 −39.63 −36.80 ± 1.2 −23.96 −43.04 −32.15 −28.94 −27.22

−0.60 −1.88♀ −2.41 ± 1.13 −1.10 −1.60 ± 0.60 −2.58 −1.10 ± 2.08 −1.43 −1.14♀ −1.71 ± 1.43 −4.31 −2.09 ± 1.06 −0.68 −1.00 ± 2.01 −3.61 −2.72 ± 0.60 −1.25 −1.52 ± 0.67 −4.75 −1.92 ± 0.46 −1.68 −1.20 −1.44 −0.50

−35.68 −34.63 −34.83 −30.14

−2.69 −5.22 0.40 −1.19

Bold denotes the experimental data.47 Superscript “♀” refers to G3MP2 (present study).

The geometry of these isomers containing the HSO4−·NH4+ ion pair is similar; however, there are subtle differences from isomer to isomer, resulting in some variations of the Gibbs free energy. The factors controlling the variations in the free energy include the number of H2SO4 molecules interacting with the ammonia, and the placement of the H2O molecules within the cluster. Out of 11 isomers, only one isomer includes three H2SO4 molecules interacting with the ammonia, while the remaining 10 isomers have two H2SO4 molecules interacting with the ammonia. The global minimum contains two H2SO4 (S1,S3) molecules interacting with the ammonia molecule, with a total of seven overstretched SO−H distances. The H2O molecules are placed closely to the center of the complex. Both water molecules interact with each other and with two other molecules as well (S1−W1−S2) and (A1−W2−S2). One of the water molecules (A1−W2−S2) interacts with the ammonia and a sulfuric acid, while the second water molecule (S1−W1−S2) interacts with two H2SO4 molecules, and both interact with the same sulfuric acid taking on a subtle twisting and tilting in toward the center of the cluster and toward both water molecules. The remaining 10 isomers do not utilize all possible O−H bonds. One case is that water has an unused O−H bond, or in the second case, both water molecules with an unused O−H bond. The first local minimum where the H2O molecules fully utilize the O−H bonds in the isomer is the only exception. This isomer is similar to that of the global minimum; however, the positioning

of the water results in the 0.49 kcal mol−l energy difference even though in this case the O−O average distance is reduced by 0.01 Å. This can be explained by the water molecule not bonding to the same oxygen of the sulfuric acid. In the first local minimum, the water bonding to the ammonia (A1−W2) is shifted to one side, allowing for a more direct O−H bond from 2.08 Å in the global minimum to 1.88 Å. However, with this adjustment in the cluster configuration, the second water is forced more to the outside edge of the cluster, and the bond increases from 2.04 to 2.16 Å. (HSO4−)(NH3)(H2SO4)3(H2O)3. Five equilibrium isomers are identified for this structure class. The average O−H bonds range from 1.680 to 1.810 Å, and O−O distances range from 5.02 to 5.21 Å, with the global minimum having an O−H length of 1.736 Å and an average O−O distance of 5.02 Å. One out of four local minima isomers are within 1 kcal mol−1 of the global minimum. The isomers have similar structures. The H2SO4 molecules (S1, S2, S3) bond on the outside edge of the sulfate ion (B1) and ammonium (A1) core. Like the isomers in the (HSO4−)(NH3)(H2SO4)3(H2O)n class, the bonding configuration varied in each isomer of this class. All five isomers are found to contain the HSO4−·NH4+ pair. The global minimum contains the ammonia interacting with one sulfuric acid (S1) and two H2O molecules (W2,W3). The positioning of the ammonium appears to be extremely important for the resulting Gibbs free energy. The placement of the ammonium is slightly inset with respect to the oxygen plane created by the H2SO4 molecules. Two out of 144



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Figure 6. Comparison of hydration free energies (kcal/mol) associated with the formation of (HSO4−)(H2SO4)m(H2O)k and (HSO4−)(NH3)(H2SO4)m(H2O)k (a) and those associated with the formation of hydrates, both neutral and charged, of similar compositions (b) Hydration free energies are represented by −ΔG. Abbreviations A, S−, S, and W refer to NH3, HSO4−, H2SO4, and H2O, respectively. Abbreviations 1, 2, and 3 refer to the hydration numbers. Calculations were carried out using PW91PW91/6-311++G(3df,3pd) at T = 298.15 K and P = 101.3 kPa.

The impacts of the cluster size and composition on the thermodynamic properties of the aforementioned clusters have been studied, and the comparison of the obtained theoretical data with experiments has been carried out. Uncertainties in both theory and experimentation have been investigated, and possible ways of their reduction have been discussed. 3.2.1. Hydration. The formation of a shell consisting of water molecules over an atmospheric molecule or ion is a fundamental process affecting the electrical mobility, stability, and lifetime of atmospheric ions and atmospheric nucleation rates. Although the only large experimental and semiexperimental data set for the negatively charged ions available at the present time47 contains thermodynamic properties of both water and sulfuric acid, the completely experimental data are available for water affinities only. The experimental enthalpies and entropies of hydration were derived from the measured reaction constants using the standard linear fit of the van’t Hoff curve. The hydration free energies of negatively charged (HSO4−)(H2SO4)m(NH3)0(H2O)n ions are expectedly low as compared

the three H2O molecules (W2, W3) are positioned in this plane. In addition, the most stable isomer contains unused hydrogen bond from the water (W3) that bonds only with a sulfuric acid (S3) and the ammonium ion (A1). The remaining H2O molecules (W1, W2) form two square-like ring structures. As the isomer energy moves farther from the global minimum, the location of the ammonium ion in the isomer moves farther away from the H2SO4 molecules allowing for the H2O molecules to move in closer to the sulfuric acids. This leads to increases in the intermolecular bonding lengths and the corresponding decrease in absolute value of the Gibbs free energy as compared to the global minimum. This suggests that the position of the ammonium ion in the global minimum, where the base-core is solvated by the sulfuric acid envelope, results in more negative Gibbs free energy as compared to other isomers. 3.2. Thermochemical Properties. In this section, the thermochemical properties for 24 classes of large pre-nucleation negatively charged ionic clusters (HSO4−)(H2SO4)m(H2O)n(NH3) and (HSO4−)(H2SO4)m(H2O)n clusters are reported. 145



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Figure 7. Comparison of theoretical enthalpies changes (a), Gibbs free energies (b), and difference between theory and experiments (c) (kcal/mol) associated with the formation of (HSO4−)(H2SO4)m(H2O) via the addition of H2O with the experimental data.54 Abbreviations A, S−, S, and W refer to NH3, HSO4−, H2SO4, and H2O, respectively. Changes in the enthalpies and Gibbs free energy changes are represented by −ΔH and −ΔG, repectively. Calculations were carried out using PW91PW91/6-311++G(3df,3pd) and G3MP2 (panels (a) and (b)) at T = 298.15 K and P = 101.3 kPa.

final conclusion about their nature and possible location of the maximum of the equilibrium cluster distribution. The (H2SO4)2(HSO4−)(H2O)n hydration exhibits a slightly different pattern. Both the theory and experimental data show the (H2SO4)2(HSO4−)(H2O)2 cluster as the maximum of the

to those of positively charged clusters and are in the order of the free energies for neutral (H2SO4)m(NH3)0−1(H2O)n clusters.57 Although the growth of the hydration strength with the hydration number is observed in the case of (H2SO4)(HSO4−)(H2O)n, the variations in the free energies are too small to draw a 146



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−4.75 kcal/mol are achieved for (H2SO4)3(HSO4−)(H2O)1 and (H2SO4)2(HSO4−)(H2O)3, respectively. This is likely to indicate a moderate growth in the cluster stability with the growing concentration of the sulfuric acid following the abrupt drop of the hydration free energies associated with the attachment of the first H2SO4 molecule to the bisulfate ion. As it may be seen from Table 3 and Figures 6 and 7, no profound effect of ammonia on the hydration free energies is observed. The effect of ammonia on the hydration of (H2SO4)− (HSO4−)−(H2O) clusters is too small to make a conclusion that this effect is certainly important. However, there exists an obvious similarity between the hydration of (H2SO4)m(HSO4−)(H2O)n and (NH3)(H2SO4)m(HSO4−)(H2O)n: the abrupt drop of the hydration free energies associated with the attachment of the first H2SO4 molecule to ions followed by a moderate growth in the cluster stability with the growing concentration of the sulfuric acid in the cluster. A similar pattern is observed in the case of positively charged ions. These considerations lead us to conclude that the hydration of (NH3)(H2SO4)m(HSO4−)(H2O)n and (H2SO4)m(HSO4−)(H2O)n is largely controlled by the concentration of the sulfuric acid. The comparison of the theoretical and experimental reaction enthalpies and Gibbs free energies is given in Table 3 and Figure 7. As it may be seen from Table 3 and Figure 7b, our theoretical results for the Gibbs free energy changes controlling the overall cluster stability are in excellent agreement with the experimental data. The average deviation of the theoretical Gibbs free energy changes from the experimental data barely exceeds 1 kcal/mol, which is considered to be the limit of chemical accuracy. While the theoretical Gibbs free energy changes agree with the experimental values within the uncertainties of the experimental data in most cases studied, the deviation in reaction enthalpies is much larger in most cases studied here. This deviation exceeds 2 kcal/mol in 9 out of 14 cases studied, being on average several times larger than the difference between theoretical and experimental Gibbs free energies. Although possible reasons for such a large discrepancy include both theoretical and experimental problems, the most probable cause is the linearization of the van’t Hoff curve applied in treatment of the experimental data, as explained in some detail in the Methods Section and references therein. It is important to note that a large deviation in (ΔH, ΔS) pairs obtained using linear and nonlinear fits from the same data set does not significantly impact the Gibbs free energy (ΔG) changes over the limited temperature range within or close to the experimental temperature interval. However, this uncertainty may become very important in the case when ΔG =ΔH − TΔS is calculated far from the experimental temperature range or/and when the derived ΔH value is used in the computations in thermodynamic calculations as an independent input parameter. 3.2.2. Sulfuric Acid Affinity. The affinity of the sulfuric acid, the key atmospheric nucleation precursor, is perhaps the most important factor controlling the growth of both binary and ternary pre-nucleation clusters. Table 4 and Figure 8 show the enthalpy, entropies, and Gibbs free energies associated with the cluster formation by the addition of the sulfuric acid molecule. As it may be seen in Table 4 and Figure 8, the presence of ammonia is favorable for the cluster growth by the attachment of H2SO4 in nearly all of the cases studied here. In most cases, the presence of ammonia leads to a substantial (2−8 kcal/mol) enhancement in the cluster stability. The strongest ammonia impact is observed in the case of clusters containing two sulfuric acid molecules in addition to the bisulfate ion. The affinities of clusters with and without ammonia differ by ∼3.5 kcal/mol on average in the

Table 4. H2SO4 Addition Reaction Enthalpies, Entropies, and Gibbs Free Energies of Hydration at Temperature of 298.15 K and Pressure of 101.3 kPa Obtained at PW91PW91/6-311+ +G(3df,3pd) Level of Theorya reaction

ΔH

ΔS

ΔG

(H2SO4) + (HSO4−) ⇔ (H2SO4)(HSO4−)

−45.7 −46.64♀ −41.8 −40.30 −41.88♀ −37.6 −25.0 −28.58♀ −27.4 −21.00 −23.80 −28.02 −28.62♀ −30.40 −36.28 −34.5 −29.36 −34.40 −22.11 −27.20 −20.12 −26.20 −25.58 −28.2 −48.49

−43.5 −45.47♀ −42.6 −36.60 −36.26♀ −40.70 −36.6 −35.60♀ −34.4 −35.90 −35.30 −42.12 −38.26♀ −46.80 −58.92 −58.80 −41.98 −58.50 −42.44 −43.3 −37.53 −41.6 −42.18 −45.2 −43.12

−32.74 −33.08♀ −29.10 −30.41 −31.04♀ −25.00 −14.10 −17.97♀ −17.14 −10.30 −13.28 −15.47 −17.22♀ −16.44 −18.72 −16.97 −16.84 −16.96 −9.46 −14.30 −8.93 −14.80 −13.01 −14.72 −35.64

−37.33

−53.03

−21.52

−19.96

−32.81

−10.18

−39.62

−30.98

−30.41

−42.62

−45.40

−29.08

−37.88

−39.04

−26.54

−37.63

−58.01

−20.34

−33.15

−42.19

−20.57

−35.45

−45.72

−21.82

−26.38

−38.50

−14.90

−26.80

−46.10

−13.06

−23.65

−40.56

−11.56

(HSO4−)(H2O) + (H2SO4) ⇔ (H2SO4) (HSO4−)(H2O) (H2SO4)(HSO4−) + (H2SO4) ⇔ (H2SO4)2(HSO4−) (H2SO4)2(HSO4−) + (H2SO4) ⇔ (H2SO4)3(HSO4−) (HSO4−)(H2SO4)(H2O) + (H2SO4) ⇔ (H2SO4)2(HSO4−)(H2O) (HSO4−)(H2SO4)(H2O)2 + (H2SO4) ⇔ (H2SO4)2(HSO4−)(H2O)2 (HSO4−)(H2SO4)(H2O)3 + (H2SO4) ⇔ (H2SO4)2(HSO4−)(H2O)3 (HSO4−)(H2SO4)2(H2O) + (H2SO4) ⇔ (H2SO4)3(HSO4−)(H2O) (HSO4−)(H2SO4)2(H2O)2 + (H2SO4) ⇔ (H2SO4)3(HSO4−)(H2O)2 (HSO4−)(H2SO4)2(H2O)3 + (H2SO4) ⇔ (H2SO4)3(HSO4−)(H2O)3 (HSO4−)(NH3) + (H2SO4) ⇔ (H2SO4) (HSO4−)(NH3) (H2SO4)(HSO4−)(NH3) + (H2SO4) ⇔ (H2SO4)2(HSO4−)(NH3) (H2SO4)2(HSO4−)(NH3) + (H2SO4) ⇔ (H2SO4)3(HSO4−)(NH3) (HSO4−)(H2O)(NH3) + (H2SO4) ⇔ (H2SO4)(HSO4−)(NH3)(H2O) (HSO4−)(H2O)2(NH3) + (H2SO4) ⇔ (H2SO4)(HSO4−)(NH3)(H2O)2 (HSO4−)(H2O)3(NH3) + (H2SO4) ⇔ (H2SO4)(HSO4−)(NH3)(H2O)3 (HSO4−)(H2SO4)(NH3)(H2O) + (H2SO4) ⇔ (H2SO4)2(NH3)(HSO4−)(H2O) (HSO4−)(H2SO4)(NH3)(H2O)2 + (H2SO4) ⇔ (H2SO4)2(NH3)(HSO4−)(H2O)2 (HSO4−)(H2SO4)(NH3)(H2O)3 + (H2SO4) ⇔ (H2SO4)2(NH3)(HSO4−)(H2O)3 (HSO4−)(H2SO4)2(NH3)(H2O) + (H2SO4)≤>(H2SO4)3(HSO4−)(NH3) (H2O) (HSO4−)(H2SO4)2(NH3)(H2O)2+(H2SO4) ⇔ (H2SO4)3(HSO4−)(NH3)(H2O)2 (HSO4−)(H2SO4)2(NH3)(H2O)3 + (H2SO4) ⇔ (H2SO4)3(NH3)(HSO4−) (H2O)3

Bold denotes the semi-experimental estimates.54 Superscript “♀” refers to G3MP2 (present study).

a

equilibrium cluster distribution. Overall, the hydration of (H2SO4)2(HSO4−)(H2O)n is a bit stronger than that of (H2SO4)1(HSO4−)(H2O)n; however, the difference in hydration free energies not exceeding 1 kcal/mol on average is too small to give us a clear indication of a considerable effect of the sulfuric acid concentration on the hydration free energies. The hydration of (H2SO4)3(HSO4−)(H2O)n is definitely stronger than that of (H2SO4)1(HSO4−)(H2O)n and (H2SO4)2(HSO4−)(H2O)n. The maximum values of the hydration free energies of −3.61 and 147



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Figure 8. Comparison of theoretical and semi-experimental54 sulfuric acid affinities, (a) difference in sulfuric acid affinity between clusters of with ammonia and those without ammonia (b), and difference between theory and semi-experimental estimates54 for changes in enthalpy and reaction free energy (kcal/mol) (c). Abbreviations A, S−, S, and W refer to NH3, HSO4−, H2SO4, and H2O, respectively. Calculations were carried out using PW91PW91/6-311++G(3df,3pd) and G3MP2 (a) panel only) at T = 298.15 K and P = 101.3 kPa.

favor of (HSO4−)(H2SO4)m(H2O)n(NH3). While both (HSO4−)(H2SO4)m(H2O)n and (HSO4−)(H2SO4)m(H2O)n(NH3) have a significant growth advantage over both neutral and positively charged binary clusters, the (HSO4−)(H2SO4)m(H2O)n(NH3)

growth by the addition of the sulfuric acid is the most favorable thermodynamically. The affinity of H2SO4 to both (HSO4−)(H2O)n and (HSO4−)(NH3)(H2O)n clusters decreases gradually with the growing 148



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and low level HF ab initio entropies (for unhydrated clusters) or a theoretical thermochemical cycle. Despite the low level of ab initio used in ref 54, the HF/6-31G(d) ab initio entropy changes are in satisfactory agreement with higher level PW91PW91/6311++G(3df,3pd) and G3MP2 methods. Although both theoretical problems and the uncertainties in the experimental data treatment may be causes of these deviations, the existence of the well-known problems associated with derivation of experimental reaction enthalpies from the measured reaction constants suggests that uncertainties in the experimental enthalpies are more likely to be the major source of the observed deviations. 3.2.3. Ammonia Affinity. Being considered as a principal stabilizer or one of the principal stabilizers of binary sulfuric acid−water clusters, ammonia is believed to play an important role in atmospheric nucleation. Table 5 and Figure 9 present the ammonia affinity to negatively charged binary clusters and the comparison of the sulfuric acid and ammonia affinities to clusters of identical composition. As it may be seen from Table 5 and Figure 9, the affinity of ammonia to negatively charged binary clusters containing one sulfuric acid in addition to the bisulfate ion is quite low. However, the attachment of the next sulfuric acid molecule leads to a large (∼5 kcal/mol) jump in the sulfuric acid affinity, indicating the growth in the sulfuric acid affinity with the concentration of the sulfuric acid. This trend is identical to those for neutral and positively charged clusters. This suggests that the ammonia affinity to binary and ternary clusters of different composition and charging state increases with the concentration of the sulfuric acid. The difference in the affinity of H2SO4 and NH3 to clusters of identical composition decreases with the number of sulfuric acid molecules in the clusters, giving us an indication of the strong dependency of the ammonia affinity on the sulfuric acid concentration. The affinity of ammonia is decreasing with the hydration number. This trend is obviously stronger for the clusters containing two sulfuric acid molecules than for those including only one H2SO4 molecule. However, no meaningful conclusion can be made on the basis of the present data and, therefore, further research is needed to verify this dependency. 3.3. Uncertainties in Theoretical Predictions and Experimental Data and Possible Ways of Their Reduction. The comparison of the theoretical high level ab initio CCSD (T)/CBS of Shields et al.59 and DFT PW91PW91/6311++G(3df, 3pd) results obtained in the present and earlier studies with the experimental data for the Gibbs free energy changes indicates a very good agreement. PW91PW91/6-311+ +G(3df, 3pd) results are in very good agreement with CCSD (T)/CBS and ab initio-based G3MP2 methods. As a matter of fact, the PW91PW91/6-311++G (3df,3pd) results for (HSO4−)(H2SO4)0(HSO4−)(H2O)n agree slightly better with the experimental data than CCSD (T)/CBS predictions. Another important detail is that the quality of the PW91PW91/6-311+ +G(3df, 3pd) predictions does not decline with the growing cluster size. This suggests that DFT PW91PW91/6-311+ +G(3df, 3pd) is a viable alternative to higher level ab initio methods in studying large binary and ternary atmospheric prenucleation clusters, for which the higher level computations are prohibitively expensive. On the other hand, application of the unharmonic correction to PW91PW91/6-311++G (3df, 3pd) and CCSD(T)/CBS, which would yield to the further improvement of the agreement of both methods with the experimental data for (HSO4−)(H2SO4)0(HSO4−)(H2O)n, would probably be more beneficial for the CCSD(T)/CBS than for PW91PW91/6-311++G

Table 5. NH3 or H2SO4 Addition Reaction Enthalpies, Entropies, and Gibbs Free Energies at Temperature of 298.15 K and Pressure of 101.3 kPa Obtained at PW91PW91/6-311+ +G(3df,3pd) Level of Theory reaction (H2SO4)(HSO4−) + (NH3) ⇔ (H2SO4) (HSO4−)(NH3) (H2SO4)(HSO4−) + (H2SO4) ⇔ (H2SO4)2(HSO4−) (H2SO4)2(HSO4−) + (NH3) ⇔ (H2SO4)2(HSO4−)(NH3) (H2SO4)2(HSO4−) + (H2SO4) ⇔ (H2SO4)3(HSO4−) (H2SO4)3(HSO4−) + (NH3) ⇔ (H2SO4)3(HSO4−)(NH3) (H2SO4)(HSO4−)(H2O) + (NH3) ⇔ (H2SO4)(HSO4−)(H2O)(NH3) (HSO4−)(H2SO4)(H2O) + (H2SO4) ⇔ (H2SO4)2(HSO4−)(H2O) (H2SO4)(HSO4−)(H2O)2 + (NH3) ⇔ (H2SO4)(HSO4−)(H2O)2(NH3) (HSO4−)(H2SO4)(H2O)2 + (H2SO4) ⇔ (H2SO4)2(HSO4−)(H2O)2 (H2SO4)(HSO4−)(H2O)3 + (NH3) ⇔ (H2SO4)(HSO4−)(H2O)3(NH3) (HSO4−)(H2SO4)(H2O)3 + (H2SO4) ⇔ (H2SO4)2(HSO4−)(H2O)3 (H2SO4)2(HSO4−)(H2O) + (NH3) ⇔ (H2SO4)2(HSO4−)(H2O)(NH3) (HSO4−)(H2SO4)2(H2O) + (H2SO4) ⇔ (H2SO4)3(HSO4−)(H2O) (H2SO4)2(HSO4−)(H2O)2 + (NH3) ⇔ (H2SO4)2(HSO4−)(H2O)2(NH3) (HSO4−)(H2SO4)2(H2O)2 + (H2SO4) ⇔ (H2SO4)3(HSO4−)(H2O)2 (H2SO4)2(HSO4−)(H2O)3 + (NH3) ⇔ (H2SO4)2(HSO4−)(H2O)3(NH3) (HSO4−)(H2SO4)2(H2O)3 + (H2SO4) ⇔ (H2SO4)3(HSO4−)(H2O)3 (H2SO4)3(HSO4−)(H2O) + (NH3) ⇔ (H2SO4)3(HSO4−)(H2O)(NH3) (H2SO4)3(HSO4−)(H2O)2 + (NH3) ⇔ (H2SO4)3(HSO4−)(H2O)2(NH3) (H2SO4)3(HSO4−)(H2O)3 + (NH3) ⇔ (H2SO4)3(HSO4−)(H2O)3(NH3)

ΔH

ΔS

ΔG

−8.09

−26.83

−0.09

−25.0

−36.6

−14.10

−18.53

−38.77

−7.00

−21.00

−35.90

−10.30

−19.21

−39.86

−7.33

−8.76

−25.45

−1.17

−28.02

−42.12

−15.47

−17.73

−40.13

−0.77

−36.28

−58.92

−18.72

−6.46

−20.99

−0.21

−29.36

−41.98

−16.84

−18.36

−41.35

−6.04

−22.11

−42.44

−9.46

−15.21

−29.93

−6.28

−20.12

−37.53

−8.93

−12.55

−24.73

−5.18

−25.58

−42.18

−13.01

−23.72

−35.70

−13.08

−17.02

−32.60

−7.30

−10.62

−23.11

−3.73

hydration number. This trend is opposite to that for the neutral clusters, whose sulfuric acid affinities typically increase with the growing water content. As it may be seen from comparison of Tables 3 and 4 and Figure 8, the difference between the theoretical results of the present study and semi-experimental estimates54 is significantly larger than that between the complete experimental data from the same data set and the present work. The deviation between the PW91PW91/6-311++G(3df,3pd) results and semi-experimental estimates in the sulfuric acid affinity (∼3 kcal/mol on average) is virtually size- and composition-independent. In most cases studied, the composite ab initio-based G3MP2 agrees with PW91PW91/6-311++G(3df,3pd) better than with semi-experimental estimates.54 As it may be seen from Figure 8c, the differences between the theoretical and semi-experimental Gibbs free energies and enthalpies typically correlate well. This is an indication that a large fraction of the uncertainties in the Gibbs free energy are associated with the uncertainties in the enthalpies. This is not surprising because the semi-experimental estimates for the sulfuric affinity in ref 54 were obtained using either the experimental values of enthalpies obtained in earlier studies 149



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Figure 9. Comparison of affinities of the sulfuric acid and ammonia to negatively charged clusters of identical composition. Abbreviations S−, S, and W refer to HSO4−, H2SO4, and H2O, respectively. Calculations were carried out using PW91PW91/6-311++G(3df,3pd) at T = 298.15 K and P = 101.3 kPa. 0 0 ⎡ K ⎤ ΔΗ − T ΔCp ⎡ 1 1 ⎤ ΔCp ⎡ T ⎤ ln⎢ 0 ⎥ = ln⎢ 0 ⎥ ⎢ 0 − ⎦⎥ + ⎣T ⎦ ⎣ ⎣K ⎦ R T R T

(3df,3pd), which replicates the experimental vibrational spectra within the uncertainty in the experimental data using the standard harmonic (RRHOA) approximation. Although the “anharmonicity issue” could be addressed using the scaling factors,61 the scaling factor values are available for small molecules only, and no data for large molecules or clusters were found in the literature. In view of these circumstances, studying the basis-set dependencies of the anharmonic corrections to the Gibbs free energies could become one of the alternative ways of the reduction of uncertainties in demanding ab initio or DFT energy computations. It is quite clear that in the case, when the basis set dependency of the unharmonic correction to energies is weak or moderately weak, unharmonic computations with affordable small basis sets would provide reasonable estimates of the unharmonic corrections to energies computed using large basis sets, for which the costs of unharmonic calculations are prohibitively high. However, further research and systematic benchmarking is needed to address this issue. The situation with the uncertainties in the experimental thermochemical data is even more difficult than that with the uncertainties in theoretical values. The “experimental uncertainties” are not limited to the statistical treatment of scattered experimental data points and precision of the instrumentation only. In this particular case, one can expect significant uncertainties related to the derivation of “experimental” ΔH and ΔS values from the measured reaction constants. As it has been mentioned in the Methods Section and references therein, the main source of uncertainties is the linearization of the van’t Hoff equation used to derive the ΔH and ΔS values. Typically, experimentalists use the linearization of the van’t Hoff equation: ln(K ) =

−ΔΗ 0 ΔS 0 + RT R

(2)

where K is the reaction constant, R is the general gas constant, ΔCp is the constant pressure heat capacity change, T is the temperature, and superscript 0 refers to the standard conditions. This linearization is believed to be the main cause of the large discrepancies between the directly experimental (calorimetric) and indirectly experimental (van’t Hoff) enthalpies (see, e.g., refs 52,60 and references therein). The main issue is that, although both linear and nonlinear fits reproduce the measured reaction constants/changes in the Gibbs free energy with virtually equal accuracy, the pairs of (ΔH, ΔS) given by different fits may differ dramatically. In a number of cases studied here, we found that the deviation between the enthalpies derived from the experimental van’t Hoff plot and ab initio and DFT values is several times larger than the difference between the measured reaction constants/Gibbs free energies and the corresponding theoretical quantities. This is likely an indication of the significant impact of the nonlinearity of the van’t Hoff plot on the indirectly experimental enthalpies. While the impact of the deviations in the ΔH obtained using different fits on the calculated Gibbs free energies is negligible within the experimental temperature range, the impact of these uncertainties far from the experimental temperature interval or in the case when the derived ΔH is used as an independent input parameter may be very large. A possible example of such error propagation can be seen from Figure 8c, where at n = 0 deviations between the theoretical and semi-experimental Gibbs free energies and deviations between the theoretical and semiexperimental enthalpies are very close. This may suggest that the uncertainties in the Gibbs free energies are largely associated with the uncertainties in indirectly derived experimental enthalpies, which were adopted from earlier experimental study. There are three main reasons behind the linearization of the van’t Hoff plot: (a) the plot appears to be linear at the first look; (b) ΔCp is not measured and its value is usually unknown; and (c) fitting the

(1)

instead of the general van’t Hoff equation: 150



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experimental curve using ΔCp and ΔH as fitting parameters is insufficiently reliable due to the large impacts of small drift or noises in the experimental data on the resulting ΔCp and ΔH. This means that the determination of the ΔCp values, which generally vary from cluster to cluster or from reaction to reaction, is critically important for the reduction of uncertainties in the experimental enthalpy and entropy changes derived from the van’t Hoff curve. It is clear that the application of the computational quantum methods could potentially be an efficient way of determining ΔCp with quite high confidence. The ΔCp value would give experimentalists clear ideas about the possible nonlinearity of the van’t Hoff curve and provide justification for its linearization in the case when ΔCp is small or negligible, or for the application of the general form of van’t Hoff’s equation, with the computed ΔCp as an input parameter, in the case when ΔCp is large. Unfortunately, no systematic benchmarks comparing theoretical and experimental values of Cp or ΔCp are available at the present time. However, in the case when the computational quantum methods would appear to be capable of predicting the ΔCp values in satisfactory agreement with experiments at reasonable computational costs, the aforementioned approach can make a significant contribution to the reduction of uncertainties in the experimental thermochemical values derived from the conventional van’t Hoff plot.

changes indicates a very good agreement. PW91PW91/6311++G(3df, 3pd) results are also in very good agreement with CCSD(T)/CBS and ab initio-based G3MP2 methods. Another important detail is that the quality of the PW91PW91/6-311++G(3df, 3pd) predictions does not seem to decline with the growing cluster size. This suggests that DFT PW91PW91/6-311++G(3df, 3pd) is a viable alternative to higher level ab initio methods in studying large binary and ternary atmospheric prenucleation clusters, for which the higher level computations are prohibitively expensive. (e) In a number of cases studied here, we found that the deviation between the enthalpies derived from the experimental van’t Hoff plot and theoretical ab initio/ DFT values is several times larger than the difference between the measured reaction constants/Gibbs free energies and the corresponding theoretical quantities. This is likely an indication of the significant impact of the nonlinearity of the van’t Hoff plot on the indirectly derived experimental enthalpies. The application of the computed values of ΔCp has been proposed as a possible way of the reduction of uncertainties in the experimental enthalpies and entropies derived from measured reaction constants using the van’t Hoff plot. It has been shown that computed ΔCp values would give experimentalists clear ideas about the possible nonlinearity of the van’t Hoff curve and provide justification for its linearization in the case when ΔCp is small or negligible, or for the application of the general form of van’t Hoff’s equation, with the computed ΔCp as an input parameter. However, further research and systematic benchmarking are needed to validate the proposed approach. The reported thermodynamic data cover a wide range of the cluster sizes and compositions, and they can be used for solving a number of important problems related to the thermochemistry, gas-phase chemistry of atmospheric species, nucleation theory, and parametrization of atmospheric nucleation rates.

4. CONCLUSION In the present study, we have investigated 20 classes of (HSO4−)(H2SO4)m(H2O)n and (HSO4−)(H2SO4)m(H2O)n(NH3) clusters and determined their thermodynamic properties. The H2SO4, H2O, and NH3 affinities, the key thermochemical properties controlling the growth of large negatively charged binary and ternary pre-nucleation clusters studied here, and their size and composition dependencies have been investigated. The present study leads us to the following conclusions: (a) No profound effect of ammonia on the hydration free energies is found. The hydration pattern for negatively charged ions is similar to that for neutral and positively charged ions. This suggests that the hydration of binary and ternary clusters of different composition and charging state is largely controlled by the concentration of the sulfuric acid. (b) The presence of ammonia is favorable for the cluster growth by the attachment of H2SO4 in most cases studied here. In most cases, the presence of ammonia leads to a substantial (2−8 kcal/mol) enhancement in the cluster stability. The affinity of H2SO4 to both (HSO4−)(H2O)n and (HSO4−)(NH3)(H2O)n clusters decreases gradually with growing hydration number. This trend is opposite to that for the neutral clusters, whose sulfuric acid affinities typically increase with the growing water content. (c) The affinity of ammonia to negatively charged binary clusters containing one sulfuric acid in addition to the bisulfate ion is quite low. However, the attachment of the next sulfuric acid molecule leads to a large (∼5 kcal/mol) jump in the ammonia affinity, indicating the growth in the ammonia affinity with the concentration of the sulfuric acid. This trend is identical to those for neutral and positively charged clusters. This suggests that the ammonia affinity to binary and ternary clusters of different composition and charging state increases with the concentration of the sulfuric acid. (d) The comparison of DFT PW91PW91/6-311++G(3df, 3pd) results obtained in the present and earlier studies with the experimental data for the Gibbs free energy

■

ASSOCIATED CONTENT

S Supporting Information *

Additional results and discussion, and Cartesian coordinates. This material is available free of charge via the Internet at http:// pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (J.H.), [email protected] (A.B.N.). Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by the U.S. National Science Foundation under grant 0942106 and the National Natural Science Foundation of China (40975073). This work was also supported by the Science Foundation of Chinese Research Academy of Environmental Sciences (Grant 2012-YSKY-15). Additional support was provided by the CRAES Supercomputing Facilities (SGI 4700 super computers).

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