Hydration of Atmospheric Molecular Clusters: Systematic

Article Cite This: J. Phys. Chem. A 2018, 122, 5026−5036

pubs.acs.org/JPCA

Hydration of Atmospheric Molecular Clusters: A New Method for Systematic Configurational Sampling Jens Vive Kildgaard,† Kurt V. Mikkelsen,*,† Merete Bilde,‡ and Jonas Elm*,‡ †

Department of Chemistry, University of Copenhagen, Copenhagen, Denmark Department of Chemistry and iClimate, Aarhus University, Aarhus, Denmark

‡

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S Supporting Information *

ABSTRACT: We present a new systematic configurational sampling algorithm for investigating the potential energy surface of hydrated atmospheric molecular clusters. The algorithm is based on creating a Fibonacci sphere around each atom in the cluster and adding water molecules to each point in nine different orientations. For the sampling of water molecules to existing hydrogen bonds, the cluster is displaced along the hydrogen bond, and a water molecule is placed in between in three different orientations. Generated redundant structures are eliminated based on minimizing the root-mean-square distance of different conformers. Initially, the clusters are sampled using the semiempirical PM6 method and subsequently using density functional theory (M06-2X and ωB97X-D) with the 6-31++G(d,p) basis set. Applying the developed algorithm, we study the hydration of sulfuric acid with up to 15 water molecules. We find that the addition of the first four water molecules “saturate” the sulfuric acid molecule and that they are more thermodynamically favorable than the addition of water molecules 5−15. Using the large generated set of conformers, we assess the performance of approximate methods (ωB97X-D, M06-2X, PW91, and PW6B95-D3) in calculating the binding energies and assigning the global minimum conformation compared to high level CCSD(T)-F12a/VDZ-F12 reference calculations. The tested DFT functionals systematically overestimate the binding energies compared to coupled cluster calculations, and we find that this deficiency can be corrected by a simple scaling factor.

1. INTRODUCTION

contribute significantly to the number concentration of ultrafine particles in polluted areas.9 Evidence exists that sulfuric acid is an important chemical compound in the new particle formation process.10 Other compounds such as bases,11−13 highly oxidized multifunctional organic compounds (HOMs),14−16 and ions17,18 might also be crucial in the initial steps. Sulfuric acid is one of the most hygroscopic substances known and its identified participation in new particle formation has led to several quantum chemical studies targeting the stabilizing effects of water on gas-phase sulfuric acid clusters.19−23 Generally, it has been found that sulfuric acid is primarily hydrated in the gas-phase, but no studies have targeted sulfuric acid hydrated by ten or more water molecules24,25 even though such hydration could represent important steps toward transition to a liquid phase particle. The stabilizing effect of water on sulfuric acid-based clusters have also been studied using computational methods.26−28 The effect of hydration on cluster formation has been identified to depend on the basicity of the base.29,30 Clusters consisting of sulfuric acid and strongly basic compounds such dimethylamine and trimethylamine show little stabilizing effect

Water molecules are ubiquitous in the lower atmosphere of Earth where they exist in solid, liquid, and gaseous states. Water molecules possess a distinct ability to form hydrogen bonds, making them candidates mediators of atmospheric transitions from gas phase to clusters to aerosol particles. The initial steps of cluster formation and the transition of an atmospheric cluster to an aerosol particle are also referred to as new particle formation and growth processes. They remain poorly understood processes, although they have important consequences for the global climate and air quality. Particles formed in nucleation events can grow to sizes of 50 nm or larger in polluted1 as well as pristine2 environments and act as cloud condensation nuclei (CCN) affecting cloud optical properties, lifetime, and thus radiative balance.3 Global modeling studies indicate that in general a significant fraction and as much as half the number concentrations of CCN in the marine boundary layer originate from particles formed via nucleation and growth.4 With respect to air quality, aerosol particles in outdoor air pollution have a documented negative impact on human health5,6 with ultrafine particles having enhanced negative effects.7 The formation of new particles in the atmosphere via nucleation and growth has been observed to happen on local as well as regional scales,8 and nucleation can © 2018 American Chemical Society

Received: March 22, 2018 Revised: May 9, 2018 Published: May 9, 2018 5026

DOI: 10.1021/acs.jpca.8b02758 J. Phys. Chem. A 2018, 122, 5026−5036

Article

The Journal of Physical Chemistry A

Figure 1. (left) Illustration of a Fibonacci sphere with 50 points and (right) a distribution of points around a sulfuric acid molecule (red points are the atoms).

from hydration. However, formation rates of clusters consisting of sulfuric acid and weaker bases such as methylamine and ammonia can be enhanced by approximately 1 order of magnitude by hydration. Obtaining the global free energy minimum cluster structure is a persisting challenge as each additional molecule in the cluster significantly complicates the configurational space. For this reason, water has been neglected in many atmospheric cluster studies or only a few water molecules have been included. Various different methods have been used for sampling atmospheric cluster configurations such as manual sampling using chemical intuition,31−33 molecular dynamics,27,34,35 Monte Carlo,36 basin hopping,37−39 and Random Sampling.40−42 In this paper, we develop a new sampling technique that can systematically sample hydrated atmospheric cluster structures. As a proof-of-concept, we utilize the method to investigate the hydration of a single sulfuric acid molecule with up to 15 water molecules. The identified (H2SO4)1(H2O)1−15 clusters present an ideal test set for assessing approximate methods and add a new dimension as they allow for the analysis of both the global free energy minimum structures as well as the relative energies.

three for each axis, yielding a total of nine different orientations of the water molecule. As water belongs to the point group C2v, the complete 2π rotation along all axes could result in similar orientations. The rotation of the molecule is applied when it has been displaced to its centroid; afterward, the centroid of the water molecule is placed at the points of the Fibonacci sphere around the cluster. Before accepting the placement, it is checked whether any atoms of the water molecule are less than 0.5 Å from any of the atoms in the cluster. The clusters are held together by hydrogen bonds, and as water is able to form two donor and two acceptor hydrogen bonds, it is likely that hydrating water molecules will also make their way and bind inside the original cluster. To create structures with water placed in between already present hydrogen bonds in the cluster, we locate the existing hydrogen bond pairs as oxygen and hydrogen atoms with an interatomic distance of 1.1−2.0 Å. For space to be allowed for the water molecule, the cluster is perturbed by moving all atoms radially away from the midpoint of the hydrogen bond (rM) according to a Gaussian function with the variable being the distance from rM to the atom ri

2. METHODS 2.1. Systematic Hydrate Configurational Sampling. Identifying the lowest free energy conformations of molecular clusters is a challenging task. A new method was developed for adding water molecules to existing unhydrated clusters in a systematic manner. The water molecules are evenly distributed around the cluster using Fibonacci spheres.43 Applying these spheres around the cluster and excluding the points too close to other atoms (≤0.5 Å), one is left with a collection of points shaped as the cluster. An example of a unit Fibonacci sphere with 50 points and a collection of points surrounding a sulfuric acid molecule can se seen in Figure 1. The next step is to place a water molecule at each point on the Fibonacci sphere in different orientations. This can be performed using a rotation matrix that rotates around the x, y and z axes by angles of ϕ, θ, and ψ, respectively, in a righthanded coordinates system. Here, the task is to sample different orientations of the water molecule, and the angles will be evenly spaced between 0 and 2π. We chose to split each angle into

(1)

⎛ |r − r |2 ⎞ r − rM ri′ = ri + αri ̂M exp⎜ − M 2 i ⎟ , ri ̂M = i |ri − rM| σ ⎝ ⎠

where σ describes how far into the cluster the perturbation should be and α is how much space is created. The water molecule is aligned such that one of the OH bonds is aligned reversely in the now stretched hydrogen bond, i.e., water will bridge the former hydrogen bond, creating two new hydrogen bonds. The water molecule is placed with the oxygen atom at the original hydrogen bond midpoint. The water molecule is aligned by rotating around the OH vector in the water molecule and the cluster. Finally, the aligned water molecule is rotated around the former hydrogen bond to have the water orientated in three different orientations (if possible by the distance requirement); otherwise, it will be lowered to 2 or 1 if required. The strength of the developed approach is that it is significantly more systematic than previously applied methods such as sampling using molecular dynamics or random sampling. This allows for a more systematic identification of low lying conformations. Furthermore, the developed code can easily be extended to other atmospherically relevant nucleation 5027


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200 cm−1) and, unless otherwise noted, were calculated at 298.15 K and 1 atm. The quasi-harmonic calculations were performed with the “GoodVibes” python script.67 The sampling technique outlined in section 2.1 generates a large amount of initial conformers (roughly 1000−2000) for each cluster. To narrow these down further, the structures were optimized, and vibrational frequencies were calculated using the semiempirical PM6 method. It was found that calculating the force constants for all steps in the optimization (opt = calcall) drastically reduced the chance that the calculation would terminate with an error. After the PM6 optimization, a Boltzmann sorting was performed with the Gibbs free energies from PM6, where any conformer less present than 1% compared to the ground state conformer at 298.15 K and 1 atm were excluded. The remaining structures were checked in pairs if any redundancies are present, excluding structures with an RMSD < 0.38 Å. This threshold was somewhat arbitrarily chosen from visual inspection of multiple structures and their RMSD. The Boltzmann and similarity sorted structures are then geometry optimized using DFT with the functional M062X and the basis set 6-31++G(d,p). After DFT optimization, new redundancies are excluded, and the obtained cluster structures are finally optimized and vibrational frequencies calculated at the ωB97X-D/6-31++G(d,p) level of theory. The combination of the ωB97X-D with the 6-31++G(d,p) basis set has been shown to be a sufficient level of theory for obtaining the thermal contribution of atmospherically relevant clusters.68,69 2.3. Method Testing. To ensure that our newly developed method is efficiently able to locate global minimum cluster structures, we tested the algorithm on the (H2O)n cluster system with n ≤ 10. The most studied systems are (H2O)n clusters with n ≤ 6, as the water hexamer corresponds to the smallest water cluster with a three-dimensional hydrogen bonding network.70−73 Using our sampling algorithm, we identified 3, 3, 3, 7, 18, 65, 17, 13, and 37 conformations of the (H2O)2−10 clusters, respectively. The obtained cluster structures at the sampling M06-2X/6-31++G(d,p) level of theory were compared to the global minimum cluster structures reported by Temelso et al.74 as depicted in Table 1. These (H2O)2−10 cluster structures have recently been confirmed to be the lowest in binding energy using higher level explicitly correlated coupled cluster methods.75 In addition to identifying numerous low lying free energy conformers, our sampling method is also able to correctly identify the global minimum cluster conformations for all of the (H2O)n clusters with n ≤ 10.

precursors, such as ammonia, sulfuric acid, and so forth. The complete python script for performing the configurational sampling is freely available at Github.44 The initial creation of a large set of molecular structures and optimization of these requires a sorting technique to eliminate redundant structures. An efficient parameter is the root-meansquare distance (RMSD), which given two sets of coordinates A and B, is RMSD =

1 N

2

N

∑

riA

−

riB (2)

i=1

where N is the number of atoms in the molecules, and ri is the spacial coordinates of the ith atom. The RMSD can be minimized by translating and rotating one set of coordinates (the other is held fixed) because the molecules are invariant under these operations. This will lead to the two molecules being superimposed but can also lead to a false RMSD value if the atoms are not ordered identically. To get around this issue, we used a newly published RMSD method by Temelso et al.45 that permutes the indices using the Kuhn−Munkers algorithm.46,47 The developed algorithm for systematic and automated placement of water molecules can be used as a step-by-step method for adding one water molecule at a time to an existing cluster. In each step, the RMSD method can be used to locate and exclude redundant structures. Using this procedure, it is possible to generate a large set of cluster conformations. 2.2. Computational Details. All geometry optimizations and vibrational frequency calculations were performed using the Gaussian0948 program package. PM649 and DFT single point energies were calculated using both the Gaussian09 and Gaussian1650 program packages. For all calculations, the Gaussian09 default values were used to ensure comparability between the results. We have chosen to utilize the M06-2X,51 PW91,52 ωB97X-D,53 and PW6B95-D354 functionals as these exhibit the lowest errors compared to higher level CCSD(T) calculations for the binding energies of atmospheric relevant cluster systems.55−57 Explicitly correlated Coupled Cluster single point energies, CCSD(T)-F12a58−60/VDZ-F12,61 were calculated using MOLPRO version 2012.1.62 The DLPNO− CCSD(T)63,64 single point energies were calculated with ORCA version 4.0.0.65 In the DLPNO calculations, we used the same auxiliary (/C) basis sets as the orbital basis set. In the Hartree−Fock part of the DLPNO calculations, we employed Coulomb and exchange fitting with the same auxiliary (/JK) basis set as the orbital basis set (except for aug-cc-pVDZ, where an aug-cc-pVTZ/JK was used). The cluster binding energies and cluster binding free energies were calculated as ΔE binding = Ecluster −

∑ Emonomer, i i

ΔG binding = Gcluster −

∑ Gmonomer, i i

Table 1. Global Minimum (H2O)n Cluster Structures from the Literature74,75

(3)

(4)

We define the thermal contribution to the binding free energy as ΔGthermal,corr = ΔG binding − ΔE binding

(5)

Thermochemical parameters were calculated using both the harmonic and quasi-harmonic approximations66 (cutoff value of

a

5028

n

global min74−76

#confsa

correct global

2 3 4 5 6 7 8 9 10 17

2Cs 3UUD 4S4 5CYC 6PR 7PR1 8D2d 9D2dd 10PP1 sphere

3 3 3 7 18 65 17 13 37 39

yes yes yes yes yes yes yes yes yes no

Number of conformations obtained using our sampling method. DOI: 10.1021/acs.jpca.8b02758 J. Phys. Chem. A 2018, 122, 5026−5036

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Figure 2. Lowest free energy (H2SO4)1(H2O)1−15 cluster structures calculated at the ωB97X-D/6-31++G(d,p) level of theory. Yellow, sulfur; red, oxygen; and white, hydrogen. Formed H3O+ ions are shown with a green circle.

+G(d,p) level of theory. Figure 2 presents the lowest free energy cluster structures. For most of the clusters, we obtain quite similar (H 2SO4 )1 (H2 O) 1−6 structures as those reported previously.19−23,77 For the different studies, there are minor variations in which minima are identified as the lowest in free energy. This is to be expected as the different studies have used different methods such as PW91/6-311++G(3df,3pd), MP2/ aug-cc-pVDZ, and in our case ωB97X-D/6-31++G(d,p). For the (H2SO4)1(H2O)1−5 clusters, we do not observe proton transfer from sulfuric acid to a water molecule in any of the global minimum structures. In all the clusters with six or more water molecules, a proton transfer is observed, as indicated by the green circle in Figure 2. This observation is consistent with the study (H2SO4)1(H2O)1−6 clusters by Temelso et al.,22 where a proton transfer was only identified in the global free energy structure of the (H2SO4)1(H2O)6 cluster. As the potential energy surfaces can be very complicated, very few studies have addressed sulfuric acid−water clusters with more than six water molecules. Ding and Laasonen studied (H2SO4)1(H2O)6−9 water clusters, where it was found that the fully deprotonated sulfate ion (SO42−) existed as a local free energy minimum for the clusters with 8 or 9 water molecules but in no cases were a fully deprotonated sulfuric acid found in the lowest free energy structure. We extended the number of hydrating water molecules from 9 to 15 and in none of the global minimum clusters do we observe the SO42− ion. We see the first evidence of a sulfate ion being formed as a local minumum in the (H2SO4)1(H2O)14 cluster, but the con-

The global minimum structure of the (H2O)17 cluster has also received significant attention as it corresponds to the first transition from a cluster that has all the atoms at the surface to one that has a fully coordinated water molecule solvated inside the cluster. This is a very difficult sampling case as the transition from the (H2O)16 to the (H2O)17 cluster is accompanied by a large molecular rearrangement of the cluster structure. Using the newly reported global minimum (Boatb) structure of the (H2O)16 cluster by Yoo et al.,76 we ran our sampling algorithm in an attempt to capture the (H2O)17 global minimum structure. Unfortunately, we were unable to obtain the reported global minimum structure but were able to identify a local minimum cluster with binding free energy within 2.8 kcal/mol of the global minimum (at the M06-2X/631++G(d,p) level of theory). For the purpose of our current study, this deficiency does not present any issues, as we build up the (H2SO4)1(H2O)1−15 clusters in a sequential manner, and thus has explicit knowledge about several of the previous cluster structures when adding another water molecule. However, for sampling highly flexible clusters where there is a large rearrangement in the cluster structure from one minimum to another, more than a single initial guess structure will be necessary.

3. RESULTS AND DISCUSSION 3.1. (H2SO4)1(H2O)1−15 Cluster Structures. A total of 243 cluster conformations were identified for the (H2SO4)1(H2O)1−15 cluster systems at the ωB97X-D/6-31+ 5029


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energies compared to the harmonic case. Using the quasiharmonic approximation, the free energy is increased by up to 4.9 kcal/mol for the largest (H2SO4)1(H2O)15 cluster. The increase in free energy due to the quasi-harmonic approximation is directly dependent on the number of water molecules and is only found to vary from 0.3 to 0.6 kcal/mol per water molecule in the clusters. Accounting for higher energy conformers slightly lowers the free energy of the clusters. The effect is found to be a lowering of the free energy by −1.2 kcal/mol or less. We have not studied the effect of anharmonicity as simple scale factors at the ωB97X-D/6-31+ +G(d,p) level of theory are not available. Temelso and coworkers have studied the effect of anharmonicity for pure water (H2O)1−10 clusters74 and (H2SO4)1(H2O)1−2 clusters.22 In both cases, the effect of anharmonicity was found to be a lowering in the free energy by ∼0.4 kcal/mol per water molecule. The binding free energy does not yield much information by itself. A much more useful measure is the stepwise reaction free energies for adding the nth water molecule

formation is 3.9 kcal/mol higher in free energy than the identified lowest free energy conformation. In the (H2SO4)1(H2O)15 cluster, a sulfate ion is formed as a local minumum, but the conformation is found to be 2.5 kcal/mol higher in free energy than the global free energy conformation. 3.2. Free Energies of the (H2SO4)1(H2O)1−15 Clusters. Figure 3 shows the calculated binding free energies of the

ΔGwater,add = ΔGn − ΔGn − 1

The reaction free energies (ΔGwater,add) are not very dependent on which of the three methods (harmonic, quasi-harmonic, or accounting for multiple conformers) is used to obtain the free energy. We find reaction free energies in the range −0.6 to −3.7 kcal/mol (harmonic), −0.5 to −3.9 kcal/mol (accounting for conformers), and −0.4 to −3.9 kcal/mol (quasi-harmonic). In all cases, we find that the first 1−4 water additions are the most thermodynamically favorable and that the addition of water molecules 5 and 12 is found to be the least favorable. Using the larger aug-cc-pVTZ basis set yields a completely different picture of the calculated binding free energies. The first four additions of water molecules are still seen to be thermodynamically favorable, whereas the remaining free energies are seen to level out more. The slope of the free energy for the (H2SO4)1(H2O)1−4 clusters is found to be −1.5 kcal/mol with an R2 = 0.99, and the slope of the free energy for the (H2SO4)1(H2O)5−13 clusters is found to be −0.5 kcal/mol with R2 = 0.81. This clearly illustrates that the first four water molecules show a much stronger interaction with sulfuric acid than the additional water molecules. This effect can easily be understood from the structures shown in Figure 2. The first four water molecules are able to directly interact with the S− OH hydrogen bond donor and the S = O hydrogen bond acceptor groups in sulfuric acid. After the first four water molecules, the sulfuric acid molecule is “saturated”, and the remaining interactions will primarily be between other water molecules. The fact that the addition of water molecule number 5 is found to be least favorable can be understood from the combination of sulfuric acid already being saturated by the first four water molecules and the fact that an H3O+ ion is not formed as in the case of the (H2SO4)1(H2O)6−13 clusters. 3.3. Effect of Temperature on the Free Energies. At lower temperature, the free energies will become more negative. Figure 4 presents the calculated binding free energies of the (H2SO4)1(H2O)1−15 clusters at 298.15, 278.15, 258.15, 238.15, and 218.15 K. These temperatures represent a large range of altitudes from the atmospheric boundary layer all the way up to the tropopause. All calculations were performed at the ωB97X-D/6-31++G(d,p) level of theory using the harmonic approximation. At 298.15 K, the reaction free energy for adding the first water molecule to sulfuric acid is found to be −3.3 kcal/mol.

Figure 3. Calculated binding free energies of the (H2SO4)1(H2O)1−15 clusters using the harmonic approximation (black), quasi-harmonic approximation (red), and accounting for multiple conformers (green) at the ωB97X-D/6-31++G(d,p) level of theory. The calculated free energies of the (H2SO4)1(H2O)1−13 clusters using the harmonic approximation (blue) at the ωB97X-D/aug-cc-pVTZ level of theory. All calculations were performed at 298.15 K and 1 atm.

(H2SO4)1(H2O)1−15 clusters at the ωB97X-D/6-31++G(d,p) level of theory with the harmonic and quasi harmonic approximations (200 cm−1 cutoff). Using the larger aug-ccpVTZ basis set, it was not possible to converge all the conformers of the (H2SO4)1(H2O)14−15 clusters; hence, the data have been excluded in Figure 3. Taking higher energy conformers into consideration might be an important contribution to the free energy. However, very few studies do this as identifying low lying conformers is rather difficult due to the complicated potential energy surfaces of hydrogen bonded molecular clusters. As our developed sampling technique generates a large set of conformers, it is easy to investigate how they influence the binding free energies. The multiconformer binding free energy (ΔGmulti‑conf) is given by78 ⎛ ⎛ −ΔGn ⎞⎞ ⎟⎟ ΔGmulti‐conf = −RT ln⎜⎜∑ exp⎜ ⎝ RT ⎠⎟⎠ ⎝ n

(6)

⎛ ⎛ −ΔGrel, n ⎞⎞ =−ΔG0 − RT ln⎜⎜∑ exp⎜ ⎟⎟⎟ ⎝ RT ⎠⎠ ⎝ n

(7)

where ΔG0 is the binding free energy of the lowest free energy conformer, ΔGn is the binding free energy of the nth conformer, and ΔGrel,n is the binding free energy of the nth conformer relative to the lowest free energy conformer. The calculations are performed at 298.15 K and 1 atm. At the ωB97X-D/6-31++G(d,p) level of theory, the binding free energy is seen to be more or less monotonically lowered as a function of the number of water molecules. Employing the quasi-harmonic approximation consistently yields higher free 5030


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reproduce the binding energies (ΔEbind) as well as the conformer relative energies (ΔErel). Using the ωB97X-D/631++G(d,p) optimized geometries, we were able to calculate CCSD(T)-F12a/VDZ-F12 single point energies on the (H2SO4)1(H2O)1−9 clusters. This gives a subset of a total of 139 clusters. On this test set, we calculated the single point energies using the DLPNO−CCSD(T) method with the augcc-pVDZ (avdz) and aug-cc-pVTZ (avtz) basis sets. We also tested the LoosePNO, NormalPNO, and TightPNO settings.79 Figure 5 shows the correlation between the calculated relative energies for the different DLPNO-CCSD(T) methods plotted against the CCSD(T)-F12a/VDZ-F12 results. It is seen that using an aug-cc-pVDZ basis set increasing the PNO threshold does not improve the correlation with the CCSD(T)-F12a/VDZ-F12 results. We observe a much better improvement by increasing the basis set size to aug-cc-pVTZ. For the aug-cc-pVTZ results, all three PNO settings yield a good correlation with the CCSD(T)-F12a/VDZ-F12 results, but there is an especially good correlation when NormalPNO or TightPNO is used. Figure 6 shows the correlation between the calculated relative energies for the different DFT methods (PW91, ωB97X-D, and PW6B95-D3) plotted against the CCSD(T)F12a/VDZ-F12 results. The result using the M06-2X functional and for may-cc-pVnZ and aug-cc-pVnZ with n = D,T are shown in the Supporting Information. In all cases, a slight improvement is seen in the correlation when increasing the basis set size from double-ζ to triple-ζ. However, there is little difference in which type of basis set is used (i.e., pople, may-cc-pVnZ, or aug-cc-pVnZ). The relative energies depend much more on the choice of DFT functional than the basis set utilized. This is a very encouraging result as it implies that sampling can safely be performed using a smaller basis set as long as an appropriate method is chosen. In general, the DFT functionals perform in the following order: PW91
may-cc-pVnZ > aug-cc-pVnZ with n being either a double- or triple-ζ basis set. The results from the PW6B95-D3 functional follow an opposite pattern for the triple-ζ basis sets. The PW6B95-D3/6-311++G(3df,3pd) level of theory exhibits a low MAE of 0.2 kcal/mol compared to the CCSD(T)-F12a/VDZF12 reference results. These values are very similar to the values 5033


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(ΔErel) compared to a high level CCSD(T)-F12a/VDZ-F12 reference. We find that DFT results with a double-ζ basis set exhibit good correlation with the CCSD(T)-F12a/VDZ-F12 results for the relative conformer energies. For all DFT methods, we find huge errors in binding energies even when the aug-cc-pVTZ basis set is employed. These findings indicate that DFT with a small 6-31++G(d,p) basis set is suitable for initial screening purposes to narrow down the amount of conformers and should not exclude any important low lying energy conformations as long as a suitable cutoff value is chosen. For the system studied here, a cutoff value of ∼3 kcal/mol should capture all conformations. For the single point energies, DLPNO-CCSD(T)/aug-cc-pVTZ appears to be the best choice for accurate binding energies in good agreement with CCSD(T)-F12a/VDZ-F12 results. We intend to expand the newly developed algorithm such that other small molecules can be added to an existing cluster, and thereby, we will be able to add small inorganic and organic bases to the cluster along with pollutants such as NOx. The generated clusters will form the starting structures for investigating chemical reactions within and on the outside of the cluster.

we calculated the scaled binding energies on the full set of (H2SO4)1(H2O)1−15 clusters, where CCSD(T)-F12a/VDZ-F12 results are not available for clusters with ten or more water molecules. Figure 9 shows the calculated binding energies using

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Figure 9. Calculated electronic binding energies (kcal/mol) of the lowest free energy (H2SO4)1(H2O)1−15 cluster structures using ωB97X-D/6-31++G(d,p) (DFT), DLPNO−CCSD(T)/aug-cc-pVTZ (DLPNO), and CCSD(T)-F12a/VDZ-F12 (F12). Scaled DFT refers to ωB97X-D/6-31++G(d,p) binding energies that has been scaled by a factor of 0.85. Note that the F12, DLPNO, and scaled DFT results are coinciding.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.8b02758. Correlation between relative energies of the PW91, M062X, ωB97X-D and PW6B95-D3 functionals plotted against CCSD(T)-F12a/VDZ-F12 results, errors normalized to the number of water molecules, and derived scaling factors (PDF) xyz files of the (H2SO4)1(H2O)n cluster structures at the ωB97X-D/6-31++G(d,p) level of theory and the (H2O)n clusters at the M06-2X/6-31++G(d,p) level of theory (ZIP)

ωB97X-D/6-31++G(d,p) (DFT), DLPNO−CCSD(T)/augcc-pVTZ (DLPNO), and CCSD(T)-F12a/VDZ-F12 (F12), and the ωB97X-D/6-31++G(d,p) energies have been scaled by a factor of 0.85 (scaled DFT). As also indicated in Figure 7, the pure DFT binding energies are clearly overestimating the binding energies compared to F12 results. It is seen that the F12, DLPNO, and scaled DFT results all agree very well for the binding energies of (H2SO4)1(H2O)1−9 clusters. For the clusters consisting of ten or more water molecules, the DLPNO and scaled DFT results yield more or less identical binding energies. This suggests that both DLPNO-CCSD(T)/aug-cc-pVTZ and ωB97X-D/6-31+ +G(d,p) scaled by a factor of 0.85 yield results in good agreement with CCSD(T)-F12a/VDZ-F12 and can safely be used to predict accurate binding energies of (H2SO4)1(H2O)n clusters. This finding might be very specific to the studied system at hand, and whether the derived scaling factors can be applied for other systems is unknown. However, previous results indicate that one should be careful about applying scaling factors to unknown systems.80

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Phone: +45 28938085. ORCID

Jens Vive Kildgaard: 0000-0002-5749-2953 Kurt V. Mikkelsen: 0000-0003-4090-7697 Jonas Elm: 0000-0003-3736-4329 Notes

The authors declare no competing financial interest.

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4. CONCLUSIONS We have developed a new systematic configurational sampling technique to sample hydrate cluster structures. Using the new algorithm we study sulfuric acid−water clusters with up to 15 water molecules. We find that the addition of the first four water molecules are more thermodynamically favorable than the remaining 5−15. This is caused by the fact that the sulfuric acid hydrogen bond donor and acceptor groups are “saturated” by interactions with four water molecules. We use the generated test set of conformers to assess the performance of approximate methods (PW91, M06-2X, ωB97X-D, and PW6B95-D3) in calculating the binding energies (ΔEbind) and correctly assigning the relative energies

ACKNOWLEDGMENTS J.E thanks the Villum foundation for financial support, and K.V.M. thanks University of Copenhagen for financial support. We thank the Danish e-Infrastructure Cooperation (DeiC) for computational resources.

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