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. 2020 Mar 23;5(13):7601–7612. doi: 10.1021/acsomega.0c00436

Assessment of the DLPNO Binding Energies of Strongly Noncovalent Bonded Atmospheric Molecular Clusters

Gunnar Schmitz †, Jonas Elm ‡,*
PMCID: PMC7144154  PMID: 32280904

Abstract

graphic file with name ao0c00436_0006.jpg

This work assesses the performance of DLPNO-CCSD(T0), DLPNO-MP2, and density functional theory methods in calculating the binding energies of a representative test set of 45 atmospheric acid–acid, acid–base, and acid–water dimer clusters. The performance of the approximate methods is compared to high level explicitly correlated CCSD(F12*)(T)/complete basis set (CBS) reference calculations. Out of the tested density functionals, ωB97X-D3(BJ) shows the best performance with a mean deviation of 0.09 kcal/mol and a maximum deviation of 0.83 kcal/mol. The RI-CC2/aug-cc-pV(T+d)Z level of theory severely overpredicts the cluster binding energies with a mean deviation of −1.31 kcal/mol and a maximum deviation up to −3.00 kcal/mol. Hence, RI-CC2/aug-cc-pV(T+d)Z should not be utilized for studying atmospheric molecular clusters. The DLPNO variants are tested both with and without the inclusion of explicit correlation (F12) in the wavefunction, with different pair natural orbital (PNO) settings (loosePNO, normalPNO, and tightPNO) and using both double and triple zeta basis sets. The performance of the DLPNO-MP2 methods is found to be independent of PNO settings and yield low mean deviations of −0.84 kcal/mol or below. However, DLPNO-MP2 requires explicitly correlated wavefunctions to yield maximum deviations below 1.40 kcal/mol. For obtaining high accuracy, with maximum deviation below ∼1.0 kcal/mol, either DLPNO-CCSD(T0)/aug-cc-pVTZ (normalPNO) calculations or DLPNO-CCSD(T0)-F12/cc-pVTZ-F12 (normalPNO) calculations are required. The most accurate level of theory is found to be DLPNO-CCSD(T0)-F12/cc-pVTZ-F12 using a tightPNO criterion which yields a mean deviation of 0.10 kcal/mol, with a maximum deviation of 0.20 kcal/mol, compared to the CCSD(F12*)(T)/CBS reference.

1. Introduction

Aerosol particles have various effects on the global climate. Particles deposited in the atmosphere can back-scatter sun light leading to a reduced warming of the Earth surface.1 Through uptake of water vapor, aerosol particles can grow to larger sizes and thereby act as cloud condensation nuclei (CCN). CCN particles influence cloud formation and properties for instance by leading to a change in cloud albedo and lifetime.2 A large source of CCN (up to 50%)3 is from new particle formation (NPF) from gas phase vapors, but high uncertainties remain about which vapors are participating in the process and what the exact mechanism is. Currently, the formation of strongly noncovalently bonded molecular clusters is believed to be the initial stage for forming new particles in the atmosphere.4 Sulfuric acid is considered an essential component in the NPF process over land,5 but additional stabilizing molecules are required to form new particles. Atmospheric bases such as ammonia,6,7 alkyl-monoamines,8−14 alkyl-diamines,15−17 and organics18−21 have been shown to significantly enhance the NPF process. For instance, Almeida et al.22 showed that trace amounts of dimethylamine (few pptv) can enhance NPF rates 3 orders of magnitude compared to ammonia. The direct involvement of organics in NPF is less certain, and in general, it is believed that highly oxygenated molecules23,24 formed from autoxidation25 of VOCs are involved in the process. Quantum chemical results have indicated that multicarboxylic acids might be the most likely organic molecules to efficiently form clusters with sulfuric acid.26 However, it is more probable that the multicarboxylic acids form new particles by themselves27 and most likely pure organic nucleation requires ionic species.28 Counterintuitively, recent evidence suggests that organics participating in nucleation actually can lead to an increase in radiative forcing, thus heating the global climate.29,30

The initial steps in the cluster formation process leading to new particles is difficult to measure using experimental techniques. State-of-the-art instrumentation such as chemical ionization atmospheric pressure interface mass spectrometry31 can yield insight into charged cluster compositions. Currently, there are no experimental techniques that measure the composition of neutral clusters. Modelling the kinetics of cluster formation based on quantum chemical calculations can elucidate which clusters can grow into aerosol particles under atmospheric conditions. The evaporation rate of a given cluster can be calculated as32

1. 1

Here, βi,j is the collision coefficient and ΔGi+j is the binding free energy of the cluster. The exponential dependence on the free energy implies that it must be calculated using very accurate methods to obtain reliable evaporation rates. It has been shown that the electronic binding energy contribution to the binding free energy is the largest source of errors in cluster formation studies,33 while the thermal contribution to the free energy only contributes a lesser extent to the errors.34 Highly accurate methods such as coupled cluster singles and doubles with perturbative triples (CCSD(T)) or Møller–Plesset second-order perturbation theory (MP2) close to the complete basis set limit are desired for obtaining reliable results. Unfortunately, because of the scaling of these methods, they can only be applied to relatively small cluster systems, and careful assessment of approximate methods is required to ensure that reliable results can be obtained when studying large clusters. In the recent years, several advances have been made in allowing more accurate calculations on larger systems. The emergence of explicitly correlated methods (F12)35−37 greatly reduces the basis set convergence problem of CCSD(T) methods.38 For instance, using explicitly correlated wavefunctions, results up to two basis set cardinal numbers higher can be expected compared to conventional CCSD(T).39 Several linear scaling approaches have also been developed, which alleviates the steep scaling of the CCSD(T) or MP2 methods with respect to system size. These approaches usually exploit the local nature of dynamic electron correlation and can loosely be divided into two categories. In the first category, the system is partitioned into fragments for which conventional calculations are carried out. The total correlation energy is then obtained as a sum over the fragment energies.40−50 In the second category, the system is treated as a whole, but electron correlation is restricted to local domains or sparsity is rigorously exploited.51−59 We use DLPNO methods, which fall into the second category. In the DLPNO or other pair natural orbital (PNO)-based methods for each pair of occupied orbitals, a private compressed set of virtual orbitals is constructed by means of PNOs60−62 to describe the dynamic correlation. The required number of PNOs per pair quickly becomes independent of the system size and allows linear scaling algorithms if combined with pair truncation schemes.

The DLPNO-CCSD(T0) method has shown excellent performance compared to the GMTKN55 superset63 and is applicable on atmospheric cluster systems ranging up to 10 molecules,64 which is completely out of reach using canonical coupled cluster methods. In recent years, DLPNO-CCSD(T0) has gained popularity in cluster formation studies and has been applied to large cluster involving sulfuric acid–bases,65−71 methanesulfonic acid–bases,72−74 and organics.75−77 Recent development of the explicitly correlated versions of the DLPNO methods78,79 represents an attractive choice of the method for studying the binding energies of atmospheric molecular clusters. However, to date, the performance of these methods has not been systematically assessed for application in cluster formation studies.

In this paper, we assess the performance of DLPNO-MP2, DLPNO-CCSD(T0), and density functional theory (DFT) methods in calculating the binding energies of atmospherically relevant molecular clusters. The performance is assessed using a representative test set of 45 cluster formation reactions, and the calculations are compared to high-level explicitly correlated CCSD(F12*)(T)1 calculations in the complete basis set limit. CCSD(F12*) shown in refs80−82 was found to be the closest approximation to the CCSD(F12)83,84 model for the calculation of reaction and atomization energies as well as the computation of harmonic frequencies as compared to other approximate explicitly correlated CCSD variants. We present a hierarchy of methods that yield the lowest deviations while being applicable to large cluster systems.

2. Results and Discussion

2.1. Cluster Test Set

As a representative test set for interactions in atmospheric molecular clusters, we evaluate the binding energetics of 45 dimer formation reactions involving organic acids [formic (fa) and acetic (aa)], inorganic acids [sulfuric (sa), methanesulfonic (msa), and nitric (na)], bases [ammonia (a), methylamine (ma), dimethylamine (dma), trimethylamine (tma), and ethylenediamine (eda)], and water (w). In particular, we look at the following cluster formation reactions

2.1. R1
2.1. R2
2.1. R3

These reactions represent some of the essential interactions in atmospheric cluster formation, and all combinations of the studied acids and bases are calculated in reactions R1–R3. We have excluded base–base interactions and interactions between bases and water as these interactions are very weak and not of relevance for cluster formation. Figure 1 presents a subset of the studied cluster structures with methanesulfonic acid used as an example for the acid in R1–R3.

Figure 1.

Figure 1

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Molecular structures of the clusters involving methanesulfonic acid (msa). All cluster structures have been optimized at the ωB97X-D/aug-cc-pVTZ level of theory. Color coding: brown = carbon, red = oxygen, blue = nitrogen, yellow = sulfur, and white = hydrogen.

For the (msa)1(ma)1, (msa)1(dma)1, (msa)1(tma)1, and (msa)1(eda)1 clusters, a proton is transferred from methanesulfonic acid to the base. The remaining clusters are held together by hydrogen bonded interactions.

2.2. Extrapolation and Choice of Reference

The binding energies of reactions R1–R3 of the clusters can be calculated as follows

2.2. 2

It should be noted that this is the pure electronic contribution and no thermal effects are taken into consideration. However, the thermal contribution only enters the calculations as a constant as the same geometries are utilized, and thus deviations in the binding energies are identical to deviations in the binding Gibbs free energies. We will use two schemes to extrapolate the CCSD(F12*)(T) energies to the complete basis set (CBS) limit based on the results in the cc-pVDZ-F12 and cc-pVTZ-F12 basis. One scheme was proposed by Friedrich85 as a composite scheme

2.2. 3

where EHF+CABSQZ-F12 is the HF energy including a CABS singles correction86 for the remaining HF + CABS basis set error at the HF level computed in the QZ-F12 basis, EMP2-F12 is the MP2-F12/QZ-F12 correlation energy, and ΔECCSD(F12*)TZ-F12 is a higher-order correction in the TZ-F12 basis defined as

2.2. 4

and E(T)CBS(23) is a Schwenke87 type extrapolation for the triples correction

2.2. 5

with F = 1.431442 to avoid that the triples correction limits the accuracy.88

In the second scheme, we use a Schwenke type extrapolation for the CCSD(F12*) correlation energy instead of estimating it from the MP2-F12 energy. Again, we need a different extrapolation for the CCSD(F12*) correlation energy and the triples correction because they show a different convergence to the CBS limit. In this scheme, we calculate the energy as

2.2. 6

where ECCSD(F12*)CBS(23) is given as

2.2. 7

Here F = 1.529817, where the parameter F is taken from Hill et al.88 Using the two extrapolation schemes, the difference of ECBS and ECBS(23) can be used to estimate the remaining basis set incompleteness error. Calculating the binding energies for the entire test set of 45 clusters, we find a mean signed deviation (MSD) of 0.06 kcal/mol between the ΔECBS and ΔECBS(23) binding energies. This implies that the results are not very sensitive to which of the two extrapolation produces that is applied, and the results can be considered quite close to the complete basis set limit. We have chosen to use the ECBS extrapolation procedure in eq (R2) as a reference in the following sections.

2.3. Statistics

We calculate the MSD, mean absolute deviation (MAD), and root mean square deviation (RMSD) as follows

2.3. 8
2.3. 9
2.3. 10

Here x is the reference CCSD(F12*)(T)/CBS binding energy values, and xi is the corresponding DLPNO-CCSD(T0), DLPNO-MP2, or DFT binding energies. A negative MSD value means that the method is overbinding (i.e., giving too stable clusters), and a positive value means that the method is underbinding.

2.4. DLPNO-MP2 Binding Energies

The binding energies of the 45 cluster reactions have been calculated using DLPNO-MP2 methods. Table 1 presents the deviations in the DLPNO-MP2 calculated binding energy using double/triple zeta basis sets, with and without explicit correlation (F12) and different PNO settings (loose, normal, and tight) relative to CCSD(F12*)(T)/CBS calculations. Figure 2 shows the corresponding box-and-whiskers plots (minimum, first quartile, median, third quartile, and maximum) of the calculated deviations.

Table 1. MSD, MAD, Maximum Deviation (Max D), and RMSD of the Calculated DLPNO-MP2 Binding Energies Compared to CCSD(F12*)(T)/CBS Calculationsa.

method MSD MAD Max D RMSD
DLPNO-MP2
ADZ—loose –0.65 0.82 –3.09 1.12
ADZ—normal –0.80 0.91 –3.42 1.26
ADZ—tight –0.84 0.93 –3.47 1.29
ATZ—loose –0.62 0.64 –1.74 0.79
ATZ—normal –0.77 0.78 –1.99 0.94
ATZ—tight –0.81 0.82 –2.05 0.98
DLPNO-MP2-F12
DZ-F12—loose 0.04 0.55 1.37 0.63
DZ-F12—normal –0.03 0.57 1.40 0.66
DZ-F12—tight –0.10 0.58 1.33 0.66
TZ-F12—loose –0.15 0.51 1.19 0.62
TZ-F12—normal –0.17 0.51 1.19 0.62
TZ-F12—tight –0.21 0.51 –1.21 0.63
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a

All values are in kcal/mol.

Figure 2.

Figure 2

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Distribution of the deviations between the binding energies calculated using DLPNO-MP2 with double and triple zeta basis sets compared to CCSD(F12*)(T)/CBS calculations. The DLPNO calculations are performed both without and with explicit correlation (F12).

In all cases, the DLPNO-MP2 performance appears to be more or less independent of the PNO truncation settings. Without using explicit correlation, the DLPNO-MP2 method slightly overpredicts the binding energies (i.e., too negative) making the clusters slightly too stable. Using double zeta basis sets, the MSDs lie in the range −0.65 to −0.84 kcal/mol. However, the maximum deviations are devastatingly high with values up to −3.47 kcal/mol. Using triple zeta basis sets, the MSDs lie in a similar range of −0.62 to −0.81 kcal/mol but with significantly lower maximum deviations between −1.74 and −2.05 kcal/mol.

Using explicitly correlated wavefunctions, the deviations are seen to be centered around zero, with maximum deviations below 1.40 kcal/mol. The results are more or less independent of the choice of basis set and the PNO settings. DLPNO-MP2-F12 calculations using either a DZ-F12 or TZ-F12 basis set is seen to be an excellent choice that leads to low errors compared to the CCSD(F12*)(T) complete basis set limit reference. Hence, DLPNO-MP2-F12/VDZ-F12 calculations might be an efficient choice to narrow down the number of relevant conformers during screening processes where many conformations are present.

2.5. DLPNO-CCSD(T0) Binding Energies

Table 2 presents the deviations in the DLPNO-CCSD(T0)-calculated binding energy using double/triple zeta basis sets, with and without explicit correlation (F12) and different PNO settings (loose, normal and tight) relative to the high level CCSD(F12*)(T)/CBS calculations. Iterative perturbative triple excitations (T) were also tested. As a comparison, RI-CC2/aug-cc-pV(T+d)Z calculations are also shown because this level of theory has been routinely applied to study atmospheric molecular clusters.32,89−92Figure 3 shows the corresponding box-and-whisker plots (minimum, first quartile, median, third quartile, and maximum) of the calculated deviations.

Table 2. MSD, MAD, Maximum Deviation (Max D), and RMSD of the Calculated DLPNO-CCSD(T0) Binding Energies Compared to CCSD(F12*)(T)/CBS Calculationsa.

method MSD MAD Max D RMSD
DLPNO-CCSD(T0)
ADZ—loose 0.57 0.74 2.21 0.91
ADZ—normal 0.04 0.61 –2.30 0.76
ADZ—tight –0.31 0.57 –2.47 0.77
ATZ—loose 0.49 0.55 1.83 0.71
ATZ—normal 0.02 0.26 1.03 0.34
ATZ—tight –0.33 0.42 –1.01 0.48
DLPNO-CCSD(T)
ADZ—loose 0.64 0.78 2.23 0.95
ADZ—normal 0.05 0.62 –2.16 0.76
ADZ—tight –0.34 0.57 –2.40 0.77
ATZ—loose 0.54 0.59 1.84 0.73
ATZ—normal 0.03 0.25 0.94 0.32
ATZ—tight –0.38 0.43 –1.01 0.50
DLPNO-CCSD(T0)-F12
DZ-F12—loose 1.74 1.74 2.86 1.79
DZ-F12—normal 1.07 1.07 1.75 1.13
DZ-F12—tight 0.51 0.51 1.16 0.59
TZ-F12—loose 1.08 1.08 1.97 1.15
TZ-F12—normal 0.47 0.47 0.84 0.52
TZ-F12—tight 0.10 0.11 0.20 0.12
RI-CC2
A(T+d)Z –1.31 1.33 –3.00 1.58
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a

All values are in kcal/mol.

Figure 3.

Figure 3

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Distribution of the deviations between the binding energies calculated using DLPNO-CCSD(T0) with double and triple zeta basis sets compared to CCSD(F12*)(T)/CBS calculations. The DLPNO calculations are performed both without and with explicit correlation (F12).

There is very little difference in the statistics between applying (T0) and (T) triples corrections. Hence, in the following, only the (T0) correction will be discussed. Using double zeta basis sets, the explicitly correlated calculations clearly follow a loosePNO > normalPNO > tightPNO pattern related to both the MSD and the RMSD. The maximum deviations are also seen to follow this pattern, with values of 2.86, 1.75, and 1.16 kcal/mol for loose, normal, and tight PNO settings, respectively. The maximum deviations are obtained for the (msa)1(tma)1, (msa)2, and (msa)1(ma)1 clusters, respectively.

The pattern is less clear for the nonexplicitly correlated calculations using double zeta basis sets. A slight improvement is seen when changing from loosePNO to normalPNO settings. However, when changing from normalPNO to tightPNO settings, a slight deterioration is observed. This is caused by an efficient cancellation of errors using normalPNO (MSD = 0.04 kcal/mol, MAD = 0.61 kcal/mol) compared to a more systematic overestimation using tightPNO (MSD = −0.31 kcal/mol, MAD = 0.57). Interestingly, the maximum deviations follow the reverse order of the PNO criteria, with values of 2.21, 2.30, and −2.47 kcal/mol for loose, normal, and tight PNO settings, respectively. The maximum deviations are found for the (msa)1(ma)1, (na)1(tma)1, and (na)1(tma)1 clusters, respectively. This indicates that DLPNO-CCSD(T0)/ADZ using a normalPNO setting comprises a good compromise between accuracy and efficiency for the test set at hand; however, large deviations up to −2.47 kcal/mol can be expected.

Utilizing triple zeta basis sets, the DLPNO binding energies are significantly improved. Without using explicit correlation, the pattern in the MSD for the triple zeta basis sets is similar to the double zeta basis sets, that is, loosePNO > tightPNO > normalPNO. The DLPNO-CCSD(T0)/ATZ level of theory using a normalPNO criterion comprises a good compromise between accuracy and efficiency that can be applied to relatively large atmospheric molecular clusters. It has a shallow distribution directly centered around zero, and a maximum deviation of 1.03 kcal/mol is obtained for the (msa)1(ma)1 cluster.

Using triple zeta basis sets, the explicitly correlated calculations also clearly follow a loosePNO > normalPNO > tightPNO pattern. The maximum deviations are also seen to follow this pattern, with values of 1.97, 0.84, and 0.20 kcal/mol for loose, normal, and tight PNO settings, respectively. The maximum deviations are found for the (msa)1(dma)1, (sa)1(aa)1, and (sa)1(na)1 clusters, respectively. Employing DLPNO-CCSD(T0)-F12/TZ-F12 using a tightPNO criterion, an excellent agreement with the reference CCSD(F12*)(T)/CBS calculations is observed, with a MSD of 0.10 kcal/mol and a maximum deviation of 0.20 kcal/mol. The explicitly correlated methods utilizing one higher basis set cardinal number show more or less same performance as tightening the PNO criteria. For instance, DLPNO-CCSD(T0)-F12/DZ-F12 with a tightPNO yields similar performance as DLPNO-CCSD(T0)-F12/TZ-F12 with a normalPNO setting.

The RI-CC2/aug-cc-pV(T+d)Z level of theory significantly overpredicts the binding energies of the clusters (i.e., making them too negative and hence too stable). A large MSD of −1.31 kcal/mol, with a maximum deviation up to −3.00 kcal/mol, is obtained. As RI-CC2/aug-cc-pV(T+d)Z is more computationally demanding than DLPNO and leads to huge errors, it should not be utilized for studying atmospheric molecular clusters.

2.6. DFT Binding Energies

When the DLPNO wavefunction methods become too expensive one will have to revert to DFT. Previously, we showed that DFT functionals that relied on the B9793 functional had the lowest errors compared to CCSD(T) complete basis set estimates.94 Here, we test a range of functionals that are built on the B97 functional (ωB97,95 ωB97X,95 ωB97X-D3,96 ωB97X-D3(BJ)97 B97M-V,98 B97M-D3(BJ),97 ωB97M-V,99 ωB97X–V,100 and ωB97M-D3(BJ)97), as well as several other popular functionals (PW91,101 PBE0,102−104 M06-2X,105 PW6B95,106 and B3LYP107,108). All DFT calculations have been performed with a def2-QZVPPD basis set, which has been shown to be large enough to suppress potential basis set superposition errors.109Table 3 presents the calculated deviations in the binding energies between the different density functionals and the CCSD(F12*)(T)/CBS reference calculations. Figure 4 shows the corresponding box-and-whiskers plots (minimum, first quartile, median, third quartile, and maximum) of the calculated deviations.

Table 3. MSD, MAD, Maximum Deviation (Max D), and RMSD of the Calculated DFT Binding Energies Compared to CCSD(F12*)(T)/CBS Resultsa.

method MSD MAD Max D RMSD
ωB97 –0.32 0.61 –1.41 0.73
ωB97X –0.04 0.62 1.91 0.76
ωB97X-D3 0.01 0.36 1.22 0.49
ωB97X-D3(BJ) 0.09 0.20 0.83 0.29
ωB97X-V 0.09 0.35 1.35 0.50
ωB97M-V 0.39 0.47 2.07 0.74
ωB97M-D3(BJ) 0.39 0.40 1.57 0.59
B97M-V 1.48 1.48 5.22 2.01
B97M-D3(BJ) 1.29 1.29 4.30 1.68
PW91 0.82 1.12 4.90 1.60
PBE0 1.11 1.17 4.76 1.66
M06-2X –0.09 0.65 2.38 0.85
PW6B95 2.22 2.22 5.51 2.51
PW6B95-D3 0.69 0.71 2.55 0.95
PW6B95-D3(BJ) 1.04 1.04 3.22 1.26
B3LYP 2.80 2.80 6.74 3.15
B3LYP-D3 –0.48 0.60 –1.25 0.65
B3LYP-D3(BJ) –0.50 0.56 –1.18 0.65
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a

All values are in kcal/mol.

Figure 4.

Figure 4

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Deviations between the binding energies calculated using DFT with a def2-QZVPPD basis set compared to CCSD(F12*)(T)/CBS calculations.

There is a clear improvement in the binding energies, when going through the hierarchy ωB97 < ωB97X < ωB97X-D3 < ωB97X-V < ωB97X-D3(BJ). Utilizing the ωB97M-V and ωB97M-D3(BJ) functionals yield very low MSD and MAD values, with a slight increase in the maximum deviations compared to ωB97X-D3(BJ). Without applying the long-range correction for these functionals [i.e. B97M-V and B97M-D3(BJ)], the deviations become substantially larger with maximum deviations up to 5.22 kcal/mol. The PW91 and PBE0 functionals also yield low MSD and MAD, but exhibit large maximum deviation of 4.90 and 4.76 kcal/mol, respectively, both in the case of the (msa)1(tma)1 dimer. The M06-2X functional performs fairly well, with a MSD of −0.09 kcal/mol, a MAD of 0.65 kcal/mol, and a maximum deviation of 2.38 kcal/mol. Low MSD and MAD is also found for the PW6B95 functionals, but with large maximum deviations in the range 2.55–5.51 kcal/mol depending on the inclusion of empirical dispersion. As also stated in previous studies,33,110 the B3LYP functional significantly underbinds, leading to a devastating maximum deviation of up to 6.74 kcal/mol. Employing dispersion corrections alleviate this deficiency yielding low, systematic deviations with maximum deviations of up to −1.25 kcal/mol. Overall, with the correct choice of functional, it is possible to obtain low MSD and MAD, but at a cost of maximum deviations between 1 and 2 kcal/mol.

2.7. Timings and Overall Recommendations

Besides the errors arising from applying less accurate methods, it is very important how long time the calculations actually take to run. Figure 5 shows the relative computational time of different levels of theory compared to PW91/aug-cc-pVTZ calculations. The computational time for the PW91 functional was 200 and 285 s for (sa)2 and (msa)2 clusters, respectively.

Figure 5.

Figure 5

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Relative timings of the tested DLPNO-MP2 (top) and DLPNO-CCSD(T0) (bottom) methods compared to PW91 for the (sa)2 and (msa)2 clusters. The DFT calculations are performed using an aug-cc-pVTZ basis set.

As expected, the M06-2X and ωB97X-D3(BJ) functionals are significantly more time consuming than the PW91 functional. A M06-2X single point energy calculation on the (sa)2 cluster will take 30 times as long a PW91 single point energy calculation and while the deviations (MSD, MAD, and RMSD) of the M06-2X and PW91 functionals are to some extent very similar, there remain a significant difference in the maximum deviations (4.90 and 2.38 kcal/mol for PW91 and M06-2X, respectively). In a similar manner, a single point energy calculation on the (sa)2 cluster using the ωB97X-D3(BJ) functional will take 54 times longer than PW91 but will also significantly reduce the errors. If possible, the M06-2X and ωB97X-D3(BJ) functionals should be applied compared to PW91. However, for very large systems, the PW91 functional, being a generalized gradient approximation, might be the only choice.

The DLPNO-MP2 timings are very insensitive to the PNO settings and a tight PNO criterion can be used without any significant loss of computational time. DLPNO-MP2 calculations using a double zeta basis set are extremely swift, but as illustrated in Table 1, deviations up to −3.47 kcal/mol can occur. In order to push DLPNO-MP2 to acceptable accuracy with maximum deviations below 2 kcal/mol, either a triple zeta basis set or explicit correlation should be used. Applying DLPNO-MP2-F12/DZ-F12 yields similar performance and timings as the ωB97X-D3(BJ) functional. Overall, it seems that there is little gain in using DLPNO-MP2 compared to DFT and DLPNO-CCSD(T0) for the systems at hand. For very large systems, the larger prefactor on the DLPNO-CCSD(T0) calculations might make DLPNO-MP2 the only choice in which case the DLPNO-MP2-F12/DZ-F12 level of theory will appear as a reasonable choice.

The DLPNO-CCSD(T0) timings are, contrary to DLPNO-MP2, very sensitive to the PNO settings. Employing a tightPNO criterion (without F12) leads to ∼2.5 times longer computations, but as evident from the distributions in Figure 3, there is no gain in accuracy. The loosePNO calculations (without F12) are in a similar manner 2 times faster than the normalPNO calculations, and in the case of using a double zeta basis set, it does not lead to significantly worse results. This could indicate that DLPNO-CCSD(T0)/ADZ calculations using a loosePNO setting could be an efficient choice in the process of narrowing down the amount of important cluster conformers when performing configurational sampling. In general, using the improved iterative (T) correction leads to approximately 50% longer computations without any gain in accuracy.

Overall, to obtain the lowest errors, DLPNO-CCSD(T0)-F12/TZ-F12 using a tightPNO should be performed if possible and will lead to binding energies in good agreement with the CCSD(F12*)(T) complete basis set limit. If this calculation is out of reach, either DLPNO-CCSD(T0)-F12/TZ-F12 using normalPNO or DLPNO-CCSD(T0)/ATZ using normalPNO are suitable choices that leads to maximum errors below ∼1 kcal/mol. If triple zeta basis sets are too computational demanding, the DLPNO-CCSD(T0)-F12/DZ-F12 using a tightPNO criterion appears as a decent level of theory for efficient calculations with low and systematic errors.

2.8. Reaction Gibbs Free Energies

Using the recommended methods, the Gibbs free energies of reactions R1–R3 can be calculated accurately. The thermal contribution to the free energy is calculated at the ωB97X-D/aug-cc-pVTZ level of theory, and the electronic binding energies are calculated at the CCSD(F12*)(T)/CBS, DLPNO-CCSD(T0)-F12/ATZ (tightPNO) DLPNO-CCSD(T0)/ATZ (normalPNO), and ωB97X-D3(BJ)/def2-QZVPPD level of theory. Tables 4–6 present the calculated free energies for reactions R1, R2, and R3, respectively.

Table 4. Reaction Gibbs Free Energies for the Studied Clusters in Reaction R1a.

cluster CCSD(F12*)(T)/CBS DLPNO-F12 DLPNO ωB97X-D3(BJ)
(aa)1(a)1 –0.15 –0.10 –0.08 –0.10
(aa)1(ma)1 –0.83 –0.73 –0.97 –0.71
(aa)1(dma)1 –1.62 –1.49 –1.83 –1.45
(aa)1(tma)1 –1.88 –1.70 –2.31 –1.63
(aa)1(eda)1 –2.23 –2.11 –2.45 –2.11
(fa)1(a)1 –0.77 –0.74 –0.75 –0.87
(fa)1(ma)1 –1.80 –1.67 –1.96 –1.83
(fa)1(dma)1 –2.43 –2.32 –2.71 –2.43
(fa)1(tma)1 –2.73 –2.56 –3.18 –2.65
(fa)1(eda)1 –2.36 –2.23 –2.61 –2.43
(msa)1(a)1 –3.95 –3.93 –3.67 –3.81
(msa)1(ma)1 –6.69 –6.53 –5.65 –5.88
(msa)1(dma)1 –8.98 –8.84 –8.28 –8.15
(msa)1(tma)1 –9.18 –9.12 –9.13 –8.55
(msa)1(eda)1 –9.35 –9.21 –8.70 –9.00
(na)1(a)1 –2.90 –2.78 –2.91 –3.15
(na)1(ma)1 –4.12 –3.96 –4.33 –4.27
(na)1(dma)1 –5.01 –4.84 –5.40 –5.13
(na)1(tma)1 –4.06 –3.91 –4.82 –4.35
(na)1(eda)1 –3.52 –3.40 –3.83 –4.18
(sa)1(a)1 –5.47 –5.46 –5.35 –5.51
(sa)1(ma)1 –8.77 –8.58 –8.04 –8.30
(sa)1(dma)1 –12.67 –12.52 –12.40 –12.19
(sa)1(tma)1 –12.23 –12.11 –12.68 –11.81
(sa)1(eda)1 –11.82 –11.62 –11.63 –11.71
Open in a new tab
a

The thermal contribution to the free energy is calculated at the ωB97X-D/aug-cc-pVTZ level of theory and the electronic binding energies are calculated using either the CCSD(F12*)(T)/CBS estimates, DLPNO-CCSD(T0)-F12/VTZ-F12 (tightPNO), DLPNO-CCSD(T)/ATZ (normalPNO) or ωB97X-D3(BJ)/def2-QZVPPD. All values are in kcal/mol.

Table 6. Reaction Gibbs Free Energies for the Studied Clusters in Reaction R3a.

cluster CCSD(F12*)(T)/CBS DLPNO-F12 DLPNO ωB97X-D3(BJ)
(aa)1(w)1 0.63 0.69 0.80 0.89
(fa)1(w)1 0.54 0.59 0.69 0.76
(msa)1(w)1 –1.73 –1.69 –1.60 –1.38
(na)1(w)1 0.09 0.20 0.11 0.18
(sa)1(w)1 –2.00 –2.03 –1.97 –1.77
Open in a new tab
a

The thermal contribution to the free energy is calculated at the ωB97X-D/aug-cc-pVTZ level of theory, and the electronic binding energies are calculated using either the CCSD(F12*)(T)/CBS estimates, DLPNO-CCSD(T0)-F12/VTZ-F12 (tightPNO), DLPNO-CCSD(T)/ATZ (normalPNO) or ωB97X-D3(BJ)/def2-QZVPPD. All values are in kcal/mol.

Table 5. Reaction Gibbs Free Energies for the Studied Clusters in Reaction R2a.

cluster CCSD(F12*)(T)/CBS DLPNO-F12 DLPNO ωB97X-D3(BJ)
(fa)1(aa)1 –4.16 –4.08 –3.93 –4.27
(msa)1(aa)1 –6.48 –6.38 –6.23 –6.34
(msa)1(fa)1 –5.72 –5.64 –5.53 –5.62
(msa)1(na)1 –2.69 –2.55 –2.89 –2.60
(na)1(aa)1 –2.86 –2.71 –2.76 –3.04
(na)1(fa)1 –1.83 –1.67 –1.71 –1.98
(sa)1(aa)1 –6.52 –6.48 –6.41 –6.53
(sa)1(fa)1 –5.62 –5.60 –5.48 –5.64
(sa)1(msa)1 –6.69 –6.69 –6.96 –6.70
(sa)1(na)1 –1.96 –1.76 –2.10 –1.92
(aa)2 –4.48 –4.36 –4.21 –4.51
(fa)2 –3.63 –3.53 –3.32 –3.79
(msa)2 –6.87 –6.76 –7.27 –7.05
(na)2 1.49 1.63 1.43 1.49
(sa)2 –6.22 –6.37 –6.27 –6.18
Open in a new tab
a

The thermal contribution to the free energy is calculated at the ωB97X-D/aug-cc-pVTZ level of theory and the electronic binding energies are calculated using either the CCSD(F12*)(T)/CBS estimates, DLPNO-CCSD(T0)-F12/VTZ-F12 (tightPNO), DLPNO-CCSD(T)/ATZ (normalPNO) or ωB97X-D3(BJ)/def2-QZVPPD. All values are in kcal/mol.

The inorganic acids (msa, na, and sa) interact significantly stronger with bases compared to the organic acids. In relation to the strength of the interactions, the inorganic acids follow the order sa > msa > na and for the organic acids, fa > aa in all cases. These trends clearly follow the acidity of the acids.

The interactions between the acids show a quite similar trend as the acid–base interactions and msa/sa are clearly displaying stronger intermolecular interaction than na, aa, and fa. The (msa)2 homodimer and the (msa)1(sa)1 heterodimer show the strongest interactions. Interestingly, the (sa)1(aa)1 and (msa)1(aa)1 clusters bind slightly stronger than the (sa)2 homodimer. Furthermore, the interaction between msa/sa with aa/fa follow the opposite trend compared to the pKa values of the organic acids. This fact has also previously been stated by Nadykto and Yu.19

The interactions between the studied acids and water are for na, aa, and fa not favorable. Sulfuric acid is found to interact slightly stronger with water than methanesulfonic acid. In both cases, there is a large deviation (up to 20%) between the CCSD(F12*)(T)/CBS best estimates and the DFT calculations. This is caused by the low numerical value of the interaction, which leads to a higher error. This indicates that DFT without higher level corrections will predict that the clusters are significantly less hydrated than they are in reality.

3. Conclusions

We have assessed the performance in calculating the binding energies of atmospheric relevant clusters using approximate DLPNO-MP2, DLPNO-CCSD(T0), and DFT methods. The performance is tested on a representative test set of 45 atmospheric relevant cluster formation reactions and compared to high level explicitly correlated CCSD(F12*)(T)a calculations extrapolated to the complete basis set limit. We identify a hierarchy of methods that can be applied to increasingly larger clusters, while keeping the introduced errors at a minimum. When CCSD(F12*)(T) calculations are out of reach, DLPNO-CCSD(T0)-F12/cc-pVTZ-F12 calculations using a tightPNO criterion can be utilized and only yield a MSD of 0.10 kcal/mol, with a maximum error of 0.20 kcal/mol. For larger systems, DLPNO-CCSD(T0)/aug-cc-pVTZ with a normalPNO setting represents an adequate level of theory yielding a MSD of 0.02 kcal/mol, with a maximum deviation of 1.03 kcal/mol. When these methods are out of reach, the ωB97X-D3(BJ) functional with a def2-QZVPPD basis set represents a good choice for calculating the binding energies of atmospheric molecular clusters, with a MSD of 0.09 kcal/mol and a maximum deviation of 0.83 kcal/mol.

4. Computational Details

The Gaussian 16 program111 was used to obtain the equilibrium structures at the ωB97X-D/aug-cc-pVTZ level of theory using default convergence criteria. Vibrational frequency analysis was carried out to ensure that all clusters and monomers are minima on the potential energy surface. The ωB97X-D112 functional was chosen because it has been shown to yield the lowest errors in the binding energies (compared to other density functionals) relative to high level coupled cluster calculations.33,94,113

Domain-based local pair natural orbital (DLPNO-CCSD(T0)114,115 and DLPNO-MP2116) single point energies (using tightSCF) and DFT single point energies (Grid4) were calculated using the ORCA 4.2.1 program.117,118 Explicitly correlated DLPNO calculations were carried out using the default approximations for the DLPNO-MP2-F1279 and DLPNO-CCSD(T0)-F1278 methods. The DLPNO calculations were performed using loosePNO, normalPNO, and tightPNO settings.119 We also tested the improved (T) correction introduced in ref (120), which solves the triples amplitudes of the perturbative triples correction iteratively. This is in contrast to earlier versions of ORCA where the so-called semi-canonical (T0) approximation was used, but also the name DLPNO-CCSD(T) was used. In our paper, we distinguish both approaches by the names (T0) and (T).

All RI-CC2 and explicitly correlated CCSD(F12*)(T) calculations were performed with a development version of the TURBOMOLE package.121 For the RI-CC2 calculations, the ricc2122 module was used, and for the CCSD(F12*)(T) calculations, the ccsdf1280,123 module was used.

For the RI-CC2 calculations, an aug-cc-pV(T+d)Z basis set was used, with an aug-cc-pVQZ auxiliary basis. For all DFT single point energies, the def2-QZVPPD basis set was used, which has been shown to be sufficiently close to the complete basis set limit when calculating the binding energies of atmospheric molecular clusters.109 For the DLPNO calculations, aug-cc-pVXZ (X = D, T) basis set was employed with complementary aug-cc-pVXZ/C124 coulomb fitting basis sets. For all explicitly correlated calculations, cc-pVXZ-F12125 basis sets were used. We will abbreviate the aug-cc-pVXZ and cc-pVXZ-F12 basis sets simply as AXZ and XZ-F12, respectively.

Acknowledgments

J.E. thanks the Independent Research Fund Denmark grant number 9064-00001B for financial support. The numerical results presented in this work were obtained at the Centre for Scientific Computing, Aarhus http://phys.au.dk/forskning/cscaa/

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.0c00436.

  • All DLPNO-MP2, DLPNO-CCSD(T), RI-CC2, CCSD(F12*)(T)/CBS(23), and CCSD(F12*)(T)/CBS single point energies (PDF)

  • xyz-files of the studied molecular clusters and monomers at the ωB97X-D/aug-ccpVTZ levels of theory (ZIP)

The authors declare no competing financial interest.

Footnotes

a

Some authors refer to CCSD(F12*) as CCSD-F12c.

Supplementary Material

ao0c00436_si_001.pdf (178.6KB, pdf)
ao0c00436_si_002.zip (22.3KB, zip)

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