On the properties and atmospheric implication of amine-hydrated clusters

Abstract

Amines have been recognized as important precursor species in the formation of new atmospheric particles. Although dimethylamine–water clusters have been the focus of a large number of theoretical studies during the last few years, some information regarding these clusters, such as the influence of temperature, the analysis of their weak interactions, and their Rayleigh scattering properties, is still lacking. In this study, the equilibrium geometric structures and thermodynamics of (CH₃)₂NH(H₂O)_n (n = 1–6) clusters were systematically investigated using density functional theory (PW91PW91) coupled with the 6-311++G(3df,3pd) basis set. To determine the most stable isomer and the order of the different isomers, single-point calculations were executed using a two-point extrapolation method in conjunction with the complete basis set for all isomers. The optimized structures show that the addition of a fifth water molecule changes the most stable configuration from a quasi-planar ring structure to a cage-like configuration. Electron density analysis shows that the interactions of these complexes are mainly medium hydrogen bonds. The dependence on temperature of the conformational population and the Gibbs free energies of the (CH₃)₂NH(H₂O)_n (n = 1–6) clusters were determined with respect to temperature (200–300 K). A weak dependence on temperature was found for the formation of (CH₃)₂NH(H₂O)_n (n = 1–6) clusters. Dimethylamine–water clusters are favorable at low temperatures, but these clusters may be difficult to form because of the combined effect of Gibbs free energies with small negative values and the low relative concentration of dimethylamine in various atmospheric conditions, and this implies that dimethylamine–water clusters are difficult to form spontaneously in the atmosphere. Finally, the Rayleigh scattering properties of (CH₃)₂NH(H₂O)_n (n = 1–6) have been investigated systematically for the first time.

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1. Introduction

Atmospheric aerosols are solid or liquid particles suspended in the air, which play a key role in direct and indirect effects on the climate^1–5 by altering cloud properties and precipitation. The airborne ultrafine particles in these aerosols adversely affect public health.^6–8 The formation of new particles (NPF) has been observed in a wide range of locations and is estimated to be an important source of aerosol particles and cloud condensation nuclei. Recently, the formation of new particles has been observed;^4,9 however, investigating the initial stage of NPF remains a challenge, although state-of-the-art instruments⁹ can measure ionic and neutral clusters. Understanding of the initial formation mechanisms of new particles is still deficient.^10–14

Neutral binary sulfuric acid–water nucleation⁴ and binary ion-induced nucleation¹⁵ have been well studied and serve as an available point of comparison,^16,17 but they cannot explain new-particle formation events in the atmospheric boundary layer, which indicates that other species must participate in stabilizing sulfuric acid particles such as ions,^15,18,19 ammonia and amines,^20–27 organics,^28–33 and iodine oxides.²⁸ Quantum chemical calculations imply that nucleation with amines is significantly favorable over ammonia.^34–37 Several experimental studies have confirmed that ammonia and alkylamines strongly influence nucleation of sulfuric acid and water particles.^{16,17,22,38–40} Moreover, recent field observations with gas-phase amines have corroborated that the presence of amines has a considerable effect on the formation of new particles.^41–44

Dimethylamine (DMA) is the strongest and most common base in the atmosphere, and it can rapidly undergo acid–base reactions, which enhance neutral and ion-induced sulfuric acid–water nucleation.^36,40,41,45 Recently, reported studies have shown a correlation between the formation of new particles and the presence of DMA.^38,46–49 Experiments³⁸ using the CLOUD chamber at CERN confirmed that concentrations of DMA exceeding three parts per trillion by volume were able to increase new-particle formation rates by more than 1000-fold compared to ammonia.

Hydrogen-bonding interactions are the driving forces for the formation of atmospheric molecular complexes, and their strength determines the thermodynamic stability of these complexes. Some quantum chemical studies of sulfuric acid, DMA and water have been performed,^34–36,50 but analysis of the weak attractions in the clusters, which are essential for their nucleation mechanism, is still absent. The theory of atoms in molecules (AIM), which was developed by Bader et al.,⁵¹ is a valuable tool for understanding hydrogen bonds in molecular complexes. There are a number of studies dealing with the sulfuric acid–DMA–water system that show how DMA will not form clusters with water.^37,50,52,53 All this research was carried out at constant temperature and 1 atm. As we all know, the dependence on temperature of thermodynamic properties is a significant parameter for understanding how a specific nucleation mechanism plays a role in NPF. Furthermore, the climate is affected by aerosols directly via scattering the light from the sun.⁵⁴ Although the Rayleigh scattering properties of large particles are well understood, the Rayleigh scattering properties of clusters, especially at the molecular level, are still not known. Considering all these, AIM analysis, correlation with temperature and the investigation of Rayleigh scattering properties were performed for the first time in this study for clusters of (CH₃)₂NH(H₂O)_1–6.

This study includes some aspects as follows: (1) basin-hopping (BH) was coupled with density functional theory (DFT) and used to determine global and local minima by sampling the potential energy surface thoroughly; (2) single-point calculations were executed using a two-point extrapolation method in conjunction with the complete basis set for all isomers; (3) electron density analysis was performed to investigate the interactions of these complexes; (4) the dependence on temperature of the conformational population and the Gibbs free energies of the (CH₃)₂NH(H₂O)_1–6 clusters were investigated; and (5) the Rayleigh scattering properties of (CH₃)₂NH(H₂O)_1–6 have been investigated systematically for the first time.

2. Computational methods

As the number of water molecules increases, the number of isomers increases enormously, and the most stable conformers become obscure and not obvious. Therefore, searching large clusters by chemical intuition is very difficult. To overcome this problem, the BH algorithm^55–58 was employed, which has been successfully applied to atomic clusters.^58–63 To effectively search molecular clusters, the BH algorithm using a compressed sampling technique was improved in previous studies by applying the structures such as (H₂O)_n (n = 4–10) clusters,⁶⁴ CH₃NH₂–H₂O clusters,⁶⁵ Cl⁻(H₂O) clusters,^66,67 and H₂SO₄–C₂H₂O₄–H₂O clusters.^68–71 The BH method includes two steps. First, a new structure is generated via the random displacement of atoms; then, the structure is optimized to a local minimum. Second, this local energy minimum is used as a criterion for accepting the structure spaces that were initially generated with Boltzmann weight at a finite temperature. The initial geometries were obtained with the BH algorithm and PW91/DND implemented in the DMol³ software package⁷² was employed in the DFT module coupled with BH. Then, the stable isomers within 10 kcal mol⁻¹ of the stable global minimum were optimized via the PW91PW91/6-311++G(3df,3pd) method. For each stationary point, frequency calculations were carried out to ensure that there were no imaginary frequencies. The convergence standards used for optimization were the default settings in Gaussian09 software.⁷³ In the standard state of 1 atm pressure and 298 K, the binding energies, enthalpies, and Gibbs free energies were computed for all the selected clusters using the harmonic-oscillator and rigid-rotor approximations. Structural informations (bond lengths) were provided by Chemcraft 1.6 (http://www.chemcraftprog.com).

Previous studies have indicated that PW91PW91/6-311++G(3df,3pd) provides better performance for predicting the vibrational spectrum of oxalic acid than MP2/6-311++G(3df,3pd) or B3LYP/6-311++G(3df,3pd).^32,33,74,75 Moreover, it has been found that the PW91PW91 method well reproduces the MP2 and CCSD(T) results of the first hydration of sulfuric acid.⁷⁶ However, one of the main sources of error in quantum chemistry calculations is that the basis set is not sufficiently large or it is far from the complete basis set limit. Using PW91PW91 with a finite basis set for the calculation of the interaction energies of hydrogen-bonded systems encounters a problem called basis set superposition error (BSSE). To reduce this error and the calculation cost, extrapolation of the energy to the complete basis set (CBS)⁷⁷ limit was performed on the single-point energy calculated using the method of DF-MP2-F12. The CBS limit was estimated via a two-point extrapolation scheme.^78,79 The equations for the corresponding calculations are as follows:

E^SCF_CBS = E^SCF_N + B × exp(−AN) (1)

E^RI-MP2_CBS = E^SCF_CBS + E^corr_CBS

where A and B are the fitting parameters and E^SCF_CBS, E^corr_CBS and E^RI-MP2_CBS are the SCF, correlation and total energies, respectively. All the calculations were performed using Molpro 2010.1 ⁸⁰ and the default convergence criteria were defined in Molpro 2010.1.

Although DFT methods can qualitatively describe geometries and vibrational frequencies, the weak interactions arising from dispersion forces are unable be accurately treated, which plays a moderately important role in hydrogen bonding. Two main pathways are provided to obtain accurate thermochemistry for hydrogen-bonded clusters. One method is to use DFT, which is sufficiently parameterized to replicate the binding energies of various benchmarking sets of weakly interacting clusters, although the treatment of dispersion is inaccurate. Another alternative is to use correlated wave function-based methods such as MP perturbation theory or coupled-cluster methods. However, the computational cost of correlated wave function-based methods is generally orders of magnitude more than that of DFT methods. Herein, extrapolation is performed on energy values calculated by DF-MP2-F12 to reduce basis set superposition errors and basis set incompleteness errors. A study by Temelso et al.⁸¹ showed that the convergence of the binding energy to the CBS is not monotonic, due to the incompleteness of the AVDE basis set resulting in an overestimation of the binding energy. Hill⁸² demonstrated that the MP2-F12 method combined with the cc-pVnZ-F12 basis set produces results that are approximately equal to those from MP2-F12/aug-cc-pV(n + 1)Z. Aug sets include additional diffuse higher angular momentum functions, which lead to significantly more basis set superposition error than the cc-pVnZ-F12 basis set. Another important problem is the calculation of energies for larger clusters wherein the computational costs prohibit the use of large basis sets. Recent study illustrated that the binding energies from DF-MP2-F12/VQZ-F12 are close to the basis set limit.⁸³ To ensure accuracy without too high calculation cost, we selected DF-MP2-F12/cc-pVnZ-F12 (n = D, T) for the extrapolation scheme.

Two-point extrapolation by a conventional scheme (eqn (1)) was carried out by applying the cc-pVDZ-F12 and cc-pVTZ-F12 basis sets coupled with DF-MP2-F12 for calculating the SCF and correlation energies in this study. To obtain the fitting parameter A, which is a constant based on eqn (1), the binding energies for the six isomers of (CH₃)₂NH(H₂O)_n=1–3 were calculated at the DF-MP2-F12/cc-pVnZ-F12 (n = D, T, Q) level of theory. A three-point extrapolation was performed to fit the constant A, which is approximately equal to 1.48.

Fig. 1 shows that the convergence of the binding energy is monotonic; systematic convergence of the energies toward the CBS limit is again a feature of these basis sets. The two-point extrapolation scheme for the binding energy deviates from the benchmark DF-MP2-F12/VQZ-F12 with the difference increasing from 1.14 to 1.77 kcal mol⁻¹ as the size of clusters grows. Thus, the deviation monotonically increases, which indicates that as the number of water molecules grows, the difference will increase correspondingly. For (CH₃)₂NH(H₂O)₂ and (CH₃)₂NH(H₂O)₅ clusters, the geometries of the most stable isomers optimized by the PW91PW91/6-311++G(3df,3pd) method were 2b and 5d, respectively. At the DF-MP2-F12/CBS//PW91PW91/6-311++G(3df,3pd) level, 2a and 5a were found to be the global minima. Furthermore, isomers with the same number of water molecules were arranged in order of increasing electronic energy. The difference in the calculated binding energies between CBS and DF-MP2-F12/cc-VTZ-F12 is several kcal mol⁻¹ and with an increase in the size of the cluster, the errors in the Gibbs free energy for the same magnitude of binding energy will generate a significant effect on quantities, such as the abundance of clusters and nucleation rates, because they depend exponentially on barriers to nucleation.

Fig. 1 Comparison of the differences between the binding energies for six isomers of (CH₃)₂NH(H₂O)_n=1,2,3 at the DF-MP2-F12/X and DF-MP2-F12/VQZ-F12 levels.

Based on the global minima of (CH₃)₂NH(H₂O)_n (n = 1–6) obtained in this study, the optical properties of the pre-nucleation clusters were evaluated. The light scattering intensities and isotropic mean polarizabilities [small alpha, Greek, macron], as well as anisotropic polarizabilities Δα, were calculated at the CAM-B3LYP/aug-cc-pVDZ level of theory and the relevant computational methods are given in our earlier study.⁶⁶ The benchmark for the smallest clusters of H₂SO₄–NH₃–H₂O was determined by Elm et al.⁸⁴ They found that CAM-B3LYP/aug-cc-pVDZ gave a good balance between efficiency and accuracy and obtained good agreement with both experimental and CCSD(T) values of polarizability. In this article, to find an appropriate methodology for calculating the optical properties of DMA–H₂O clusters, five different DFT functionals with the aug-cc-pVDZ basis set were investigated. The results are shown in Table 1; by analyzing the results, CAM-B3LYP/aug-cc-pVDZ was confirmed to be a suitable method, which yielded good agreement with MP2 values of polarizability.

Table 1 Calculated isotropic mean polarizabilities of H₂O, CH₃NHCH₃, (CH₃NHCH₃)(H₂O), and (CH₃NHCH₃)(H₂O)₂ using different DFT functionals with the aug-cc-pVDZ basis set

Functional	H2O	CH3NHCH3	(CH3NHCH3)(H2O)	(CH3NHCH3)(H2O)2
B3LYP	9.472	39.481	49.087	58.787
PW91PW91	10.051	41.143	56.648	61.809
M06-2X	8.893	38.351	47.072	56.437
ωB97x-D	9.185	38.431	47.553	56.925
CAM-B3LYP	9.326	38.477	47.781	57.283
MP2	9.301	38.486	47.699	57.320

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3. Results and discussion

To understand the initial steps of the formation of new particles when DMA participates in forming new particles, we first performed quantum chemical calculations for (CH₃)₂NH(H₂O)_n (n = 1–6) to search for the global minimum and several local minima. All the hydrated clusters with the same number of water molecules were arranged according to the binding energy with the ZPE correction from lowest to highest. The complexes were marked by n_a, n_b, n_c, etc., where n is the number of water molecules in the (CH₃)₂NH(H₂O)_n clusters.

3.1 Structures

Monohydrates. In our search for the global and low-lying local minima of (CH₃)₂NH(H₂O) clusters, the two structures shown in Fig. 2 were found to be the global minimum and local minimum in energy. According to Nadykto et al.,³⁵ isomer 1b was reported to be the most stable isomer. High-level calculation at the DF-MP2-F12/CBS//PW91PW91/6-311++G(3df,3pd) level showed that 1a was the lowest structure.

Fig. 2 Global minimum and local minima for (CH₃)₂NH(H₂O)_n=1–3 optimized at the PW91PW91/6-311++G(3df,3pd) level. The relative energies (kcal mol⁻¹) with respect to the most stable structure are given in parentheses.

Dihydrates. In our search for the global minimum of the (CH₃)₂NH(H₂O)₂ cluster, the ground-state geometry of isomer 2b shown in Fig. 2 was optimized at the PW91PW91/6-311++G(3df,3pd) level, which agrees with the previous result.³⁵ We calibrated the single-point energy at the DF-MP2-F12/CBS//PW91PW91/6-311++G(3df,3pd) level. Configuration 2b is the structure with the second lowest energy, which is higher in energy by 0.24 kcal mol⁻¹ than 2a.

Trihydrates. Eight isomers were taken into account (in Fig. 2). According to the arrangement, there are two types of pattern in the cyclic forms. The three isomers with lowest energy (3a, 3b and 3c) are similar and have four-membered ring structures with four HBs. The atoms forming the HB network nearly lie in the same plane, and the difference in the orientation of the three isomers is due to the free hydrogen atoms. Therefore, the binding energies are very similar. For the other five isomers, three water molecules form a coplanar three-membered ring structure, except for the free hydrogen atoms. The orientation of the free hydrogen atoms and the three-membered ring gives rise to a difference in the structure and binding energies. For the first type, the computed binding energies with the zero-point energy correction for each structure exhibit small gaps; compared with the most stable structure, the differences in energy are 0.12 kcal mol⁻¹ and 0.39 kcal mol⁻¹. In contrast, a wide gap exists between the two types on the binding energies, ranging from 2.79 kcal mol⁻¹ to 3.80 kcal mol⁻¹. This gap means that the four-membered ring structure is more stable than the three-membered ring structure. The structure of 3a is similar to the global minimum of the (H₂O)₄ cluster.⁸⁵

Tetrahydrates. The most stable configuration (4a) for (CH₃)₂NH(H₂O)₄ clusters is a ring structure with five H-bonds, which is similar to the cluster of (H₂O)₅.⁸⁵ The HB length of N–H⋯O is 1.909 Å, which is weaker than the others (1.643 Å, 1.644 Å, 1.667 Å, and 1.702 Å). Interestingly, with a change in the orientation of the free hydrogen, the configurations labeled 4b and 4e became distorted and are no longer planar. The clusters change from planar to contorted planar to minimize repulsions from the adjacent hydrogens. For 4h and 4i, four water molecules form a four-membered ring with similar features to the trihydrates. In addition to the single-ring structures, four bicyclic structures (4d, 4f, 4g and 4j) are observed. Detailed informations about these structures are shown in Fig. 3.

Fig. 3 Global minimum and local minima for (CH₃)₂NH(H₂O)₄ optimized at the PW91PW91/6-311++G(3df,3pd) level. The relative energies (kcal mol⁻¹) with respect to the most stable structure are given in parentheses.

Pentahydrates. For the case of the (CH₃)₂NH(H₂O)₅ cluster in Fig. 4, eighteen low-lying isomers within 4 kcal mol⁻¹ were selected. Compared with the trihydrate and tetrahydrate clusters, the fifth water molecule produces a significant change in the most stable structure, changing it from a planar ring to a three-dimensional form. For the pentahydrate clusters, the most stable structure is a prism structure lacking one side with eight H-bonds, whereas it is a complete prism structure for the (H₂O)₆ cluster.⁸⁵ The geometry of 5b is closest to that of 5a, although its binding energy lies higher by 0.85 kcal mol⁻¹. The orientation of the free hydrogen and the relative orientation between DMA and hydrated ring structures make the conformation different. The binding energies of these similar isomers display small gaps that are within a few kcal mol⁻¹. Many configurations are derived from (CH₃)₂NH(H₂O)₄ clusters. For instance, 5i is a derivative of the existing structure 4h with the addition of H₂O and minor rearrangements to the existing network of HBs. Similarly, 5e is a derivative of 4d. The frequency calculation, which was performed on 5k shows that it has one imaginary frequency of 3.15. This imaginary frequency is so small that the geometry can be considered as a local-minimum structure.

Fig. 4 Global minimum and local minima for (CH₃)₂NH(H₂O)₅ optimized at the PW91PW91/6-311++G(3df,3pd) level. The relative energies (kcal mol⁻¹) with respect to the most stable structure are given in parentheses.

Hexahydrates. Among the stable isomers of the (CH₃)₂NH(H₂O)₆ cluster in Fig. 5, isomer 6a is the structure with lowest energy and was derived from isomer 5a with one water molecule inserted into the existing HB network and then forming a non-planar four-membered ring. The second most stable structure (6b) has a slightly distorted cube-like structure with a corner missing and is similar to the most stable (H₂O)₇ cluster.⁸⁵ Although structures 6b and 6d are similar to 6a, which are derived from 5a, a considerable alteration was generated, because the sixth water molecule is located at a different position compared to isomer 5a. Our calculations of the binding energies for 6a and 6b without the ZPE correction (in Table 3) show that 6a lies slightly higher than 6b by 0.27 kcal mol⁻¹. Due to the formation of HBs between amino hydrogen atoms and the oxygen atoms of water molecules, 6c, 6e and 6h are all polycyclic structures with nine H-bonds. When the amino hydrogen atom is free, different types of polycyclic structures (6i, 6k and 6l) are composed of six water molecules. In addition to the polycyclic structures, there are many bicyclic structures such as 6j with eight H-bonds.

Fig. 5 Global minimum and local minima for (CH₃)₂NH(H₂O)₆ optimized at the PW91PW91/6-311++G(3df,3pd) level. The relative energies (kcal mol⁻¹) with respect to the most stable structure are given in parentheses.

Table 2 Comparison of changes at the DF-MP2-F12/CBS level in terms of ΔE (kcal mol⁻¹), ΔE + ZPE (kcal mol⁻¹), ΔH (kcal mol⁻¹), and ΔG (kcal mol⁻¹) associated with formation of (CH₃)₂NH(H₂O)_n=1–4. The zero-point energy, thermal correction to enthalpy and Gibbs free energy were calculated from vibrational analysis at the PW91PW91/6-311++G(3df,3pd) level (1 atm, 298 K)

Reaction	Isomer	ΔE	ΔE + ZPE	ΔH	ΔG
(CH3)2NH + H2O ↔ (CH3)2NH·H2O	1a	−7.72	−5.63	−5.99	1.60
	1b	−7.45	−5.43	−5.74	1.46
(CH3)2NH·H2O + H2O ↔ (CH3)2NH·2H2O	2a	−9.04	−6.64	−7.41	2.25
	2b	−8.95	−6.53	−7.36	2.49
(CH3)2NH·2H2O + H2O ↔ (CH3)2NH·3H2O	3a	−10.48	−8.07	−9.04	1.02
	3b	−10.15	−7.95	−8.78	0.67
	3c	−10.02	−7.68	−8.61	1.25
	3d	−7.97	−5.28	−6.14	2.93
	3e	−7.76	−5.08	−5.94	3.07
	3f	−7.4	−4.77	−5.58	2.51
	3g	−7.11	−4.53	−5.31	2.81
	3h	−6.92	−4.27	−5.07	3.28
(CH3)2NH·3H2O + H2O ↔ (CH3)2NH·4H2O	4a	−8.69	−6.64	−7.26	1.42
	4b	−8.17	−6.23	−6.76	1.65
	4c	−8.08	−6.16	−6.7	1.44
	4d	−8.41	−6.11	−6.85	3.11
	4e	−8.13	−6.1	−6.7	1.77
	4f	−8.16	−5.98	−6.56	2.75
	4g	−8.62	−5.88	−6.76	3.14
	4h	−8.52	−5.81	−6.61	1.14
	4i	−8.34	−5.67	−6.48	2.15
	4j	−7.81	−5.66	−6.17	2.72

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Table 3 Comparison of changes at the DF-MP2-F12/CBS level in terms of ΔE (kcal mol⁻¹), ΔE + ZPE (kcal mol⁻¹), ΔH (kcal mol⁻¹), and ΔG (kcal mol⁻¹) associated with formation of (CH₃)₂NH(H₂O)_n=5–6. The zero-point energy, thermal correction to enthalpy and Gibbs free energy were calculated from vibrational analysis at the PW91PW91/6-311++G(3df,3pd) level

Reaction	Isomer	ΔE	ΔE + ZPE	ΔH	ΔG
(CH3)2NH·4H2O + H2O ↔ (CH3)2NH·5H2O	5a	−10.36	−7.68	−8.66	2.64
	5b	−9.91	−6.83	−7.92	3.35
	5c	−9.41	−6.77	−7.69	2.67
	5d	−9.51	−6.72	−7.62	2.18
	5e	−9.31	−6.47	−7.44	2.87
	5f	−8.91	−6.3	−7.22	2.94
	5g	−8.53	−5.96	−6.85	3.26
	5h	−8.38	−5.94	−6.84	3.32
	5i	−8.54	−5.72	−6.62	2.4
	5j	−8.28	−5.55	−6.36	2.68
	5k	−8.25	−5.48	−6.88	3.89
	5l	−8.23	−5.44	−6.27	2.01
	5m	−7.94	−5.14	−5.96	2.36
	5n	−7.24	−4.85	−5.52	3.85
	5o	−7.58	−4.54	−5.27	2.79
	5p	−7.3	−4.34	−5.21	3.82
	5q	−7.26	−4.31	−5.14	3.55
	5r	−7.22	−3.99	−4.85	3.79
(CH3)2NH·5H2O + H2O ↔ (CH3)2NH·6H2O	6a	−11.64	−10.88	−9.64	0.80
	6b	−11.37	−10.23	−9.09	1.58
	6c	−9.68	−9.18	−7.68	1.52
	6d	−8.74	−8.88	−7.26	2.18
	6e	−9.40	−8.63	−7.33	2.55
	6f	−8.93	−8.48	−7.76	4.48
	6g	−7.56	−8.40	−6.78	2.59
	6h	−8.25	−8.14	−6.67	3.16
	6i	−8.54	−7.66	−6.12	1.85
	6j	−7.61	−7.51	−5.83	1.46
	6k	−8.44	−7.45	−5.98	2.20
	6l	−8.01	−7.30	−5.75	2.15
	6m	−7.17	−7.08	−5.33	1.66
	6n	−7.39	−7.07	−5.38	2.98
	6o	−7.71	−7.02	−5.44	2.31

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3.2 Electron density analysis

As intermolecular interaction distances are just below the sum of the van der Waals radii of atoms naturally, they cannot be used to determine the strength of hydrogen bonds and the nature of any possible bonding between methyl groups and water molecules, or even for revealing steric interactions. Therefore, AIM theory provided by topological analysis of the electron density was performed here to deepen understanding of the intermolecular hydrogen bonding of the most stable conformations. The basis of the topological parameters at the bond critical points (BCPs) was calculated based on Bader's atoms in molecules theory.⁵¹ Some representative parameters for this type of interaction were chosen such as the electron density ρ, the Laplacian of the electron density ∇²ρ, the kinetic electron energy density G, the potential density V, the electron energy density H, and the value of G/|V| at the bond critical point. The values of ∇²ρ and H indicate the nature of the interaction and a negative value of ∇²ρ shows that there is a shared interaction as in a covalent bond, whereas a positive value indicates closed-shell system interactions such as HB and van der Waals forces. Furthermore, the energy density has proved to be a more appropriate and sensible criterion than ∇²ρ to characterize the strength of hydrogen bonds. Thus, both ∇²ρ and H > 0 represent weak HB, ∇²ρ > 0 and H < 0 indicate medium HB, whereas both ∇²ρ and H < 0 show strong HB. Moreover, G/|V| taken as the ratio between a positive value of G and a negative value of V may indicate bonds corresponding to covalent or non-covalent interaction. Thus, if G/|V| is greater than 1, the interaction has the character of a closed-shell non-covalent interaction. If the ratio is between 0.5 and 1, the interaction is partly covalent in nature and if it is less than 0.5, the interaction is a shared covalent interaction.

From Table S1,† it is observed that the values of ∇²ρ of the most stable conformations at the BCPs are all positive except for O5–H14⋯O2 of isomer 6a. All the O–H⋯N linkages display medium HBs with H < 0. And the corresponding values of ∇²ρ decrease with an increase in the number of water molecules up to five in the range of 0.0735–0.0051 a.u., whereas ∇²ρ exhibits a large step increase for 6a. For isomer 2a, C–H6⋯O2 and C–H7⋯O2 are both non-covalent because the values of G/|V| are greater than 1. In the cases of 3a and 6a, the HBs formed between an amino hydrogen atom and the oxygen atom of a water molecule are both non-covalent. For 4a, all the HBs are medium HBs. For 5a, except for the HBs in C–H13⋯O3, O2–H11⋯O6 and O4–H15⋯O6, the other HBs in this structure display the characteristics of medium HBs. As the number of water molecules increases to six, a strong HB in O5–H14⋯O2 was observed.

The non-covalent interactions (NCI) analysis method, which is derived from the correlation between the reduced density gradient and the electron density, has been studied by Yang et al.^86,87 The reduced density gradient

is a fundamental dimensionless quantity in DFT that is presented to describe the deviation from a homogeneous electron distribution.⁸⁶ To a certain extent, NCI analysis can be regarded as an extension of AIM.⁸⁷ Not only can the location of pairwise atoms that are connected along the bond path be identified, the properties around BCPs can also be visualized using NCI. The reduced density gradient is able to be used to confirm covalent interactions and non-covalent interactions in real space.⁸⁸

Fig. 6 shows the reduced gradient isosurfaces (s = 0.5 a.u.) using the Multiwfn program⁸⁹ and VMD program.⁹⁰ The green to blue-colored regions of the binding isosurface indicate that the HB interaction is becoming stronger. HBs that consist of ammonia nitrogens and hydrogen atoms of water molecules are the strongest HBs for every conformation and their strength becomes greater as the number of water molecules increases. At the same time, steric hindrance cannot be neglected because of the two methyl groups. Furthermore, repulsions located at the center of an H-bond ring are very weak and the more H-bonds form a ring, the weaker becomes the repulsion. However, there is medium hydrogen-bonding interaction building up between DMA and water molecules, which is not substantial enough to overcome the energy barrier for the reaction of hydration spontaneously in the standard state of 1 atm pressure and 298 K.

Fig. 6 Lowest-energy structures and isosurfaces (s = 0.5 a.u.) of the (CH₃)₂NH(H₂O)₆ clusters at the PW91PW91/6-311++G(3df,3pd) level.

3.3 Thermodynamics of (CH₃)₂NH(H₂O)_n formation

The binding energies ΔE (kcal mol⁻¹) of the clusters were calculated using the following equation. The ZPE-corrected binding energies ΔE + ZPE (kcal mol⁻¹), interaction enthalpies ΔH (kcal mol⁻¹) and changes in Gibbs free energy ΔG (kcal mol⁻¹) were calculated in the same way:

ΔE_{(CH3)2NH(H2O)n} = E_{(CH3)2NH(H2O)n} − E_{(CH3)2NH(H2O)n−1} − E_H2O

Tables 2 and 3 show the thermodynamics of the most stable configurations at the DF-MP2-F12/CBS level of theory upon adding a water monomer to the most stable (CH₃)₂NH(H₂O)_n−1 cluster to form (CH₃)₂NH(H₂O)_n clusters. Furthermore, ΔE (kcal mol⁻¹), ΔE + ZPE (kcal mol⁻¹), ΔH (kcal mol⁻¹) and ΔG (kcal mol⁻¹) were calculated.

Fig. 7 presents how the binding energy, enthalpy and Gibbs free energy of the global minimum change with an increase in the cluster size. The change in the binding energy is non-monotonic, and the binding energy with the ZPE correction ranges from −5.63 to −10.88 kcal mol⁻¹ during the stepwise addition of water monomers. As one to three water molecules are added, the binding energy gradually becomes more negative because new ring hydrogen bonds are formed. In the case of the (CH₃)₂NH(H₂O)₄ cluster, this tendency is reversed and there appears to be a maximum energy rise in the stepwise hydration. By adding the fifth water molecule, the relative stabilization is increased as the conformation changes from 2-dimensional to 3-dimensional. In the case of the (CH₃)₂NH(H₂O)₆ cluster, the binding energy rapidly decreases and there appears to be an energy minima in the stepwise hydration. This discrepancy resulted from the differences between the most stable conformations. The results showed that the stability of the clusters varies, and the discrepancies depend on (a) the number of H-bonds, (b) the network of H-bonds, (c) the configuration of the isomers (e.g. quasi-planar, distortion-planar and three-dimensional), (d) whether the amino nitrogen atom is involved in hydrogen bonding and (e) the orientation of free hydrogen atoms. These factors determine which isomer is the global minimum. The non-monotonic change in enthalpy is similar to the variation of binding energy with the ZPE correction.

Fig. 7 Comparison of the binding energies, enthalpies and Gibbs free energies of the global minimum for the stepwise hydration of DMA at the DF-MP2-F12/CBS (limit DZ, TZ) level [(CH₃)₂NH(H₂O)_n−1 + H₂O → (CH₃)₂NH(H₂O)_n].

The non-monotonic change in ΔG with the addition of each successive water molecule (Fig. 7) is different from the change in the binding energy. For (CH₃)₂NH(H₂O)₃, the most stable structure corresponds to ΔG being minimized and the maximum Gibbs energy of interaction is obtained when the fifth water molecule is added. Tables 2 and 3 show that all the reactions involving the addition of water monomers are slightly exothermic. Due to the types of HB networks and the length of HBs, there is no absolute correlation between the contribution of enthalpy and the number of hydrogen bonds. The addition of more water molecules results in a cage of more H-bonds and in most cases the structures with lowest free energy have the fewest H-bonds because of entropic cost. Up to this point, clusters with up to six water molecules have been discussed. Thermodynamically, the formation of these structures is clearly unfavorable. The magic number of water molecules that is thermodynamically favored remains unknown. Because of the positive free energy, the equilibrium hydrate distributions are very small and these clusters cannot form spontaneously at 298 K and 1 atm. However, this study can act as a reference for the study of ternary nucleation systems containing DMA, sulfuric acid and water molecules. The study by Xu et al.³³ shows that, although the formation of DMA hydrates is not favorable, the formation of H₂SO₄–DMA–(H₂O)_n clusters is more favorable via the collision between H₂SO₄ and DMA–(H₂O)_n clusters than between H₂SO₄–(H₂O)_n clusters, DMA, and more water molecules.

3.4 Temperature dependence of the conformational population and Gibbs free energy

The contribution from the global minimum is important for the ensemble of energetically accessible conformations because it has the largest population. However, as the cluster systems become larger and the configurations more complex, the difference in energy between the global minimum and low local minima becomes smaller due to the flexible HB networks. These results indicate that the local minima might be as significant as those with larger weighting. The dependence on temperature of the thermodynamics is also an important parameter that could change the order of stability of the global minimum and the local minimum as well as the weighting of the isomers. Thus, the effects of temperature on the contributions of lower local minima were investigated in this study, which could give insight into the relative stabilities of the different isomers. The relevant computational methods are given in our earlier study.⁷⁰

The variation of the population of the isomers versus the temperature is presented in Fig. 8. For (CH₃)₂NH(H₂O), the proportion of the global minimum of 1a is less than that of isomer 1b, and as the temperature increases, the weighting of 1b increases monotonically. The difference between the two isomers gradually increases. Fig. 8a shows the results for (CH₃)₂NH(H₂O)₂ clusters. The weighting of the global minimum of 2a decreases slightly from 212.6 to 249.9 K, reaching a maximum of 62.92% and then decreasing to a minimum of 60.01% at 298.15 K. In the case when n = 3, 3b has the largest proportion and this increases slowly for the entire time. Moreover, the total population of 3a, 3b and 3c is approximately 100%, which indicates that a cyclic structure with four H-bonds is the most stable structure for trihydrates. Fig. 8c presents the conformational population of isomers for n = 4. The populations of the local minima increase, whereas that of the global minimum decreases monotonically as the temperature increases. A strong entropy effect can be observed in 4h for the population with the largest entropy. For the case when n = 5, given in Fig. 8d, our results predict a monotonic increase for 5i, 5l, and 5m with respect to temperature. The growth rate of 5l is clearly higher than that of the other isomers and 5l has the largest proportion above a temperature of 259.3 K, which indicates that planar structures play a leading role at high temperatures. The populations of 5d and 5a fall slightly and significantly, respectively. However, 5d is more heavily populated than 5a as the temperature increases. The weightings for other isomers almost remain constant and their proportions are so small as to be negligible. In the case of n = 6, the weighting of the global minimum is the largest among all the isomers in the temperature range from 212.6 to 298.15 K. Similarly, the population of 6a decreased dramatically from 72.44% to 35.39%. The local minima, except for isomer 6b, increase as the temperature increases. In conclusion, the weightings of the global minima and local minima change as the temperature increases. The effects of temperature could contribute to the alteration of the order of stability of the isomers.

Fig. 8 Conformational population changes for the low-energy isomers of (CH₃)₂NH(H₂O)_1–6versus temperature: (8a) (CH₃)₂NH(H₂O)_1,2; (8b) (CH₃)₂NH(H₂O)₃; (8c) (CH₃)₂NH(H₂O)₄; (8d) (CH₃)₂NH(H₂O)₅; and (8e) (CH₃)₂NH(H₂O)₆.

The dependence on temperature of Gibbs free energy for the formation of (CH₃)₂NH(H₂O)_n=1–6 clusters with respect to different temperatures (200–300 K) was investigated. Fig. 9 shows the minimum thermodynamics for stepwise hydration of (CH₃)₂NH(H₂O)_n=1–6. There is a strong dependence on temperature for the formation of (CH₃)₂NH(H₂O)_n=1–6 clusters. The lowest free energy shifts from n = 6 below 223.7 K to n = 3 around 230 K. As the temperature falls below 223.7 K, the stepwise addition from one to six water molecules into DMA clusters is favorable. As the temperature increases, the clusters become less favorable and as the temperature reach 298.15 K, all the clusters become unfavorable. According to Kim et al.,⁹¹ it is easy to change the stabilities of clusters containing multiple hydrogen bonds when the temperature rises, which is a result of the entropy effect. Although hydrated DMA clusters are favorable at low temperatures, these clusters could not form because of the combined effect of Gibbs free energies with small negative values and the low relative concentration of DMA in various atmospheric conditions.

Fig. 9 Gibbs free energy changes (in kcal mol⁻¹) for the minimum thermodynamics for stepwise hydration of (CH₃)₂NH(H₂O)_n=1–6 clusters at different temperatures.

3.5 Optical properties

Aerosols weaken light by absorbing and scattering and extinction properties have a great effect on atmospheric visibility and radiative forcing.^92,93 Rayleigh scattering is the principal form of scattering for molecules, clusters, and small particles with diameters much smaller than the wavelength of the light. Although the Rayleigh scattering properties of large particles are well understood, the Rayleigh scattering properties of clusters, especially at the molecular level, are still not known. In this study, the Rayleigh scattering properties of (CH₃)₂NH(H₂O)_n=1–6 clusters were investigated for the first time.

Small clusters with up to six water molecules exhibit slightly higher binding isotropic mean polarizabilities [small alpha, Greek, macron]_binding in the range of −0.02–2.43 a.u., as shown in Table S2,† which is attributed to the small hydrogen-bonding network when the number of water molecules is less than 6. The isotropic mean polarizabilities [small alpha, Greek, macron] are quite dependent on the number of water molecules and vary linearly, as shown in Fig. 10a, similarly to the study of methanol clusters⁹⁴ and chloride ion hydration systems,⁶⁶ and increase sharply from 47.78 to 96.86 a.u. From Fig. 10b, it is quite clearly observed that the Rayleigh scattering intensities of natural light [scr R, script letter R]_n dramatically increase with an increase in the number of water molecules, following the trend of a second-order polynomial with correlation coefficient 0.99919. This increasing trend can be attributed to the gradually increasing isotropic mean polarizabilities [small alpha, Greek, macron], which will dominate the Rayleigh scattering.

Fig. 10 Rayleigh light scattering and polarizability properties of clusters: (a) isotropic mean polarizabilities as a function of number of water molecules; (b) Rayleigh light scattering intensities as a function of number of water molecules; (c) anisotropic polarizabilities as a function of number of water molecules; (d) depolarization ratios as a function of number of water molecules.

From Fig. 10c and d, the anisotropic polarizabilities Δα and depolarization ratios of natural light σ_n of (CH₃)₂NH(H₂O)_n=1–6 clusters can be observed as a function of the number of water molecules in the clusters. It is observed that the anisotropic polarizability Δα decreases from a monomer to a dimer and then increases from a dimer to a tetramer. In the case of (CH₃)₂NH(H₂O)_5–6, Δα decreases slightly then decreases by a large step from a pentamer to a hexamer. The calculated depolarization ratio of natural light σ_n has a similar variation trend to the anisotropic polarizability Δα, which is due to an increase in the isotropic mean polarizability [small alpha, Greek, macron] in combination with the anisotropic polarizability Δα; this plays the leading role and has a positive correlation with σ_n. In Fig. 2–5, it is clearly observed that clusters become quasi-planar ring structures with the addition of a second water molecule; however, the conformation changes to be a cage-like three-dimensional structure for pentamers. Therefore, this is consistent with what is to be expected as the cluster changes from a molecular cluster into a spherical isotropic particle.

Observations by Elm et al.,⁸⁴ who studied the Rayleigh scattering properties of atmospheric pre-nucleation systems, found that all different conformations within each cluster yield very similar isotropic mean polarizabilities [small alpha, Greek, macron] within 2 a.u. Moreover, from other studies^94,95 of Rayleigh scattering properties, remarkable differences between different conformations have been studied, which indicate that the anisotropic polarizability Δα and depolarization ratio σ could be alternative parameters depending on the constituents and conformations. In our study of the ten isomers of (CH₃)₂NH(H₂O)₅, it was observed that even though 5a, 5c, 5e, 5h, 5i, and 5j exhibit different hydrogen-bonding networks with various numbers of N–H and O–H hydrogen bonds, the difference in [small alpha, Greek, macron]_binding among these isomers is rather small, within 3 a.u., as shown in Table S2,† which is smaller for the isomers 5c, 5d, 5f, and 5g with the same bonding pattern. For the depolarization ratios of (CH₃)₂NH(H₂O)₅, the case is similar. Therefore, this could imply that the anisotropic polarizability and depolarization ratio cannot be regarded as alternative parameters, at least for hydrated DMA clusters, which is similar to the cases of methanol⁹⁴ and chloride ion hydration systems.⁶⁶ However, the anisotropic polarizabilities Δα_binding are highly isomer-dependent and fluctuate from −3.49 to 0.79 a.u. for the ten isomers of (CH₃)₂NH(H₂O)₅, which have been observed in sulfuric acid hydration system.⁸⁴

4. Conclusions

We have investigated the structural evolution, non-covalent interactions, temperature effects, and optical properties of hydrated DMA clusters. The global and low-lying local minima of isomers were optimized at the PW91PW91/6-311++G(3df,3pd) level. The thermodynamics properties were further corrected using DF-MP2-F12/CBS//PW91PW91/6-311++G(3df,3pd). Electron density analysis shows that the interactions of these complexes are mainly medium hydrogen bonds, but are not substantial enough to overcome the energy barrier for the reaction of hydration spontaneously in the standard state of 1 atm pressure and 298 K. The dependence on temperature of the conformational population and the Gibbs free energy for the formation of (CH₃)₂NH(H₂O)_1–6 clusters were investigated. A weak dependence on temperature was found for the formation of (CH₃)₂NH(H₂O)_n=1–6 clusters. Hydrated DMA clusters are favorable with negative energy values at low temperatures, especially for (CH₃)₂NH(H₂O)₃, but these clusters may be difficult to form because of the combined effect of Gibbs free energies with small negative values and the low relative concentration of DMA in various atmospheric conditions. This implies that hydrated DMA clusters are difficult to form spontaneously in the atmosphere. Finally, the Rayleigh scattering properties have been investigated. It appears that the isotropic mean polarizabilities display a linear relation, whereas the Rayleigh scattering intensities of natural light follow a second-order polynomial trend as the size of the clusters increases. Furthermore, the anisotropic polarizability and depolarization ratio cannot be regarded as alternative parameters, at least for hydrated DMA clusters. Large hydrated DMA clusters need further study and this study could also provide help for the study of ternary nucleation systems containing DMA, sulfuric acid and water molecules.

Acknowledgements

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

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c5ra11462d

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