DFT study on the mechanism of trimolecular radical reactions: isomerisation in small clusters

ABSTRACT Structural and thermodynamic properties of 48 trimolecular clusters containing one radicl and two protic molecules (H2O, NH3, H2O2, CH3OH, HOCl) were studied at B3LYP/6-311++G(3df,3pd) level of theory. These radical-clusters have non-cyclic structures and are stabilised via two inter-molecular hydrogen bonding interactions. The calculated enthalpies of formation of the radical-clusters were generally in the range of −30 to −50 kJ/mol. The calculated activation energies (Ea) of the intra-cluster hydrogen transfers were smaller than 70 kJ/mol. Also, structures and thermodynamics of 15 cyclic molecular clusters as well as multi-hydrogen transfers in them were investigated. The results showed that the stability of the cyclic clusters and activation energies of the multi-hydrogen transfers depend on the cluster size.

Water molecules interact with different anions and cations and form clusters such as OH − (H 2 O) n [14,15], (HOO − )(H 2 O) n [16], F − (H 2 O) n [17], Cl − (H 2 O) n [18], H + (H 2 O) n [19,20]  bonding interactions between water molecules and open shell radicals cause formation of radical-water clusters in earth's atmosphere [21]. The most important radical-waterThe most important radical-water clusters are HO x -H 2 O, HOCO-H 2 O, NO x -H 2 O and ClO-H 2 O which have been studied, previously [22][23][24][25]. Also, theoretical and experimental studies show that hydration of radicals catalyses their atmospheric reactions [26][27][28] [26]. In the both cases, the reactants form pre-reactive complexes containing three molecules, H 2 O, H 2 O 2 and OH. Then, these trimolecular complexes or clusters undergo hydrogen transfer reactions. It has been reported that hydration of molecules can also increase the energy barriers of hydrogen transfers, depending on the reaction mechanism [26]. In this work, thermodynamics of formation of 48 trimolecular clusters containing some important radicals and H 2 O, CH 3 OH, H 2 O 2 , HOCl and NH 3 molecules are studied and stability of the clusters and catalytic effect of these protic molecules are investigated.

Computational details
Two types of clusters were studied: the clusters which contain radical species and clusters formed from neutral molecules such as H 2 O, CH 3 OH, NH 3 and HF. The cluster structures were fully optimised employing DFT method using B3LYP functional [29]. DFT methods have been used to study mechanism and kinetics of many radical reactions [30][31][32]. The studies show that the results of these DFT calculations are satisfactory [32]. In a study on the self-reactions of OOH radicals using CCSD(T), MP2, B3LYP and G2M methods, it was found that the energy barriers calculated by the B3LYP are in better agreement with those obtained by the CCSD(T) method [32]. The calculations were performed using the 6-311++G(3df,3pd) and 6-311++G(d,p) basis sets. The frequency calculations were performed at the same level of theory and basis sets to obtain the thermodynamic data. All the thermodynamic and kinetic data including H, G, activation energies (E a ) and G # , were corrected for zero point energy (ZPE), however, both ZPEcorrected and uncorrected data were reported. The transition state (TS) structures were simply optimised using OPT = TS keyword, then an intrinsic reaction coordinate [33] calculation was performed to confirm the accuracy of the TS structure. Gaussian 09 software was used for all the calculations [34]. Figure 1 shows the optimised structures of the trimolecular radical-clusters formed form H 2 O, H 2 O 2 , HOCl, CH 3 OH and NH 3 molecules and OH, OOH, OCl, CH 3 O and NH 2 radicals. Geometrical parameters of the optimised radical-clusters have been collected in Supplementary Material section.

Non-cyclic radical-clusters
We considered radical-clusters with only three species, two molecules and a radical. The clusters that can convert to each other have been shown by a and b letters. For example, the cluster 1a is NH 2 …H 2 O …HOCl which can be converted to NH 3 …H 2 O …OCl (1b) via intra-cluster proton transfer. In some clusters, 4, 5 and 6 for example, the intra-cluster hydrogen transfers do not result in different cluster (isomer), therefore, these clusters are shown without a and b letters. Furthermore, the clusters were optimised so that there are only two hydrogen bonding interactions in each cluster, hence, they do not have cyclic structures.
The radical-clusters are mainly formed from two molecules and a free radical which are stabilised through two hydrogen bonding interactions. However, the clusters 2b and 28b contain two radical in their structures. The enthalpy ( H), Gibbs free energy ( G) and entropy ( S) of the formation of the radical-clusters at 298.15 K have been tabulated in Table 1. In fact, the calculated data explain thermodynamics of formation of the two hydrogen bonds among the molecules and the radical. In other words, the H, G and S are the stabilisation enthalpy, free energy and entropy for the radicalclusters. Both the ZPE-corrected and uncorrected energies are reported. Since the S values were computed from the G = H − T S equation the ZPE-corrected and uncorrected values were the same. The enthalpies of the cluster formation are negative (exothermic) except for the clusters 2b and 28b whose formations are endothermic. Interestingly, the clusters 2b and 28b are the only clusters containing two radicals. Therefore, presence of the radicals decreases the stability of the clusters, probably, because of their low tendency in participation in the formation of hydrogen bonds. The G values are positive indicating that the formation of the clusters is not favoured, thermodynamically. The positive values of G are due to decrease in the entropy during the cluster formation from corresponding free molecules. The change in entropies is almost the same (about −220 J/K mol) for the formation of the all clusters.
The clusters a and b in Figure 1 can be considered as two isomers of a cluster. Therefore, these isomers can be converted to each other via two synchronous intra-cluster hydrogen transfers. In other words, the centric molecule can be considered as a catalyst for the hydrogen transferring from the molecule to the radical. Figure 2 shows the TS structures of these intra-cluster hydrogen transfers. Geometrical parameters of the TS structures have been collected as Supplementary Materials.
The activation energies (E a ), activation free energies ( G # ) and entropies ( S # ) of the intra-cluster proton transfers (isomerisation) in the radical-clusters have been summarised in Table 2. These activation energies were calculated from the energy differences between the TS structures in Figure 2 and the clusters shown in Figure 1. Also, Table 2 represents the imaginary frequencies (ν) of the TS structures. The ν values are generally in the range of 1500-1800 cm −1 which correspond with stretching vibrations. The activation energies are generally more than 40 kJ/mol indicating that the isomerisations or hydrogen transfers occur but hardly. However, for some isomerisations, such as 3b→3a, the energy barriers are small and the proton transfers take place more easily. The TS structures ( Figure 2) are more compact than the structures of the clusters (Figure 1), in other words, the bond lengths in the TS structures are shorter than those in the cluster (see Supplementary Materials). This change in the structures during the isomerisation is the reason of the negative values of the calculated S # in Table 2.
Activation energies of keto-enol and imine-amine tautomerisms, which are intramolecular proton transfer reactions, are usually larger than 120 kJ/mol [35]. However, when the tautomerisms are catalysed by a protic molecule (H 2 O mainly), the activation energy decreases about 100 kJ/mol [35]. The water-catalysed tautomerisms are very similar to the intra-cluster proton transfers ( Figure 2) because in both reactions a protic molecule acts as a bridge to transfer a hydrogen atom from the donor site to the acceptor site. Do the middle protic molecules in the radical-clusters of Figure 1 catalyse the hydrogen transfer between the outer molecules? To explore it, the energy barriers of hydrogen transfers between the molecules H 2 O, CH 3 OH, HOCl, NH 3 and H 2 O 2 were calculated. The calculated energy barriers have been summarised in Table 3. As seen, the energy barriers of the direct hydrogen transfers are smaller than those for the protic molecule-assisted hydrogen transfers.In other words, the protic molecules in the considered systems not only do not catalyse the hydrogen transfers but also increase the energy barriers of the hydrogen transfers. These results are in agreement with those reported by Buszek et al. [26]. They found that water molecule actually slows down the hydrogen abstraction form H 2 O 2 by OH radical [26]. Other than hydration, protonation also increases the enthalpy of C-H dissociation in alcohols [36]. Therefore, although protic molecules catalyse the keto-enol tautomerisms, it cannot be generalised to the hydrogen abstractions in the radical reactions. In a keto-enol tautomerism in the absence of water molecules, the proton is transferred through a tetragonal ring in the TS structure which is very unstable. When a water molecule catalyses the tautomerism, a TS structure containing a hexagonal ring is formed which is more stable and proton is transferred through this ring [35]. In the direct hydrogen   transfer between a molecule and a radical, only one hydrogen is transferred and one bond is broken while in the trimolecular reactions two hydrogens are transferred and two bonds are broken in the same time. Therefore, the energy barriers of the trimolecular reactions are higher than those for the direct hydrogen transfers. It should be mentioned that we cannot use the H values in Table 1 to compare the stability of two isomers (a and b), because these values are enthalpies of formation of isomers a and b from different reactants. For instance, the H of formation of the cluster 3b from H 2 O, OH and HOCl is more negative than that for formation of the cluster 3a from 2H 2 O and OCl, however, comparison of the energies shows that the cluster 3a is more stable than 3b. We do not report the energies of the clusters because these values and consequently the relative stability of the clusters a and b can be obtained from the difference of activation energies for the forward and backward reactions. The E a values of 3a→3b and 3b→3a are 106.48 and 21.13 kJ/mol, respectively, therefore, the cluster 3a is more stable than the 3b by about 85 kJ/mol.

Molecular cyclic clusters
Protic molecules such as H 2 O, NH 3 , CH 3 OH and HF can form clusters through inter-molecular hydrogen bonds. Many clusters can be formed with different structures and these structures, mainly water clusters, have been well studied [1][2][3][4][5][6][7][8][9][10][11][12][13]. Here, we investigate isomerisation and multi-proton transfers in some of these clusters and compare them with the radical-clusters. Isomerisations in the (H 2 O) 2-6 , (NH 3 ) 2-4 and (HF) 2-6 clusters have been studied in detail by Karton et al. [37]. Figure 3 shows the optimised structures of small cyclic clusters of H 2 O, NH 3 , CH 3 OH and HF. The geometrical parameters of the structure of the clusters shown in Figure 3 are brought about as Supplementary Materials. The letters a and b are two isomers of each cluster which have very similar structures (comparable with enantiomers) and can be converted to each other via multi-proton transfers.
The enthalpies ( H), free energies ( G) and entropies ( S) of formation of the clusters shown in Figure 3 have been tabulated in Table 4. The formation enthalpies obtained by the 6-311++g(d,p) and 6-311++G(3df,3pd) basis sets differ a few kJ/mol for the H 2 O, NH 3 and CH 3 OH clusters, while the results obtained by the two basis sets are the same, in the case of the HF clusters. As the size of the clusters increases the H values increase, because of increase in the number of the inter-molecular hydrogen bonds. The enthalpies of formation of the trigonal, tetragonal, pentagonal and hexagonal clusters of waters, calculated at the B3LYP/6-311++G(d,p) level, are −53.5, −104.6, −139.4 and −176.9 kJ/mol, respectively. The H values for trigonal clusters of H 2 O, NH 3 , CH 3 OH and HF are −53.5, −31.2, −53.9 and −54.9 kJ/mol, respectively, which are the same except for NH 3 . The same behaviour is observed for the tetragonal structures. Since the number of the hydrogen bonds in the clusters is the same, it is concluded that the hydrogen bonding interaction between NH 3 molecules is weaker than that for the other molecules. The cluster VIII is a heterogeneous trigonal cluster of two H 2 O and one NH 3 whose H is −55.4 kJ/mol which is comparable with pure trigonal clusters of H 2 O, CH 3 OH and HF. The Table . The calculated activation energies (E a ), free energies ( G # ) and entropy ( S # ) of the intracluster hydrogen transfers in the radical-clusters and imaginary frequencies (ν) of the TS structures of Figure . E ) and free energies ( G # ) of the isomerisation in the small clusters shown in Figure  and imaginary frequencies (ν) of the TS structures of Figure .  Therefore, it is expected that the entropies of the TS structures be smaller than those for the molecular clusters. All the S # values in Table 5 are negative and are in the range of −35 to −70 J/K mol. Furthermore, there are no relationship between the size of the clusters and their S # values.
The activation energies (E a ) and free energies ( G # ) of the multi-proton transfers (isomerisation) in the molecular clusters of Figure 3 are summarised in Table 5. The trend of the activation energies of the proton transfers in the water clusters is as: trigonal > tetragonal pentagonal < hexagonal, i.e. the tetragonal and pentagonal clusters have the lowest activation energies.

Conclusion
Generally, water molecules are employed as catalyst to assist the hydrogen transfer in the study of radicalmolecule reactions. This study showed that catalytic effects of CH 3 OH, HOCl, H 2 O 2 and NH 3 are as well as water, therefore, these protic molecules can be considered for clustering with the radicals and assisting the hydrogen transfers in the radical-molecule systems.
During the hydrogen transfer, the distance of acceptor and donor atoms in the TS structures becomes shorter, hence, the TS structures are more compact than the clusters and consequently, the S # values are negative. In the cyclic clusters, the activation energies depend on the cluster size and geometrical parameters of the TS structures. Hydrogen transfers in tetragonal and pentagonal water clusters occur more easily than trigonal and hexagonal clusters.