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\title{Supporting information for \\ Decomposition pathways of isoprene-derived hydrotrioxides and their clustering in the atmosphere}
\author{Emelda Ahongshangbam, Lauri Franzon, Thomas Golin Almeida,\\ Galib Hasan, Benjamin N. Frandsen, Nanna Myllys}
\date{}

\begin{document}
\maketitle

\subsection*{Benchmarking of computational methods: Detailed balance approach}

Another way to benchmark the accuracy of our computations is to use the experimental decomposition rate of $1.1\times 10^{-4}\rate$ at 300 K with a temperature-dependence of \\ $k(T)\approx 1.9\times10^{10} T^{1.35}\exp{(-\frac{12000~\mathrm{K}}{T})}\rate$ provided by \citet{assaf2018experimental}. Seeing which of the decomposition energies in Table 2 ( in main text) is able to best reproduce this value would provide additional hints as to where the observed systematic error in the decomposition energies comes from. Computational decomposition rates were determined using canonical detailed balance:

\begin{equation}
     k_d = k_a\frac{[\ce{RO}][\ce{HO2}] }{[\ce{ROOOH}]} = k_a\frac{p}{k_BT}\exp{\left(\frac{G_{\ce{ROOOH}}-G_{\ce{HO2}}-G_{\ce{RO}}}{k_BT}\right)} 
\end{equation}

where $\frac{p}{k_BT}$ is the ideal gas concentration, $G_X$ are the Gibbs free energies of the system $X$, and $k_a$ is the association rate of the two radicals, which was calculated using the Long-Range Transition State Theory dipole-dipole capture rate\cite{georgievskii2005long} (presented in atomic units as in the source):

\begin{equation}
    k_a = \frac{C}{\sqrt{\mu}}(d_{\ce{HO2}}d_{\ce{RO}})^{\frac{2}{3}} T^{-\frac{1}{6}}
\end{equation}

where $C$ is a constant provided by \citet{georgievskii2005long}, $\mu=\frac{m_{\ce{RO}}m_{\ce{HO2}}}{m_{\ce{RO}}+m_{\ce{HO2}}}$ is the reduced mass, and $d_X$ are the dipole moments. Values for the dipole moments used were $d_{\ce{HO2}}=2.20020$ Debye and $d_{\ce{RO}}=2.07234$ Debye, determined using using B3LYP\cite{stephens1994ab}/aug-cc-pVTZ, which has been proved to be a reliable method when it comes to dipole moments\cite{hickey2014benchmarking}. The resulting association rate at 300 K was $k_a = 4.92\times10^{-10}\bimole$. The $T$-dependence of the Gibbs free energy for each individual system may be expressed as $G(T)= E + k_BT(1-\ln{Q(T)})$, where the M06-2X list of frequencies was used for the $T$-dependence of the partition functions $Q$ at all levels of theory, as this quantity is only a negligible source of error. The resulting decomposition rates are presented in Figure \ref{Decrates}.

\begin{figure}[H]
    \centering
    \includegraphics[width=0.75\textwidth]{SI_Fig1_decompRatebenchmark_detailed_ balance.jpg}
    \caption{Comparison of the \citet{assaf2018experimental} experimental $k_d(T)$ with detailed balance rates calculated using the UCCSD(T) and CASPT2(14,12) energies from \citet{berndt2022hydrotrioxide}, the UCCSD(T)//M06-2X-D3 energies from \citet{muller2016reaction} and our DLPNO//DFT energies.}
    \label{Decrates}
\end{figure}

\subsection*{Proposed decomposition channel of isoprene-derived hydrotrioxides.}
We also investigate an alternative decomposition channel pathway, say pathway (3) which involves the cleavage of the oxygen-oxygen bond of trioxide moiety and subsequent transfer of the -OOH group to the nearest carbon center followed by intra-molecular hydrogen shift to form R(OH)(OOH) as seen in the Figure \ref{pathway3_reactions}. The intermediate [R-(OH)(OHO)] generated in the pathway possesses two opposite charges and is more stable than their respective reactant system. It is noteworthy to mention that among all the isoprene-derived hydrotrioxides investigated, only three isomers namely; \ce{R^{1}}-OOOH, \ce{R^{4}}-OOOH and \ce{R^{5}}-OOOH  as well as methyl-trioxide \ce{CH3OOOH} exhibit to undergo this decomposition channel which was confirmed by the IRC calculations. While the rest of the isomers deviate to form respective products with C=O complexes and hydrogen peroxides (\ce{H2O2}), their initial barrier heights are found to be extremely large. Therefore, we abstain from further calculation for this particular mechanism in this study.\\

\begin{equation}\label{rx:R1}
\ce{ROOOH -> R(OH)(OHO)  }
\end{equation}
\begin{equation}\label{rx:R2}
\ce{R(OH)(OHO)   ->R(OH)(OOH)} 
\end{equation}
      
\begin{figure}[H]
    \centering
    \includegraphics[width=0.75\textwidth]{SI_Fig2_pathway3_reactions.jpg}
    \caption{Proposed decomposition mechanism (3) of selected isomers of isoprene derived hydrotrioxides.}
    \label{pathway3_reactions}
\end{figure}

         
Table \ref{pathway_3_values} provides the barrier energies (TS - reactant), reaction energy (final product - reactant), and unimolecular rate coefficients of both the pathway (3.1) and pathway (3.2). The energetics were calculated in the same workflow as described in the method section of the main text at the $\omega$B97X-D/6-31+G* and single-point energy corrections at the DLPNO-CCSD(T)/aug-cc-pVTZ level. For investigating the rates, we employed traditional transition state theory with a quantum-chemical tunneling coefficient taking into account (the equation given in the main text). As we can see from Table \ref{pathway_3_values}, the barriers to form the intermediate (first step itself) are so high, although the subsequent intramolecular H-transfer is quite fast. This infers that the reaction is unlikely to occur in the gas phase, however, perhaps the reaction could catalyzed in an aqueous phase or in a particle surface. On top of this pathway, it is worth noting that we briefly calculated ring-closing reactions of isoprene-derived hydrotrioxide (in particular, \ce{R^{1}}-OOOH) proceeding via intra-molecular H-shift transfer to form either five or six-membered ring structures at DFT level theory. Preliminary results indicate that this is infeasible in the atmosphere, due to extremely high energy barriers. Therefore, we cease to investigate from further study nor present its energetics in this manuscript.
\begin{table}[htp]
    \centering
\caption{Zero-point energy corrected energy barriers ($\Delta E^{TS}$ in kcal mol$^{-1}$) and unimolecular rate coefficients $k_{uni}$ (in s$^{-1}$) with the reaction energy (in kcal mol$^{-1}$).\\}    
    \begin{tabular}{l|cc|cc|c}
\hline
Molecules& Pathway (3.1)	& &Pathway (3.2) & & Reaction Energy\\
& $\Delta E^{TS}$& $k_{uni}$& $\Delta E^{TS}$& $k_{uni}$& (Final product - reactant) \\
\hline  
C\ce{H_{3}}OOOH	& 55.1	&	$2.4\times10^{-28}$	&	6.4	&	$1.4\times10^{9}$	&	-54.7\\
\ce{R^{1}}-OOOH	& 62.7	&	$6.8\times10^{-34}$	&	7.7	&	$1.5\times10^{8}$	&	-46.4\\
\ce{R^{4}}-OOOH	& 52.0	&	$4.8\times10^{-26}$	&	6.4	&	$2.5\times10^{9}$	&	-49.5\\
\ce{R^{5}}-OOOH	& 54.6	&	$5.5\times10^{-28}$	&	9.6	&	$1.1\times10^{7}$	&	-53.8\\

\hline            
    \end{tabular}
    \label{pathway_3_values}
\end{table}

\subsection*{Thermodynamic data for the two decomposition pathways}
Table \ref{decom_reaction-energy dlpno} represents the unimolecular reaction energy (product-reactant) in terms of electronic energy and Gibbs-free energy (both in kcal mol$^{-1}$) associated with two main decomposition pathways studied in the work. They were calculated using DLPNO–CCSD(T)/aug-cc-pVTZ//$\omega$B97X-D/6-31+G*. As pathway (1) deals with the fragmentation process leading to radical generation, the formation reaction energy is positively high due to higher product energy than that of the reactant. Although, thermodynamically unfavorable via a non-spontaneous channel, this pathway has favorable kinetics with rate coefficients competitive in the atmosphere. Considering the evolution of the molecular oxygen either on the triplet or singlet potential energy surface, the dissociation energy in the case of the pathway (2) shows distinguishable ranges both in terms of electronic energy and Gibbs free energy. The results exhibit that the reaction proceeding via a triplet potential energy surface is considerably more favorable than that of a singlet potential energy surface with a stark difference of over 25 kcal mol$^{-1}$. It is to be noted that the single-point corrected electronic energies of some of the products on the singlet potential energy surface have energies higher than those of the reactant energies.

\begin{table}[htp]
    \centering
\caption{Zero-point energy corrected decomposition reaction energy (\ce{$\Delta E_d$} in kcal mol$^{-1}$) and Gibbs free energy (\ce{$\Delta G_d^{\#}$} in kcal mol$^{-1}$) for all the isomers. Note: the superscripts (1) and (3) under pathway (2) signify the electronic state of molecular oxygen taken into consideration while calculating the total product energy .\\}
    \begin{tabular}{l|cc|cccl}
\hline
Molecules	&	Pathway (1) 	&		&	Pathway (2)		\\
	& $\Delta E_d$	& $\Delta G_d^{\#}$ &	$^{3}[\Delta E_d]$ & $^{3}[\Delta G_d^{\#}]$&	$^{1}[\Delta E_d]$	&$^{1}[\Delta G_d^{\#}]$  \\
\hline
HOOOH		&	29.4	&	20.4	&	-38.8	&	-38.8	&	-7.9	&	-15.3	\\
C\ce{H_{3}}OOOH		&	25.1	&	13.9	&	-29.9	&	-29.9	&	1.0	&	-8.0	\\
\ce{R^{1}}-OOOH		&	26.7	&	13.7	&	-31.7	&	-31.7	&	-0.8	&	-11.1	\\
\ce{R^{2}}-OOOH		&	26.6	&	14.5	&	-29.2	&	-29.2	&	1.8	&	-7.6	\\
\ce{R^{3}}-OOOH		&	27.0	&	14.2	&	-28.9	&	-28.9	&	2.0	&	-7.9	\\
\ce{R^{4}}-OOOH		&	26.1	&	13.2	&	-29.6	&	-29.6	&	1.4	&	-9.1	\\
\ce{R^{5}}-OOOH		&	26.3	&	14.8	&	-31.4	&	-31.4	&	-0.4	&	-9.4	\\
\ce{R^{6}}-OOOH		&	26.7	&	14.3	&	-27.8	&	-27.8	&	3.2	&	-6.9	\\
\ce{R^{7}}-OOOH		&	30.7	&	16.7	&	-30.1	&	-30.1	&	0.9	&	-9.8	\\
\ce{R^{8}}-OOOH		&	26.8	&	14.2	&	-32.0	&	-32.0	&	-1.1	&	-11.6	\\
\hline
    \end{tabular}
    \label{decom_reaction-energy dlpno}
\end{table}

Table \ref{augccqz dlpno} presents the ZPE corrected barriers using augmented correlation consistent triple and quadrupole basis sets in DLPNO-CCSD(T) single point energy calculation.\cite{Woon1993} Our benchmark results confirm that aug-cc-pVTZ basis set with DLPNO-CCSD(T) method is suitable when calculating transition state energies.
\begin{table}[htp]
    \centering
\caption{Comparison of aug-cc-pVTZ and aug-cc-pVQZ basis sets with DLPNO-CCSD(T) method to calculate energy barrier (in kcal mol$^{-1}$) for \ce{CH3OOOH} system for the decomposition pathway (2). $E^{(TS-ROOOH)}$ is the transition state for one-step reaction and $E^{(TS-Complex)}$ is the transition state via complex (\ce{CH3O}...\ce{HO2}) intermediate. Note: zero-point energy correction has been calculated at the $\omega$B97X-D/6-31+G* level.\\}
    \begin{tabular}{l|cc|c}
\hline
	Single-Point  & $E^{(TS-ROOOH)}$ & $E^{(TS-Complex)}$ & Reference	\\  
\hline
DLPNO-CCSD(T)/aug-cc-pVTZ & 43.1 & -&(Table 3) \\
DLPNO-CCSD(T)/aug-cc-pVQZ & 43.6 & -&\\
DLPNO-CCSD(T)/aug-cc-pVTZ & - & 3.0 & (Table 4)\\
DLPNO-CCSD(T)/aug-cc-pVQZ & - & 2.7 &\\
\hline
    \end{tabular}
    \label{augccqz dlpno}
\end{table}



\subsection*{Clusters of hydrotrioxides consisting of multiple monomeric units}
Molecular structures of hydrotrioxide clusters with three monomeric units are shown in Figure \ref{trimer_ch3oooh} and \ref{trimer_roooh} and their respective single-point energy corrected Gibbs binding energies are presented in Table \ref{trimer-energy} along with the one computed at the DFT level of theory is presented at the Table \ref{trimer- energy-dft}. Table \ref{trimer-energy} in which the values computed at the high-level CCSD(T), quantum chemical method illustrates that the larger organic trioxide of ROOOH has a stronger binding capability than the methyl-trioxide moiety. This can be explained by the higher stabilization energy possessed by the larger trioxide, contributed by the high excess energy in the formation of ROOOH. However, it is very unlikely that the reaction cross section will be high enough to have reactions of trimer formation to be feasible in the atmosphere. At the same time, the trend of binding capabilities is in alignment with that of a cluster consisting of two molecular units ( presented in the main text ), where the ability to bond is higher in the case of acidic molecules than with the bases and neutral molecules.

\begin{table}[htp]
    \centering
\caption{Gibbs free binding energy ($\Delta G ^{Cluster}_{D}$ in kcal mol$^{-1}$) calculated at the DLPNO–CCSD(T)/aug-cc-pVTZ//$\omega$B97X-D/6-31++G** level at 298 K of C\ce{H_{3}}OOOH, and ROOOH with two monomeric units of other atmospheric relevant molecules. Note: MA : methylamine (\ce{CH3NH2}), DMA : dimethylamine (\ce{(CH3)2NH}), TMA : trimethylamine (\ce{(CH3)3N}), SA : sulfuric acid (\ce{H2SO4}), NA : nitric acid (\ce{HNO3}), and FA : formic acid (\ce{HCOOH}). \\}    
    \begin{tabular}{l|c|c}
\hline
Cluster type	& C\ce{H_{3}}OOOH	&	ROOOH	\\
\hline  
water	&	3.66	&	0.71	\\
\ce{NH3}	&	3.85	&	1.82	\\
MA	&	2.17	&	-0.14	\\
DMA	&	0.34	&	-1.35	\\
TMA	&	3.52	&	1.95	\\
SA	&	-9.21	&	-14.96	\\
FA	&	-9.38	&	-13.37	\\
NA	&	2.99	&	-2.99	\\

\hline            
    \end{tabular}
    \label{trimer-energy}
\end{table}

\begin{table}
    \centering
\caption{Gibbs free binding energy ($\Delta G ^{Cluster}$ in kcal mol$^{-1}$) calculated at the $\omega$B97X-D/6-31++G** level at 298 K at 1 atm of C\ce{H_{3}}OOOH, and ROOOH with two monomeric units of other atmospheric relevant molecules. Note: MA : methylamine (\ce{CH3NH2}), DMA : dimethylamine (\ce{(CH3)2NH}), TMA : trimethylamine (\ce{(CH3)3N}), SA : sulfuric acid (\ce{H2SO4}), NA : nitric acid (\ce{HNO3}), and FA : formic acid (\ce{HCOOH}).\\}    
    \begin{tabular}{l|c|c}
\hline
Cluster type	& C\ce{H_{3}}OOOH	&	ROOOH	\\
\hline  
water	&	0.48	&	-4.10	\\
\ce{NH3}	&	0.38	&	-3.17	\\
MA	&	-0.64	&	-4.10	\\
DMA	&	-1.82	&	-4.85	\\
TMA	&	1.97	&	-0.67	\\
SA	&	-9.52	&	-17.26	\\
FA	&	-12.48	&	-17.51	\\
NA	&	1.66	&	-5.72	\\

\hline            
    \end{tabular}
    \label{trimer- energy-dft}
\end{table}

\begin{figure}[htp]
\centering
{\includegraphics[width=1.0\textwidth]{SI_Fig3_trimer_ch3oooh.png}}
  \caption{Global minimum molecular structures of clusters of C\ce{H_{3}}OOOH with two other monomeric units of atmospheric relevant molecules viz water, N\ce{H{_3}}, MA, DMA, TMA, SA, FA, NA, and self clustering. Color coding: brown is carbon, red is oxygen, blue is nitrogen, yellow is sulfur and white is hydrogen.}
  \label{trimer_ch3oooh}
\end{figure}

\begin{figure}[htp]
\centering
{\includegraphics[width=1.0\textwidth]{SI_Fig4_timer_roooh.png}}
  \caption{Global minimum molecular structure of clusters of ROOOH with two other monomeric units of atmospheric relevant molecules viz water, N\ce{H{_3}}, MA, DMA, TMA, SA, FA, NA, and self clustering. Color coding: grey is carbon, red is oxygen, blue is nitrogen, yellow is sulfur and white is hydrogen.}
  \label{trimer_roooh}
\end{figure}

\newpage
Additionally, we perform clustering analysis of (a) C\ce{H_{3}}OH, (b) C\ce{H_{3}}OO)H, (c) C\ce{H_{3}}OOOH and (d) ROOOH with two sets of sulfuric acid and ammonia cluster compositions viz \ce{(SA)}\ce{(NH3)} and \ce{(SA)4}\ce{(NH3)4}.  We can see from Table \ref{energy 4sa-4am} that the trioxide ( in general) has better binding capabilities than those of methanol and methyl-peroxide. Global-minimum molecular representations are illustrated in the Figure \ref{1sa_1amm_dimer}. Interestingly, the large difference in binding energy between \ce{CH3OOOH} and ROOOH, indicates the stability of the cluster is resulted from the size of the R- group dominantly. Due to limited computation sources, the binding energy of ROOOH and \ce{(SA)4}\ce{(NH3)4} is not presented here.

\begin{table}[htp]
    \centering
\caption{Gibbs free binding energy ($\Delta G ^{Cluster}_{D}$ in kcal mol$^{-1}$) calculated at the DLPNO–CCSD(T)/aug-cc-pVTZ//$\omega$B97X-D/6-31++G** level of (1) C\ce{H{_3}}OH, (2) C\ce{H{_3}}OOH, (3) C\ce{H{_3}}OOOH (4) ROOOH with (\ce{SA})(\ce{NH3}) and \ce{(SA)4}\ce{(NH3)4} clusters. Note: SA is sulfuric acid (\ce{H2SO4}).\\}    
    \begin{tabular}{l|c|c|c|c}
\hline
Cluster type	&	\ce{CH3OH} 	&	\ce{CH3OOH}	&	\ce{CH3OOOH} & ROOOH	\\
\hline  
(SA)(\ce{NH3})	&	-0.95	&	-1.22	&	-2.2	&	-10.39	\\
\ce{(SA)4}\ce{(NH3)4}	&	1.94	&	0.12	&	-1.56	&		------\\

\hline            
    \end{tabular}
    \label{energy 4sa-4am}
\end{table}

\begin{figure}[htp]
\centering
{\includegraphics[width=1.0\textwidth]{SI_Fig5_1sa_1amm_dimer.png}}
  \caption{Global minimum molecular structure of clusters of \ce{CH3OOOH} and ROOOH with dimeric clusters of \ce{(SA)}\ce{(NH3)}and \ce{(SA)4}\ce{(NH3)4}. Color coding: grey is carbon, red is oxygen, blue is nitrogen, yellow is sulfur and white is hydrogen.}
  \label{1sa_1amm_dimer}
\end{figure}

In Table \ref{comparison-dft-energies}, we present Gibbs binding free energies (calculated at $\omega$B97X-D/6-31++G** level of theory) of all the atmospheric precursors taken into account for clustering analysis namely; C\ce{H_{3}}OH, C\ce{H_{3}}OOH, C\ce{H_{3}}OOOH, and ROOOH with other atmospheric relevant systems. 

\begin{table}
    \centering
\caption{Gibbs free binding energy of dimers (in kcal mol$^{-1}$) consisting of C\ce{H_{3}}OH, C\ce{H_{3}}OOH, C\ce{H_{3}}OOOH, or ROOOH with other atmospheric relevant systems.  \\}    
    \begin{tabular}{l|c|c|c|c}
\hline
Cluster type	&  C\ce{H_{3}}OH & C\ce{H_{3}}OOH &  C\ce{H_{3}}OOOH	&	ROOOH	\\
\hline  
water	&	2.24	&	2.45	&	1.37	&	-1.57	\\
\ce{NH3}	&	0.60	&	-0.78	&	-1.42	&	-3.35	\\
MA	&	-0.69	&	-0.52	&	-2.09	&	-4.17	\\
DMA	&	0.34	&	-0.37	&	-2.45	&	-4.88	\\
TMA	&	1.12	&	-0.78	&	-2.39	&	-4.52	\\
SA	&	-4.23	&	-2.74	&	-3.04	&	-7.51	\\
NA	&	-2.51	&	-0.59	&	-0.35	&	-5.00	\\
FA	&	-5.84	&	-6.07	&	1.21	&	-4.70	\\
self	&	1.43	&	1.50	&	1.07	&	-6.65	\\
\ce{(SA)}\ce{(NH3)}	&	-2.94	&	-2.38	&	-2.52	&	-8.31	\\
\ce{(SA)4}\ce{(NH3)4}&	-1.19	&	-0.99	&	-1.84	&	-6.30	\\

\hline           
    \end{tabular}
    \label{comparison-dft-energies}
\end{table}


\subsection*{Lennard-Jones parameters for Master Equation calculations}
\begin{table}[H]
    \centering
\caption{Lennard-Jones parameters $\sigma$ and $\varepsilon$ used for MESMER calculations for the pathway (1), and the relative electronic energies (R.E) calculated at the DLPNO–CCSD(T)/aug-cc-pVTZ//$\omega$B97X-D/6-31+G* level of all the associated systems .\\}    
    \begin{tabular}{l|c|c|cc}
\hline
Molecules	& $\sigma$ (Å) & $\varepsilon$/$k_{B}$ (K) &  R.E (in kcal mol$^{-1}$)\\
  &  &  & (ROO + OH) & (RO + \ce{HO2})\\
\hline  
HOOOH	&	4.5	&	437	&	29.4	&	29.4	\\
C\ce{H3OOOH}	&	4.9	&	403	&	28.8	&	25.1	\\
\ce{R^{1}}-OOOH	&	5.9	&	541	&	29.7	&	26.7	\\
\ce{R^{2}}-OOOH	&	5.9	&	541	&	29.6	&	26.6	\\
\ce{R^{3}}-OOOH	&	5.9	&	541	&	29	&	27	\\
\ce{R^{4}}-OOOH	&	5.9	&	541	&	29.6	&	26.1	\\
\ce{R^{5}}-OOOH	&	5.9	&	541	&	29.4	&	26.3	\\
\ce{R^{6}}-OOOH	&	5.9	&	541	&	30.9	&	26.7	\\
\ce{R^{7}}-OOOH	&	6	&	629	&	29.9	&	30.7	\\
\ce{R^{8}}-OOOH	&	6	&	629	&	28.7	&	26.8	\\
\hline            
    \end{tabular}
    \label{mesmer_inputs}
\end{table}

\newpage
\bibliography{rsc_ea}
\end{document}