Dissociation Time, Quantum Yield, and Dynamic Reaction Pathways in the Thermolysis of trans-3,4-Dimethyl-1,2-dioxetane

The thermolysis of trans-3,4-dimethyl-1,2-dioxetane is studied by trajectory surface hopping. The significant difference between long and short dissociation times is rationalized by frustrated dissociations and the time spent in triplet states. If the C–C bond breaks through an excited state channel, then the trajectory passes over a ridge of the potential energy surface of that state. The calculated triplet quantum yields match the experimental results. The dissociation half-times and quantum yields follow the same ascending order as per the product states, justifying the conjecture that the longer dissociation time leads to a higher quantum yield, proposed in the context of the methylation effect. The populations of the molecular Coulomb Hamiltonian and diagonal states reach equilibrium, but the triplet populations with different Sz components fluctuate indefinitely. Certain initial velocities, leading the trajectories to given product states, can be identified as the most characteristic features for sorting trajectories according to their product states.

−3 The process is called bioluminescence when it occurs in living organisms, e.g., fireflies.Chemiluminescence or bioluminescence has wide-ranging applications, such as treating cancers via photodynamic therapy, 4−6 acting as a bioluminescence imaging tool, 7,8 and tracking environmental pollutants. 9The thermolysis of the four-membered heterocyclic peroxides, e.g., 1,2-dioxetane, generates electronically excited state carbonyl compounds that emit light to release the excess energy 1 (see Figure 1).The enhancement of the quantum yields of singlet and triplet chemiexcitation was achieved experimentally by the methylation of the 1,2-dioxetanes. 1,10−18 The dynamical quantities, e.g., dissociation time and branching ratios, however, cannot be analyzed in the context of critical points and PESs.−33 As the full quantum-mechanical methods are expensive and limited to a few degrees of freedom, 33−35 the trajectory surface hopping (TSH) 19 method has emerged as one of the most popular tools for the NAMD simulations, which was used recently to study the chemiluminescent processes for the four-membered heterocyclic peroxides. 1he thermal decomposition of 1,2-dioxetane derivatives was discussed via the TSH approach 36,37 implemented in the Newton-X package 38 interfaced with Molcas 39 (Newton-X/ Molcas).The electronic structure method employed in Molcas 39 was the state-averaged complete active space selfconsistent field (SA-CASSCF) 40 with the ANO-RCC-VTZP basis set 41 and state averaging over the four lowest singlet states in the active space of 12 electrons and 10 orbitals, i.e., SA4-CASSCF(12e,10o)/ANO-RCC-VTZP.It inferred that the 1,2-dioxetane molecule is trapped in the singlet excited states, and the decomposition process is deferred. 36The methyl substitution of the 1,2-dioxetane increased the dissociation time via the mass effect in which the dissociation time was reproduced by the substitution groups with the same moment of inertia of the methyl group. 37With the Newton-X/ Molcas combined package, the quantum yield of trans-3,4dimethyl-1,2-dioxetane (TDMD) 10 was estimated by the SA8-CASSCF(8e,6o)/STO-3G+6-31G* mixed basis set, including the four lowest singlet states and four lowest triplet states. 42he obtained data revealed that the calculated quantum yield of the first triplet excited state (T 1 ) was comparable to the experimental result, 10 and the quantum yield of the second singlet excited state (S 2 ) was greater than that of the first singlet excited state (S 1 ).Including the triplet excited states in the TSH simulation increases the dissociation half-time enormously, and this is the reason that the small active space and mixed basis set were selected. 42Even with these achievements, some questions are still unclear.(1) As the fragmentation product of the trajectories, i.e., the acetaldehyde molecule (Figure 1), can be in different energy states (e.g., T 1 , S 0 , or S 1 ), what is the order of the dissociation times as per the product energy states, or what state has the longest dissociation half-time?(2) Why do some trajectories have short dissociation times, some long, and is it related to the initial conditions of trajectories?(3) Why is the quantum yield of the S 2 product state greater than that of the S 1 product state, and what mechanism is behind this anomaly?(4) After the molecule escapes the entropic trap region 12,36 in which the four lowest singlet states and four lowest triplet states are nearly degenerated, do the surface hops still take place, and can the populations of the triplet states with different z components of the spin reach the constant values after a certain time period?(5) Are there some traits that make it most obvious for them to assort the trajectories according to their product energy states?
Guided by these questions, in this work we study the dynamic thermolysis of the TDMD molecule by the TSH method 19 implemented in the SHARC package 43−45 interfaced with OpenMolcas. 46One of the strengths of the SHARC is that it processes internal conversion and intersystem crossing on the same footing by combining the spin−orbit-free input energies, gradients, and nonadiabatic couplings with the spin− orbit matrix elements in a spin−orbit-free electronic basis.As the decomposition of the TDMD can be characterized by the O1−O2−C3−C4 dihedral angle (Figure 1), we evaluate the distribution of the trajectories over this dissociation dihedral angle and dissociation time, minimum and maximum dissociation times, and dissociation half-time to demonstrate the allocation of the trajectories, and the order of the dissociation times as per their product energy states.The time evolutions of the C3−C4 distance, the O1−O2−C3−C4 dihedral angle, the potential energy of the active state, the number of frustrated dissociations, and the time spans in the singlet and triplet states along the short and long trajectories are calculated to explain the significant dissociation time difference between the short and long trajectories and address the cause of delay of the decomposition process.We compute the time evolutions of the ensemble-averaged populations of the MCH (molecular Coulomb Hamiltonian) and diagonal states 44 to examine the quantum yields for the respective product energy states and check the population equilibrium of the triplet excited states with different z components of the spin.To explore how the product energy state of the trajectories stochastically correlates with the initial velocities, we evaluate the distribution of the number of the trajectories over the initial velocities and dissociation time to determine if in the initial velocities there are the most characteristic features that are easiest to use to separate the trajectories as per their product energy states. 47or the evaluation of electronic structure of the biradicals in the thermolysis of the TDMD, the SA-CASSCF approach 40 with state averaging over states S 0 −S 3 and T 1 −T 4 was applied to optimize the transition states, evaluate the normal modes at the transition states, and compute the potential energy and its first derivative with respect to nuclear coordinates along the trajectories.The selected active space consists of the 12 electrons distributed in 10 orbitals, i.e., the four σ and four σ* orbitals of the four-membered ring and the two oxygen lonepair orbitals perpendicular to the ring. 36,37The 6-31G basis set 48 was employed in the optimization and NAMD simulation, i.e., SA8-CASSCF(12e,10o)/6-31G. The dynamic process starting from the O−O transition state (Figure 1 and Figure S1) can be simulated by the SA-CASSCF method (see Figure S2).
−45 The nuclear equations of motion were integrated by the velocity-Verlet algorithm 52 with a time step size of 0.5 fs and propagated by using the projected nonadiabatic couplings as the direction to adjust the momentum after the trajectory hops. 53The electronic structure is handled quantum mechanically by solving the time-independent Schrodinger equation, and the nuclei propagate in trajectories governed by multiple potential energy surfaces.The total Hamiltonian for intersystem crossing dynamics in the SHARC is written as H total = H MCH + H SOC , where H MCH is the nonrelativistic molecular Coulomb Hamiltonian (MCH) and the H SOC is the spin−orbit coupling operator. 54The eigenstates of H MCH are denoted as the MCH states, e.g., states S 0 and T 1,−1 , in which the total electron spin S and its z component S z are good quantum numbers.If the spin−orbit couplings (SOCs) are included, 54 there are two drawbacks in the MCH representation.(1) The off-diagonal couplings in H total are usually delocalized over the PES, which results in the non-zero transition probability even far from crossing regions; thus, a much larger number of trajectories are needed to sample the process correctly because surface hops may occur in a much larger phase space volume. 44(2) The sum of the transition probabilities over all multiple components depends on the rotation of the molecule in the laboratory frame. 55To overcome these shortcomings, the diagonal basis is selected, which is the set of eigenstates of H total , i.e., diagonal states. 44All couplings between the diagonal states are described by a nonadiabatic coupling vector.To simulate the nonadiabatic dynamics of the thermal decomposition of the TDMD effectively, all sampling trajectories start from the ratecontrolling O−O bond-breaking transition state (TS O−O ) in the S 0 state (Figure 1 and Figure S1) and have the same initial coordinates of TS O−O , but the initial velocities are generated by the random sampling algorithm proposed by Sellner, Barbatti, and Lischka (SBL). 56 The Journal of Physical Chemistry Letters energy (see the Supporting Information).First, the initial velocities are sampled by the Boltzmann distribution at 300 K, and then the initial kinetic energies of all of the trajectories at TS O−O are rescaled to the same energy (3.96 eV).In the O−O reaction coordinate with the imaginary frequency, the sampled velocities, forward to the product direction, are selected for the NAMD simulation.We have coded the SBL sampling algorithm in SHARC.The C−C van der Waals bond length is 2.4 Å, and the trajectories are forced to be terminated if the distance between the C3 and C4 atoms is >3.7 Å.−45 In the NAMD simulation, we consider the S 0 −S 3 and T 1 −T 4 states among which the surface hopping to state T 3 or T 4 is turned off to reduce the computational cost and simulation time.Of the 310 trajectories, 302 trajectories terminated successfully with a C3−C4 distance of >3.7 Å, and eight trajectories failed because of the lack of conservation of energy.All trajectories end in the T 1 , S 0 , and S 1 product state, and no trajectory was found in the other states.The MCH state of the trajectory labeled at the time of the cleavage of the C3−C4 bond (it just becomes greater than 2.4 Å) remains unchanged after the C3−C4 bond breaks.For example, if the MCH state of the trajectory is T 1 around 2.4 Å of the C3−C4 distance, the trajectory follows the T 1 state until it reaches 3.7 Å (the S z components still change).The distribution of the number of the completed trajectories over the dissociation time (the time needed when the C−C distance evolves to 2.4 Å) and O1− O2−C3−C4 dissociation dihedral angle in Figure 2 shows that (1) the maximum dissociation times for the T 1 , S 0 , and S 1 trajectories, which have corresponding T 1 , S 0 , and S 1 product state, are ∼1600, ∼4500, and ∼1200 fs respectively; (2) the most prevalent dissociation dihedral angles are almost the same for the T 1 , S 0 , and S 1 trajectories and are around 150.0°(the initial dihedral angle is 135.8°; the most populated angle variation between the final and initial dihedral angle of the trajectories is 14.2°) and the most populated trajectories have short dissociation times; (3) the O1−O2−C3−C4 dissociation dihedral angle of the trajectories is distributed from −180°to 0°to 180°(−180°is equal to 180°in a dihedral angle); and (4) most S 1 trajectories have short dissociation times.
The minimum and maximum dissociation times, dissociation half-times, means, and standard deviations are listed in Table 1 for the T 1 , S 0 , and S 1 product states.The dissociation half-time is the time required for half of the trajectories to dissociate.Table 1 indicates that the S 0 trajectories have the longest dissociation half-time of 253.25 fs, and the S 1 trajectories have the shortest dissociation half-time of 64.50 fs.Both orders of the dissociation half-times and maximum dissociation times are the same for the T 1 , S 0 , and S 1 product states.The maximum dissociation time perhaps provides the time scale for the chemiluminescent decomposition.For the S 0 trajectories, the shortest trajectory takes only 39.00 fs for dissociation to occur, but the longest one needs 4329.00 fs to break the C3−C4 bond.The dissociation time difference between the long and short S 0 trajectories is significant, and similar conclusions hold for the T 1 and S 1 trajectories.
To see how the short and long trajectories behave differently in their dissociation times, we study the time evolutions of the C3−C4 distance and the O1−O2−C3−C4 dihedral angle (Figure 3a,b).For the T 1 short trajectory (trajectory 69, a representative of the T 1 short trajectories) whose product state is T 1 , the C3−C4 distance increases almost monotonously with time; however, the dihedral angle changes from 135.8°to 128.9°, and the angle variation is 6.9°, which demonstrates that this T 1 short trajectory dissociates without any frustrated dissociation. 36At 44.0 fs, the T 1 short trajectory is on the PES of the T 1 state, its potential energy reaches the maximum [3.27 eV (Figure 3c)], and the corresponding C3−C4 distance is 2.16 Å (Figure 3a), which is greater than 2.08 Å, which is the C3−C4 distance of the C−C transition state on the T 1 PES (Figure S1c).At 42.5 fs, the dihedral angle arrives at its local maximum.Before 42.5 fs, the dihedral angle increases, and at 42.5 fs, the dihedral angle begins to decrease to prepare the configuration to pass over the ridge of the T 1 PES between TS O−O and the dissociated TDMD, which is termed the T 1 ridge (the crossing point over this T 1 ridge is the potential energy peak at 44.0 fs).Most peaks (e.g., at 18.5 fs) along the  The Journal of Physical Chemistry Letters potential energy curve of the active state represent the surface hops between the diagonal states, in which the dihedral angles at the corresponding times of the potential energy peaks do not display the locally minimal or maximal values.The nonadiabatic transitions between the MCH states in this T 1 short trajectory are listed in Table 2.The diagonal element represents the number of transitions within a MCH state (step i to step i + 1 within a MCH state is counted as one transition), and the off-diagonal element shows the number of surface hops between two different MCH states.The total steps in a MCH state are the diagonal element of this state plus the number of the surface hops from all other states to this state, which is the sum of all of the off-diagonal elements along the column in which this state resides (Table 2) (see the Supporting Information for details).The C3−C4 bond in this T 1 short trajectory takes 159 steps, i.e., 79.5 fs (step size of 0.5 fs), to reach 3.7 Å.From Table 2, we find that the T 1 short trajectory spends 30.5 fs (61 steps), 5.5 fs, 34.0 fs, and 9.5 fs in states S 0 , S 1 , T 1 , and T 2 , respectively, and stays in the T 1 state with the longest time.This T 1 short trajectory (1) starts from TS O−O on the S 0 PES; (2) passes through the entropic trap region by hopping among the S 0 , S 1 , T 1 , and T 2 ; and (3) climbs on the T 1 PES and passes over the T 1 ridge.In other words, the molecule is dissociated with the T 1 product state.In general, in the early stage the trajectory is mainly in the S 0 state; later, it spends its most time in the T 1 state with the surface hops among the T 1 states with different z components of the spin [S z = −1, 0, or 1; i.e., T 1,−1 , T 1,0 , and T 1,1 (the energy gaps among T 1,−1 , T 1,0 , and T 1,1 are small even out of the entropic trap region), and finally, it overcomes the T 1 ridge.
For the T 1 long trajectory (trajectory 23), the dihedral angle changes from 135.8°to 180°(−180°) to −90°to 0°to 32.9°a nd the angle variation is 257.1°.A dihedral angle takes a value in the ranges of 0°to 180°and 0°to −180°instead of 0°→ The Journal of Physical Chemistry Letters distance is that of the frustrated dissociations (see Figure 3b)]. 36The average oscillation period is 40.6 fs (975.5/24) which matches the C−C vibration in the biradical region.In Figure 3d, there are ∼24 noticeable potential energy peaks, which means that each frustrated dissociation can be visualized as the oscillation between the two adjacent local maxima of the potential energy curve.The global maximal potential energy along the T 1 long trajectory is 3.66 eV located at 105.5 fs; however, it is on the T 2 PES, and the molecule does not dissociate by overcoming the T 2 ridge.The T 1 long trajectory  The Journal of Physical Chemistry Letters hops among states S 0 −S 3 , T 1 , and T 2 in the entropic trap region and stochastically seeks its dynamic reaction path.At 966.5 fs, the trajectory is on the T 1 PES, and the potential energy reaches the second highest peak [3.22 eV (Figure 3d)], which is identified as the other crossing point over the T 1 ridge.At the same time, the dihedral angle attains its peak (∼966.5 fs), which means that the C3−C4 bond starts to break upon adjustment of the dihedral angle from increasing to decreasing (Figure 3b).After 966.5 fs, the C3−C4 distance rapidly increases to 3.7 Å.From Table 3, we find that the T 1 long trajectory spends 30.5, 113.5, 24.5, 12.5, 546.0, and 317.0 fs in states S 0 −S 3 , T 1 , and T 2 , respectively.For the most time, the T 1 long trajectory stays in the T 1 and T 2 states and hops among states T 1,−1 , T 1,0 , T 1,1 , T 2,−1 , T 2,0 , and T 2,1 ; however, the molecule does not dissociate through the T 2 ridge, and the C− C bond breaks by passing over the T 1 ridge.
For the S 0 short trajectory (trajectory 138), at 28.5 fs the potential energy arrives at its global maximum peak [2.86 eV (Figure S4a)], but it is on the S 3 PES.One femtosecond later (at 29.5 fs), the trajectory jumps to the S 0 PES and the C3−C4 bond starts to break (there is no local maximum on the S 0 PES when the C3−C4 distance is between 2.1 and 2.4 Å); i.e., the dissociation does not encounter the S 0 ridge.From Table S1, we discern that the S 0 short trajectory spends 60.1%, 21.0%, 1.5%, 15.9%, and 1.5% of the simulation time in states S 0 −S 3 and T 2 , respectively; in other words, it stays mainly in state S 0 .From Table S2, we find that the S 0 long trajectory (trajectory 4) spends its most time in states T 1 and T 2 (46.0% and 32.3%, respectively) and 10.1%, 7.2%, 2.7%, and 1.7% in states S 0 −S 3 , respectively.Just before 4298.5 fs, the trajectory was wandering in the T 2 state and then jumps to S 3 → S 2 → S 1 → S 0 in the short time period, and the molecule dissociates on the S 0 PES without overcoming any local S 0 maximum along the potential energy curve (Figures S3b and S4b).The molecule that breaks the C−C bond through the S 0 PES does not encounter the S 0 ridge, which is consistent with the reaction coordinate analysis that after TS O−O , the C−C bond breaking transition states on the S 1 −S 3 and T 1 −T 4 PES exist, but there is no TS C−C on the S 0 PES. 12,36Between 0.0 and 4300.0 fs, there are 108 frustrated dissociations in the C3−C4 bond oscillation (Figure S4b), and the period is 39.8 fs, which is consistent with the C−C vibration.The 108 frustrated dissociations make this trajectory take a long time to dissociate.
At 38.0 fs, the C3−C4 distance is 2.16 Å, and the S 1 short trajectory (trajectory 88) is in the S 1 state and reaches its global maximum (3.19 eV) of the potential energy (Figures S3c and S4c).Thus, the molecule dissociates by crossing the S 1 ridge located at 38.0 fs.From Table S3, we find that the S 1 short trajectory spends 37.9% of the time in state S 0 , 2.1% in state T 1 , and the highest percentage, 60.0%, in state S 1 and dissociates along the S 1 PES.The S 1 long trajectory (trajectory 253) spends 16.4%, 15.2%, 3.4%, 6.0%, 37.4%, and 21.6% in states S 0 −S 3 , T 1 , and T 2 , respectively (Table S4); in other words, for the most time, it stays in the triplet excited states.At 1147.0 fs, the S 1 long trajectory is on the S 1 PES and reaches its local maximum of the potential energy (2.85 eV) and the corresponding the C3−C4 distance is 2.16 Å (Figures S3d and  S4d); thus, we infer this is the other crossing point over the S 1 ridge.After 1147.0 fs, the C3−C4 bond rapidly breaks.Within 1147.0 fs, the C3−C4 bond oscillates 24 times, and the oscillation period is 47.8 fs, which is comparable with that of the C−C bond vibration.
The average total number of hops among all states is 98.9 per trajectory, but the average total number of nonadiabatic transitions among states T 1,−1 , T 1,0 , T 1,1 , T 2,−1 , T 2,0 , and T 2,1 states is 61.2 per trajectory, which indicates that most hops in trajectories are among the triplet states because of the small energy gaps among states T 1,−1 , T 1,0 , and T 1,1 (or T 2,−1 T 2,0 , and T 2,1 ).The short trajectories spend a small amount of time in the triplet states, and the C−C bond breaks directly; however, the long trajectories stay in the triplet states for the most time with multiple frustrated dissociations.
The ensemble-average time evolutions of the MCH state populations for states S 0 −S 3 , T 1 , and T 2 are illustrated in Figure 4a, which shows the population equilibrium after 4200 fs.The equilibrium populations for states T 1 , S 0 , and S 1 are 23.2%, 59.6%, and 17.2%, respectively, and are zero for other states.The previous research 42 found that the quantum yield of state S 2 was greater than that of state S 1 .This anomalous phenomenon is not observed here.One possible reason is the use of the small active space and low-level basis set in the previous work, 42 which underestimates the transition probability from state S 2 to lower states.Panels a and b of Figure 4 show that the state S 0 population (1) decreases from 100% to 29.8% within 95.5 fs, (2) fluctuates between 95.5 and 190.5 fs, and (3) increases between 190.5 and 4200.0 fs and reaches a constant value 59.6% after 4200.0 fs.The S 1 population increases and arrives at its maximum of 30.1% at 71.0 fs, then decreases to 17.2% at 3979.5 fs, and finally reaches its equilibrium.The state S 2 and S 3 populations increase and achieve their maximum values of 10.3% at 48.0 fs and 7.6% at 49.5 fs, respectively, and then go to zero after 2186.5 fs.The T 2 population increases and reaches its maximum of 18.5% at 132.0 fs and then decreases and vanishes after 3668.0 fs.The T 1 population increases and reaches its maximum of 34.8% at 365.5 fs, decreases, and reaches its equilibrium 23.2% after 4200.0 fs.States S 0 −S 3 and T 1 −T 4 are not degenerated around the TS O−O structure, and the S 0 state jumps more easily to state S 1 than to state T 1 , which explains that before 95.5 fs, the S 1 population grows faster than the T 1 population.Later, the trajectories enter the entropic trap region, hopping to state T 1 becomes easier than before, and the trajectories accumulate favorably on the T 1 PES (see Table 3), which elucidates that the T 1 population exceeds the S 1 population after 157.0 fs.
The fragmentation product of the trajectories retains its MCH state (ignoring the difference among triplet states with different S z values; in other words, T 1,−1 , T 1,0 , and T 1,1 are termed T 1 ) as that labeled at the C−C bond cleavage.If the product is in state S 0 or S 1 , after scission of the C−C bond, there is no surface hopping in the trajectory.However, if the product is in state T 1 , the surface hops among states T 1,−1 , T 1,0 , and T 1,1 continue after the C−C bond breaks.Two types of the surface hops occur: (1) between different diagonal states belonging to different MCH states, e.g., second diagonal state (T 1,0 ) → third diagonal state (T 1,1 ) (the second diagonal state is approximated as the T 1,0 state in the MCH representation, and the third diagonal state corresponds to the T 1,1 state), and (2) between different diagonal states belonging to the same MCH state, e.g., fourth diagonal state (T 1,0 ) → third diagonal state (T 1,0 ). Figure 4c indicates that the populations of states T 1,−1 , T 1,0 , and T 1,1 fluctuate around 4.3%, 10.3%, and 8.6%, respectively; in other words, they cannot reach population equilibrium because of the small energy differences among them.T 1,0 is the most populated and the T 1,−1 the least populated among states T 1,−1 , T 1,0 , and T 1,1 .

The Journal of Physical Chemistry Letters
Comparing panels a and c of Figure 4 with panel d, we find that the first diagonal state, D1, which is the lowest state in the diagonal representation, mainly contributes to state S 0 and the major component of state D5 comes from state S 1 .Furthermore, the major parts of states D2−D4 correspond to states T 1,−1 , T 1,0 , and T 1,1 , respectively, but the populations of states D2−D4 can reach equilibrium after 3500 fs.At the TS O−O configuration, state S 0 has the lowest energy among states S 0 −S 3 and T 1 −T 4 .Between 6.0 and 15.0 fs, most trajectories (77% of the total number of the trajectories) jump from the first diagonal state (S 0 ) to the fourth diagonal state (S 0 ) via internal conversion, which implies that state S 0 evolves from the lowest state (D1) to the fourth lowest state (D4) and the three z components of state T 1 possess energies that are lower than that of state S 0 in this part of the entropic trap region.This explains why there is a population peak (77.2%) of state D4 (S 0 ) around 11.0 fs (see Figure 4d).Here we point out that for most of the time, state D4 corresponds to state T 1,1 and state D1 matches state S 0 .
The number of trajectories, the quantum yields, and the dissociation half-times for the product MCH states are listed in Table 4.The quantum yields for states S 2 , S 3 , and T 2 are zero, and the calculated quantum yield of state T 1 is 23.2%, which matches the experimental value (20.0%).The simulated quantum yield of the singlet excited states (S 1 +S 2 ) is 17.2%, which is comparable with the value of 10.1% obtained by the SA8-CASSCF(8e,6o)/STO-3G+6-31G* method. 42Table 4 shows that for states S 0 , S 1 , and T 1 , the order of the quantum yields is the same as that of the dissociation half-times, which supports the argument that the longer dissociation time should cause the higher quantum yield observed in the mass effect of the methylation. 37o examine whether the product energy states of the dissociated TDMD are related to the initial velocities, first we evaluate the distributions of the S 0 , S 1 , and T 1 trajectories over the dissociation time and the initial angular velocity of the O1−O2−C3−C4 (or C3−C4−C5−C6) dihedral angle, denoted as ω 0,1234 (or ω 0,3456 ) in Figure 5.The ω 0,1234 coordinates of the most populated T 1 , S 0 , and S 1 trajectories are ∼0°/fs (Figure 5a−c); in other words, the highest peaks of product states T 1 , S 0 , and S 1 cannot be distinguished by ω 0,1234 .The range of ω 0,1234 for all trajectories is −3.0 ≤ ω 0,1234 ≤ 2.0; i.e., its range length is 5.0.The subranges are −3.0 ≤ ω 0,1234 ≤ 1.5, −2.0 ≤ ω 0,1234 ≤ 2.0, and −2.5 ≤ ω 0,1234 ≤ 1.0 for the T 1 , S 0 , and S 1 trajectories, respectively.In the subrange adjacent to the negative end point, −3.0 ≤ ω 0,1234 < −2.0, whose length (1.0) is 20.0% of the range length (5.0) [this ratio characterizes the relative length of the subrange near an end point (RLSE)], there is no S 0 trajectory; thus, the trajectories with the ω 0,1234 close to the negative end point tend to evolve into product states S 1 and T 1 .In the subrange 1.0 ≤ ω 0,1234 < 2.0 with a 20.0%RLSE, there is no S 1 state; thus, the trajectories with the ω 0,1234 neighboring the positive end develop most likely to states T 1 and S 0 .From the ω 0,1234 side view, the number of trajectories on the left side of the highest peak is greater than that on the right side of the highest peak for states T 1 , S 0 , and S 1 , which is attributed to the normal mode along the O1−O2 reaction coordinate being assigned with positive initial velocities to ensure molecular dissociation.
The ω 0,3456 coordinates of the highest peaks of the T 1 , S 0 , and S 1 trajectories are 0.5°/fs, 1.5°/fs, and 1.0°/fs, respectively; in other words, the most populated T 1 , S 0 , and S 1 trajectories can be distinguished by the ω 0,3456 coordinate (Figure 5d−f).In the subrange −2.0 ≤ ω 0,3456 < −1.0 with a 20.0%RLSE, there is no T 1 trajectory; thus, the trajectories with the ω 0,3456 vicinal to the negative end tend to product states S 0 and S 1 .Furthermore, Figure S5 shows that (1) the trajectories with the v 0,12 (initial velocity of O1−O2) near the lower end (25.0%RLSE) incline to state S 0 , (2) the trajectories with the v 0,12 adjacent to the upper end (12.5% RLSE) evolve least likely to state T 1 , (3) the trajectories with the v 0,34 (initial velocity of C3−C4) vicinal to the negative end (23.1% RLSE) are apt to the singlet states, (4) the trajectories with the v 0,34 close to the positive end (23.1% RLSE) tend to states T 1 and S 0 , and (5) the trajectories with the v 0,56 (initial velocity of C5−C6) neighboring to the positive end (33.3%RLSE) lean toward states S 0 and S 1 .The initial dihedral angular velocities and/or initial velocities that take values near certain end points with considerable RLSE, e.g., the v 0,56 adjacent to the positive end (33.3%RLSE), can be identified as the most characteristic features in trajectories, which incline to the given energy state, and their final energy states are "easiest" to classify. 47n conclusion, the thermal decomposition of trans-3,4dimethyl-1,2-dioxetane has been studied by the TSH dynamic approach.The significant difference in the dissociation time between the long and short trajectory has been elucidated by the time spent in the triplet states and the number of frustrated dissociations, which can be regarded as the C−C bond oscillation between two adjacent local maxima of the potential energy of the active state.Adding the singlet excited states to the NAMD simulation postpones the dissociation by introducing several frustrated dissociations.However, inclusion of the triplet states with different S z components results in more than a hundred frustrated dissociations, which delays the chemiluminescent decomposition remarkably.When the molecule ruptures its C−C bond through the excited state channel, e.g., T 1 or S 1 , it has to pass over the T 1 or S 1 ridge to reach the corresponding energy product.The ground state ridge has not been found.The simulated quantum yield of state T 1 matches the experimental result well.The dissociation half-times have been classified by the product energy states, and the order of the dissociation half-times is the same as that of the quantum yields for states T 1 , S 0 , and S 1 .This result provides evidence for the conjecture that the longer decomposition time leads to the higher quantum yield observed in the context of the mass effect of methylation.After the dissociation, the degeneracy of the ground and excited states has been lifted, and the label of the product energy state in terms of states T 1 , S 0 , and S 1 has remained unchanged; thus, the ensemble-averaged populations of the MCH and diagonal states can reach equilibrium.Due to small energy gaps among states T 1,−1 , T 1,0 , and T 1,1 (or T 2,−1 , T 2,0 , and T 2,1 ), surface hops still exist even after the dissociation, and the populations of the triplets with different S z components oscillate around certain values.The distribution of the trajectories over the initial velocities has revealed that the initial velocities near the end points with considerable relative sizes of the subranges make the product molecule tend to certain energy states, and these special initial velocities can be identified as the most characteristic features for classifying the trajectories as per their final energy states.Because of the stochastic nature of the TSH method, it would be interesting to check if there are fuzzy boundaries, 57 which separate the trajectories by their product energy states, by applying a Bayesian support vector machine model. 58,59We will discuss these issues in the future.

■ ASSOCIATED CONTENT
* sı Supporting Information The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpclett.3c03578.■ AUTHOR INFORMATION The initial kinetic energy at TS O−O (3.96 eV) is the zero-point vibrational energy of TS O−O (3.35 eV, computed via SA-CASSCF) plus the CASPT2 correction

Figure 3 .
Figure 3.Time evolutions of the C3−C4 distance (angstroms) and O1−O2−C3−C4 dihedral angle (degrees) for (a) a T 1 short trajectory and (b) a T 1 long trajectory.The peaks along the C3−C4 curve represent the frustrated dissociations, and the vertical arrows indicate two frustrated dissociations.Time evolution of the O1−O2−C3−C4 dihedral angle and the potential energy (electronvolts) of the active state for (c) a T 1 short trajectory and (d) a T 1 long trajectory.
pubs.acs.org/JPCL180°→360°; in other words, 180°equals −180°.If an angle oscillates between 178°and 183°, however, for the dihedral angle, it acts as if it is oscillating between 178°and −177°, so it looks discontinuous but continuous geometrically, which explains why the time evolution of the dihedral angle in Figure3bappears discontinuous between 180°and −180°.The C3−C4 distance oscillates around 1.6 Å, and there are 24 frustrated dissociations within 975.5 fs, which are identified by the local maxima in the time evolution of the C3−C4 bond length [the number of the peaks along the curve of the C3−C4

Figure 4 .
Figure 4. Time evolutions of (a) the populations of the MCH states in 5000 fs, (b) the populations of the MCH states in 500 fs, (c) the populations of states T 1,−1 , T 1,0 , and T 1,1 , and (d) the populations of the diagonal states.
Ground state, TS O−O , and TS C−C structures of the TDMD and their coordinates optimized by the SA-CASSCF; comparison of the potential energy gaps among SA-CASSCF/6-31G, SA-CASSCF/6-31G*, and CASPT2/ANO-RCC-VDZP; time evolutions of the C3−C4 distance, O1−O2−C3−C4 dihedral angle, and potential energy of the active state for the S 0 and S 1 trajectories; nonadiabatic transitions between the MCH states for the S 0 and S 1 trajectories and for the ensemble; spin−orbital coupling matrix elements over spin components of spin-free eigenstates; and distributions of the trajectories with product states T 1 , S 0 , and S 1 over the dissociation time and the initial velocity of O1−O2, C3−C4, and C5−C6, respectively (PDF)

Table 1 .
Minimum and Maximum Dissociation Times, Dissociation Half-Times, Means, and Standard Deviations (femtoseconds) for the T 1 , S 0 , and S 1 Product States

Table 2 .
Numbers of Nonadiabatic Transitions among the MCH States in the T 1 Short Trajectory

Table 3 .
Numbers of Nonadiabatic Transitions among the MCH States in the T 1 Long Trajectory

Table 4 .
Numbers of Trajectories, Quantum Yields, and Dissociation Half-Times (femtoseconds) for the Product Energy States