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. 2020 Sep 9;5(37):24057–24063. doi: 10.1021/acsomega.0c03483

Photochemical Ring Opening of Oxirane Modeled by Constrained Density Functional Theory

Marvin Krenz 1,*, Uwe Gerstmann 1, Wolf Gero Schmidt 1
PMCID: PMC7513336  PMID: 32984727

Abstract

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A constrained density functional theory/classical trajectory surface hopping study of the photochemical dissociation of oxirane (CH2)2O is presented. The calculations confirm the Gomer–Noyes mechanism for the initial reaction and agree largely with experimental photolysis data including reaction yields. The calculated yields, however, depend both on temperature and its modeling. The timescales of the various reaction steps are well below 100 fs, similar to previous time-dependent density functional calculations. At variance with those, however, the present calculations obey Kasha’s rule, i.e., the photoreaction is initiated in the energetically lowest excited state.

1. Introduction

Photoinduced nuclear reorganization processes such as photoisomerization and light-stimulated reactions are common in nature, e.g., for photosynthesis, as well as of immense importance for technical applications such as photocatalysis. They may also be destructive, as seen by the photodegradation of many materials. Therefore, it is highly desirable to thoroughly understand photochemical reactions and to be able to model them computationally with predictive accuracy. Photochemical reaction paths access high-energy intermediates and thus allow to overcome large activation barriers in a short period of time. Thereby, the Born–Oppenheimer approximation, i.e., the assumption that the nuclei move on a single potential energy surface (PES) and that the electrons adjust immediately to any change in the nuclear position, may no longer hold, and non-adiabatic effects can become relevant. Moreover, the computational description of excited-state potential energy surfaces is a serious computational challenge.

Many-body perturbation theory provides a systematic and well-founded approach to calculate electronic excitation energies.1,2 Here, one solves in the first step the equation of motion for the one-particle propagator using Hedin’s GW approximation to self-energy.3 In the second, step the electron–hole pair excitation energies are obtained from the poles of the interacting two-particle Greens’ function. A practical scheme of solving for the poles—known as the Bethe–Salpeter equation (BSE) approach4—is given by a diagrammatic expansion of the electron–hole propagator in terms of the scattering kernel. This GW-BSE approach allows for the accurate modeling of excited-state PES.5,6 However, this method is associated with considerable numerical effort, limiting its application to relatively small systems and short-dynamics time intervals. This holds as well for correlated wave function methods such as multireference configuration interaction (MR-CI)7 or the complete active state self-consistent field method.8 Time-dependent density functional theory (TD-DFT),9 most frequently used within the Tamm–Dancoff approximation,10 is a numerically more efficient approach to the calculation of excited-state PES.11,12 It works usually well for geometries near their equilibrium, but, within the typically applied approximations, has some limitations as, e.g., the underestimation of charge transfer and Rydberg excitations,13 an underestimation of triplet excitation energies,14 and difficulty to describe high-lying excited states.15 Constrained density functional theory (cDFT) is another numerically highly efficient and robust scheme to calculate optical excitation energies.16−19 Excited-state PES calculated within cDFT were recently successfully used to model the dynamics of optically driven surface phase transitions.20,21 In these studies, however, the occupation of the excited states was either frozen throughout the entire phase transition20 or adjusted to match the experimentally measured electronic temperature.21 Moreover, the structural reorganization corresponding to the phase transition was limited to geometries close to the equilibrium, and the hopping between different potential energy surfaces (PES) has not been taken into account. A reliable direct assessment of photochemical reactions, however, requires a study of the entire reaction mechanisms including excess energy leading to PES surface hopping. Such a study is reported in the present work, where we model the oxirane ring opening using cDFT in conjunction with Tully’s fewest switches surface hopping (FSSH) algorithm22 as implemented in the Libra-X package.23

Oxirane (ethylene oxide, (CH2)2O), a three-member ring consisting of one oxygen atom and two carbon atoms, has been serving as a model for photochemistry for a long time.11,24−28 Gomer and Noyes24 found methane, ethane, hydrogen, and carbon monoxide to be the principal products of the photochemical decomposition of oxirane and suggested

(CH2)2O + ℏω → CH3 + HCO (H + CO) to be the primary process of the reaction, cf. Figure 1. This mechanism was confirmed experimentally by Kawasaki et al.27 Interestingly, in this theoretical work, it was suggested that the photochemical ring opening is facilitated by occupation of the second excited state S2, while the first excited state S1 was found to be a nonreactive channel, which can be energetically tapped, however, by the reactive S2 state. This seems to contradict Kasha’s rule29 that suggests that the first excited state is the most likely candidate for the initiation of photochemical reactions. The present cDFT/FSSH study aims at identifying the most relevant states for the oxirane photochemical reactions and at exploring the influence of the temperature modeling. To that end, we perform non-adiabatic molecular dynamics (NAMD) calculations both within the microcanonical NVE ensemble and within the canonical NVT ensemble.

Figure 1.

Figure 1

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Gomer–Noyes mechanism24 for the photochemical reaction of oxirane. The initial ring opening is followed by proton transfer before the reaction is finalized by C–C bond breaking.

2. Methodology

The present calculations are based on spin-polarized density functional theory within the generalized-gradient approximation using the PBE functional30 in the Quantum Espresso31 implementation. Norm-conserving Troullier–Martins pseudopotentials32 are used to describe the electron–ion interaction. The electron states are expanded in a plane-wave basis up to an energy cutoff of 70 Ry. Oxirane is modeled within an 11×11×11 Å3 supercell.

Excited-state PES are calculated using cDFT. While the term “cDFT” can be used to describe various constraints like charge, spin, or occupation numbers,16 we refer to the latter case, i.e., occupying the Kohn–Sham orbitals in a non-Aufbau manner, using occupation constraints.17−19,33 A schematic of this cDFT method is shown in Figure 2. This procedure goes by different names, and besides cDFT, it can be also referred to as ΔSCF,34 ΔDFT,35 or excited-state DFT.36

Figure 2.

Figure 2

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Constraint DFT cycle used for excited-state calculations. In contrast to DFT, cDFT uses the weights fi, also referred to as occupation numbers, for the calculation of the electronic density n(r). These weights are fixed during the cDFT calculation, and therefore, the Kohn–Sham orbitals can be occupied in a non-Aufbau manner to simulate an excited electronic state.

Tully’s FSSH algorithm22 has been employed to accommodate for non-adiabatic effects. The NAMD calculations were performed using the Libra-X package,23 which uses the Kohn–Sham orbitals from the cDFT calculations to approximate the wave functions of the excited systems via Slater determinants. The active electronic state of the system determined by FSSH is used for structural relaxation, which means that only one electronic state contributes to the evolvement of the NAMD. The non-adiabatic character of the MD is given by the possibility of swapping between PES and by this swapping between the electronic states used to evolve the system. Time steps of 5 as and 0.25 fs have been chosen for the integration of the Schrödinger equation in the FSSH algorithm and the molecular dynamics, respectively. The atomic temperature was controlled in the NVE ensemble by using randomly generated and appropriately scaled initial velocities. Additionally, calculations were performed that model an NVT ensemble with the Nosé–Hoover chain thermostat37 using a chain length of 5 and an effective relaxation time of 24.4 fs. Thereby, the chain length determines the accuracy of the thermostat concerning oscillation in the temperature,37 and the effective relaxation time determines the coupling strength to the heat bath.37 Usually, the frequency corresponding to the effective relaxation time should have the same order of magnitude as characteristic frequencies of the nuclear degrees of freedom.23 With most of the characteristic frequencies of oxirane being between 1000 and 3000 cm–138 and the frequency corresponding to the effective relaxation time being 1367 cm–1, this criteria is fulfilled. The NAMD calculations consider, in addition to the electronic ground state (S0), the first three single electron excited states (HOMO→LUMO (S1), HOMO→LUMO+1 (S2), and HOMO→LUMO+2 (S3)). While the character of the excited states changes during the evolution of the system (as seen for example in Figure 9), the terms S0 for the ground state, S1 for the first excited state, etc. will still be used for the sake of clarity. The excited states are modeled with cDFT assuming no spin flips. Smearing of the occupation numbers based on the Fermi distribution function was employed in order to mimic a multiconfigurational behavior in the case of (quasi-)degeneracy of states.23 Thereby, a broadening parameter of 0.01 Ry is used, and the smearing is updated after 30 electronic iterations. We restrict our simulations to reactions initiated in the second excited state (S2) and from slightly randomized C2H4O ground-state geometries, i.e., each ion displaced by max. 0.05 Å in each direction. A number of 50 trajectories are calculated and averaged to model photoreactions taking place at temperatures of 30, 77, 100, 150, 300, and 400 K.

Figure 9.

Figure 9

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Calculated electronic density corresponding to the excited orbital of the first excited state (S1) during the ring opening reaction path after 5, 15, and 30 fs integration time.

The time evolution of the photoreaction is monitored by evaluating the occupation numbers of the electronic states, the charge distribution, and the molecular geometries. With respect to the latter, we use thresholds listed in the Supporting Information in order to classify the reaction products. In the case of state crossing, which includes the active state, which can either happen permanently (which, however, was not observed) or temporarily due to the smearing, the indexing of the states swaps with the states. If, for example, the active state Sv crosses S0, due to indexing changes, then S1 becomes S0 and vice versa. The character of the active state is, however, preserved by the FSSH algorithm as the probability of a hop to the new S0 state is large since it has the character of the old S1 state. Since no permanent state crossing is observed, in the analysis, the swaps of the indices are undone by comparing the PES to avoid confusion. Therefore, in this example, the active state would be S1 before and after the crossing.

3. Results and Discussion

In Figure 3, the calculated oxirane electron densities corresponding to specific orbitals relevant for the lowest optically excited states are shown. The ground-state orbital character of states obtained here for HOMO, LUMO, LUMO+1, and LUMO+2 confirm previous DFT-PBE calculations.11 The character of these states is roughly conserved in the S0 and S2 excitations, while we observe some mixing of the LUMO and LUMO+1 states upon S1 excitation.The S3 excitation calculated here resembles S2 calculated in ref (28). Moreover, we observe that the excited states depend sensitively on the molecular geometry. This can be seen in the lower part of Figure 3, where the lowest excitations for slightly deformed molecules are shown.

Figure 3.

Figure 3

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Calculated electron densities corresponding to the orbitals for the states most relevant for the photoreaction of oxirane. The top row shows the densities for HOMO, LUMO, LUMO+1, and LUMO+2 in the structural and electronic ground state. The middle row shows the respective densities corresponding to the orbitals with one electron removed or added, calculated within cDFT, i.e., the highest occupied orbitals for S0, S1, S2, and S3. The bottom row shows the same densities for a slightly disturbed geometry with a maximum deviation of 0.05 Å from the ground state.

Table 1 shows the cDFT excitation energies of the states S1, S2, and S3 and compares them with the singlet excitation energies calculated by other methods as well as with experimental data. It can be seen that the cDFT excitation energies slightly underestimate the measured excitation energies.

Table 1. Oxirane Singlet Excitation Energies (in eV) for cDFT in Comparison with Other Computational Methods as well as with Experimental Data.

cDFT exp. TDHF39 TDLDA39 TDB3LYP39
6.69 7.2440−42 9.14 6.01 6.69
6.76 7.4541 9.26 6.73 7.14
7.26 7.88,40 7.8941 9.36 6.78 7.36
    9.56 7.61 7.85
    9.90 7.78 8.37
    9.93 8.13 8.39
    8.15 8.40  
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In the following, we present NVE calculations initiated in the second excited state S2, resulting from an electron transfer HOMO→LUMO+1. As seen from Figure 4, within a few femtoseconds, the excitation is mostly transferred to S1 and in some cases to S3 before it is quenched in S0. However, even after 250 fs, there is still a sizable number of molecules in an electronically excited configuration.

Figure 4.

Figure 4

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State populations (averaged over up to 50 trajectories) vs. time for NVE calculations modeling an initial temperature of 100 K (left) and 300 K (right).

As can be seen from Figure 5, where the formation probabilities of the reaction products are plotted vs. time for different temperatures, the Gomer–Noyes mechanism is the most dominant process in the NAMD simulations, in agreement with measured data.27 After 30···40 fs, the ring opening (CH2COH2 formation) is completed in the majority of the excited oxirane molecules. The third reaction step according to Gomer and Noyes, the CH3COH formation, is essentially completed for most molecules after about 80 fs. The following dissociation of the molecule into CH3 + COH can also be observed in some cases but not in the quantity observed experimentally. This is the same finding as in the previous work of ref (11). The processes above compete with oxygen and hydrogen abstraction. The influence of the initial temperature on the occupation numbers of the respective states and on the reaction kinetics is limited. The latter can be seen from the nearly constant position of the peak maxima in Figure 5. The calculated reaction times for the C2H4O → CH2COH2 reaction with 18···45 fs and for the CH2COH2 → CH3COH with 43···85 fs match largely the results of ref (11) (20···40 and ∼70 fs).

Figure 5.

Figure 5

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Reaction products vs. time in the microcanonical NVE ensemble for temperatures of 100 K (left) and 300 K (right). With the peaks at nearly the same time, we see little influence of the temperature on the reaction times.

The situation changes when the calculations are performed in the NVT ensemble using the Nosé–Hoover thermostat. This is shown in Figure 6 for the occupation numbers. Both calculations modeling a temperature of 100 and 300 K show a rapid quenching of the electronic excitation. Already after 80 fs, the majority of molecules has returned to the electronic ground state S0. Apart from the ground state, S1 is then the only other relevant excitation. This contrasts with the NVE calculations, where additionally S2 and S3 excited molecules occur throughout the simulation time.

Figure 6.

Figure 6

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State populations (averaged over 50 trajectories) vs. time for NVT calculations modeling a temperature of 100 K (left) and 300 K (right).

The different excitation scenarios observed in the NVE and NVT ensembles lead also to different reactions as shown in Figure 7. The energy dissipation due to the thermostat causes a slowing of the reactions, which inhibits C–C bond breaking and may even lead to a reversal of the first Gomer–Noyes reaction step

3.

after about 70 fs. Additionally, we find the temperature itself to have a much more pronounced influence on the NVT calculation than that observed for NVE. Higher temperatures, for example, considerably prolong the time period where CH2COH2 molecules constitute the majority of species.

Figure 7.

Figure 7

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Reaction products vs. time in the canonical NVT ensemble for temperatures of 100 K (left) and 300 K (right).

Figure 8 summarizes the influence of the temperature on the relative yield of the reaction products available after 250 fs. As expected, the influence of the temperature is much more pronounced in the case of NVT calculations than that observed for the NVE ensemble. In the latter case, there occur slightly less O abstractions and some more H abstractions at higher temperature. In the case of the O abstractions, this is due to a faster loss of symmetry of the molecule. As shown by ref (11) and verified in this work (see below), the O abstraction occurs if the O and C ions form an isosceles triangle. With a higher initial velocity, this symmetry is broken faster, reducing the probability of O abstractions. In the case of the H abstractions, an increase with temperature is observed since the C–H bond can break with a higher kinetic energy. The slowing of the reactions as modeled by the NVT ensembles also influences the end products as seen in Figure 8 (right). Ion abstraction is hindered in NVT ensembles, lowering the amount of O + C2H4 and (2)H + H2C2O(H) products at every temperature. A clear dependence on the temperature can be seen for the C2H4O and CH3+COH products, which prevail at low and high temperatures, respectively.

Figure 8.

Figure 8

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Percentage of reaction products vs. temperature after 250 fs integration time calculated in the microcanonical NVE (left) and canonical NVT ensembles (right). In the interest of clarity of presentation, reactions with and without C–C bond breaking are not discriminated.

A quantitative comparison with the experimental data27 measured at 77 K is presented in Table 2. Both NVE and NVT calculations overestimate the probability of CH2+COH2 formation at low temperature, while the room temperature data are closer to the experimental findings. In the case of the oxygen abstraction, the NVE calculations seem to predict a realistic probability, while NVT ensembles in some cases do not show any O abstraction, in contrast to experiment. In the case of hydrogen abstraction, all NVE calculations overestimate the measured yield, while the low-temperature NVT calculations do not find any H abstraction. This is consistent with the expectation that the complete neglect of energy dissipation in NVE calculations leads to an overestimation of kinetic excess energy, while due to the unrealistically fast energy dissipation to the NVT thermostat, there is not enough energy available for bond breaking. The fact that NVE ensembles provide a more realistic description of the O abstraction compared to the H abstraction may be related to the fact that the former reaction occurs soon after the excitation when energy dissipation effects are not yet important. Altogether, the present findings suggest that neither NVE nor NVT calculations are ideally suited to describe the energy dissipation on the timescales investigated here. Possibly, the microscopic description of the energy dissipation by means of larger atomistic ensembles, unfortunately beyond our computational limitations, could resolve these issues.

Table 2. Calculated Reaction Yields after 250 fs Simulation Time in Comparison to Measured Data after Excitation with 174–147 nm Light27a.

  NVE
 NVT
 exp.
temperature (K) 100 150 300 400 77 100 150 300 400 77
CH3+COH 1 1 1 1 1 1 1 1 1 1
CH2+COH2 0.42 0.29 0.08 0.28 0.50 0.18 0.17 0.08 0.11 0.00–0.17
O+C2H4 0.50 0.33 0.20 0.22 0.25 0.00 0.00 0.04 0.00 0.10–0.58
(2)H+H2C2(H) 0.78 0.38 0.60 1.06 0.00 0.27 0.12 0.08 0.18 0.08
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a

All yields are normalized to the CH3+COH formation. Reactions with and without C–C bond breaking are not discriminated.

Finally, based on the averaged occupation numbers and the electronic densities, we address the question of which states drive the photochemical reactions of oxirane. The ring opening, the first step in the Gomer–Noyes mechanism, is found to be correlated to the occupation of the S1 state. This can clearly be seen in Figures 4, 5, 6, and 7. Moreover, as demonstrated in the Supporting Information by means of curve fits, there is even a quantitative correlation between S1 occupation and the ring opening. The electronic density corresponding to S1 during the reaction can be seen in Figure 9. The present calculations thus, in deviation from the findings in ref (11), indicate the validity of Kasha’s rule for the Gomer–Noyes reaction: While Tapavicza et al. found that the reaction is triggered by the excitation of the S2 state, which switches later on in the reaction to the S1 state via orbital crossing, we do not observe such an orbital crossing. The reaction starts after an actual hop from the S2 to the S1 state. We attribute this deviation to the sensitivity of the oxirane excited states to changes in the occupation numbers and structural distortions as discussed above, cf. Figure 3.

In the case of the O abstraction, both the occupations of the S1 and the S2 state are found to trigger the reaction. In most of the cases, both states participate in the reaction, first, the S2 state and then the S1 state, though both states can individually cause the reaction. The reaction requires that the molecule is nearly symmetric, i.e., in its relaxed ground state. With no preference to each side, both states pushes the O ion away from the rest of the molecule. This finding is in accordance with ref (11). Electronic densities during this reaction can be seen in Figure 10.

Figure 10.

Figure 10

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Electronic densities during the O abstraction reaction. The oxirane molecule is shown at 0, 10, 20, and 35 fs. The first two images show the density of the highest occupied orbital of the excited S2 state; the latter two densities reflect the highest orbital of the S1 state.

The proton transfer (second step in the Gomer–Noyes mechanism) and the following C–C bond breaking (last step in the mechanism) occur in the S0 ground state, which is again in accordance with the findings in ref (11). Additionally, we found that while this reaction happens due to excess kinetic energy, the reaction can be stopped by occupation of the S1 state, which prohibits the proton transfer. The electronic densities during this inhibited reaction can be seen in Figure 11. The PES surface hopping necessary for this process occurs naturally via the FSSH algorithm, in contrast to ref (11). The identifications of the states responsible for the reactions via electronic densities are supported by means of the averaged occupation numbers, which can be found in the Supporting Information.

Figure 11.

Figure 11

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Electronic densities corresponding to the excited orbitals of the respective electronic states during an inhibited H transfer. The H transfer is initiated by the S0 state (left, at 135 fs). Further 10 fs later, the H ion is repelled by the S1 state (middle, at 145 fs), and finally, the molecule returns to S0 (right, at 155 fs).

4. Conclusions

The photoreaction of oxirane has been modeled with constrained DFT in conjunction with Tully’s FSSH algorithm. The present results confirm the Gomer–Noyes mechanism, i.e., a ring opening followed by an internal proton transfer concluding with a C–C bond breaking. After about 30···40 fs, the ring opening is completed. This part of the reaction is driven by the lowest excited state S1, in accordance with Kasha’s rule. Further molecular decompositions take place in the ground state S0 by means of excess kinetic energy on a similar timescale. If the molecule is nearly symmetric, then an oxygen abstraction can occur, which is driven by the S2 or S1 state. The computationally predicted reaction yields are in qualitative agreement with measured data. Remaining discrepancies are partially due to the description of the energy dissipation. On the one hand, after several hundred femtoseconds NVE ensembles are characterized by excess energy, which leads, e.g., to an overestimation of hydrogen abstraction reactions. NVT ensembles based on a Nosé–Hoover thermostat, on the other hand, experience too high cooling rates, which lead to an underestimation of dissociative reactions. Our calculations thus suggest that (i) occupation constrained DFT is a viable alternative to computationally more expensive schemes to describe photoreactions and (ii) underline the importance of a realistic modeling of the energy dissipation in the sub picosecond range, ideally by ensemble calculations, which contain a reasonable, statistically relevant number of molecules.

Acknowledgments

The Paderborn Centre for Parallel Computing (PC2) and the Höchstleistungs-Rechenzentrum Stuttgart (HLRS) are acknowledged for grants of high-performance computer time. We gratefully acknowledge funding from the Deutsche Forschungsgemeinschaft (DFG) under project numbers SCHM 1361/25, SCHM 1361/26, and TRR 142 project number 231447078.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.0c03483.

  • Numerical parameters for structure classification, the identification of the responsible occupied excited states for the reactions via occupation numbers, and a short discussion about conical intersections of the S1 and S0 state (ZIP)

The authors declare no competing financial interest.

Supplementary Material

ao0c03483_si_001.pdf (2.5MB, pdf)

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