UV photodissociation and population dynamics of some important Criegee intermediates

Abstract

The influence of Criegee intermediates [CH₂OO, (CH₃)₂COO, syn- and anti-CH₃CH₂CHOO] on atmospheric chemistry depends significantly on their photodissociation dynamics under irradiation by sunlight. In this review, we highlight examples of electronic structure (multireference electronic wave functions) and nuclear quantum dynamics calculations relying on the wavepacket propagation technique for studying photodissociation dynamics of the Criegee intermediates in the visible or near UV region. In general, two interacting electronic states and three nuclear degrees of freedom are considered for construction of model Hamiltonians in our investigation. The UV absorption spectra are found to agree very well with available experimental recordings when accounting for broadening effects due to vibrational and rotational congestion and lifetime effects. This result serves to validate the Hamiltonian model built within the quasi-diabatic representation. It is found that the vibronic coupling is a vital ingredient to provide accurate insight into the photodissociation of these atmospheric species along the O–O bond and for reproducing the experimental absorption spectra, especially for the larger species (CH₃)₂COO, syn- and anti-CH₃CH₂CHOO. Time-dependent electronic populations reveal a faster decay than for the smaller system CH₂OO. This is interpreted in terms of the stronger coupling between the B and C states in the larger systems (CH₃)₂COO, syn- and anti-CH₃CH₂CHOO leading to the shorter lifetime for the B state compared to CH₂OO.

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1 Introduction

In this review article, we concentrate on molecular species which are especially important for the Earth's lower atmosphere. These are in particular the carbonyl oxides or Criegee intermediates (CH₂OO, (CH₃)₂COO, syn- and anti-CH₃CH₂CHOO). Criegee intermediates, which constitute a significant fraction of volatile organic compounds in the Earth's atmosphere, originate from the ozonolysis of alkenes via carbonyl oxide intermediates.¹ Ozonolysis has been considered a well-known mechanism in the tropospheric oxidation of alkenes.^2–4 The ozonolysis reaction involves cycloaddition of ozone across the double bond of the alkene to constitute a primary ozonide, which is generated in a highly internally excited state and subsequently decays to a primary carbonyl species and a Criegee intermediate. Alkenes that perform atmospheric ozonolysis reactions can be categorized into two groups, i.e. those that are generated by vegetation in clean air environments and those that arise from anthropogenic sources in polluted environments.^5–7 Clean air environments tend to be richer in alkenes such as terpenes and isoprenes. Ozonolysis of these larger alkenes produces semi-volatile oxygenated species that can finally generate secondary organic aerosols.³ In urban areas, combustion emissions generate high levels of short chain alkenes including ethene, propene, butene, and pentene;⁸ these unsaturated hydrocarbons are primary pollutants, accounting for about 12% of total volatile organic compounds.⁸ These smaller alkenes are only slowly oxidized in atmospheric reactions with OH and O₃.^9,10 The Criegee intermediates produced from these smaller alkenes have 4 or less carbon atoms, starting with CH₂OO, (CH₃)₂COO and CH₃CH₂CHOO, which are in the focus of this review, see Fig. 1.

Fig. 1 The ground state equilibrium structures of CH₂OO, (CH₃)₂COO, anti-CH₃CH₂CHOO and syn-CH₃CH₂CHOO shown in (a), (b), (c) and (d), respectively. The method UCCSD(T)-F12b/cc-pVTZ-F12 is used for CH₂OO, while for the other Criegee intermediates DFT/B3LYP is employed. For all atoms, the aug-cc-pVTZ basis set is used. The relevant bond lengths and angles are in Å and degrees, respectively.

In 1949 for the first time, Rudolf Criegee proposed that the “Criegee intermediates” are formed in the atmosphere during ozonolysis, the reaction of ozone O₃ with alkenes, or unsaturated hydrocarbons.^1,11 Criegee intermediates generated in this way are unstable and thus the direct measurements of the Criegee intermediates have been a challenge for a few decades.^1,12,13 To overcome this difficulty, it was suggested that one should generate directly Criegee intermediates in the laboratory and thus they do not elude direct detection.¹ One of the ways for its direct production in the gas phase (in the laboratory) relies on the role of iodine in the troposphere.^11,14–16 As suggested in these aforementioned references, a new synthetic method was used. In this method, they considered CH₂I₂ and used the photolysis process (which is the dominant loss process in the atmosphere) to break one of the carbon–iodine bonds, then one can generate CH₂I and iodine^11,14–16

CH₂I₂ + hν → CH₂I + I. (1)

A Criegee intermediate is generated in the laboratory by the following reaction,

CH₂I + O₂ → CH₂OO + I. (2)

Direct studies, which have monitored the decay of a stabilized Criegee intermediate (SCI) or a proxy of the SCI (e.g. HCHO, OH), return rate coefficients that are considerably larger than previous indirect estimates based on end product analysis.¹⁷ These new kinetic data suggest a greater role for SCI species in the atmospheric oxidation of SO₂ and NO₂ in particular. Field measurements support a role for the SCI assisted production of H₂SO₄¹⁵ and although model studies disagree on the extent, they do agree that there is a non-negligible impact of Criegee intermediates on the oxidation of SO₂.¹⁸ If the gas-phase oxidation of SO₂ to SO₃ (and subsequently H₂SO₄) by SCI competes with, or even dominates in regions of the lower troposphere, over the oxidation by OH, the formation of H₂SO₄ may be accelerated and aerosol nucleation rates be affected.¹⁸

CH₂OO + SO₂ → HCHO + SO₃ (3)

OH + SO₂ → HOSO₂. (4)

An experimental study was performed by using the Fourier-transform microwave spectroscopy technique providing valuable information about the electronic and geometric structure of CH₂OO.¹⁹ The C–O bond length is close to that of a typical double bond and is shorter than that of the O–O bond. This argument proves that CH₂OO has a more zwitterionic character H₂C = O^⊕–O, rather than a biradicaloid H₂C˙–O–O˙. For this molecule, the time-resolved UV absorption spectrum corresponding to the B(¹A′) ← X(¹A′) electronic transition was measured.^20–23 The B(¹A′) ← X(¹A′) transition of CH₂OO was assigned to an intense absorption band, and the confirming evidence for this assignment originated from chemical kinetic measurements of formation and decay time scales for this band.^20–22 CH₂OO was produced by using the reaction (2) and the authors measured the absorption spectrum within the wavelength range, 280–440 nm.^20,21 In the experimental work of Foreman et al., they reported room temperature spectrum of the B(¹A′) ← X(¹A′) electronic absorption band of CH₂OO, acquired at higher resolution, using both single-pass broadband absorption and cavity ring-down spectroscopy. The new absorption spectra confirm the vibrational structure on the red edge of the band that is absent from ionization depletion measurements. They confirmed that the vibrational structure is not due to hot bands.²²

In ref. 24, the potential energy surfaces (PESs) along O–O bond length corresponding to the electronic states X¹A′, A¹A′′ and B¹A′ at the EOM-CCSD level using the 6-311++G(d,p) basis set were computed. They found that the B state is crossed near the Franck–Condon region by the C state potential surface. Due to this crossing it can be inferred that the dynamics after excitation to the B state depends on the strength of the interactions between these two B and C excited states. The precise determination of the coupling strength between these two states is important for a thorough understanding of the dynamics after excitation to the B state. This coupling can be zero, weak or strong. Zero coupling between the two states C and B would produce bound, potentially fluorescent vibrational levels in the B state. Weak coupling between these two states would result in predissociation of the B and C states, with potential for vibrational structure in the spectrum. Strong coupling would cause these two C and B states to avoid each other (repel) significantly in the vicinity of the crossing point. Lee et al. simulated the absorption spectrum using the double-harmonic approximation.²⁵ However, anharmonicity was not taken into account; consequently no agreement between their simulation and the latest experimental results was achieved (see ref. 21). Dawes et al. computed the absorption spectrum corresponding to the B(¹A′) ← X(¹A′) electronic transition using the Chebyshev real wave packet method, where they used the 1D model Hamiltonian depending on only the O–O stretching coordinate and obtained a spectrum in good agreement with the corresponding experimental one.²⁶ However, they were not able to produce the vibronic structure appearing in the experimental spectrum and neither the considerable shift between the computed and experimental spectra specially at longer wavelengths. In the experimental work of Chhantyal-Pun et al.,²⁷ the authors reported that CH₂OO oxidizes SO₂ to SO₃ and hence may contribute to atmospheric sulphuric acid production. They measured the bimolecular reaction rate of CH₂OO + SO₂ under low pressure and ambient temperature conditions via direct study. These authors measured a rate coefficient of (3.80 ± 0.04) × 10⁻¹¹ cm³ molecule⁻¹ s⁻¹ for the reaction of CH₂OO + SO₂ at low SO₂ concentrations. They mentioned that at low SO₂ concentrations, the formation of the triplet state in the biradical adduct of CH₂OO and SO₂ via intersystem crossing (ISC) may play an important role in the reaction of CH₂OO + SO₂. Other theoretical investigations showed that there is a small difference between the singlet and triplet states (0.4 kJ mol⁻¹).^12,13

Regarding larger Criegee intermediates, (CH₃)₂COO, as well as its structural isomer CH₃CH₂CHOO were generated by synthesizing of (CH₃)₂CI₂ and (CH₃)CH₂CHI₂ precursors, respectively in the laboratory.²⁸ Experimental evidence showed that these species have a more zwitterionic character rather than a biradicaloid nature.²⁸ The authors of ref. 28 and 29 measured the absorption spectra corresponding to the B̃(¹A′) ← X̃(¹A′) electronic transition for these species in the wavelength range 280–380 nm. In ref. 29, authors also computed the spectrum and observed a good agreement between the theoretical and experimental spectra for (CH₃)₂COO. Authors in ref. 28 observed the significant depletions of the ion signals when UV laser radiations with wavelengths near the peak of the absorption spectrum (323 and 322 nm for (CH₃)₂COO and CH₃CH₂CHOO, respectively) are introduced prior to the VUV photoionization laser. They argued that these large depletions are indicative of rapid dynamics in the excited B state, most likely due to direct dissociation.²⁸

Motivated by these studies, we have investigated the photodissociation dynamics and UV absorption spectra of the Criegee intermediates mentioned above.^30–33 We have primarily focused on two aspects. First, the PESs of the excited states with A′ symmetry are studied by employing multireference methods, and thus the role of electron correlation is taken into account by selecting a suitable active space. This is followed by constructing model Hamiltonians for each molecule in which the vibronic coupling between PESs of the two lowest excited A′ states is estimated by the minimum energy difference at the avoided crossing geometries. Second, the photodissociation dynamics is studied by employing a fully quantal description of the nuclear motion using the model Hamiltonians built in the first step and the multiconfiguration time-dependent Hartree (MCTDH) approach.^34–37 As will be mentioned, this results in computed spectra and time-dependent populations of the electronic states of each molecular species involved in the dynamics. This analysis can provide an accurate and quantitative innht into the photodissociation dynamics of these important atmospheric molecules under irradiation by sunlight.^30–33,38

In the following section (Section 2), the detailed information regarding the construction of the Hamiltonian model will be addressed. In Section 3, the diabatization by ansatz method adopted will be introduced. In Section 4, the details of electronic structure methods for computing optimized geometries, and potential energy surfaces (PESs) will be outlined. Section 5 is devoted to the discussion of the absorption spectra of each molecular species, where they are compared with available experimental spectra. The time-dependent population analysis is documented in Section 6. Finally, Section 8 is devoted to summary and outlook.

2 Vibronic Hamiltonian and quantum dynamical approach

In our previous works,^31–33 vibronic model Hamiltonians for the description of the UV absorption process for these Criegee intermediates were suggested. These models yield a good agreement between the computed spectra and the corresponding experimental ones indicating that the models can express reliably the physical processes underlying the UV absorption spectroscopy and ultrafast radiationless decay. Here, we recap the main idea for the construction of the model Hamiltonian for these molecular species. The vibronic Hamiltonian H is constructed in a (quasi-)diabatic electronic basis (see e.g. ref. 37 and 39) for the B¹A′ and C¹A′ states of the Criegee intermediates, while the X¹A′ state is assumed to be decoupled in the usual adiabatic picture,

H = T_N1 + W(S). (5)

Here H contains the kinetic part T_N (1 stands for the 3 × 3 unit matrix) and the potential part W(S). As we showed previously for the relevant Criegee intermediates,^31–33 a 3D Hamiltonian model including the O–O, C–O stretching and C–O–O bending modes is sufficient to produce the UV absorption spectra in a very satisfactory agreement with the corresponding experimental results. The Hamiltonian model includes the O–O and C–O stretching and the C–O–O bending modes. The three modes are totally symmetric, i.e. of a′ symmetry for planar conformations (C_s point group). In the diabatic representation for the excited states, W(S) reads

In eqn (6), the vector S refers to the three relevant internal coordinates mentioned above. V_X(S) refers to the adiabatic PES for the ground state X¹A′, while W_B(S) and W_C(S) correspond to the PESs of the relevant B¹A′ and C¹A′ electronic states in the diabatic representation along the relevant internal coordinates. W_BC(S) is the potential coupling between the B and C states. It is known that these Criegee intermediates are dissociated via the O–O bond cleavage in the presence of UV light (see Section 4). Since in the wavepacket (WP) propagation calculation, we need to consider periodic boundary conditions (e.g. FFT), the WP along the O–O stretching coordinate can be reflected or it penetrates the other side of the grid leading to the depreciation of the wavefunction. A complex absorbing potential (CAP) along the O–O stretching coordinate will be used in the model Hamiltonian of eqn (6) to avoid the convergence issue in the WP propagation calculation. This includes an negative imaginary potential (introduced by Leforestier and Wyatt,⁴⁰ and later Kosloff⁴¹), which is used to absorb the wavepacket as it approaches the end of the grid. It thus helps in the propagation to converge better along the O–O stretching coordinate. The CAP is usually defined as follows³⁵

iW(Q) = −iη(Q − Q_c)^bθ(Q − Q_c) (7)

In this equation, Q_c, b and η are the starting point, order, and strength of the CAP, respectively. The parameters Q_c, η and b used in the current work are 2.1 Å, 3.1 × 10⁻⁴ a.u., 3, respectively.^30–33 The symbol θ(x) refers to Heaviside's step function. The distance between the starting point of the CAP and the end of the grid determines the length of the CAP, which is usually specified by visualizing the topology of potential energy cuts (see Section 4). The optimal CAP order is 2 or 3 as suggested in ref. 42. In our study, the length of the CAP is ∼1.4, resulting in a sufficiently smaller CAP strength η to avoid unwanted CAP reflections in the quantum dynamical calculation (see ref. 42 for more information for determining of the CAP parameters). We thus modify the potential part of the Hamiltonian in eqn (6) by adding iW(Q) according to eqn (7).

We compute the kinetic part T_N of the Hamiltonian using the familiar G-matrix method of Wilson et al.^43,44 as indicated in our previous works.^31–33 Since we are considering the O–O stretch, C–O–O bend and C–O stretch in the evaluation of W(S) in eqn (6), the C and O atoms play roles in the evaluation of the kinetic operator. Thus, the kinetic operator reads,^30,43

The variables R, R′ and θ refer to the O–O and C–O stretching as well as C–O–O bending coordinates, respectively. In eqn (8), the coefficients are defined as follows,

μ_O = 1/m_O μ_C = 1/m_C ρ_OO = 1/R_OO ρ_CO = 1/R_CO, (9)

where m_O = 15.999 and m_C = 12.011 in atomic mass units. For the R_OO and R_CO stretching coordinates and the θ_COO bending coordinate, we use the values for the ground state equilibrium structure of the Criegee intermediates shown in Fig. 1.

We use the multi-configuration time-dependent Hartree (MCTDH) method^34–36 for the nuclear quantum dynamics calculations. More detailed information regarding the MCTDH approach can be found in the literature (e.g. ref. 34–36). The relevant information as employed in this work is given in our previous papers^30,31 and thus we skip it in the current work. We use the POTFIT method for fitting the ab initio data to the model introduced in eqn (6). We refer the reader to ref. 31 for more details relevant to this aspect. For more details on the POTFIT method, the reader is referred to ref. 45.

3 Diabatization of excited-state potential energy surfaces

It is well known that the Born–Oppenheimer separation of electronic and nuclear motions fails when two or more electronic states are close in energy within a certain range of the nuclear coordinate space.⁴⁶ Within this condition, the Born–Oppenheimer approximation can be still used if one can develop an efficient method to deal with the near-degeneracy of electronic states in localized regions of coordinate space. A suitable choice to circumvent this issue is the introduction of suitable “diabatic” electronic states that may cross as a function of the internuclear distances, whereas the adiabatic electronic states are subject to the noncrossing rule.^{37,39,47–49} There are a number of ways to diabatize the adiabatic PESs (e.g. regularized diabatic states).^50,51

For the purpose of the excited-state PES diabatization of these Criegee intermediates, we have used a diabatization-by-ansatz approach.^30–33 This is a well-known method in the diabatization context and well developed in the literature.^37,39 For the Criegee intermediates discussed in this study, the relevant excited states are B¹A′ and C¹A′ and the relevant coordinate is the O–O distance, that is a single nuclear coordinate. Illustrations are provided by Fig. 2 (see also Section IV for the abinitio methods adopted).

Fig. 2 Adiabatic PECs of the Criegee intermediates along the O–O bond length for the three states X¹A′ (black), B¹A′ (green) and C¹A′ (cyan). The vertical excitation energy, minimum energy of the B state, energy at the avoided crossing and the energy at the barrier maximum are shown by red, blue, black and gray circles, respectively. In the insets, the energy gaps at the avoided crossing geometries are shown. The other molecular coordinates are fixed at the equilibrium structure values.

We apply the diabatization-by-ansatz approach in the spirit of the well-established vibronic coupling model where only the coupling terms of leading order are retained. For such one-dimensional same-symmetry problems the leading coupling terms are of order R⁰_OO or constant. This allows to extract the diabatic potentials in a simple way from the adiabatic quantities by equating the off-diagonal coupling W_BC with the minimum energy gap along the avoided crossing, as follows.

Diagonalizing the block of the potential matrix in eqn (6) results in the energies of the B¹A′ and C¹A′ states in the adiabatic picture. The diagonalization of each block can be represented as follows:

where i ∈ {B¹A′} and j ∈ {C¹A′}. The off-diagonal term W_ij(S) refers to the vibronic coupling between the coupled states (B¹A′, C¹A′) and is shown as W_BC(S), in the Hamiltonian model. In this equation, U_ij(S) is a unitary transformation and readswhere α(S) is the adiabatic to diabatic (ATD) transformation angle and given by

We define the matrix elements of the unitary transformation using the sin and cosine functions depending on the ATD angle (see eqn (11) and (12)). Taking the eigenvalue problem in eqn (10) to be solved, we compute the diabatic PESs in terms of adiabatic energies V_i(S) and V_j(S) obtained from the electronic structure methods discussed in the next section. Thus, the diabatized surfaces can be computed using the following relations,

The diabatic energies W_B,C can thus be obtained from the adiabatic counterparts V_B,C once the quantity W_BC is known. This is determined from the minimum energy difference between states B and C, see next section.

4 Quantum chemistry methods and potential energy surfaces

We used the optimized structures reported in the previous works.^30,32,33 The UCCSD(T)-F12b/cc-pVTZ-F12 level of theory was employed for the optimization of the CH₂OO ground state structure,^52,53 while the semiemprical density functional theory (DFT) method with the B3LYP functional^54,55 and the aug-cc-pVTZ basis set for all atoms⁵³ was used for the (CH₃)₂COO and CH₃CH₂CHOO molecules. The resulting equilibrium ground state structures of for these molecular systems are shown in Fig. 1.

For the CH₂OO, the potential energy surfaces (PESs) were computed in our previous work³¹ by employing the CASPT2-F12 (hereafter RS2-F12)⁵² method using the cc-pVTZ-F12 basis set for all atoms.^53,56 We employ the full-valence complete active space (CAS), with 18 electrons distributed in 14 orbitals (18e,14o). For (CH₃)₂COO molecule and CH₃CH₂CHOO with its molecular conformers,^32,33 we used the RS2C method (second-order multireference perturbation theory with a more contracted configuration space)⁵⁷ with the basis set 6-311++G(2d,2p) for all atoms.⁵⁸ For this calculation, the active space with eight electrons distributed in eleven orbitals, CAS(8e,11o), is employed. During the evaluation of the PESs, state averaging is used in order to avoid root-flipping problems. In the state averaging, we consider the three lowest states of a′ symmetry and the two lowest states of a′′ symmetry. For all molecules, we compute the PESs along R_OO, θ_COO and R_CO for the X¹A′, B¹A′ and C¹A′ states within the ranges specified in Table 1. For the evaluation of PESs along the relevant coordinates, we keep the other coordinates fixed at the equilibrium values mentioned in Fig. 1. The geometry optimizations and PES computations are carried out by using the MOLPRO 2019.2 suite of programs.⁵⁹ In Fig. 2, the adiabatic potential energy cuts (PECs) along R_OO by fixing all other coordinates, for the X (black line), B (green line) and C (cyan line) states are shown for all Criegee intermediates relevant to this work.^31–33 It can be seen that the PECs of B and C states represent avoided crossings for all four cases shown in this figure. The avoided crossings for CH₂OO, (CH₃)₂COO, syn-CH₃CH₂CHOO and anti-CH₃CH₂CHOO occur at 3.05, 3.43, 3.20, 3.33 eV, respectively, while locations of the avoided crossing along R_OO are 1.74, 1.68, 1.66 and 1.67 Å, respectively. The vertical excitation energies of the B states are indicated by red circles in each panel. We also compare our values with the available ones in the literature (see Table 2). For all molecular cases, in the asymptotic limit of O–O stretch, CH₂OO, (CH₃)₂COO and syn- and anti-CH₃CH₂CHOO are dissociated via the O–O bond cleavage to H₂CO(X¹A) + O(¹D), (CH₃)₂CO + O(¹D) and CH₃CH₂CHO + O(¹D) products, respectively. The estimated dissociation energies for these Criegee intermediates are 2.55, 2.76, 2.22 and 2.49 eV, respectively.

Table 1 The grid used for molecular coordinates of all Criegee intermediates

Molecule	ROO	Interval	θCOO	Interval	RCO	Interval
CH2OO	1.09–4.0 Å	0.05 Å	92°–120°	2°	1.12–1.32 Å	0.05 Å
(CH3)2COO	1.38–3.48 Å	0.1 Å	102°–120°	2°	1.06–1.46 Å	0.1 Å
syn-CH3CH2CHOO	1.30–4.0 Å	0.1 Å	110°–122°	2°	1.16–1.46 Å	0.1 Å
anti-CH3CH2CHOO	1.30–4.0 Å	0.1 Å	110°–124°	2°	1.15–1.45 Å	0.1 Å

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Table 2 Reported vertical excitation energies of the B and C states and their character for the relevant Criegee intermediates in this work. The corresponding reference for each value is mentioned in the table footnote

Molecule	Transition	Theory	Experiment
a Ref. 22.b Ref. 30.c Ref. 31.d Ref. 60.e Ref. 26.f Ref. 20.g Ref. 32.h Ref. 61.i Ref. 28.j Ref. 33.
CH2OO	π → π*	3.66^a, 3.83^b, 3.65^c, 3.81^d, 3.75^e, 4^f	3.65^a, 3.704^f
	n → σ*	6.99^c	—
(CH3)2COO	π → π*	3.78^g, 3.82^h	3.82^h
	n → σ*	5.62^g	—
syn-CH3CH2CHOO	π → π*	4.00^j	3.85^i
	n → σ*	6.15^j	—
anti-CH3CH2CHOO	π → π*	3.63^j	—
	n → σ*	5.85^j	—

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It can be seen that the minimum energy of the B state (shown by blue color circle in each panel of Fig. 2) for CH₂OO is located slightly before the avoided crossing while for the rest of Criegee intermediates it is after the avoided crossing. In the inset of each panel in Fig. 2, the PECs for the B and C states are replotted in a smaller energy range allowing to observe clearly the energy gap between the minimum energies of each curve. This indicates that it has the smallest value for CH₂OO (0.05 eV) and the largest value for (CH₃)₂COO (0.2 eV). Thus, for the specific PECs shown in Fig. 2, one may conclude that the PECs of the B and C states are more strongly avoided in the case of (CH₃)₂COO compared with other Criegee intermediates.

In Fig. 3, we plot – for R_CO fixed at the ground state equilibrium – the minimum energy difference between the B and the C state along the R_OO stretching coordinate for various values of θ_COO. This is done for the various Criegee intermediates. Afterwards we average over θ_COO and repeat this procedure for different values of R_CO, as shown in Fig. 4a. An overview of the selected values for R_CO and R_OO is presented in Table 1. Finally we average over R_CO to get the average values presented in Fig. 4b for the various Criegee intermediates. This is equated with twice the off-diagonal potential coupling W_BC. Thus, within this 3D PESs, anti-CH₃CH₂CHOO shows the largest vibronic coupling between the B and the C state indicating that the B and the C state show a stronger repulsion compared with the other Criegee intermediates. Please note that this finding is not the same as the one in Fig. 2, where the PECs of B and C state are computed only along R_OO.

Fig. 3 Minimum energy difference of the R_OO PE curves of the B and C states for all Criegee intermediates. The dependence of the minimum energy difference on θ_COO is taken into account by carrying out the calculation for different values of θ_COO in the range of 110° < θ_COO < 124°. In this figure, the values of R_CO for each molecule are fixed at their ground state equilibrium structures specified in Fig. 1.

Fig. 4 (a) (average of the minimum energy difference ΔE_min over the bending angle for all Criegee intermediates) in terms of R_CO within the molecular coordinate range specified in Table 1. (b) The black dots refer to the average of over the range of R_CO specified in Table 1 for each molecule. These doubly averaged values are used as twice the coupling constants between the B and C states or twice the off-diagonal element W_BC in the 3D model for all Criegee intermediates.

In the following, the PECs are diabatized based on the method described in Section 3. In a one-dimensional treatment the coupling W_BC is assumed to be constant (as indicated above). The other degrees of freedome θ_COO and R_CO are treated as spectator modes and are accounted for by the simple averaging procedure described above, thus modifing the constant coupling W_BC. The resulting diabatic PECs are shown in Fig. 5. The use of a diabatic representation is a consequence of the weakly avoided crossing between the B¹A′ and C¹A′ adiabatic states. Due to the nature of the diabatization process, the diabatic PECs are smoother than the corresponding adiabatic ones in Fig. 2. Thus, as can be seen in Fig. 2 and 5, changing the avoided crossing features to intersection points is a consequence of the diabatization process removing potential couplings between the adiabatic states. Within these diabatized PESs, we can estimate the barrier energy of the B state for each molecule. Fig. 5 shows that barrier heights of 0.38, 0.01, 0.57 and 0.32 eV are estimated for CH₂OO, (CH₃)₂COO, syn-CH₃CH₂CHOO and anti-CH₃CH₂CHOO, respectively.

Fig. 5 Diabatic PECs of the B and C states as function of R_OO obtained by employing the diabatization by ansatz method. The dashed green lines refer to the CAP as defined by using eqn (7). It is used in the diagonal elements of the model Hamiltonian of eqn (6) to let the propagation calculation converge faster along the O–O stretching coordinates for all Criegee intermediates.

In Fig. 5, the imaginary part of the complex absorption potential (CAP) along the O–O stretching coordinate relevant to the photodissociation along this coordinate is also included. The CAP starts at ∼2.11 Å, right after the barrier of the diabatic B state.

5 Absorption spectroscopy

5.1 CH₂OO

The absorption spectroscopy of CH₂OO has been studied experimentally and theoretically since a decade ago.^{20–22,24,28,62–64} The early measurements have been carried out by Lester and co-workers, who reported the action spectra corresponding to the Criegee intermediates and identified a very strong UV absorption of CH₂OO, due to the B ← X transition within the UV region.²⁰ The velocity and angular distributions of O(¹D) photofragments arising from UV excitation of CH₂OO by the B¹A′ ← X¹A′ transition indicate that the corresponding electronic transition has π* ← π character.⁶² Sheps also measured the B¹A′ ← X¹A′ electronic transition of formaldehyde oxide, CH₂OO, produced by the reaction of CH₂I radicals with O₂.²¹ In contrast to its UV photodissociation action spectrum, the absorption spectrum of formaldehyde oxide extends to longer wavelengths and exhibits resolved vibrational structure on its low-energy side.²¹ In another attempt carried out by Foreman et al., a high-resolution absorption spectrum of CH₂OO was reported in which the authors confirmed the presence of vibrational structure on the red edge of the band for the first time.²² There have been significant discrepancies among the reported measurements of the strong B¹A′–X¹A′ electronic transition in the visible/near-UV spectral range. The peak cross section measured by Ting et al. is (1.23 ± 0.18) × 10⁻¹⁷ cm² at 340 nm.²³ This number is significantly smaller than the previous values reported e.g. by Sheps.²¹ Ting et al. mentioned that this intense B¹A′–X¹A′ excitation spectrum overlaps with the incoming solar spectrum, thus yielding efficient photolysis of CH₂OO.²³ Furthermore, the reported spectrum by Ting et al. shows stronger vibrational structures in the long-wavelength range (λ > 360 nm) compared with Sheps' work. Foreman et al.²² performed a similar measurement and their result gave significant vibrational structure in the low-energy part of the B¹A′–X¹A′ excitation spectrum of CH₂OO.

In ref. 30 and 31, we have also focused on the photodissociation dynamics of CH₂OO using the nuclear quantum dynamics approach (see Table 3). We performed a quantum dynamical calculation considering three vibrational modes, the R_OO and R_CO stretching modes as well as the θ_COO bending mode, where the PESs corresponding to the X, B and C states were computed by using the MRCI-F12 and RS2-F12 methods as mentioned in Section 4. The coupling between the B and C states was taking into account in our Hamiltonian model. We used the Hamiltonian of eqn (5) in the propagation calculation and computed the absorption spectrum within different wavelength ranges. In Fig. 6, we present the B¹A′–X¹A′ excitation spectrum of CH₂OO computed using the 3D model including the O–O and C–O stretching as well as C–O–O bending modes (shown by the solid green line in Fig. 6). The corresponding PESs were computed by using the RS-F12 method.³⁰ In this figure, we also present the absorption spectrum (shown by the solid blue line in Fig. 6) obtained by using the 2D PESs depending on the O–O stretching and C–O–O bending coordinates. These 2D PESs were evaluated by employing the MRCI-F12b method.³⁰ We further compared our results for CH₂OO with the available theoretical results in the literature.^26,30 The dark blue line in Fig. 6 was computed in our previous work using the 2D model (depending only on the r_OO and θ_COO coordinates) with PESs evaluated by the MRCI-F12b method,³⁰ while the solid red line was calculated by Dawes et al.,²⁶ who used a 1D model with the uncoupled adiabatic PES of the B state alone as a function of the O–O stretching coordinate. The dephasing time in the spectrum shown by dark blue and green lines is 22 fs. It can be seen that there is agreement between the green line spectrum and the experimental result of Ting et al., see Fig. 6. In fact, the main peak of both spectra occurs at ∼3.65 eV (or 339 nm) and the shape of the computed spectrum is close to the experimental result of the Ting et al. especially for λ < 340 nm. Another important feature of the green line spectrum is that it produces nicely the vibronic structures in the low-energy part of the spectrum apeared in the corresponding experimental ones measured by Ting et al. and Sheps, while the action spectrum measured by Beames et al. shows no vibronic structure (see below).

Table 3 Number of basis functions for the primitive harmonic and SPF basis used for the propagation calculations

Modes	DVR	Primitive basis	SPF basis
ROO	FFT	128	9, 10, 9
RCO	HO	88	9, 10, 9
θCOO	HO	80	9, 10, 9

---

Fig. 6 The computed excitation spectrum of CH₂OO for the B¹A′–X¹A′ transition compared with the available experimental and theoretical results from ref. 20, 21, 23 and 26, respectively. The dark blue line is taken from our previous work,³⁰ based on the 2D PES using MRCI-F12 method. The full green line (this work) is obtained from the 3D model (r_OO, r_CO and θ_COO) based on the RS2-F12 method.

In the experimental work of Foreman et al.,²² a high resolution of the absorption spectrum (λ > 360 nm) was reported and vibrational progressions were identified. The authors attributed observed vibrational progressions to the excitation of bound levels of the B state, while it was mentioned in ref. 28 that the progressions originate from hot bands.²⁸ To further investigate on the low energy part of the absorption spectrum (λ > 360 nm), we set up comprehensive nuclear quantum dynamical calculations in ref. 31, in which we carried out propagation calculations employing the 3D, 2D and 1D model Hamiltonians. In Fig. 7, the spectra evaluated with dephasing times 40 and 1000 fs are depicted by red and green lines, respectively, and compared with experiment. In the inset, the spectrum within the wavelength range of 376–406 nm is visualized (red line) and compared with the experimental spectrum measured by Foreman et al. in ref. 22 (black line). From this result, one can see that the positions of observed peaks are very well reproduced by the 3D calculation, and thus an excellent agreement between theory and experiment is achieved. In ref. 31, we further investigated the abosorption spectum using the 2D and 1D model diabatic PES depending the O–O stretching and C–O–O bending coordinates. Detailed nuclear quantum dynamical calculations from these calculations revealed that the main spectral progression represents the O–O stretching and C–O–O bending modes.³¹

Fig. 7 Comparison of the computed low-energy part of the excitation spectrum employing the 3D model using the RS2-F12 method with the experiment from ref. 22. The spectrum with green color has no dephasing time, while the spectrum depicted in red has a dephasing time τ = 40 fs (also in inset), which depicts the spectrum on an expanded scale.

So far, we have treated the B and C states are coupled states. The question arises, whether the vibronic coupling plays an important role in reproducing the spectrum or not. To address it, we compute the spectrum using the uncoupled 1D, 2D and 3D models separately. The corresponding results are shown in Fig. 8. Although not directly visible from the figures, a closer analysis reveals that the coupled and uncoupled 3D spectra are crudely similar in the appropriate energy range. Thus, the vibronic coupling plays a less important role in the spectrum. In Fig. 8, it can be seen that the computed spectrum shows quasi-discrete and continuous features below and above the barrier energies (E_b) of the diabatic B state, respectively using the 1D, 2D and 3D model Hamiltonians. Furthermore, increasing the dimensiality in the Hamiltonian models shifts the energy barrier of the diabatic B state towards the high-energy part of the B¹A′ ← X¹A′ absorption spectrum. As a result, we can observe more quasi-discrete structure in the spectrum obtained from the 3D model than in the spectra obtained via the 2D and 1D models. In principle, the continuous part of the spectrum within the 3D model is located in the spectrum tail beyond ∼4 eV.

Fig. 8 The B¹A′ ← X¹A′ absorption spectrum (with no dephasing time) using the uncoupled 1D, 2D and 3D Hamiltonian models in CH₂OO molecule. Spectrum changes its feature below and above the barrier energies E_b 3.43, 3.80 and 4.20 eV within the 1D, 2D and 3D Hamiltonian models, respectively from quasi-discrete to continuous structure. Due to the small width (slow tunneling) the narrow spectral features look discrete to the eye.

These continuous and quasi-discrete features of the spectrum can be understood by looking at the behavior of WP at the barrier energy. The WP is partly reflected by the barrier to the right of the shallow well (at ∼1.7) resulting in the discrete structure in the spectrum as shown in the lower panel of Fig. 16. Some parts of the WPs penetrate the barrier (by tunneling) and reach the CAP, which is activated at R_OO ∼ 2.11 (after the barrier of the diabatic B state potential energy curve, shown in Fig. 5). This part of the WP is absorbed in the CAP resulting in a different spectral feature, namely a small width of the quasi-discrete structures below the barrier and an continuous structure above the barrier.

5.2 (CH₃)₂COO

The B¹A′–X¹A′ absorption spectrum (CH₃)₂COO was measured by Peltola et al. within the wavelength range of 320–340 nm.⁶³ The spectrum shows clearly strong oscillatory structures in this wavelength range, which have not been detected in the other experimental results (see for e.g. ref. 64).

In ref. 32, we studied the B¹A′ ← X¹A′ absorption spectrum using the nuclear qunatum dynamics approach. The 3D PESs were obtained by employing the RS2C method as mentioned in Section 4. Similar to CH₂OO, we used the relaxation scheme to compute the initial wavepacket corresponding to the ground state PESs. Then, we lift the wavepacket from ground state to the excited state B¹A′ in the propagation calculation. The computed spectrum (using the 3D coupled Hamiltonian model) is shown in Fig. 9 (shown by the solid red line) and compared with the experimental recording taken from ref. 64 (shown by the solid black and blue lines). The agreement between theory and experiment is satisfactory; oscillatory structures rather than noise can be seen in both computed and measured spectrum, the width of the experimental and theoretical spectra is quite similar and finally the main peak of the computed spectrum is 328 nm compared to the experimental value (330 nm).^9,28,64 Peltola et al. obtained a value of 340 nm for the main peak of the B¹A′–X¹A′ absorption spectrum.⁶³

Fig. 9 The computed B¹A′–X¹A′ absorption spectrum of (CH₃)₂COO using the coupled 3D model (shown by red line with dephasing time, τ = 25 fs) compared with the available experimental spectrum (shown by black and blue lines) taken from ref. 64. The computed spectrum is obtained by employing the coupled 3D model depending the stretching modes R_OO, R_CO and the bending mode θ_COO.

In Fig. 10, we present the spectrum with no phenomenological dephasing time in the wavelength range of 400–320 nm (shown by the red line) using the coupled 3D model.³² In this figure, the computed spectrum is compared with the measured spectrum of Peltola et al.⁶³ represented by the blue line. It can be seen that the simulation can generate the spectrum with strong oscillatory structures similar to what is observed in the experiment. The locations of stong peaks in both theory and experiment are reasonably close to each other, except for peaks located beyond 370 nm in the spectrum. As observed for CH₂OO, where its photodissociation occurs upon the B–X electronic excitation,³¹ (CH₃)₂COO upon electronic excitation from 280 to 400 nm undergoes rapid dissociation due to the repulsive PES of the excited B state. At the lowest dissociation asymptote, the photolysis products are (CH₃)₂CO + O(¹D).

Fig. 10 The computed B¹A′–X¹A′ absorption spectrum of (CH₃)₂COO using the coupled 3D model (shown by red line with no dephasing time) compared with a high resolution experimental spectrum (shown by blue line) taken from ref. 63.

Similar to CH₂OO, we want to investigate the role of vibronic coupling in generating spectrum and photodissociation dynamics. For this task, we compute spectrum using uncoupled 1D, 2D and 3D Hamiltonian models. The corresponding results are shown in Fig. 11. Comparing the spectrum computed by the uncoupled 3D Hamiltonian model in Fig. 11 with the spectrum computed with the coupled 3D Hamiltonian model in Fig. 9 leads us the conclusion that the vibronic coupling plays an important role in the absorption spectrum, while this was less the case for CH₂OO. For this molecule, we can see that the spectrum has continuous and quasi-discrete features above and below the barrier energy, respectively, for spectra computed using the 1D, 2D and 3D Hamiltonian models. Furthermore, Fig. 11 reveals that the barrier energy moves toward to the high-energy part of the absorption spectrum, when we enhance the dimensionality of the Hamiltonian model, similar to CH₂OO. The continuum part of spectrum goes to the spectral tail (above barrier energy of 4.4 eV). Most parts of the spectrum remain below the energy barrier within the uncoupled 3D model, so that it shows a quasi-discrete structur (also observed in CH₂OO). For understanding the continuous and quasi-discrete features of the spectrum in this molecule, the same argument mentioned for CH₂OO holds. In other words, some parts of the WP are reflected by the barrier resulting in the quasi-discrete structure in the spectrum (shown in the upper panel of Fig. 16). Other parts of the WPs penetrate the barrier by tunneling and reach the CAP, which is activated at R_OO ∼ 2.11, shown in Fig. 5.

Fig. 11 The B¹A′ ← X¹A′ absorption spectrum (with no dephasing time) using the uncoupled 1D, 2D and 3D Hamiltonian models in (CH₃)₂COO molecule. Spectrum changes its feature below and above the barrier energies E_b 3.80, 4.1 and 4.40 eV within the 1D, 2D and 3D Hamiltonian models, respectively from quasi-discrete to continuous structure.

5.3 CH₃CH₂CHOO

As shown in Fig. 1, this molecule has two conformers which are termed as syn- and anti- CH₃CH₂CHOO conformers. The experimental absorption spectroscopy for CH₃CH₂CHOO was carried out nearly a decade ago.²⁸ This relatively large Criegee intermediate was photolytically generated from diiodo precursors, detected by VUV photoionization at 118 nm, and spectroscopically characterized via UV-induced depletion of the m/z = 74 signals under jet-cooled conditions.²⁸ As a result, the strong UV absorption assigned to the B¹A′ ← X¹A′ transition with π* ← π character and peaked at 320 nm was measured within wavelength window 280–440 nm.²⁸

In ref. 33, we conducted a nuclear quantum dynamical study relying on the 3D PESs computed by using the RS2C methods for each conformer as mentioned in Section 4. Similar to other Criegee intermediates, we used the relaxation scheme, computed the ground state wavefunction and lifted the ground state wavefunction to the excited B state in the propagation calculation. The propagation time was chosen to be 1000 fs with a time step of 0.5 fs.

The syn-, anti- as well as the cumulative spectra of CH₃CH₂CHOO employing the coupled 3D model with the dephasing time 10 fs and no dephasing time in the left and and right panels of Fig. 12. As can be seen in this figure, we use the following color coding for the spectra; the black, red and blue lines are for syn-, anti- and the cumulative spectra, respectively. In the left panel of Fig. 12, we compared our result with experiment shown by black dots taken from ref. 28. The cumulative spectrum, which is the sum over the spectra of each conformer, is computed assuming that the excitation to the B state from the ground state X takes place with the same probability for both conformers.

Fig. 12 The computed B¹A′ ← X¹A′ absorption spectrum of CH₃CH₂CHOO using the coupled 3D model for the two conformers syn- and anti- shown by black and red lines with dephasing time (left panel) τ = 10 fs, which is compared with the experimental absorption spectrum (black points) of CH₃CH₂CHOO taken from ref. 28. The spectra for both conformers with no dephasing time are shown in the right panel. The cumulative spectrum shown by a bold blue line.

Comparing the computed spectrum with the experiment indicates that the spectrum obtained from syn- conformer is very close to the experimental spectrum in the left panel of Fig. 12 taken from ref. 28. As a consequence, the spectrum measured in ref. 28 may be attributed to the B¹A′ ← X¹A′ transition of the syn- conformer. In Fig. 12, the spectrum shown by a red line is assigned to the B¹A′ ← X¹A′ transition for the anti-conformer.

For CH₃CHOO molecule, experimentally and theoretically separate spectra of the two possible structural isomers were suggested.^65,66 Sheps et al. were able to detect the distinct spectra of the two conformers using their different reactivity towards H₂O and SO₂.⁶⁵ As a consequence, it was found that the spectra of the two conformers are enough separated to allow direct conformer-specific probing of the reactions of CH₃CHOO with other important tropospheric species.⁶⁵ Furthermore, it was shown that syn-CH₃CHOO peaks at the shorter wavelength range of the absorption spectrum, while the peak of anti-CH₃CHOO is located in the longer wavelength range of the spectrum. We also reported similar observation for CH₃CH₂CHOO in ref. 33, where the syn- and anti-CH₃CH₂CHOO conformers are sufficiently separated; the band maximum of the syn-conformer spectrum (the black line in Fig. 12) is moved toward higher energy, while the band maximum of the anti-conformer spectrum is at the low energy side of the spectrum (red-shifted from syn-CH₃CH₂CHOO absorption).

In the following, we recalculate the absorption spectra for these conformers using the 3D model Hamiltonian ignoring the vibronic coupling between the B and C states (the uncoupled Hamiltonian model). The corresponding results are shown in Fig. 13. It can be seen that the prominent features are quasi-discrete though irregular line structures in Fig. 13, which are replaced by broad peaks in the right panel of Fig. 12, due to the coupling to the lower, repulsive energy surface (seen in Fig. 2 and 5). Thus, one infers that the vibronic coupling is essential for the Hamiltonian model to correctly produce the relevant physical observables, especially in the larger Criegee intermediates as opposed to the smallest Criegee intermediate CH₂OO.

Fig. 13 The B¹A′ ← X¹A′ absorption spectrum (with no dephasing time) using the uncoupled 3D Hamiltonian models in syn- and anti-CH₃CH₂CHOO conformers. Spectrum changes its feature below and above the barrier energy E_b 4.40 eV within this model from quasi-discrete to continuous structure.

6 Time-dependent quantities

We calculate the time-dependent populations of the B and C states in the diabatic representation of eqn (6). As indicated in the previous section, employing the wavepacket propagation method results in the nuclear wavefunction Ψ_α(t) describing the nuclear quantum dynamics of a molecular system composed of several electronic states and f nuclear degrees of freedom.^34–36 For the Criegee intermediates studied in this work, we choose three nuclear degrees of freedom (f = 3), which are the O–O and C–O stretching and C–O–O bending modes. For each molecule, the diabatic electronic populations P_α(t) (with α being B or C state in our case) can be computed as 〈Ψ_α(t)|Ψ_α(t)〉.³⁵

In Fig. 14, the diabatic populations of the B and C states for the Criegee intermediates are presented.^31–33 At t = 0, the populations of the diabatic B and C states in each molecule are one and zero, respectively, because the B state is populated by lifiting wavepacket from the ground to the B state in the propagation calculation. At a time of ∼20 fs, there is a rapid and efficient transfer of B state population to the C state for syn, anti conformers and (CH₃)₂COO, while for CH₂OO, the population transfer is moderately slower than for the larger Criegee intermediates. In the inset of Fig. 14, the squared norm of the wavepacket within the 3D model Hamiltonian is also shown for each molecule. It can be concluded that also the norm of the wavepacket in CH₂OO initially decays slowly compared to the other, larger Criegee intermediates. In Fig. 15, the diabatic populations of the B state for the relevant molecular systems are depicted by using the uncoupled Hamiltonian model. Comparing the norm of the wavepacket in Fig. 15 and the inset of Fig. 14 shows that the behavior of the norm within the propagation time is similar for CH₂OO, while there are considerable changes for the heavier Criegee intermediates. This conclusion is in line with our findings in the previous section, where the computed spectra using the coupled and uncoupled Hamiltonian models are similar for CH₂OO, while we could see considerable differences for the larger systems. Thus, it is important to take into account the vibronic coupling between the B and C states for the heavier Criegee intermediates.

Fig. 14 The diabatic populations of the B¹A′ and C¹A′ states for the syn- and anti-CH₃CH₂CHOO, (CH₃)₂COO as well as CH₂OO. For all molecules, we use the 3D coupled model Hamiltonian. The populations of the B and C states are shown by the solid and dashed lines, respectively.

Fig. 15 The diabatic populations of the B¹A′ state or squared norm for the syn- and anti-CH₃CH₂CHOO, (CH₃)₂COO as well as CH₂OO. We use the 3D uncoupled model Hamiltonian.

Based on one-dimensional reduced density results along the dissociation coordinate O–O, at time ∼20 fs, the wavepacket has reached the avoided crossing region as shown in Fig. 16 and 17. For these Criegee intermediates, the WP gets largely reflected within the shallow well near 1.7 Å. The WPs moving out are suppressed by the CAP and the norm decreases, which represents photodissociation of these molecular species (see inset of Fig. 14). Consequently, at ∼40 fs, the C state population also decreases quickly. For times beyond ∼60 fs the squared norm decreases more slowly to zero. From the decay of the squared norm the lifetime of both conformers can be estimated via fitting the results in the inset of Fig. 14 to f(t) ≃ e^−t/τ. This results in τ to be ∼44 fs for both syn and anti conformers, while the lifetimes of the squared norms for CH₂OO and (CH₃)₂COO are 84 and 49 fs, respectively. These values can be compared with the lifetime of the B state (by fitting the populations of the B state to f(t) in Fig. 14), which results in ∼22 fs for both syn- and anti-conformers. For CH₂OO and (CH₃)₂COO, the lifetime of the B state is obtained as 75 and 27 fs, respectively.

Fig. 16 One-dimensional reduced densities for the B state of CH₂OO (upper panel) and (CH₃)₂COO (lower panel). Conformers along R_OO are computed using the coupled 3D model. In the inset of the upper panel, we zoom in on the one-dimensional reduced densities for t > 25 fs, where the WP reaches the CAP.

Fig. 17 One-dimensional reduced densities for the B state of the syn- (upper panel) and anti- (lower panel) conformers along R_OO are computed using the coupled 3D model. In the inset, we zoom in on the one-dimensional reduced densities for t > 25 fs, where the WP reaches the CAP.

7 Impact of triplet states and spin–orbit coupling

Having studied photodissociation dynamics using the Hamiltonian model of eqn (5) relying on the coupled singlet B and C states, it is tempting to ask whether including several low-lying triplet states in the nuclear quantum dynamics calculation might reveal previously undetected photodissociation channels in these molecular species. To address it, we set up a nuclear quantum dynamical mechanisms such as calculation for CH₂OO in ref. 38 and included the triplet states a³A′, b³A′, a³A′′ and b³A′′ in our calculation. As spin–orbit couplings (SOC) between the singlet and the triplet states are taken into account, PESs of triplet states with different spin components M_s = 0, ±1 and their SOCs with the other singlet states B and C are explicitly treated in the Hamiltonian model.³⁸ In Fig. 19, the potential energy cuts for the aforementioned triplet and singlet states are shown, where we use the RS2C-F12 method with cc-pVTZ-F12 basis set for all atoms. From this figure, several avoided crossings can be observed between the PE curves of the excited singlet and triplet states with the same symmetry. As shown in the lower panel of Fig. 19, avoided crossings occur between the two singlet states B¹A′ and C¹A′ at 3.06 eV with R_OO = 1.741, between the excited triplet states a³A′ and b³A′ at 2.35 eV with R_OO = 1.591 and finally between the excited triplet states a³A′′ and b³A′′ at 2.57 eV with R_OO = 1.591. The dissociation channel along the O–O stretching coordinate after excitation to the B state leads to the H₂CO(X¹A₁) and O¹D products. The electronic states X¹A′, B¹A′ and C¹A′ correlate with this same asymptotic limit generating the H₂CO(X¹A₁) and O¹D products as shown in the upper panel of Fig. 19. The dissociation energy for this channel is estimated to be ≈57 kcal mol⁻¹ or 2.47 eV, which is in close agreement with the experimental value ≈54 kcal mol⁻¹ or 2.34 eV.⁶⁷ In the upper panel of Fig. 19, the three triplet states a³A′, a³A′′ and b³A′′ correlate adiabatically with the same state in the asymptotic O–O stretching region and this represents a dissociation channel, which is energetically lower than the aforementioned dissociation channel. In this new channel, CH₂OO(X¹A′) is dissociated into H₂CO(X¹A₁) and O³P products. As shown in ref. 38, the dissociation energy attributed to this channel is estimated as ≈13 kcal mol⁻¹ or 0.56 eV.

For the purpose of the dynamics, we diabatized PESs and SOCs matrix elements which are primarily evaluated within the adiabatic picture.³⁸ In principle, the diabatization is one way to incorporate PESs and SOCs data allowing to describe both IC and ISC using a fully quantal approach.³⁸ The advantage of using PESs and SOC matrices in the diabatic picture is to remove not only the singular derivative couplings and discontinuities appearing in PESs (owing to the presence of avoided crossings in the adiabatic picture) but also the discontinuities in SOCs as a function of the nuclear coordinates. Based on the 3D PESs including SOCs (in the diabatic picture), the spectrum in the energy range below 2.8 eV was computed in ref. 38. This spectral band is due to contributions of the two lowest triplet states with a³A′ and a³A′′ symmetry including their SOCs with the B¹A′ and C¹A′ states. As shown in that reference, the spectral band shows an irregular and discrete structure which is ∼0.9 eV above the minimum energy of the 3D PESs of a³A′ (E_min = 1.45 eV). Furthermore, this spectral band is below the energy barrier of the a³A′′ state (3.09 eV). In ref. 60, the dynamics of the the triplet states of Criegee intermediates using trajectory surface-hopping (TSH) molecular dynamics simulations was reported. However, they could not observe any population transfer to the triplet states in their calculation even for large Criegee intermediates. Our finding in ref. 38 shows that there is a population transfer among triplet and singlet states. Thus, it is required to consider the triplet states to capture correct photodissociation dynamics for all Criegee intermediates. Our finding is aligned with other findings for molecular systems with a similar size to formaldehyde oxide, such as formaldehyde and acetaldehyde,⁶⁸ wherein photodissociation is driven by eventual formation of a triplet state manifold mediated by efficient intersystem crossing from an initially excited bright singlet state. Van veen et al. reported the absorption spectra of formaldehyde and acetaldehyde including spectral bands with important contributions from the lowest triplet states and possible SOCs between the singlet and triplet states in these aldehydes.⁶⁸

As explained here and in ref. 38, the contributions of the triplet states and the spin–orbit couplings (SOCs) between relevant singlet and triplet states are crucial for accurately capturing the photodissociation dynamics of CH₂OO. The same reasoning applies to larger Criegee intermediates, where we believe that the involvement of triplet states and their SOCs with several low-lying singlet states will offer valuable insight into the various photodissociation pathways.

8 Concluding remarks

In this article we have surveyed salient features of some representative Criegee intermediates, the parent system CH₂OO and three alkyl derivatives with two carbon atoms in the alkyl moiety (i.e. not counting the parent-system C-atom, see Fig. 1). These species share important joint features, in particular a rapid O–O bond cleavage in the presence of UV light accompanied by a broad and partly structureless spectral band in the 280–400 nm wavelength range. Further joint features are femtosecond internal conversion processes caused by the near-degenerate B and C excited states, leading to a rapid population transfer between the singlet states (for the triplet states see the remark below). Nevertheless also important differences deserve mentioning.

The B–C energetic proximity differs substantially between the systems treated here, the gap being smallest (0.05 eV) for the parent system and considerably larger (0.18 eV) for the alkyl derivatives, see Fig. 4b. This impacts on the stability of these species as evidenced by the norm of the wavepacket (see inset of Fig. 14). After the rapid (incomplete) decay at 20–50 fs, the norm and B-state population of the parent system CH₂OO evolve more slowly than those of the larger systems because the asymptotic regime (with O–O dissociation being complete) is less easily accessible, see panels of Fig. 2. This is made more quantitative in CH₂OO Fig. 18 in which the lifetime is plotted against the vibronic coupling 2W_BC, estimated as the avoided crossing energy gap according to Section IV. An interesting correlation is evidenced, which is nearly linear with the exception of syn-CH₃CH₂CHOO, to be analysed more deeply in future works. In particular, we have confirmed for the first time the presence of vibrational structures in the absorption spectra of both CH₂OO and (CH₃)₂COO molecules (as shown in Fig. 7 and 10), which are consistent with the reported experimental spectra. These accurate computational results allow us to conclude that the dominant vibrational modes responsible for the observed features are the O–O and C–O stretching modes, as well as the C–O–O bending mode. Another important finding concerns the simulation of absorption spectra for different conformers of CH₃CH₂CHOO shown in Fig. 12. Our results indicate that the spectrum reported in ref. 28 should in fact be assigned explicitly to the syn-CH₃CH₂CHOO conformer-a novel assignment not previously recognized in the literature.

Fig. 18 The inverse lifetime are plotted against twice the vibronic coupling W_BC for the Criegee intermediates.

Fig. 19 (Upper)the adiabatic PE curves along the OO stretching mode for the X¹A′, B¹A′, C¹A′, a³A′, b³A′, a³A′′ and b³A′′ states. (Lower) the adiabatic PE curves along O–O stretching mode, zooming in geometries in the range 1.55 < R_OO < 1.80 (indicated by red circles in the upper panel). The bonding angle θ_COO and the stretching mode R_CO are fixed at the equilibrium structure values 117.905° and 1.27, respectively.

In a more complete study also spin–orbit coupling and spin-forbidden processes should be considered. So far this has been done for CH₂OO³⁸ and sometimes on a classical dynamical (SHT) level only⁶⁹ and was pointed out to lead to new, lower-energy disscociation channels.^38,69 In combination with the present approach it will allow for a unified description of spectral features and nonradiative phenomena in these systems in a broad sense.

Finally, in this work we have attempted to achieve a balanced accuracy for the different system sizes treated. If this can be achieved also for larger systems, for example with three or four C-atoms in the alkyl moiety, this is hoped to lay the ground for a comprehensive understanding of these elusive species.

Conflicts of interest

There are no conflicts of interest to declare.

Data availability

All raw data used in this review are available from the corresponding author upon reasonable request.

Acknowledgements

We would like to express our gratitude to Professor Andreas Dreuw for a very productive and insightful discussion. We are indebted to the Deutsche Forschungsgemeinschaft (DFG) for financial support (KO 945/22-1, KO 945/23-1). The authors acknowledge support by the state of Baden-Württemberg through bwHPC (JUSTUS 2 cluster).

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Footnote

† It is our pleasure to dedicate this paper to Prof. Christel Marian on the occasion of her 70th birthday.

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