Examining the ground and first excited states of methyl peroxy radical with high-level coupled-cluster theory1

Peroxy radicals (RO2) are intermediates in fuel combustion, where they engage in efficiency-limiting autoignition reactions. They also participate in atmospheric chemistry leading to the formation of unwanted tropospheric ozone. Advances in spectroscopic techniques have allowed for the possibility of employing the lowest () electronic transition of RO2 as a tool to selectively monitor these species, enabling accurate kinetic values to be obtained. Herein, high-level ab initio methods are employed to systematically refine spectroscopic predictions for the methyl peroxy radical (CH3O2), one of the most abundant peroxy radicals in the atmosphere. In particular, vibrationally corrected geometries and anharmonic vibrational frequencies for both the ground () and first excited () state are predicted using coupled-cluster theory with up to perturbative triples [CCSD(T)] and large atomic natural orbital basis sets. Equation-of-motion coupled-cluster theory is utilised to compute vertical transition properties; a radiative lifetime of 4.7 ms is suggested for the excited state. Finally, we predict the adiabatic excitation energy (T0) via systematic extrapolation to the complete basis limit of coupled-cluster with up to full quadruples (CCSDTQ). After accounting for several approximations, and including an anharmonic zero-point vibrational energy correction, we match experiment for this transition to within 9 cm−1.


Introduction
The chemistry of peroxy radicals (RO 2 ) appears in the combustion and consumption of hydrocarbon fuels, as well as the oxidation of volatile organic compounds in the atmosphere [1][2][3][4]. Understanding the reactivity of these radicals is necessary for modelling the vast network of reactions that ensue in either case [5]. For example, the propensity of RO 2 species to promote autoignition via isomerisation and further oxidation impacts combustion efficiency, and quantifying the kinetic rates of these reactions allows for the design of optimal fuel mixtures. In the atmosphere, the interaction of peroxy radicals with nitric oxides yields unwanted tropospheric ozone [6] -determining the kinetics of these low temperature pathways aids in the understanding of smog formation in polluted areas. In this research we examine the simplest RO 2 species, methyl peroxy radical (CH 3 O 2 ).
Rate constants for the aforementioned processes are often determined in the laboratory using ultraviolet (UV) or infrared (IR) absorption as a tool to monitor the depletion of reactants, formation of products, and, in some cases, the generation of intermediates [7,8]. All peroxy radicals exhibit a strong absorbance in the UV region (200-300 nm), which corresponds to anB ←X excitation; studies employing this transition dominate the literature [9]. Unfor-tunately, this absorption is broad and lacks features specific to R, masking underlying absorbers and making the differentiation of individual RO 2 compounds difficult.
For CH 3 O 2 , the lowest excited states correlate adiabatically with ground-state CH 3 (X 2 A 2 ) and the lowest three states of O 2 (X 3 − g ,ã 1 g ,b 1 + g ), which gives rise to the five states shown in Figure 1. The ground Figure 1. Correlation diagram showing the low-lying states of the methyl peroxy radical, which arise adiabatically from the association of ground-state methyl radical with molecular oxygen in its three lowest electronic states. Next to the term symbols for CH 3 and O 2 , their C s projections are given in parentheses. electronic state (X 2 A ) has an electron configuration of (1a ) 2 . . .(2a ) 2 (10a ) 2 (3a ) 1 and correlates with theX 3 − g state of O 2 at the C-O dissociation limit. The first (Ã 2 A ) and second (B 2 A ) excited states arise from the two components of theã 1 g state on O 2 at the dissociation limit. These two states correspond to 10a → 3a (n → π * ) and 2a → 3a (π → π * ) transitions from the ground state, respectively. Early theoretical work on the excited states of CH 3 O 2 [27] found that it shares many of these general features with the simplest peroxy radical, HO 2 [29,32], including the ordering of states and the structure of the R-O and O-O dissociation limits. In fact, the general picture in Figure 1 is common to most peroxy radicals, including the abundant peroxyacetyl radical [3].
Herein, we employ coupled-cluster (CC) methods to study several key spectroscopic features of methyl peroxy radical. An earlier study of ours probed the ground-state spectroscopy of this species [34], and we now report spectroscopic parameters for the first excited state. Vibrationally corrected equilibrium geometries, adiabatic excitation energies, and anharmonic vibrational frequencies are computed for both the ground (X 2 A ) and first excited (Ã 2 A ) states using CC methods with perturbative triples or higher. The full NASA Ames atomic natural orbital (ANO) basis set is employed to obtain equilibrium structures, and effects due to orbital relaxation are evaluated using Brueckner CC theory. Additionally, transition dipole moments and oscillator strengths for theÃ 2 A ←X 2 A transition are predicted using equation-of-motion CC theory. Comparisons to experiment are made where possible.
Anharmonic frequency values were obtained by appending anharmonic corrections (δν) determined with CCSD(T)/ANO1 to harmonic frequencies (ω) determined with CCSD(T)/ANO2. Second-order vibrational perturbation theory (VPT2) was employed to yield the anharmonic corrections, using analytic second derivatives at displaced geometries along the normal coordinates [48]. We have also included a correction for orbital relaxation (δω), which is calculated as the difference between CCSD(T)/ANO1 harmonic frequencies computed with Brueckner [49,50] and UHF reference determinants. For modes perturbed by a Fermi resonance, the offending terms were excluded from VPT2 analysis [51]; the correction for neglecting these terms was estimated using first-order couplings and included post facto [52,53]. VPT2 analysis was performed using PyVPT2 [54].
Transition properties were computed using equationof-motion CC [69,70] with singles and doubles (EOM-CCSD). These and CC computations with a Brueckner reference were computed using PSI4 [71]. Computations with full triple excitations and higher were obtained with MRCC [55,72], as interfaced with CFOUR.

Stationary points
Methyl peroxy radical possesses C s point-group symmetry, with the peroxy terminus trans to one of the hydrogen atoms (see Figure 2). We have determined the 0 K vibrationally averaged equilibrium geometries (r g,0 K ) for thẽ X 2 A andÃ 2 A states using CCSD(T)/ANO2 with vibrational corrections predicted at CCSD(T)/ANO1. The r g,0 K parameters were obtained by expanding each distance, r, in the leading terms of its normal coordinate Taylor Figure 2. Optimised geometric parameters for the ground (X 2 A ) and first excited (Ã 2 A ) electronic states of the methyl peroxy radical (CH 3 O 2 , C s ). Listed distances (inÅ) are r g,0 K values (i.e., r e distances optimised at CCSD(T)/ANO2 with a zeropoint vibrational correction computed at CCSD(T)/ANO1). See Supplemental data for a complete set of coordinates, and Refs [73,74] for r g,0 K definition. and averaging with respect to the ground vibrational state [73,74].
In Equation (1), the linear term, Q s , is averaged with respect to the VPT2 anharmonic vibrational wavefunction, whereas the quadratic term, Q 2 s , is averaged with respect to the harmonic vibrational wavefunction.
Compared to the ground state (X 2 A ), the first excited state (Ã 2 A ) features a contracted C-O bond and an elongated O-O bond. This geometric difference reflects the fact that the ground and first excited states correlate with thẽ , which has the same bond order as CH 3 [36,75].
With regard to the CH 3 moiety, the in-plane C-H bond remains essentially unchanged between the ground and first excited states, whereas the out-of-plane C-H bonds are elongated by theÃ 2 A ←X 2 A transition. This 0.02Å lengthening is a hyperconjugation effect in which doubly occupying the 3a orbital, which is partially π -bonding over C-O, enhances the p-character of the π CH 3 orbital. The barrier to torsional rotation of the methyl group is small -prior computational work suggests a value of 0.72 kcal mol −1 [19].

Excitation energy and transition properties
To monitor CH 3 O 2 using the vibronicÃ 2 A ←X 2 A transition, an accurate transition energy is required. To this end, several experimental investigations have refined this value, often employing theoretical work to confirm assignment. However, the theory data include computations that utilise a wide array of approaches, including density-functional and composite methods, which are not subject to systematic improvement or error assessment. In this work, we determine the transition energy using focal point analysis [56][57][58][59] in order to rigorously approach the correlation and basis set limits. Using this prescription we arrive at a transition energy within 9 cm −1 of the best experimental value.
The data from our focal point analysis are shown in Table 1. At the CBS limit, contributions from full dou- ble, triple, and quadruple excitations are 2370, 306, and 35 cm −1 , respectively, indicating convergence to better than sub-chemical accuracy (0.1 kcal mol −1 ≈ 35 cm −1 ). As expected, we also observe convergence with increasing basis set size in the total CCSDTQ energies (increments of 155, 138, 34, and 10 cm −1 for TZ, QZ, 5Z, and 6Z, respectively), instilling confidence in our final value. Further inclusion of core ( core ), relativistic ( rel ), and non-adiabatic ( DBOC ) corrections refines the extrapolated electronic energy ( E e ) to 7477 cm −1 . The small magnitude of these corrections indicates that the corresponding approximations are appropriate for this system. In particular, the small DBOC correction implies negligible non-adiabatic effects, which can sometimes play a role in peroxy radical systems [3].
The E e value measures the energy separation between the lowest points of theÃ 2 A andX 2 A potential energy surfaces, whereas reported experimental values correspond to the distance between the lowest vibrational energy levels (T 0 ) on these surfaces. As such, our final value for the adiabatic excitation energy (T 0 ) includes an anharmonic zero-point vibrational energy correction ( ZPVE ). Our predicted transition energy of 7374 cm −1 , which carries an uncertainty of approximately 15 cm −1 , compares very well with available experimental values in the literature (see Table 2). Density-functional and composite methods give a broad variety of predictions [76] for this transition that sometimes fall fortuitously close to the experimental value. Our predicted value is the right answer for the right reasons.
As an additional spectroscopic characterisation, we computed vertical transition properties for theÃ 2 A ← Table 2. Comparison ofÃ 2 A ←X 2 A transition origins (T 0 ) in cm −1 .

Experiment
Ref. [11] Ref. [13] Ref. [17] Ref. [16] Ref. [ X 2 A transition using EOM-CCSD/ANO2. These results, which are summarised in Table 3 , are significantly dependent on the geometry; therefore, we report properties for both equilibrium structures. Properties computed using theX 2 A equilibrium structure are most relevant for absorption processes, whereas properties using thẽ A 2 A geometry are relevant for emission. For the absorption process, we predict an Einstein B coefficient of BX →Ã = 1 6 2 ε 0 | X |μ|Ã | 2 = 3.1 × 10 16 m 3 J −1 s −2 . Similarly, for spontaneous emission, we predict an Einstein A coefficient of which indicates a lifetime of τÃ ≈ 4.7 ms for the excited state.

Anharmonic frequencies
Further spectral identification and assignment of CH 3 O 2 using IR spectroscopy may be achieved through comparison with reported fundamental transitions. Indeed, there are several experimental investigations that report these values, but none that observe all 12 modes -10 of the 12 modes have been reported for the ground state, while only 3 of the 12 modes have been identified for the first excited state. In the latter case, the difficulty in preparingÃ 2 A CH 3 O 2 in the laboratory has limited analysis to those modes deduced from measuring the vibronic spectrum. This is a more complicated approach, since it involves disentangling spectra Table 4. Fundamental vibrational transitions for theX 2 A andÃ 2 A state equilibrium geometries of methyl peroxy radical. The final predicted values (ν) are given by ν = ω + δω + δν, where ω is the harmonic, CCSD(T)/ANO2 vibrational frequency, δω is a correction for orbital relaxation at the CCSD(T)/ANO1 level of theory using Brueckner coupled-cluster, and δν is an anharmonic correction at the CCSD(T)/ANO1 level of theory using VPT2. Intensities are relative (%) CCSD(T)/ANO2 values corrected by the inclusion of anharmonic effects at the CCSD(T)/ANO1 level of theory.  Table 4. The final anharmonic values (ν) are a sum of the harmonic frequencies (ω), a correction for orbital relaxation (δω), and an anharmonic correction (δν). Anharmonic corrections were determined using second-order vibrational perturbation theory (VPT2) and coupled-cluster theory [CCSD(T)] with the ANO1 basis set. These corrections are less sensitive to the level of theory than the underlying harmonic frequencies, and are in general agreement with previous work on the ground state using CCSD(T)/cc-pVTZ [34]. To improve the quality of our predictions, harmonic frequencies were computed with the larger ANO2 basis and were corrected for deficiencies due to the UHF reference via Brueckner CC computations.
The accuracy of this approach has been examined in prior work [3].
For the ground state (X 2 A ), we compare to five experimental reports: two argon matrix isolation investigations, one from Ase, Bock, and Snelson (1986) [20] and another from Ellison and co-workers (2002) [77]; two gas-phase studies, including work by Lineberger and coworkers (2001) [17] and Lee and co-workers (2007) [18]; and a recent He-matrix study by Douberly and co-workers (2012) [19]. The gas-phase and He-matrix values are most amenable to comparison with the present theoretical computations because they are least perturbed by environmental effects [78]. Compared to the values reported by Lee and coworkers [18], our predicted transitions differ on average by 5 cm −1 , and agreement with ν 1 is improved from our previous report (3026.7 vs. 3022 cm −1 ) [34]. The largest disparity exists with ν 6 (11 cm −1 ), but there is some disagreement between the reported values for this mode: Lineberger and co-workers report 1124 cm −1 [17] compared to Lee and coworker's value of 1117 cm −1 [18], noting error bars of 5 and 2 cm −1 , respectively. Our predictions differ on average by 8 cm −1 from the values for ν 6 and ν 8 reported by Lineberger and co-workers. With regard to the He-nanodroplet fundamentals observed by Douberly and co-workers, we find excellent agreement with ν 2 (< 1 cm −1 ), and qualitative agreement with ν 1 and ν 9 , with our values shifted lower by 7-8 cm −1 . Overall, the close correspondence between our predicted transitions and the available data gives confidence in our computed frequencies for the first electronic excited state.

Conclusions
Accurate characterisation of the methyl peroxy radical A 2 A ←X 2 A transition along with spectral data for both states is required for selective monitoring in kinetics experiments. We have determined the zero-point corrected geometries for the equilibrium structures on the ground (X 2 A ) and first excited (Ã 2 A ) states. Upon excitation, the O-O bond elongates concomitant with contraction of the C-O bond and lengthening of the out-of-plane C-H bonds. The latter effect is ascribed to hyperconjugation between π CH 3 and π * O 2 . We predict a transition energy of 7374 cm −1 from extrapolation to the complete basis set limit at CCSDTQ, including anharmonic zero-point vibrational energies plus corrections to account for methodological approximations. This final theoretical T 0 value is within 9 cm −1 of the best experimental results (7383 cm −1 ); it has an estimated uncertainty of 15 cm −1 . Additionally, we predict Einstein A (214 Hz) and B (3.1 × 10 16 m 3 J −1 s −2 ) coefficients for spontaneous emission and for absorption, respectively. These data suggest a lifetime of τÃ ≈ 4.7 ms for the excited state. Anharmonic vibrational frequencies are also reported for the ground and first excited states, the latter for the first time, using VPT2 theory and computations at the CCSD(T) level of theory with ANO basis sets. Our ground state values differ on average by approximately 5 cm −1 from available gas-phase values, and good agreement is observed for the few available excited state transitions.

Disclosure statement
No potential conflict of interest was reported by the authors.