Rotation Couplings in Methyl Peroxy

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Modeling the CH Stretch/Torsion/Rotation Couplings in Methyl Peroxy (CHOO) Meng Huang, Terry A. Miller, Anne B. McCoy, Kuo-Hsiang Hsu, Yu-Hsuan Huang, and Yuan-Pern Lee J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b10784 • Publication Date (Web): 30 Nov 2017 Downloaded from http://pubs.acs.org on December 5, 2017

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The Journal of Physical Chemistry

Modeling the CH Stretch/Torsion/rotation Couplings in Methyl Peroxy (CH3OO) Meng Huang,† Terry A. Miller,⇤,† Anne B. McCoy ,⇤,‡ Kuo-Hsiang Hsu,¶ Yu-Hsuan Huang,¶ and Yuan-Pern Lee⇤,¶,§ Department of Chemistry and Biochemistry, The Ohio State University, Columbus, OH 43210, USA, Department of Chemistry, University of Washington, Seattle, WA 98195, USA, and Department of Applied Chemistry and Institute of Molecular Science, National Chiao Tung University, 1001 Ta-Hsueh Rd., Hsinchu 30010, Taiwan E-mail: [email protected]; [email protected]; [email protected]

⇤

To whom correspondence should be addressed Department of Chemistry and Biochemistry, The Ohio State University, Columbus, OH 43210, USA ‡ Department of Chemistry, University of Washington, Seattle, WA 98195, USA ¶ Department of Applied Chemistry and Institute of Molecular Science, National Chiao Tung University, 1001 Ta-Hsueh Rd., Hsinchu 30010, Taiwan § Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 10617, Taiwan †

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Abstract The manifestations of CH stretch/torsion/rotation coupling in the region of the CH stretch fundamentals are explored in the CH3 OO radical. Following our earlier study of the fundamental in the totally symmetric CH stretch (the ⌫2 fundamental), this work focuses on the other two CH stretch fundamentals, ⌫1 and ⌫9 , which would be degenerate in the absence of a barrier in the potential along the methyl torsion coordinate. The simplest model, which assumes a decoupling of the CH stretch vibrations from the torsion, fails to reproduce several important features of the spectrum. Specifically the absence of a strong peak around the origin of the ⌫1 fundamental, and broadening of the strong peak near the origin in the observed spectrum of the ⌫9 fundamental are not captured by this model. The origins of these features are explored through two more sophisticated treatments of the torsion/CH stretch couplings. In the first, a four-dimensional potential based on the three CH stretches and the torsion is developed, and shown to reproduce both of these features. Based on the results of these calculations, the calculated parameters are adjusted to simulate the recorded spectrum. To further explore the torsion/CH stretch couplings in CH3 OO, a 9-state model Hamiltonian is developed and discussed. The implications of various types of couplings on the observed energy level patterns are also discussed.

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Introduction Organic peroxy radicals (ROO) are important intermediates in the oxidation of hydrocarbons, both in the atmosphere 1–4 and under combustion conditions. 5,6 Usually, ROO radicals are formed through intermolecular reactions of R· and O2 , in the presence of a third body, typically N2 . In polluted areas, the formation of ozone in the troposphere is largely the result of reactions of ROO radicals with NO to produce NO2 . Subsequently, NO2 is photolyzed by sunlight to produce oxygen atoms, which in turn combine with O2 to form ozone. In combustion a chain-branching sequence of reactions of ROO leads to the necessary regeneration of OH radicals. The reactions of organic radicals with O2 to form ROO radicals, rather than with other organic species, are critical to the control of the sooting process. 7 Methyl peroxy (CH3 OO) is the simplest and arguably one of the most important ROO radicals in atmospheric and combustion chemistry. The spectra of CH3 OO are obviously relevant for its diagnostics during these reactions. However, the principal motivation of the present work is to explore the molecular interactions, which determine its spectra. In particular, we consider the low-frequency methyl torsion in methyl peroxy, and how it a↵ects the experimentally observed CH stretch fundamental bands and various sequence bands involving the torsion. The low-frequency, large amplitude methyl torsion in methanol has received considerable attention, 8–12 and much of our understanding of the couplings between the torsion and the high frequency CH stretches is based on work on this molecule. More recently, highresolution, rotationally-resolved spectra of methyl mercaptan, CH3 SH, have been studied in the CS stretch region, 13 the asymmetric methyl bending region, 14 and the CH stretch region. 15 The rotational structure in these vibrational bands is strongly a↵ected by the couplings between the low-frequency torsion and other high-frequency vibrations. For example, there are surprisingly large K-type splittings in the asymmetric methyl-bending bands. The vibration/torsion/rotation structure of methacrolein also has been studied 16 by millimeterwave spectroscopy, focusing on the fundamentals of the methyl torsion and the skeleton 3



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torsion. This study revealed strong couplings between these two large-amplitude torsion motions. In addition to these asymmetric top molecules, the rotational and infrared spectra of a symmetric top molecule, CH3 CN, in the regions of the fundamental and overtones of the CCN bend have been studied. 17 The methyl torsion provides another example of a large amplitude vibration, which is coupled to smaller-amplitude vibrational motions. In methyl peroxy (CH3 OO), this vibration corresponds to ⌫12 and will be identified as ⌫tor in the following discussion. In recent years considerable experimental and theoretical work 18–23 has been done on the spectroscopy of CH3 OO. The first study that focused on the methyl torsion involved an analysis of the e A

e near-IR electronic spectrum of methyl peroxy. 21 This study included an exploration X,

e and A e electronic of how the height of the barriers along the torsion coordinate on the X

states is reflected in the spectrum. Recently, we reported the mid-IR spectrum of CH3 OO.

Specifically we investigated the totally symmetric CH3 stretch fundamental, ⌫2 , which has its origin at 2954.4 cm

1

along with its sequence bands with the methyl torsion. In that

study we found that the torsion/CH stretch sequence bands were shifted from the ⌫2 origin due to couplings between states with one quantum in ⌫2 and n quanta in ⌫tor (⌫2 + n⌫tor ) and torsion/CH stretch combination bands involving excitation in one of the other CH stretches and one fewer quanta of excitation in the torsion (⌫9 +(n 1)⌫tor ). 24 We are now extending this work to the other two CH stretch fundamentals: the antisymmetric CH2 stretch, ⌫9 , and outof-phase combination of the symmetric CH2 and CH stretches, ⌫1 , and their accompanying torsional structure. These bands, have overlapping rotational structure, which lies in the 3000 3080 cm

1

range, as shown in the upper red trace of Figure 1.

The focus of the present study is on an exploration of the implications of the torsion/rotation couplings, which were shown to a↵ect the band contour of the ⌫2 fundamental, on the band contour for the ⌫1 /⌫9 fundamentals in the 3000 3080 cm

1

region of the spec-

trum. Here, we investigate the relative influence of three types of coupling. The first is the torsion/CH stretch vibrational coupling involving these two CH stretches. In the absence of

4


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torsion, the two asymmetric CH stretch vibrations would be degenerate. 9 With the introduction of a barrier along the methyl torsion coordinate, the splitting of the energy levels in the torsion and the di↵erences between the ⌫1 and ⌫9 fundamental frequencies based on the equilibrium structure become similar in size. When there is no torsion excitation, the splitting of the CH stretch frequencies is larger than the torsion splitting, while once several quanta of excitation are placed in the torsion, the torsion splittings exceed the splitting between the ⌫1 and ⌫9 fundamentals. The second coupling of interest involves the coupling between states with one quantum of excitation in ⌫9 and n quanta of excitation in the torsion (⌫9 + n⌫tor ) and states with one quantum of excitation in the lower frequency ⌫2 fundamental and one more quantum of excitation in the torsion (⌫2 + (n + 1)⌫tor ). These two sets of states are accidentally degenerate due to the similarity of the torsion frequency and the splitting between the ⌫2 and the ⌫1 /⌫9 fundamentals. As several torsion levels are populated in the room temperature spectrum, this system a↵ords the opportunity to probe the implications of these couplings on the measured spectrum. A third type of interaction that is considered involves the vibrational Jahn-Teller e↵ect in which the ⌫1 and ⌫9 frequencies become degenerate at certain values of the COX bond angle in CH3 OX. This e↵ect was identified by Dawadi et al. in their studies of CH3 OH. 25 In the present study we will consider possible implications of this e↵ect in the spectrum of CH3 OO radical.

Experiment A more detailed description of the experimental procedure was provided previously, 24 and a brief summary follows. An ArF excimer laser beam at 193 nm is passed through a White-cell reactor three times to photodissociate a flowing mixture of (CH3 )2 CO (0.6-1.6 Torr) and O2 (99-200 Torr) to produce CH3 that subsequently reacted with O2 to form CH3 OO. In this setup, O2 served as both a reactant and an efficient quencher. The average efficiency of photolysis of acetone is estimated to be ⇠ 2.4% at 193 nm according to an absorption cross 5



section 26 of ⇠ 2.36 ⇥ 10

18

cm2 molecule

1

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and a laser fluence ⇠ 30 mJ cm 2 ; the yield for

production of CH3 is ⇠ 1.9. 27 We employed a step-scan FTIR spectrometer coupled with a multi-reflection White cell (total path length 3.6 m) to record temporally resolved IR spectra. We recorded interferograms with an InSb detector and an internal 24-bit analogue-to-digital converter with temporal resolution of 12.5 µs. The signal was typically averaged over 15-20 laser shots at each scan step. Thirteen spectra at resolution 0.15 cm

1

were recorded at 298 K in three

sets: six at total pressure 100.6 Torr (acetone/O2 = 1.6/99.0), two at 100.2 Torr (acetone/O2 = 1.2/99.0), and five at 199.6 Torr (acetone/O2 = 0.6/199.0). As the spectra, after spectral stripping of weak absorption of (CH3 )2 CO, CH4 and C2 H6 in this region, in these three sets are nearly identical, we averaged them to improve the ratio of signal to noise. At instrumental resolution of 0.15 cm 1 , the FWHM (full width at half maximum) of the line is 0.19 cm

1

after apodization with the Blackman-Harris three-term function.

Theory To analyze the observed spectrum in the red trace in Figure 1, we initially consider how well the standard asymmetric rotor model Hamiltonian, Hrot = ANa2 + BNb2 + CNc2

(1)

describes the rotational structure of the ⌫1 and ⌫9 bands in terms of the rotational constants A, B and C in principal axes system. The parameters used in the simulations, which are determined by combining the results of electronic structure calculations and ground state rotational constants obtained from fits to microwave spectra, 28 are listed in Table 1. Consistent with our earlier study of the ⌫2 band (see the inset in Figure 1), the simple asymmetric top model fails to reproduce the detailed rotational structure observed in the experimental spectrum. 24 In particular, the simulation fails to reproduce the broadening of 6


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the rotational structure around the origin of the ⌫9 band. Unlike the ⌫1 and ⌫2 fundamentals, the ⌫9 fundamental is purely perpendicular, and the large feature near the origin of this transition results from | Ka | = 1 transitions with Ka = 1

0 and Ka = 0

1, which

merge at large values of J to form the largest peak in the calculated spectrum. In addition, the simulation predicts a relatively strong peak around the origin of the ⌫1 band, which is not seen in the experimental spectrum. This is surprising. Based on the transition moments evaluated at the equilibrium structure (see Table 1), this band should have both parallel and perpendicular components in approximately a 1:2 ratio. In the previous analysis of the ⌫2 band, the simulation of rotational structure of this band in the experimental spectrum required a convolution of the rotational contours for the torsional sequence bands that originate from the excited torsion states that are populated at room temperature. Comparison of the frequency of the torsion to the frequency di↵erence in the origins of the ⌫2 and ⌫9 fundamentals led us to expect there to be an accidental degeneracy between the combination levels involving ⌫2 with one or more quanta of excitation in the torsion (⌫2 + n⌫tor ), and the combination levels involving ⌫9 with one fewer quantum in torsion (⌫9 + (n

1)⌫tor ). This coupling results in small shifts of the torsional sequence

bands built on ⌫2 , which are sufficient to account for the observed breadth of the Q-branch in the ⌫2 band. In the present study, we apply a similar approach to analyze the ⌫9 and ⌫1 bands to that used to analyze the ⌫2 band.

Model Hamiltonian Since the details of this theoretical model have been described previously, 24 we provide only a brief overview of the theoretical methods employed in the present study. We first calculate the frequencies and intensities of the ⌫1 and ⌫9 fundamentals along with their torsional sequence bands. We refer to this part of the calculation as the torsion/CH stretch calculation, and the associated Hamiltonian as HT . The values of the energy levels, intensities, and tunneling splittings obtained from the torsion/CH stretch calculation are then used as parameters in a 7



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calculation of the rotational contour of these bands of interest. We refer to this second part of the calculation as the torsion/rotation calculation. For the initial torsion/CH stretch calculation, a four dimensional model including three local CH stretches and the CH3 torsion is employed. The corresponding Hamiltonian, HT , is separated into the contributions from i) the torsion and ii) a harmonically coupled anharmonic oscillator 29 treatment of the three CH stretches along with iii) the couplings between them. Specifically,

HT = Hunc + Hcpl = Htor + Hstr + Hcpl

(2)

These terms are discussed in more detail in the Supporting Information. In the second step, we calculate the rotational contour for each of the vibrational transitions of interest, using the results of the torsion/CH stretch calculations to determine the parameters in the torsion/rotation Hamiltonian, Htor,rot . In order to take into account the couplings between the CH3 torsion and the molecular rotations, we rotate the molecule to the ⇢-axis frame. This treatment was previously used to study acetaldehyde, methanol and its isotopologues 21,30–35 as well as CH3 OO. 21,24 In this approach, the principal axes of the molecule are rotated so that the a-axis is aligned with the ⇢~ vector (see Figure 2), and relabeled as the z-axis. The direction of ⇢~ is derived from the rotational constants in the principal axis system and the angles between the C-O bond axis and the principal axes. This change of axis system ensures that the only Coriolis coupling involving the torsion comes from the z-component of its angular momentum. Because we are no longer in a principal axis system, an o↵-diagonal term is introduced to the rotational Hamiltonian. To simplify the discussion that follows, we divide the transformed torsion/rotation Hamiltonian, 0 , into two parts as Htor,rot

0 0 0 = Htor + Hrot Htor,rot

8


(3)

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where the forms of these terms are provided in the Supporting Information. 0 with n quanta of excitation As shown by Hougen, 33 the K-dependent eigenvalues of Htor

in the torsion are described well by

(K) En,

= Btor



an0

+

an1

cos

✓

2⇡(⇢K 3

)

◆

(4)

Here, h ¯ K is an eigenvalue of Nz , the z-component of the angular momentum arising from molecular rotation, while with A or E symmetry,

denotes the symmetry under the G6 group. Specifically, for states = 0 or ±1, respectively.

It should be noted that because 21

(K=0) =±1

n Etunn = En,

(K=0) =0

En,

=

3 Btor an1 2

(5)

the values of Btor an1 for the nth torsional level, with either zero or one quantum of CH stretch excitation, can be trivially obtained from the tunneling splittings between the

= 0 and

±1 states when K = 0. These energy di↵erences are obtained from the torsion/CH stretch calculation described above. Combining the energies obtained using Eq 4 with the eigenvalues of the corresponding 0 pure rotational Hamiltonian, Hrot , we are able to calculate the rotational contours of the

torsional sequence bands of the CH stretch fundamentals. This is the approach that is employed in our rotational contour simulation.

Basis Functions In order to facilitate the analysis described in the following sections, it is useful to ensure that the basis functions, as well as the degenerate eigenstates of HT , transform as irreducible representations of the appropriate G6 symmetry group. When the methyl torsion is feasible, as is the case in CH3 OO, the appropriate symmetry group for the molecule is the G6 permutation-inversion (PI) group rather than the Cs point group that describes one of the 9



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three equivalent minimum energy geometries. In the G6 PI group, ⌫1 and ⌫9 transform under the E irreducible representation, while ⌫2 transforms under A1 . Likewise, when the torsion wave functions are expanded in a free rotor basis, linear combinations of these functions can be generated that transform as either A1 , A2 or E± . Using this basis set, the eigenstates of Htor in Eq 2 are expressed as |n itor . Here, n represents the number of quanta in the torsion. States with E± , while states with

= ±1 transform as

= 0 transform under A1 when n is even and A2 when n is odd.

The eigenstates of Hstr , |mA , mlE istr , are described in terms of the number of quanta in the vibration with A1 symmetry (mA ), the number of quanta in the vibrations with E symmetry (mE ) and the associated vibrational angular momentum, l, where l ranges from

mE to mE

in increments of two. These basis functions can be correlated with the traditional Cs labels for the CH stretch vibrations, ⌫2 , ⌫1 and ⌫9 . The ⌫2 = 1 level is denoted as |1, 00 istr , while the ⌫1 = 1 and ⌫9 = 1 levels are linear combinations of the |0, 1 1 istr and |0, 11 istr states. The torsion and CH stretch basis functions are combined to generate torsion/CH stretch basis functions that transform under specific irreducible representations of G6 , and are E |l| . A more detailed discussion of this basis and the associated denoted as n| | ; mA , mE ; ts

CH stretch coordinates is provided in the Supporting Information.

Results As discussed above, in order to simulate the experimental spectrum, the eigenvalues and eigenvectors of HT need to be obtained from the torsion/CH stretch calculations. The procedure used to obtain these solutions is similar to the one described previously, 24 and the results are numerically identical. In both studies, we first obtain the eigenvalues and eigenvectors of

Hunc = Htor + Hstr

10


(6)

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Then we evaluate the e↵ect of torsion/CH stretch coupling, Hcpl , using the eigenvectors of Hunc as the basis functions. Finally, the results of the calculation involving Hcpl are 0 in Eq 3. Within this treatment, we assume a used to generate the parameters in Htor,rot

partial separation of the CH stretching vibrations and the rotational structure as the CH 0 contribution to Htor,rot is included through these parameters.

Calculation of the Torsion/CH Stretch Energies from HT In order to obtain the eigenvalues and eigenvectors of the torsion/CH stretch Hamiltonian, the force constants as well as the e↵ective rotational constant, Btor , are determined using electronic structure calculations. For the results reported in this section, the parameters in HT are obtained at the B3LYP level of electronic structure theory evaluated using Gaussian 09, 36 and are listed in Tables S1 and S2. The details of the calculation are described in the Supporting Information. The calculated eigenvalues of HT for the states involving n quanta in the torsion and | |,|l|

⌫CH = 1, En,mA ,mE , as well as their corresponding eigenfunctions are reported in Table 2. | |,|l|

The eigenfunctions are represented in terms of the expansion coefficients, Cn,mA ,mE , for the symmetrized basis functions. These energies agree numerically with the ones reported in our earlier study in which we considered the ⌫2 fundamental of CH3 OO 24 (see Table S5 of that paper) using on the same HT . For the eigenvectors, we report the two leading contributions to the states with one quantum of excitation in the CH stretches with E symmetry (⌫1 or ⌫9 ) along with n quanta in the torsion. The leading contribution from the basis functions that represent the combination levels involving torsional excitations and the ⌫2 = 1 levels | |,|l|

2

is also listed in cases where the squared coefficient, Cn,mA ,mE , exceeds 0.03. The

Ecpl

terms in Table 2 provide the di↵erences between the eigenvalues of HT and the corresponding eigenvalues of Hunc . The magnitude of this quantity provides a way to determine the e↵ect of Hcpl on the energy of this state. The parameters derived from the torsion/CH stretch calculation and used in the tor11



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sion/rotation calculation are reported in Table S3. Interestingly, with the exception of Btor a10 1 (⌫9 ) (the value of which shows a large sensitivity to the level of electronic structure theory used to evaluate it), the Btor an0 1 (⌫(1/9) ) parameters have the opposite sign compared to the corresponding Btor an00 1 parameters. This contrasts the situation for the ⌫2 fundamenn00 tal, where the Btor an0 1 (⌫2 ) parameters have the same sign as the Btor a1 parameters. As

the tunneling splittings of the torsion levels are proportional to Btor an1 , this change in sign indicates a reversal of the energy ordering between the torsional levels with A1/2 and E symmetries. Such inverted torsion structure has been previously discussed in the context of similar behavior in methanol. 8–10,37 Table S3 also provides the calculated values of the shifts of torsional sequence bands relative to the corresponding fundamentals,

Edif,n = E n (⌫1/9 ) and the energy di↵erence,

E¯(⌫1

⌫9 ) ,

E¯(⌫1

E n (GS)

E 0 (⌫1/9 )

E 0 (GS)

(7)

between the ⌫1 and ⌫9 fundamentals.

⌫9 )

= E 0 (⌫1 )

E 0 (⌫9 )

(8)

where

E n (⌫1/9 ) =

1 1 X | |,`=1 E (⌫1/9 ) 3 = 1 n,0,1

(9)

In Eq 7, “GS” is used to represent torsion levels without any CH stretch excitation, while | |,`=1

En,0,1 (⌫1/9 ) are reported in Table 2. We equate the states with excitation in ⌫9 with the lower energy A and E levels with a specified value of n, while the higher energy states are identified as ⌫1 for the purpose of this analysis. The near degeneracies among the combination levels involving the ⌫9 fundamental and n quanta of excitation in the torsion (⌫9 + n⌫tor ) and combination levels with one quantum of excitation in ⌫2 and one more quantum of excitation in the torsion (⌫2 + (n + 1)⌫tor ) lead 12


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to a strong sensitivity of the values of

E¯(⌫1

⌫9 )

and E n to the level of electronic structure

theory that is used. This is illustrated by the values of these parameters, reported in Table S3. Even greater variability is observed in some of the tunneling splittings, as reflected in n00 the values of the Btor an0 1 (⌫1/9 ) and Btor a1 parameters that are also reported in Table S3. For

example, the calculated values of the Btor a10 1 (⌫9 ) parameters range from

3.3 to 8.8 cm 1 .

Interestingly, when the summed tunneling splittings are considered by evaluating

n = Etunn,tot

n Etunn (⌫9 ) +

n Etunn (⌫1 ) +

n+1 Etunn (⌫2 )

n n+1 + Etunn (GS) Etunn (GS) 3⇥ n0 (⌫2 ) + Btor an0 Btor an+10 = 1 1 (⌫9 ) + Btor a1 (⌫1 ) 2 ⇤ +Btor an00 Btor an+100 1 1

=

(10)

3 Btor an1 ⇡ 0 2

as shown in the results reported in the last two rows of Table S3 for n = 0 and 1. The fact that n ⇡ 0 is insensitive to the level of theory while there is considerable variability in the Etunn,tot

values of the parameters used to evaluate

n is somewhat surprising. The origins of Etunn,tot

this relationship among the tunneling splittings are explored using reduced dimension 6+3state and 9-state model Hamiltonians, which are described in the Supporting Information and discussed in the following section. 0 are approximated by Eq 4. As To simulate the spectrum, the energy eigenvalues of Htor

such, before continuing, we need to confirm that Eq 4 accurately reproduces the eigenvalues 0 of Htor . To do this, the eigenvalues of HT

0 Htor + Htor are calculated for K = 0 to 4. These

eigenvalues along with the approximated eigenvalues calculated based on Eq 4 for n = 0 and 1 using the adjusted B3LYP parameters in Table S2 are plotted in Figures S2 and S3 in the Supporting Information. The agreement between the energies obtained by the two approaches is nearly exact when n = 0, while slight deviations are noted when n = 1. These 13



di↵erences reflect small shifts in the calculated energies of the torsion/rotation states due to the accidental degeneracy between the ⌫9 +n⌫tor and ⌫2 +(n+1)⌫tor levels. At the resolution of the observed spectrum such small shifts will not produce detectable changes to the simulated spectrum. As such, this analysis confirms that the torsion/rotation model that we used for simulating the rotational contour around the ⌫2 fundamental region is also appropriate for the simulation of the rotational contour around the ⌫9 and ⌫1 fundamental region. Before concluding this section, it should be noted that the adjusted B3LYP force constants were developed from the force constants obtained at the B3LYP level of theory in order to obtain a good simulation of the CH3 OO spectrum in the region of the ⌫2 fundamental. 24 They lead to weaker couplings between the torsion and the CH stretches compared to the parameters obtained using the unadjusted B3LYP force constants. When the unadjusted B3LYP force constants are used in the simulation of the spectrum, the larger couplings between the torison and CH stretch vibrations lead to larger discrepancies between the calculated K-dependent energy eigenvalues of HT

0 Htor + Htor and those obtained usig Eq 4. These results are

shown in Figure S4.

Simulated spectrum Once the eigenvalues of HT are obtained, the SpecView package is used to calculate the rotational contour of the various relevant bands: 910 , 910 1211 , 910 1222 , 110 and 110 1211 , where mode 12 is the torsion. Similar to our previous ⌫2 simulation, in some cases the values of the parameters are constrained to their calculated values; others are varied to obtain the best agreement between the simulated and experimental spectra subject to the constraint that the fit values fall within the range of the values obtained at various levels of electronic structure theory. In the discussion that follows we describe the sources of the parameters used to develop the simulated spectrum presented in Figures 3 and 4 and provided in Table 3. The first set of parameters to consider are the rotational constants. For the ground state, we adopt 14


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the parameters obtained from the analysis of the microwave spectrum by Endo. 28 The excited state rotational constants are obtained by scaling the ground state constants by the ratio of the excited and ground state values calculated at the B3LYP/aug-cc-pVTZ level of theory/basis set using VPT2 as implemented in Gaussian 09, 36 and are listed in Table 1. The relative values of the transition moments are evaluated at the same level of electronic structure theory and are rotated into the ⇢-axis frame. The values of ⇢ and ✓ in Table 3 were evaluated using the optimized structure from the B3LYP calculation and the rotational constants that are reported in Table 1. Based on the force constants obtained using several levels of electronic structure theory, as reported in Table S2, we calculated the tunneling splittings for the ground and excited levels. These are reported in Table S3. The tunneling splittings for the vibrationless level and for the fundamentals in ⌫1 and ⌫9 show little sensitivity to the level of electronic structure theory used. Consequently in the simulation, the values of these parameters were constrained to equal the values obtained using the adjusted B3LYP level of theory (reported in Tables S1 and S2). For the torsional sequence bands built on ⌫9 , the rotational contours are significantly di↵erent from one another as well as that of the ⌫9 fundamental. In the case of the simulation of the 910 1211 band, the splitting of the two dominant features reflects the magnitude of the change in the tunneling splitting between the lower and upper levels, which can be equated to

3 2

Btor a11 , where

Btor an1 = |Btor an00 1 We find that a value of Btor a11 = 3.1 cm

1

Btor an0 1 (⌫9 )|

(11)

provides a good simulation of the ⌫9 origin feature

in the measured spectrum. In our study of the ⌫2 fundamental region of the spectrum, we 100 1 used parameters that are consistent with Btor a10 1 (⌫9 ) ⇡ 0. As Btor a1 = 3.1 cm , this choice

leads to the desired result that Btor a11 = 3.1 cm 1 . 1 Interestingly, the parameters from the B3LYP calculation yield Btor a10 1 (⌫9 ) = 6.2 cm ,

15



which also corresponds to Btor a11 = 3.1 cm 1 . In Figure S5, we report the simulated spectrum that is obtained when this larger value of Btor a10 1 (⌫9 ) is used. Overall, the agreement with experiment is comparable to that shown in Figure 4. On the other hand, as is illustrated in the plots in Figure S4, when the B3LYP parameters are used to calculate the energies of the torsion/rotation levels for the state with one quantum in ⌫9 and one quantum in the torsion, the energy levels are not cleanly reproduced by Eq 4. As this equation is used in the SpecView model, which was used to simulate the spectrum, the poor agreement between the results of the full four-dimensional calculation and the expression in Eq 4 leads us to question the validity of the simulated spectrum when the larger tunneling splitting is used. For this reason, Btor a10 1 (⌫9 ) = 0.0 cm

1

is used in the spectral simulation in Figure 3. Higher

resolution spectra would be needed to provide additional constraints on these parameters. As n is increased, the value of Btor an1 increases. A consequence of this can be seen in the red trace of Figure 3, which is the calculated spectrum for the second torsional sequence band of ⌫9 , 910 1222 , using the B3LYP parameters. Here the change in the tunneling splitting leads to a broadening of the rotational structure to the point of being basically unobservable. This points to the critical role that the value of Btor an1 plays in the simulated spectrum. The torsional sequence bands built o↵ the ⌫9 fundamental with n > 2 are not included in the simulated spectrum, shown in Figure 4, since they do not contribute to the rotational structure. For the torsional sequence bands built o↵ of the ⌫1 fundamental, the calculated spectrum for the 110 1211 band, obtained using the B3LYP parameters is also shown in Figure 3. Following similar reasoning to that used for the torsional sequence bands built o↵ of the ⌫9 fundamental, the torsional sequence bands of ⌫1 with n > 1 are not included in the simulated spectrum. Finally, the values of Edif,1 and

E¯(⌫1

⌫9 )

were adjusted within the reported range of

calculated values to provide the best agreement between the measured and simulated spectra. As can be seen by comparing the results reported in Tables 3 and S3, the values of these parameters fall within the ranges of the calculated parameters obtained using quadratic force

16


Page 16 of 42

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constants in HT evaluated at di↵erent levels of electronic structure theory and anharmonic force constants obtained at B3LYP level of theory reported in Tables S1 and S2. In contrast to the ⌫2 fundamental, which is mainly a-type, the ⌫1 fundamental is mainly btype, while the ⌫9 fundamental is c-type. Therefore, the rotationally resolved structure in the ⌫9 and ⌫1 bands is primarily K-structure rather than the J-structure resolved in the ⌫2 band. This K-structure results in resolved peaks for di↵erent K 00 with the high frequency side and

K =

K = |K 0

K 00 | = +1 on

1 on the low frequency side, and the corresponding

values of |K 00 | are provided above the trace. Some broadening of the K-structure is provided by unresolved J-structure, where the Q-subbranches provide most of the intensity while resolved R- and P -branch lines nearly reach observable intensity. Interestingly, the strong peak at the center of the trace for the ⌫9 fundamental is caused by overlapping K = 0 and K = 1

1

0 transitions. Additional discussion of the intensity distributions for di↵erent

|K 00 |, especially for lines designated by the smallest |K 00 | values, is provided in Ref. 38. For the ⌫1 fundamental, the center peak reflects a small a-type contribution to the transition moment. The intensity of the peak around the origin of the ⌫1 fundamental, which is labeled “origin” in Figure 3, is suppressed compared to the simulation in Figure 1. The apparent absence of a strong peak around the origin of the ⌫1 fundamental reflects the dif00 ference between the values of Btor a000 1 and Btor a1 . It also reflects the fact that rotation to

the ⇢-axis system decreases the a-type component of this transition from 37% to 15%. The sum of the traces in Figure 3 and the experimental spectrum are compared in Figure 4. In summing the traces, the relative intensities of the sequence bands, compared to the corresponding fundamental, are determined by the population of the J = K = 0 level of the torsion at 300 K. The relative intensities of the transitions where n = 0 are determined by the calculated ratio of the intensities of the ⌫1 : ⌫9 fundamentals. As seen in the results reported in Table S3, the ⌫1 fundamental has roughly 60% of the intensity of the ⌫9 fundamental. Overall, this simulation provides much better agreement with the experimental spectrum than the one shown in Figure 1. The rotational structure around the origin region of the ⌫9

17



Page 18 of 42

band in this simulation is significantly broadened, although not quite to the extent found in the experiment. The simulation of the K-structure of the ⌫9 bands matches the experiment nearly quantitatively. The peak around the origin of the ⌫1 fundamental is relatively weak, and provides only a small contribution to the convoluted spectrum.

Possible Rotation/CH Stretch Interactions As mentioned at the start of this section, the approach taken in calculating the spectrum assumes a partial separation of the rotational structure and the CH stretching motions. In the absence of torsion motion this approximation is appropriate. On the other hand, the large amplitude torsion motion can cause the CH stretch vibrations to become e↵ectively equivalent. In some studies of CH3 OH, 10,12,39 the cos(⌧ ) and cos(2⌧ ) dependences of Hcpl are likened respectively to a linear and a quadratic vibrational Jahn-Teller (or Renner-Teller) e↵ect. Further, as recent work of Dawadi and Perry on methanol 25 has shown, the energy di↵erences between the CH stretch frequencies depend on the value of the COX angle (X = O in CH3 OO and H in methanol). When these atoms are colinear, the molecule has C3v symmetry and the CH stretches will transform as A1

E. It is not obvious that this would

also occur for a near prolate symmetric top. Degeneracies can also be been found at other values of the COX angle, leading to a vibrational analogy to the Jahn-Teller e↵ect. In this analogy, the E-type CH stretch vibrations in the methyl rotor systems take the role of the degenerate electronic state, while the torsion and the COX bending take the role respectively of the angular and radial motion of the Jahn-Teller active, symmetry-breaking vibration. In the case of CH3 OH, the energies of the conical intersections that correspond to an accidental degeneracy of two of the CH stretch frequencies are well above the potential minimum. In contrast, in CH3 OO, the harmonic frequencies of two of the CH stretches become degenerate at COO angles that will be sampled by the ground state vibrational wave function since, as is shown in Figure 5, this “conical intersection” is only 120 cm

1

above the minimum in the potential. Similar observations have been made by Dawadi et 18


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al. 40 in their study of methyl mercaptan, CH3 SH, and Lees et al. 15 obtained the vibrational spectrum of CH3 SH in the CH stretch region in which they observed evidence of this type vibrational Jahn-Teller e↵ect. This near-degeneracy of two of the CH stretch frequencies in CH3 OO leads to the question of how the Coriolis coupling between the nearly degenerate CH stretches and rotation will be manifested in the spectrum. Following the work of Lees et al., 15 this Coriolis coupling is incorporated through the introduction of an additional term in the Hamiltonian used to simulate the spectrum:

H cc = Here p =

i¯h@/@

2Ae↵ ⇣Nz p

(See Eq S20 for a discussion of

(12) ), and ⇣ is the Coriolis coupling

coefficient. Performing an analysis based on the rotational constants provided in Table 1, we find ⇣ < 0.02 for CH3 OO. Based on this analysis and the contributions of the traces in Figure 3 to the simulated spectrum, we expect that H cc would primarily a↵ect the fundamental in ⌫9 . Specifically, while the side-bands on either side of the ⌫9 origin will remain equally spaced, the inclusion of H cc will alter the spacing between these features. Based on these results, the CH3 OO radical appears to provide an excellent new candidate to further explore manifestations of the vibrational Jahn-Teller e↵ect in molecules with a methyl rotor, although such an investigation would require higher resolution spectral results than are presently available.

Discussion With the experimental spectrum successfully simulated, we turn our attention to a set of models that focus on the couplings among ⌫1/9 + n⌫tor levels and the ⌫2 + (n + 1)⌫tor levels with J = K = 0. We use these parameters to explore the e↵ects of various couplings on the energy level structure and the origins of the observation in Eq 10 that 19


Etunn,tot ⇡ 0.


Page 20 of 42

A summary of the results of this analysis is provided in Figures 6, 7 and 8 for states with n = 0, 1, and 4, respectively. Figures S6 and S7 provide the results when n = 2 and 3. Each of the first four columns of these figures are labeled by a 9 ⇥ 9 matrix representation of an increasing complicated Hamiltonian: Hunc , Hunc+ , H6+3 , H9 . The basis set consists of linear combination of direct products of eigenfunctions of the torsional Hamiltonian, Htor and the CH stretch Hamiltonian, Hstr . They transform as the

irreducible representation

of the G6 group and are defined by Eqs S25 to S33.

The Hunc and Hunc+ Models The columns labeled Hunc in Figures 6, 7, 8, S6 and S7 display the energy eigenvalues of Hunc , which corresponds to a one-dimension torsion and a three-dimension CH stretches that are not coupled. The local CH stretch basis states are 3-fold degenerate, and with the introduction of a bilinear coupling term in Hstr the three CH stretches split into an identifiable ⌫2 and a degenerate ⌫1 /⌫9 pair. In these figures, each of the red/blue dashed lines represents a pair of doubly degenerate levels with ⌫1 /⌫9 = 1 and n quanta in the torsion. These are represented by either a single line or a pair of lines, based on the symmetry of the torsion contribution to the wave function, lines represent the four degenerate states with transform as A1 with

=0(

t

A2

t.

Specifically, the pairs of red/blue dashed = ±1 (

t

= e). As such, these four states

E under G6 . The single red/blue dashed lines represent pairs of levels

= a) and

= E. The black lines represent the ⌫2 + (n + 1)⌫tor levels. Since the

⌫2 fundamental has A1 symmetry, these levels transform as E

A1/2 depending on whether

n is odd or even. Based on this, the splitting between the states with

t

= a and

t

=e

is the splitting of the eigenvalues of Htor . At the far-right side of these plots, the levels are identified by n, , where n provides the number of quanta in the torsion and

provides the

symmetry of irreducible representation of G6 according to which the levels in the right-most column transform. There are two types of couplings introduced by Hcpl . The first arises from the terms in 20


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Hcpl that are proportional to cos 3⌧ . Within this model, they provide diagonal corrections to the uncoupled energies and are included in Hunc+ . The matrix elements of these terms are denoted by hn, 3

t s

. The second type of couplings, which arise from terms in Hcpl , are

proportional to cos ⌧ and cos 2⌧ . These terms are added to Hunc+ to form H6+3 . These terms break the degeneracy of the CH stretch levels, and the matrix elements of these terms are n,

denoted h1/2t

0 0 s| t s

. Definitions of these matrix elements are provided in Eqs S37 and S43

and their values are provided in Table S4 . As is seen by comparing the results provided in the Hunc and Hunc+ columns of Figure 6, the addition of hn, 3

t s

shifts the energies of all nine levels, but will not break any of the

degeneracies. In the free rotor limit, hn, 3

t s

= 0. For torsional levels that are well below

the torsion barrier, these corrections are positive. As is seen in Table S4, the values of the hn, 3

t s

parameters for the states with n = 0 to 2 decrease with increasing n, becoming

negative for n = 2. For larger values of n the values of the hn, 3

t s

parameters fluctuate,

but for all states explored in this study they remain relatively small. Examination of the Hunc+ column of Figure 6 also shows that the values of the hn, 3 the symmetry of the torsion,

t,

number, n, but di↵erent values of

t s

parameters depend on

and levels with the same value of the torsion quantum t

will experience slightly di↵erent shifts. This introduces

a slight perturbation to the apparent tunneling splittings. We will refer to the tunneling splittings between the levels in the Hunc column in the figures as the uncoupled torsional splittings, while the splittings between the levels in the Hunc+ column will be referred to as the perturbed tunneling splittings. As is seen in Figure 6, this change in the tunneling splittings from diagonal perturbations is small compared to the shifts due to the o↵-diagonal perturbations, which are discussed next.

The 6+3-state model n,

When we include the h1/2t with

s

0 0 s| t s

terms that couple combination levels involving CH stretches

= e and the same number of quanta in the torsion, Hunc+ becomes H6+3 . Based on 21



Page 22 of 42

n,ee|ee

the definition of the basis states, the introduction of the h1/2

terms breaks the degeneracy

of the states with A1 and A2 symmetries. This splitting is indicated by the black dashed arrows in the H6+3 column of Figure 6, while the black solid arrows show the contributions n,ae|ee

from the h1/2

matrix elements that couple the two pairs of states with E symmetry. These

terms will not a↵ect the energies of the states with states with

s

s

= a, but will shift the energies of the

= e. As such, when these o↵-diagonal couplings are included, the Hamiltonian

matrix consists of five 1 ⇥ 1 blocks and two identical 2 ⇥ 2 blocks. The e↵ect of including these couplings is seen by comparing the energies in the Hunc+ and H6+3 columns of the Figures 6, 7, 8, S6 and S7. Closer inspection of Figure 6 shows that the introduction of the o↵-diagonal couplings in H6+3 splits the red/blue dashed levels, while leaving the magnitude of the splittings within the resulting red and blue sets of states nearly constant. Compared with the Hunc+ column, the tunneling splittings change sign (blue) or decrease to roughly zero (red). This is consistent with the observation based on the parameters in Table S3 that the tunneling splittings of the torsion combination bands built o↵ of the ⌫1 and ⌫9 fundamentals and the corresponding tunneling splitting for the torsion level with no CH stretch excitation sum to zero (see Eq 10 and the discussion following Eq S52). Further, the values of the o↵-diagonal couplings are much larger than the tunneling splittings, and the energy level pattern reflects the expectations for a molecule with Cs symmetry and no large amplitude torsion motion. Based on this interpretation, the levels shown in red correspond to the fundamental in ⌫1 while those in blue correspond to the fundamental in ⌫9 . A similar energy level structure is also shown for n = 1 in Figure 7. As the tunneling splitting further increases, particularly for states with torsion energies that are near to or slightly above the torsion barrier on the potential, the energy level structure of the ⌫1 and ⌫9 fundamentals ceases to resemble two sets of nearly triply degenerate torsion levels. As this occurs, the zero-order energy splitting between the two pairs of levels n,ae|ee

with E symmetry increases. As the values of the h1/2 22


terms do not display strong n

Page 23 of 42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


dependences, their e↵ect on the energies diminishes with increased n. As a result, the splitting between the A levels of ⌫1 + n⌫tor and ⌫9 + n⌫tor decreases as n increases. This is consistent with the result that if there is no barrier along the torsion coordinate the A1 and A2 levels become degenerate. For the n = 2 levels shown in the H6+3 column of Figure S6, the shifts in the energies due n, e| t0 e

to the addition of the h1/2t

coupling terms is smaller than the tunneling splitting. This

leads to an energy level pattern with two states that are lower in energy than the remaining four. These states can be interpreted as the torsion levels for a pair of nearly degenerate CH stretch vibrations. For the n = 3 (Figure S7) and n = 4 (Figure 8) levels, where the torsion excited states have energies that exceed the height of the torsion barrier, the perturbed tunneling splittings are much larger than the o↵-diagonal coupling terms in the 6+3 state model. Therefore, the originally degenerate e ⌦ e levels and a ⌦ e torsion/CH stretch levels are slightly shifted into the nearly degenerate levels with A1

A2

E symmetries and the E

levels that are separated from the other four levels by the tunneling splitting of the torsion. In this case, the assignment of levels to ⌫1 and ⌫9 is no longer appropriate. These trends, which are obtained from the calculations for CH3 OO, are consistent with the conclusions from Hougen 10 in his study of methanol using a similar model.

The H9 and the HT Models In the 6+3-state model, we only considered terms of Hcpl that couple states with the e. We next investigate the terms that couple states with

s

s

=

= a and e by considering the

fully coupled 9 ⇥ 9 model Hamiltonian, H9 . With these terms added, the 9 ⇥ 9 Hamiltonian contains into one 1 ⇥ 1 block, one 2 ⇥ 2 block and two identical 3 ⇥ 3 blocks. The e↵ects of introducing these additional coupling terms on the eigenvalues is illustrated by the di↵erences between the H6+3 and H9 columns of Figures 6, 7, 8, S6 and S7. As is seen, the inclusion of the terms that couple the ⌫1/9 + n⌫tor and ⌫2 + (n + 1)⌫tor levels has a larger e↵ect on the energies of the levels shown with red and black lines than the ones shown in blue, 23



Page 24 of 42

moving the red and black levels further apart. This is consistent with symmetry expectations for a rigid CH3 OO radical with Cs symmetry. While ⌫1 and ⌫2 both have A0 symmetry, the torsion has A00 symmetry. Therefore, states built o↵ of ⌫1 and ⌫2 that di↵er by one in n will have di↵erent symmetries and cannot be coupled. As the torsion barrier becomes finite, the symmetry properties are better represented by the G6 permutation inversion group, and, as is seen in the results shown in Figure 8 for larger n, levels depicted by the blue lines shift between the H6+3 and H9 columns. As with the 6+3-state model, analysis of 9-state model shows that the only coupling terms that contribute to

Etunn,tot are the diagonal terms, which are incorporated in Hunc+ n,

as the contributions from all of the h1/2t

0 0 s| t s

terms cancel. In addition the hn, 3

are generally small, and the contributions of the hn,e 3

s

and hn,a 3

s

t s

terms

terms to the tunneling

splittings to a large extent cancel. As a result both the 6+3- and 9-state models yield values for

Etunn,tot that are close to zero, consistent with the observation reported in Eq 10. Finally, comparison of the H9 and HT columns in Figures 6, 7 and 8 illustrates the

di↵erence in the eigenvalues between the 9-state model Hamiltonian and a fully coupled treatment of the torsion/CH stretch system. The small di↵erence found between these two columns indicates that the 9-state model nearly quantitatively represents all the torsion/CH stretch couplings in the molecule.

Summary and Conclusions In this study, we have investigated the e↵ects of torsion/CH stretch couplings on the spectrum of CH3 OO in the region of the ⌫1 and ⌫9 fundamental transitions. In contrast to the spectrum anticipated by a rotational contour simulation that assumed a rigid molecule with Cs symmetry and the associated A, B and C rotational constants, the feature near the origin of ⌫9 is broadened, and the ⌫1 origin feature is suppressed. The broadening at first may appear to be similar to that found for the ⌫2 fundamental. 24

24


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In the case of the ⌫2 fundamental, the broadening was attributed to shifts in the Q-branches of a series of sequence bands. These shifts result from couplings between the ⌫2 + (n + 1)⌫tor and ⌫9 + n⌫tor combination levels. Since the ⌫9 band arises from a perpendicular c-type transition, we expect the structure of the band contour to reflect K rather than J structure. As such, the large intensity near the origin of the ⌫9 fundamental results from the K = 1 and K = 0

0

1 transitions. When the tunneling splittings for the upper and lower levels

are roughly equal, these two features merge into a single strong peak near the origin in the room temperature spectrum. When the size of the tunneling splitting is increased, the two peaks become resolvable, as is the case when n = 1 (see the 910 1211 trace in Figure 3). It is the convolution of the traces associated with sequence bands with n = 0 and 1 that leads to the observed broadening. The loss of intensity near the origin of the ⌫1 band reflects a reorientation of the transition moment due to the treatment of torsion/rotation coupling in a ⇢-axis system. This leads to a significantly smaller parallel component in this mixed aand b-type transition. CH3 OO has shown itself to be an interesting radical in which much of the CH stretch/torsion/rotation mixing previously investigated for CH3 OH and CH3 SH is also at play. While the resolution of the present spectrum does not allow us to fully explore the role of Corilois couplings involving the vibrational angular momentum of the CH stretches, analysis of the potential surface and vibrational frequencies lead us to expect that these couplings will introduce additional spectroscopic richness at higher resolution.

25



Acknowledgement Support from the Chemistry Division of the National Science Foundation (A.B.M.: CHE1619660), the Ministry of Science and Technology, Taiwan (Y.-P.L.: MOST106-2745-M-009001-ASP) and Ministry of Education, Taiwan (Y.-P.L.: “Aim for the Top University Plan” of National Chiao Tung University) are gratefully acknowledged. Support has also been provided by the Office of Basic Sciences Department of Energy (T.A.M.: Grant DE-FG02-01ER14172). M.H. acknowledges fellowship support from the Graduate School at the Ohio State University. This work was supported in part by allocations of computing time from the Ohio Supercomputer Center to A.B.M. and T.A.M. and the National Center for High-performance Computing provided computer time to Y.-P.L.

Supporting Information Available Additional details about the torsion/CH stretch and torsion/rotation Hamiltonians and the basis sets used to expand their solutions; Analysis of the 6+3- and 9-state models; Parameters in the total Hamiltonian HT calculated at B3LYP level of electronic structure theory; Parameters used to evaluate the 6+3- and 9-state model Hamiltonians; Eigenvalues of the 6+3- and 9-state model Hamiltonians for n  4; Raw spectrum of CH3 OO in the region of the ⌫1 and ⌫9 fundamentals; K-dependence of the torsional energies evaluated based on HT and the approximation in Eq 4 based on the adjusted B3LYP force constants for n = 0 and 1; K-dependence of the torsional energies evaluated based on HT and the approximation in Eq 4 based on the B3LYP force constants for n = 0 and 1; Simulated spectrum obtained when 1 Btor a10 1 = 6.2 cm ; Energy levels obtained using various model Hamiltonians when n = 2

and 3. This material is available free of charge via the Internet at http://pubs.acs.org/.

26


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(17) Mueller, H. S.; Brown, L. R.; Drouin, B. J.; Pearson, J. C.; Kleiner, I.; Sams, R. L.; Sung, K.; Ordu, M. H.; Lewen, F. Rotational Spectroscopy as a Tool to Investigate Interactions Between Vibrational Polyads in Symmetric Top Molecules: Low-lying States v8 2 of Methyl Cyanide, CH3 CN. J. Mol. Spect. 2015, 312, 22 – 37. (18) Thomas, J. K. Pulse Radiolysis of Aqueous Solutions of Methyl Iodide and Methyl Bromide. The Reactions of Iodine Atoms and Methyl Radicals in Water. J. Phys. Chem. 1967, 71, 1919–1925. (19) Hunziker, H. E.; Wendt, H. R. Electronic Absorption Spectra of Organic Peroxyl Radicals in the Near Infrared. J. Chem. Phys. 1976, 64, 3488–3490. (20) Pushkarsky, M. B.; Zalyubovsky, S. J.; Miller, T. A. Detection and Characterization of Alkyl Peroxy Radicals Using Cavity Ringdown Spectroscopy. J. Chem. Phys. 2000, 112, 10695–10698. (21) Just, G. M. P.; McCoy, A. B.; Miller, T. A. Computation of Barriers to Methyl Rotation for the Methyl Peroxy Radical. J. Chem. Phys. 2007, 127, 044310. (22) Chung, C.-Y.; Cheng, C.-W.; Lee, Y.-P.; Liao, H.-Y.; Sharpe, E. N.; Rupper, P.; Miller, T. A. Rovibronic Bands of the A˜

˜ Transition of CH3 OO and CD3 OO X

Detected with Cavity Ringdown Absorption near 1.2-1.4 µm. J. Chem. Phys. 2007, 127, 044311. (23) Fu, H. B.; Hu, Y. J.; Bernstein, E. R. Generation and Detection of Alkyl Peroxy Radicals in a Supersonic Jet Expansion. J. Chem. Phys. 2006, 125, 014310. (24) Hsu, K.-H.; Huang, Y.-H.; Lee, Y.-P.; Huang, M.; Miller, T. A.; McCoy, A. B. Manifestations of Torsion-CH Stretch Coupling in the Infrared Spectrum of CH3 OO. J. Phys. Chem. A 2016, 120, 4827–4837.

29



(25) Dawadi, M. B.; Perry, D. S. Communication: Conical Intersections between Vibrationally Adiabatic Surfaces in Methanol. J. Chem. Phys. 2014, 140, 161101. (26) Nobre, M.; Fernandes, A.; Ferreira da Silva, F.; Antunes, R.; Almeida, D.; Kokhan, V.; Ho↵mann, S. V.; Mason, N. J.; Eden, S.; Limao-Vieira, P. The VUV Electronic Spectroscopy of Acetone Studied by Synchrotron Radiation. Phys. Chem./Chem. Phys. 2008, 10, 550–560. (27) Lightfoot, P. D.; Kirwan, S. P.; Pilling, M. J. Photolysis of Acetone at 193.3 nm. J. Phys. Chem. 1988, 92, 4938–4946. (28) Y. Endo private communication K. Katoh, Ph.D. thesis, University of Tokyo, 2007; relevant results are also provided in Ref. 22. (29) Child, M. S.; Lawton, R. T. Local and Normal Vibrational States: a Harmonically Coupled Anharmonic Oscillator Model. Faraday Discuss. Chem. Soc. 1981, 71, 273– 85. (30) Herbst, E.; Messer, J.; DeLucia, F. C.; Helminger, P. A New Analysis and Additional Measurements of the Millimeter and Submillimeter Spectrum of Methanol. J. Mol. Spect. 1984, 108, 42 – 57. (31) DeLucia, F. C.; Herbst, E.; Anderson, T.; Helminger, P. The Analysis of the Rotational Spectrum of Methanol to Microwave Accuracy. J. Mol. Spect. 1989, 134, 395 – 411. (32) Anderson, T.; Herbst, E.; DeLucia, F. A New Analysis of the Rotational Spectrum of CH3 OD. J. Mol. Spect. 1993, 159, 410 – 421. (33) Hougen, J. T.; Kleiner, I.; Godefroid, M. Selection Rules and Intensity Calculations for a C2 Asymmetric Top Molecule Containing a Methyl Group Internal Rotor. J. Molec. Spect. 1994, 163, 559–86.

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(34) Xu, L.-H. X.; Hougen, J. T. Global Fit of the Torsion-Rotation Transitions in the Ground and First Excited Torsional States of Methanol. J. Mol. Spectrosc. 1995, 173, 540–551. (35) Wu, S.; Dupre, P.; Rupper, P.; Miller, T. A. The Vibrationless A˜

˜ Transition of X

the Jet-Cooled Deuterated Methyl Peroxy Radical CD3 O2 by Cavity Ringdown Spectroscopy. 2007, 127, 224305. (36) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A. et al. Gaussian09 Revision D.01. Gaussian, Inc., Wallingford, CT 2009. (37) Lees, R. M.; Xu, L.-H. Dark State Illuminated: Infrared Spectrum and Inverted Torsional Structure of the ⌫11 Out-of-Plane CH3 -Rocking Mode of Methanol. Phys. Rev. Lett. 2000, 84, 3815–3818. (38) Huang, M. Spectroscopy Studies of Free Radicals and Ions Containing Large Amplitude Motions. Ph.D. thesis, The Ohio State University, 2017. (39) Xu, L.-H.; Hougen, J.; Fisher, J.; Lees, R. Symmetry and Fourier Analysis of the ab Initio-Determined Torsional Variation of Structural and Hessian-Related Quantities for Application to Vibration-Torsion-Rotation Interactions in CH3 OH. J. Mol. Spect. 2010, 260, 88 – 104. (40) Dawadi, M. B.; Thapaliya, B. P.; Perry, D. S. An Extended E⌦e Jahn-Teller Hamiltonian for Large-Amplitude Motion: Application to Vibrational Conical Intersections in CH3 SH and CH3 OH. J. Chem. Phys. 2017, 147, 044306.

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Page 32 of 42

Table 1: The Constants Used for the Rotational Analysis Based on the Asymmetric Top Model.

Parameters A B C a:b:c c a b

c

Vibrationless 1.730 cm 1a 0.379 cm 1a 0.330 cm 1a –

⌫9 1.725 cm 0.379 cm 0.330 cm 0:0:1

1b 1b 1b

⌫1 1.730 cm 1b 0.379 cm 1b 0.330 cm 1b 0.37:0.63:0

Ground state rotational constants are from microwave spectroscopy. 22,28 Excited state rotational constants are determined by multiplying the ground state constants by the ratio of excited and ground state rotational constants calculated at the B3LYP/aug-cc-pVTZ level of theory and basis set using second order vibrational perturbation theory, as implemented in Gaussian 09. 36 a:b:c provides the ratio of the squares of transition dipole moment components along the a-, b- and c-axes in the molecule based on calculation at the B3LYP/aug-cc-pVTZ level of theory and basis set.

32


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Table 2: The Eigenvalues and Eigenfucntions of the Total Hamiltonian, HT , for the States with ⌫9 = 1 and ⌫1 = 1.

Eigenstate | |,|l| n| | ; mA , mE |l| ; En,mA ,mE

⌫1 = 1 and ⌫9 = 1 contributions n| | ,

00 ;0, 11 ;E

3061.3

00 ,E

01 ;0, 11 ;A2

3061.5

01 ,A2

1

1

0 ;0, 1 ;E

1

| |,|1|

|Cn,0,1 |2, a

Ecpl b

0.48

-7.4

0.94

-7.3

3078.1

0 ,E

0.50

01 ;0, 11 ;A1

3078.3

01 ,A1

1.00

9.4

11 ;0, 11 ;E

3168.8

11 ,E

0.44

-8.9

1 ;0, 1 ;A1

3178.0

1

1 ,A1

0.86

0.3

11 ;0, 11 ;A2

3180.5

11 ,A2

0.98

2.8

1

1

n| | , 01 ,E

| |,|1| |Cn,0,1 |2, a

0.46

⌫2 = 1 contributions Ecpl b -7.6

n| | , 11 ,E

9.2

0 ,E

0.48

9.4

10 ,E

0.17

-13.3

Ecpl b

0.06

-24.3

0.05

-28.4

21 ,E

0.37

-11.5

20 ,A1

0.13

18.7

10 ,A2 0

| |,|0|

|Cn,1,0 |2, a

10 ;0, 11 ;E

3185.3

10 ,E

0.81

3.2

11 ,E

0.08

7.6

21 ,E

0.10

5.0

20 ;0, 11 ;E

3252.4

20 ,E

0.77

1.1

21 ,E

0.06

-20.0

31 ,E

0.14

13.3

21 ;0, 11 ;A2

3264.9

21 ,A2

0.94

-7.5

30 ,A2

0.04

-42.2

31 ,E

0.03

33.8

40 ,A1

0.04

-34.2

40 ,A1

0.06

23.8

41 ,E

0.14

3.9

1

1

1

2 ;0, 1 ;E

3272.9

2 ,E

0.92

0.6

21 ;0, 11 ;A1

3276.8

21 ,A1

0.94

4.4

31 ;0, 11 ;A2

3328.5

31 ,A2

0.96

-2.7

31 ;0, 11 ;E

3330.9

31 ,E

0.96

-0.3

31 ;0, 11 ;A1

3334.8

31 ,A1

0.90

3.6

0

1

0

0

2 ,E

0

0.04

21.6

3 ;0, 1 ;E

3400.0

3 ,E

0.75

0.9

40 ;0, 11 ;E

3404.0

40 ,E

0.81

0.9

41 ,E

0.15

7.9

41 ;0, 11 ;E

3486.8

41 ,E

0.92

-1.5

51 ,E

0.08

-11.5

41 ;0, 11 ;A2

3488.0

41 ,A2

0.98

-0.2

41 ;0, 11 ;A1

3488.5

41 ,A1

0.98

0.3

51 ;0, 11 ;A2

3589.5

51 ,A2

0.98

-0.8

51 ;0, 11 ;A1

3589.5

51 ,A1

0.98

-0.8

51 ;0, 11 ;E

3590.4

51 ,E

1.00

0.1

50 ;0, 11 ;E

3706.3

50 ,E

0.50

-0.6

60 ,E

0.49

-0.6

60 ;0, 11 ;E

3706.3

60 ,E

0.51

0.1

50 ,E0

0.49

0.1

4 ,E

0.09

-3.1

Energies (in cm 1 ) calculated using the parameters obtained at the B3LYP/aug-cc-pVTZ level of theory and basis set (Tables S1 and S2), and reported relative to the zero-point energy of 4716.8 cm 1 . | |,|l| | | |l| b Ecpl = En,mA ,mE En EmA ,mE . | |,|l| c |Cn,mA ,mE |2 provides the contribution of the corresponding basis state to the eigenfunction of HT . a

33



Table 3: Parameters Used in the Simulation of the Experimental Spectrum Shown in Figures 3 and 4.

Parameter A00e↵ b A0e↵ 00 a,b Be↵ 00 a,b Ce↵ 00 a,b De↵ ⇢00a ✓ x :y:z c,d e Btor a000 1 e Btor a100 1 Btor a00 1 Btor a10 1 Edif,1 E¯(⌫ ⌫ ) 1

I

d,f

9

Value(⌫9 ) 1.706 cm 1 1.701 cm 1 0.403 cm 1 0.330 cm 1 0.177 cm 1 0.299 7.64 0.00:1.00:0.00 0.1 cm 1 3.1 cm 1 0.1 cm 1 0.0 cm 1 0.4 cm 1 11.3 cm 1 1.0

Value(⌫1 ) 1.706 cm 1 1.706 cm 1 0.403 cm 1 0.330 cm 1 0.177 cm 1 0.299 7.64 0.85:0.00:0.15 0.1 cm 1 – 0.1 cm 1 – – – 0.59

a The corresponding values of Be↵ , Ce↵ , De↵ , ✓ and ⇢ constants for the states with ⌫CH = 1 are the same as those for the ground state. b Calculated using the rotational constants reported in Table 1. c The ratio of the transition dipole moment components squared, evaluated in the ⇢-axis system. d For comparison, the ratio of the squares of the transition dipole moment components for the ⌫2 fundamental is 0.03:0:0.97, and the intensity of the ⌫2 fundamental is a factor of 1.85 times larger than the intensity of the ⌫9 fundamental. e Calculated using the parameters evaluated at the B3LYP/aug-cc-pVTZ level of theory and basis set, provided in Tables S1 and S2. f Calculated relative intensities of the ⌫1 and ⌫9 fundamentals.

34


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Figure 1: The experimental spectrum (red) of the ⌫9 and ⌫1 fundamentals of the CH3 OO radical and the simulated spectrum (black (⌫9 ), blue (⌫1 )), evaluated using the simple asymmetric rotor model in Eq 1. The experimental trace reflects the spectrum obtained after spectral stripping of weak absorptions of (CH3 )2 CO, CH4 and C2 H6 , as shown in Figure S1. For comparison, the spectrum of CH3 OO in the region of the ⌫2 fundamental and its simulation based on an asymmetric rotor mode are shown in the inset. 24

35



Figure 2: The equilibrium geometry of the CH3 OO radical with the atoms labeled as described in the text. Inertial a- and b-axes are shown with the c inertial axis lying perpendicular to the symmetry plane. ✓ is the angle between the ⇢-vector and the a-axis. The torsional angle ⌧ is measured from this reference geometry (⌧ =0), which corresponds to the H(1) COO plane bisecting the H(2) C H(3) angle. The x-, y-, z-axes are defined such that the z-axis lies along ⇢~, while the x-axis is in the ab plane. The y-axis corresponds to the c-inertial axis and is perpendicular to the COO plane.

36


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Figure 3: The simulated spectra of the ⌫9 and ⌫1 fundamentals for the CH3 OO radical at 300 K and various torsional sequence bands based on the results of the coupled torsion/rotation, HT , model Hamiltonian using the parameters reported in Table 3. For the 110 1211 and 910 1222 bands, the parameters used in this simulation are based on the values obtained from the adjusted B3LYP calculation, as described in the text. The bands in the 110 and 910 simulations are labeled by approximately good quantum numbers, K 00 , for the near-prolate symmetric top. Bands moving to higher frequency with increased |K 00 | correspond to K = K 0 K 00 = +1, while bands moving to lower energy with increased |K 00 | correspond to transitions with K = 1.

37



Figure 4: The experimental spectrum of the ⌫9 and ⌫1 fundamental regions of the CH3 OO radical and the simulated 300 K spectrum (black), which is the sum of the torsional sequence band simulations shown in Figure 3.

38


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Figure 5: Scans of the harmonic frequencies of the ⌫1 (blue) and ⌫9 (red) CH stretches as functions of the COO angle for the minimum (0 , open) and transition state (180 , solid) structures. These curves were evaluated at the B3LYP/aug-cc-pVTZ level of theory and basis set. The black dashed line is placed at ✓COO,e , and the dot dash lines represent the geometries where the ⌫1 and ⌫9 frequencies are equal. The electronic energies (in cm 1 ) associated with these geometries are provided above the black lines.

39



Figure 6: The energies of the ⌫2 +⌫tor (black), ⌫9 (red) and ⌫1 (blue) levels obtained by solving (from left to right) the uncoupled Hamiltonian, Hunc , the perturbed uncoupled Hamiltonian, Hunc+ , the 6+3-state Hamiltonian, H6+3 , the 9-state Hamiltonian, H9 , and the total Hamiltonian, HT , as described in the text. The eigenstates are labeled at the far right based on the number of quanta in the torsion, n, and the symmetry of the level in the HT column, . The red/blue dashed lines represent degenerate levels with s = e. Two of these pairs levels have t = e and = A1 A2 E. They are represented by pairs of red/blue dashed lines shown at the same energy. The remaining red/blue dashed lines represent pairs of states with t = a and = E. Each solid colored line represents a single energy level. The pairs of solid colored lines, which have the same energy, represent levels with = E, while remaining solid colored lines represent states with = A1 or A2 . The colored matrices above each column illustrate the non-zero matrix elements in the corresponding model, and the colors represent the symmetries of the levels, with green, yellow, purple and orange boxes representing A1 , A2 , E+ , E symmetry states, respectively. For HT the open squares each represent individual 9 ⇥ 9 blocks given by H9 , and the grey shading represents the coupling between these blocks. The grey dashed lines in the H6+3 column illustrate the corresponding n,ee|ee levels in Hunc+ in order to show the magnitude of the shifts due to the h1/2 (black dashed n,ee|ea

arrows) and h1/2

(black solid arrows) terms. 40


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Figure 7: Same as Figure 6 for the ⌫2 + 2⌫tor levels (black), ⌫9 + ⌫tor (red) and ⌫1 + ⌫tor (blue) levels.

41



Figure 8: Same as Figure 6 for the ⌫2 + 5⌫tor levels (black), ⌫9 + 4⌫tor (red) and ⌫1 + 4⌫tor (blue) levels.

42


Page 42 of 42

Rotation Couplings in Methyl Peroxy

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