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Manifestations of Torsion-CH Stretch Coupling in the Infrared Spectrum of CHOO 3

Kuo-Hsiang Hsu, Yu-Hsuan Huang, Yuan-Pern Lee, Meng Huang, Terry A. Miller, and Anne B. McCoy J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.5b12334 • Publication Date (Web): 22 Feb 2016 Downloaded from http://pubs.acs.org on February 27, 2016

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The Journal of Physical Chemistry

Manifestations of Torsion-CH Stretch Coupling in the Infrared Spectrum of CH3OO Kuo-Hsiang Hsu,† Yu-Hsuan Huang,† Yuan-Pern Lee,∗,†,§ Meng Huang,‡ Terry A. Miller,∗,‡ and Anne B. McCoy

∗,¶

Department of Applied Chemistry and Institute of Molecular Science, National Chiao Tung University, 1001 Ta-Hsueh Rd., Hsinchu 30010, Taiwan, Department of Chemistry and Biochemistry, The Ohio State University, Columbus, OH 43210, USA, and Department of Chemistry, University of Washington, Seattle, WA 98195, USA E-mail: [email protected]; [email protected]; [email protected]

∗

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

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Abstract With a step-scan Fourier-transform spectrometer we recorded temporally resolved infrared absorption spectra of CH3 OO radicals that were produced upon irradiation of CH3 COCH3 and O2 at 193 nm in a flowing mixture. At a resolution of 0.15 cm−1 , the rotational structure of the ν2 band of CH3 OO near 2954.4 cm−1 is partially resolved, and shows an unexpectedly broadened, and somewhat distorted, Q-branch. A four-dimensional model Hamiltonian, consisting of three CH stretches and the methyl torsion, was developed to explore the origins of this broadening. The vibrational progressions predicted by the model Hamiltonian and the rotational contours of the ν2 band, based on experimental ground state rotational parameters and their values scaled by their calculated ratios for the upper state, produced simulations in satisfactory agreement with the observed spectrum. These results provide new insight into the vibrational couplings in CH3 OO.

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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 ROO radicals are formed typically via intermolecular reactions of R· and O2 . In polluted air, the subsequent reactions of ROO radicals with NO produce NO2 ; NO2 is photolyzed by sunlight to produce O atoms, which in turn lead to the formation of ozone in the troposphere. Self reactions of ROO and the reactions of ROO with HO2 are also important in the HOx catalytic cycle of the atmosphere. 7 In a combustion system, the reactions of ROO are involved in a chain-branching sequence of reactions that ultimately lead to the regeneration of OH radicals, which are essential for combustion. The competition between the reactions of organic radicals with O2 to form ROO radicals and with themselves or other organic species to form larger hydrocarbons is critical to the production of soot. 8 The ultraviolet (UV) absorption of ROO radicals is typically probed in experiments on ˜ ←X ˜ electronic the chemical kinetics of ROO. This UV absorption associated with the B transition is diffuse, with no vibrational or rotational structure because the upper state is unbound; this probe hence provides little information about the structure of ROO, which precludes selective detection among various ROO radicals. The much weaker absorption in ˜ electronic transition is highly the near infrared (NIR) region associated with the A˜ ← X structured; it consequently provides a highly selective diagnostic tool. Detailed information about the relation between the spectra and structures of ROO is reviewed by Sharp et al. 9 The methylperoxy (CH3 OO) radical is the simplest ROO radical; its spectra have been ˜ A′′ ← X ˜ 2 A′′ transition near extensively investigated. An intense UV absorption of the 2 B ˜ 2 A′′ of CH OO was 240 nm was reported. 10 Weak absorption corresponding to A˜ 2 A′ ← X 3 reported initially by Hunziker and Wendt, 11 and subsequently by Pushkarsky et al. 12 and by Chung et al. 13 who applied the cavity ringdown (CRD) technique in the 1.2-1.4 µm region to determine the wavenumbers of several vibrational levels of the A˜ state of CH3 OO and CD3 OO. Similar vibronic bands of CH3 OO and CD3 OO were reported by Fu et al. 3



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who employed near-infrared and vacuum-ultraviolet lasers to record photofragmentationionization spectra. 14 This sensitive CRD detection technique was employed to investigate the kinetics involving CH3 OO. 15 Analysis of the room temperature spectra of these radicals showed evidence of sequence bands in the CH3 torsion as well as strong couplings between the torsion and rotation. 16 The geometry and vibrational frequencies of the electronic ground state have been investigated quantum-chemically using various methods, 17–25 and experimental values have been obtained from microwave spectra. 13,26 These results agree satisfactorily with those ˜ 2 A′′ reported by Pushkarsky et al. 12 and those predicted quantum-chemically for the X state. Blanksby et al. 27 employed photodetachment to record the photoelectron spectrum of gaseous CH3 OO. They reported the wavenumbers of two vibrational states of CH3 OO, which they assigned as ν6 = 1124 ± 5 and ν8 = 482 ± 9 cm−1 . A low-resolution infrared (IR) spectrum of gaseous CH3 OO was recorded by Huang et al. 28 by coupling a step-scan Fourier-transform infrared (FTIR) spectrometer with a multipass absorption cell. Based on this work, the wavenumbers for the fundamentals for the three CH stretches are 2954 ± 1 cm−1 (totally symmetric stretch, ν2 ), 3020 ± 2 cm−1 (CH2 asymmetric stretch, ν9 ), and 3033 ± 1 cm−1 (CH2 symmetric stretch, ν1 ) . The wavenumbers of the observed bands agree well with those of matrix-isolated CH3 OO reported previously, 24,29 but no information on the rotational structure of these bands was available. We have implemented a second-generation FTIR spectrometer that has improved resolution and signal to noise ratio, as demonstrated by detection of the important Criegee intermediates and related compounds. 30–32 Here we report the spectral analysis of the partially rotationally resolved symmetric CH3 -stretching (ν2 ) band of CH3 OO recorded at a resolution of 0.15 cm−1 with this spectrometer. To interpret the spectrum, a four-dimensional Hamiltonian based on the three CH stretches and the methyl torsion is developed. The approach is similar to one recently employed by Mosley et al. in a study of CH3 OH+ ·Ar. 33 Combining these experimental and theoretical approaches, the role of the methyl torsion in

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the CH stretch region of the spectrum of CH3 O2 is investigated.

Photolysis of (CH3)2CO in O2 We used a step-scan FTIR spectrometer (Bruker, Vertex 80v) coupled with a multi-reflection White cell to record temporally resolved IR spectra as described previously. 30,34 An ArF excimer laser (Coherent, Compex Pro 102F, 193 nm, 13 Hz, ∼100 mJ pulse−1 , beam size 3×1 cm2 ) passed through the reactor and was reflected three times with two external laser mirrors to photodissociate a flowing mixture of (CH3 )2 CO (acetone) and O2 to produce CH3 that subsequently reacted with O2 to form CH3 OO. We recorded interferograms consecutively from the preamplified ac- and dc-coupled signals of an InSb detector with an internal 24-bit analogue-to-digital converter (ADC) with a resolution of 12.5 µs. The signal was typically averaged over N (= 15-20) laser shots at each scan step. With appropriate optical filters to define a narrow spectral region, we performed undersampling to decrease the number of points in the interferogram, hence the duration of data acquisition. For spectra in the range 2830-3380 cm−1 at a resolution of 0.15 cm−1 , 7616 scan steps were completed within ∼3 h. (In the text hereafter the indicated resolution is instrumental, unless otherwise noted.) The effective full width at half maximum (FWHM) after apodization with the Blackman-Harris three-term function is ∼128% of the listed instrumental resolution. At a resolution of 0.15 cm−1 , the FWHM of the line is hence 0.19 cm−1 . Thirteen spectra with 0.15 cm−1 resolution were recorded under similar experimental conditions and averaged to yield a spectrum with a satisfactory ratio of signal to noise. Such data were recorded at 298 K in three sets − six at a total pressure of 100.6 Torr (acetone/O2 = 1.6/99.0) and N = 15, two at a total pressure of 100.2 Torr (acetone/O2 = 1.2/99.0) and N = 15, and five at 199.6 Torr (acetone/O2 = 0.6/199.0) and N = 20. As the spectra in these three sets, after spectral stripping of weak absorption of (CH3 )2 CO, CH4 and C2 H6 in

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this region, are nearly identical, we averaged them to decrease the noise. O2 served as both a reactant and as an efficient quencher. Experimental conditions were as follows: flow rates Facetone ∼ = 0.4 and FO2 ∼ = 24.6 STP cm3 s−1 (STP denotes standard temperature 273 K and pressure 1 atm), and T = 298 K. To enhance the formation of CH3 OO we performed the reaction at a moderate pressure (100-200 Torr). The average efficiency of photolysis of acetone is estimated to be ∼2.4% at 193 nm according to an absorption cross section (Ref. 35) of ∼ 2.36 × 10−18 cm2 molecule−1 and a laser fluence ∼30 mJ cm−2 ; the yield 36 for production of CH3 is 1.9. (CH3 )2 CO (99.8%, J.T. Baker) and O2 (99.999%, Chiah-Lung) were used without further purification.

Experimental Results The ν2 spectrum (resolution 0.15 cm−1 ) of CH3 OO averaged from three sets of experiments as described above, upon 193-nm laser irradiation of a flowing mixture of (CH3 )2 CO and O2 at 298 K is shown in Figure 1(a). In this ac-coupled spectrum, features pointing upward indicate production, whereas those pointing downward indicate destruction. The downward feature near 2970 cm−1 is due to loss of (CH3 )2 CO upon irradiation. A spectrum of (CH3 )2 CO recorded separately is shown in Figure 1(b) for comparison. New features with a band near 2954 cm−1 and several sharp lines between 2970 and 2990 cm−1 appeared immediately after irradiation. The latter are due to product C2 H6 . A spectrum of C2 H6 recorded separately is shown in Figure 1(c) for comparison. A very weak downward band near 2979 cm−1 is due to CH4 . Because these absorption bands of the parent molecule (CH3 )2 CO and byproducts C2 H6 and CH4 interfered with the ν2 band of CH3 OO, we performed spectral stripping by adding the absorption of (CH3 )2 CO and subtracting the absorption of C2 H6 and CH4 to the observed difference spectrum. The resultant corrected spectrum is shown in Figure 1(d). However, the acetone has a greater temperature after irradiation so that its band shape in Figure 1(a) is

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slightly different from that in Figure 1(b), so we could not remove completely the interference from the absorption bands of (CH3 )2 CO near 2970 cm−1 . Furthermore, the weak band near 2964 cm−1 might be due to absorption of vibrationally excited acetone; this band became more prominent in experiments with greater laser power. These two interfering bands of acetone are indicated with arrows in Figure 1(d); small remnant lines of C2 H6 are indicated with *.

Theory The experimental spectrum of the CH3 OO radical at near room temperature, with a band peaked at 2954.4 ±0.1 cm−1 in Figure 1(d), is assigned to the totally symmetric CH stretch, ν2 . The rotationally resolved spectrum of the ν2 fundamental is plotted in red in Figure 2, which is an expanded view of the region around 2955 cm−1 . When analyzing any observed spectrum, it is usually reasonable to start with the simplest model. We first calculated the rotational structure, using an asymmetric top rotational Hamiltonian, Hrot = ANa2 + BNb2 + CNc2

(1)

with the mainly-experimentally-based parameters listed in Table 1. (In Eq 1 Na , Nb , Nc are ~ onto the a, b and c principal axes of the projections of the rotational angular momentum N CH3 O2 , as depicted in Figure 3.) While the simulation in Figure 2 shows good agreement with most of the features in the experimental spectrum, it clearly does not account for the breadth of the Q-branch in the experimental spectrum. Multiple overlapping spectra are sometimes the source of spectral broadening. Indeed several low frequency torsion bands of CH3 OO would be significantly populated at room temperature and might provide the source of the broadening of the Q-branch. However, in the harmonic approximation the 210 12nn torsional sequence bands will have the same frequency as the 210 band. Any changes in the rotational constants among these vibrational levels should 7



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be so small that they would not affect the spectrum at the resolution displayed in Figure 2. In the discussion that follows, we will explore whether a treatment of the vibrations, including CH stretch/torsion coupling and rotation/torsion coupling, can account for the observed spectrum. The calculation of the spectrum is performed in two steps. First, the torsion/vibration energy levels are evaluated based on a four-dimensional model Hamiltonian, which is expressed as power-series expansions in the CH bond length displacements and a Fourier expansion in the methyl torsion coordinate. The parameters for this Hamiltonian are evaluated based on electronic structure calculations, performed at the MP2 and DFT (with M062X, M052X and B3LYP functionals) levels of theory with an aug-cc-PVTZ basis set using the Gaussian 09 37 program package. Because of the reduced-dimensional nature of these calculations, the parameters obtained from electronic structure theory will be used as a guide in determining the physically relevant ranges of parameters that are appropriate for modeling the experimental spectrum, rather than with an expectation that the calculated parameters will yield exact agreement between experiment and calculation. After the positions of the stretch/torsional band origins and the corresponding eigenfunctions are obtained, the rotational band contours are evaluated using a rotation-torsion Hamiltonian, previously described by DeLucia, 38–40 Hougen 41,42 and their co-workers, and used by some of us in earlier studies of the electronic spectra of CH3 OO and CD3 OO. 16,43 These two parts of the calculation will be described separately, below.

CH Stretch/Torsion Calculations A four dimensional model Hamiltonian involving three CH stretches and the CH3 torsion is developed to describe the vibrational spectrum in the CH stretch region and to semiquantitatively understand the broadening of the Q-branch of the ν2 fundamental in the spectrum of the CH3 OO radical. To describe the geometry of the molecule unambiguously, the O atom that is directly bonded to the carbon atom is labeled as O(1) and the terminal 8


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O atom is labeled as O(2) . The H atom that is in the C-O(1) -O(2) plane in the reference structure is labeled as H(1) , and the other two H atoms are labeled as H(2) and H(3) . This is illustrated in Figure 3. The torsion angle τ is then defined as the dihedral angle between the C-O(1) -O(2) plane and the H(1) -C-O(1) plane, with τ = 0◦ corresponding to the displayed reference structure. The total Hamiltonian is divided into two contributions, cpl HT = Htor + Hstr

(2)

cpl describes the CH stretch where H tor provides the Hamiltonian for the pure torsion, while Hstr

vibrations and the coupling between the CH stretches and the torsion. The torsion contribution to HT is given by

Htor = Btor p2tor + V3 [1 − cos(3τ )]

(3)

where Btor is the effective rotational constant for the CH3 torsion, and ptor is the momentum of this torsion. The potential energy contribution is expressed as an expansion in cos(3kτ ) to ensure the C3 symmetry of this potential along τ . In the present study, this potential energy function is truncated at the cos 3τ term since fits of the electronic energies show that the higher order terms in the expansion of the potential are small. cpl The general form of Hstr is

cpl Hstr

3 3 3 X 3 X 3 X 1 XX 2(i − 1)π (k) k = pi Gil pl + Fi,j ∆ri cos j τ − 2 i=1 l=1 3 j=0 i=1 k=2 X i−1 3 X 3 X 3 X 2(5 − i − l)π (2) (4) + + Fil,j ∆ri ∆rl cos j τ − Fi,0 ∆ri4 3 j=0 i=1 l=1 i=1

(4)

The three CH stretches are described using the harmonically coupled anharmonic oscillator model of Child and Lawton. 44 The potential surface for each CH stretch is represented as

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a quartic expansion in the displacement of the local CH bond length with respect to the reference geometry where all three CH bond lengths are equal to rref . (For computational purpose, rref is taken as 1.0875 ˚ A, which is the average of the three CH bond lengths at τ = 0◦ , obtained from the B3LYP calculation. The range of values of the optimized CH distances over all values of τ is smaller than 5 × 10−4 ˚ A, allowing us to treat the reference (k)

value of r as its equilibrium value in the approach described below.) The Fi,j is (k!)−1 times the kth order derivative of the potential energy with respect to ∆ri at the reference geometry (k)

for a given value of the torsional expansion index, j. Finally, the Fil,j provide the mixed second derivative of the potential energy with respect to ri and rl at the reference geometry for a given value of the torsional expansion index, j. The CH stretch Hamiltonian is given by the j = 0 contribution to Eq 4, 3

Hstr

3

3

4

i−1

3

i−1

X X (k) X X (2) 1X 1 XX = pi Gii pi + pi Gil pl + Fi,0 ∆rik + Fil,0 ∆ri ∆rl 2 i=1 2 i=1 l=1 i=1 k=2 i=1 l=1

(5)

where the ∆ri represents the displacement of one of the bond lengths, C-H(i) , from rref . The Gil are the Wilson G-matrix elements, 45

Gii =

Gil =

1 µCH

cos θH(i) CH(l) mC

(6)

(7)

where µCH is the reduced mass of a CH bond and mC represents the mass of the carbon atom. The first two terms in Eq 5 represent, respectively, the kinetic and potential energy of an anharmonic oscillator. The coupling between pairs of the local CH stretches are provided by the remaining two bilinear terms in the Hamiltonian. As the CH3 group rotates around the C-O bond, the force constants for the CH stretches will depend on τ . This introduces couplings between the stretches and torsion, which are

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cpl given by the j > 0 terms of Hstr , thereby creating a cosine series expansion of the force

constants with respect to the torsional angle τ . These couplings, H cpl , are given by

Hcpl

3 X 3 X 3 X

2(i − 1)π = cos j τ − 3 j=1 i=1 k=2 i−1 3 X 3 X X 2(5 − i − l)π (2) Fil,j ∆ri ∆rl cos j τ − + 3 j=1 i=1 l=1 (k) Fi,j ∆rik

(8)

The τ -dependence of the Wilson G-matrix elements, Gil , and the quartic force constants, (4)

Fi,0 , were found to be weak and are not included in Eq 8. To calculate the spectral line positions, the eigenvalues of Hstr , Htor and HT are all determined, using values of parameters obtained from electronic structure calculations. The details of the calculation are given in the Theoretical Results section. In order to predict the spectrum, we also need to calculate the intensities of the transitions. This requires an expression for the dipole moment function. For this purpose, the dipole moment is expanded to first order in the local CH stretch displacements in the same manner as the potential energy.

(0)

µ ~ (τ ) = µ ~ (τ ) +

3 X

(1)

µ ~ i (τ )∆ri

(9)

i=1

(1)

At each point of the scan along τ , the dipole moment, µ ~ (0) (τ ), and its first derivatives, µ ~ i (τ ), are obtained from the electronic structure calculations. The resulting data are interpolated using cubic splines to allow for the calculation of the matrix element of the dipole moment operator. With this in hand, the intensities of the transitions are calculated using

Ig→e

Eg ∝ |hΨe |~µ(τ )|Ψg i| (Ee − Eg ) exp − kb T v 2

(10)

where the |Ψg i, Eg and |Ψe i, Ee are the eigenvectors and eigenvalues of HT for the ground and excited vibrational states, respectively. Here, kb is the Boltzmann constant and Tv = 298 11



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K is the vibrational temperature used in the calculation.

Rotation/Torsion Calculations The rotational contour itself is modified by the torsion-rotation coupling in the molecule. To include these interactions in our model, the CH stretch/torsional vibrational energies and wave functions, evaluated using the Hamiltonian described above, are used to evaluate the parameters for rotational contour simulations of the fundamental and associated torsional sequence bands using a model Hamiltonian that includes the torsion-rotation couplings. Following the previous treatment of the same molecule, we simulate the rotational contour of the origin and each of the torsional sequence bands independently, focusing on the couplings of the rotations and torsion. The Hamiltonian that is used for this analysis is given by 16,38–43

~ )2 + V3 [1 − cos(3τ )] + AN 2 + BN 2 + CN 2 Htor,rot = Btor (ptor − ρ~ · N a b c

(11)

With one exception, Htor,rot is the simple sum of Eq 3 for the torsion and an asymmetric top Hamiltonian, in the principal axis system, (a,b,c), to describe the overall rotation. The ~ term, which describes the coupling between the two motions. The one exception is the ρ~ · N calculation of the components of ρ~, which lies in the ab plane, has been described by Hougen and co-workers 41 and involves the ratio of the rotational constants of the molecule and Btor , and the angle, θ, between the CO axis and the a-principal axis of CH3 OO, as shown in Figure 3. Rotating Eq 11 into the ρ-axis system, (z,x,y), the Hamiltonian is separated into two parts,

′ ′ Htor,rot = Htor + Hrot

(12)

′ Htor = Btor (ptor − ρNz )2 + V3 [1 − cos(3τ )]

(13)

where

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and

′ Hrot = Aeff Nz2 + Beff Nx2 + Ceff Ny2 + Deff (Nz Nx + Nx Nz )

(14)

Here, ρ is the magnitude of ρ~. The effective rotational constants in Eq 14 are

Aeff = A cos2 θ + B sin2 θ

(15)

Beff = A sin2 θ + B cos2 θ

(16)

Ceff = C

(17)

Deff = (B − A) sin θ cos θ

(18)

in terms of the same angle θ. The Deff rotational constant represents the off-diagonal contribution to the inertial tensor about the x and z axes of the molecule in the ρ-axis system. ′ remains an asymmetric top rotational Hamiltonian; however, it is now expressed in the Hrot

ρ-axis system rather than the principal axis system. ′ The nth set of three nearly degenerate eigenvalues of Htor in Eq 13 can be expressed

as 41,46 (K) En,σ

=

X

Btor anj

cos

j

2πj(ρK − σ) 3

(19)

As the expression implies, the torsion splitting depends on the value of K . (K)

In the analysis that follows, we truncate the expansion of En,σ at j = 1. This choice is based on earlier studies of CH3 OO, which showed that Btor an1 is at least an order of magnitude larger than the j = 2 term. 16 Within this approximation, Btor an1 can be obtained from the splitting between the σ states of the K = 0 torsion levels, with

Btor an1

2 (K=0) (K=0) = En,σ=0 − En,σ=±1 3

(20)

where the torsional energies are obtained from the vibrational calculation previously de13



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scribed. The resulting values of an1 for both ground state and the ν2 excited state in the CH stretch are used to simulate the rotational contours. The values of Btor an1 , calculated at several levels of electronic structure calculations are provided in Table S1 of the supporting information.

Theoretical Results Evaluation of CH Stretch/Torsion Vibrational Levels Based on the above discussion, to obtain the vibrational eigenvalues and eigenfunctions we first evaluate the effects of couplings of the torsion and stretch vibrations; thereafter we will investigate the effects of torsional and rotational coupling. For the first step, the potential energy surface was calculated at a variety of levels of theory, and was fit to obtain the parameters in Eqs 3, 5 and 8. To obtain the parameters in Hstr and Hcpl , we performed one dimensional scans of the B3LYP potential energy surface along the torsional angle τ in increments of 5◦ . At each point along the torsional scan, potential energy scans over the three CH bond lengths were evaluated with all of the other internal coordinates constrained to their equilibrium values. Then the force constants of the CH stretch at different values of τ were determined by finite difference, and the calculated values for symmetry equivalent structures were averaged for subsequent fits. For the other electronic structure calculations, this was achieved by using the Hessian, which was evaluated as a function of τ , to extract the quadratic force constants for the three CH stretch vibrations. The results are reported in Tables 2 and S2. Because the higher order terms in the expansions do not have a significant effect on the spectrum, these terms are only evaluated from the B3LYP calculations. For this reason, only the harmonic parameters are reported for the other electronic structure calculations in Table 2. The reported values of these parameters have been truncated to facilitate comparisons. The complete values of the parameters from the B3LYP calculation that were used in the spectral calculations (including the anharmonic terms) are summarized 14


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in Table S2 of the supporting information. In the discussion below, we focus on the calculated energy levels and spectrum that are obtained using the parameters from the the B3LYP calculations. Results based on parameters from other levels of electronic structure theory are shown in Figures S1 and S2 of the supporting information. Comparison of the results for different levels of electronic structure theory provides information about the sensitivity of the reported results to variations in the values of these parameters that are consistent with electronic structure calculations. For the torsional Hamiltonian Htor in the present study, converting from atomic units in Table S2, we obtained V3 = 133.2 cm−1 . Following the study of Just et al., 16 Btor is evaluated for the equilibrium structure of CH3 OO and scaled by 0.97, using the procedure described in Ref. 41 thereby yielding a value of 6.93 cm−1 . The eigenvalues and eigenfunctions of Htor are obtained using discrete variable representation, 47 where τ ranged from −180◦ to 180◦ in increments of 3.6◦ . The 9 lowest energy states (vtor ≤ 2) are below this barrier while the remaining ones have energies that are above the barrier. As the energy increases, the states transform from being nearly triply degenerate, with a small tunneling splitting, to pairs of degenerate states characteristic of a one dimensional free rotor. The eigenfunctions show the expected trend, transforming from being localized in the wells to delocalized over the full range of τ as would be expected for a free rotor. The values of the torsional energies, obtained by calculating the eigenvalues of Eq 3 using these B3LYP parameters, are reported in Table S3. Harmonic oscillator eigenfunctions with up to 7 quanta in each of the local CH stretch modes are used as the basis for calculations of the eigenvalues and eigenvectors of Hstr . For the evaluation of Gil in Eq 7, θH(i) CH(l) = 111.06◦ , which is the average of all of the HCH angles obtained in the scan over the B3LYP surface. The stretch states, reported in Table S4 are characterized by the total number of quanta in the stretch as well as a mode assignment. In CH3 OO, the totally symmetric CH stretch is ν2 , while the anti-symmetric stretch involving H(2) and H(3) in the reference structure (see Figure 3) is ν9 and the remaining CH stretch

15



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Page 16 of 42

vibration (ν1 ) is the out-of-phase combination of the CH stretch involving H(1) and the symmetric stretch of hydrogen atoms (2) and (3). The eigenfunctions that correspond to the energies reported in Tables S3 and S4, along with the ten states that have three quanta distributed among the CH stretches, are used to develop a product basis for the evaluation of the energies and eigenfunctions that are solutions to HT . The eigenfunctions of the H T can be written as a linear combination of the basis functions,

|ΨT i =

1 20 X 6 X X

l Cn,σ |li|n, σi

(21)

l=1 n=0 σ=−1

where |n, σi is the eigenket with torsional quantum number, n, and symmetry, σ. For states with A symmetry, σ is equal to 0 while for states with E symmetry, σ = ± 1. The ket l |li is an eigenfunction of H str with l denoting states of increasing energy. The Cn,σ is the

expansion coefficient for a certain product basis set. The summation limits in Eq 21 lead to convergence of the eigenvalues of H T to better than 0.1 cm−1 . The two-step diagonalization approach allows us to evaluate the effect of terms in H cpl on the energy level pattern. The resulting energies of excited torsion levels built off of the CH ν2 =m stretch ground state and the ν2 = 1 excited state are reported in the columns labeled En,σ

in Table 3. The corresponding results for transitions built off of the ν1 and ν9 fundamentals are reported in Table S5. In these tables, the differences between the reported energies and the sum of the stretch and torsion energies reported in Tables S3 and S4 provide a measure of how the energies of these zero-order states are shifted by introduction of the H cpl term. ν2 =m 2 In Table 3, the value of |Cn,σ | , which provides the leading contribution of a zero-order 2 =m basis function to the Ψνn,σ eigenfunction of H T , illustrates how a given state mixes with

ν2 =m 2 2 =m other states. When |Cn,σ | < 0.9 states that contribute significantly to Ψνn,σ are listed

explicitly.

16


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Sequence Band Structure Based the parameters from the B3LYP level of theory, calculated eigenvalues of H T are reported in Table 3. A stick diagram of the vibrational spectrum is presented in Figure 4 along with the experimental spectrum for comparison. As can be seen in the figure, there are several torsional sequence bands on the blue side of the origin and others are on the red side of the origin. This is similar to what is observed in the experimental spectrum, both in terms of the direction and the magnitude of these shifts. To aid in the discussion that follows, we will characterize the size and direction of the shifts of the sequence bands by

(ν =1) (ν =1) (ν2 =0) (ν2 =0) 2 2 Edif,n = E n − E0 − En − E0

(22)

where the superscript indicates the stretch state, while the n subscript provides the number of quanta in the torsion, with

(ν2 =m) En

1 1 X (ν2 =m) E = 3 σ=−1 n,σ

(23)

In the equations above, E is the average energy of the symmetry components of the nth (ν2 =1)

torsional state and E n

(ν2 =0)

− En

represents the frequency of the 210 12nn transition. Values

of Edif,n obtained from the calculations described above are provided in the right-most column of Table S1 of the supporting information. As indicated, these range from +2.6 cm−1 for n = 2 to −1.0 cm−1 for n = 4. Values of Edif,n based on other electronic structure calculations are also provided in Table S1. The basis of the rather unusual combination of the blue and red shifts of the sequence bands can be rationalized using second-order perturbation theory. Specifically H cpl is treated as a perturbation to the uncoupled Hamiltonian, H unc = H str + H tor . In the uncoupled representation all the of the torsional sequence transitions have the same frequencies as the ν2 fundamental. In other words Edif,n = 0 for all values of n. We consider the first and second order corrections separately. 17



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The first order corrections to each of the energies are given by (ν2 =m),(1) En,σ =

(0) Ψ(0) mν2 ,n,σ |Hcpl |Ψmν2 ,n,σ

3 3 X 3 X X

(24)

2(i − 1)π n, σ cos j τ − = n, σ + 3 j=1 i=1 k=2 i−1 3 X 3 X X 2(5 − i − l)π (2) n, σ Fil,j hm|∆ri ∆rl |mi n, σ cos j τ − 3 j=1 i=1 l=1 (k) Fi,j

m|∆rik |m

where m represents the number of quanta in the ν2 stretch, respectively. In the harmonic (2)

limit, the contribution from the Fil,j term vanishes as h∆ri i = 0. Likewise, in this same (3)

limit, the Fi,j term vanishes because h∆ri3 i is also zero. Based on this, both of these terms are expected to be small for these weakly anharmonic CH stretch states. The torsion (k)

contribution to the Fi,j term is composed of two parts. The hm|∆ri2 |mi components for i = 1, 2 and 3 are the same because ν2 is the totally symmetric stretch. Considering this, the P3 i=1 hn, σ| cos(j(τ + δ))|n, σi integral vanishes unless j is an integer multiple of three, while the constant contribution will not affect the value of Edif,n . This leads to the conclusion that the first order contribution to Edif,n can be approximated by

(1) Edif,n

≈

1 X

(2)

Fi,3

σ=−1

m = 1|∆ri2 |m = 1 − m = 0|∆ri2 |m = 0

(25)

× [hn, σ| cos(3τ )|n, σi − hn = 0, σ| cos(3τ )|n = 0, σi] We find that hm|∆ri2 |mi is larger for m = 1, the ν2 CH stretch fundamental, than when P m = 0. Further, the 1σ=−1 hn, σ| cos(3τ )|n, σi term decreases as n increases. This leads to (2)

the conclusion that when Fi,3 > 0, as all of the electronic structure calculations performed (1)

in this study indicate, Edif,n < 0. As a result, based only on the first order corrections to the energy, the torsional sequence bands are expected to be red-shifted with respect to the origin transition, as is typically observed spectroscopically. The size of the contributions from second-order perturbation depends on the proximity of other states to the state of interest. As the ground state is well isolated from other 18


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vibrational states, the effects of second-order perturbation theory can be neglected for this state. However, we need to consider second-order shifts in the energies of states with one quantum of excitation in ν2 . Figure 5 shows the relative energies of the torsional excited states built off of the three CH stretch fundamentals. As we can see, the ν2 + nνtor levels are close in energy to the ν9 + (n − 1)νtor and ν1 + (n − 1)νtor levels. Couplings among these nearly degenerate states will lead to additional shifts in the energies of the torsion excited states that are built off of the ν2 fundamental. Such mixing is noted in the analysis of the vibrational wave functions reported in Table 3 for the ν2 fundamental and in Table S5 for the fundamentals in ν1 and ν9 . The value of Edif,n contains contributions from both the first and second order corrections to the energies of the states involved in the transition. The shift in the components of the ν2 +νtor state is illustrated in Figure 6. The reason why the 210 1211 transition is to the blue side of the origin is because couplings between the ν2 + νtor state and the ν9 and ν1 fundamentals raise the energy of this combination band. The size of this shift exceeds the lowering of this energy from first order perturbation theory, resulting in a net blue shift. For states with one quantum of excitation in ν2 and three or four quanta in the torsion, the contributions to the energy shift from the second order corrections are smaller than the contributions from the first order corrections, and these bands are shifted to the red.

Limitations of the Model In order to understand the effect of the vibrational parameters in HT of Eq 2 on the spectrum, MP2, M062X, M052X as well as B3LYP levels of theory with aug-cc-pVTZ basis set are used to generate these parameters, which are listed in the Table 2. Since the calculations are based on a reduced dimensional treatment, contributions from vibrational modes other than three CH stretches and torsion are not considered. In particular, the model does not include possible Fermi resonances with HCH bend overtones. To

19



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Page 20 of 42

obtain acceptable agreement between the experimental and calculated spectra, the quadratic (2)

(2)

force constants and the bilinear force constants, F i,0 and F il,0 , in Hstr are adjusted so that the ν2 fundamental frequency and the average value of the frequencies of the ν9 and ν1 fun(2)

damentals reported in Ref. 28 match the experimental values. The revised values of F i,0 (2)

and F il,0 are provided in footnote b of Table 2. The large adjustment to the bilinear force constant in H str was required to decrease the energy difference between the anharmonic frequencies of the ν2 and ν9 fundamentals. The magnitude of this adjustment is likely due to neglect of the couplings between the CH stretch fundamentals and other degrees of freedom in the molecule. For instance, Fermi resonance between the ν2 and the overtones of HCH bending modes shifts the ν2 fundamental by about 30 cm−1 to the blue, according to the VPT2 analysis performed using the Gaussian 09 package. 37 Spectra calculated using the adjusted quadratic parameters in Hcpl from B3LYP and MP2 calculations are provided in Figures S1 and S2. The higher order terms in the expansion in stretch displacements are calculated at the B3LYP level of theory, and their values are provided in Table S2. These calculated spectra can be compared to the spectrum that was calculated using the unadjusted B3LYP force constants, shown in Figure 4. These results illustrate a strong sensitivity of the blue or red shift of the sequence band transitions, relative to the ν2 origin transition, to the values of the parameters that are used. Based on this, rather than trying to adjust parameters in HT to obtain the best fit to experiment, we use the calculated relative intensities and shift the position of the sequence band contours within the ranges of values of the Edif,n obtained by using the various sets of parameters in Table 2 to obtain the best agreement between the calculated and experimental spectra for CH3 OO.

20


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Simulation of ν2 Spectrum with HT We are now in a position to perform a more realistic simulation of the observed spectrum. ′ n′′ The eigenvalues of Htor are evaluated using Eq 19, with Btor an′ 1 = Btor a1 , where the ground

state Btor an′′ 1 values were evaluated using the calculated torsion splitting obtained with the adjusted parameters evaluated at the B3LYP/aug-cc-pVTZ level of theory and basis set. We ′ solve Eq 14 to obtain the rotational contours for each eigenvalue of Htor . The parameters in ′ Hrot are determined using Eqs 15–18. The values of the parameters used in the simulation

are summarized in Table 4. The SpecView package 48 is used to calculate the rotational contour of each of the 210 12nn band (n = 0 to 4). The relative positions of the band origins, i.e. the values of the four Edif,n parameters, are varied according to the procedure described above, and the resulting best fit values are reported in Table 4, with the intensity of each band being weighted by a Boltzmann distribution at 298 K. These results are displayed in Figure 7. This figure provides the rotational structure associated with the vibrational band origins shown in Figure 4 with several important adjustments. In particular, the torsional splittings have been recalculated using the adjusted parameters given in footnote b of Table 2. As noted above, the values of the Edif,n are also adjusted, and the values that provide the best fit to the experimental spectrum are used to determine relative positions of the sequence band origins. Figure 8 shows a comparison between the experimental spectrum and the sum of the traces in Figure 7. Because of the relative weakness of the R and P branches, the appearance of the P and R branches in the sum is little altered from those of the 210 band, shown in Figure 2. However the Q branch clearly is broadened. Given the assumptions that were made, the agreement between the experimental and simulated spectra is clearly acceptable. Obviously this simulation is much better than that in Figure 2, and the Edif,n values are within the range predicted by the various electronic structure calculations provided in Table S1.

21



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Conclusion The experimental spectrum of the methyl peroxy radical has been obtained at room temperature with partial rotational resolution using a step-scan Fourier-transform spectrometer. Analysis of the rotational structure shows that it is not consistent with a simple 1-0 vibrational transition in the ground electronic state of CH3 OO, with the largest discrepancy occurring in the Q-branch region of the spectrum. Including the sequence bands in the torsional mode that should be populated at room temperature improves the agreement, but the simulation is still not sufficient the couplings between the torsion and other degrees of freedom are neglected. A satisfactory simulation can be obtained by including rotational/torsional coupling and using a four-dimensional model consisting of the torsion and three CH stretching modes with coupling among them. The values of the shifts of the band origins of the sequence bands, Edif,n , determined by fitting the spectrum to the model, can be compared to their values obtained from various levels of electronic structure theory. There is a good deal of variation among the parameters obtained from these calculations, and the physical significance of the empirical parameters determined from the fit is limited by the model. Nonetheless their consistency provides an adequate explanation of the observed spectrum, and its assignment to CH3 OO.

Acknowledgement Support from the Chemistry Division of the National Science Foundation (A.B.M.: CHE1213347 and CHE-1465001), the Ministry of Science and Technology, Taiwan (Y.-P.L.: MOST104-2745-M-009-001-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-FG-02-01ER14172). 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

22


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for High-performance Computing provided computer time to Y.-P.L.

Supporting Information Available Parameters in the rotational contour simulation calculated from different levels of electronic structure theory; Parameters in the total Hamiltonian HT calculated at B3LYP level of electronic structure theory; Eigenvalues of the torsional Hamiltonian, Htor ; Eigenvalues of the CH stretch Hamiltonian, Hstr ; Eigenvalues of HT for the ν9 = 1 and ν1 = 1 excited states; Calculated stick spectrum using the scaled B3LYP parameters; Calculated stick spectrum using the scaled MP2 parameters. This material is available free of charge via the Internet at http://pubs.acs.org/.

23



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(45) Wilson, E. B.; Decius, J. C.; Cross, P. C. Molecular Vibrations; Dover: New York, 1955. (46) Lin, C. C.; Swalen, J. D. Internal Rotation and Microwave Spectroscopy. Ref. Mod. Phys. 1959, 31, 841–892. (47) Colbert, D. T.; Miller, W. H. A Novel Discrete Variable Representation for Quantum Mechanical Reactive Scattering via the S-matrix Kohn Method. J. Chem. Phys. 1992, 96, 1982–1991. (48) Stakhursky, V. L. Vibronic Structure and Rotatinal Spectra of Radicals in Degenerate Electronic State. Case of CH3 O and Asymmetrically Deuterated Isotopomers (CHD2 O and CH2 DO). Ph.D. thesis, The Ohio State University, 2005.

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

A′′a B′′a C′′a A′b B′b C′b a:b:c c a

b

c

Constants(cm−1 ) 1.730 0.379 0.330 1.726 0.379 0.330 0.91:0.09:0

Ground state rotational constants are from microwave spectroscopy. 13,26 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. 37 a:b:c provides the calculatedratio of the squares of transition dipole moment components for the a-type, b-type and c-type transitions in the molecule , evaluated at the B3LYP/augcc-pVTZ level of theory and basis set.

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Table 2: Harmonic Parameters in HT Calculated at Different Levels of Electronic Structure Theory With an aug-cc-pVTZ Basis Constantsa (2) F i,0 (2) F i,1 (2) F i,2 (2) F i,3 (2) F il,0 (2) F il,1 (2) F il,2 (2) F il,3 V3 a

b

MP2 0.17884 -0.00060 0.00138 0.00016 0.00086 -0.00022 0.00052 0.00001 0.000530

M062X 0.17514 -0.00123 0.00117 0.00021 0.00194 -0.00034 0.00033 -0.00008 0.000668 (2)

M052X B3LYP 0.17823 0.17151b -0.00191 -0.00083 0.00034 0.00219 0.00019 0.00012 0.00260 0.00249b -0.00019 -0.00018 0.00047 0.00058 -0.00002 0.00005 0.000630 0.000608 (2)

All the force constants, F il,j and F i,j , are in units of Hartree/Bohr2 ; V3 is reported in Hartrees. (2) (2) Adjusted values of F i,0 and F il,0 are 0.16799 and 0.00373 (see text). These values are used in calculating the spectra shown in Figures S1 and S2.

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Table 3: The Eigenvalues of the Total Hamiltonian, HT , for the Ground and ν2 = 1 Excited States

n, σ 0,0 0,±1 1,±1 1,0 2,0 2,±1 3,±1 3,0 4,0 4,±1 5 6 a b c d

ground state (ν =0) ∆Ecpl b |C n,σ2 |2,c 1.2 1.00 1.2 1.00 0.3 1.00 0.5 1.00 -0.6 1.00 -0.3 0.99 -0.5 1.00 -0.2 1.00 -0.4 1.00 -0.2 0.98 -0.2 1.00 -0.2 1.00

(ν =0) E n,σ2 a

(ν =1) E n,σ2 a

0.0 0.2 108.1 112.7 180.9 202.2 260.9 329.0 332.8 418.1 520.3 636.9

2978.3 2978.5 3087.3 3091.8 3154.3 3185.0 3234.8 3308.0 3309.7 3394.3 3499.5 3615.4

∆Ecpl 1.6 1.7 1.6 1.8 -5.0 4.7 -4.4 0.9 -1.3 -2.0 1.1 0.5

b

ν2 = (ν2 =1) 2,c |C n,σ | 0.99 0.99 0.93 0.95 0.86 0.58 0.83 0.95 0.90 0.68 0.91 0.98

1 other contributions

d

(ν1 /ν9 ,1,±1); (ν1 /ν9 ,1,±1) (ν1 /ν9 ,1,±1); (ν1 /ν9 ,1,±1) (ν1 /ν9 ,2,0)

(ν1 /ν9 ,3,0); (ν1 /ν9 ,4,0)

Energies (in cm−1 ) calculated using the parameters obtained at the B3LYP/aug-cc-pVTZ level of theory and basis set, and reported relative to the zero-point energy of 4716.8 cm−1 . (ν =m) ∆Ecpl = E n,σ2 − En,σ − Eν2 =m . (ν2 =m) 2 |C n,σ | provides the contribution of the corresponding basis state to the eigenfunction of HT . (ν =1) (Stretch, n, σ) states that also contribute to the eigenstate when |C n,σ2 |2 < 0.9.

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Table 4: Parameters Used in the Simulation of the Experimental Spectrum Shown in Figures 7 and 8. Parametera Value ′′ b Aeff 1.706 cm−1 ′′ b Beff 0.403 cm−1 ′′ b Ceff 0.330 cm−1 ′′ b Deff −0.177 cm−1 ρ′′b 0.299 ′′b θ 7.64◦ x :y:z c,d 0.03:0:0.97 d Btor a0′′ −0.1 cm−1 1 d Btor a1′′ 3.1 cm−1 1 d Btor a2′′ −14.3 cm−1 1 d Btor a3′′ 45.5 cm−1 1 d Btor a4′′ −57.0 cm−1 1 Edif,1 e 0.5 cm−1 Edif,2 e −0.9 cm−1 Edif,3 e −1.5 cm−1 Edif,4 e −2.1 cm−1 a

b

c

d

e

Values of all of the parameters except for the intensity ratios and the Edif,n values are reported for the ground state. The corresponding values for the ν2 = 1 state are identical to the ground state values except for the A′eff = 1.702 cm−1 , Calculated using the rotational constants reported in Table 1 . The intensity ratio (square of transition dipole moment components) evaluated in the ρ-axis system. Calculated using the adjusted parameters evaluated at the B3LYP/aug-cc-pVTZ level of theory and basis set. Adjusted to obtain agreement with the experimental spectra.

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Figure 1: Difference absorption spectrum of CH3 OO in the 2910-2990 cm−1 region (resolution 0.15 cm−1 ). (a) Spectra recorded and averaged from three sets of data upon 193-nm photolysis of a flowing mixture of CH3 C(O)CH3 and O2 ; see text. (b) Absorption spectrum of CH3 C(O)CH3 . (c) Absorption spectrum of C2 H6 . (d) Spectrum in (a) with absorption of CH3 C(O)CH3 , C2 H6 and CH4 stripped and baseline corrected. Interfering downward bands of acetone are indicated with arrows, and small remnant lines from C2 H6 are indicated with *. Based on an analysis of the rotational contour, the band origin is at 2954.4 ± 0.1 cm−1 . 34


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Figure 2: The experimental spectrum (red, baseline shifted by an absorbance of 2 × 10−3 ) of the ν2 fundamental of the CH3 OO radical and the simulated spectrum, evaluated using the simple asymmetric rotor model (black).

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Figure 3: 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-axis lying perpendicular to the symmetry plane. θ is the angle between the ρ-vector (see text) and the a-axis. The torsional angle τ is measured from this reference geometry (τ =0), which corresponds to the H(1) COO plane bisecting the H(2) −C−H(3) angle.

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Figure 4: The calculated (black) vibrational band origins for the torsional sequence bands obtained using the constants calculated with B3LYP level of theory, as listed in Table 2. The experimental spectrum (red, offset as in Figure 1) is also shown for comparison.

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Figure 5: Energy level diagram of the calculated eigenvalues of Hunc , which use parameters from calculations at the B3LYP/aug-cc-pVTZ level of theory and basis set. The black, red and blue lines represent the combination bands involving torsion and one quantum of CH stretch excitation in the ν2 , ν9 and ν1 CH stretch vibrations, respectively. The levels are labeled by number of quanta, n, in νtor and symmetry, A for σ = 0 and E for σ = ±1.

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Figure 6: Expanded view of energy level diagram in Figure 5 over the 2960 - 3010 cm−1 region. The eigenvalues of Hunc are indicated by the solid lines on the left side of the diagram. The eigenvalues of the HT are indicated with dashed lines on the right side. Here the purple lines provide the energies of states with predominantly ν1 and ν9 character, while the black lines correspond to the states with predominantly ν2 + nνtor character.

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Figure 7: The simulated spectra of the ν2 fundamental for the CH3 OO radical and various torsional sequence bands based on the results of the coupled torsion-rotation, HT , model Hamiltonian in Eq 2.

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Figure 8: The experimental spectrum (red, shifted as in Figure 1) of the ν2 fundamental region of the CH3 OO radical and the simulated spectrum (black), which is the sum of the torsional sequence band simulations shown in Figure 7.

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Manifestations of Torsion-CH Stretch Coupling in ... - ACS Publications

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