Article pubs.acs.org/JPCA
Photoionization Cross Sections and Asymmetry Parameters for the Valence Shell of Methanol C. Lavín,* M. V. Vega, and A. M. Velasco Departamento de Química Física y Química Inorgánica, Facultad de Ciencias, Universidad de Valladolid, E-47005 Valladolid, Spain
ABSTRACT: Theoretical cross sections for photoionization of the methanol valence orbitals in covering a region up to 80 eV beyond the first ionization potential are reported. The molecular quantum defect orbital, MQDO, method, which has proved to be reliable in previous applications to molecular photoionization, has been used. To our knowledge, predictions of electronic partial cross section profiles on this molecule are made here for the first time, and we are not aware of any reported experimental data. Partial cross sections for production of parent and fragment ions of methanol have also been calculated and compared with previous measurements. In addition, the MQDO method has been used to calculate the angular distribution of photoelectrons for the valence orbitals of methanol over the 11−50 eV photon energy range. Our results are compared with experimental data, showing a good agreement in most cases. We hope that the present results might be of use in atmospheric and interstellar chemistry, where this molecule plays an important role.
1. INTRODUCTION Molecular methanol, CH3OH, plays an important role in atmospheric and interstellar chemistry. After methane, CH3OH is the second most abundant organic trace gas in the atmosphere,1 where it is a significant source of reactive species such as formaldehyde, carbon monoxide, ozone, and hydrogen radicals.2,3 CH3OH is also one of the most abundant and chemically important molecules in the interstellar medium. It is a useful tracer of the dense interstellar gas phase.4 Observations of Gottlieb et al.5 (1979) show that methanol is an abundant molecule found in most galactic molecular clouds. CH3OH has also been observed in comets6 and high-mass protostars.7 In such environments, ionization processes are relatively common.8 Thus, it is apparent that the modeling of the role of the methanol in the physical and chemical processes occurring in the above environments requires accurate cross sections for total photoionization, partial channel photoionization and fragmentation processes over wide spectral ranges. Over the last several decades, there have been numerous experimental investigations on the photoionization of methanol. Much valuable spectroscopic information on the energy states of the methanol ion has been obtained from photoelectron spectroscopy.9−11 The ionic dissociation of CH3OH has been investigated by using photoion−photoelectron coincidence spectroscopy.12,13 Berkowitz14 has reported relative © 2012 American Chemical Society
photoionization yield curves for the major fragments from methanol. Focusing on cross sections, the object of the present study, earlier measurements of total photoabsorption cross sections were carried out by Ogawa and Cook,15 Person and Nicole,16 and Reilhac and Damany.17 Relative photoionization cross sections for parent and fragment ions of methanol over the photon energy range from threshold up to 14.0 eV have been determined by Refaey and Chupka.18 Absolute oscillator strengths for the molecular and all dissociative photoionization channels of methanol were measured by Burton et al.,19 from the first potential ionization up to 80 eV, using dipole (e, e + ion) spectroscopy. Recently, Douglas and Price8 determined relative partial ionization cross sections for fragment ions, formed by electron ionization of methanol, by using time-offlight mass spectrometry coupled with a two-dimensional ion coincidence technique. However, as far as we know, neither experimental nor theoretical data of partial photoionization cross sections for the production of electronic states of the CH3OH+ ion have been reported to date, despite the fact that tunable synchrotron radiation photoelectron spectroscopy and Received: October 19, 2012 Revised: November 15, 2012 Published: November 15, 2012 11913
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potential, which includes the effects of electron screening, of the form
dipole (e, 2e) coincidence spectroscopy have made possible such measurements in a variety of molecular ions.20 In view of the alluded lack of electronic partial photoionization cross section data for methanol, we felt motivated to perform calculations in order to gain a better insight into the photoionization of this molecule. Consequently, partial photoionization cross sections for the production of the six lowest electronic states of the CH3OH+ molecular ion as a function of the photon energy have been presently determined. In our calculations, we have used the molecular quantum defect orbital (MQDO) method, which has proved to be both reliable and efficient for predicting electronic partial cross sections in different molecular systems.21−25 In order to assess the quality of the MQDO results, we have calculated the partial cross sections for molecular and dissociative photoionization of CH3OH, for which previous data have been reported. Besides the cross section, a key quantity in studies of photoionization is the asymmetry parameter β, which characterizes the angular distribution of photoelectrons. To the best of our knowledge, the only study of the angular distribution of photoelectrons on methanol is that of Keller et al.26 These authors obtained the asymmetry parameter for valence orbitals of CH3OH over the 10−30 eV photon energy range using synchrotron radiation as the excitation source. Recently, the first application27 of the MQDO method to deal with asymmetry parameters for valence molecular orbital photoionization has yielded results that agree fairly well with the comparative data found in the literature. In this work, we have determined the photoelectron asymmetry parameters as a function of photon energy for the ionization of the valence orbitals of the CH3OH molecule. The knowledge of this property is very useful because it contains a wealth of information about molecular electronic structure and dynamics.28,29
V (r ) =
(1)
The parameter λ is related to both the orbital angular moment quantum number, l, and the quantum defect, δ. The MQDO radial continuum and bound-state wave functions are related to Whittaker functions. The angular part of the MQDO functions is expressed as a symmetry-adapted linear combination of the spherical harmonics. This allows us to formulate separately the radial and angular contributions to the transition moment, both as closed-form analytic expressions. Hence, the square of the transition moment between a bound state i and a continuous state f is expressed as M if2 = Q if R if2
(2)
where Qif results from the angular integration and Rif is the radial transition moment. The present calculations for boundcontinuum transitions have all been made in the dipole length approximation. The photoionization cross section and the asymmetry parameter, β, as a function of photon energy are magnitudes of fundamental importance in photoionization studies. The photoionization cross section for a transition between a bound state i and a continuous state f may be expressed, in units of megabarns (Mb), as ⎤1 ⎡ 1 σ = 2.6891N ⎢ + k 2 ⎥ Q if |R if |2 2 ⎦ 2k ⎣ (n − δ )
(3)
where N is the number of equivalent electrons in the molecular orbital from which the ionization originates. In the MQDO method, the partial photoionization cross section, defined as the probability for ionization to a particular electronic state of the ion, is obtained as a sum of all compatible channels leading to that ionic state. In the central potential model approximation, the expression of the asymmetry parameter for the photoionization of an electron with orbital angular quantum number l is given by the Cooper−Zare formula31
2. METHOD OF CALCULATION The MQDO formalism has been described in detail elsewhere.21,30 We shall, thus, only mention here some aspects of the theory that are relevant to the present calculations. The MQDO radial wave functions are the analytical solutions of a one-electron Schrödinger equation that contains a central field β=
λ(λ + 1) − l(l + 1) 1 − 2 r 2r
l(l − 1)R l2− 1 + (l + 1)(l + 2)R l2+ 1 − 6l(l + 1)R l − 1R l + 1 cos(ξl + 1 − ξl − 1) (2l + 1)[lR l2− 1 + (l + 1)R l2+ 1]
where Rl±1 are the radial dipole matrix elements and ξl±1 are the phase shifts of the respective scattered waves. The phase shift ξl is presently calculated as the sum of a Coulomb shift and a nonCoulomb shift. The latter is represented by πδl as proposed by Burgess and Seaton.32 One of the main advantages of the MQDO method is that it leads to closed-form analytical expressions for the transition integrals and, hence, of both the photoionization cross sections and asymmetry parameters, thus avoiding the convergence problems that many ab initio calculations have to face.
(4)
[(1a′)2 (2a′)2 ](3a′)2 (4a′)2 (5a′)2 (1a″)2 (6a′)2 (7a′)2 (2a″)2
where the core orbitals appear in brackets. For the 14-electron valence shell, a distinction can be made between MOs that are built mainly with 2p atomic orbitals, outer-valence MOs, and MOs built with 2s atomic orbital, denoted as inner-valence orbitals. The 5a′, 1a″, 6a′, 7a′, and 2a″ orbitals correspond to the outer-shell orbitals. The two outermost orbitals, 2a″ and 7a′, have been described as being largely nonbonding 2p orbitals of oxygen, while 1a″, 5a′ and 6a′ are combinations of C(2p), O(2p), and H(1s) atomic orbitals.33 In contrast, the 3a′ and 4a′ MOs, constituents of the inner-valence shell, are mainly O(2s) and C(2s) atomic orbitals.34 The one-electron dipole series corresponding to individual excitations of the six outermost molecular orbitals of methanol are (2a″−1 ka′), (2a″−1 ka″), (7a′−1 ka′), (7a′−1 ka″), (6a′−1
3. RESULTS AND DISCUSSION At the ground state equilibrium geometry, the CH3OH molecule belongs to the Cs point group; its electronic configuration can be written as 11914
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Figure 1. MQDO partial photoionization cross sections for production of the six lowest-lying electronic states of the CH3OH+ ion and for the individual channels associated with the final symmetries of each electronic ionic state. Experimental cross sections for the production of the CH3OH+ ion are also shown (see the text for details).
ka′), (6a′−1 ka″), (1a″−1 ka′), (1a″−1 ka″), (5a′−1 ka′), (5a′−1 ka″), (4a′−1 ka′), and (4a′−1 ka″). Given the one-center character of the MQDO wave functions, the Laporte selection rule Δl = ±1 applies to electric dipole transitions, in addition to restrictions imposed by molecular symmetry. This selection rule is approximately valid in transitions to Rydberg molecular states because they possess mainly one-center character. In order to calculate the radial dipole matrix elements with the MQDO method, data of ionization energies and quantum defects of the continuum orbitals involved in the transition are required as input. Vertical ionization energies of 10.96, 12.62, 15.21, 15.64, 17.62, and 22.65 eV for the six outermost orbitals, reported by Robin and Kuebler,9 have been used. It should be mentioned that the 1a″ and 6a′ bands have only been partially resolved even at high resolution. In the Cs point group, s, dz2, dx2−y2, and dxy orbitals possess a′ symmetry, whereas dxz and dyz are orbitals of a″ symmetry. Hence, the ks, kdz2, kdyz, and kdxz Rydberg series constitute different possible photoionization channels from the 2a″ and 1a″ valence orbitals of methanol. On
the other hand, the photoionization channels for the 7a′, 6a′, and 5a′ valence electrons are ks, kdz2, kdx2−y2, kdxy, and kdyz. The most strongly contributing channel to the partial photoionization for the 4a′ inner-valence orbital of CH3OH is the kp Rydberg series. Because no data of the quantum defects for the continuum states of methanol have been found in the literature, we have used δ values deduced from bound Rydberg state energies through the well-known Rydberg formula. This assumption is approximately valid because the quantum defect mainly depends on the angular momentum quantum number and remains nearly constant as the principal quantum number, n, varies in a given Rydberg series. Quantum defects of 1.2 for the s Rydberg series converging to the first and fifth ionization potentials and of 1.1 for the s series converging to the fourth potential ionization are obtained from experimental vertical excitation energies.9,35−37 No energy data are available for the s Rydberg series converging to the second and third ionization potentials. We assume a δ value of 1.2 for the former and 1.1 for the latter on the basis that the quantum 11915
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Table 1. Cross Sections (Mb) for the Molecular and Dissociative Photoionization of CH3OHa photon energy (eV)
CH3OH+
CH3O+
CH2O+
COH+
11.5
13.67 8.44 11.66 10.00 8.59 10.39 12.40 10.75 10.77 11.10 9.53 10.55 8.45 10.37 7.49 10.13 8.67 10.15 9.74 10.14 9.44 10.21 8.98 10.02 8.56 10.05 9.29 9.92 9.03 9.76 8.94 10.03 8.86 10.12 8.56 9.72 8.42 9.41 8.26 8.78 8.11 8.26 7.88 7.69 7.58 6.98 7.22 6.42 6.96 6.04 6.60 5.55 6.29 5.13
1.41 0.87 3.26 2.79 6.03 7.28 14.69 12.74 15.60 16.08 15.89 17.58 14.95 18.35 13.41 18.14 16.69 19.53 18.36 19.10 16.87 18.25 15.08 16.82 14.01 16.46 14.89 15.90 13.90 15.03 13.55 15.18 13.01 14.85 12.40 14.08 11.92 13.31 11.45 12.16 11.11 11.31 10.57 10.32 10.32 9.51 9.61 8.54 9.19 7.97 8.62 7.26 8.24 6.71
0.02 0.01 0.08 0.09 0.38 0.33 0.54 0.56 0.68 0.75 0.65 0.80 0.64 0.86 0.86 1.00 1.00 1.04 1.03 1.11 0.95 1.06 1.00 1.17 1.29 1.38 1.30 1.40 1.31 1.54 1.38 1.57 1.31 1.49 1.22 1.36 1.13 1.20 1.12 1.14 1.05 1.02 0.99 0.91 1.01 0.90 0.95 0.82 0.90 0.76 0.83 0.68
0.02 0.01 0.11 0.11 0.21 0.23 0.60 0.74 1.21 1.64 2.69 3.14 5.06 5.26 6.04 6.54 7.25 8.09 8.01 9.41 10.05 10.73 10.70 11.56 10.52 12.14 10.18 11.62 9.76 11.09 9.21 10.28 9.28 9.86 8.80 8.97 8.60 8.40 8.14 7.50 8.02 7.13 7.69 6.66 7.42 6.25 7.03 5.72
12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 a
CO+
0.01 0.01 0.07 0.07 0.09 0.10 0.07 0.08 0.09 0.10 0.14 0.15 0.14 0.15 0.08 0.13 0.11 0.13 0.16 0.18 0.10 0.11 0.14 0.15 0.12 0.12 0.13 0.13 0.13 0.12 0.18 0.16 0.20 0.18 0.27 0.22 0.34 0.27
OH+
CH3+
CH2+
0.23 0.26 0.20 0.23 0.25 0.27 0.22 0.24 0.19 0.23 0.19 0.21 0.20 0.23 0.23 0.26 0.19 0.20 0.18 0.18 0.16 0.16 0.17 0.16 0.17 0.15 0.18 0.15 0.21 0.17 0.23 0.19
0.03 0.03 0.27 0.30 1.31 1.61 2.56 3.46 4.70 5.51 7.03 7.31 7.16 7.75 7.32 8.17 7.19 8.45 8.05 8.60 7.87 8.51 7.72 8.45 7.01 8.00 6.68 7.58 6.26 6.99 6.14 6.53 5.78 5.89 5.46 5.33 5.29 4.87 4.98 4.42 4.74 4.10 4.52 3.80 4.31 3.51
0.05 0.05 0.03 0.03 0.09 0.11 0.12 0.16 0.21 0.25 0.41 0.43 0.38 0.41 0.41 0.46 0.48 0.56 0.62 0.66 0.66 0.71 0.71 0.68 0.65 0.74 0.62 0.70 0.58 0.65 0.61 0.64 0.56 0.57 0.60 0.58 0.58 0.54 0.60 0.53 0.64 0.55 0.73 0.62 0.72 0.59
CH+
0.04 0.04 0.08 0.07 0.13 0.11 0.15 0.12
C+
0.05 0.04 0.05 0.04
H2+
H+
0.06 0.05 0.06 0.05 0.04 0.03 0.08 0.06
0.09 0.10 0.22 0.24 0.35 0.36 0.42 0.41 0.45 0.42 0.59 0.52 0.65 0.57 0.77 0.65 0.93 0.76
Present results (first entry); dipole (e, e + ion) measurementsb (second entry). bBurton et al.19
been reported in the literature to date. A δ value of 0.7 has been adopted for this series, which has been derived from the energy reported by Robin35 for the first member, n = 3, of the Rydberg series 1a″3p. It should be pointed out that the above quantum
defects are very nearly independent of the originating molecular orbital.9 For the d Rydberg series, we have used a δ value of 0 as deduced from the vertical energy of the 3d Rydberg state of methanol.35 No energy data for the 4a′np Rydberg series have 11916
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al.19 These authors have suggested that the onset observed at about 20 eV in the partial cross section curve for CH3OH+ can be due to an additional contribution from the 4a′ state of the CH3OH+ ion. In addition, we have calculated the cross sections for the production of molecular and fragment ions of methanol. Investigations of the ionic decomposition of methanol by means of photoelectron−photoion coincidence spectroscopy13 show that the major ions resulting from the ionization of methanol in the energy region below 20 eV are CH3OH+, CH3O+, CHO+, and CH3+. In a more extended energy range, up to 80 eV, Burton et al.19 detected also CH2O+, CO+, OH+, CH2+, CH+, C+, H2+, and H+ species from the analysis of the time-of-flight mass spectra of CH3OH. According to Gallagher et al.,20 the total photoionization cross section of a molecule can be partitioned into the electronic partial cross sections or alternatively into the parent and fragment ion cross sections. These partitions can be written as
defects correspond to typical values for s, d, and p Rydberg orbitals in molecules composed of second-row atoms. The MQDO partial photoionization cross sections for the production of the (2a″−1 ) 2A″, (7a′ −1) 2A′, (6a′−1) 2A′, (1a″−1) 2A″, (5a′−1) 2A′, and (4a′−1) 2A′ electronic states of the CH3OH+ ion for photon energies from near threshold to 80 eV are displayed in Figure 1. We have also plotted the profiles for the individual channels associated with the two final-state symmetries from the excitation of each valence orbital in the ground state of methanol. An inspection of Figure 1 reveals that MQDO cross sections for the photoionization from the 2a″, 7a′, 6a′, 1a″, and 5a′ orbitals decrease rapidly with increasing photon energy, whereas cross sections for the 4a′ orbital decrease much more slowly. It is as expected because the transition moment is generally much smaller for s electrons than that for p electrons at the ionization threshold, while it becomes progressively larger compared with p electrons with increasing photon energy.38 It can also be seen that for orbitals of a′ symmetry, the bulk of the photoionization cross section comes from ionization into the ka′ channel, whereas for orbitals of a″ symmetry contributions of ka′ and ka″ channels are similar in magnitude. As mentioned in the Introduction section, no direct experimental measurements of electronic state partial photoionization cross sections for methanol have been reported before. Using dipole (e, e) spectroscopy, Burton et al.19 have carried out measurements of absolute photoabsorption oscillator strengths (df/dE) for methanol. They also determined the photoionization efficiency, ionic photofragmentation branching ratios, and absolute partial photoionization oscillator strengths for the production of the molecular and fragment ions of CH3OH by dipole (e, e + ion) spectroscopy. Partial photoionization cross sections (Mb) may be obtained by multiplying df/dE (eV−1) by a factor 109.75. Because CH3OH+ is derived primarily from the 2a″ orbital,13 dipole (e, e + ion) measurements19 for the production of the CH3OH+ ion are plotted in Figure 1 for comparison. Earlier measurements of Ogawa and Cook15 have also been included in Figure 1. It should be mentioned that their reported absorption coefficients for methanol have been presently converted into photoionization cross sections by using photoionization efficiencies and branching ratios given by Burton et al.19 Inspection of Figure 1 reveals a significant discrepancy between dipole (e, e + ion) measurements and the results of Ogawa and Cook15 from threshold up to a photon energy of ∼20 eV. Recently, Berkowitz33 has carried out a critical review of photoabsorption cross section data on methanol using the sum rules as a guide. In his analysis, he concludes that the cross section data of Burton et al.19 are low for photon energies from the first ionization potential to 21.21 eV. Our results compare rather well with those of Ogawa and Cook15 except in the region close to the threshold. It can be noticed that these authors estimated the reproducibility of their results within 3%, although they claimed that some adjustment of the values could be necessary because they did not correct their pressure readings for possible condensation of the vapor. It can also be seen from Figure 1 that the shape of the MQDO data is in excellent agreement with that from the dipole (e, e + ion) measurements in the range above ∼30 eV and in quite good agreement in the 20−30 eV region. In this latter region, MQDO values are lower than the experimental results. These differences could be attributed to the contribution of other electronic states, besides the 2a″, to the production of methanol cation, as pointed out by Burton et
T σion =
frag j ≅ ∑ σion ∑ σion frag
σTion
(5)
j
σfrag ion
where is the total photoionization cross section, is the partial photoionization cross section for production of an ion of any type, and σjion is the partial photoionization cross section for production of a molecular ion in the electronic state j. Making use of this expression, we have determined the total photoionization cross section of methanol through the six ionization channels represented in Figure 1. In order to obtain the molecular and fragment ion cross sections, we have multiplied the resulting values by the ionic branching ratios previously reported.19 The MQDO partial cross sections for the molecular and dissociative photoionization of methanol so calculated are displayed in Table 1 together with the measurements performed by Burton et al.19 An overview of Table 1 shows that our results are comparable to the experimental ones. Most discrepancies are found for photon energies from 11.5 to 15.5 eV. At this point, it is worth noting the relatively close spacing of the outer valence ionization potentials of methanol that lie in that energy range. This fact together with the low resolution at which dipole (e, e + ion) measurements were performed might explain, in part, the differences between both data sets. Finally, we have calculated the asymmetry parameters for the outer valence shell orbitals of methanol at photon energies up to 50 eV. Angular distribution measurements for this molecule over the 10−30 eV photon energy range have been carried out by Keller et al.26 by using dispersed polarized synchrotron as the radiation source. These authors present their results in graphical form. For comparative purposes, we have digitized the figures corresponding to their asymmetry parameter β data. In Figure 2, the MQDO asymmetry profiles for ejection of photoelectrons from the 2a″ and 7a′ orbitals together with the experimental data are plotted as functions of the incident photon energy. As can be observed in Figure 2, the 2a″ and 7a′ asymmetry parameters have negative values just above the ionization threshold and show a quite rapid increase with the increasing photoelectron energy followed by flat behavior at high photoelectron energies. Figure 2 also shows that the asymmetry parameter curve corresponding to the 7a′ orbital is shifted to lower values as compared with the 2a″ curve. The same trend is followed by the experimental results. Although both orbitals have been described as nonbonding 2pπ orbitals 11917
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Figure 2. Asymmetry parameters for the photoionization of the 2a″ and 7a′ orbitals of CH3OH in its ground state.
of oxygen, the 7a′ orbital contributes partially to binding in the O−H bond according to Cheng et al.39 Our results support such a bonding character of the 7a′ molecular orbital because the effect of bonding is to lower the asymmetry parameters with respect to their values for the nonbonding orbital. The MQDO angular distribution profiles for the 6a′, 1a″, and 5a′ orbitals are shown in Figure 3 and are compared with experimental measurements.26 As mentioned above, the spectrum of methanol shows that the (6a′−1) and (1a″−1) bands overlap substantially. Keller et al.26 determined the β values as a function of photon energy for the envelope composed of both bands. These authors also reported the asymmetry parameters obtained by point-by-point analysis for the 6a′ and 1a″ orbitals. As occurs in the cases of 2a″ and 7a′ orbitals, our calculated β values for photoionization from the 1a″ orbital agree quite well in both trend and magnitude with the experimental results. The calculated asymmetry parameters for the 6a′ and 5a′orbitals are in very good accord with the experimental values for energies higher than 20 eV, whereas they are lower at low photon energies. Overall, the presently calculated asymmetry parameters of five valence ionization processes in methanol can be considered satisfactory, more so if we take into account the uncertainties by which the experimental measurements can be affected. Further experimental measurements of photoelectron angular distribution profiles on methanol would be desirable.
Figure 3. Asymmetry parameters for the photoionization of the 6a′, 1a″, and 5a′ orbitals of CH3OH in its ground state.
CH3OH, for which experimental data are available. The general good agreement between MQDO cross sections for production of parent and fragment ions and the experimental ones can be taken as a proof of the correctness of our electronic partial cross section data. This work also represents the first theoretical calculation of asymmetry parameters of methanol. The reliability of our results has been assessed by a comparison with the experimental data. To our knowledge, values of the angular distribution parameter for methanol at photon energies beyond 30 eV are reported here for the first time. The reasonably good agreement between our results and experimental data at photoelectron energies around 30 eV makes us confident that MQDO predictions at higher energies are correct. Despite its simplicity, the MQDO method has the potential to be very efficient for the estimation of key parameters in studies of molecular photoionization. Thus, its use may serve as an alternative to ab initio procedures, which find difficulties in calculating the continuum wave functions for the final state of the electron. Finally, we hope that the present data might be of help in the interpretation of future experimental measurements and may be useful in atmospheric and interstellar models.
4. CONCLUSIONS Photoionization cross sections for specific ionic states of methanol have been calculated up to a photon energy of 80 eV with the MQDO method. These data are supplied, to our knowledge, for the first time. In the absence of comparative data for electronic partial cross sections, we have calculated cross sections for molecular and dissociative photoionization of 11918
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(31) Cooper, J.; Zare, R. N. Photoelectron Angular Distribution. In Lectures in Theoretical Physics: Atomic Collision Processes; Geltman, S., Mahanthappa, K. T., Brittin, W. E., Eds.; Gordon and Breach: New York, 1969; Vol 11C. (32) Burgess, A.; Seaton, M. J. Mon. Not. R. Astron. Soc. 1960, 120, 121−151. (33) Berkowitz, J. Atomic and Molecular Photoabsorption; Academic Press: London, 2002. (34) Al-Joboury, M. I.; Turner, D. W. J. Chem. Soc. B 1967, 373−376. (35) Robin, M. Higher Excited States of Polyatomic Molecules; Academic Press: New York, 1974, Vol. 1. (36) Tam, W. C.; Brion, C. E. J. Electron Spectrosc. Relat. Phenom. 1974, 3, 263−279. (37) Yoshidone, T.; Kawazumi, H.; Ogawa, T. J. Electron Spectrosc. Relat. Phenom. 1990, 53, 185−192. (38) McMaster, B. N. Theory and Energetics of Mass Spectra. In Mass Spectrometry; Johnstone, R. A. W., Ed.; The Chemical Society: London, 1975; Vol 3. (39) Cheng, B.-M.; Bahou, M.; Chen, W.-C.; Yui, C.; Lee, Y.-P. J. Chem. Phys. 2002, 117, 1633−1640.
AUTHOR INFORMATION
Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work has been supported by the Spanish MICINN (Project No. CTQ2010-17892). M.V.V. acknowledges a research grant from the Universidad de Valladolid.
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