Absorption Oscillator Strengths for Vibronic Transitions of n pπ

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Absorption Oscillator Strengths for Vibronic Transitions of npπ Rydberg Series in NO A. M. Velasco,*,† C. Lavı´n,† E. Bustos,† I. Martı´n,† G. Granucci,‡ and M. Persico‡ Departamento de Quı´mica Fı´sica y Quı´mica Inorga´nica, Facultad de Ciencias, UniVersidad de Valladolid, E-47005 Valladolid, Spain, and Dipartimento di Chimica e Chimica Industriale, UniVersita´ di Pisa, V. Risorgimento 35, I-56126 Pisa, Italy ReceiVed: May 18, 2010; ReVised Manuscript ReceiVed: July 8, 2010

The vibronic intensities for band systems of NO corresponding to transitions with origin in both the X2Π ground and the 3sσ(A2Σ+) Rydberg states, and ending in the npπ Rydberg series with n ) 3-5, have been determined. The description of the Rydberg states has been made with the molecular quantum defect orbital methodology. The Rydberg-valence interaction of the 2Π symmetry states involved in the studied transitions has been analyzed through a vibronic matrix. The present results have been compared with experimental and theoretical data available in the literature. Additionally, predictions for a number of unknown intensities have been made, which may be useful for the interpretation of the spectrum of NO. Introduction The NO molecule plays an important role in the physics and chemistry of the upper atmosphere, in photochemical smog as a major atmospheric pollutant, in human physiology, and also in combustion processes. Moreover, it has been detected in the interstellar medium.1,2 The NO radical constitutes the classical example of Rydberg-valence interactions,3 and consequently it has been extensively studied. However, its spectroscopy is not as well understood as the spectroscopy of other diatomic molecules.4 The emission and absorption spectra of NO were studied by Miescher and co-workers,5,6 and a detailed interpretation of the electronic excited states was given by Jungen.7 In particular, the δ(C2Π-X2Π) band system of NO has been subjected to considerable attention. It was first observed in absorption spectra in the vacuum ultraviolet (VUV) region by Herzberg et al.8 δ bands of NO have been detected in the upperatmospheric nightglow of the Earth9,10 and Venus,11,12 and recently it has also been observed in the atmosphere of Mars.13 The major contribution to the dissociation of NO in the mesosphere and stratosphere is the predissociation of the δ(0,0) and δ(1,0) bands.14 The δ system results from a transition between the ground state X2Π and the first member of the npπ Rydberg series of NO. The upper state of this transition, the 3pπ(C2Π), is homogeneously perturbed by configuration interactions with valence and Rydberg states of 2Π symmetry.6,15 These interactions between 2Π excited states observed in the high-resolution absorption spectra were first described as mutual perturbations between the B2Π valence and the C2Π Rydberg states.6,16 Later studies showed that the Rydberg-valence homogeneous perturbations also affect the n ) 4 and n ) 5 members of the npπ Rydberg series and the first two valence states, B2Π and L2Π.15 Apart from these pioneering works, extensive studies of the NO 2Π valence-Rydberg interaction have also been reported.17,18 Another less studied but rather important band system of NO, lying in the near-infrared, is the C2Π-A2Σ+, which is also known as the Heath band system. Besides the δ(0,V′′) progres* To whom correspondence should be addressed. E-mail: amvelasco@ qf.uva.es. † Universidad de Valladolid. ‡ Uiversita` di Pisa.

sion, the C2Π(0)-A2Σ+(0) band is an important deactivating channel for the electronically excited molecule.19 This band has been observed experimentally by several authors, and it has been detected, for the first time, in the upper atmosphere of Venus at 1.224 µm very recently.19 Other than the (0,0) band of the C2Π-A2Σ+ system, Amiot and Verges20 have also observed the (1,1) band in a high-resolution spectrum obtained with a Connes-type Fourier transform interferometer. The Heath system is basically a Rydberg-Rydberg transition, originating in the lowest excited state of NO, the 3sσ(A2Σ+), which is of Rydberg character, and ending in the perturbed 3pπ(C2Π) Rydberg state. As regards absorption oscillator strengths, or f values, of the δ band system, most of the investigations have been focused on the (0,0) band. A summary of the experimental oscillator strength data of the δ band system reported in the literature before 2006 can be found in the paper of Yoshino et al.21 These authors derived optical oscillator strengths for the δ(V′,0) bands with V′ ) 0-3 from their cross-section measurements carried out with a combination of vacuum-ultraviolet Fourier transform spectrometry (FTS) and synchrotron radiation. Very recently, Kato et al.22 have also derived optical oscillator strengths for the δ(V′,0) bands with V′ ) 0-5, using electron energy loss spectroscopy. From a theoretical point of view, only two works have reported vibronic intensities for the δ system. Galluser and Dressler17 have calculated oscillator strengths for the δ(V′,0) bands with V′ ) 0-8 in a basis of electronically coupled diabatic states. Vivie and Peyerimhoff23 have studied the progression (0,V′′) of the δ system, using a multireference configuration interaction method. Regarding the Heath band system, absorption oscillator strengths for the C2Π-A2Σ+ transition have been derived from shock tube experiments24 and from infrared emissions detected by a dry ice cooled PbS cell.25 As reported above, most of the work of both experimentalists and theoreticians has focused on transitions to Rydberg states with n ) 3. Transitions to high-lying electronic states have been less investigated, as well as those for bands with V′′ > 0. This situation calls for further studies; thus, for a better understanding of the spectrum of NO, we have calculated vibronic intensities, expressed in the form of oscillator strengths, for different bands involving the npπ Rydberg series. These include the abovementioned δ(C2Π-X2Π) and Heath (C2Π-A2Σ+) band systems,

10.1021/jp1045113  2010 American Chemical Society Published on Web 07/23/2010

Vibronic Transitions of npπ Rydberg Series in NO

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as well as the K2Π-X2Π and Q2Π-X2Π transitions, which occur in the UV, and other transitions between Rydberg states, the K2Π-A2Σ+ and the Q2Π-A2Σ+ systems, that happen in the visible region. The K2Π and Q2Π states correspond to the n ) 4 and n ) 5 members of the npπ Rydberg series, respectively. It should be mentioned that some bands of these last two Rydberg-Rydberg transitions, in particular, the (2,1) band of the K2Π-A2Σ+ system and the (0,1) and (1,1) bands of the Q2Π-A2Σ+ transition, have been experimentally observed using optical-optical double resonance-multiphoton ionization spectroscopy.26,27 We have undertaken the present work in connection with a previous study in which we applied an extension of the molecular quantum defect orbital (MQDO) methodology, which takes into account the Rydberg-valence interaction, to the description of the β band system of the NO.28 Moreover, in an earlier paper we have reported oscillator strengths for some δ bands of NO, in particular those corresponding to V′ ) 0-2 and V′′ ) 0-5. However, in that work we did not take into account the perturbation of the 3pπ(C 2Π) Rydberg state. 29 Thus, it is also the aim of the present work to revise these results. To our knowledge, the values of transition probabilities for some bands of the δ and Heath systems, as well as for most of the studied bands corresponding to transitions involving npπ high-lying Rydberg states, are first reported here. It should be mentioned that the studied spectroscopic properties of NO are of fundamental importance in several areas of research, such as trace gas analysis, combustion, and atmospheric and astrophysical chemistry. In this context, Kato et al.22 have recently claimed that optical oscillator strengths are one of the many essential inputs for a quantitative description of the atmospheric chemistry. Method of Calculation In a diatomic molecule, the oscillator strength for a vibronic transition can be expressed as follows:30

fV'V''

8π2mca20 ) νV'V''qV'V''Re2 3h

(1)

where V′ and V′′ refer to the upper- and lower-state vibrational quantum numbers, respectively, qV′V′′ is the Franck-Condon factor, νV′V′′ is the wavenumber of the band origin in cm-1, and Re is the electronic transition moment, which runs over all degenerate components of the upper state. This relationship is appropriate for transitions between unperturbed vibronic levels. In the present work, we study different bands of NO, in which the vibronic levels are perturbed. Thus, an adequate model that describes correctly the vibronic absorption spectrum requires to take into account the perturbation between the Rydberg and valence states of 2Π symmetry. To this end, we have followed a theoretical model of Rydberg-valence mixing which has been used in a previous work,28 and which has demonstrated to be a useful tool to estimate transition intensities for mixed states. We described this computational procedure in detail,28 so we give here just a brief summary. The wave functions of the perturbed states are the appropriate linear combinations |e0, ν0〉 of the diabatic unperturbed vibronic wave functions |e, νe〉:

|e0, V0〉 )

∑ ∑ |e, Ve〉CV ,V e

Ve

e 0

(2)

where Ve is the vibrational level of the electronic state e. The perturbation coefficients are obtained by diagonalization of the interaction matrix. So the vibronic transition moment matrix element from a generic nonperturbed state |X, 0〉 to the mixed |e0, ν0〉 state, can be written as follows:

〈MV0,0〉 )

∑ ∑ C*V ,V ReSVe,X,0 e

0 e

Ve

(3)

e

Consequently, the expression for the vibronic oscillator strength when the states involved in the transition are perturbed is

fV'V'' )

8π2mca02 νV'V''[ 3h

∑ ∑ C*V ,V ReSVe,X,0]2 e

Ve

0 e

e

(4)

The vibrational overlap integrals, Sνe,e,0X, were computed using the vibrational wave functions obtained by solving the Schro¨dinger equation with the Numerov algorithm, from the potential energy surfaces evaluated with a Rydberg Klein Ress (RKR) potential. The electronic transition moment involving Rydberg states, Re, has been determined with the MQDO methodology, described in detail elsewhere.31 The MQDO method, based on a model potential, is appropriate in dealing with molecular Rydberg states, because these possess a quasihydrogenic character with the spherical core replaced by a molecular cation which imposes internal crystal field splitting according to its symmetry. In this method, the radial parts of the molecular quantum defect orbitals are the analytical solutions of a oneelectron Schro¨dinger equation that contains a parametric potential. The transition moment can be factorized into radial and angular parts, so the square of the transition moment between two electronic states is expressed as

Re2 ) Q{iff}Rif(r)2

(5)

where Q{iff}, referred to as the angular factor, results from the integration of the angular parts of the initial and final molecular orbitals involved in the transitions, and that of the transition operator. The radial transition moments, Rif, within the MQDO approach, result in closed form analytical expressions, independently of the nature of the molecular system under study. Results and Discussion NO is a stable radical with an open-shell structure which gives rise to either Rydberg or valence states. The ground-state electronic configuration, in the C∞ν symmetry, may be expressed as follows:

...(4σ)2(1π)4(5σ)2(2π)1X2Π where the outermost molecular orbital, 2π, closely resembles a “united atom” 3dπ orbital with a smaller 2pπ contribution.7 As mentioned above, the npπ Rydberg states are subject to homogeneous interaction with valence states. In particular, we have studied the interactions between the B2Π(ν)0-37) and L2Π(ν)0-11) valence and 3pπ(C2Π)(ν)0-9), 4pπ(K2Π)(ν)0-4), and 5pπ(Q2Π)(ν)0-3) Rydberg states. The Rydberg-valence interactions of the 2Π vibronic levels involved in the studied transitions are

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TABLE 1: Main Components of Wave Functions for the npπ Vibronic States (Coefficients g |0.10|) ev C0 C1 C2 C3 C4 C5 C6 C7 K0 K1 K2 K3 K4 Q0 Q1 Q2 Q3

0.18 (B 7) -0.10 (B 9) 0.65 (B 12) -0.12 (B 13) 0.14 (B 17) 0.12 (B 19) -0.10 (B 23) -0.10 (B 26) 0.22 (B 21) -0.17 (B 24) -0.13 (B 28) 0.14 (B 34) 0.99 (K 4) 0.21 (B 28) -0.12 (B 33) 1.00 (Q 2) 1.00 (Q 3)

0.98 (C 0) 0.12 (B 10) -0.14 (B 13) -0.30 (B 14) -0.24 (B 18) 0.33 (B 20) 0.23 (B 24) -0.20 (B 27) 0.96 (K 0) 0.47 (B 25) 0.65 (B 29) -0.43 (B 35)

0.10 (B 26) -0.73 (K 2) 0.85 (K 3)

0.22 (Q 1)

-0.12 (B 29) -0.35 (B 34)

0.96 (Q 0) 0.39 (B 35)

0.12 (B 36)

0.98 (C 1) 0.73 (C 2) 0.64 (B 15) 0.94 (C 4) -0.48 (B 21) -0.10 (L 3) 0.35 (B 28)

treated through a vibronic interaction matrix, as we have described in the previous section. We have constructed this matrix for the spin component Σ ) 1/2, adopting the interaction energies given by Galluser and Dressler.17 In Table 1 we collect the main eigenvector components of the npπ states in the interaction matrix representation. Contributions less than an absolute value of 0.10 are not included. As can be seen from the table, the interaction between 2Π states is so strong that it globally affects all vibronic levels. For the calculation of the vibrational overlap integrals with the RKR model, we have used the molecular constants given by Engleman et al.32 and by Galluser and Dressler.17 The nonperturbed transition moments have been derived using the MQDO methodology. To this end, we need the ionization energy of the NO molecule and the electronic energies of the Rydberg states as input. We have adopted the ionization energy value of 74 721.7 cm-1, obtained by Reiser et al.33 using zero kinetic energy (ZEKE) spectroscopy. For the excitation energies, we have used the experimental data given by Brunger et al.4 For the B2Π-X2Π, also named as β system, and the L2Π-X2Π system we have adopted the electronic transition moments derived from a multireference configuration interaction method, as reported by Vivie and Peyerimhoff.23 Finally, transitions with the origin in the Rydberg 3sσ(A2Σ+) state and ending in valence states have been considered to have a negligible transition probability. In Tables 2 and 3, the calculated unperturbed and perturbed MQDO oscillator strengths for the δ-band system of NO are collected, together with experimental and theoretical data found in the literature. The experimental band oscillator strengths of Bethke34 were deduced from pressure-broadened absorption spectra and have a maximum error of 10%. Cieslik35 derived the f values by the curve of growth method, and Brzozowski et al.36 derived them using high-frequency deflection technique. Other experimental data included in the table were obtained using synchrotron radiation by Guest and Lee37 with an uncertainty of 20%, and by Mandelman,38 who measured the absolute intensity of recombination emission. Chan et al.39 measured the band oscillator strengths with high-resolution dipole (e,e) spectroscopy, and their estimated uncertainties are 5-10% for strong and partially resolved bands, and about 10-20% for the remaining peaks. The most recent experimental data are those derived by Yoshino et al.21 from ultra-highresolution measurements, and those of Kato et al.22 obtained from measurements of differential cross sections. The estimated uncertainties of the previous measurements are 5% for the values of Yoshino et al.,21 and about 18% for the data of Kato et al.22

0.15 (B 16)

0.67 (C 3)

0.77 (C 5) 0.67 (L 4) -0.13 (L 6)

0.66 (C 6) 0.34 (L 7)

0.14 (C 6)

0.83 (K 1)

0.81 (C 7)

0.82 (Q 1)

TABLE 2: Oscillator Strengths (×103) for W′ ) 0-2, W′′ ) 0-10 Vibronic Transitions Belonging to the δ(C2Π-X2Π) Band System of NO V′′

MQDOa

MQDOb

0

V′ ) 0 2.59

2.54

1 2 3 4 5 6 7 8 9 10

4.18 3.88 2.73 1.61 0.836 0.397 0.175 0.0726 0.0284 0.0105 V′ ) 1 6.38

4.36 3.48 2.63 1.63 0.739 0.389 0.190 0.0550 0.0259 0.0163

1.92 0.0104 0.879 1.77 1.94 1.38 0.831 0.507 0.244 0.0882

0

2.20 0.00253 0.887 1.9 2.00 1.52 0.953 0.522 0.259 0.118 V′ ) 2 6.24

1 2 3 4 5 6 7 8 9 10

0.313 2.83 1.37 0.0251 0.449 1.24 1.50 1.27 0.870 0.513

0.00972 1.78 1.16 0.00000 0.370 0.527 0.751 0.905 0.508 0.200

0 1 2 3 4 5 6 7 8 9 10

7.15

2.69

expt

theork

theorl

3.32,c 2.34,d 2.29,e 3.52,f 2.14,g 2.84,h 2.04,i 2.2j

2.20

2.49

5.68 6.36 4.63 2.51 1.19 0.453

6.84,c 5.85,d 6.01,e 8.53,f 5.78,g 5.63h

6.1

4.37,c 2.79,d 3.08,e 2.94,f 2.74g

2.59

a Unperturbed MQDO. b Perturbed MQDO. c Kato et al.22 Yoshino et al.21 e Chan et al.39 f Guest and Lee.37 g Bethke.34 h Cieslik.35 i Brzozowski et al.36 j Mandelman et al.38 k Galluser and Dressler.17 l Vivie and Peyerimhoff.23 d

Vibronic Transitions of npπ Rydberg Series in NO

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TABLE 3: Oscillator Strengths (×103) for W′ ) 3-7, W′′ ) 0-10 Vibronic Transitions Belonging to the δ(C2Π-X2Π) Band System of NO V′′ 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 a

MQDOa V′ ) 3 3.10 4.66 0.690 0.895 1.97 0.744 0.00111 0.407 1.02 1.23 1.06 V′ ) 5 0.121 2.23 5.66 0.191 2.52 0.315 0.470 1.32 0.760 0.0664 0.117 V′ ) 7 0.00026 0.0362 0.894 4.84 3.55 0.633 1.20 1.44 0.0396 0.504 1.00

MQDOb

expt

theorf

1.21 1.76 0.502 0.483 1.40 0.292 0.0329 0.232 0.418 0.682 0.703

0.930,c 1.20,d 0.976e

0.98

0.0429 1.26 2.87 0.0620 1.36 0.330 0.234 0.987 0.592 0.0260 0.129

0.120c

0.02

MQDOa V′ ) 4 0.835 5.30 1.70 2.17 0.0102 1.30 1.33 0.312 0.0227 0.470 0.932 V′ ) 6 0.00869 0.426 3.68 4.81 0.0760 1.98 0.970 0.0445 0.941 1.01 0.309

MQDOb

expt

theorf

0.742 4.65 1.26 1.83 0.00066 0.973 1.25 0.347 0.0167 0.409 0.725

0.190c

0.540

0.0583 0.117 1.54 2.58 0.00323 0.860 0.324 0.0525 0.469 0.400 0.021

0.09

0.0140 0.0164 0.582 2.09 1.82 0.440 0.695 0.982 0.0640 0.241 0.562

Unperturbed MQDO. b Perturbed MQDO. c Kato et al.22 d Yoshino et al.21 e Chan et al.39 f Galluser and Dressler.17

Regarding the theoretical data available, we have included, for comparative purposes, those calculated by Vivie and Peyerimhoff,23 and the perturbed band oscillator strengths of Galluser and Dressler.17 It should be mentioned that the unperturbed MQDO oscillator strengths for the δ bands with V′ ) 0-3 and V′′ ) 0-5 do not exactly agree with our previously published values.29 This fact is due to the differences in the Franck-Condon factors (FFC) used in both calculations. In our previous work, we used the FFC calculated by Ory et al.40 from a Morse potential, while in the present study we have determined them by using the RKR approximation. An inspection of Table 2 reveals that the MQDO values obtained with and without perturbed electronic moments are similar for the V′ ) 0 and V′ ) 1 progressions. This is because these levels are only weakly perturbed by the ν ) 7,9,10 levels of the valence state B2Π. In addition, the vibrational overlap integrals corresponding to the β(0,7), β(0,9), and β(0,10) vibronic transitions are small. It is evident from Table 1 that the mutual perturbation between the C2Π Rydberg and the B2Π valence states increases for V′ ) 2 and V′ ) 3 levels. The C2Π(2) and B2Π(12) vibrational levels perturb each other strongly as well as the C2Π(3) and B2Π(15) levels. A comparison between MQDO oscillator strengths obtained with nonmixed and mixed transition moments, and both experimental and theoretical results, reveals that our results adequately account for the valence-Rydberg mixing for the V′ ) 2 and V′ ) 3 progressions. A consequence of the greater perturbing effect of the valence

state is a weakening of the intensity of the δ(V′)2,3,V′′) bands. MQDO f values confirm earlier predictions of Bethke34 in the sense that the perturbations range from almost negligible for the δ(0,0) and β(7,0) bands to about 50% mixture for the δ(3,0) and β(15,0) bands. For δ(4,0) and δ(5,0) bands, there exist important discrepancies between the f values of Galluser and Dressler17 and those of Kato et al.22 MQDO f values are closer to the calculations of Galluser and Dressler.17 This situation demands further experimental studies of these bands. An overview of Tables 2 and 3 reveals that the reported MQDO perturbed values and the experimental and theoretical data are in good agreement, more so if we take into account the uncertainties of the experimental data. For δ(0,V′′>0) bands, the only comparative data are the calculations of Vivie and Peyerimhoff,23 which are slightly higher in magnitude than our results. We would like to remark that the pertubed MQDO oscillator strengths agree well with the measurements of Yoshino et al.,21 which present minor uncertainties. In Tables 4 and 5 we collect the MQDO band oscillator strengths for the K2Π-X2Π and Q2Π-X2Π systems of NO. The only comparative data available in the literature are the theoretical oscillator strengths calculated by Galluser and Dressler.17 Their perturbed results have also been included in the tables. The main differences between the unperturbed and perturbed MQDO f values for the most intense band in each V′ progression belonging to the K2Π-X2Π system occur for V′ ) 1-3. This fact can be explained by the strong perturbation

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TABLE 4: Oscillator Strengths (×103) for W′ ) 0-4, W′′ ) 0-10 Vibronic Transitions Belonging to the (K2Π-X2Π) Band System of NO V′′ 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 a

MQDOa V′ ) 0 0.413 0.668 0.646 0.479 0.298 0.163 0.0807 0.0362 0.0148 0.00550 0.00181 V′ ) 3 0.387 1.20 0.00014 0.292 0.257 0.0363 0.0190 0.126 0.198 0.191 0.139

MQDOb

theorc

0.357 0.721 0.492 0.423 0.351 0.200 0.0901 0.0129 0.00003 0.00033 0.00388

0.83

0.458 0.742 0.00067 0.172 0.0754 0.0230 0.0362 0.158 0.170 0.138 0.105

0.56

MQDOa V′ ) 1 1.20 0.359 0.00039 0.138 0.300 0.322 0.249 0.157 0.0852 0.0404 0.0169 V′ ) 4 0.0332 0.708 1.07 0.0403 0.146 0.253 0.0803 0.00054 0.0709 0.156 0.177

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10

theorc

0.863 0.366 0.0484 0.00282 0.225 0.364 0.245 0.131 0.0446 0.00023 0.00346

1.45

0.0633 0.749 0.993 0.0271 0.114 0.279 0.0480 0.00543 0.0764 0.148 0.166

0.05

MQDOa

MQDOb

theorc

V′ ) 2 1.14 0.169 0.486 0.161 0.00016 0.102 0.220 0.242 0.191 0.122 0.0652

0.542 0.0748 0.364 0.0577 0.0284 0.0661 0.176 0.124 0.0533 0.0412 0.0682

1.41

Unperturbed MQDO. b Perturbed MQDO. c Galluser and Dressler.17

TABLE 5: Oscillator Strengths (×103) for W′ ) 0-3, W′′ ) 0-10 Vibronic Transitions Belonging to the (Q2Π-X2Π) Band System of NO V′′

MQDOb

MQDOa V′ ) 0 0.236 0.346 0.298 0.196 0.109 0.0536 0.0239 0.00981 0.00370 0.00127 0.00039 V′ ) 2 0.445 0.0826 0.235 0.0567 0.00496 0.0754 0.126 0.119 0.0847 0.0492 0.0244

MQDOb

Theorc

0.139 0.339 0.414 0.0890 0.209 0.0524 0.00285 0.00004 0.00002 0.00137 0.00295

0.26

0.400 0.0510 0.308 0.0594 0.00609 0.111 0.126 0.108 0.0766 0.0414 0.0174

0.95

a Unperturbed MQDO. Dressler.17

b

MQDOa V′ ) 1 0.534 0.121 0.00557 0.107 0.169 0.152 0.104 0.0584 0.0287 0.0125 0.00483 V′ ) 3 0.171 0.445 0.00434 0.146 0.131 0.0134 0.0167 0.0780 0.109 0.0963 0.0653

MQDOb

Theorc

0.394 0.0330 0.00216 0.0360 0.127 0.0815 0.0533 0.0538 0.0432 0.0238 0.00793

0.42

0.165 0.431 0.00433 0.142 0.132 0.0127 0.0158 0.0753 0.110 0.0981 0.0625

0.62

Perturbed MQDO.

c

Galluser and

experienced by the V′ ) 1-3 vibrational levels of the K2Π Rydberg state in contrast to the weak interaction suffered by the V′ ) 0 and V′ ) 4 levels. Concerning the Q2Π-X2Π system, the intensities of the bands are also affected by the interaction between the Q2Π Rydberg and the B2Π valence states. For instance, the V′ ) 1 level of the Q2Π state is perturbed by the V ) 34 and V ) 35 levels of the B2Π valence state. As a consequence of the interaction, the intensity of the Q2Π(1)-X2Π(0) band decreases, probably because this band

lends intensity to the β(34,0) and β(35,0) bands. As can be seen from Tables 4 and 5, the f values of Galluser and Dressler17 agree rather well with our perturbed values for the K2Π(3)-X2Π(0), K2Π(4)-X2Π(0), and Q2Π(1)-X2Π(0) bands. For the remaining bands, their values are greater than the MQDO ones. We would like to stress at this point that Galluser and Dressler17 calculated the moments for the K2Π-X2Π and Q2Π-X2Π transitions by using the Rydberg-series relationship according to which the electronic dipole transition moment decreases as n*-3/2, where n* is the effective principal quantum number. It is worth mentioning the generally good accord between our results and those of the above-mentioned authors for the transitions to the first member of this series, 3pπ(C2Π) (see Tables 2 and 3). For this transition, Galluser and Dressler17 obtained the electronic dipole transition moment from the absolute f value measurement by Bethke.34 Another reason for the discrepancies between both calculations can be the choice of the relative signs of the electronic transition moments and of the electronic interaction energies. In our calculations, the MQDO electronic moments for the transitions between the ground state and the npπ Rydberg states result to be all positive, and the electronic interaction energies between the valence B2Π state and the npπ Rydberg states have also been taken to be positive, as derived by the calculations of Galluser and Dressler.17 It should be noticed that, as pointed out,17,41 it is reasonable to choose the same sign for the electronic transition moment and for the electronic interaction energy when the states belong to the same Rydberg series. However, Galluser and Dressler have chosen a negative sign for the moment of the Q2Π-X2Π transition, in order to obtain a slightly better overall fit of the oscillator strength values, but, as they have claimed, this is a result whose significance is questionable. Nonetheless, both calculations predict the (1,0) and the (2,0) bands to be the strongest bands in the K2Π(V′)-X2Π(0) and Q2Π(V′)-X2Π(0) progressions, respectively. In Tables 6-8 we present the band oscillator strengths for transitions to the npπ Rydberg series with n ) 3-5 from the

Vibronic Transitions of npπ Rydberg Series in NO

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TABLE 6: Oscillator Strengths (×103) for W′ ) 0-7 Progressions Belonging to the (C2Π-A2Σ+) Band System of NO V′′ 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 a

MQDOa

MQDOb

MQDOa

MQDOb

MQDOa

MQDOb

V′ ) 0 533 1.27 0.0157 0.00346

516 0.985 0.0243 0.00205

V′ ) 1 2.30 533 0.558 0.00998 0.0187

1.51 516 0.00203 0.108 0.0295

V′ ) 2 0.0309 1.03 535 0.0165 0.00009 0.0583

0.0510 0.232 284 0.315 0.312 0.00793

V′ ) 4 0.0173 0.146 0.0130 2.97 521 8.71 0.0563 0.267

0.0444 0.00278 0.304 0.125 465 5.76 0.0774 0.281

V′ ) 5 0.00516 0.0956 0.359 0.0366 12.8 494 23.0 0.103

0.00266 0.179 0.0583 0.0632 10.2 297 10.3 0.392

V′ ) 6 0.00192 0.0379 0.309 0.745 0.539 30.9 447 47.2

0.0154 0.0559 0.00038 0.609 0.314 12.3 185 27.3

V′ ) 3 0.0388 0.0479 0.0196 533 1.91 0.0130 0.137 V′ ) 7 0.00090 0.0166 0.154 0.759 1.42 2.31 55.8 379

MQDOb 0.0579 0.375 1.95 238 0.684 0.312 0.00000 0.00006 0.0335 0.0995 0.571 1.27 0.839 37.1 262.

Unperturbed MQDO. b Perturbed MQDO.

TABLE 7: Oscillator Strengths (×103) for W′ ) 0-4, W′′ ) 0-5 Vibronic Transitions Belonging to the (K2Π-A2Σ+) Band System of NO V′′

MQDOa V′ ) 0 6.99 0.0441 0.00116 0.00154 0.00060 0.00014 V′ ) 3 0.00157 0.00528 0.162 5.98 0.658 0.0170

0 1 2 3 4 5 0 1 2 3 4 5 a

MQDOa

MQDOb 6.51 0.0453 0.0288 0.0408 0.142 2.20 0.00060 0.0606 0.0371 4.53 0.487 0.0141

MQDOa V′ ) 1 0.0561 6.96 0.00000 0.00097 0.00765 0.00175 V′ ) 4 0.00118 0.00314 0.00022 0.774 4.16 1.57

MQDOb

MQDOa

MQDOb

0.00646 4.20 0.0379 0.0981 0.162 0.866

V′ ) 2 0.00260 0.00001 6.83 0.131 0.00250 0.0227

0.0470 0.0351 3.31 0.221 0.0883 0.100

0.00016 0.0251 0.00185 0.683 4.51 1.95

the correctness of the present MQDO transition moments, we have determined the electronic oscillator strength for the C2Π-A2Σ+ transition, for which previous experimental results have been found in the literature. Wray24 obtained an f value for the C2Π-A2Σ+ transition equal to 0.7 by comparing highresolution spectra with calculated spectra. Groth et al.25 derived an oscillator strength of 0.61 for this transition from intensity measurements of the δ and γ bands of NO. With the MQDO method we have obtained an electronic oscillator strength of 0.53 which is close to the comparative experimental data. Such agreement seems to confirm the reliability of our transition moment, and the adequacy of our methodology in dealing with Rydberg-Rydberg transitions. Conclusions

Unperturbed MQDO. b Perturbed MQDO.

3sσ(A2Σ+) Rydberg state of NO. It can be noticed that the most intense bands for the three Rydberg-Rydberg transitions correspond to those that have V′ ) V′′. This is because there is an almost vertical alignment of the potential energy surfaces for the Rydberg states. For the most intense bands of each progression in the C2Π-A2Σ+ system, the oscillator strengths diminish when the interaction Rydberg-valence is taken into account, mainly for the progressions with V′ ) 2,3,5-7, where the perturbation of the C2Π levels is strongest. The effect of the Rydberg-valence interaction is also noticeable for the K2Π-A2Σ+ and Q2Π-A2Σ+ bands. In most cases, the perturbation increases the intensity of bands that would, otherwise, be largely irrelevant. As far as we know, no comparative data have been found in the literature. Consequently, in order to establish

Absorption oscillator strengths for vibronic transitions of the δ(C2Π-X2Π), K2Π-X2Π, and Q2Π-X2Π band systems have been calculated with the MQDO method. The Rydberg-valence interaction has been taken into account by diagonalizing the interaction matrix in a vibronic basis. The reliability of our results has been assessed by a comparison with experimental and theoretical data. Our oscillator strengths show a good agreement with the most accurate experimental data.21 The present work represents, to the best of our knowledge, the first theoretical calculation of band oscillator strengths for excitations from the first excited Rydberg state, the 3sσ(A2Σ+), to the npπ Rydberg series with n ) 3-5. The influence of the 2Π Rydberg-valence interaction in the Rydberg-Rydberg band intensities has been studied. In summary, we have determined vibronic intensities for transitions lying the UV, visible, and nearinfrared regions of the spectrum of NO. It should be noticed that contributions to the knowledge of the air spectrum in the near-

TABLE 8: Oscillator Strengths (×103) for W′ ) 0-3, W′′ ) 0-4 Vibronic Transitions Belonging to the (Q2Π-A2Σ+) Band System of NO V′′

MQDOa

MQDOb

MQDOa

MQDOb

MQDOa

MQDOb

MQDOa

MQDOb

0 1 2 3 4

V′ ) 0 0.774 0.00007 0.00000 0.00009 0.00003

0.756 0.00773 0.0125 0.0124 0.0185

V′ ) 1 0.00008 0.768 0.00490 0.00003 0.00043

0.00440 0.775 0.0436 0.0690 0.00638

V′ ) 2 0.00000 0.00536 0.739 0.0249 0.00012

0.00181 0.00170 0.975 0.0803 0.0304

V′ ) 3 0.00009 0.00011 0.0249 0.670 0.0701

0.00024 0.00006 0.0182 0.768 0.115

a

Unperturbed MQDO. b Perturbed MQDO.

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J. Phys. Chem. A, Vol. 114, No. 32, 2010

infrared region are important in different fields as atmospheric transmission studies, solar spectra, and auroral investigations. Acknowledgment. This work has been supported by the National Research Division of the Spanish Ministry for Education, within Research Grant No. CTQ2007-67143-C02/BQU, and by Junta de Castilla y Leon (VA059A09) as well as by European FEDER funds. A.M.V. acknowledges her research agreement awarded, with the Ramon y Cajal program, by the Spanish MEC and FSE. References and Notes (1) Liszt, H. S.; Rurner, B. E. Astrophys. J. 1978, 224, L73–L76. (2) Ziurys, L. M.; McGonagle, D.; Minh, Y.; Irvine, W. M. Astrophys. J. 1991, 373, 535–542. (3) Giusti-Suzor, A.; Jungen, C. H. J. Chem. Phys. 1984, 80, 986– 1000. (4) Brunger, M. J.; Campbell, L.; Cartwright, D. C.; Middleton, A. G.; Mojarrabi, B.; Teubner, P. J. O. J. Phys. B: At. Mol. Phys. 2000, 33, 783– 808. (5) Miescher, E.; Huber, K. P. International ReView of Science, Physical Chemistry, Series 2, Vol. 3, Spectroscopy; Butterworths: London, 1976. (6) Largerqvist, A.; Miescher, E. HelV. Phys. Acta 1958, 31, 221–262. (7) Jungen, C. J. Chem. Phys. 1970, 53, 4168–4182. (8) Herzberg, G.; Lagerqvist, A.; Miescher, E. Can. J. Phys 1956, 34, 622–624. (9) Cohen-Sabban, J.; Vuillemin, A. Astrophys. Space Sci. 1973, 24, 127–132. (10) Feldman, P. D.; Takacs, P. Z. Geophys. Res. Lett. 1974, 1, 169– 171. (11) Steward, A. I.; Barth, C. A. Science 1979, 205, 59–62. (12) Feldman, P. D.; Moos, H. W.; Clarke, J. T.; Lane, A. L. Nature 1979, 279, 221–222. (13) Bertaux, J. L.; Leblanc, F.; Perrier, S.; Quemerais, W.; Korablev, O.; Dimarellis, E.; Reberac, A.; Forget, F.; Simon, P. C.; Stern, S. A.; Sandel, B. Science 2005, 307, 566–569. (14) Cieslik, S.; Nicolet, M. Planet. Space Sci. 1973, 21, 925–938. (15) Dressler, K.; Miescher, E. Astrophys. J. 1965, 141, 1266–1283. (16) Lagerqvist, A.; Miescher, E. Can. J. Phys. 1966, 44, 1525–1539. (17) Galluser, R.; Dressler, K. J. Chem. Phys. 1982, 76, 4311–4327.

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