Theoretical Study of the Rotational Structure of the c4′1Σu + (6)–X1Σg +(0–9) Absorption Bands of N2

We have theoretically determined the absorption oscillator strengths and wavenumbers for rotationally resolved transitions of the c4′1Σu + (6)-X1Σg +(0–9) bands of N2, which are relevant to analyze the spectra of planetary atmospheres. The Molecular Quantum Defect Orbital method has been used in our calculations. The interaction between the c4′1Σu + (6) Rydberg state and the b′1Σu + valence states has been considered using an adequate rovibronic energy matrix. In addition, we have calculated the lifetimes of the rotational levels of the c4′1Σu + (6) state. We hope that the reported data, most of them for the first time, can be useful in the interpretation of planetary atmospheres where N2 is present.


Introduction
The c 4 ′ 1 Σ u + -X 1 Σ g + Rydberg band system of molecular nitrogen, also named the Carroll-Yoshino bands, plays an essential role in the interpretation of extreme ultraviolet (EUV) emission observations in the airglow upper atmosphere of the Earth and other nitrogen-rich planetary atmospheres. Several bands of the c 4 ′ 1 Σ u + (0, 3, 4, 6)-X 1 Σ g + (v″) progressions have been observed in Earthʼs airglow spectra (Feldman et al. 2001) obtained with the Far Ultraviolet Spectroscopic Explorer (FUSE) and in the EUV spectrum of Titanʼs atmosphere (Ajello et al. 2007;Stevens et al. 2011) achieved using the Ultraviolet Imaging Spectrometer (UVIS) on Cassini. Most of the research so far has been focused on the c 4 ′ 1 Σ u + (0, 3, 4)-X 1 Σ g + (v″) progressions (see, for example, Ajello et al. 1998;Walter et al. 2000;Liu et al. 2008). The c 4 ′ 1 Σ u + (6)-X 1 Σ g + (v″) bands have been investigated less frequently, despite their observation in spectra from atmospheres dominated by molecular nitrogen. Feldman et al. (2001) identified the c 4 ′ 1 Σ u + (6)-X 1 Σ g + (4, 5, 8) bands in the terrestrial airglow spectra obtained with FUSE. These bands, together with the c 4 ′ 1 Σ u + (6)-X 1 Σ g + (7) band, have been observed in the EUV spectra of Titanʼs atmosphere obtained by UVIS (Ajello et al. 2007;Stevens et al. 2011). The c 4 ′ 1 Σ u + (6)-X 1 Σ g + (8) band has also been observed in the spectrum of Mars, obtained with FUSE (Krasnopolsky & Feldman 2002). As claimed by Stark et al. (2005) and Liu et al. (2009), the quantitative interpretation of the observations in such atmospheres and the modeling of the upper atmospheric processes require reliable spectroscopic data, including line positions, oscillator strengths, and lifetimes with rotational effects.
To achieve a better understanding of planetary observations, where the c 4 ′ 1 Σ u + (6)-X 1 Σ g + (v″) bands of N 2 are present, in this work, we have calculated transition energies and line intensities, expressed as oscillator strengths or f-values, for R and P branches of the c 4 ′ 1 Σ u + (6)-X 1 Σ g + (0-9) bands. It is well known that the vibrational levels of the c 4 ′ 1 Σ u + Rydberg state of N 2 interact with the appropriate vibrational levels of the b′ 1 Σ u + valence state (Yoshino et al. 1979;Stahel et al. 1983). Hence, in our calculations, we have taken into account this homogeneous perturbation between states of 1 Σ u + symmetry through an interaction matrix for each value of the rotational quantum number J. The description of the c 4 ′ 1 Σ u + Rydberg state has been made with the molecular quantum defect orbital (MQDO) method, which has proven to be a reliable approach in dealing with rovibronic transitions of N 2 in earlier applications (Lavín et al. 2008(Lavín et al. , 2010Lavín & Velasco 2011;Velasco & Lavín 2020). We have also determined the radiative lifetimes of the rotational levels of the c 4 ′ 1 Σ u + (6) state, which can be helpful to elucidate the competition between predissociation and radiation in N 2 emissions.

Method of Calculation
In a diatomic molecule, the dimensionless absorption oscillator strength for a rotational line can be defined in the following form (Larsson 1983): are the line wavenumbers in cm −1 and ¢  S J J is the rotational line strength in atomic units. Primed quantities refer to the upper electronic state and double primes designate the lower state. This standard notation is used throughout the paper.
The wave function for a diatomic molecule can be separated into independent electronic and nuclear factors, and the nuclear term can be approximated to the product of the rotational and vibrational wave functions. Then, the rotational line strength can be written as the multiplication of independent vibrational, electronic, and rotational factors (Whiting & Nicholls 1974) in the following way: q v v is the Franck-Condon factor, R e is the electronic transition moment expressed in atomic units, and ¢  J J I is the Hönl-London factor.
The factorization of Equation (2) is no longer appropriate when rotational perturbation happens. In this case, interactions between the rotational excitation states should be considered to adequately describe the rovibronic structure of the absorption spectra. To this end, each perturbed electronic state of a rovibrational level should be characterized by an appropriate linear combination of diabatic unperturbed basis states (Walter et al. 2000). Therefore, the expression for the line strength is transformed as follows: is the vibrational overlap integral, and R k e is the electronic moment between the lower electronic state and the upper k th diabatic state. The overlap integrals were determined using vibrational wave functions obtained by solving the Schödinger equation with the Numerov algorithm from potential energy curves calculated with a Rydberg-Klein-Rees (RKR) approach (Rydberg 1931;Klein 1932;Rees 1947).
Ck's perturbation components have been obtained by diagonalization of an interaction matrix for each J value. The diagonal and off-diagonal elements are the unperturbed rotational energies of the states and the interaction parameters, respectively. The Hönl-London factors have been estimated following the equations reported by Kovács (1969). The electronic transition moment of the transitions involving the Rydberg states has been determined with the MQDO method.
The MQDO methodology was formulated to deal with molecular Rydberg transitions and has been described in detail in previous papers (Martín et al. 1996). In this approach, the radial part of the wave functions is the analytical solution of a one-electron Schrödinger equation containing a model potential. On the other hand, the angular part of the molecular Rydberg wave functions is a symmetry-adapted linear combination of spherical harmonics. In such a way, the transition moment for a transition between two unperturbed electronic states i and j is expressed as the product of a radial and angular contribution: where Q{i → j} is the angular factor that results from the integration of the angular part of the MQDOs and that of the transition dipole moment operator, and R ij (r) is the radial part of the electronic transition moment. One of the advantages of the MQDO approach is that the transition integrals result in closed-form analytical expressions, which avoids numerical errors and convergence problems.

Results and Discussion
The c 4 ′ 1 Σ u + (6)-X 1 Σ g + (0-9) bands are the result of rovibronic absorption transitions from the v″ = 0-9 vibrational levels of the ground state X 1 Σ g + to the v́= 6 vibrational level of the lowest member of the Rydberg npσ series (with n = 3). Stahel et al. (1983) demonstrated that the Rydberg-valence homogeneous electronic interaction is the dominant perturbation in singlet ungerade states, such as the c 4 ′ 1 Σ u + state. Accordingly, we have considered the interaction between the c 4 ′ 1 Σ u + Rydberg and b′ 1 Σ u + valence states in our model. The associated diabatic potential energy curves calculated with the RKR approach are shown in Figure 1. Given that the Rydberg-valence interaction causes every vibrational level to be homogeneously perturbed by several of its neighbors, in our calculations, we have taken into account the coupling of c 4 ′ 1 Σ u + (6) and its nearby levels: b´(18) 1 Σ u + , b´(19) 1 Σ u + , b´(20) 1 Σ u + , and b´(21) 1 Σ u + . The eigenvector components for the five interacting diabatic states are obtained by diagonalizing the 5 × 5 symmetric Hamiltonian matrix for each J value. In a previous study (Velasco & Lavín 2020) involving the c 4 ′ 1 Σ u + (4) state, we used the expression given by Kovács (p.207) for singlet states to determine the offdiagonal coupling parameters. This procedure is not possible for the c 4 ′ 1 Σ u + (6) state because of the limited number of rotational levels observed. According to Carroll & Hagim (1988), the off-diagonal elements may be written as: where H ee′ , a constant for two given states, is the electronic interaction parameter and ¢ ¢ S ee vv is the vibrational overlap integral. In our mentioned work (Velasco & Lavín 2020), a value of 1100 cm −1 was estimated for the interaction parameter between the c 4 ′ 1 Σ u + and b′ 1 Σ u + electronic states. This value has been adopted in the present calculations and the vibrational overlap integrals calculated from the RKR method. The diagonal elements, which are the unperturbed rotational energies of the c 4 ′ 1 Σ u + and b′ 1 Σ u + vibrational states, are obtained by fitting the eigenvalues to experimentally determined energy terms for the rotational levels of the c 4 ′ 1 Σ u + (6), b′ 1 Σ u + (18), b′ 1 Σ u + (19), b′ 1 Σ u + (20), and b′ 1 Σ u + (21) states. As pointed out by Walter et al. (2000), a first assessment of the quality of the deperturbation scheme can be made by comparing the calculated rovibrational term energies to the observed values. Experimentally, term values have been Roncin et al. (1998) and Harvard-Smithsonian Center for Astrophysics). The deperturbation procedure was performed quite satisfactorily with a mean absolute deviation of less than 1.2 cm −1 . The eigenvector components of the c 4 ' 1 Σ u + (6) state are given in Table 1. The Rydberg-valence mixing is small at low values of J but are not negligible (91.1% of Rydberg character and 8.9% of valence character). The percentage of b′ 1 Σ u + character, primarily b´(20) 1 Σ u + , increases with J until it approaches 48.4 % at J = 20.
The transition wavenumbers, along with the perturbed absorption oscillator strengths for the lines of the R and P branches of the c 4 ′ 1 Σ u + (v́= 6)-X 1 Σ g + (v″ = 0-9) bands, are given in Tables 2-6. The experimental transition wavenumbers are also shown for comparative purposes. We subtracted the appropriate ground state level (Edwards et al. 1993) from each perturbed rovibronic term presently calculated to obtain line transition energies. Comparison with the observed absorption and emission line positions (CfA Molecular Data) shows a mean absolute deviation of 0.41 cm −1 . It should be mentioned that about half of the comparative rotational lines are blended.
In the calculations of the oscillator strengths, we have used the MQDO formalism to determine the unperturbed c 4 ′ 1 Σ u + -X 1 Σ g + transition moment. To this end, we needed the ionization energy of N 2 and the electronic energy of the Rydberg state c 4 ′ 1 Σ u + as input. For the ionization energy, we have used the experimental value obtained by Huber & Jungen (1990), and for the Rydberg electronic energy, the experimental value reported by Huber & Herzberg (1979). The MQDO vertical electronic transition moment for c 4 ′ 1 Σ u Stahel et al. (1983) reported a value of −0.5946 au, derived by fitting the vibronic band strength, calculated based on electronically coupled diabatic states, to the experimental band strength. A reasonable agreement exists between both values. For the b′ 1 Σ u + -X 1 Σ g + transition moment, we have used that reported by Spelsberg & Meyer (2001).
As can be observed from Tables 2-6, our calculations predict that the most intense absorption bands of the c 4 ′ 1 Σ u bands. The f -values of the P branch of both transitions increase up to J ≈ 15, falling off thereafter, while the R branch values of both bands decrease with J.
In the absence of perturbations, the f-values for lines of the (1) bands are of the order of 10 −9 and 10 −7 , respectively, owing to the small vibrational overlap integral between the c 4 ′ 1 Σ u + (6) state and the X 1 Σ g + (0) and X 1 Σ g + (1) states, 0.00018 and 0.0021, respectively. The increase in the oscillator strengths in these bands, and its consequent observation in the N 2 EUV spectra, is understable in terms of the valence character acquired by the c 4 ′ 1 Σ u + (6) level through the perturbative interaction of the c 4 ′ 1 Σ u + -b′ 1 Σ u + states. In fact, for these bands, the contribution to the rovibronic transition moment comes entirely from the b′ 1 Σ u + states. The perturbed oscillator strengths calculated by the combinations of Equations (1) and (3) involve either the sum or the difference of terms containing the unperturbed rovibronic transition moments. So, the homogeneous interaction can lead to constructive or destructive interference in the line f -value. Lines of both branches of the c 4 ′ 1 Σ u + (6)-X 1 Σ g + (3) bands are very weak at low and medium J values due to destructive interference. However, a noticeable increase in line f-values is shown at high J values because the transition moment is dominated by b′ 1 Σ u + , mainly b′ 1 Σ u + (20). Destructive interference is also found in the R and P branches of the c 4 ′ 1 Σ u + (6)-X 1 Σ g + (6) band. In contrast, the analysis of the c 4 ′ 1 Σ u + (6) and b′ 1 Σ u + contributions to the transition moments for the c 4 ′ 1 Σ u + (6)-X 1 Σ g + (4) and c 4 ′ 1 Σ u + (6)-X 1 Σ g + (8) bands reveals that the interference is constructive for these bands. For example, oscillator strengths for lines of the P and R branches of the c 4 ′ 1 Σ u + (6)-X 1 Σ g + (4) band are at twice the unperturbed values.
No comparative data, to our knowledge, have been reported in the literature for the f-values of lines studied here, despite the fact that the line positions of the bands belonging to the c 4 ′ 1 Σ u + (6)-X 1 Σ g + (v″) progression have been known for many years. However, band f-values have been reported for the c 4 ′ 1 Σ u + (6)-X 1 Σ g + (0) band from high-resolution absorption spectra (Huber et al. 2009;Heays 2010). Therefore, to assess the accuracy of the present results, we have calculated the oscillator strength for this band. To do so, we first calculated the integrated absorption cross-sections for both the P and R lines of the c 4 ′ 1 Σ u + (4)-X 1 Σ g + (0) band through the expression given by Nicholls (1969): where ν is the frequency in cm −1 and  N J N tot is the relative population of the rotational level J″ of the lower electronic state. We assumed a Boltzmann distribution for the rotational population calculations. The results were obtained at a temperature of 300 K.
We then derived the band f-value, ¢  f v v , by adding up the integrated absorption cross-section contributions from all the rotational lines and, finally, the resulting value was converted into band oscillator strength by using the equation ( No comparison with other works can be made, since there are no experimental or theoretical band f-values for the remaining bands studied in this work. Nevertheless, it should be mentioned that the bands in the c 4 ′ 1 Σ u + (6) progression were identified in discharge measurements at high spectral resolution by Roncin et al. (1987). These researchers noted that the c 4 ′ 1 Σ u + (6)-X 1 Σ g + (1) and c 4 ′ 1 Σ u + (6)-X 1 Σ g + (2) bands are weak, the c 4 ′ 1 Σ u + (6)-X 1 Σ g + (3) band is very weak, the c 4 ′ 1 Σ u + (6)-X 1 Σ g + (4) and c 4 ′ 1 Σ u + (6)-X 1 Σ g + (8) bands are strong, and the c 4 ′ 1 Σ u + (6)-X 1 Σ g + (5) and c 4 ′ 1 Σ u + (6)-X 1 Σ g + (7) bands are very strong. Our results display a similar behavior as can be seen from Tables 2-6.
We have also calculated the radiative lifetimes (τ v′J′ ) of rotational levels of the c 4 ′ 1 Σ u + (6) state using the expression given by Larsson (1983): where A v′J′,v″J″ is the transition probability for a spontaneous radiative transition from an excited rovibronic state (v′, J′) to a state (v″, J″), and it is related to the absorption line oscillator strength through the expression: Table 2 Transition Wavenumbers (in cm −1 ) and Absorption Oscillator Strengths for the R and P Branches of the c′ 4 1 Σ u + (6)-X 1 Σ g + (v″=0,1) Bands of N 2 c′ 4 1 Σ u + (6)-X 1 Σ g

Table 4
Transition Wavenumbers (in cm −1 ) and Absorption Oscillator Strengths for the R and P Branches of the c′ 4 1 Σ u + (6)-X 1 Σ g + (v″=4,5) Bands of N 2   where ν v′J′,v″J″ is the wavenumber for the transition in cm −1 and A v′J′ , v″J″ is expressed in s −1 .
In the calculation of the lifetime, we have considered transitions from rotational levels of the c 4 ′ 1 Σ u + (6) state to the appropriate rotational levels of the X 1 Σ g + and a″ 1 Σ g + states since transitions to the latter are dipole allowed. The electronic transition moment for the c 4 ′ 1 Σ u + -a″ 1 Σ g + transition calculated with the MQDO formalism is -3.0 au. No comparative data have been found in the literature. Although the c 4 ′ 1 Σ u + -a″ 1 Σ g + electronic transition moment is large when compared to that of c 4 ′ 1 Σ u + -X 1 Σ g + , the contribution of the emission to the a″ 1 Σ g + states to the global transition probability turns out to be less than 0.2% because the frequencies of the c 4 ′ 1 Σ u + (6)-a″ 1 Σ g + (v″) transitions are very small. In fact, the c 4 ′ 1 Σ u + -a″ 1 Σ g + and c 4 ′ 1 Σ u + -X 1 Σ g + systems fall into the near-infrared and the EUV, respectively.
Radiative lifetimes for the individual rotational levels (J = 0-20) of the c 4 ′ 1 Σ u + (6) state are given in Table 7. Our model predicts a small rotational dependence on the radiative lifetime of the c 4 ′ 1 Σ u + (6) state for J 19. Thereby, radiative lifetimes remain constant with a value of 0.89 ns over the J = 0-10 range and then smoothly decrease until 0.84 ns as the interaction c 4 ′ 1 Σ u + -b′ 1 Σ u + increases. These values are somewhat larger than the only value reported, 0.61 ± 0.13 ns, which is a rotationally averaged fluorescence lifetime of the c 4 ′ 1 Σ u + (6) state obtained by Moise et al. (2011). Taking into account the fact that predissociation and spontaneous emission are the two primary decay mechanisms for the excited states of N 2 (Liu et al. 2009), the shorter experimental lifetime, compared to our values of radiative lifetimes, could be indicative of the predissociation of the c 4 ′ 1 Σ u + (6) state through its coupling with the b′ 1 Σ u + states which in turn are predissociated by continuum states (Helm et al. 1993).
To summarize, in this study, we have calculated transition energies and oscillator strengths for the rotational lines of the c 4 ′ 1 Σ u + (6)-X 1 Σ g + (v″ = 0-9) bands in molecular nitrogen, and the radiative lifetimes of rotational levels of the c 4 ′ 1 Σ u + (6) state. In the calculations, we have used a model that includes the perturbations between the c 4 ′ 1 Σ u + (6) Rydberg state and b′ 1 Σ u + Table 6 Transition Wavenumbers (in cm −1 ) and Absorption Oscillator Strengths for the R and P Branches of the c′ 4 1 Σ u + (6)-X 1 Σ g + (v″=8,9) Bands of N 2  valence states. The comparison between R-and P-line positions presently calculated and experimental results, when available, suggest that the procedure used properly takes into account the valence-Rydberg perturbation. To the best of our knowledge, the line f-values for the bands of the c 4 ′ 1 Σ u + (6)-X 1 Σ g + (v″) progression and radiative lifetimes at the rotational level are reported here for the first time. As mentioned, several of these bands have been observed in the airglow spectra from Titan. From their spectral analysis of Cassini UVIS observations, Stevens et al. (2011) found that the N 2 c 4 ′ 1 Σ u + (3, 4, 6) progressions are blended and considered all three of these progressions as one spectral component in their airglow radiance model. We hope that the new data reported here may aid to identify the contribution from each fine structure feature in the mentioned blended progressions, thus leading to understanding of the EUV spectra of atmospheres containing molecular nitrogen.