Transition Energies and Absorption Oscillator Strengths for ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$, ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$, and ${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ Band Systems in N2

C. Lavín; A. M. Velasco

doi:10.3847/1538-4365/aa656e

1. Introduction

In the last decades, there has been growing interest in the extreme ultraviolet (EUV) spectrum of molecular nitrogen, which arises from electronic excitations from its ground state to the lowest excited states. This interest is closely related to its important role in the understanding of the physics and chemistry of planetary atmospheres rich in ${{\rm{N}}}_{2}$ . Electronic transitions of ${{\rm{N}}}_{2}$ have been detected in the airglow spectra from Earth, Titan, and Triton atmospheres, where ${{\rm{N}}}_{2}$ is the major constituent. In particular, several bands belonging to the ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ and ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ systems, which are the object of the present work, have been detected in such atmospheres. The ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(v^{\prime} =0)-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'' = 0–2, 6–9), ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(v^{\prime} =3)-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}({v}^{\prime\prime }=4)$ , and ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(v^{\prime} =4)\ -{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}({v}^{\prime\prime }=5)$ bands have been identified in terrestrial airglow spectra obtained with the Far Ultraviolet Spectroscopic Explorer (FUSE; Feldman et al. 2001; Bishop et al. 2007). In addition, Bishop et al. (2007) have suggested that the ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(v^{\prime} =0)-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}({v}^{\prime\prime })$ bands with v'' > 9 may be present in the far-ultraviolet (FUV) terrestrial airglow. The ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ $(v^{\prime} =0)-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'' = 0–1) bands have also been detected in the Titan and Triton atmospheres (Strobel et al. 1991). Strobel & Shemansky (1982) identified the ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v' = 3, 17, 9) − ${{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'' = 0), ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ $(v^{\prime} =0)-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'' = 0–2), ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ $(v^{\prime} =3)-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'' = 0, 2, 3), and ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ $(v^{\prime} =4)-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'' = 0) bands in the EUV emission spectra from Titans atmosphere obtained by Voyager 1. More recently, from analysis of the spectra of Titan obtained by the Ultraviolet Imaging Spectrometer (UVIS) on board Cassini, several bands belonging to the ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v' = 3, 9, 11, 12, 16) and ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v' = 0, 1, 3, 4, 6) progressions have been identified as strong Titan emission features (Ajello et al. 2007; Stevens et al. 2011). On the other hand, the ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ $(v^{\prime} =9)-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'' = 5, 6), ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v' = 0) − ${{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'' = 0–2), ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ $(v^{\prime} =3)-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'' = 2–4), ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ $(v^{\prime} =4)-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'' = 3–5), and ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ $(v^{\prime} =6)-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'' = 8) bands have been observed in the spectrum of Mars obtained with FUSE (Krasnopolsky & Feldman 2002). The interpretation of the observations of such atmospheres and, in particular, the planetary-atmospheric modeling requires reliable spectroscopic data.

Spectral properties of the ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ Rydberg and valence ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ systems of ${{\rm{N}}}_{2}$ have been the subject of several experimental and theoretical studies. Spectra of these band systems have been extensively investigated in the laboratory both in absorption and emission (Carroll et al. 1970; Yoshino & Tanaka 1977; Yoshino et al. 1979; Verma & Jois 1984; Roncin et al. 1987, 1991). Emission cross-sections for a great number of bands have been measured at medium- (Ajello et al. 1989) and high-resolution (Heays et al. 2014). Lifetimes of ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ and ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ vibrational states have been derived from high-resolution laser experiments (Ubachs et al. 2001; Sprengers & Ubachs 2006), from synchrotron radiation experiments (Oertel et al. 1981), and from linewidths (Kam et al. 1989; Helm et al. 1993; Sprengers & Ubachs 2006). Many of these states show significant predissociation (Walter et al. 1994; Shemansky et al. 1995; Ajello et al. 1998; Walter et al. 2000). An ab initio study of internuclear distance-dependent dipole transition moments for ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ and ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ transitions have been performed by Spelsberg & Meyer (2001). Concerning absorption oscillator strengths, or f -values, most of the research so far performed has been focused on the ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ $(v^{\prime} )-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'' = 0) and ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ $(v^{\prime} )-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'' = 0) bands (Ajello et al. 1989; Chan et al. 1993; Stark et al. 2005, 2008; Heays et al. 2009; Huber et al. 2009). Although the existence of bands from ground-state levels with v'' > 0 is known for many years, f-values have only been reported by Ajello et al. (1989) for the ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ $(v^{\prime} =0)-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'' = 1, 2, 5, 7) bands and by Lavin & Velasco (2016) for the ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v' = 3, 4, 6) − ${{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'' = 0–12) and ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v' = 10, 13, 20) − ${{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'' = 0–12) bands, as far as we know. Similar studies concerning the ${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ Rydberg band system, also investigated in this work, appear to be more scarce. We have only found experimental f-values for the ${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ $(v^{\prime} =0)-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'' = 0) band, those reported by Chan et al. (1993) and Heays et al. (2009), in spite of several bands of this system being observed in high-resolution emission (Roncin et al. 1987; Heays et al. 2014) spectra of molecular nitrogen.

Despite considerable experimental and theoretical efforts, there is still a substantial lack of spectroscopic data for molecular nitrogen. In particular, oscillator strengths for many bands observed in the EUV spectrum have not yet been measured. In this work, we have calculated transition energies and absorption oscillator strengths for the ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v' = 0–2, 5, 7, 8) − ${{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'' = 0–14), ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v' = 0–9, 11, 12, 14–19, 21, 22) − ${{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'' = 0–14), and ${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v' = 0, 2) − ${{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'' = 0–14) bands of ${{\rm{N}}}_{2}$ . In calculations of oscillator strengths, we have used the Molecular Quantum Defect Orbital (MQDO) method to calculate electronic transition moments for Rydberg transitions while for the transition to the b' valence state, we have adopted the electronic moment reported by Stahel et al. (1983). The MQDO method has proven to work well in dealing with electronic and rovibronic transitions involving Rydberg states in previous studies on the EUV spectrum of ${{\rm{N}}}_{2}$ (Lavín et al. 2004, 2008, 2010; Lavin & Velasco 2011).

It is well established that strong Rydberg-valence interactions within the ${}^{1}{{{\rm{\Sigma }}}_{u}}^{+}$ manifold complicate the spectroscopy of ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v'), ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v'), and ${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v') states (Lefevbre-Brion 1969; Yoshino et al. 1979; Stahel et al. 1983). In order to take into account the coupling between Rydberg and valence states, in the present calculations, we have used a theoretical model (Bustos et al. 2004; Velasco et al. 2010), which gives accounts of homogeneous interactions through an interaction matrix that includes vibrational coupling. Recently, this model has been successfully applied to the determination of band f-values for some progressions of the ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ , ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ , and ${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ systems of molecular nitrogen (Lavin & Velasco 2016). Encouraged by these results, here we extend our calculations to other progressions with the aim of providing more complete information on the vibrational structure of such band systems.

2. Method of Calculation

For a transition to a given bound vibrational state v' of the final electronic state from a bound vibrational level v'' of the initial electronic state, the absorption oscillator strength is defined as (Larsson 1983)

$\begin{eqnarray}&&{f}_{{v}^{\prime }{v}^{\prime\prime }}=\tfrac{8{\pi }^{2}{{mca}}_{0}^{2}}{3h}{\nu }_{{v}^{\prime }{v}^{\prime\prime }}{| \langle {v}^{\prime }| {R}_{e}| {v}^{\prime\prime }\rangle | }^{2},\end{eqnarray} \tag{ 1 }$

where ${\nu }_{{v}^{\prime }{v}^{\prime\prime }}$ is the wavenumber of the band origin in cm⁻¹, and R_e is the electric dipole transition moment (in atomic units) for the transition between the initial and final electronic states.

If both the lower and upper states are unperturbed, the integral in Equation (1) can be written as a product of independent vibrational and electronic factors:

$\begin{eqnarray}&&{| \langle {v}^{\prime }| {R}_{e}| {v}^{\prime\prime }\rangle | }^{2}={q}_{{v}^{\prime }{v}^{\prime\prime }}{R}_{e}^{2}.\end{eqnarray} \tag{ 2 }$

Here ${q}_{{v}^{\prime }{v}^{\prime\prime }}$ is the Franck–Condon factor, which is given by the square of the overlap integral of the vibrational wave functions of the two electronic states involved in the transition. In this work, the Rydberg–Klein–Rees (RKR) method has been used to determine the potential energy curves of the states involved in the transition. From these potential energies, the vibrational wave functions are obtained by solving the Schrdinger equation with the Numerov algorithm.

When the upper state is perturbed by interaction with other vibrational levels, as occurs in transitions studied here, the factorization assumption in Equation (2) is no longer possible. In this case, an appropriate model of the vibronic structure requires us to consider the coupling between the excited states. At this end, we have followed a perturbation model that has been described in detail previously (Bustos et al. 2004; Velasco et al. 2010); hence only a brief mention of its major features will be given here. In the model, the wave function of a perturbed state, $| {e}_{0},{v}_{0}\rangle$ , is represented by a linear combination of the unperturbed wave functions ∣ $e,{v}_{e}\rangle$ ,

$\begin{eqnarray}&&| {e}_{0},{v}_{0}\rangle =\displaystyle \sum _{e=1}^{{ne}}\displaystyle \sum _{{v}_{e}=0}^{{v}_{e,\max }}| e,{v}_{e}\rangle {C}_{{v}_{e},{v}_{0}},\end{eqnarray} \tag{ 3 }$

where ne is the number of electronic states considered in the interaction, ${v}_{e}$ is the vibrational level of the electronic state e, and ${v}_{e,\max }$ is the highest vibrational level of the electronic state e. The vibronic energies and the perturbation coefficients, ${C}_{{v}_{e},{v}_{0}}$ , are obtained by diagonalization of a vibronic interaction matrix whose diagonal elements are the energies of the unperturbed states and whose off diagonal elements are the coupling parameters. In this work, the molecular constants reported by Edwards et al. (1993) for the ${{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ ground state and by Stahel et al. (1983) for the ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ , ${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ , and ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ excited states have been used for the noninteracting vibrational states. The electronic interaction energies given by Stahel et al. (1983) were adopted in our calculations.

The expression for the oscillator strength of a vibronic transition from a nonperturbed $| {\rm{X}},0\rangle$ to the mixed $| {e}_{0},{v}_{0}\rangle$ state is

$\begin{eqnarray}&&{f}_{{v}^{\prime }{v}^{\prime\prime }}=\tfrac{8{\pi }^{2}{{mca}}_{0}^{2}}{3h}{\nu }_{{v}^{\prime }{v}^{\prime\prime }}{\langle {M}_{{v}_{0}0}\rangle }^{2},\end{eqnarray} \tag{ 4 }$

where $\langle {M}_{{v}_{0}0}\rangle$ , the vibronic transition moment, is defined as

$\begin{eqnarray}&&\langle {M}_{{v}_{0}0}\rangle =\displaystyle \sum _{e=1}^{{ne}}\displaystyle \sum _{{v}_{e}=0}^{{v}_{e,\max }}{C}_{{v}_{0},{v}_{e}}^{* }{R}_{e}{S}_{{v}_{e}0}^{{eX}}.\end{eqnarray} \tag{ 5 }$

Here ${S}_{{v}_{e}0}^{{eX}}$ is the vibrational overlap integral.

For the transition from the ground state to the ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ valence state, we have used the electronic transition moment reported by Stahel et al. (1983) and derived by fitting the band vibronic strength, calculated in a basis of electronically coupled diabatic states, to the experimental band strength. The electronic transition moments for the ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ and ${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ Rydberg transitions has been calculated with the MQDO method (Martín et al. 1996, 2001), formulated to study molecular Rydberg transitions. Briefly, in the MQDO approach, the molecular Rydberg electron is described by a wave function that is expressed as the product of the radial part and the angular part. The radial part of the MQDO wave function is obtained through a one-electron Schrdinger equation that contains an effective potential of the molecular core of the form

$\begin{eqnarray}&&V(r)=\tfrac{(c-\delta )(2l+c-\delta +1)}{2{r}^{2}}-\tfrac{1}{r},\end{eqnarray} \tag{ 6 }$

where δ is the quantum defect for a given molecular state and c is an integer with a narrow range of values that ensure the normalizability of the wave functions. The angular part of the MQDO wave functions is a linear combination of spherical harmonics adapted to the molecular symmetry, ${D}_{\infty h}$ in the case of ${{\rm{N}}}_{2}$ . The MQDO orbitals lead to analytical expressions for the transition's integrals. This feature of the method is one of its advantages because it allows the calculation of oscillator strength data free from the convergence problems.

3. Results and Discussion

The ${{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ ground-state electronic configuration of the molecular nitrogen at its equilibrium geometry is $1{{\sigma }_{g}}^{2}1{{\sigma }_{u}}^{2}2{\sigma }_{g}^{2}2{{\sigma }_{u}}^{2}1{{\pi }_{u}}^{4}3{{\sigma }_{g}}^{2}$ . The ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ and ${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}\ -{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ band systems arise from the excitation of the outermost valence electron of ${{\rm{N}}}_{2}$ to the first two members of the $\mathrm{np}{\sigma }_{u}$ Rydberg series that converges into the lowest ionic ground state (Carroll & Yoshino 1972). The calculations of the electronic transition moments for both Rydberg transitions with the MQDO method require the ionization energy of ${{\rm{N}}}_{2}$ and the electronic energies of the Rydberg states. These data have been taken from Huber & Jungen (1990). The MQDO electronic transition moments for ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ and ${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ are 0.6568 and 0.3250 au, respectively, in fairly good agreement with those reported by Stahel et al. (1983). As already mentioned, there is a strong homogeneous interaction between ${}^{1}{{{\rm{\Sigma }}}_{u}}^{+}$ vibronic levels. We have taken into account the interactions between the ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(v=0\mbox{--}27)$ valence states and the ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(v=0\mbox{--}8)$ and ${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(v=0\mbox{--}2)$ Rydberg states through a vibronic interaction matrix, as we have described in the previous section. The main eigenvector components of the ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(v=0\mbox{--}9,11,12,14\mbox{--}19,21,22)$ , ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(v=0\mbox{--}2,5,7,8)$ , and ${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(v=0,2)$ vibronic states in the interaction matrix representation are given in Table 1. As can be seen from the table, the mixing between ${}^{1}{{{\rm{\Sigma }}}_{u}}^{+}$ states is so strong that it globally affects all vibronic levels. In Table 2, Franck–Condon factors calculated from RKR potential curves are reported for the ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v' = 0–2, 5, 7, 8) − ${{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'' = 0–14), ${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v' = 0, 2) − ${{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'' = 0–14), and ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v' = 0–9, 11, 12, 14–19, 21, 22) − ${{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'' = 0–14) vibronic transitions.

Table 1. Main Components of Wave Functions for the ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v'), ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v'), and ${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v') Vibronic States

Vibronic state	Wave Function Components
b' 0	1.00 (b' 0)	0.01 (b' 1)	−0.01 ( ${{c}_{4}}^{\prime }8$ )	0.01 ( ${{c}_{4}}^{\prime }7$ )	−0.01 ( ${{c}_{4}}^{\prime }$ 6)
b' 1	1.00 (b' 1)	0.07 ( ${{c}_{4}}^{\prime }$ 0)	0.02 (b' 2)	0.02 ( ${{c}_{4}}^{\prime }$ 7)	−0.02 ( ${{c}_{4}}^{\prime }$ 6)
b' 2	1.00 (b' 2)	0.03 (b' 3)	−0.03 (b' 1)	0.03 ( ${{c}_{4}}^{\prime }$ 1)	−0.02 ( ${{c}_{4}}^{\prime }$ 4)
b' 3	0.99 (b' 3)	0.12 ( ${{c}_{4}}^{\prime }$ 1)	0.06 (b' 4)	−0.05 ( ${{c}_{4}}^{\prime }$ 2)	−0.04 (b' 2)
b' 4	0.91 (b' 4)	−0.40 ( ${{c}_{4}}^{\prime }$ 1)	−0.06 ( ${{c}_{4}}^{\prime }$ 2)	0.05 ( ${{c}_{4}}^{\prime }$ 3)	−0.03 ( ${{c}_{4}}^{\prime }$ 4)
b' 5	0.97 (b' 5)	−0.16 ( ${{c}_{4}}^{\prime }$ 1)	−0.16 ( ${{c}_{4}}^{\prime }$ 2)	0.07 ( ${{c}_{4}}^{\prime }$ 3)	−0.06 (b' 4)
b' 6	0.92 (b' 6)	−0.33 ( ${{c}_{4}}^{\prime }$ 2)	−0.12 (b' 5)	−0.11 ( ${{c}_{4}}^{\prime }$ 1)	0.09 (b' 7)
b' 7	0.78 (b' 7)	0.55 ( ${{c}_{4}}^{\prime }$ 2)	−0.19 (b' 8)	0.12 (b' 6)	0.09 ( ${{c}_{4}}^{\prime }$ 3)
b' 8	0.96 (b' 8)	0.23 ( ${{c}_{4}}^{\prime }$ 2)	−0.08 ( ${{c}_{4}}^{\prime }$ 1)	0.07( ${{c}_{4}}^{\prime }$ 3)	0.05 (b' 7)
b' 9	0.98 (b' 9)	0.10 ( ${{c}_{4}}^{\prime }$ 2)	−0.07 ( ${{c}_{4}}^{\prime }$ 1)	−0.06 ( ${{c}_{4}}^{\prime }$ 3)	0.06 ( ${{c}_{4}}^{\prime }$ 4)
b' 11	0.97 (b' 11)	0.19 ( ${{c}_{4}}^{\prime }$ 3)	0.10 ( ${{c}_{4}}^{\prime }$ 4)	−0.06 ( ${{c}_{5}}^{\prime }$ 0)	0.03 ( ${{c}_{4}}^{\prime }$ 0)
b' 12	0.98 (b' 12)	0.11 ( ${{c}_{4}}^{\prime }$ 3)	−0.09 ( ${{c}_{5}}^{\prime }$ 0)	0.07 ( ${{c}_{4}}^{\prime }$ 4)	0.05 ( ${{c}_{4}}^{\prime }$ 5)
b' 14	0.97 (b' 14)	0.16 ( ${{c}_{4}}^{\prime }$ 4)	−0.13 ( ${{c}_{5}}^{\prime }$ 0)	0.09 ( ${{c}_{4}}^{\prime }$ 5)	−0.06 (b' 13)
b' 15	0.97 (b' 15)	−0.17 ( ${{c}_{5}}^{\prime }$ 0)	0.11 ( ${{c}_{4}}^{\prime }$ 4)	−0.08 (b' 14)	0.07 ( ${{c}_{4}}^{\prime }$ 5)
b' 16	0.80 (b' 16)	−0.47 ( ${{c}_{4}}^{\prime }$ 5)	−0.27 ( ${{c}_{5}}^{\prime }$ 0)	0.15 (b' 17)	0.09 (b' 18)
b' 17	0.87 (b' 17)	−0.33 ( ${{c}_{5}}^{\prime }$ 0)	0.20 ( ${{c}_{4}}^{\prime }$ 5)	−0.20 (b' 16)	−0.10 (b' 15)
b' 18	0.77 (b' 18)	0.53 ( ${{c}_{5}}^{\prime }$ 0)	−0.19 (b' 19)	0.16 ( ${{c}_{4}}^{\prime }$ 6)	0.15 (b' 17)
b' 19	0.94 (b' 19)	0.24 ( ${{c}_{5}}^{\prime }$ 0)	−0.20 ( ${{c}_{5}}^{\prime }$ 1)	0.09 ( ${{c}_{4}}^{\prime }$ 6)	0.07 ( ${{c}_{4}}^{\prime }$ 5)
b' 21	0.75 (b' 21)	−0.50 ( ${{c}_{5}}^{\prime }$ 1)	−0.31 (b' 20)	0.15 (b' 22)	0.14 ( ${{c}_{4}}^{\prime }$ 6)
b' 22	0.86 (b' 22)	0.36 ( ${{c}_{5}}^{\prime }$ 1)	0.24 ( ${{c}_{4}}^{\prime }$ 7)	−0.18 (b' 23)	0.12 (b' 21)
b' 23	0.94 (b' 23)	0.21 ( ${{c}_{5}}^{\prime }$ 1)	−0.20 ( ${{c}_{5}}^{\prime }$ 2)	0.12 ( ${{c}_{4}}^{\prime }$ 7)	0.06 ( ${{c}_{4}}^{\prime }$ 8)
b' 24	0.91 (b' 24)	−0.34 ( ${{c}_{5}}^{\prime }$ 2)	0.15 ( ${{c}_{4}}^{\prime }$ 7)	0.10 ( ${{c}_{5}}^{\prime }$ 1)	−0.09 (b' 23)
${{c}_{4}}^{\prime }$ 0	0.99 ( ${{c}_{4}}^{\prime }$ 0)	−0.07 (b' 1)	−0.04 (b' 10)	−0.04 (b' 11)	−0.04 (b' 12)
${{c}_{4}}^{\prime }$ 1	0.87 ( ${{c}_{4}}^{\prime }$ 1)	0.40 (b' 4)	0.15 (b' 5)	−0.14 (b' 3)	0.11 (b' 6)
${{c}_{4}}^{\prime }$ 2	0.69 ( ${{c}_{4}}^{\prime }$ 2)	−0.60 (b' 7)	0.34 (b' 6)	−0.14 (b' 8)	0.11 (b' 5)
${{c}_{4}}^{\prime }$ 5	0.83 ( ${{c}_{4}}^{\prime }$ 5)	0.51 (b' 16)	−0.10 (b' 15)	−0.09 (b' 17)	−0.09 (b' 14)
${{c}_{4}}^{\prime }$ 7	0.92 ( ${{c}_{4}}^{\prime }$ 7)	−0.20 ( ${{c}_{5}}^{\prime }$ 1)	−0.16 (b' 22)	−0.13 (b' 21)	0.11 ( ${{c}_{5}}^{\prime }$ 2)
${{c}_{4}}^{\prime }$ 8	0.89 ( ${{c}_{4}}^{\prime }$ 8)	−0.28 ( ${{c}_{5}}^{\prime }$ 2)	−0.26 (b' 26)	−0.16 (b' 25)	−0.13 (b' 24)
${{c}_{5}}^{\prime }$ 0	0.61 ( ${{c}_{5}}^{\prime }$ 0)	−0.60 (b' 18)	0.40 (b' 17)	0.17 (b' 16)	−0.14 (b' 19)
${{c}_{5}}^{\prime }$ 2	0.61 (b' 25)	0.53 ( ${{c}_{5}}^{\prime }$ 2)	−0.48 (b' 26)	0.18 (b' 24)	−0.17 (b' 27)

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Table 2. Franck–Condon Factors for ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v') − ${{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v''), ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v') − ${{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v''), and ${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v') − ${{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'') Transitions of ${{\rm{N}}}_{2}$

State	v'/v''	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14
b'	0	0.000	0.000	0.000	0.000	0.001	0.002	0.004	0.009	0.018	0.033	0.052	0.074	0.095	0.112	0.120
	1	0.000	0.000	0.000	0.001	0.004	0.010	0.021	0.038	0.058	0.076	0.083	0.075	0.052	0.024	0.004
	2	0.000	0.000	0.001	0.005	0.014	0.029	0.049	0.067	0.072	0.059	0.032	0.007	0.001	0.019	0.045
	3	0.000	0.001	0.005	0.015	0.032	0.054	0.067	0.062	0.037	0.009	0.001	0.018	0.043	0.048	0.027
	4	0.000	0.003	0.012	0.031	0.053	0.065	0.053	0.024	0.002	0.007	0.032	0.044	0.027	0.003	0.005
	5	0.001	0.008	0.025	0.049	0.064	0.051	0.020	0.000	0.013	0.036	0.036	0.012	0.000	0.019	0.038
	6	0.003	0.017	0.042	0.062	0.054	0.021	0.000	0.014	0.036	0.030	0.005	0.004	0.027	0.032	0.010
	7	0.007	0.030	0.058	0.060	0.029	0.001	0.012	0.035	0.028	0.003	0.007	0.030	0.025	0.003	0.008
	8	0.013	0.045	0.065	0.042	0.006	0.006	0.032	0.029	0.004	0.007	0.029	0.021	0.001	0.012	0.029
	9	0.023	0.059	0.060	0.018	0.001	0.026	0.032	0.007	0.004	0.027	0.021	0.000	0.013	0.027	0.009
	11	0.051	0.069	0.020	0.002	0.031	0.024	0.000	0.017	0.026	0.003	0.008	0.025	0.008	0.003	0.023
	12	0.067	0.059	0.004	0.016	0.034	0.006	0.007	0.027	0.009	0.003	0.023	0.011	0.001	0.021	0.013
	14	0.092	0.022	0.009	0.036	0.006	0.010	0.025	0.003	0.010	0.021	0.001	0.011	0.019	0.001	0.013
	15	0.098	0.007	0.024	0.027	0.000	0.023	0.014	0.002	0.022	0.008	0.004	0.020	0.004	0.007	0.019
	16	0.107	0.000	0.041	0.012	0.010	0.027	0.001	0.016	0.017	0.000	0.019	0.010	0.003	0.020	0.004
	17	0.092	0.004	0.039	0.002	0.020	0.015	0.002	0.022	0.004	0.008	0.017	0.000	0.014	0.011	0.001
	18	0.081	0.016	0.032	0.001	0.027	0.004	0.012	0.016	0.000	0.018	0.006	0.005	0.017	0.000	0.012
	19	0.069	0.032	0.020	0.009	0.024	0.000	0.020	0.005	0.008	0.016	0.000	0.015	0.006	0.005	0.015
	21	0.045	0.066	0.000	0.030	0.003	0.019	0.009	0.006	0.018	0.000	0.016	0.005	0.007	0.013	0.000
	22	0.031	0.066	0.002	0.027	0.000	0.023	0.001	0.012	0.009	0.004	0.014	0.000	0.014	0.004	0.006
	23	0.022	0.066	0.010	0.020	0.003	0.021	0.001	0.016	0.001	0.012	0.007	0.004	0.013	0.000	0.013
	24	0.015	0.062	0.023	0.010	0.009	0.014	0.007	0.012	0.000	0.015	0.001	0.011	0.005	0.005	0.010
${{c}_{4}}^{\prime }$	0	0.909	0.087	0.004	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
	1	0.086	0.737	0.166	0.011	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
	2	0.005	0.159	0.577	0.234	0.023	0.001	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
	5	0.000	0.000	0.003	0.049	0.293	0.209	0.357	0.082	0.007	0.000	0.000	0.000	0.000	0.000	0.000
	7	0.000	0.000	0.000	0.001	0.010	0.095	0.307	0.065	0.365	0.138	0.017	0.001	0.000	0.000	0.000
	8	0.000	0.000	0.000	0.000	0.001	0.016	0.120	0.293	0.024	0.347	0.169	0.026	0.002	0.000	0.000
${{c}_{5}}^{\prime }$	0	0.917	0.080	0.003	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
	2	0.004	0.148	0.601	0.229	0.016	0.001	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000

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Transition wavelengths along with perturbed absorption oscillator strengths obtained in this study for the Rydberg ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v' = 0–2, 5, 7, 8) − ${{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'' = 0–14) and ${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ $(v^{\prime} =0,2)-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}(v^{\prime\prime} =0\mbox{--}14)$ bands are listed in Tables 3 and 4. In order to analyze the perturbation effects on oscillator strengths, unperturbed band f-values are included in the tables. Also shown are the experimental transition wavelengths found in the literature (Carroll et al. 1970; Roncin et al. 1987, 1991; Heays et al. 2014) for comparative purposes. As can be seen, a good agreement between theoretical and experimental transition energies is found for the ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(v^{\prime} =0\mbox{--}2)$ and ${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v' = 0) progressions. Regarding the remaining bands of Rydberg systems here studied, we have only found the transition energies reported by Carroll et al. (1970) for the ${{c}_{4}}^{\prime 1}$ ${{{\rm{\Sigma }}}_{u}}^{+}(5,7,8)-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}(0)$ and ${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(2)$ − ${{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (0) bands, which are in good agreement with our results. From now on, the lower state ${{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ and the symmetry of the upper electronic state will be omitted to denote bands; for instance, we will use the notation ${{c}_{4}}^{\prime }(0,0)$ to denote the ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(0)-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}(0)$ band.

Table 3. Transition Wavelengths (in Å) and Absorption Oscillator Strengths for the Rydberg ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v' = 0, 1, 2, 5) − ${{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'') Progressions of ${{\rm{N}}}_{2}$

	${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(0)$ Progression				${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(1)$ Progression				${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(2)$ Progression				${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(5)$ Progression
v''	λ^a	λ^b	f^c	f^d	λ^a	λ^b	f^c	f^d	λ^a	λ^b	f^c	f^d	λ^a	λ^e	f^c	f^d
0	958.55	958.18	1.24E–1	1.54E–1	940.15	940.03	1.19E–2	9.21E–3	921.42	921.23	7.13E–4	1.20E–3	870.84	870.76	1.21E–7	5.03E–5
1	980.44	980.46	1.17E–2	1.87E–2	961.21	961.18	1.00E–1	1.10E–1	941.64	941.49	2.22E–2	8.62E–3	888.88	⋯	9.63E–6	6.90E–6
2	1003.08	1003.10	4.72E–4	1.55E–3	982.95	982.95	2.21E–2	4.62E–2	962.50	962.34	7.86E–2	6.13E–2	907.44	⋯	4.40E–4	5.10E–4
3	1026.47	1026.49	1.13E–5	2.08E–4	1005.41	1005.38	1.45E–3	1.27E–2	984.02	983.85	3.11E–2	2.70E–2	926.55	⋯	6.94E–3	4.19E–3
4	1050.67	1050.69	1.29E–7	7.99E–5	1028.61	1028.59	4.78E–5	4.62E–3	1006.24	1006.07	2.94E–3	7.66E–4	946.22	⋯	4.06E–2	2.69E–2
5	1075.71	1075.73	4.56E–10	5.04E–5	1052.59	1052.57	7.88E–7	1.94E–3	1029.17	1029.00	1.27E–4	2.87E–3	966.47	⋯	2.83E–2	2.46E–2
6	1101.61	1101.64	9.13E–11	5.36E–5	1077.39	1077.36	4.30E–9	3.25E–4	1052.86	1052.68	2.87E–6	5.57E–3	987.34	⋯	4.74E–2	2.96E–2
7	1128.43	1128.46	1.19E–10	5.52E–5	1103.03	1102.79	1.28E–9	4.11E–5	1077.33	1077.15	2.16E–8	3.37E–3	1008.82	⋯	1.06E–2	1.58E–3
8	1156.20	1156.23	5.58E–11	6.73E–5	1129.54	1129.51	1.54E–10	8.27E–4	1102.62	1102.42	5.43E–9	1.51E–4	1030.96	⋯	8.85E–4	2.19E–4
9	1184.97	1185.00	1.42E–10	6.52E–5	1156.98	1156.95	4.61E–12	1.56E–3	1128.75	1128.53	3.90E–10	1.06E–3	1053.77	⋯	4.12E–5	3.59E–5
10	1214.77	1214.80	8.72E–11	5.44E–5	1185.38	1185.34	2.85E–11	1.47E–3	1155.76	1155.54	4.85E–11	3.44E–3	1077.27	⋯	7.70E–7	1.31E–3
11	1245.66	1243.50	1.06E–10	3.71E–5	1214.77	1214.73	1.70E–9	6.66E–4	1183.68	1183.45	1.85E–8	2.73E–3	1101.50	⋯	8.20E–8	7.87E–4
12	1277.68	1275.22	7.48E–12	1.90E–5	1245.21	1244.77	1.20E–11	3.17E–5	1212.56	1212.32	1.16E–11	3.03E–4	1126.46	⋯	3.50E–10	1.37E–4
13	1310.90	1308.08	3.25E–11	4.67E–6	1276.73	1276.24	1.28E–11	2.66E–4	1242.44	1242.07	3.07E–12	7.31E–4	1152.20	⋯	2.80E–10	1.40E–3
14	1345.35	1342.14	2.56E–12	1.21E–8	1309.39	1308.83	2.34E–12	9.96E–4	1273.34	1272.94	1.83E–11	2.89E–3	1178.73	⋯	4.40E–10	3.37E–4

Notes.

^aThis work. ^bHeays et al. (2014), Roncin et al. (1987), Roncin et al. (1991). ^cThis work, with "non-mixed" vibronic transition moments. ^dThis work, with "mixed" vibronic transition moments. ^eCarroll et al. (1970).

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Table 4. Transition Wavelengths (in Å) and Absorption Oscillator Strengths for the Rydberg ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v' = 7, 8) − ${{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'') and ${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v' = 0, 2) − ${{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'') Progressions of ${{\rm{N}}}_{2}$

	${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(7)$ Progression				${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(8)$ Progression				${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(0)$ Progression				${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(2)$ Progression
v''	λ^a	λ^b	f^c	f^d	λ^a	λ^b	f^c	f^d	λ^a	λ^e	f^c	f^d	λ^a	λ^b	f^c	f^d
0	842.12	841.91	4.18E–10	2.76E–3	828.71	828.54	9.02E–11	7.85E–4	863.26	863.16	3.41E–2	8.87E–3	832.01	831.98	1.65E–4	3.12E–4
1	858.97	⋯	3.49E–8	4.72E–3	845.03	⋯	4.46E–10	3.18E–3	880.98	880.91	2.92E–3	5.39E–3	848.46	⋯	5.61E–3	2.22E–5
2	876.29	⋯	3.51E–6	2.14E–3	861.79	⋯	3.89E–7	7.49E–4	899.21	899.13	1.05E–4	1.08E–6	865.35	⋯	2.23E–2	4.52E–3
3	894.10	⋯	7.38E–5	1.74E–3	879.00	⋯	5.74E–6	1.44E–6	917.97	917.89	2.97E–6	4.25E–3	882.71	⋯	8.31E–3	1.06E–4
4	912.40	⋯	1.46E–3	1.31E–3	896.69	⋯	1.55E–4	7.97E–4	937.28	937.19	9.13E–9	5.08E–4	900.55	⋯	5.60E–4	2.54E–4
5	931.22	⋯	1.34E–2	1.83E–2	914.86	⋯	2.36E–3	4.96E–3	957.15	957.06	7.12E–11	3.74E–4	918.88	⋯	2.11E–5	8.42E–4
6	950.58	⋯	4.23E–2	3.20E–2	933.53	⋯	1.69E–2	1.95E–2	977.61	977.51	3.43E–11	1.13E–3	937.71	⋯	1.51E–7	1.19E–3
7	970.48	⋯	8.76E–3	5.48E–3	952.72	⋯	4.03E–2	2.23E–2	998.67	998.52	1.04E–10	1.91E–4	957.08	⋯	3.48E–8	4.46E–3
8	990.95	⋯	4.83E–2	3.20E–2	972.44	⋯	3.25E–3	2.10E–3	1020.36	1020.20	1.77E–10	3.53E–3	976.98	⋯	1.90E–8	1.99E–5
9	1012.00	⋯	1.79E–2	1.42E–2	992.71	⋯	4.58E–2	2.35E–2	1042.69	1042.54	1.73E–10	1.83E–3	997.44	⋯	1.86E–8	3.65E–3
10	1033.66	⋯	2.21E–3	2.37E–3	1013.54	⋯	2.19E–2	1.50E–2	1065.70	1065.53	1.27E–10	1.81E–4	1018.47	⋯	2.82E–8	2.85E–3
11	1055.94	⋯	1.53E–4	2.26E–4	1034.95	⋯	3.31E–3	4.70E–3	1089.40	1089.22	1.24E–10	2.05E–3	1040.09	⋯	2.76E–8	6.20E–5
12	1078.86	⋯	5.48E–6	5.55E–6	1056.96	⋯	2.80E–4	6.89E–4	1113.81	1113.62	4.66E–11	3.42E–4	1062.33	⋯	1.01E–8	8.37E–4
13	1102.45	⋯	4.97E–7	3.60E–6	1079.59	⋯	1.36E–5	1.55E–4	1138.97	1138.77	1.34E–11	8.92E–4	1085.18	⋯	5.50E–9	2.91E–7
14	1126.71	⋯	8.64E–9	1.83E–7	1102.85	⋯	1.22E–6	1.40E–4	1164.89	1164.67	4.93E–12	1.64E–3	1108.69	⋯	1.15E–9	1.18E–3

Notes.

^aThis work. ^bCarroll et al. (1970). ^cThis work, with "non-mixed" vibronic transition moments. ^dThis work, with "mixed" vibronic transition moments. ^eHeays et al. (2014), Roncin et al. (1987).

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The potential curves of the ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ and ${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ Rydberg states and the ground state ${{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ exhibit almost identical equilibrium internuclear (Lofthus & Krupenie 1977). It is, thus, expected that the intensity distribution in the ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v') and ${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v') progressions give rise to a strong enhancement of absorption when v' ≈ v'' and to bands that decrease rapidly in intensity when v' and v'' are very different. The oscillator strengths calculated with non-mixed transition moments (Equation (2)) behave in this way. However, as it is known, bands other than those with v' ≈ v'' of the ${c}_{4}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v') and ${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v') progressions are abnormally strong due to interaction with nearby lying levels. Our calculations predict the ${{c}_{4}}^{\prime }(0,0)$ and ${{c}_{4}}^{\prime }(1,1)$ as the most intense bands in absorption in the ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ system. It can also be inferred that, if perturbations are not considered, the oscillator strengths are extremely weak for the ${{c}_{4}}^{\prime }(0,v^{\prime\prime} )$ bands with v'' ≥ 3, owing to the poor Franck–Condon overlap between the ${{c}_{4}}^{\prime }(0)$ and ${\rm{X}}({v}^{\prime\prime }\geqslant 3)$ states. A consequence of the interaction between the ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (0) and ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ vibrational states, mainly ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (1), is an increasing of the oscillator strength in all members of the ${{c}_{4}}^{\prime }(v^{\prime} =0)$ progression. In fact, for ${{{\rm{c}}}_{4}}^{\prime }(0,v^{\prime\prime} )$ bands with v'' = 3–14, the largest contribution to the vibronic transition moment comes, not from the unperturbed ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (0) state, but from the ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ states due to the better vibrational overlap of the latter with the vibrational levels of the ground state. It seems, thus, that the observation of ${{c}_{4}}^{\prime }(0,v^{\prime\prime} )$ bands with v'' = 6–9 in the terrestrial airglow spectra obtained with FUSE (Bishop et al. 2007) can be explained in terms of the homogeneous interaction ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ .

The largest perturbation between the ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ and ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ states occurs at ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (2) vibrational level, as can be seen in Table 1. The fractional diabatic electronic character, ${P}_{e}$ , of a perturbed state, $| {e}_{0},{v}_{0}\rangle$ , can be obtained by using the expression given by Stahel et al. (1983).

$\begin{eqnarray}&&{P}_{e}=\displaystyle \sum _{{v}_{e}}{| {C}_{{v}_{0},{v}_{e}}| }^{2}.\end{eqnarray} \tag{ 7 }$

The sum in Equation (7) is extended to all vibrational levels of the individual diabatic electronic state. According to our calculations, for the conventionally named ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(2)$ state, the percentages of ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ and ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ character are 51.6% and 48.4%, respectively. In contrast to what is expected on the basis Franck–Condon factors, the ${{c}_{4}}^{\prime }(2,5)$ , ${{c}_{4}}^{\prime }(2,6)$ , ${{c}_{4}}^{\prime }(2,10)$ , and ${{c}_{4}}^{\prime }(2,11)$ bands have been reported as being strong in emission (Roncin et al. 1987, 1991). The present results obtained with mixed transition moments reveal that the b' contribution to the vibronic transition moment completely dominates the oscillator strengths for the members of the ${{c}_{4}}^{\prime }(2)-{\rm{X}}({v}^{\prime\prime })$ progression with ${v}^{\prime\prime }\geqslant 5$ .

The ${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ band system is weaker than the ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ system. This feature can be understood on the grounds of a decreasing electronic transition moment as the energy of the upper state increases along a given Rydberg series, which is not compensated by the increase in transition energy. A comparison between unperturbed and perturbed oscillator strengths reveals that the ${{c}_{5}}^{\prime }(0)-{\rm{X}}({v}^{\prime\prime })$ and ${{c}_{5}}^{\prime }(2)-{\rm{X}}(v^{\prime\prime} )$ progressions show strong deviations from considerations based on the Franck–Condon principle. Such deviations can be attributed to the fact that the ${{c}_{5}}^{\prime }(0)$ and ${{c}_{5}}^{\prime }(2)$ vibronic states are strongly perturbed, as shown in Table 1. In fact, the electronic character of the ${{c}_{5}}^{\prime }$ state is predominantly b'. According our calculations, ${{c}_{5}}^{\prime }(0)$ has 61.1% of b' character, 37.5% of ${{c}_{5}}^{\prime }$ character, and 1.4% of ${{c}_{4}}^{\prime }$ character, and ${{c}_{5}}^{\prime }(2)$ has 68.8% of b' character, 28.3% of ${{c}_{5}}^{\prime }$ character, and 2.9% of ${{c}_{4}}^{\prime }$ character.

Regarding the ${{c}_{5}}^{\prime }(0)-{\rm{X}}({v}^{\prime\prime })$ transitions, the ${{c}_{5}}^{\prime }(0,0)$ is the most intense band in absorption. Bands with v'' > 2 are relatively strong as a consequence of the perturbation; these bands borrow their intensities from b' states, mainly b'(18) and b'(17). The perturbation is also responsible of the great weakening in the intensity of the ${{c}_{5}}^{\prime }(0,2)$ band. For this band, the contributions of the ${{c}_{5}}^{\prime }$ and b' vibrational states to the transition moment (Equation (5)) are similar in magnitude but opposite sign resulting in a destructive interference and therefore in a decrease in the band f -value. Bands belonging to the ${{c}_{5}}^{\prime }(0)-{\rm{X}}({v}^{\prime\prime })$ progression have been observed in emission (Roncin et al. 1987; Heays et al. 2014). The perturbation effects on the band intensities are also noticeable in the ${{c}_{5}}^{\prime }(2)-{\rm{X}}({v}^{\prime\prime })$ progression. Our calculations predict that several bands of this progression have appreciable absorption intensities. On the other hand, the weakening of the f -value of the ${{c}_{5}}^{\prime }(2,1)$ can be attributed to the destructive interference caused by the coupling ${{c}_{5}}^{\prime }(2)-{\text{}}b^{\prime} (25)-{\text{}}b^{\prime} (26)$ .

In contrast to Rydberg systems, large progressions are expected for the valence ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ system since the electronic ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ state shows a potential curve with an equilibrium internuclear distance much larger than that of the ground state. In Tables 5–9, the calculated unperturbed and perturbed absorption oscillator strengths, and transition wavelengths for the valence ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v' = 0–9, 11, 12, 14–19, 21, 22) − ${{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'' = 0–14) bands are shown. Experimental transition wavelengths (Wilkinson & Houk 1956; Tilford & Wilkinson 1964; Carroll et al. 1970; Verma & Jois 1984; Roncin et al. 1987, 1991; Heays et al. 2014) are also included in the tables. The transition energies presently obtained show a very good agreement with the experimental ones whenever available. For bands belonging to v' = 21 and 22 progressions, we have only found in the literature transition energies for the b'(21, 0) and b'(22, 0). Experimental studies of the ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ transition performed by Ajello et al. 1989 indicate strong predissociation for v' > 19.

Table 5. Transition Wavelengths (in Å) and Absorption Oscillator Strengths for the Valence ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v' = 0–3) − ${{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'') Progressions of ${{\rm{N}}}_{2}$

	${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(0)$ Progression				${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(1)$ Progression				${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(2)$ Progression				${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(3)$ Progression
v''	λ^a	λ^b	f^c	f^d	λ^a	λ^e	f^c	f^d	λ^a	λ^f	f^c	f^d	λ^a	λ^g	f^c	f^d
0	964.61	964.56	5.87E–8	1.08E–6	957.81	957.66	7.96E–7	6.56E–4	951.13	950.99	5.52E–6	3.82E–5	944.63	944.54	2.63E–5	1.74E–4
1	986.79	⋯	8.12E–7	2.96E–6	979.68	979.52	9.52E–6	1.34E–5	972.68	⋯	5.68E–5	1.93E–4	965.89	⋯	2.32E–4	2.84E–3
2	1009.72	⋯	4.49E–6	1.06E–5	1002.27	1002.10	4.88E–5	7.05E–5	994.95	⋯	2.65E–4	7.50E–4	987.84	⋯	9.57E–4	4.72E–3
3	1033.43	⋯	2.72E–5	4.62E–5	1025.63	1025.46	2.39E–4	3.84E–4	1017.97	1017.84	1.05E–3	1.95E–3	1010.53	1010.43	3.03E–3	7.02E–3
4	1057.97	⋯	1.01E–4	1.51E–4	1049.79	1049.60	7.55E–4	1.10E–3	1041.77	1041.60	2.75E–3	4.16E–3	1033.97	1033.90	6.48E–3	1.05E–2
5	1083.35	⋯	3.06E–4	4.10E–4	1074.79	1074.63	1.90E–3	2.54E–3	1066.37	1066.23	5.65E–3	7.58E–3	1058.21	1058.14	1.05E–2	1.40E–2
6	1109.63	⋯	7.76E–4	1.04E–3	1100.65	1100.49	3.97E–3	5.12E–3	1091.83	1091.68	9.32E–3	1.16E–2	1083.27	1083.19	1.28E–2	1.48E–2
7	1136.85	1136.74	1.70E–3	2.01E–3	1127.42	1127.25	6.97E–3	8.24E–3	1118.17	1118.01	1.24E–2	1.43E–2	1109.19	1109.10	1.15E–2	1.19E–2
8	1165.04	1164.93	3.27E–3	4.10E–3	1155.14	1154.96	1.04E–2	1.25E–2	1145.43	1145.26	1.31E–2	1.44E–2	1136.01	1135.92	6.65E–3	5.66E–3
9	1194.25	1194.13	5.63E–3	6.71E–3	1183.85	1183.66	1.32E–2	1.50E–2	1173.65	1173.48	1.04E–2	1.05E–2	1163.77	1163.67	1.56E–3	8.10E–4
10	1224.53	1224.41	8.70E–3	9.55E–3	1213.59	1213.40	1.42E–2	1.48E–2	1202.88	1202.70	5.44E–3	4.86E–3	1192.50	⋯	1.05E–4	4.48E–4
11	1255.92	1255.7	1.21E–2	1.26E–2	1244.42	1244.17	1.24E–2	1.22E–2	1233.16	⋯	1.12E–3	7.17E–4	1222.25	1222.15	3.08E–3	3.94E–3
12	1288.48	1288.3	1.52E–2	1.55E–2	1276.38	1276.11	8.39E–3	7.79E–3	1264.54	⋯	1.51E–4	3.65E–4	1253.07	⋯	7.04E–3	7.37E–3
13	1322.27	1322.0	1.75E–2	1.75E–2	1309.53	1309.24	3.74E–3	3.21E–3	1297.06	⋯	2.95E–3	3.55E–3	1285.00	1284.8	7.65E–3	7.05E–3
14	1357.33	⋯	1.83E–2	1.82E–2	1343.91	1343.60	5.40E–4	3.46E–4	1330.78	⋯	7.01E–3	7.46E–3	1318.09	⋯	4.23E–3	3.33E–3

Notes.

^aThis work. ^bWilkinson & Houk (1956), Roncin et al. (1991), Verma & Jois (1984). ^cThis work, with "non-mixed" vibronic transition moments. ^dThis work, with "mixed" vibronic transition moments. ^eHeays et al. (2014), Roncin et al. (1987), Roncin et al. (1991). ^fCarroll et al. (1970), Tilford & Wilkinson (1964), Roncin et al. (1987), Roncin et al. (1991). ^gCarroll et al. (1970), Tilford & Wilkinson (1964), Roncin et al. (1987), Roncin et al. (1991), Verma & Jois (1984).

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Table 6. Transition Wavelengths (in Å) and Absorption Oscillator Strengths and Transition Energies for the Valence ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v' = 4–7) − ${{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'') Progressions of ${{\rm{N}}}_{2}$

	${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(4)$ Progression				${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(5)$ Progression				${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(6)$ Progression				${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(7)$ Progression
v''	λ^a	λ^b	f^c	f^d	λ^a	λ^e	f^c	f^d	λ^a	λ^e	f^c	f^d	λ^a	λ^f	f^c	f^d
0	937.78	937.65	9.48E–5	2.42E–3	931.84	931.72	2.81E–4	9.27E–4	926.04	925.91	7.02E–4	1.38E–3	917.98	917.80	1.56E–3	1.48E–4
1	958.73	⋯	7.08E–4	1.87E–2	952.52	952.42	1.75E–3	1.63E–3	946.46	⋯	3.63E–3	1.19E–3	938.04	937.89	6.52E–3	1.53E–2
2	980.36	⋯	2.53E–3	4.62E–4	973.86	973.76	5.28E–3	6.82E–3	967.53	967.42	8.90E–3	2.61E–2	958.73	958.58	1.25E–2	1.28E–2
3	1002.70	⋯	6.34E–3	4.09E–3	995.90	995.79	1.02E–2	1.82E–2	989.28	989.17	1.29E–2	3.09E–2	980.09	979.93	1.27E–2	1.67E–6
4	1025.78	⋯	1.07E–2	1.06E–2	1018.67	1018.56	1.29E–2	1.84E–2	1011.74	1011.62	1.10E–2	1.42E–2	1002.13	1001.96	5.94E–3	5.52E–3
5	1049.62	⋯	1.28E–2	1.31E–2	1042.18	1042.05	1.01E–2	1.13E–2	1034.93	1034.82	4.24E–3	2.17E–3	1024.88	1024.70	2.21E–4	3.44E–3
6	1074.28	⋯	1.03E–2	1.05E–2	1066.48	1066.36	3.79E–3	3.11E–3	1058.89	⋯	3.20E–5	4.43E–4	1048.37	1048.19	2.30E–3	4.67E–6
7	1099.76	⋯	4.50E–3	4.83E–3	1091.60	⋯	3.61E–5	2.45E–5	1083.64	⋯	2.64E–3	5.11E–3	1072.63	1072.43	6.81E–3	3.16E–3
8	1126.12	⋯	2.85E–4	3.00E–4	1117.56	1117.43	2.32E–3	3.63E–3	1109.23	1109.09	6.78E–3	7.24E–3	1097.69	1097.49	5.24E–3	5.37E–3
9	1153.39	1153.24	1.28E–3	1.21E–3	1144.42	1144.27	6.56E–3	7.57E–3	1135.68	1135.53	5.46E–3	3.64E–3	1123.58	1123.37	5.83E–4	2.30E–3
10	1181.61	⋯	5.53E–3	4.93E–3	1172.19	1172.04	6.37E–3	5.85E–3	1163.02	⋯	9.33E–4	8.14E–5	1150.34	1150.05	1.23E–3	1.53E–7
11	1210.82	1210.6	7.44E–3	6.46E–3	1200.93	1200.77	2.05E–3	1.29E–3	1191.31	1191.15	6.85E–4	1.88E–3	1178.00	1177.77	5.19E–3	1.96E–3
12	1241.05	⋯	4.49E–3	3.88E–3	1230.66	⋯	5.87E–5	3.06E–4	1220.56	1220.40	4.58E–3	5.12E–3	1206.60	1206.36	4.35E–3	3.73E–3
13	1272.36	⋯	5.61E–4	5.06E–4	1261.45	⋯	3.11E–3	3.71E–3	1250.84	⋯	5.33E–3	3.82E–3	1236.18	1235.93	4.50E–4	1.70E–3
14	1304.80	⋯	7.52E–4	6.18E–4	1293.32	⋯	6.01E–3	5.79E–3	1282.17	⋯	1.59E–3	4.39E–4	1266.77	1266.40	1.23E–3	1.69E–5

Notes.

^aThis work. ^bCarroll et al. (1970), Roncin et al. (1991), Verma & Jois (1984). ^cThis work, with "non-mixed" vibronic transition moments. ^dThis work, with "mixed" vibronic transition moments. ^eCarroll et al. (1970), Roncin et al. (1987), Roncin et al. (1991). ^fHeays et al. (2014), Roncin et al. (1987), Roncin et al. (1991).

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Table 7. Transition Wavelengths (in Å) and Absorption Oscillator Strengths for the Valence ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}({v}^{\prime }=8,9,11,12)-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}(v^{\prime\prime} )$ Progressions of ${{\rm{N}}}_{2}$

	${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(8)$ Progression				${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(9)$ Progression				${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(11)$ Progression				${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(12)$ Progression
v''	λ^a	λ^b	f^c	f^d	λ^a	λ^b	f^c	f^d	λ^a	λ^b	f^c	f^d	λ^a	λ^b	f^c	f^d
0	912.98	912.85	2.97E–3	2.38E–3	907.54	907.45	5.14E–3	5.89E–3	896.31	896.19	1.16E–2	1.01E–2	891.06	890.94	1.54E–2	1.64E–2
1	932.82	932.71	9.85E–3	1.24E–2	927.15	927.07	1.31E–2	1.35E–2	915.42	915.33	1.55E–2	1.26E–2	909.95	909.85	1.33E–2	1.16E–2
2	953.29	953.17	1.41E–2	9.33E–4	947.36	947.28	1.30E–2	9.32E–3	935.12	935.02	4.50E–3	2.70E–4	929.41	929.29	9.12E–4	7.56E–6
3	974.40	974.27	8.96E–3	5.75E–3	968.20	968.12	3.91E–3	2.17E–3	955.43	955.32	4.90E–4	1.96E–3	949.47	949.35	3.46E–3	1.90E–4
4	996.18	996.04	1.18E–3	2.37E–3	989.70	989.62	1.82E–4	9.74E–4	976.36	976.25	6.53E–3	3.45E–3	970.13	970.02	7.22E–3	6.49E–3
5	1018.65	1018.52	1.27E–3	5.65E–4	1011.89	1011.80	5.22E–3	7.89E–3	997.94	997.82	5.00E–3	6.17E–3	991.44	991.32	1.31E–3	1.48E–3
6	1041.86	1041.71	6.33E–3	6.32E–3	1034.78	1034.70	6.47E–3	8.28E–3	1020.20	1020.04	4.26E–5	1.70E–4	1013.41	1013.29	1.49E–3	2.09E–3
7	1065.81	1065.66	5.62E–3	7.23E–3	1058.41	1058.30	1.31E–3	1.62E–3	1043.16	1043.03	3.28E–3	3.79E–3	1036.06	1035.89	5.36E–3	7.04E–3
8	1090.55	1090.36	7.02E–4	1.32E–3	1082.80	1082.66	8.38E–4	9.71E–4	1066.84	1066.71	4.93E–3	5.67E–3	1059.42	1059.28	1.70E–3	1.80E–3
9	1116.11	1115.91	1.24E–3	7.85E–4	1107.99	1107.84	4.98E–3	5.28E–3	1091.29	1091.11	6.26E–4	7.91E–4	1083.52	1083.34	6.33E–4	8.15E–4
10	1142.51	1142.30	5.18E–3	4.63E–3	1134.00	1133.85	3.74E–3	3.70E–3	1116.51	1116.32	1.49E–3	1.40E–3	1108.38	1108.21	4.31E–3	4.54E–3
11	1169.79	1169.58	3.76E–3	3.98E–3	1160.87	1160.71	7.98E–5	6.72E–5	1142.55	1142.36	4.49E–3	4.47E–3	1134.04	1133.85	2.04E–3	1.92E–3
12	1197.99	1197.76	1.27E–4	3.31E–4	1188.64	1188.47	2.28E–3	2.20E–3	1169.44	1169.24	1.42E–3	1.45E–3	1160.52	1160.32	2.38E–4	3.06E–4
13	1227.14	1226.90	1.99E–3	1.33E–3	1217.33	1217.15	4.62E–3	4.30E–3	1197.20	1196.99	5.18E–4	4.92E–4	1187.86	1187.65	3.58E–3	3.62E–3
14	1257.28	1257.02	4.80E–3	4.30E–3	1246.99	1246.79	1.44E–3	1.28E–3	1225.87	1225.64	3.83E–3	3.72E–3	1216.08	1215.85	2.29E–3	2.10E–3

Notes.

^aThis work. ^bHeays et al. (2014), Roncin et al. (1987). ^cThis work, with "non-mixed" vibronic transition moments. ^dThis work, with "mixed" vibronic transition moments.

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Table 8. Transition Wavelengths (in Å) and Absorption Oscillator Strengths for the Valence ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (v' = 14–17) − ${{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ (v'') Progressions of ${{\rm{N}}}_{2}$

	${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(14)$ Progression				${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(15)$ Progression				${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(16)$ Progression				${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(17)$ Progression
v''	λ^a	λ^b	f^c	f^d	λ^a	λ^b	f^c	f^d	λ^a	λ^b	f^c	f^d	λ^a	λ^e	f^c	f^d
0	880.86	880.72	2.16E–2	2.16E–2	876.01	875.87	2.31E–2	2.63E–2	871.59	871.40	2.53E–2	3.96E–2	866.87	866.77	2.18E–2	1.97E–2
1	899.32	899.18	5.02E–3	2.81E–3	894.26	894.12	1.51E–3	3.06E–4	889.65	889.49	1.87E–7	1.07E–3	884.74	⋯	9.04E–4	3.61E–3
2	918.32	918.20	2.06E–3	3.61E–3	913.05	912.92	5.43E–3	6.72E–3	908.25	908.08	9.23E–3	9.29E–3	903.13	903.03	8.95E–3	4.34E–3
3	937.90	937.74	7.87E–3	1.09E–3	932.40	932.24	6.07E–3	1.34E–3	927.39	927.21	2.72E–3	3.78E–3	922.05	⋯	3.98E–4	1.15E–3
4	958.06	957.92	1.25E–3	5.01E–3	952.33	952.16	1.66E–5	4.54E–4	947.10	946.91	2.14E–3	1.89E–2	941.53	⋯	4.44E–3	1.32E–4
5	978.83	978.67	2.13E–3	1.07E–3	972.85	972.68	4.89E–3	4.34E–3	967.39	967.16	5.78E–3	2.03E–4	961.58	961.48	3.30E–3	4.64E–3
6	1000.23	1000.06	5.26E–3	5.97E–3	993.99	993.81	2.85E–3	2.46E–3	988.29	988.09	2.31E–4	1.24E–2	982.23	⋯	3.85E–4	7.39E–5
7	1022.29	1022.13	5.43E–4	3.54E–4	1015.77	1015.58	3.72E–4	1.34E–3	1009.82	1009.61	3.35E–3	1.40E–2	1003.50	⋯	4.42E–3	2.86E–3
8	1045.03	1044.84	2.07E–3	2.89E–3	1038.22	1038.02	4.29E–3	5.22E–3	1032.00	1031.74	3.46E–3	3.01E–3	1025.40	⋯	8.83E–4	8.57E–7
9	1068.47	1068.31	4.08E–3	4.29E–3	1061.35	1061.14	1.48E–3	9.85E–4	1054.86	1054.57	4.97E–5	5.15E–4	1047.96	⋯	1.57E–3	3.03E–3
10	1092.65	1092.44	2.70E–4	1.51E–4	1085.20	1084.98	7.95E–4	1.30E–3	1078.41	1078.12	3.72E–3	2.85E–3	1071.20	⋯	3.35E–3	2.15E–3
11	1117.57	1117.36	2.07E–3	2.34E–3	1109.78	1109.56	3.76E–3	3.54E–3	1102.69	1102.38	1.85E–3	3.83E–4	1095.15	⋯	4.50E–5	1.59E–4
12	1143.28	1143.05	3.39E–3	3.08E–3	1135.13	1134.89	7.14E–4	3.01E–4	1127.71	1127.38	5.44E–4	1.14E–3	1119.83	⋯	2.52E–3	2.68E–3
13	1169.80	1169.56	1.11E–4	2.00E–5	1161.27	1161.02	1.26E–3	1.79E–3	1153.50	1153.16	3.65E–3	2.10E–3	1145.26	⋯	2.04E–3	7.65E–4
14	1197.16	1196.90	2.16E–3	2.47E–3	1188.22	1187.96	3.25E–3	2.90E–3	1180.09	1179.73	7.21E–4	2.72E–5	1171.47	⋯	1.85E–4	8.53E–4

Notes.

^aThis work. ^bHeays et al. (2014), Roncin et al. (1987). ^cThis work, with "non-mixed" vibronic transition moments. ^dThis work, with "mixed" vibronic transition moments. ^eCarroll et al. (1970), Roncin et al. (1987).

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Table 9. Transition Wavelengths (in Å) and Absorption Oscillator Strengths for the Valence ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(v^{\prime} =18,19,21,22)-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}(v^{\prime\prime} )$ Progressions of ${{\rm{N}}}_{2}$

	${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(18)$ Progression				${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(19)$ Progression				${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(21)$ Progression				${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(22)$ Progression
v''	λ^a	λ^b	f^c	f^d	λ^a	λ^e	f^c	f^d	λ^a	λ^f	f^c	f^d	λ^a	λ^f	f^c	f^d
0	860.62	860.51	1.95E–2	1.61E–3	857.13	857.01	1.65E–2	1.17E–2	849.87	849.74	1.09E–2	3.67E–3	844.04	843.98	7.64E–3	3.67E–3
1	878.23	878.12	3.74E–3	2.67E–4	874.59	874.49	7.60E–3	1.24E–2	867.04	⋯	1.57E–2	2.54E–2	860.97	⋯	1.58E–2	1.72E–3
2	896.35	896.26	7.43E–3	6.35E–3	892.56	892.45	4.55E–3	5.01E–3	884.69	⋯	7.45E–5	8.06E–4	878.37	⋯	3.83E–4	2.70E–5
3	914.98	914.89	2.25E–4	5.90E–4	911.04	⋯	2.10E–3	1.54E–3	902.84	⋯	6.85E–3	3.09E–3	896.26	⋯	6.16E–3	4.82E–3
4	934.16	934.04	5.99E–3	1.45E–4	930.05	⋯	5.25E–3	1.36E–3	921.51	⋯	7.16E–4	1.21E–3	914.66	⋯	1.31E–7	1.97E–4
5	953.90	953.8	7.83E–4	6.73E–3	949.61	949.43	4.64E–5	8.38E–4	940.71	⋯	4.23E–3	1.16E–3	933.57	⋯	5.05E–3	2.31E–7
6	974.22	974.08	2.51E–3	6.24E–4	969.75	969.62	4.25E–3	4.22E–3	960.46	⋯	1.99E–3	3.47E–4	953.02	⋯	3.21E–4	8.73E–3
7	995.13	995.02	3.41E–3	7.52E–3	990.47	990.33	1.10E–3	1.74E–3	980.79	⋯	1.17E–3	1.13E–3	973.03	⋯	2.60E–3	2.13E–3
8	1016.67	1016.51	6.24E–5	1.37E–3	1011.80	⋯	1.67E–3	1.57E–3	1001.70	⋯	3.64E–3	2.05E–4	993.61	⋯	1.84E–3	8.36E–3
9	1038.84	1038.68	3.52E–3	1.17E–3	1033.76	⋯	3.13E–3	3.27E–3	1023.22	⋯	3.50E–6	2.47E–3	1014.78	⋯	8.53E–4	3.71E–4
10	1061.68	1061.50	1.17E–3	2.14E–3	1056.37	⋯	3.55E–6	5.42E–6	1045.36	⋯	3.22E–3	2.96E–3	1036.56	⋯	2.85E–3	1.74E–3
11	1085.20	1085.01	1.01E–3	8.87E–7	1079.65	⋯	2.92E–3	2.32E–3	1068.16	⋯	9.06E–4	2.06E–5	1058.96	⋯	3.29E–7	3.72E–4
12	1109.42	1109.23	3.08E–3	1.93E–3	1103.62	⋯	1.18E–3	1.02E–3	1091.62	⋯	1.37E–3	2.50E–3	1082.02	⋯	2.58E–3	1.18E–3
13	1134.38	1134.17	9.05E–5	8.54E–4	1128.31	⋯	8.25E–4	6.51E–4	1115.77	⋯	2.48E–3	4.88E–4	1105.74	⋯	7.43E–4	1.50E–3
14	1160.08	1159.86	2.18E–3	4.82E–4	1153.75	⋯	2.77E–3	2.38E–3	1140.63	⋯	3.90E–5	1.17E–3	1130.16	⋯	1.12E–3	1.69E–4

Notes.

^aThis work. ^bHeays et al. (2014), Roncin et al. (1987). ^cThis work, with "non-mixed" vibronic transition moments. ^dThis work, with "mixed" vibronic transition moments. ^eCarroll et al. (1970), Roncin et al. (1987). ^fCarroll et al. (1970).

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According to our calculations, the strongest b' progression in absorption is b'(16), followed by b'(6) and b'(9). The b'(16, 0) transition is the most intense band in the ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ system and it is also the strongest b' emission band (Ajello et al. 1989). The ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (16) vibronic state is strongly perturbed by the ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(5)$ and ${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (0) Rydberg states, as can be seen in Table 1. In general, bands belonging to the v' = 16 progression gain intensity when interaction is considered in the model. The perturbation effect is particularly significant for the b'(16, 1) band, which has a measurable intensity despite an unfavorable Franck–Condon factor. On the other hand, our calculations reveal that the b'(16, 4) band, observed in the spectrum of Titan (Ajello et al. 2007), gains intensity at the expenses of the ${{c}_{4}}^{\prime }(5)-{\rm{X}}(4)$ transition. Regarding the v'' progression from v' = 9, our calculations indicate that the ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}(9)$ state is essentially unperturbed and, so, the intensity distribution along the progression shows a normal Franck–Condon intensity pattern. The most intense absorption bands of this progression, that is, the b'(9, 0), b'(9, 1), b'(9, 2), b'(9, 5), b'(9, 6), and b'(9, 9) vibronic bands, have been detected from Titan's emissions (Ajello et al. 2007) with the exception of the b'(9, 0) band. Our calculations predict that the f-value for the b'(9, 0) band at 907 Å is similar in magnitude to that for the b'(9, 9) band. This suggests that an emission feature at 907 Å could be present in the EUV spectrum of Titan. Concerning the b'(6) − X(v'') progression, bands are relatively strong in absorption; in fact, according to the present results, the b'(6, 3) is the second most intense band in the b'-X system. However, experimental studies performed by Roncin et al. (1987, 1991), showed that bands of v'' progression from v' = 6 are weak in emission. This could be indicative of a large predissociation rate for the ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ (6) level.

The normal intensity distribution expected within the b'(4) progression is such that the b'(4, 5) band would be the strongest; instead, the b'(4, 1) is the most intense transition as a result of the perturbation of the b'(4) level by the ${{c}_{4}}^{\prime }$ (1). Important deviations from predictions based on the Franck–Condon principle are also observed in b'(7) and b'(8) progressions. The b'(7) and b'(8) levels are perturbed mainly by ${{c}_{4}}^{\prime }$ (2). In absence of perturbation, it is expected that the most intense bands of these progressions are the b'(7, 3) and b'(8, 2) bands. However, if the interaction between vibronic states is taken into account, our calculations predict b'(7, 1) and b'(8, 1) as the most intense bands; this fact is consistent with emission measurements from the v' = 7 and v' = 8 levels of the ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ valence state performed by Heays et al. (2014). The noticeable weakening in the intensities of b'(7, 3) and b'(8, 2) bands is caused by the destructive interference between b'-X and ${{c}_{4}}^{\prime }-{\rm{X}}$ transition moments.

In order to estimate the reliability of our results, the perturbed oscillator strengths presently calculated are displayed in Table 10 together with experimental values found in the literature. These include band f-values derived from electron impact measurements (Ajello et al. 1989), from electron energy loss measurements (Chan et al. 1993), and "rotationless" band f-values derived from high-resolution photoabsorption measurements of the rotational lines (Stark et al. 2005, 2008; Heays et al. 2009; Huber et al. 2009). A comparison with experimental data reveals that f-values obtained with mixed transition moments are in better agreement with high-resolution photoabsorption measurements.

Table 10. Comparison of Oscillator Strengths for Vibronic Transitions to the ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ valence state, ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ , and ${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ Rydberg States from the Ground State of ${{\rm{N}}}_{2}$ .

Transition	f^a	f^b	f^c	f^d	f^e	f^f
X(0)-b'(1)	0.0007	0.00045(9)	⋯	⋯	⋯	⋯
X(0)-b'(4)	0.0024	0.0018(2)	⋯	⋯	0	⋯
X(0)-b'(5)	0.0009	0.0011(1)	⋯	⋯	0.000135	⋯
X(0)-b'(6)	0.0014	0.0017(2)	⋯	⋯	0.002649	0.00216
X(0)-b'(7)	0.0001	0.00051(5)	⋯	⋯	0.003409	⋯
X(0)-b'(8)	0.0024	0.004(4)	⋯	⋯	0.022107	⋯
X(0)-b'(9)	0.0059	0.0087(9)	⋯	⋯	0.009529	0.0128
X(0)-b'(11)	0.0101	0.013(1)	⋯	⋯	0.003504	0.00654
X(0)-b'(12)	0.0164	⋯	0.016(2)	⋯	0.030297	0.0303
X(0)-b'(14)	0.0216	⋯	0.018(2)	⋯	0.041825	0.0341
X(0)-b'(15)	0.0263	⋯	0.024(3)	⋯	0.054902	0.0409
X(0)-b'(16)	0.0396	⋯	0.034(4)	⋯	0.067792	0.0626
X(0)-b'(17)	0.0197	⋯	0.016(2)	⋯	0.037148	0.0318
X(0)-b'(18)	0.0016	⋯	0.0024(3)	0.00452(40)	0.003366	0.00326
X(0)-b'(19)	0.0117	⋯	⋯	0.0131(23)	0.021500	0.0166
X(0)-b'(21)	0.0037	⋯	⋯	0.00565(42)	0	⋯
X(0)-b'(22)	0.0037	⋯	⋯	0.00482(41)	0.005532	0.00455
X(0)- ${c}_{4}^{\prime }$ (0)	0.1543	0.138(14)	⋯	⋯	0.1567	0.195
X(0)- ${c}_{4}^{\prime }$ (1)	0.0092	0.0052(6)	⋯	⋯	0.0038	0.00147
X(0)- ${c}_{4}^{\prime }$ (2)	0.0012	0.0012(1)	⋯	⋯	0.0027	⋯
X(0)- ${c}_{4}^{\prime }$ (5)	0.0001	⋯	0.00018	⋯	0	0.0006
X(0)- ${c}_{4}^{\prime }$ (7)	0.0029	⋯	⋯	⋯	0	⋯
X(1)- ${c}_{4}^{\prime }$ (0)	0.0187	⋯	⋯	⋯	0.0273	⋯
X(2)- ${c}_{4}^{\prime }$ (0)	0.0016	⋯	⋯	⋯	0.0027	⋯
X(3)- ${c}_{4}^{\prime }$ (0)	0.0002	⋯	⋯	⋯	0.00064	⋯
X(0)- ${{c}_{5}}^{\prime }$ (0)	0.0089	⋯	0.0052(6)	⋯	⋯	0.0104

Notes.

^aThis work, with "mixed" vibronic transition moments. ^bStark et al. (2005), Stark et al. (2008). The experimental error is given in parentheses. ^cHeays et al. (2009). The experimental error is given in parentheses. ^dHuber et al. (2009). The experimental error is given in parentheses. ^eAjello et al. (1989). ^fChan et al. (1993).

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4. Conclusion

In the present work, wavelengths and absorption oscillator strengths for a large number of bands belonging to the ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ and ${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ Rydberg systems and the ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ valence system of molecular nitrogen have been calculated taking into account the homogeneous interaction among ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ , ${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ , and ${b}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}$ states. The band f-values obtained with mixed transition moments are consistent with the scarce experimental data, which indicates that the interaction model used here is valid for predicting absorption intensities. Nonetheless, in cases where rotational mixing is important, a marked J-dependence of band f-values would be expected. Our results show that the vibronic interactions play an important role in modifying the oscillator strengths for many bands of the above systems. The vibrational perturbation effects are particularly important for band f-values of ${{c}_{4}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ and ${{c}_{5}}^{\prime 1}{{{\rm{\Sigma }}}_{u}}^{+}-{{\rm{X}}}^{1}{{{\rm{\Sigma }}}_{g}}^{+}$ Rydberg systems. Due to Rydberg-valence interaction, many bands of these systems have measurable intensity, so they should be included in planetary atmosphere models. We hope that the new ${{c}_{4}}^{\prime }-{\rm{X}}$ , ${{c}_{5}}^{\prime }-{\rm{X}}$ , and b'-X band data of ${{\rm{N}}}_{2}$ reported here may aid in the identification of features in the very complex and blended EUV spectra of planetary atmospheres.