ANALYSIS OF ELECTRON-IMPACT EXCITATION AND EMISSION OF THE npσ¹Σ⁺_u AND npπ¹Π_u RYDBERG SERIES OF H₂

Steady-state photon emission intensity resulting from direct excitation by a continuous electron beam has been described in detail by Jonin et al. (2000). A brief review will be given here. The volumetric photon emission rate (I) from electron-impact excitation is proportional to the excitation rate and emission-branching ratio:

$$\begin{eqnarray} {I(}v_{j},v_{i};J_{j},J_{i}{)} &=& {g(}v_{j};J_{j}{)}\frac{{A(}v_{j},v_{i};J_{j},J_{i}{)}}{{A(}v_{j};J_{j}{)}} (1-\eta (v_{j}, J_{j}))\nonumber \\ &&\times (1-\kappa (\epsilon _{ij},\zeta _{i})) \end{eqnarray} \tag{ 1 }$$

where the indexes j and i refer to the upper and lower electronic-state vibrational and rotational levels, respectively, and summation over the missing index is assumed. J and v refer to rotational and vibrational quantum numbers, respectively. A(v_j, v_i;J_j, J_i) is the Einstein spontaneous transition probability for emission from level (v_j, J_j) to level (v_i, J_i), and A(v_j, J_j) is the total radiative transition probability for level (v_j, J_j) (including transition to lower singlet-gerade states and to the continuum levels of the X¹Σ⁺_g state). The yield of nonradiative processes is represented by η(v_j, J_j). Under the present experimental conditions, collision deactivation is 10⁴–10⁵ times slower than radiative decay, and is, therefore, negligible. The sum of appropriate predissociation, dissociation, and autoionization yields is denoted by η(v_j, J_j). Since the experimental condition for certain strong resonance transitions to the X¹Σ⁺_g(0) levels is optically thick, the parameter κ(_ij, ζ_i) in Equation (1) accounts for the self-absorption for those resonance transitions. The calculation of κ(_ij, ζ_i) from H₂ foreground column density and transition probability has been given by Jonin et al. (2000).

The excitation rate, g(v_j, J_j), represents the sum of the excitation rates from the rotational and vibrational levels of the X¹Σ⁺_g state. It is proportional to the population of the molecules in the initial level, N(v_i, J_i), the excitation cross section (σ_ij), and the electron flux (F_e):

$$\begin{equation} {g(}v_{j};J_{j}{)} = F_{e}\sum _{i}N(v_{i},J_{i}) \sigma (v_{i},v_{j};J_i, J_{j}). \end{equation} \tag{ 2 }$$

The excitation cross section σ_ij is calculated from the analytical function (Liu et al. 1998)

$$\begin{eqnarray} \frac{\sigma (v_i,v_j;J_i,J_j)}{\pi a_{0}^{2}} &=& 4 f (v_i,v_j;J_i,J_j) \frac{Ry}{E_{ij}} \frac{Ry}{E} \bigg[ \frac{C_{0}}{C_{7}}\left(\frac{1}{X^{2}} - \frac{1}{X^{3}} \right) \nonumber \\ && + \sum _{k =1}^{4} \frac{C_{k }}{C_{7}} (X - 1) \exp (-k C_{8}X) \nonumber \\ && + \frac{C_{5}}{C_{7}} \left(1- \frac{1}{X} \right) + \ln (X) \bigg] \end{eqnarray} \tag{ 3 }$$

where a₀ and Ry are the Bohr radius and Rydberg constant, respectively, and f_ij is the optical absorption oscillator strength, given by Equation (16). E_ij is the threshold energy for the (v_i, J_i) → (v_j, J_j) excitation, E is the excitation energy, and X = E/E_ij. The coefficients C_k/C₇ (k = 0 to 6) and C₈ are determined by fitting the experimentally measured relative excitation function. For the present work, C_k/C₇ (k = 0 to 6) and C₈ determined by Liu et al. (1998) for the B¹Σ⁺_u − X¹Σ⁺_g and C¹Π_u − X¹Σ⁺_g band systems of H₂ are used for the direct excitation of the other npσ¹Σ⁺_u and npπ¹Π_u − X¹Σ⁺_g band systems. The absolute value of the collision strength parameter, C₇, of Equation (3), is fixed to the absorption oscillator strength f_ij by the relation

$$\begin{equation} C_7 = {{4 \pi a_0^2 (2 J_i + 1) Ry} \over E_{ij}} f(v_i, v_j; J_i, J_j). \end{equation} \tag{ 4 }$$

In addition to the direct excitation, indirect excitation from the higher levels of singlet-gerade states is also possible. While the cascade from higher levels of the singlet-gerade states results in the indirect excitation of many rovibronic levels of the singlet-ungerade states, a recent time-resolved study of Liu et al. (2002) has shown that it preferentially contributes to the low v_j levels of the B¹Σ⁺_u and B'¹Σ⁺_u states, and the indirect excitation to the $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$, D'¹Π_u, or higher states is negligible.

The excitation cross section for a band system in the present study will be defined as the statistical average of the rovibrational cross section components:

$$\begin{equation} \sigma _{ex} = \frac{1}{N_{T}}\sum _{i,j} \sigma (v_{i},v_{j};J_i, J_{j})N_{i} \end{equation} \tag{ 5 }$$

where N_T is the total H₂ population of the X¹Σ⁺_g state. The corresponding emission cross section is then given by

$$\begin{equation} \sigma _{em} = \frac{1}{N_{T}}\sum _{i,j} \sigma (v_{i},v_{j};J_i, J_{j}) (1- \eta _{j})N_{i}. \end{equation} \tag{ 6 }$$

Since transition probabilities of npσ¹Σ⁺_u and npπ¹Π_u states to the excited singlet-gerade (such as the EF¹Σ⁺_g and GK¹Σ⁺_g) states are negligible when compared with those to the X¹Σ⁺_g state, the emission cross sections of the npσ¹Σ⁺_u and npπ¹Π_u − X¹Σ⁺_g band systems can be considered identical to the emission cross sections of the npσ¹Σ⁺_u and npπ¹Π_u states, given by Equation (6).

The present model utilizes the B¹Σ⁺_u, C¹Π_u, and D¹Π⁻_u − X¹Σ⁺_g transition probabilities of Abgrall et al. (1994) and the presently calculated adiabatic npσ¹Σ⁺_u and npπ¹Π_u − X¹Σ⁺_g transition probabilities of the higher Rydberg n ⩾  4 states. Significant coupling exists among some of the v_j levels of the B'¹Σ⁺_u, $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$, 5pσ¹Σ⁺_u, D¹Π⁺_u, and D'¹Π⁺_u states. The calculated nonadiabatic transition probabilities are used for these levels. Wherever possible, the experimentally measured term values of Roncin & Launay (1994), Dabrowski (1984), and Takezawa (1970) are utilized to calculate the transition frequencies. To compare with the observed spectrum, the calculated spectrum is convoluted with a triangular instrument function with an appropriate FWHM.

3.2.1. Adiabatic Calculation of Higher npσ¹Σ_u+ and npπ¹Π_u −X¹Σ⁺_g Band Systems

The potential curves of the B¹Σ⁺_u, B'¹Σ⁺_u, $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$, and 5pσ¹Σ⁺_u states (Staszewska & Wolniewicz 2002), the C¹Π_u, D¹Π_u, and D'¹Π_u states (Wolniewicz & Staszewska 2003b), and the X¹Σ⁺_g state (Wolniewicz 1993), are known theoretically with adiabatic and nonadiabatic corrections. The dipole transition moment functions of the transitions to the X¹Σ⁺_g state have been tabulated by Wolniewicz & Staszewska (2003a, 2003b). For higher members of the Rydberg series, the ab initio potential energy curves are unavailable. The quantum defect theory developed by Jungen & Atabek (1977) allows the determination of the Born–Oppenheimer potential energy with large adiabatic corrections through the classical formula

$$\begin{equation} V(R) = V_{H_2^+}(R) - {R_{H_2} \over { (n - \delta)^2}} \end{equation} \tag{ 7 }$$

where δ is the quantum defect, and $V_{H_2^+}(R)$ is the potential energy curve of H₂ X²Σ⁺_g, to which the Rydberg series converge.

When the calculated dipole transition moment function, D_nΛ(R), is not available, it can be approximated from the known function of the 4pΛ− X transition, using the quantum defect δ_Λ:

$$\begin{equation} D_{n \Lambda }(R)=D_{4 \Lambda }(R) \left[ { {(4 - \delta _{\Lambda })} \over {(n - \delta _{\Lambda })} } \right]^{3/2}. \end{equation} \tag{ 8 }$$

Numerical integration using the Numerov method is employed to solve the Schrödinger equation to obtain eigenvalues and eigenfunctions. The spontaneous transition probability, A(v_j, v_i; J_j, J_i), is given by (Hilborn 1982)

$$\begin{eqnarray} {A(}v_{j},v_{i};J_{j},J_{i}{)} &=& {{2 \omega _{ij}^3} \over {3 \epsilon _0 h c^3}} \left< \chi _{v_j J_j} (R) \left| D(R) \right| \chi _{v_i J_i}(R) \right> ^2\nonumber \\ &&\times {{\mathcal{H}_{ji}(J_j,J_i)} \over {2J_j+1}} \end{eqnarray} \tag{ 9 }$$

where ω_ij$\big[\!\!=\!\!\big(E_{v_j, J_j} - E_{v_i, J_i}\big)\big/\hbar\big]$ is the angular frequency for the transition j → i, χ(R) is the vibrational wavefunction with the rotational energy correction, and $\mathcal{H}_{ji}(J_j,J_i)$ is the Hönl–London factor as defined by Hansson & Watson (2005).

Equation (9) can be rewritten as

$$\begin{eqnarray} {A(}v_{j},v_{i};J_{j},J_{i}{)} &=& 2.142 \times 10^{10} \left(E_{v_j, J_j} - E_{v_i, J_i} \right) ^3\nonumber \\ &&\times\left < \chi _{v_j J_j} (R)\left| D(R) \right| \chi _{v_i J_i}(R) \right> ^2\nonumber \\ &&\times {{\mathcal{H}_{ji}(J_j,J_i)} \over {2J_j+1}} \end{eqnarray} \tag{ 10 }$$

where A(v_j, v_i; J_j, J_i) is in s⁻¹, $\big(E_{v_j, J_j} - E_{v_i, J_i}\big)$ is in hartree, and the dipole transition moment function, D(R) is in bohr.

3.2.2. Nonadiabatic Calculation of the B'¹Σ⁺_u, $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$, 5pσ¹Σ⁺_u, D¹Π⁺_u, and D'¹Π⁺_u − X¹Σ⁺_g Band Systems

The small mass of the H₂ often prevents a satisfactory reproduction of the experimental measurement using the Born–Oppenheimer approximation even if the adiabatic and diagonal nonadiabatic coupling corrections are applied. Coupling between the different electronic states must be included. While nonadiabatic coupling often appears as positional shifts of observed energy levels, the deviation of spectral intensities is even more apparent. Nonadiabatic coupling is of two types: rotational interaction mixing the ¹Σ⁺_u and ¹Π⁺_u states, and vibrational interaction mixing states with the same symmetry. Following the work and notation of Wolniewicz et al. (2006), the matrix elements between two coupling states for rotational mixing can be written:

$$\begin{equation} \big< \Pi ^+ \big| D_{i,k}^{1,0} \big| \Sigma ^+ \big> = \sqrt{2 J(J+1)} { {\left< L^+ \right>} \over {R^2}}. \end{equation} \tag{ 11 }$$

Likewise, for vibrational coupling,

$$\begin{equation} C_{i,k}^{\Lambda, \Lambda } = D_{i,k}^{\Lambda, \Lambda }(R) + G_{i,k}^{\Lambda, \Lambda }(R) + 2B_{i,k}^{\Lambda, \Lambda }(R) \frac{d}{dR}. \end{equation} \tag{ 12 }$$

These matrix elements for the B'¹Σ⁺_u, $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$, D¹Π_u, and D'¹Π_u states have been tabulated as functions of the internuclear distance R by Wolniewicz et al. (2006).

In the present work, nonadiabatic coupling is assumed to be so weak that it can be treated as a perturbation. Consequently, the coupling matrix elements can be evaluated in an adiabatic vibrational basis set. The present adiabatic vibrational basis set, having 122 terms, includes the full vibrational progressions of the B'¹Σ⁺_u, $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$, 5pσ¹Σ⁺_u, D¹Π⁺_u, and D'¹Π⁺_u states. Distinguishing between ¹Σ⁺_u and ¹Π⁺_u symmetry, the nonadiabatic eigenfunction, Φ_j, expressed in terms of the adiabatic vibrational basis, ϕ^Λ_k, is

$$\begin{equation} \Phi _j = \sum _{k} \alpha _{jk} \phi _k^{\Sigma } + \sum _{l}\beta _{jl} \phi _l^{\Pi }. \end{equation} \tag{ 13 }$$

The transition probability from the level j of the singlet-ungerade state to the level i of the X¹Σ⁺_g state is then

$$\begin{eqnarray} {A(}v_{j},v_{i};J_{j},J_{i}{)} &=& {{2 \omega _{ij}^3} \over {3 \epsilon _0 h c^3}} \bigg\langle \sum _k \alpha _{jk} D_{ki} \sqrt{{ {\mathcal{H}_{\Sigma \Sigma }(J_j,J_i)} \over {2J_j+1}}}\nonumber\\ && + \epsilon \sum _{l} \beta _{jl} D_{li} \sqrt{{{\mathcal{H}_{\Pi \Sigma }(J_j, J_i)} \over {2J_j+1}}} \bigg\rangle ^2 \end{eqnarray} \tag{ 14 }$$

where D_ji is the dipole matrix element of the transition from level j to the ground state i and is given by

$$\begin{equation} D_{ji} = \left< \phi _j\left| D(R) \right|\chi _{v_i,J_i}(R) \right>. \end{equation} \tag{ 15 }$$

In Equation (14) is 1 for a P-branch transition and −1 for an R-branch transition (Vigué et al. 1983; Glass-Maujean & Beswick 1989).

The dipole moment functions of the D¹Π_u − X¹Σ⁺_g, $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$ − X¹Σ⁺_g, and 5pσ¹Σ⁺_u − X¹Σ⁺_g band systems have been calculated and tabulated by Wolniewicz & Staszewska (2003a, 2003b).

The absorption line oscillator strength, f(v_i, v_j; J_i, J_j), of Equations (3) and (4) is related to the line transition probability, A(v_j, v_i; J_j, J_i), of Equations (9) and (14) by Abgrall & Roueff (2006)

$$\begin{equation} f(v_i, v_j; J_i, J_j) = 1.4992 \frac{2J_j+1}{2J_i+1} {{A(}v_{j},v_{i};J_{j},J_{i}{)} \over v^2(v_i, v_j; J_i,J_j)} \end{equation} \tag{ 16 }$$

where v(v_i, v_j; J_i, J_j) refers to the transition wavenumber in reciprocal centimeters.

In addition to spontaneous emission, nonradiative processes such as predissociation and autoionization occur for some levels. Figure 1 shows the experimental energy term values of the lowest J_j levels of some npσ¹Σ⁺_u(v_j) and npπ¹Π⁺_u(v_j) states, along with the positions of the H(1s)+H(2ℓ) dissociation limit and the first H₂ ionization potential. In general, any npσ¹Σ⁺_u or npπ¹Π⁺_u levels above the dissociation limit can be predissociative. Any npσ¹Σ⁺_u or npπ¹Π_u levels above the ionization potential can autoionize.

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The extensive experimental and theoretical work of Glass-Maujean (1986) and coworkers (Glass-Maujean et al. 1978, 1979, 1984, 1985a, 1987) have shown that the predissociation of the npσ_u¹Σ⁺_u (n > 3) and npπ_u ¹Π_u (n ⩾  3) states primarily takes place via coupling to the continuum levels of the B'¹Σ⁺_u state. The rate of predissociation differs drastically depending on the orbit symmetries and relative energy separations. For instance, the D¹Π⁺_u and B'¹Σ⁺_u states are strongly coupled by Coriolis interaction. Thus, the D¹Π⁺_u rovibrational levels that lie above the H(1s)+H(2ℓ) limit are predissociated very rapidly. The lifetimes of the J_j = 2 of the v_j = 3 − 11 levels of the D¹Π⁺_u state were determined to be (3.7−5.9) × 10⁻¹³ s (Glass-Maujean et al. 1979, 1985b), which are 10,000 times shorter than their expected fluorescence lifetime, (2−4) × 10⁻⁹ s (Glass-Maujean et al. 1985c; Abgrall et al. 1994). In contrast, the D¹Π⁻_u state and other higher npπ¹Π⁻_u states are not coupled to the B'¹Σ⁺_u or other npσ ¹Σ⁺_u states. These states can only couple to a dissociating ¹Π⁻_u state. For the npπ¹Π⁻_u states below the H(1s)+H(n = 3) limit, C¹Π⁻_u is the only dissociative ¹Π⁻_u state. Since npπ_u¹Π⁻_u states are only weakly coupled to the C¹Π⁻_u state, their predissociation rates are negligibly small. Above the H(1s)+H(n = 3) limit, npπ ¹Π⁻_u levels can also be dissociated by the D¹Π⁻_u continuum. Glass-Maujean et al. (2007c) recently found that the coupling between the D'¹Π⁻_u state and D¹Π⁻_u continuum is fairly efficient. Moreover, the predissociation of the $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$ and npσ ¹Σ⁺_u (n > 4) states, in general, takes place by homogeneous coupling with the B'¹Σ⁺_u continuum levels (Glass-Maujean et al. 1978). Due to the difference in the npσ ¹Σ⁺_u − B'¹Σ⁺_u continuum Franck–Condon overlap integrals, variations in the predissociation rates are expected. The predissociation of the npπ ¹Π⁺_u (n > 3) arises from either npπ¹Π⁺_u − D¹Π⁺_u homogeneous coupling followed by D¹Π⁺_u − B'¹Σ⁺_u Coriolis coupling or npπ¹Π⁺_u − npσ ¹Σ⁺_u Coriolis coupling followed by npσ ¹Σ⁺_u − B'¹Σ⁺_u homogeneous coupling (Glass-Maujean 1979). Levels below the H(1s)+H(2ℓ) limit cannot be predissociated. In particular, the v_j = 0 levels of the $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$ and D'¹Π_u states lie below the limit, and spontaneous emission is the only decay mechanism.

The ionization energy of the v_i = 0 and J_i = 0 level of the X¹Σ⁺_g state of H₂ is 124,417.507 cm⁻¹. For the low J_j levels of the of $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$ and D'¹Π_u states, autoionization is energetically not possible unless v_j is ⩾4 for the $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$ state and v_j⩾ 3 for the D'¹Π_u state (see Figure 1). Furthermore, vibrational autoionization of H₂ has a tendency to proceed with a small change in vibrational quantum number (i.e., Δv = v₊ − v_j) and the autoionization rate rapidly decreases for processes with large Δv changes (Dehmer & Chupka 1976; O'Halloran et al. 1988, 1989). Autoionization of the $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$ state requires a fairly large change in Δv and its efficiency is low (⩽ 5%; Dehmer & Chupka 1976). Autoionization rates for the v_j > 4 levels of the D'¹Π_u state is significant and the efficiency for the low J_j levels of these bands has been measured by Dehmer & Chupka (1976).

The examination of the relative accuracy of the calculated transition probabilities rests on the relationship of the photoemission intensity to the emission-branching ratio via Equation (1) and the oscillator strength via Equation (3). The first set of spectra with an H₂ foreground column density of (2.3± 0.6) × 10¹³ cm⁻² is optically thin for all except a few strong Werner-band resonance transitions. In principle, it gives accurate relative intensities. In practice, spectral intensities and signal-to-noise ratios for most transitions on the blue side of 850 Å are too weak for reliable relative intensity measurement. In the present analysis, examination of the relative accuracy of the transition probabilities and derivation of nonradiative yields of the npσ ¹Σ⁺_u (n > 3) and npπ ¹Π_u (n > 3) states are primarily carried out in the 790−900 Å region of the second set of spectra, although transitions in 900−1180 Å region of the first set are also utilized to confirm the derived nonradiative yields. The second set of spectra has a foreground column density of ∼15 × 10¹³ cm⁻². The largest self-absorption in the 790−900 Å region is ∼28%, for the Q(1) and R(1) transitions of the D¹Π_u(2)−X¹Σ⁺_g(0) band. Comparison of relative intensities between the first and second sets of spectra in the (2, 0) and (1, 0) band regions of the D¹Π_u − X¹Σ⁺_g band system verified that the self-absorption model described in Jonin et al. (2000) reliably accounts for the intensity.

Analysis was initiated by adding the D'¹Π⁻_u − X¹Σ⁺_g transition to an existing model of the B¹Σ⁺_u − X¹Σ⁺_g, C¹Π_u − X¹Σ⁺_g, B'¹Σ⁺_u − X¹Σ⁺_g, and D¹Π_u − X¹Σ⁺_g band systems (Jonin et al. 2000). Since predissociation of the D'¹Π⁻_u levels below the H(1s)+H(n = 3) dissociation limit is negligibly small, only autoionization was considered. Figure 1 shows that autoionization of D'¹Π⁻_u is negligible for v_j⩽ 3 vibrational levels. So, a comparison of the relative spectral intensity of calculated and observed D'¹Π⁻_u − X¹Σ⁺_g transitions can be performed straightforwardly. It was found that the calculated spectrum can reproduce the relative intensities of observed D'¹Π⁻_u − X¹Σ⁺_g transitions from 790 Å to 1100 Å if the emission yields of 0.62, 0.62, and 0.35 are applied to the J_j = 1, 2, and 3 levels of the D'¹Π⁻_u(4) state and the calculated adiabatic transition probabilities of the Q(1) transitions are reduced by 48%, as suggested by the measurement of Glass-Maujean et al. (2007c). Since the autoionization yield of the Q(1) transition is 8%–17% (Dehmer & Chupka 1976; M. Glass-Maujean et al. 2008, in preparation), the emission yield implies a dissociation yield of 0.2–0.3 for the J_j = 1 level of the D'¹Π⁻_u(4) state, which is consistent with an upper limit of 0.3 obtained by Glass-Maujean et al. (1987). In particular, emission in the 790−850 Å wavelength region is dominated by the Q-branch transitions of the D¹Π_u and D'¹Π_u states. The experimental spectrum provides a good test of consistency among the transition probabilities of the D¹Π⁻_u − X¹Σ⁺_g and D'¹Π⁻_u − X¹Σ⁺_g band systems. The good agreement in the relative intensities between the model and observed spectra shows that the calculated transition probabilities of the D'¹Π⁻_u − X¹Σ⁺_g band system are consistent with those of D¹Π⁻_u − X¹Σ⁺_g.

Emission features on the blue side of 750−790 Å are generally very weak, and can only be practically measured with 80 μm slit widths (∼0.24 Å resolution). Most of the lines are from the Q-branch lines of the D¹Π⁻_u − X¹Σ⁺_g and D'¹Π⁻_u − X¹Σ⁺_g bands. The line at ∼784.04 Å, which arises from the Q(1) transition of the D'¹Π⁻_u(5)−X¹Σ⁺_g(0) band, is the strongest feature in the 750–790 Å region. As noted by Liu et al. (2000), the observed emission intensities of the Q(2) and Q(3) transitions are much weaker than those expected from a simple population difference of J_i = 1, 2, and 3 levels at room temperature. A recent combination of absorption, ionization, and dissociation measurements by M. Glass-Maujean et al. (2008, in preparation) have shown a sharp increase in ionization efficiency going from the J_j = 1 to the 2 and 3 levels. The R(1) and P(3) transitions of the D'¹Π⁺_u(5) level, with a 7%–8% emission yield, are weak but observable.

The establishment of transition probabilities for the D'¹Π⁻_u − X¹Σ⁺_g band allows the determination of variation of the overall relative sensitivity of the spectrometer. The inclusion of the D'¹Π⁻_u − X¹Σ⁺_g emission in the model spectrum and utilization of a higher-resolution experimental spectrum (0.095 Å versus 0.115 Å) with significantly better signal-to-noise ratio produced a somewhat more accurate and flatter sensitivity curve in the 800–850 Å region than that reported by Jonin et al. (2000). After the improved sensitivity curve is applied to the observed spectrum, the Q-branch transitions of the 5pπ¹Π⁻_u, 6pπ¹Π⁻_u, 7pπ¹Π⁻_u, 8pπ¹Π⁻_u, and 10pπ¹Π⁻_u states are introduced into the model. Glass-Maujean et al. (2007c) have shown that the measured transition-probability values of a number of Q-branch lines differ significantly from calculated values. Except for the Q(1) transitions of D¹Π⁻_u(6) and D'¹Π⁻_u(3) levels, the transition probabilities of all other Q-branch lines used in the model have been adjusted to their experimental values. The P- and R-branch transitions of the $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$, 5pσ¹Σ⁺_u, 6pσ¹Σ⁺_u, 7pσ¹Σ⁺_u, and npπ¹Π⁺_u (n = 4–8) state are then added into the model. Because of the adiabatic nature of the calculation for many of these states, the transition probabilities of the P- and R-branches are essentially obtained from the calculated band transition probabilities partitioned by Hönl–London factors. Such an approximation is not expected to be valid in the presence of rovibronic coupling. For low v_j levels of the $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$ and D'¹Π⁺_u states and all discrete v_j level of the B'¹Σ⁺_u and D¹Π⁺_u states, the nonadiabatic calculation based on the ab initio results of Wolniewicz et al. (2006) described in Section 3.2.2 was used (see Section 5.1). The emission yields of some rovibrational levels of these states can be determined by comparing synthetic and calibrated experimental spectra.

Nonadiabatic coupling among B'¹Σ⁺_u(4), $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$(0), D¹Π⁺_u(2), and D'¹Π⁺_u(0) levels results in significant differences in the calculated spectrum. Figure 3 compares the relative intensities of experimental and calculated spectra in the region of B'¹Σ⁺_u(4), $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$(0), and D¹Π_u(2)−X¹Σ⁺_g(0) transitions. Although experimental spectrum (solid trace) has been calibrated using the procedures described by Liu et al. (1995) and Jonin et al. (2000), the variation of instrumental sensitivity, which is less than 1.7% over the region shown, is completely negligible. The model spectrum (dot trace, Model 1) in the top panel of Figure 3 was calculated from the B¹Σ⁺_u, C¹Π_u, B'¹Σ⁺_u, and D¹Π_u − X¹Σ⁺_g transition probabilities of Abgrall et al. (1994) and the present adiabatic transition probabilities of higher (n ⩾  4) npσ¹Σ⁺_u and npπ¹Π_u series. Apart from the B'¹Σ⁺_u − D¹Π⁺_u coupling that was taken into account in the calculation of Abgrall et al. (1994), coupling among B'¹Σ⁺_u, $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$, D¹Π⁺_u, and D'¹Π⁺_u levels was not considered.

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Transitions in Figure 3 are labeled in terms of J_i(v_j, v_i)ΔJ β, where i and j refer to the lower and upper states, β is the electronic designation of singlet-ungerade states, and ΔJ = −1, 0 and +1 correspond to P, Q, and R transitions, respectively. The ab initio calculation by Wolniewicz et al. (2006) indicates that the eigenvalues of the J_j = 1 levels of the B'¹Σ⁺_u(4) and $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$(0) states in Abgrall et al. (1994) and Takezawa (1970) need to be interchanged. The model calculation and labeling shown in the figure conform to the indication. The alternative assignment would result in more significant difference at the P(2) and R(0) transitions. In any case, the top panel of Figure 3 shows that the adiabatic model generally underestimates the intensity of the $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$(0) level. Moreover, the relative intensity between the R(J_j − 1) and P(J_j + 1) transitions also differs significantly from experiment. Similar discrepancy is apparent in the spectral regions involving transitions to the v_i = 1 and 2 levels of the X¹Σ⁺_g state. The calculated intensities for the R(1) line of the D'¹Π⁺_u(0)−X¹Σ⁺_g(1) band at 879.13 Å (not shown) is also too strong.

The bottom panel of Figure 3 compares the observed spectrum (solid trace) with model spectrum obtained from nonadiabatic coupling calculation (dot trace, Model 2). Specifically, the transition probabilities of the P and R branches of the B'¹Σ⁺_u, D¹Π⁺_u, $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$, D'¹Π⁺_u, and 5pσ¹Σ⁺_u states, used in the top panel, have been replaced by their counterparts obtained from nonadiabatic calculation. The use of nonadiabatic transition probabilities clearly results in better agreement between the calculated and observed spectra. Except for the R(0) and P(2) lines of the $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$(0) and B'¹Σ⁺_u(4) levels, and the R(1) and P(3) lines from of the D¹Π_u(2) level, all other discrepancies shown in the top panel have been removed. The disagreement between the nonadiabatic model and observation in the R(1) and P(3) lines of the D¹Π_u(2)−X¹Σ⁺_g(2) band, and the R(0), R(1), and P(2) lines of the B'¹Σ⁺_u(4) and $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$(0)−X¹Σ⁺_g(1) bands, while reduced significantly, is still larger than the experimental error. The nonadiabatic transition probabilities also give rise to much better agreement for the R(1) transition of the D'¹Π_u(0)−X¹Σ⁺_g(1) band.

The top entries of the Tables 1 and 2 list predissociation yields for the J_j = 0–3 of v_j = 1–4 levels of the $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$ and J_j = 1–3 of v_j = 1–5 levels of the D'¹Π⁺_u states. In Table 1, v_j refers to the vibrational quantum number of the B''¹Σ⁺_u (inner well) state. The v_j = 4 level of the $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$ state lies above the first ionization potential of H₂. While autoionization is possible, the autoionization rate is apparently negligibly slow in comparison with predissociation (Dehmer & Chupka 1976). The v_j = 0 levels of both $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$ and D'¹Π_u states are below the B'¹Σ⁺_u continuum; their emission yields are unity. The error in the predissociation yields in Tables 1 and 2 are estimated to be ∼8% (i.e., ± 0.08). Within the experimental error, the emission yields of the v_j = 1, 2, and 3 levels of the D'¹Π⁻_u state are found to be near unity, consistent with the negligible predissociation yields determined by Glass-Maujean et al. (1987). Predissociation yields for other higher rovibrational levels of the $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$ and D'¹Π⁺_u state cannot be reliably determined in the present work. In the final analysis, the emission yields for these higher levels used in the synthetic spectrum were calculated from autoionization yields obtained by Dehmer & Chupka (1976) and predissociation yields reported by Glass-Maujean et al. (1987) or from the Glass-Maujean et al. (2007a, 2007b, 2007c, 2008a, 2008b) line-width measurements. The transition from the D'¹Π_u and the inner well of the $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$ states to the lower excited singlet-gerade states and the continuum levels of the X¹Σ⁺_g are very weak. Predissociation yields for the levels listed in these tables are thus derived considering only the contribution of the discrete transition to the X¹Σ⁺_g state to the total transition probability, A(v_j, J_j), of Equation (1).

Nonradiative yields for other higher Rydberg states were obtained after the yields of the $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$ and D'¹Π_u were established. Perturbations among the Rydberg series results in significant differences between the measured and calculated transition probabilities of many Q-branch excitations (Glass-Maujean et al. 2007c, 2008b). The calculated transition probabilities for the perturbed levels used in the model are adjusted to the measured values with the assumption that emission-branching ratios are insignificantly affected by the perturbation. Additionally, the transition probabilities of the P and R branches of the higher Rydberg states are generated from those of the Q branches with Hönl–London factors, even though perturbation will invariably introduce significant deviation. In absence of the nonadiabatic calculation, these two assumptions are necessary. The emission yields, obtained by assuming that the utilized transition probabilities are accurate, may thus deviate somewhat from the real values. Nevertheless, the fact that the most of the nonradiative yields in Tables 1 and 2 agree with the measured predissociation yields of Glass-Maujean et al. (1987) and the autoionization yields of Dehmer & Chupka (1976) indicates that the present approach is valid. More importantly, the use of the present emission yields and adjusted transition probabilities leads to the correct model intensities.

Predissociation yields of many rovibrational levels of the $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$, D'¹Π_u, and many other higher npσ¹Σ⁺_u and npπ¹Π_u states have also been determined by Glass-Maujean et al. (1987) from a simultaneous experimental measurement of H₂ absorption and H Lyman-α excitation spectra, and from measurements of Fano profiles of absorption transitions. For comparison purposes, their predissociation yields are listed in parentheses in Tables 1 and 2. It can be noted that Glass-Maujean et al. (1987) were only able to measure the lower limits of the predissociation yields of the $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$ state. In this sense, the present study has obtained more definitive predissociation yields for the v_j = 1 to 4 levels of the $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$ state. However, the two sets of values, in general, agree within error limits, though a slight difference can be noted for the J_j = 1 and v_j = 1 level of the D'¹Π⁺_u state (0.88± 0.08 versus 0.65± 0.15). The predissociation yield for the J_j = 2 of the 5pσ¹Σ⁺_u(1) is also significantly different from that given by Glass-Maujean et al. (1987). The difference, however, arises from problems in both the adiabatic and nonadiabatic calculations. Based on the predissociation yield of Glass-Maujean et al. (1987) and the calculated P(3) and R(1) branch transition probabilities, the model shows that the calculated emission intensities of the R(1) transition are too weak in a number of locations (e.g., 820.44 Å, 849.39 Å, 878.65 Å, 908.09 Å, and 937.57 Å) while those of the P(3) transitions roughly agree with observation. Based on Hönl–London factors, the P(3) transitions should be approximately 50% stronger than the R(1) transitions. The nonadiabatic calculation also predicts stronger P(3) branch transitions to v_i = 0 − 5 levels of the X¹Σ⁺_g state. The observed intensities of the R(1) transitions in many of these bands, however, are stronger than the corresponding P(3) lines. It is unclear which perturbing state is responsible for the deviation in P- and R-branch relative intensities. The nearest known npσ¹Σ⁺_u state is the 7pσ¹Σ⁺_u(0) level, which is about 249 cm⁻¹ higher in energy than the J_j = 2 level of the 5pσ¹Σ⁺_u(1) state. The R/P branches relative intensities of the 7pσ¹Σ⁺_u(0)−X¹Σ⁺_g transitions are also significantly stronger than those implied by Hönl–London factors (see below). Simple homogeneous coupling between the 5pσ¹Σ⁺_u(1) and 7pσ¹Σ⁺_u(0) states cannot increases the R-branch intensities of both states at the expense of the P branch. The deviation of the R/P branch relative intensities requires one or more interacting npπ¹Π_u levels such as 5pπ¹Π_u and 7pπ¹Π_u states. In any case, the intensities of the P(3) and R(1) lines of the 5pσ¹Σ⁺_u(1)−X¹Σ⁺_g bands can be approximately reproduced by partitioning their transition probabilities in a 2:3 ratio. The predissociation yield for J_j = 2 of the 5pσ¹Σ⁺_u(1) state is correspondingly lowered to ∼30%. The measured predissociation yield, 0.6± 0.1, is obviously more accurate than the derived predissociation yield. The large difference between the two results shows that neither absolute band transition probabilities nor P/R branch relative values are calculated reliably.

In addition to the 5pσ¹Σ⁺_u(1) level, the R(J_i−1)/P(J_i+1) relative intensities of the 6pσ¹Σ⁺_u(0), 6pσ¹Σ⁺_u(1), and 7pσ¹Σ⁺_u(0)−X¹Σ⁺_g transitions are also abnormally strong. The emission originating from the 5pσ¹Σ⁺_u(1), 6pσ¹Σ⁺_u(0), and 6pσ¹Σ⁺_u(1) levels can be reconciled with the essential features of observed spectrum by repartitioning the P- and R-branch transition probabilities without a change in the calculated band transition probabilities. For the 7pσ¹Σ⁺_u(0) level, however, the band transition probabilities need to increase by a factor of ∼2.5, in addition to the adjustment of the P- and R-branch Hönl–London factors. Even with the adjustment in band transition probabilities, intensities of a few 7pσ¹Σ⁺_u(0)−X¹Σ⁺_g transitions are not well reproduced. In any case, while the predissociation yields obtained with these adjustments are not expected to be reliable, the inferred values for the 6pσ¹Σ⁺_u(0), 6pσ¹Σ⁺_u(1), and 7pσ¹Σ⁺_u(0) levels are consistent with those determined by Glass-Maujean et al. (1987). All other higher v_j levels of the 6pσ¹Σ⁺_u and 7pσ¹Σ⁺_u states have negligible emission yields.

The calculated transition probabilities of npσ¹Σ⁺_u and npπ¹Π_u band systems, along with Equations (3)−(6) and collision strength coefficients of Liu et al. (1998), permit a good estimation of the cross sections of the npσ¹Σ⁺_u and npπ¹Π_u, especially those of the $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$ and D'¹Π_u states. The solid lines of Figures 4 and 5 show the estimated excitation cross section of the $B^{\prime \prime }\bar{B}\,{^{1}\Sigma _{u}^{+}}- X\,{^{1}\Sigma _{g}^{+}}$ and D'¹Π_u − X¹Σ⁺_g band system as a function of excitation energy. The calculation of corresponding emission cross sections requires appropriate emission yields for the rovibrational levels. For the v_j = 1 − 4 levels of the $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$ state and v_j = 1 − 5 levels of the D'¹Π_u state, the nonradiative yields listed in Tables 1 and 2 have been applied. For higher vibrational levels, the predissociation yields listed by Glass-Maujean et al. (1987) are utilized. For some rovibrational levels, lower limits of the predissociation yields were given by Glass-Maujean et al. (1987). In these cases, lower limits are used. For autoionization, the autoionization yields reported by Dehmer & Chupka (1976) have been applied. At room temperature, emission yields for rotational levels up to the J_j = 4 are required. Alternatively, the emission yields can be estimated from the calculated spontaneous transition probabilities and photoabsorption line width measurements of Glass-Maujean et al. (2007a, 2007b, 2007c, 2008a). While the experimental uncertainties of the width, typically ± 0.3 cm⁻¹, are somewhat too large, they can provide a rough estimate of the yields for the high v_j levels when other data are unavailable. Because emissions from these levels are generally very weak, the error in line width does not lead to significant change in the band emission cross sections, at least, for the low n Rydberg states. The emission cross sections of the $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$ and D'¹Π_u states are indicated by dotted lines in Figures 4 and 5.

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Table 3 compares excitation and emission cross sections of singlet-ungerade states obtained in several studies at 100 eV and 300 K. The present cross sections for the B¹Σ⁺_u and C¹Π_u states differ from those reported by Liu et al. (1998) in two ways. First, excitation to H(1s)+H(2ℓ) continuum levels, neglected in Liu et al. (1998), is taken into account in the present work using oscillator strengths derived from photodissociation cross sections calculated by Glass-Maujean (1986). Second, while Liu et al. (1998) used the B¹Σ⁺_u − X¹Σ⁺_g and C¹Π_u − X¹Σ⁺_g transition probabilities of Abgrall & Roueff (1989), the present work used transition probabilities from Abgrall et al. (1993a, 1993b, 1994). The emission cross sections of the B¹Σ⁺_u, C¹Π_u, and B'¹Σ⁺_u states do not include emission originating from the H(1s)+H(2ℓ) continuum, but include continuum emission originating from the discrete levels of the B¹Σ⁺_u, C¹Π_u, and B'¹Σ⁺_u states into the X¹Σ⁺_g continuum, H(1s)+H(1s). At 100 eV and 300 K, Abgrall et al. (1997) have shown that transitions to the X¹Σ⁺_g continuum contributes about 27.5% and 1.5% to the B¹Σ⁺_u and C¹Π_u emission cross sections, respectively. The transition probabilities of some high v_j levels of the D¹Π⁻_u state have been adjusted to the experimental values of Glass-Maujean et al. (2007c). The present cross sections of the D¹Π⁻_u state differs slightly from those obtained by Jonin et al. (2000). In case of the $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$ and D'¹Π_u states, Table 3 shows that the present cross sections are significantly larger than those estimated by Jonin et al. (2000). Possible reasons for the large difference will be discussed in Section 6.

Table 3. Electronic-Band Cross Sections and Emission Yields of H₂ Singlet-ungerade States^a

State	Present σex	Previous σex	Present σem	Previous σem	Present Em. Yield	Previous Em. Yield
B1Σ+u	264b	262c	263	262c	99%b	100%
C1Πu	244b	241c	240b	241c	98%b	100%
B'1Σ+u	40b	38d,e	21	21d	53%	56%
D1Π+u	25	24d	11	11d	43%	46%
D1Π−u	21	18d	21	18d	100%	100%
$B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$	11	>4d	2.2	1.6d	20%	<40%
D'1Π+u	9.3	7.1d	1.6	1.0d	18%	14%
D'1Π−u	7.3	⩾5.3d	5.7	5.3d	78%	⩽ 100%
D''1Πu	3.2	>0.6	0.9	0.6	28%	...
5pσ1Σ+u	...	...	1.1	...	...	...
6pσ1Σ+u	...	...	0.6	...	...	...
6pπ1Πu	...	...	0.9	...	...	...
7pσ1Σ+u	...	...	0.6	...	...	...

Notes. ^aE = 100 eV and T = 300 K. Unit is 10⁻¹⁹ cm². σ_ex and σ_em denote excitation and emission cross sections, respectively. Certain numbers may not add up due to roundings. See Section 5.3 for estimated errors in cross sections. ^bExcitation cross sections include the excitation into the H(1s)+H(2ℓ) continuum, which is estimated from the calculation of Glass-Maujean (1986). Emission cross sections exclude emission from the H(1s)+H(2ℓ) continuum levels, but include continuum emission from the excited discrete levels into the continuum levels of the X¹Σ⁺_g state. Transitions to the X¹Σ⁺_g continuum contribute 27.5% and 1.5%, respectively, to total emission cross sections of B¹Σ⁺_u − X¹Σ⁺_g and C¹Π_u − X¹Σ⁺_g (Abgrall et al. 1997). ^cFrom Liu et al. (1998). ^dFrom Jonin et al. (2000). ^eInclude excitations into the continuum levels of the B'¹Σ⁺_u state.

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The estimated errors of the electronic-state cross sections listed in Table 3 are as follows. The errors for the B¹Σ⁺_u, C¹Π_u, and D¹Π_u states, which primarily arise from the uncertainty in the excitation function, are no greater than 10%. For the B'¹Σ⁺_u and D'¹Π_u states, uncertainty is less than 12%–13%. The $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$(0)−X¹Σ⁺_g emission makes a substantial contribution to the total $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$ state emission cross section. As the calculated nonadiabatic transition probabilities are unable to satisfactorily reproduce the observed relative intensities of the R(0) and P(2) of the B'¹Σ⁺_u(4) and $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$(0) levels, there is an additional uncertainty in the $B^{\prime \prime }\bar{B}\,^{1}\Sigma _{u}^{+}$ state cross section, approaching to 15% for the excitation cross section and ∼18% for the emission cross section. The error limit for the 5pσ¹Σ⁺_u, 6pσ¹Σ⁺_u, and 7pσ¹Σ⁺_u state emission cross sections can be as high as 25%–35% because of weak emission intensities and, more importantly, rovibronic coupling noted in Section 5.2.