A Theoretical Study of Temperature-dependent Photodissociation Cross Sections and Rates for O₂

Implementing the high-level ab initio calculations in the MOLPRO 2015 software package (Werner et al. 2015, 2020), potential energy curves (PECs) and transition dipole moments (TDMs) of O₂ have been obtained. For a homonuclear diatomic molecule like O₂ with D_∞h symmetry, MOLPRO cannot take advantage of the full symmetry of the non-Abelian group, so the Abelian subgroup D_2h is chosen. The relationships of the irreducible representations from D_∞h to D_2h are as follows: ${{\rm{\Sigma }}}_{{\rm{g}}}^{+}$ → A_g, ${{\rm{\Sigma }}}_{{\rm{g}}}^{-}$ → B_1g, ${{\rm{\Sigma }}}_{{\rm{u}}}^{+}$ → B_1u, ${{\rm{\Sigma }}}_{{\rm{u}}}^{-}$ → A_u, Π_u → B_2u/B_3u, Π_g → B_2g/B_3g, Δ_g → A_g/B_1g, and Δ_u → A_u/B_1u. The computational steps are conventional. First of all, the Hartree–Fock calculation was used for the ground X ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$ state of O₂ to generate the initial single-configuration wave function and energy. Then, the complete active space self-consistent field (CASSCF) method (Knowles & Werner 1985; Werner & Knowles 1985) was used to optimize the initial wave function to obtain the multiconfiguration wave function. Finally, the dynamic correlation effect of O₂ was calculated using the internally contracted multireference configuration interaction (icMRCI) method (Knowles & Werner 1988, 1992; Werner & Knowles 1988; Shamasundar et al. 2011) based on the CASSCF wave function, and the Davidson correction (+Q) was also contained to take into account the size-consistency error (Langhoff & Davidson 1974). All the calculations of the PECs and TDMs for O₂ were performed with the augmented correlation-consistent polarized weighted core–valence aug-cc-pwCV5Z-DK basis set (Peterson & Dunning 2002).

The electronic arrangement of O is 1s²2s²2p⁴. For O₂, the electrons in the 1s shell were treated as closed, and the electrons in the remaining shells were put into the active space. Extra virtual orbitals were also added for better relaxation of the wave functions of high-lying electronic states. The set of orbitals is composed of four A_g orbitals, two B_3u orbitals, two B_2u orbitals, zero B_1g orbitals, four B_1u orbitals, one B_2g orbital, one B_3g orbital, and zero A_u orbitals and it is denoted as (4, 2, 2, 0, 4, 1, 1, 0). For the singlet and ground X ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$ states, we consider the internuclear distances between 0.9 and 6.0 Å. For the remaining triplet states, the internuclear distances from 1.0 to 6.0 Å are chosen. The step sizes are 0.02 Å for the internuclear distances from 1.0 to 2.5 Å and 0.05 Å for other internuclear distances.

The theory of photodissociation for diatomic molecules has been described in detail in previous works (Kirby & Van Dishoeck 1989; El-Qadi & Stancil 2013; Heays et al. 2017; Pattillo et al. 2018). Here we present a brief overview for the calculation of photodissociation cross sections and rates.

In units of cm² molecule⁻¹, the state-resolved cross section for a bound → free transition from the initial rovibrational level υ''N'' is

$$\begin{eqnarray}\begin{array}{rcl}{\sigma }_{\upsilon ^{\prime\prime} N^{\prime\prime} }\left({E}_{\mathrm{ph}}\right) & = & 2.689\times {10}^{-18}\,\times \,{E}_{\mathrm{ph}}\left(\displaystyle \frac{2-{\delta }_{0,{\rm{\Lambda }}^{\prime} +{\rm{\Lambda }}^{\prime\prime} }}{2-{\delta }_{0,{\rm{\Lambda }}^{\prime\prime} }}\right)\\ & \times & {\sum }_{N^{\prime} }\left(\displaystyle \frac{1}{2N^{\prime\prime} +1}{S}_{J^{\prime} ,J^{\prime\prime} }\left(N^{\prime} J^{\prime} ,N^{\prime\prime} J^{\prime\prime} \right){\left|{D}_{E^{\prime} N^{\prime} ,\upsilon ^{\prime\prime} N^{\prime\prime} }\right|}^{2}\right),\end{array}\end{eqnarray} \tag{ 2 }$$

where E_ph is the photon energy in atomic units, ${S}_{J^{\prime} ,J^{\prime\prime} }\left(N^{\prime} J^{\prime} ,N^{\prime\prime} J^{\prime\prime} \right)$ are the Hönl–London factors (Kovács & Nemes 1969), the term $\tfrac{2-{\delta }_{0,{\rm{\Lambda }}^{\prime} +{\rm{\Lambda }}^{\prime\prime} }}{2-{\delta }_{0,{\rm{\Lambda }}^{\prime\prime} }}$ is the degeneracy factor, and Λ is the angular momentum projections along the molecular axis. J, υ, and N are the total angular, vibrational, and rotational momentum quantum numbers, respectively. The initial state is denoted with a single prime superscript and the final state with a double prime.

In Equation (2), ${D}_{E^{\prime} N^{\prime} ,\upsilon ^{\prime\prime} N^{\prime\prime} }$ is the electron transition dipole matrix element, given by

$$\begin{eqnarray}&&{D}_{E^{\prime} N^{\prime} ,\upsilon ^{\prime\prime} N^{\prime\prime} }=\left\langle {\chi }_{E^{\prime} N^{\prime} }\left(R\right)\left|D\left(R\right)\right|{\chi }_{\upsilon ^{\prime\prime} N^{\prime\prime} }\left(R\right)\right\rangle ,\end{eqnarray} \tag{ 3 }$$

where D(R) is the electronic TDM in atomic units, ${\chi }_{E^{\prime} N^{\prime} }$ (R) is the continuous wave function of the final state, and χ_υ''N'' (R) is the bound wave function of the initial state.

For the case in which a Boltzmann distribution is assumed for the rovibrational levels of the initial electronic state, the corresponding total photodissociation cross section is a function of both temperature T and wavelength λ, and can be expressed as

$$\begin{eqnarray}&&\sigma \left(\lambda ,T\right)=\displaystyle \frac{{\sum }_{\upsilon ^{\prime\prime} }^{{\upsilon }_{\max }}{\sum }_{N^{\prime\prime} }^{{N}_{\max }\left(\upsilon \right)}(2-{\delta }_{{\rm{\Lambda }}^{\prime\prime} })(2S+1)(2N^{\prime\prime} +1)\exp \left(-\left|{E}_{\upsilon ^{\prime\prime} N^{\prime\prime} }-{\varepsilon }_{0}\right|/{k}_{B}T\right){\sigma }_{\upsilon ^{\prime\prime} N^{\prime\prime} }}{Q\left(T\right)},\end{eqnarray} \tag{ 4 }$$

where Q(T) is the rovibrational partition function, expressed as

$$\begin{eqnarray}&&\begin{array}{l}Q\left(T\right)=\displaystyle \sum _{n}^{{n}_{\max }}\displaystyle \sum _{\upsilon }^{{\upsilon }_{\max }\left(n\right)}\displaystyle \sum _{N}^{{N}_{\max }\left(n,\upsilon \right)}\left(2-{\delta }_{{\rm{\Lambda }}}\right)\left(2S+1\right)\left(2N+1\right)\\ \times \,\exp \left(-\left|{E}_{n\upsilon N}-{\varepsilon }_{0}\right|/{k}_{{\rm{B}}}T\right),\end{array}\end{eqnarray} \tag{ 5 }$$

where E_{n υ N} is the energy of the nth electronic state with quantum numbers υ, N and ε₀ refers to the energy of the lowest energy level. S is the spin quantum number. h, k_B, and c are the Planck constant, Boltzmann constant, and the speed of light in vacuum, respectively.

The photodissociation rate k of a molecule exposed to a UV radiation field can be estimated using the photodissociation cross sections σ(λ), given by

$$\begin{eqnarray}&&k=\int \sigma \left(\lambda \right)I\left(\lambda \right)d\lambda ,\end{eqnarray} \tag{ 6 }$$

where I (λ) is the sum of the photon intensities from the radiation field at all angles of incidence. The photon radiation intensity surrounded by a blackbody at the temperature of T_rad is expressed as

$$\begin{eqnarray}&&I\left(\lambda ,{T}_{\mathrm{rad}}\right)=\displaystyle \frac{8\pi c/{\lambda }^{4}}{\exp \left({hc}/{k}_{b}{T}_{\mathrm{rad}}\lambda \right)-1},\end{eqnarray} \tag{ 7 }$$

where h is the Planck constant and c is the speed of light. As inspired by Heays et al. (2017) and Pezzella et al. (2022), we take into account the blackbodies of three distinct temperatures to define various sorts of stars. T Tauri stars and stars in their early stages (Appenzeller & Mundt 1989; Natta 1993) are modeled using the blackbody of T_rad = 4000 K. To model the Herbig Ae stars and young A stars still encased in gas and dust (Vioque et al. 2018), the blackbody of T_rad = 10,000 K was selected. The brilliant and fleeting B stars (Habets & Heintze 1981) are modeled using the blackbody of T_rad = 20,000 K. It is worth noting that the blackbody radiation fields are normalized in this work to match the ISRF's energy intensity between 91.2 and 200 nm, as treated by Heays et al. (2017) and Pezzella et al. (2022). The scaling factors of 3.627 × 10⁸, 5.786 × 10¹³, and 6.109 × 10¹⁵ are used for the photodissociation rates in the radiation fields of the blackbody bodies for T_rad = 4000 K, T_rad = 10,000 K, and T_rad = 20,000 K, respectively.

For the standard ISRF, the photodissociation rate is computed using the wavelength dependence UV intensity defined by Draine (1978) at wavelengths of 91.2 < λ < 200 nm and extended by van Dishoeck & Black (1982) for λ > 200 nm. The solar radiation field was drawn from Heays et al. (2017) and its intensity was originally compiled from the data that were measured by Woods et al. (1996) and Curdt et al. (2001). A scaling factor of 37,700 should be used to increase the solar photodissociation rates calculated here to values appropriate for the approximate solar intensity at a distance of 1 au from the Sun.

In this work, six electronic states of O₂ correlating to the lowest two dissociation limits have been calculated and shown in Figure 1 as a function of the internuclear distance R, including three singlet states (i.e., a ¹Δ_g, b ${}^{1}{{\rm{\Sigma }}}_{{\rm{g}}}^{+}$, and 1 ¹Π_u) and three triplet states (i.e., X ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$, B ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}$, and E ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}$). The X ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$, a ¹Δ_g, b ${}^{1}{{\rm{\Sigma }}}_{{\rm{g}}}^{+}$, and 1 ¹Π_u states converge to the first dissociation limit O (2s²2p^{4 3}P) + O (2s²2p^{4 3}P) and the B ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}$ and E ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}$ states correlate to the second dissociation limit O (2s²2p^{4 3}P) + O (2s²2p^{4 1}D). The computed energy of the second dissociation limit relative to the first one is 15,745.11 cm⁻¹, which is only 122.75 cm⁻¹ (0.77%) smaller than the experimental value (Kramida et al. 2022). Liu et al. (2014) pointed out the existence of a double potential well in the B ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}$ state. We guess that the double potential well obtained by Liu et al. (2014) may come from the root flipping, i.e., the calculational energy points of the B ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}$ and 2 ³Δ_u states interchange with each other (at about 2.0 Å). To avoid root flipping, we performed state-averaged calculations of the first four electronic states with the same symmetry as the B ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}$ state. The results show that the B ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}$ state has a single potential well. Moreover, adiabatic B ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}$ and E ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}$ states avoid crossing at about 1.2 Å near the equilibrium internuclear of the ground state, as shown in Figure 1(a). These interactions produce large gradients in the coupling curves connecting these states within the region of the avoided crossing, as shown for the TDMs in Figure 2.

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Table 1 presents the spectroscopic constants of the X ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$, a ¹Δ_g, b ${}^{1}{{\rm{\Sigma }}}_{{\rm{g}}}^{+}$, and B ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}$ states, including the dissociation energy D_e , the electronic excitation energy relative to the ground state T_e , the equilibrium internuclear distance R_e , the harmonic frequency ω_e , the first-order anharmonic constant ω_e χ_e , the rotational constant B_e , and the rovibrational coupling constant α_e . These spectroscopic constants were obtained by fitting the rovibrational levels determined by solving the nuclear motion equation over the PECs. Previous experimental and theoretical results were also provided for comparison.

Table 1. Spectroscopic Constants of the X ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$, B ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}$, a ¹Δ_g, and b ${}^{1}{{\rm{\Sigma }}}_{{\rm{g}}}^{+}$ States for O₂ along with Available Experimental and Theoretical Values

State	Source	De	Te	Re	ωe	ωe χe	Be	αe
		(eV)	(cm−1)	(Å)	(cm−1)	(cm−1)	(cm−1)	(cm−1)
X ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$	This work	5.2070	0	1.2089	1578.72	12.5967	1.4438	0.0161
	Exp. a	5.2132	0	1.2075	1580.19	11.98	1.4456	0.0159
	Exp. b	5.213	0	1.208	1580.2	11.98	1.45	0.0159
	Calc. c	5.2203	0	1.2068	1581.61	10.039	1.4376	0.0125
	Calc. d	4.957	0	1.236	1498.8	9.87	1.38	0.0141
B ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}$	This work	1.0087	49,502.50	1.6071	701.49	7.5962	0.8123	0.0103
	Exp. a		49,793.28	1.6042	709.31	10.65	0.8190	0.0121
	Exp. b	1.007	49,794.33	1.604	709.1	10.61	0.819	0.0119
	Calc. c	0.8417	51,028.73	1.5978	723.60	10.756	0.8256	0.0136
	Calc. d	1.136	49,030.42	1.627	724.9	7.04	0.791	0.0077
a 1Δg	This work	4.2538	7800.12	1.2187	1515.08	13.1273	1.4237	0.0169
	Exp. a		7918.1	1.2156	1483.5	12.9	1.4263	0.0171
	Exp. b	4.231	7918.11	1.216	1509.3	12.9	1.43	0.0171
	Calc. c	4.2258	7776.43	1.2147	1491.07	8.1245	1.3814	0.0042
	Calc. d	3.857	8855.96	1.250	1403.4	8.74	1.35	0.0158
b 1Σ+ g	This work	3.6658	12,951.19	1.2273	1449.98	16.4295	1.3981	0.0182
	Exp. a			1.2269	1432.77	14.00	1.4004	0.0182
	Exp. b	3.577	13,195.31	1.227	1432.7	13.93	1.40	0.0182
	Calc. c	3.6058	13,099.92	1.2258	1438.65	12.723	1.4030	0.0180
	Calc. d	3.168	14,324.40	1.267	1310.8	10.44	1.31	0.0172

Notes.

^a Huber & Herzberg (1979). ^b Krupenie (1972). ^c Liu et al. (2014). ^d Saxon & Liu (1977).

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For the ground X ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$ state, the equilibrium internuclear distance R_e of 1.2089 Å is slightly larger than the experimental values of 1.2075 Å (Huber & Herzberg 1979) and 1.208 Å (Krupenie 1972) and agrees well with the theoretical results with relative errors of 0.17% (Liu et al. 2014) and 2.19% (Saxon & Liu 1977). The calculated harmonic frequency ω_e is 1578.72 cm⁻¹, which differs from the experimental value by 1.47 cm⁻¹ with a relative error of 0.09% (Huber & Herzberg 1979) and differs from the recent theoretical value by 2.89 cm⁻¹ with a relative error of 0.18% (Liu et al. 2014). For the rotational constant B_e and rovibrational coupling constant α_e , the errors relative to the experimental values (Huber & Herzberg 1979) are 0.12% and 0.94%, respectively. For the a ¹Δ_g and b ${}^{1}{{\rm{\Sigma }}}_{{\rm{g}}}^{+}$ states, their spectroscopic constants are in good agreement with the experimental values (Huber & Herzberg 1979) and the theoretical ones (Liu et al. 2014). For the B ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}$ state, our calculations obviously improve the spectroscopic constants relative to those calculated by Liu et al. (2014), by comparing with the experimental values (Huber & Herzberg 1979). Such improvement may be attributable to the high-level icMRCI/aug-cc- pwCV5Z-DK calculation.

Dipole-allowed TDMs between the abovementioned six electronic states are shown in Figure 2 as a function of the internuclear distance R. Figure 3 compares the TDMs of the B ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}\leftarrow {\rm{X}}$ ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$ transition with previous results. The TDM for the B ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}\leftarrow {\rm{X}}$ ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$ transition shows the same trend as those computed by Allison et al. (1986) and Liang et al. (2020) for R larger than 1.24 Å. For R smaller than about 1.24 Å, our TDM exhibits a similar trend to that given by Allison et al. (1986), but different from that presented by Liang et al. (2020). Such difference corresponds to different PECs of the B ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}$ state, in which we considered the avoided crossing with the E ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}$ state.

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To better reproduce the spectrum of the Schumann–Runge band, we use the CHIPR program (Rocha & Varandas 2019, 2020, 2021; Chen et al. 2022, 2023; Li et al. 2022, 2023) in combination with experimental energy levels (Krupenie 1972) to refine the PECs of the B ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}$ and X ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$ states. Here, we present a brief overview for the theory of this method.

In the CHIPR method, the diatomic PEC assumes the following form:

$$\begin{eqnarray}&&V(R)=\displaystyle \frac{{Z}_{{\rm{A}}}{Z}_{{\rm{B}}}}{R}\displaystyle \sum _{k=1}^{L}{C}_{k}{y}^{k},\end{eqnarray} \tag{ 8 }$$

where Z_A and Z_B are the nuclear charges for the atoms A and B, and y^k is expanded as

$$\begin{eqnarray}&&{y}^{k}=\displaystyle \sum _{\alpha =1}^{M}{c}_{\alpha }{\phi }_{p,\alpha },\end{eqnarray} \tag{ 9 }$$

where c_α are contraction coefficients, with α establishing the primitive functions' indexes ϕ_p,α . ϕ_p,α have the following two expressions:

$$\begin{eqnarray}&&{\phi }_{p,\alpha }={{\rm{sech}} }^{{\eta }_{\alpha }}\left({\gamma }_{p,\alpha }{\rho }_{p,\alpha }\right),\end{eqnarray} \tag{ 10 }$$

and

$$\begin{eqnarray}&&{\phi }_{p,\alpha }={\left[\displaystyle \frac{\tanh \left({\beta }_{\alpha }{R}_{p}\right)}{{R}_{p}}\right]}^{{\sigma }_{\alpha }}{{\rm{sech}} }^{{\eta }_{\alpha }}\left({\gamma }_{p,\alpha }{\rho }_{p,\alpha }\right),\end{eqnarray} \tag{ 11 }$$

where ρ_p,α is the deviation of the coordinate R_p from the primitive origin ${R}_{p,\alpha }^{\mathrm{ref}}$, ${\rho }_{p,\alpha }={R}_{p}-{R}_{p,\alpha }^{\mathrm{ref}}$. γ_p,α are nonlinear parameters, η_α = 1, β_α = 6, and σ_α = 1/5. The distributed origins ${R}_{p}^{\mathrm{ref}}$ can be expressed by

$$\begin{eqnarray}&&{R}_{p,\alpha }^{\mathrm{ref}}=\zeta {\left({R}_{p}^{\mathrm{ref}}\right)}^{\alpha -1},\end{eqnarray} \tag{ 12 }$$

where ζ and ${R}_{p}^{\mathrm{ref}}$ need to be carefully chosen while fitting.

The refined PECs for the X ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$ and B ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}$ states are shown in Figures 4(a) and 5(a), respectively. The errors between the vibrational-level energies obtained from our experimentally refined PECs and previously measured ones are shown in Figures 4(b) and 5(b) for the X ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$ and B ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}$ states, respectively. The maximum errors for the X ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$ and B ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}$ states are 0.99 cm⁻¹ and 2.35 cm⁻¹, respectively. The experimental vibrational levels come from Krupenie (1972). For the E ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}$ state, its electronic excitation energy T_e was first shifted to be the estimated value of 79,800 cm⁻¹ from Huber & Herzberg (1979), but the resulting line position for the peak of the cross sections for the E ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}\leftarrow {\rm{X}}$ ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$ transition slightly deviates from that of the more recent measurement by Lu et al. (2010). Hence, we adjusted the T_e of the E ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}$ state to be 79,643 cm⁻¹ to better reproduce the observation by Lu et al. (2010).

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To calculate the photodissociation cross sections and rates, ab initio PECs and TDMs are needed to be interpolated and extrapolated. For short-range internuclear distances at R< 0.9 Å, an exponential function is used for extrapolation, given by

$$\begin{eqnarray}&&V\left(R\right)=A\exp \left(-{BR}\right)+C,\end{eqnarray} \tag{ 13 }$$

where A, B, and C are fitting parameters. For long-range internuclear distances at R > 6 Å, the following formula is used for extrapolation:

$$\begin{eqnarray}&&V\left(R\right)=-\displaystyle \frac{{C}_{5}}{{R}^{5}}-\displaystyle \frac{{C}_{6}}{{R}^{6}}+V\left(R\to \infty \right),\end{eqnarray} \tag{ 14 }$$

where C₅ and C₆ are fitting coefficients, which are approximately estimated in this work. C₆ was calculated using the London formula:

$$\begin{eqnarray}&&{C}_{6}=\displaystyle \frac{3}{2}\displaystyle \frac{{{\rm{\Gamma }}}_{{\rm{O}}}{{\rm{\Gamma }}}_{{\rm{O}}}}{{{\rm{\Gamma }}}_{{\rm{O}}}+{{\rm{\Gamma }}}_{{\rm{O}}}}{\alpha }_{{\rm{O}}}{\alpha }_{{\rm{O}}},\end{eqnarray} \tag{ 15 }$$

where Γ_O and α_O are the ionization energy and static dipole polarizability, respectively, for a specific electronic state of the O atom. Γ_O can be obtained from the NIST Atomic Spectroscopic Database (Kramida et al. 2022). The dipole polarizabilities of the oxygen atoms in the ³P and ¹D states are 5.35 and 5.43 au, respectively, computed by Medveď et al. (2000). C₅ was estimated by fitting ab initio points while keeping C₆ and the dissociation limits fixed. A cubic spline was used to interpolate the ab initio points. A similar treatment was used in previous publications (Pattillo et al. 2018; Babb et al. 2019; Meng et al. 2022; Zhang et al. 2022).

State-resolved cross sections are the basis for a detailed study of the temperature-dependent photodissociation cross sections and rates. State-resolved photodissociation cross sections from the initial rovibrational energy level (υ'', N'') = (0, 1) of the ground X ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$ state to the excited B ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}$ and E ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}$ states, and from the initial rovibrational energy level (υ'', N'') = (0, 0) of the a ¹Δ_g and b ¹Σ⁺ _g states to the 1 ¹Π_u state, are calculated and shown in Figure 6 for photon wavelengths from 500 Å to the corresponding thresholds. The well-known Schumann–Runge continuum band of B ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}\leftarrow {\rm{X}}$ ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$ plays a major role at larger wavelengths between about 1200 and 1800 Å, in which the 1 ¹Π_u ←a ¹Δ_g and 1 ¹Π_u ←b ¹Σ⁺ _g transitions exhibit some peaks, while the populations of the a ¹Δ_g and b ¹Σ⁺ _g states are smaller relative to that of the ground X ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$ state in a general condition.

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Assuming the populations of the initial rovibrational levels for the X ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$, a ¹Δ_g, and b ${}^{1}{{\rm{\Sigma }}}_{{\rm{g}}}^{+}$ states satisfy a Boltzmann distribution, temperature-dependent cross sections were calculated for four electronic transitions of O₂ at temperatures from 0 to 10,000 K in intervals of 100 K. Note that we also consider the discrete progressions for the B ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}\leftarrow {\rm{X}}$ ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$ and E ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}\leftarrow {\rm{X}}$ ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$ transitions, whose cross sections are calculated by the DUO and EXOCROSS programs (Yurchenko et al. 2016, 2018) and then smoothed using a normalized Gaussian function proposed by the ExoMol group (Pezzella et al. 2021, 2022). Figure 7 shows the photodissociation cross sections of four transitions of O₂ at 300 K, along with those compiled by Heays et al. (2017). For wavelengths from 490 to 1080 Å, the photodissociation cross sections come from transitions into Rydberg states (Holland et al. 1993). Ogawa & Ogawa (1975) pointed out the cross sections at wavelengths from 1080 to 1150 Å might come from the absorption of the a ¹Δ_g state. The positions of the peaks show the final state should lie higher than the 1 ¹Π_u state. For wavelengths from 1150 to 1790 Å, the cross sections come from the absorption spectra at 303.7 K measured by Lu et al. (2010), which results from the E ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}\leftarrow {\rm{X}}$ ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$ and B ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}\leftarrow {\rm{X}}$ ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$ transitions. Both Lu et al. (2010) and Ogawa & Ogawa (1975) assigned the peak value at 120.54 nm to be the transition from the a ¹Δ_g state. Our calculations guess that this peak may be due to the E ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}\leftarrow {\rm{X}}$ ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$ transition. For wavelengths from 1790 to 2030 Å, the Schumann–Runge band is dominant and its cross sections are chosen from Yoshino et al. (1992). For wavelengths from 2050 to 2400 Å, the cross sections come from the Herzberg continuum presented by Yoshino et al. (1988). Overall, our photodissociation cross sections show reasonable agreement with the experimental ones for wavelengths from 1150 to 2030 Å. For wavelengths below 1150 Å, except for the direct photodissociation to high electronic states, the predissociation from nonadiabatic couplings and absorption to the Rydberg states may be also important, but these are not considered in this work.

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In DUO calculations of the discrete transitions for the B ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}\leftarrow {\rm{X}}$ ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$ and E ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}\leftarrow {\rm{X}}$ ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$ systems, 60, 20, and 3 vibrational basis functions are considered. The generated line lists for the B ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}\leftarrow {\rm{X}}$ ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$ and E ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}\leftarrow {\rm{X}}$ ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$ transitions are given in Tables 2 and 3, respectively. Based on the generated line lists, several spectra were calculated and compared to available laboratory measurements. Figure 8 compares the absorption spectra for the Schumann–Runge bands of O₂ with the experimental ones from Yoshino et al. (1987) and Yoshino et al. (1992), showing a satisfactory agreement. The spectra are simulated at the effective temperatures of 79 and 300 K, respectively, in Figures 8(a) and (b). In Figure 9, we simulate the absorption spectra for the E ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}\leftarrow {\rm{X}}$ ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$ bands of O₂ at temperatures of 38 and 303.7 K, respectively, to provide direct comparison with the experimental spectra from Lu et al. (2010). The overall agreement can be observed. A comparison of the E ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}\leftarrow {\rm{X}}$ ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$ spectra to the experimental one from Metzger & Cook (1964) is also given in Figure 10. Note that the line lists in Tables 2 and 3 cannot be used to simulate an observed spectrum at high resolution and at low temperatures, because the spin rotation structure is not resolved in the computations.

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Table 2. The Line List for the B ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}\leftarrow {\rm{X}}$ ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$ Transition

J'	J''	Typ	E'	E''	${\nu }_{J^{\prime} J^{\prime\prime} }$	${A}_{J^{\prime} J^{\prime\prime} }$	State'	υ'	Λ'	Σ'	Ω'	State''	υ''	Λ''	Σ''	Ω''
1	0	R	56,739.2163	2.8749	56,736.3414	6.11E+03	2	16	0	−1	−1	1	0	0	0	0
1	0	R	56,341.3890	2.8749	56,338.5141	5.6758E+03	2	14	0	−1	−1	1	0	0	0	0
103	104	P	57,396.9447	40,714.1649	16,682.7798	2.9022E+02	2	0	0	−1	−1	1	27	0	1	1
103	104	P	57,396.9447	41,500.6402	15,896.3045	3.4896E+01	2	0	0	−1	−1	1	28	0	1	1

Notes.

' : upper state.

'' : lower state.

J: total angular momentum.

Typ: transition type.

E: rovibrational energy level.

ν: transition wavenumber.

A: Einstein coefficient.

State: electronic state—2 stands for the B ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}$ state and 1 stands for the X ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$ state.

υ: state vibrational quantum number.

Λ: projection of the electronic angular momentum along the internuclear axis.

Σ: projection of the electronic spin along the internuclear axis.

Ω: projection of the total angular momentum along the internuclear axis, Ω = Λ+Σ.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Table 3. The Line List for the E ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}\leftarrow {\rm{X}}$ ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$ Transition

J'	J''	Typ	E'	E''	${\nu }_{J^{\prime} J^{\prime\prime} }$	${A}_{J^{\prime} J^{\prime\prime} }$	State'	υ'	Λ'	Σ'	Ω'	State''	υ''	Λ''	Σ''	Ω''
1	0	R	85,660.0515	2.8749	85,657.1766	1.3838E+06	2	2	0	−1	−1	1	0	0	0	0
1	0	R	90,138.5898	2.8749	90,135.7149	7.3700E+04	2	4	0	−1	−1	1	0	0	0	0
103	104	P	95,504.1824	40,714.1649	54,790.0175	3.7216E+01	2	0	0	−1	−1	1	27	0	1	1
103	104	P	95,504.1824	41,500.6402	54,003.5423	8.0825E+01	2	0	0	−1	−1	1	28	0	1	1

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Figure 11 compares the cross sections for the Schumann–Runge continuum to previous experimental ones from Metzger & Cook (1964), Hudson et al. (1966), Ogawa & Ogawa (1975), Gibson et al. (1983), Yoshino et al. (2005), and Lu et al. (2010). The differences between these experiments are within 10%. Our calculational results agree well with the recent measurements by Lu et al. (2010).

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A comparison of temperature-dependent cross sections for four electronic transitions as a function of photon wavelength at T = 0, 500, 3000, and 10,000 K is shown in Figure 12. For the cross sections at T = 0 and T = 500 K, there is no significant difference, but obvious alterations are seen above T = 3000 K. At long wavelengths, the tails of the cross sections grow because of the more excited rovibrational states at high temperatures. Moreover, the 1 ¹Π_u ←a ¹Δ_g and 1 ¹Π_u ←b ¹Σ⁺ _g transitions become more and more important with the temperature increasing.

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Interstellar, solar, and blackbody radiation fields have widespread applications in astrochemistry. We calculated the photodissociation rates of O₂ in these three radiation fields. The photodissociation rates of O₂ for each transition in the ISRF were calculated using temperature-dependent cross sections and are presented in Table 4. Our calculated photodissociation rate of O₂ at T = 0 K is 7.48 × 10⁻¹⁰ s⁻¹, which is slightly lower than that of 7.7 × 10⁻¹⁰ s⁻¹ provided by Heays et al. (2017). Figure 13 shows the photodissociation rates of O₂ in the blackbody radiation fields of 4000, 10,000, and 20,000 K, as well as in the solar radiation field. The total photodissociation rates at 0 K are 7.57 × 10⁻¹¹ s⁻¹, 5.66 × 10⁻¹⁰ s⁻¹, and 6.81 × 10⁻¹⁰ s⁻¹ for the blackbody at 4000, 10,000, and 20,000 K, respectively, versus 7.50 × 10⁻¹¹ s⁻¹, 5.60 × 10⁻¹⁰ s⁻¹, and 7.23 × 10⁻¹⁰ s⁻¹ reported by Heays et al. (2017). For the blackbody at 4000 K, the photodissociation rate of O₂ increases by nearly 3 orders of magnitude at temperatures from 0 to 10,000 K. However, the photodissociation rates are flattening for the blackbody at 10,000 and 20,000 K, because the radiation fields produced by high-temperature stars tend to flatten the temperature effects on the rates (Pezzella et al. 2022). Our photodissociation rate of O₂ in solar radiation fields is 5.87 × 10⁻¹¹ s⁻¹ at T = 0 K, versus 6.10 × 10⁻¹¹ s⁻¹ reported by Heays et al. (2017).

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Table 4. Photodissociation Rates (s⁻¹) of O₂ Obtained from Temperature-dependent Cross Sections under the Standard ISRF (Draine 1978; Heays et al. 2017)

	0 K	500 K	1000 K	2000 K	3000 K	5000 K	10,000 K
1 1Πu ←a 1Δg	0	5.26E–21	4.48E–16	1.20E–13	7.20E–13	2.57E–12	4.27E–12
1 1Πu ← b ${}^{1}{{\rm{\Sigma }}}_{{\rm{g}}}^{+}$	0	4.21E–28	6.16E–20	1.08E–15	3.35E–14	5.37E–13	3.00E–12
B ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}\leftarrow {\rm{X}}$ ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$	6.71E–10	6.72E–10	6.85E–10	6.65E–10	5.95E–10	4.83E–10	2.86E–10
E ${}^{3}{{\rm{\Sigma }}}_{{\rm{u}}}^{-}\leftarrow {\rm{X}}$ ${}^{3}{{\rm{\Sigma }}}_{{\rm{g}}}^{-}$	7.71E–11	7.56E–11	7.73E–11	8.11E–11	8.01E–11	6.88E–11	3.82E–11
Total	7.48E–10	7.48E–10	7.62E–10	7.46E–10	6.76E–10	5.55E–10	3.31E–10

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