STATE RESOLVED DATA FOR RADIATIVE ASSOCIATION OF H AND H⁺ AND FOR PHOTODISSOCIATION OF ${\rm H}_2^+$

Radiative association of H and H⁺ produces ${\rm H}_2^+$ in various vibrational-rotational states. Recent studies of H₂ chemistry in the early universe indicated the necessity of accounting for the vibrational-rotational state distribution of the ${\rm H}_2^+$ population in the absence of local thermodynamic equilibrium, which accordingly requires "state-resolved," i.e., vibrational-rotational state resolved, calculations of data for the H + H⁺ radiative association process and for the inverse process, photodissociation of ${\rm H}_2^+$. The absence of such data in the literature was noted by Hirata & Padmanabhan (2006), Coppola et al. (2011), and Glover et al. (2014). In the present paper, the necessary data to calculate radiative association and photodissociation cross sections for each vibrational-rotational level of ${\rm H}_2^+$ are presented. Cross sections so assembled can then be summed or averaged as required for particular applications, e.g., rate coefficients for non-local thermal equilibrium models. In addition to being useful for early universe chemistry models, ${\rm H}_2^+$ photodissociation cross sections enter in models of the solar atmosphere (Stancil 1994; Mihajlov et al. 2007), where ${\rm H}_2^+$ is vibrationally excited, and in models of the atmospheres of DA white dwarfs (Santos & Kepler 2012), and of chemistry in the winds of young stellar objects (Glassgold 1998, p. 168).

The radiative association reaction

$$\begin{equation} {\rm H}+ {\rm H}^+\rightarrow {\rm H}_2^+(v,N) + h\nu \end{equation} \tag{ 1 }$$

and its inverse process photodissociation

$$\begin{equation} {\rm H}_2^+(v,N) + h\nu \rightarrow {\rm H}+ {\rm H}^+ \end{equation} \tag{ 2 }$$

for a photon of energy hν are related through microscopic reversibility (Moses & Wu 1951a 1951b; Light et al. 1969; Hirata & Padmanabhan 2006), where (v, N) denotes the vibrational v and rotational N quantum number of the molecular ion.

The rate coefficients for Equation (1) calculated by Bates (1951), Ramaker & Peek (1976), and Stancil et al. (1993) are in agreement. The cross section summed over all bound states (v, N) of ${\rm H}_2^+$ was given by Stancil et al. (1993).

The photodissociation cross sections for Equation (2) were calculated by Oksyuk (1967), Dunn (1968a), Dunn (1968b), Argyros (1974), Stancil (1994), Lebedev & Presnyakov (2002), and Mihajlov et al. (2007). Where individual data for select (v, N) levels and photon energies hν is given there is general agreement between their calculations. A tabulation of the photodissociation cross sections for each of the vibrational states (v, 1) is given in the report of Dunn (1968b). Recently, calculations of the photodissociation cross sections were carried out for applications to DA white dwarfs by Santos & Kepler (2012).

The cross section for radiative association (1) is (Stancil et al. 1993)

$$\begin{equation} \bar{\sigma }^\mathrm{ra}_N(v,E) = \frac{8}{3}\frac{\pi ^2}{\hbar ^3c^3} \frac{h^3\nu ^3}{k^2} p \left[ N M^2_{v,N-1;k,N} + (N+1) M^2_{v,N+1;k,N} \right], \end{equation} \tag{ 3 }$$

and M_{v, N − 1; k, N} is the matrix element of the transition dipole moment between the continuum wave function of energy E and angular momentum N and the bound vibrational-rotational wave function, k² = 2μE/ℏ², μ is the reduced mass of the colliding system, hν is the photon energy, and p = 1/2 is the probability of approach in the 2pσ_u electronic state.

The cross section for photodissociation is (Lebedev & Presnyakov 2002)

$$\begin{equation} \bar{\sigma }^\mathrm{pd}_{v,N} (h\nu) = \frac{8\pi ^3\nu }{3c (2N+1)} \left[ (N+1) M^2_{v,N;k,N+1} + N M^2_{v,N;k,N-1} \right]. \end{equation} \tag{ 4 }$$

In Equations (3) and (4), the nuclear spin symmetry weighting is omitted and it can be included later if the cross sections are being folded into a temperature distribution (Stancil 1994; Mihajlov et al. 2007). The effect of centrifugal distortion is negligible for the repulsive 2pσ_u state and it is a good approximation to replace N + 1 and N − 1 by N in Equations (3) and (4). Then,

$$\begin{equation} \sigma ^\mathrm{ra}_N(v,E) = \frac{8}{3} \frac{\pi ^2}{\hbar ^3c^3} \frac{h^3\nu ^3}{k^2} p (2N+1) M^2_{v,N;k,N} \end{equation} \tag{ 5 }$$

and the cross section for photodissociation is

$$\begin{equation} \sigma ^\mathrm{pd}_{v,N} (h\nu) = \frac{4\pi ^2 h \nu }{3\hbar c} M^2_{v,N;k,N} \quad. \end{equation} \tag{ 6 }$$

A comparison of Equations (5) and (6) shows that

$$\begin{equation} \sigma ^\mathrm{ra}_N(v,E) = p \frac{(h\nu)^2}{\mu c^2 E} (2N+1) \sigma ^\mathrm{pd}_{v,N} (h\nu), \end{equation} \tag{ 7 }$$

in accordance with microscopic reversibility (Light et al. 1969). Thus, tabulation of v, N, E_{v, N}, E, and $M^2_{v,N;k,N}$ provides the necessary information for calculations of $\sigma ^\mathrm{ra}_N(v,E)$ and $\sigma ^\mathrm{pd}_{v,N}(h\nu)$, subject to the requirement δ(hν − (|E_{v, N}| + E)) for either Equation (5) or Equation (6), where E_{v, N} is the vibrational-rotational eigenvalue in the 1sσ_g state measured with respect to the dissociation limit (taken to be zero here).

The Born-Oppenheimer potential energy surfaces for the ground 1sσ_g state and excited 2pσ_u state were calculated using the methods described by Madsen & Peek (1971) and extended asymptotically as in Stancil et al. (1993). The transition dipole moment was calculated using a variational method (Babb 1994) and extended using the asymptotic formula of Ramaker & Peek (1973). Energy differences and oscillator strengths are in excellent agreement with the recent calculations of Tsogbayar & Banzragch (2010).

There are 423 bound vibrational-rotational levels for ${\rm H}_2^+$ in the 1sσ_g state. In the present work, the Born-Oppenheimer potential is used and adiabatic, relativistic, and radiative corrections are ignored. It is worth noting that more precise approaches yield the same number of bound levels (Hunter et al. 1974; Moss 1993), but due to the inclusion of higher-order corrections the resulting eigenvalues differ from those obtained using only the Born-Oppenheimer potential. For the present purposes, the corrections are not needed.

The energy-normalized continuum wave functions on the 2pσ_u state were evaluated as in Stancil et al. (1993).

The quantities are calculated at a range of energies from the threshold energy |E_{v, N}| for photodissociation of level (v, N) to a maximum photon wavelength of 55 nm (22.5 eV or 0.828 a.u.). Only values of $M^2_{v,N;k,N} > 1\times 10^{-6}$ are listed.

Table 1 gives the vibrational quantum number v, the rotational quantum number N, the relative energy E in atomic units of energy (27.2114 eV), the eigenvalue |E_{v, N}| in cm⁻¹, the photon wavelength λ = c/hν in nm, and the value of $M^2_{v,N;k,N}$ in atomic units.

Table 1. Calculated Values of the Squared Matrix Element, $M^2_{v,N;k,N}$, for Bound Vibrational-rotational Levels (v, N) of the 1sσ_g State of ${\rm H}_2^+$ with Eigenvalue |E_{v, N}| to a Continuum Level of the 2pσ_u State of Relative Kinetic Energy E for a Photon of Energy hν

v	N	E	|Ev, N|	c/hν	$M^2_{v,N;k,N}$
0	0	0.7310E+00	21375.95	55.00	0.1333E-03
0	0	0.7249E+00	21375.95	55.41	0.1687E-03
0	0	0.7188E+00	21375.95	55.83	0.2126E-03
0	0	0.7128E+00	21375.95	56.24	0.2667E-03
0	0	0.7069E+00	21375.95	56.65	0.3329E-03
0	0	0.7011E+00	21375.95	57.06	0.4138E-03
0	0	0.6953E+00	21375.95	57.48	0.5122E-03
0	0	0.6897E+00	21375.95	57.89	0.6312E-03
0	0	0.6841E+00	21375.95	58.30	0.7747E-03
0	0	0.6786E+00	21375.95	58.72	0.9469E-03
0	0	0.6732E+00	21375.95	59.13	0.1153E-02
0	0	0.6678E+00	21375.95	59.54	0.1398E-02
0	0	0.6626E+00	21375.95	59.95	0.1689E-02
0	0	0.6574E+00	21375.95	60.37	0.2033E-02
0	0	0.6523E+00	21375.95	60.78	0.2438E-02
0	0	0.6472E+00	21375.95	61.19	0.2914E-02
0	0	0.6422E+00	21375.95	61.61	0.3470E-02
0	0	0.6373E+00	21375.95	62.02	0.4118E-02
0	0	0.6324E+00	21375.95	62.43	0.4872E-02
0	0	0.6276E+00	21375.95	62.84	0.5743E-02
0	0	0.6229E+00	21375.95	63.26	0.6750E-02
0	0	0.6182E+00	21375.95	63.67	0.7907E-02
0	0	0.6136E+00	21375.95	64.08	0.9235E-02
0	0	0.6091E+00	21375.95	64.49	0.1075E-01
0	0	0.6046E+00	21375.95	64.91	0.1248E-01
0	0	0.6001E+00	21375.95	65.32	0.1445E-01
0	0	0.5958E+00	21375.95	65.73	0.1668E-01
0	0	0.5914E+00	21375.95	66.15	0.1920E-01
0	0	0.5872E+00	21375.95	66.56	0.2204E-01
0	0	0.5829E+00	21375.95	66.97	0.2523E-01

Notes. In the table, E is in atomic units, |E_{v, N}| is in cm⁻¹, c/hν is in nm, and $M^2_{v,N;k,N}$ is in atomic units.

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.

Download table as:  DataTypeset image

The photodissociation cross section in units of cm² is (el-Qadi & Stancil 2013)

$$\begin{equation} \sigma ^\mathrm{pd}_{v,N} (h\nu) = 2.689 \times 10^{-18} (45.563/\lambda) M^2_{v,N;k,N} \quad {\rm cm}^2, \end{equation} \tag{ 8 }$$

with λ in nm and $M^2_{v,N;k,N}$ in atomic units. The radiative association cross section in units of cm² is (Stancil et al. 1993)

$$\begin{eqnarray} \sigma^\mathrm{ra}_N(v,E) &=& 1.475\times 10^{-20} (45.563/\lambda)^3\nonumber\\ &&\times { (2N+1)} E^{-1} M^2_{v,N;k,N} \quad {\rm cm}^2, \end{eqnarray} \tag{ 9 }$$

with λ in nm, E in atomic units, and $M^2_{v,N;k,N}$ in atomic units.

Dunn (1968a, 1968b) calculated the photodissociation cross section for v = 0 to v = 18 using the potential energy surfaces calculated by Bates et al. (1953). There are slight differences between the accurate potential energy surfaces of the present work and the early calculations of Bates et al. (1953). Nevertheless, agreement is generally good between the current calculations and the tabulated values of Dunn (1968a, 1968b). In Figure 1 for v = 10, N = 0, a plot of $\sigma ^\mathrm{pd}_{v,N}(h\nu)$ calculated from the data in Table 1 is presented. The representation is very good and the tabulated data reflect the peaks arising from the v = 10 bound state wave function. The present tabulated data can be interpolated using cubic splines, but due to the oscillations several intervals should be selected.

Zoom In Zoom Out Reset image size

ITAMP is supported in part by a grant from the NSF to the Smithsonian Astrophysical Observatory and Harvard University.