ERRATUM: "H₂ FORMATION ON GRAIN SURFACES" (2004, ApJ, 604, 222)

 As noted by Bai & Goodman (2009), the equations for the transmission coefficients to go from a site i to a site j in Cazaux & Tielens (2004) erroneously omitted a key factor and also contained a typo. The correct equations are

$$\begin{equation} T_{ij}^{(1)}=4\times \sqrt{\frac{E-B_{ij}}{E}} \left[ \left(1+\sqrt{\frac{E-B_{ij}}{E}} \right)^2 +\frac{B_i B_j \left(\sinh \left(\sqrt{{\frac{2m(B_i-E)}{\hbar ^2}}}Z\right)\right)^2}{(B_i-E)E}\right]^{-1} \qquad {\rm if}\; E<B_i\,, \end{equation} \tag{ 1 }$$

$$\begin{equation} T_{ij}^{(2)}=4\times \sqrt{\frac{E-B_{ij}}{E}} \left[ \left(1+\sqrt{\frac{E-B_{ij}}{E}} \right)^2 -\frac{B_i B_j \left(\sin \left(\sqrt{{\frac{2m(E-B_i)}{\hbar ^2}}}Z\right)\right)^2}{(B_i-E)E}\right]^{-1} \qquad {\rm if}\; E>B_i\,. \end{equation} \tag{ 2 }$$

These equations differ from Equations (1) and (2) of the original paper in the additional factor $\sqrt{\frac{E-B_{ij}}{E}}$, which is the ratio of the transmitted over the incident wave numbers, with a change of variable in order to define the energy E as being zero at the bottom of the potential well i (e.g., for T_pc, the energy of the atom is defined from the physisorbed site). The relevant energies B_i, B_j, and B_ij are summarized in Table 1 where—compared to the original paper—a small typo in the expression for the B_ij has been corrected as well. Note that B_ij is negative with these definitions and the omitted factor can greatly increase the values of the transmission coefficient for an H atom to go from a physisorbed site to a chemisorbed site. As a result, the probability for a physisorbed H atom to become chemisorbed increases. By the same token, the H₂ formation efficiency will also increase. Using the parameters for the potential energy curves as described in Figure 15 of Cazaux & Tielens (2004), Figure 1 shows the corrected efficiencies for H₂ formation on olivine and carbonaceous surfaces. For grain temperatures higher than 25 K, these efficiencies are ∼3.5 times higher than without the correction. For lower temperatures, where H₂ formation from physisorbed H atoms dominates, the efficiencies remain unchanged.

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The general expressions for the H₂ formation efficiency obtained in Cazaux & Tielens (2002, Equation (9)) and Cazaux & Tielens (2004, Equation (19)) remain valid. However, the expression for α_pc is calculated from the transmission coefficients that were previously underestimated. Therefore, the corrected expression for α_pc is

$$\begin{eqnarray} {\alpha _{\mbox{pc}}}=&8\sqrt{\pi T} \, \nu _{H_P} \frac{\sqrt{E_{H_C}-E_{H_P}}}{E_{H_C}-E_S} \exp \left(-2a\frac{\sqrt{2 m_H k_B (E_{H_P}-E_S)}}{\hbar }\right)\; \nonumber \\ &+4\nu _{H_P}\, \sqrt{\frac{E_{H_P}-E_S}{E_{H_C}-E_{S}}}\exp \left({\frac{-(E_{H_P}-E_S)}{T}}\right), \end{eqnarray} \tag{ 3 }$$

where the first term is the mobility through tunneling and the second term through thermal hopping. In the original paper, the tunneling term was not taken into account because tunneling was only important at low dust temperatures and, for those conditions, direct recombination of physisorbed atoms dominates H₂ formation. With the corrected expression, this term has to be taken into account.