The total cross sections for reaction (1) computed in this work are presented in Fig. 1 along with all the available experimental data, previous calculations, and the recommended cross section of Janev et al. ([1995]). The quantal MOCC (QMOCC) calculations are in good agreement with the measurements of Stebbings et al. ([1960]) and Fite et al. ([1962]). Between $\sim 0.2$ and $\sim 1$ eV/u a rapid oscillation is apparent in the QMOCC cross section (see also Fig. 2). This may be due to orbiting effects in the very shallow bound potential of the OH⁺ $2~^3\Sigma^-$ state which is further discussed in Krajcar-Bronic et al. ([1999]). The current QMOCC calculation included three electronic channels with the energy difference of 115 K (9.91 meV), obtained from the centroid of the O fine-structure levels (see Table 1). It was found that radial coupling was the dominant charge transfer mechanism with rotational coupling giving a negligible contribution.

$\begin{figure}\epsfxsize=12cm \epsfbox{ds9003f1.eps} \end{figure}$

Figure 1: Total charge transfer cross sections for reaction (1), O⁺ + H $\to$ O + H⁺. This work: QMOCC [thick solid line], QMOCC-FS [thick dotted line], SCMOCC [thick dashed line], CTMC [thick long dashed line], CDW [thick dot-dashed line]. Previous experiment: Stebbings et al. (1960) [pluses], Fite et al. (1962) [filled left triangles], Phaneuf et al. (1978) [filled circles], Meyer et al. (1979) [filled square]. Previous theory: Rapp & Ortenburger (1960), SC [thin dot-dashed line]; Eichler et al. (1981), OBK [thin long dashed line]; Janev & McDowell, CTMC (1984) [open diamond]. Recommended cross section: Janev et al. (1995) [thin dashed line]

$\begin{figure}\epsfxsize=12cm \epsfbox{ds9003f2.eps} \end{figure}$

Figure 2: Fine-structure resolved charge transfer cross sections for reaction (1), O⁺ + H $\to$ O + H⁺. Symbols same as Fig. 1, with additions. Fine-structure resolved: this work, QMOCC-FS [thin dotted lines]; Rapp & Ortenburger (1960), SC [thin dot-dashed lines]

$\begin{figure}\epsfxsize=12cm \epsfbox{ds9003f3.eps} \end{figure}$

Figure 3: Total charge transfer cross sections for reaction (2), H⁺ + O $\to$ H + O⁺. This work: QMOCC [thick solid line], QMOCC-FS [thin dotted line, J-resolved; thick dotted line, statistical averaged sum], SCMOCC [thick dashed line]. Previous experiment: Stebbings et al. (1964) [pluses], Rutherford & Vroom (1974) [X's], Van Zyl & Stephen (1992) [filled up triangles], Lindsay et al. (1996) [filled circles]. Previous theory: Chambaud et al. (1980), QMOCC-FS [thin long dashed lines, J-resolved and statistical averaged]; Kimura et al. (1997), QMOCC [open left triangles]; Hedström et al. (1998), END [open diamonds]. Recommended cross section: Janev et al. (1995), by detailed balance [thin dashed line]

$\begin{figure}\epsfxsize=12cm \epsfbox{ds9003f4.eps} \end{figure}$

Figure 4: Same as Fig. 3, but with additions. This work: CTMC (all shells) [thick long dashed line], CDW (all shells) [thick dot-dash line]. Previous experiment: Stiers & Barnett (1956) [open down triangle], de Heer et al. (1966) [filled squares], Schryber (1967) [stars], Toburen et al. (1968) [open left triangles], Cocke et al. (1977) [open squares], Williams et al. (1984) [filled diamonds], Thompson et al. (1996) [open circles]. Previous theory: Hamre et al. (1999) [thin solid line], Lin et al. (1978) [thin long dashed line], Tan & Lee (1981) [thin dot-dash line], Miraglia (1984) [thin dot-dash line], Saha et al. (1985) [thin solid line]

Figure 1 also shows our semiclassical MOCC (SCMOCC) which is in excellent agreement with the measurements of Stebbings et al. ([1960]), Fite et al. ([1962]), and Phaneuf et al. ([1978]), while the semiclassical calculation of Rapp & Ortenburger appears to overestimate the cross section by a factor of $\sim 2$ between 0.1 eV/u and 1 keV/u and has the wrong energy dependence for lower energies.

For higher energies, Fig. 1 displays new CTMC and CDW results compared to the experiments of Phaneuf et al. ([1978]) and Meyer et al. ([1979]), the Oppenheimer-Brinkman-Kramers (OBK) approximation calculation of Eichler et al. ([1981]), and the CTMC calculation of Janev & McDowell ([1984]). For collision energies greater than 20 keV/u and 500 keV/u, the current CTMC and CDW results, respectively are in good agreement with the previous measurements and calculations. As mentioned earlier, CTMC is not reliable at lower energies because it neglects quantum tunneling effects.

For lower energies (< 0.5 eV/u), the fine-structure of the oxygen atom becomes important. Our new QMOCC-FS fine-structure resolved and total cross sections for reaction (1) are displayed in Fig. 2. The only data available for comparison are the semiclassical calculations of Rapp & Ortenburger ([1960]) which are in significant disagreement with the current results. The current total QMOCC-FS is in good agreement with the current QMOCC and SCMOCC between 0.1 and 1 eV/u.

Cross sections for reaction (2), H⁺ + O, are presented in Figs. 3 and 4. In Fig. 3 the new QMOCC computations are compared to the experiments of Stebbings et al. ([1964]), Rutherford & Vroom ([1974]), Van Zyl & Stephen ([1992]), and Lindsay et al. ([1996]), the QMOCC-FS calculations of Chambaud et al. ([1980]), the QMOCC results of Kimura et al. ([1997]), the electron nuclear dynamics (END) calculations of Hedström et al. ([1998]), and the cross section deduced by detailed balance from the recommended reaction (1) cross section of Janev et al. ([1995]). The current QMOCC is in good agreement with the experimental data for energies less than about 200 eV/u. The discrepancy at higher energies may be related to our neglect of ETFs. For energies greater than 10 eV/u, the current SCMOCC results, which include ETFs, are in excellent agreement with the experimental data, while the END calculation underestimates the data. The previous QMOCC calculation of Kimura et al. ([1997]) neglected the initial approach degeneracy factors and apparently used an integration step size which was too large.

At lower energies, Fig. 3 displays the current fine-structure resolved QMOCC-FS results and those of Chambaud et al. Near 0.1 eV/u the two calculations are in reasonable agreement, but appear to diverge for lower energies with the current results for J=0 and J=1 following the expected 1/v behavior. The Chambaud et al. cross sections have a much steeper low-energy dependence. The statistically averaged sum of the fine-structure cross sections agree between 0.1 and 0.5 eV/u, but again diverge for lower energies. While the differences in the magnitudes of the cross sections may be related to differences in the radial coupling matrix element (not explicitly given by Chambaud et al.), we cannot explain the lower energy discrepancies.

$\begin{figure}\epsfxsize=8.8cm \epsfbox{ds9003f5.eps} \end{figure}$

Figure 5: Recommended total charge transfer cross sections. Reaction (1), O⁺ + H $\to$ O + H⁺: this work [thick solid line], Janev et al. (1995) [thin dashed line]. Reaction (2), H⁺ + O $\to$ H + O⁺: this work [thick dotted line]

Figure 4 shows reaction (2) cross sections for energies greater than 100 eV/u. The current CTMC and CDW calculations (which include the sum of capture from the O( $1{\rm s}$ ), O( $2{\rm s}$ ), and O( $2{\rm p}$ ) subshells computed using the independent electron model) are compared with the experiments of Stier & Barnett ([1956]), de Heer et al. ([1966]), Schryber ([1967]), Toburen et al. ([1968]), Williams et al. ([1984]), and Thompson et al. ([1996]), and the calculations of Tan & Lee ([1981]), and the atomic-orbital close-coupling (AOCC) results of Hamre et al. ([1999]). Additionally, the K-shell capture (i.e., removal of an O(1s) electron) measurements of Cocke et al. ([1977]) are compared to the theoretical calculations of Lin et al. ([1978]), Miraglia et al. ([1984]), and Saha et al. ([1985]). Other K-shell capture calculations (not shown for clarity) were performed by Ghosh et al. ([1987]), Belkic ([1988]), Dunseath et al. ([1988]), Gravielle & Miraglia ([1988]), and Kuang ([1991]). The CTMC results are in fair agreement with the experimental data. Unlike for reaction (1), CTMC is in good agreement with the measured cross sections from 1 to 300 keV/u. For E> 300 keV/u, it overestimates the total capture cross section because the K-shell contribution is calculated to be about a factor of three larger than the measurements of Cocke et al. ([1977]) (see Schultz & Stancil [1999]). The current CDW results are in fair agreement with the measurements and the calculations of Tan & Lee ([1981]) for E> 100 keV/u, but are about a factor of two larger. However, we note that all of the high energy measurements were performed on O₂ targets except for Williams et al. and Thompson et al., with the O target assumed to be one-half of the O₂cross section.

$\begin{figure}\epsfxsize=12cm \epsfbox{ds9003f6.eps} \end{figure}$

Figure 6: Total charge transfer rate coefficients for a) Reaction (1), O⁺ + H $\to$ O + H⁺ and b) reaction (2), H⁺ + O $\to$ H + O⁺. This work: total for reaction (1) and collisional equilibrium for reaction (2) [thick solid lines with pluses]. Previous experiment: Fehsenfeld & Ferguson (1972) [open diamond], Federer et al. (1984) [filled up triangle]. Previous theory: Field & Steigman (1971) [squares with thin dashed lines], Chambaud et al. (1980) [circles with thin dotted lines], Kimura et al. (1997) [left triangles]. Recommended: Kingdon & Ferland (1996) [thin long dashed lines], Millar et al. (1997) [thin dot-dashed lines]

A new recommended cross section for reaction (1) was constructed through a non-linear fit of all the available data, but with exclusion of the calculations of Rapp & Ortenburger ([1960]) and the lower energy portions of the Phaneuf et al. and CDW results. The new recommended cross section is shown in Fig. 5 with fit coefficients for the relation

$\displaystyle \sigma (E_3)$

$\textstyle = a_1 \biggl [\frac{\exp(-a_2/E_3)}{1+a_3E_3^2+a_4E_3^{4.5}} + a_5\frac{\exp(-a_6E_3)}{E_3^{a_7}}$

$\textstyle + a_8\frac{\exp(-a_9E_3)}{E_3^{a_{10}}} \biggr ]~~{cm}^2$

(3)

Assuming detailed balance, we deduced a "recommended'' cross section for process (2) from the Janev et al. reaction (1) cross section as shown in Figs. 3 and 4. The deduced cross section does not reproduce the most recent results. Using a consistent set of experimental and theoretical data, we also constructed a recommended cross section for process (2) shown in Fig. 5 with fit coefficients given in Table 2.

**Table 2:** Cross section fit coefficients for Eq. (3)
Coefficient	Reaction (1)	Reaction (2)
a₁	3.93 10^-16	1.20 10^-15
a₂	2.05 10^-3	32.0
a₃	7.16 10^-2	4.15 10^-3
a₄	2.95 10^-8	2.35 10^-9
a₅	0.812	0.792
a₆	3.01 10^-2	2.94 10^-2
a₇	0.150	7.26 10^-3
a₈	0.183	8.45 10^-4
a₉	1.04 10^-2	8.35 10^-4
a₁₀	0.236	0.563

Figure 6 displays various theoretical and experimental rate coefficients for reactions (1) and (2) with the current results obtained by averaging the new recommended cross sections over a Maxwellian velocity distribution. Using a Langevin orbiting approximation, Field & Steigman ([1971]) estimated the rate coefficients for temperatures up to 10000 K, while using their QMOCC-FS cross sections, Chambaud et al. obtained rate coefficients up to only 1000 K. There is significant discrepancy between these two calculations. Further, the O⁺ + H measurement of Federer et al. ([1984]) is in agreement with Chambaud et al. while the H⁺ + O measurement of Fehsenfeld & Ferguson ([1972]) is consistent with Field & Steigman ([1971]). For reaction (1), our new total rate coefficients are consistent with Chambaud et al., but generally are slightly larger. The new results are also a factor of two larger than Field and Steigman at 10000 K. For reaction (2), the rate coefficients for a population of oxygen atoms in collisional equilibrium is only slightly larger than for the rate coefficients with all atoms in the J=2 level for T<1000 K. Both are only slightly smaller than the rate coefficients determined by Field and Steigman except at T= 10000 K, where the current results are a factor of 1.8 larger. The current reaction (2) rate coefficients are somewhat larger than the Chambaud et al. results. At high temperatures, the current rate coefficients are in fair agreement with Kimura et al. ([1997]) for reaction (2) for T>50000 K. J-resolved rate coefficients are also displayed in Fig. 6.

**Table 3:** Fine-structure resolved charge transfer rate coefficients $\alpha$ (cm³ s^-1) for process (1) and fit coefficients for Eq. (4)
$T({\rm K})$	J=0	J=1	J=2	Total
10	9.46-15^a	9.59-11	2.97-10	3.93-10
20	1.22-12	9.46-11	2.82-10	3.77-10
30	5.94-12	9.10-11	2.52-10	3.49-10
50	2.11-11	8.94-11	2.42-10	3.52-10
100	5.73-11	9.15-11	2.66-10	4.15-10
200	1.04-10	9.93-11	3.13-10	5.17-10
500	1.71-10	1.20-10	4.01-10	6.91-10
1000	2.27-10	1.47-10	5.00-10	8.74-10
2000	2.93-10	1.85-10	6.31-10	1.11-9
5000	4.02-10	2.59-10	8.66-10	1.53-9
10000	5.28-10	3.44-10	1.14-9	2.01-9
b₁	4.47-10^b	3.29-10	1.14-9	2.08-9
c₁	2.57-1	4.55-1	3.97-1	4.05-1
d₁	-5.49+4	$\infty$	$\infty$	$\infty$
b₂	1.03-11	1.97-11	1.38-11	1.11-11
c₂	-3.65-1	-2.09-1	-2.98-1	-4.58-1
d₂	85.7	3.22+4	3.64+4	$\infty$

^a: The notation A-B corresponds to $A\times 10^{-B}.$
^b: Multiply the fit by the endothermicity factor $\exp(- 99~{\rm K/T}).$

**Table 4:** Fine-structure resolved charge transfer rate coefficients $\alpha$ (cm³ s^-1) for process (2) and fit coefficients for Eq. (4)
$T({\rm K})$	J=0	J=1	J=2	Equil.
10	9.17-10	2.71-10	3.17-20	5.21-20
20	8.21-10	2.56-10	3.59-15	5.33-15
30	7.89-10	2.48-10	1.70-13	2.48-13
50	7.65-10	2.42-10	3.70-12	5.41-12
100	7.83-10	2.45-10	4.25-11	5.94-11
200	8.67-10	2.65-10	1.59-10	1.98-10
500	1.04-9	3.21-10	4.06-10	4.29-10
1000	1.25-9	3.92-10	6.37-10	6.19-10
2000	1.53-9	4.95-10	8.97-10	8.34-9
5000	2.08-9	6.95-10	1.30-9	1.19-9
10000	2.74-9	9.32-10	1.75-9	1.59-9
b₁	2.39-9	7.91-10	1.57-9^a	1.26-9^a
c₁	3.27-1	3.11-1	2.98-1	5.17-1
d₁	-2.06+5	-5.97+5	-7.51+4	$\infty$
b₂	3.54-11	1.96-11	1.62-7	4.25-10
c₂	-4.26-1	-3.29	1.13	6.69-3
d₂	5.27+3	2.21+2	19.4	$\infty$

^a: Multiply the fit by the endothermicity factor $\exp(- 227~{\rm K/T}).$

**Table 5:** Total charge transfer rate coefficients $\alpha$ (cm³ s^-1)
$T({\rm K})$	Reaction (1)		Reaction (2)
	a	b	c	d	b
10000	2.01-9	1.4-9	1.61-9	6.1-11	9.1-10
20000	2.69-9		2.19-9
50000	3.97-9		3.37-9	2.91-9
80000	4.83-9		4.22-9
100000	5.30-9		4.71-9	7.02-9
200000	7.04-9		6.59-9	1.39-8
500000	1.02-8		1.03-8
800000	1.24-8		1.29-8
1 10⁶	1.36-8		1.44-8
2 10⁶	2.14-8		2.02-8
5 10⁶	2.61-8		3.11-8
8 10⁶	3.09-8		3.86-8
1 10⁷	3.31-8		4.27-8

^a: This work.
^b: Field & Steigman (1971).
^c: This work, determined from the fine-structure cross section as given in Table 2.
^d: Kimura et al. (1997).

3 Results and discussion