In this paper we develop an approximate quantum scattering method capable of determining cross sections for reactive A+BC collisions, with A being an open shell atom and BC being a closed shell diatomic molecule. This method is based on time-independent coupled channel calculations, and absorbing potentials are used to describe reaction. The coupled channel expansion includes all electronic states of the atom that correlate to a selected atomic term, and a converged set of rotational states of the diatomic. Diatomic vibration is approximated as an adiabatic degree of freedom. The method is used to study the title reaction, including all five of the electronic surfaces that correlate to O(D-1)+H-2 as well as terms in the Hamiltonian that couple these surfaces. These couplings include: electronic and rotational Coriolis coupling, and electrostatic nonadiabatic coupling. Coriolis coupling causes all five states to interact and is most important at long range, while electrostatic coupling produces strong interactions between the 1(1)Sigma and 1(1)Pi states at short range (where these states have a conical intersection) and weak but non-negligible interactions between these states at long range. The most important three of the five surfaces (1(1)Sigma and 1(1)Pi, or 1(1)A('), 1(1)A" and 2(1)A(')) and the electrostatic nonadiabatic coupling between them are taken from the recent ab initio calculations of Dobbyn and Knowles [A. J. Dobbyn and P. J. Knowles, Mol. Phys. 91, 1107 (1997); Faraday Discuss. 110, 247 (1998)], while the other surfaces (1(1)Delta or 2(1)A" and 3(1)A(')) are based on a diatomics- in-molecules potential. Our results for the fully coupled problem indicate that Coriolis coupling is significant between the electronic fine structure levels so that electronic alignment is not strongly preserved as the reactants approach. However, the fine structure averaged reaction probability is relatively insensitive to the electronic Coriolis mixing. Averaged reaction probabilities from a centrifugal decoupled calculation where both electronic and rotational Coriolis interactions are neglected are in good agreement (10% or better) with the results of the fully coupled calculations. We find that electrostatic nonadiabatic coupling between the lowest Sigma and Pi states is significant, even at energies below the Pi barrier where only the long-range nonadiabatic coupling between these states is important. As a result, the low energy cross section summed over electronic states receives a approximate to 10% contribution from the Pi state. We find that the total cross section decreases with energy for energies below approximate to 3.5 kcal/mol and increases slightly at higher energies, with the increase due to reaction over the Pi barrier. We find that the Pi barrier contribution to the cross section is about twice that obtained by treating the reaction adiabatically, with the difference due to nonadiabatic dynamics on the 2(1)A(') state.