This article presents a continuum-level constitutive model of a single crystal deforming by dislocation climb, accounting for crystallography and dislocation/point-defect interactions. The proposed constitutive description represents an improvement over a previous recent model, which extended the rate-sensitivity approach for single crystal plasticity by dislocation glide to consider the deformation geometry of dislocation climb under the assumption of instantaneous restoration of equilibrium concentration of vacancies near climbing dislocations. The key element of the new model is a chemical stress parameter, which represents the additional driving force on climbing dislocations due the excess vacancy concentration. The original and new versions of the crystallographic model of climb are compared through a simple example of a strongly anisotropic single crystal that illustrates the differences in response due to the consideration or not of the chemical driving force for climb.