The structure, energetics, and interconversion of isomers of Rh-6 and Rh-6(+) are studied by using density functional theory with Gaussian basis sets, using guess structures derived from basin-hopping simulations, and obtained by using the Sutton-Chen potential. A large range of spin multiplicities is considered for each isomer. Our calculations suggest two low-lying structures as possible structural isomers: a square bipyramid and a trigonal prism. The reactivity of these two candidate structural isomers with respect to adsorption of nitric oxide is studied via location of reaction transition states and calculation of reaction barriers. Similarities and differences with surface reaction studies are highlighted. These data provide powerful evidence that structural isomerism, and not different spin states, is responsible for the observed biexponential reaction kinetics.