The Born-Oppenheimer potential energy hypersurfaces of copper and gold trimers were calculated using density functional theory with an analytic potential. The calculated Jahn-Teller distortion energies, pseudorotation barriers, dissociation, and isomerization energies for the two trimers are discussed. Global minima from the surfaces were optimized using the density functional theory method as well as the coupled cluster-singles-doubles-with-triples energies technique. The agreement of the optimized structures with the analytic potential was very good. The Mulliken population analysis compared favorably with the experimental electron spin resonance results. Spin-orbit coupling was subsequently included and the effect was significant for gold, but negligible for copper. The spin-orbit effect suppressed the Jahn-Teller distortion of the gold trimer, and the potential surface with the spin-orbit effect included was also obtained. The spin-orbit splitting splitting for the D-3h geometry of the gold trimer was in excellent agreement with the most recent infrared spectroscopic results.