First-principles density-functional theory studies have reported open structures based on the formation of double simple-cubic (DSC) arrangements for Ru₁₃, Rh₁₃, Os₁₃, and Ir₁₃, which can be considered an unexpected result as those elements crystallize in compact bulk structures such as the face-centered cubic and hexagonal close-packed lattices. In this work, we investigated with the projected augmented wave method the dependence of the lowest-energy structure on the local and semilocal exchange correlation (xc) energy functionals employed in density-functional theory. We found that the local-density approximation (LDA) and generalized gradient formulations with diffrent treatment of the electronic inhomogeneities (PBE, PBEsol, and AM05) confirm the DSC configuration as the lowest-energy structure for the studied TM₁₃ clusters. A good agreement in the relative total energies are obtained even for structures with small energy differences, e.g., 0.10 eV. The employed xc functionals yield the same total magnetic moment for a given structure, i.e., the differences in bond lengths do not affect the moments, which can be attributed to the atomic character of those clusters. Thus, at least for those systems, the differences among the LDA, PBE, PBEsol, and AM05 functionals are not large enough to yield qualitatively different results.