The subtlety of resolving orbital angular momenta in calculating Hubbard U parameters in the density functional tight-binding theory and its delicacy is illustrated by the calculated magnetic properties of carbon clusters

We study magnetic properties of carbon clusters C_n by combining the spin-polarized parametrized density functional tight-binding (SDFTB) theory with an unbiased modified basin hopping (MBH) optimization algorithm. With an intent to develop a physically self-consistent technique, we deliberately treat valence electronic charges, spin charges and ionic charges on equal footing. Within the density functional tight-binding (DFTB) theory, we examine the effect of using the orbital angular-momentum unresolved and resolved schemes in calculating the on-site Coulombic energy which, we judge, will have subtle but significant influence on the C_n’s magnetism, their topologies and the change with size n their conformational structures. As a concrete means to substantiate our conjecture, we apply the SDFTB/MBH method to C_ns and determine their stable magnetic structures within the angular-momentum resolved scheme. Our calculations show that the lowest energy C_n changes from a linear-chain shape for n = 3–9, turns over to a singlet monocyclic ring for n = 10–18, becomes a polycyclic ring for n = 19–25 and finally assumes a cage-like geometry at n = 26. Except for n = 4, 6 and 8, all other C_ns are found unmagnetized. Accordingly, the newly discovered n that marks the first occurrence of a bi- to tridimensional transition occurs at n = 26, and this size is in contrast to n = 24 which was predicted by similar calculations using the unresolved scheme. Our calculations reveal furthermore two different features. The first one is that the predicted optimized geometries for all of the C_ns, except for C₂₄, are structurally the same as the size n = 3–23 of C_n calculated by the DFTB/MBH employing the unresolved orbital angular-momentum scheme. As a result, the present calculations which employ the resolved angular-momentum scheme thus showed that the latter affects only the larger size C_n starting at C₂₄; this finding redefines therefore the turnover transition point of C_n from a bidimensional planar at n = 25 to a tridimensional cage-like at n = 26. The second feature is that the SDFTB/MBH method yields only triplet C₄, C₆ and C₈, whereas in our previous work employing DFTB/MBH, not only C₄ and C₆, all of C₁₃, C₁₅, C₁₇, C₁₉, C₂₂ and C₂₃ were predicted to carry a magnetic moment of 2 μ_B. These differences in the magnetism obtained are attributed to the combined use of both the SDFTB/MBH procedure and the orbital angular-momentum resolved scheme.