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 Cn 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 Cn’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 Cns and determine their stable magnetic structures within the angular-momentum resolved scheme. Our calculations show that the lowest energy Cn 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 Cns 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 Cns, except for C24, are structurally the same as the size n = 3–23 of Cn 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 Cn starting at C24; this finding redefines therefore the turnover transition point of Cn 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 μB. 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.
Magnetic carbon cluster Spin-polarized DFTB theory Structural optimization Topological transition
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This work is financially supported by the Ministry of Science and Technology (MOST103-2112-M-008-015-MY3) and (MOST106-2112-M-008-015), Taiwan. S.K.L. is grateful to the National Center for High-performance Computing for computer time and facilities.