Theoretical investigation of noncollinear magnetism for Mn$_{5}$ and Mn$_{6}$

The presence of odd-number rings in systems marked by nearest -neighbor antiferromagnetic (AFM) coupling can lead to situations where it is not possible to satisfy all interactions to find the ground state. This leads to so called ``frustration.'' Since atomic clusters tend to take structures containing triangles or tetrahedra, this can cause magnetic frustration. It is therefore possible that some transition metals like chromium or manganese, whose crystals tend to be AFM, show noncollinear (NCL) magnetic configuration in small clusters. In the present study, we carry out first-principles calculations with NCL magnetism for Mn$_{5}$ and Mn$_{6}$ clusters. We adopt the projected augmented-wave method of the program called VASP (Vienna \textit{ab initio} simulation package) for this calculation. From the results, we can see Mn$_{5}$ may take NCL magnetic configuration as one of the ground states isomers, while other isomers have collinear configurations. For the Mn$_{6}$, we found the NCL magnetic configuration is highly expected as the ground state.