We prescribe general rules to predict the existence of edge states and zero-energy flat bands in graphene nanoribbons and graphene edges of arbitrary shape. No calculations are needed. For the so-called {it{minimal}} edges, the projection of the edge
translation vector into the zigzag direction of graphene uniquely determines the edge bands. By adding extra nodes to minimal edges, arbitrary modified edges can be obtained. The edge bands of modified graphene edges can be found by applying hybridization rules of the extra atoms with the ones belonging to the original edge. Our prescription correctly predicts the localization and degeneracy of the zero-energy bands at one of the graphene sublattices, confirmed by tight-binding and first-principle calculations. It also allows us to qualitatively predict the existence of $E e 0$ bands appearing in the energy gap of certain edges and nanoribbons.
Using calculations from first principles, we herein consider the bond made between thiolat e with a range of different Au clusters, with a particular focus on the spin moments inv olved in each case. For odd number of gold atoms, the clusters show a
spin moment of 1.~ $mu_B$. The variation of spin moment with particle size is particularly dramatic, with t he spin moment being zero for even numbers of gold atoms. This variation may be linked w ith changes in the odd-even oscillations that occur with the number of gold atoms, and is associated with the formation of a S-Au bond. This bond leads to the presence of an extra electron that is mainly sp in character in the gold part. Our results sugg est that any thiolate-induced magnetism that occurs in gold nanoparticles may be locali zed in a shell below the surface, and can be controlled by modifying the coverage of the thiolates.
