Ring currents in the clar goblet calculated using configurational state averaging

The closed-shell Hückel-London-Pople-McWeeny formalism for ring currents is extended to Aufbau configurations with open shells calculated as configurational averages. The method is applied to the non-Kekulean benzenoid known as the Clar goblet, recently synthesized on the Au(111) surface. Multiplicity of the ground state is a complication: for the Clar goblet, Hund’s rule of maximum multiplicity implies a triplet whereas Ovchinnikov’s rule implies a singlet. This disagreement has little effect on the predicted ring currents. Ring-current maps are calculated for the 36π dication, 40π dianion, and low-lying states of the 38π neutral, using Hückel-London and Hubbard-London models. All show twin diatropic perimeter currents on separate halves of the molecule. These are compared with ipsocentric pseudo-π and ab initio maps of induced π-current for closed-shell singlet configurations of dianion, dication, and neutral. Configurationally averaged Hückel-London calculations give a good account of the consistent diatropic ring currents in the Clar goblet for the three charge states

On the eigenvalues and eigenvectors of certain finite, vertex-weighted, bipartite graphs

AbstractThe established, spectral characterisation of bipartite graphs with unweighted vertices (which are here termed homogeneous graphs) is extended to those bipartite graphs (called heterogeneous) in which all of the vertices in one set are weighted h1 , and each of those in the other set of the bigraph is weighted h2. All the eigenvalues of a homogeneous bipartite graph occur in pairs, around zero, while some of the eigenvalues of an arbitrary, heterogeneous graph are paired around 12(h1 + h2), the remainder having the value h2 (or hl). The well-documented, explicit relations between the eigenvectors belonging to “paired” eigenvalues of homogeneous graphs are extended to relate the components of the eigenvectors associated with each couple of “paired” eigenvalues of the corresponding heterogeneous graph. Details are also given of the relationships between the eigenvectors of an arbitrary, homogeneous, bipartite graph and those of its heterogeneous analogue