Theoretical Study on Magnetic Interaction in Pyrazole-Bridged Dinuclear Metal Complex: Possibility of Intramolecular Ferromagnetic Interaction by Orbital Counter-Complementarity

A possibility of the intramolecular ferromagnetic (FM) interaction in pyrazole-bridged dinuclear Mn(II), Fe(II), Co(II), and Ni(II) complexes is examined by density functional theory (DFT) calculations. When azide is used for additional bridging ligand, the complexes indicate the strong antiferromagnetic (AFM) interaction, while the AFM interaction becomes very weak when acetate ligand is used. In the acetate-bridged complexes, an energy split of the frontier orbitals suggests the orbital counter-complementarity effect between the dxy orbital pair, which contributes to the FM interaction; however, a significant overlap of other d-orbital pairs also suggests an existence of the AFM interaction. From those results, the orbital counter-complementarity effect is considered to be canceled out by the overlap of other d-orbital pairs.

Stability of the co-orbital resonance under dissipation

Despite the existence of co-orbital bodies in the solar system, and the prediction of the formation of co-orbital planets by planetary system formation models, no co-orbital exoplanets (also called trojans) have been detected thus far. In this paper we investigate how a pair of co-orbital exoplanets would fare during their migration in a protoplanetary disc. To this end, we computed a stability criterion of the Lagrangian equilibria L4 and L5 under generic dissipation and slow mass evolution. Depending on the strength and shape of these perturbations, the system can either evolve towards the Lagrangian equilibrium, or tend to increase its amplitude of libration, possibly all the way to horseshoe orbits or even exiting the resonance. We estimated the various terms of our criterion using a set of hydrodynamical simulations, and show that the dynamical coupling between the disc perturbations and both planets have a significant impact on the stability: the structures induced by each planet in the disc perturb the dissipative forces applied on the other planets over each libration cycle. Amongst our results on the stability of co-orbitals, several are of interest to constrain the observability of such configurations: long-distance inward migration and smaller leading planets tend to increase the libration amplitude around the Lagrangian equilibria, while leading massive planets and belonging to a resonant chain tend to stabilise it. We also show that, depending on the strength of the dissipative forces, both the inclination and the eccentricity of the smaller of the two co-orbitals can be significantly increased during the inward migration of the co-orbital pair, which can have a significant impact on the detectability by transit of such configurations.