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A ring-shaped carbon allotrope was recently synthesized for the first time,reinvigorating theoretical interest in this class of molecules. The dual $\pi$structure of these molecules allows for the possibility of novel electronicproperties. In this work we use reduced density matrix theory to study theelectronic structure and conductivity of cyclo[18]carbon and its boron nitrideanalogue, B\textsubscript{9}N\textsubscript{9}. The variational 2RDM methodreplicates the experimental polyynic geometry of cyclo[18]carbon. We use acurrent-constrained 1-electron reduced density matrix (1-RDM) theory withHartree-Fock molecular orbitals and energies to compute the molecularconductance in two cases: (1) conductance in the plane of the molecule and (2)conductance around the molecular ring as potentially driven by a magnetic fieldthrough the molecule's center. In-plane conductance is greater than conductancearound the ring, but cyclo[18]carbon is slightly more conductive thanB\textsubscript{9}N\textsubscript{9} for both in-the-plane and in-the-ringconduction. The computed conductance per molecular orbital provides insightinto how the orbitals---their energies and densities---drive the conduction.

Non-equilibrium Steady State Conductivity in Cyclo[18]carbon and Its Boron Nitride Analogue