  While mesoscopic conducting loops are sensitive to external magnetic fields,
as seen by observations of the Aharonov-Bohm (AB) effect in such structures,
the field needed to observe the AB periodicity in small molecular rings is
unrealistically large. The present study aims to identify conditions under
which magnetic field dependence can be observed in electronic conduction
through such molecules. We consider molecular ring structures modeled both
within the tight-binding (H\"uckel) model and as continuous rings. In fact,
much of the observed qualitative behavior can be rationalized in terms of a
much simpler two-state model. Dephasing in these models is affected by two
common tools: the B\"uttiker probe method and coherence damping within a
density matrix formulation. We show that current through a benzene ring can be
controlled by moderate fields provided that (a) conduction must be dominated by
degenerate (in the free molecule) molecular electronic resonances, associated
with multiple pathways as is often the case with ring molecules; (b)
molecular-leads electronic coupling must is weak so as to affect relatively
distinct conduction resonances; (c) molecular binding to the leads must be
asymmetric (e.g., for benzene, connection in the meta or ortho, but not para,
configurations) and, (d) dephasing has to be small. Under these conditions,
considerable sensitivity to an imposed magnetic field normal to the molecular
ring plane is found in benzene and other aromatic molecules. Interestingly, in
symmetric junctions (e.g. para connected benzene) a large sensitivity of the
transmission coefficient to magnetic field is not reflected in the
current-voltage characteristic. Although sensitivity to magnetic field is
suppressed by dephasing, quantitative estimates indicate that magnetic field
control can be observed under realistic condition.
