  The ground states of few electrons confined in two vertically coupled quantum
rings in the presence of an external magnetic field are studied systematically
within the current spin-density functional theory. Electron-electron
interactions combined with inter-ring tunneling affects the electronic
structure and the persistent current. For small values of the external magnetic
field, we recover the zero magnetic field molecular quantum ring ground state
configurations. Increasing the magnetic field many angular momentum, spin, and
iso-spin transitions are predicted to occur in the ground state. We show that
these transitions follow certain rules, which are governed by the parity of the
number of electrons, the single particle picture, the Hund's rules and
many-body effects.
