  Doubly polar molecules, possessing an electric dipole moment and a magnetic
dipole moment, can strongly couple to both an external electric field and a
magnetic field, providing unique opportunities to exert full control of the
system quantum state at ultracold temperatures. We propose a method for
creating a purely long-range doubly polar homonuclear molecule from a pair of
strongly magnetic lanthanide atoms, one atom being in its ground level and the
other in a superposition of quasi-degenerate opposite-parity excited levels
[Phys.~Rev.~Lett.~\textbf{121}, 063201 (2018)]. The electric dipole moment is
induced by coupling the excited levels with an external electric field. We
derive the general expression of the long-range, Stark, and Zeeman interaction
energies in the properly symmetrized and fully-coupled basis describing the
diatomic complex. Taking the example of holmium, our calculations predict
shallow long-range wells in the potential energy curves that may support
vibrational levels accessible by direct photoassociation from pairs of
ground-level atoms.
