  Wannier functions are real-space representations of Bloch orbitals that
provide a useful picture for chemical bonding and offer a localized description
of single-particle wave functions. There is a unitary freedom in the
construction of Wannier functions from Bloch orbitals, which can be chosen to
produce Wannier functions that have advantageous properties. A popular choice
for this freedom is the minimization of spatial variance, which leads to
maximally localized Wannier functions. We minimize a weighted sum of the
spatial and energy variances, yielding what we call dually localized Wannier
functions. Localization in energy results in Wannier functions that are
associated with a particular energy and allows for dually localized Wannier
functions to be comprised of occupied and unoccupied spaces simulatneously.
This can lead to Wannier functions that are fractionally occupied, which is a
key feature that is used in correcting the delocalization error in density
functional approximations. We show how this type of localization results in
bonding (antibonding) functions for the occupied (unoccupied) spaces around the
frontier energy of silicon in the diamond lattice and of molecular ethylene.
Dually localized Wannier functions therefore offer a relevant description for
chemical bonding and are well suited to orbital-dependent methods that
associate Wannier functions with specific energy ranges without the need to
consider the occupied and unoccupied spaces separately.
