  We describe an efficient approximation for the electron-electron interaction
in the determination of the low-energy effective interaction in multiband
lattice systems. By using ideas for channel decomposition, form-factor
expansion and the truncated-unity technique we describe the interaction as
arising from the non-local and orbital-dependent coupling of particle-hole and
particle-particle bilinears formed by fields residing in the same one or two
orbitals. This allows us to employ the constrained functional renormalization
group (cfRG) with a suitable momentum and frequency discretization. The
approach gives insights into the non-local screening of spin and charge
interactions when bands away from the Fermi level are integrated out.
Specifically, we compute the effective low-energy interactions in the
low-energy target band of a three-band model with onsite and non-local bare
interactions. We show that the cfRG adds important features to the effective
target-band interaction that cannot be found using the constrained random phase
approximation (cRPA).
