  In this work, we address the question of calculating the local effective
Coulomb interaction matrix in materials with strong electronic Coulomb
interactions from first principles. To this purpose, we implement the
constrained random phase approximation (cRPA) into a density functional code
within the linearized augmented plane wave (LAPW) framework.
  We apply our approach to the 3d and 4d early transition metal oxides SrMO3
(M=V, Cr, Mn) and (M=Nb, Mo, Tc) in their paramagnetic phases. For these
systems, we explicitly assess the differences between two physically motivated
low-energy Hamiltonians: The first is the three-orbital model comprising the
t2g states only, that is often used for early transition metal oxides. The
second choice is a model where both, metal d- and oxygen p-states are retained
in the construction of Wannier functions, but the Hubbard interactions are
applied to the d-states only ("d-dp Hamiltonian"). Interestingly, since -- for
a given compound -- both U and J depend on the choice of the model, so do their
trends within a family of these compounds. In the 3d perovskite series SrMO3
the effective Coulomb interactions in the t2g Hamiltonian decrease along the
series, due to the more efficient screening. The inverse -- generally expected
-- trend, increasing interactions with increasing atomic number, is however
recovered within the more localized "d-dp Hamiltonian". Similar conclusions are
established in the layered 4d perovskites series Sr2MO4 (M=Mo, Tc, Ru, Rh).
Compared to their isoelectronic and isostructural 3d analogues, the 4d 113
perovskite oxides SrMO3 (M=Nb, Mo, Tc) exhibit weaker screening effects.
Interestingly, this leads to an effectively larger U on 4d shells than on 3d
when a t2g model is constructed.
