Constrained Random Phase Approximation: the spectral method

We present a constrained Random Phase Approximation (cRPA) method, termed spectral cRPA (s-cRPA), and compare it to established cRPA approaches for Scandium and Copper by varying the 3d shell filling. The s-cRPA method generally produces larger Hubbard U interaction values compared to conventional approaches. When applied to the realistic system CaFeO$_3$ , s-cRPA yields interaction parameters that align more closely with those required within DFT+U to reproduce the experimentally observed insulating state, addressing the metallic behaviour predicted by standard density functionals. We examine the issue of negative interaction values encountered in the projector cRPA method for filled d-shells. We show that s-cRPA provides improved numerical stability by preserving electron number conservation, a constraint that is violated in the projector cRPA method. The s-cRPA approach addresses some limitations of standard cRPA methods, particularly the tendency to underestimate U values, suggesting its potential utility for the community. Additionally, we have enhanced our implementation to include computation of multi-centre interactions for analysing spatial decay and developed an efficient low-scaling variant employing a compressed Matsubara grid to obtain full frequency-dependent interactions.