  We present a mapping of correlated multi-impurity Anderson models to a
cluster model coupled to a number of effective conduction bands capturing its
essential low-energy physics. The major ingredient is the complex
single-particle self energy matrix of the uncorrelated problem that encodes the
influence to the host conduction band onto the dynamics of a set of correlated
orbitals. While the real part of the self-energy matrix generates an effective
hopping between the cluster orbitals, the imaginary part determines the
coupling to the effective conduction bands in the mapped model. The rank of the
imaginary part determines the number of independent screening channels of the
problem, and allows the replacement of the phenomenological exhaustion
criterion by a rigorous mathematical statement. This rank provides a
distinction between multi-impurity models of first kind and of second kind. For
the latter, there are insufficient screening channels available, so that a
singlet ground state must be driven by the inter-cluster spin correlations.
This classification provides a fundamental answer to the question of why
ferromagnetic correlations between local moments are irrelevant for the spin
compensated ground state in dilute impurity models, whereas they compete with
the Kondo-scale in dense impurity arrays, without evoking a spin density wave.
The low-temperature physics of three examples taken from the literature are
deduced from the analytic structure of the mapped model, demonstrating the
potential power of this approach. NRG calculations are presented for up to five
site cluster. We investigate the appearance of frustration induced non-Fermi
liquid fixed points in the trimer, and demonstrate the existence of several
critical points of KT type at which ferromagnetic correlations suppress the
screening of an additional effective spin-$1/2$ degree of freedom.
