  We present a generalized framework, renormalized density matrix downfolding
(R-DMD), to derive systematically improvable effective models from ab initio
calculations. This framework moves beyond the common role of ab initio
calculations as calculating the parameters of a proposed Hamiltonian. Instead,
R-DMD provides the capability to decide whether a given effective Hilbert space
can be identified from the ab initio data and assess the relative quality of
ansatz Hamiltonians. Any method of ab initio solution can be used as a data
source, and as the ab initio solutions improve, the resultant model also
improves. We demonstrate the framework in an application to the downfolding of
a hydrogen chain to a spin model, in which we find the interatomic separations
for which a nonperturbative mapping can be made, and compute a renormalized
spin model Hamiltonian that quantitatively reproduces the ab initio dynamics.
