  Embedding calculations that find approximate solutions to the Schr\"{o}dinger
equation for large molecules and realistic solids are performed commonly in a
three step procedure involving (i) construction of a model system with
effective interactions approximating the low energy physics of the initial
realistic system, (ii) mapping the model system onto an impurity Hamiltonian,
and (iii) solving the impurity problem. We have developed a novel procedure for
parametrizing the impurity Hamiltonian that avoids the mathematically
uncontrolled step of constructing the low energy model system. Instead, the
impurity Hamiltonian is immediately parametrized to recover the self-energy of
the realistic system in the limit of high frequencies or short time. The
effective interactions parametrizing the fictitious impurity Hamiltonian are
local to the embedded regions, and include all the non-local interactions
present in the original realistic Hamiltonian in an implicit way. We show that
this impurity Hamiltonian can lead to excellent total energies and
self-energies that approximate the quantities of the initial realistic system
very well. Moreover, we show that as long as the effective impurity Hamiltonian
parametrization is designed to recover the self-energy of the initial realistic
system for high frequencies, we can expect a good total energy and self-energy.
Finally, we propose two practical ways of evaluating effective integrals for
parametrizing impurity models.
