  We demonstrate an algebraic construction of frequency-dependent bath orbitals
which can be used in a robust and rigorously self-consistent DMFT-like
embedding method, here called $\omega-$DMFT, suitable for use with
Hamiltonian-based impurity solvers. These bath orbitals are designed to exactly
reproduce the hybridization of the impurity to its environment, while allowing
for a systematic expansion of this bath space as impurity interactions couple
frequency points. In this way, the difficult non-linear fit of bath parameters
necessary for many Hamiltonian-formulation impurity solvers in DMFT is avoided,
while the introduction of frequency dependence in this bath space is shown to
allow for more compact bath sizes. This has significant potential use with a
number of new, emerging Hamiltonian solvers which allow for the embedding of
large impurity spaces within a DMFT framework. We present results of the
$\omega-$DMFT approach for the Hubbard model on the Bethe lattice, a 1D chain,
and the 2D square lattice, which show excellent agreement with standard DMFT
results, with fewer bath orbitals and more compact support for the
hybridization representation in the key impurity model of the method.
