  We derive a density matrix (DM) theory for quantum cascade lasers (QCLs) that
describes the influence of scattering on coherences through a generalized
scattering superoperator. The theory enables quantitative modeling of QCLs,
including localization and tunneling effects, using the well-defined energy
eigenstates rather than the ad hoc localized basis states required by most
previous DM models. Our microscopic approach to scattering also eliminates the
need for phenomenological transition or dephasing rates. We discuss the
physical interpretation and numerical implementation of the theory, presenting
sets of both energy-resolved and thermally averaged equations which can be used
for detailed or compact device modeling. We illustrate the theory's
applications by simulating a high performance resonant-phonon terahertz (THz)
QCL design which cannot be easily or accurately modeled using conventional DM
methods. We show that the theory's inclusion of coherences is crucial for
describing localization and tunneling effects consistent with experiment.
