Steven Valone Los Alamos National Laboratory Many kinds of defects in ceramics and metal-ceramic composites imply the rebalancing of charge on an atom (numbers of electrons) compared to the charge on that atom in the corresponding pristine bulk crystal. The ability to rebalance the charge at the defects (also referred to as chemical potential or electronegativity equalization) is therefore an essential component of atomistic-scale models for these problems. Recently the "fragment'' Hamiltonian (FH) model was devised to achieve the required rebalance charges by constructing a quantum Hamiltonian that has been coarsened to the atomic scale (to be differentiated from the "one-electron scale"). Through the FH model, we have been able to unify atomistic models for ceramics and metals. The FH model describes each atom in a system as a function of two variables. One variable is the traditional net charge. The second variable is new and has been termed the ionicity. Ionicity measures the fraction of time that an atom spends in a non-neutral state. The development of the FH model Hamiltonian is described here and some of its basic properties are demonstrated. For instance, the local gap matches the definition of the "Hubbard U" parameter as a system dissociates. The FH model also obeys certain well-known theoretical limits that come from nonlinearities to be described, as the volume of a crystal is varied. It reduces to successful models for metals, such as the embedded atom method, and to commonly-used atomistic models focused around net charge, such as ReaxFF, COMBS, and split charge models. The variables describing the state of an atom in a material are captured in a density matrix of occupation numbers for its charge states. An important extension of the concept of the density matrix is that of a spectral density matrix that is related to the density of states of the system at given energies. The FH model is applied to several defective systems including one-dimensional chain systems, the analog of a single Anderson impurity, Ni surfaces, stacking faults, and grain boundaries. In addition to the basic structure of the model, changes in the spectral density matrix will be tracked with changes in defect properties. ???????§Work was performed under the auspices of the U. S. Department of Energy, National Nuclear Security Agency, by the Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396. Work was performed for the Los Alamos National Laboratory, Laboratory-Directed Research and Development Program. Support from the National Science Foundation during the early stages of this work, and from the Defense Threat Reduction Agency CB Basic Research Program under Grant No.\ HDTRA1-09-1-008 is gratefully acknowledged.