  We develop an energy density matrix that parallels the one-body reduced
density matrix (1RDM) for many-body quantum systems. Just as the density matrix
gives access to the number density and occupation numbers, the energy density
matrix yields the energy density and orbital occupation energies. The
eigenvectors of the matrix provide a natural orbital partitioning of the energy
density while the eigenvalues comprise a single particle energy spectrum
obeying a total energy sum rule. For mean-field systems the energy density
matrix recovers the exact spectrum. When correlation becomes important, the
occupation energies resemble quasiparticle energies in some respects. We
explore the occupation energy spectrum for the finite 3D homogeneous electron
gas in the metallic regime and an isolated oxygen atom with ground state
quantum Monte Carlo techniques implemented in the QMCPACK simulation code. The
occupation energy spectrum for the homogeneous electron gas can be described by
an effective mass below the Fermi level. Above the Fermi level evanescent
behavior in the occupation energies is observed in similar fashion to the
occupation numbers of the 1RDM. A direct comparison with total energy
differences shows a quantitative connection between the occupation energies and
electron addition and removal energies for the electron gas. For the oxygen
atom, the association between the ground state occupation energies and particle
addition and removal energies becomes only qualitative. The energy density
matrix provides a new avenue for describing energetics with quantum Monte Carlo
methods which have traditionally been limited to total energies.
