Abstract
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 imple-
mented 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 demonstrates a quantita-
tive 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.
