Energy density matrix formalism for interacting quantum systems: Quantum Monte Carlo study

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 an avenue for describing energetics with quantum Monte Carlo methods which have traditionally been limited to total energies.

Figure 1

Occupation energies for the 3D homogeneous electron gas at $r_s=3$ from diffusion Monte Carlo calculations vs crystal momentum (solid shapes). The top panel contains results for the noninteracting system with the black dashed line representing the expected dispersion of ${\hslash }^2{k}^2/2m_e$. Results for the interacting system are in the lower panel. The dashed green line is a ${\hslash }^2{k}^2/2m^{*}$ fit with $m^{*}=0.84m_e$ and the black dashed line is the standard dispersion shifted to match at $k=0$. $N$ is the number of electrons in each finite simulation.

Figure 2

Similar to Fig. 1 but for occupation numbers of the 1RDM. Simulations of interacting systems deviate from one (zero) below (above) $k_F$. The dashed green lines are exponential fits of the deviation from ideal behavior about $k_F$. For comparison, the solid black curves are polynomial fits to the DMC data of Ortiz and Ballone [14, 15].

Figure 3

DMC occupation energies for a 13-electron HEG vs state index (solid shapes) resolved by spin with spin up in blue and spin down in red. State 1 corresponds to $k=0$ and states 2 through 7 belong to the same shell $k\in \{\pm \Delta k\widehat{x},\pm \Delta k\widehat{y},\pm \Delta k\widehat{z}\}$. A single electron has been removed from the spin-down channel of state 7.

Figure 4

DMC occupation energies for a 14-electron HEG spin-flip excitation vs state index (solid shapes) resolved by spin with spin up in blue and spin down in red. A single electron has been removed from the spin-down channel of state 7 and added to the spin-up channel of state 8, which belongs to the third shell, $k=2\Delta k\widehat{x}$.

Figure 5

DMC occupation energies for the ground state of a neutral oxygen atom with 5 up and 3 down electrons (solid triangles). Experimental binding energies (negative of ionization potential) of the first through the eighth electron in oxygen are shown as dashed horizontal lines as a comparative energy scale.