Towards a practical pair density-functional theory for many-electron systems

Abstract

In pair density-functional theory, the only unknown piece of the energy is the kinetic energy $T$ as a functional of the pair density $P({x}_{1},{x}_{2})$. Although $T\phantom{\rule{0.1em}{0ex}}[P]$ has a simpler structure than the Hohenberg-Kohn functional of conventional density-functional theory, computational requirements are still moderate. In the present work, a particularly convenient model system to represent many-electron pair densities is introduced. This ``boson pair model'' (BPM) approximately treats electron pairs as noninteracting bosons. The resulting explicit model for the kinetic energy ${T}_{2}[P]$ is shown to be exact for two-electron systems and a lower bound to $T\phantom{\rule{0.1em}{0ex}}[P]$ for more than two electrons. The one- and two-particle density matrices obtained from the BPM yield upper bounds for the corresponding many-electron quantities. This suggests a partitioning $T\phantom{\rule{0.1em}{0ex}}[P]={T}_{2}[P]+{T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]$, where only the remainder ${T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]\ensuremath{\geqslant}0$ needs to be approximated. If the BPM is constrained to yield the exact ground-state pair density, a two-electron Schr\"odinger equation with an effective local two-particle potential results; the latter is identified as a sum of the bare Coulomb interaction and the functional derivative of ${T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]$. This self-consistent scheme to minimize the energy with respect to $P$ is more efficient than previous procedures. Further information on the functional derivative of ${T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]$ is derived from a contracted Schr\"odinger equation. Since ${T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]$ is explicitly known in the two-electron and noninteracting (Hartree-Fock) limits, the present method provides an alternative to density-matrix functional theories, which can be exact in the same limits and are similar in computational cost.