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

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 $T\left[P\right]$ 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 $T_2\left[P\right]$ is shown to be exact for two-electron systems and a lower bound to $T\left[P\right]$ 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 $T\left[P\right]=T_2\left[P\right]+T_{\mathrm{eff}}\left[P\right]$, where only the remainder $T_{\mathrm{eff}}\left[P\right]⩾0$ needs to be approximated. If the BPM is constrained to yield the exact ground-state pair density, a two-electron Schrödinger 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 $T_{\mathrm{eff}}\left[P\right]$. 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 $T_{\mathrm{eff}}\left[P\right]$ is derived from a contracted Schrödinger equation. Since $T_{\mathrm{eff}}\left[P\right]$ 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.