Bypassing the Energy Functional in Density Functional Theory: Direct Calculation of Electronic Energies from Conditional Probability Densities

Density functional calculations can fail for want of an accurate exchange-correlation approximation. The energy can instead be extracted from a sequence of density functional calculations of conditional probabilities ($CP$ DFT). Simple $CP$ approximations yield usefully accurate results for two-electron ions, the hydrogen dimer, and the uniform gas at all temperatures. $CP$ DFT has no self-interaction error for one electron, and correctly dissociates ${\mathrm{H}}_2$, both major challenges. For warm dense matter, classical $CP$ DFT calculations can overcome the convergence problems of Kohn-Sham DFT.

Figure 1

CP (blue curve) and exact (black curve). (a) XC energy per particle in uniform gas at increasing Wigner-Seitz radii ($r_s$) and $T=0$, (b) binding energy curve for ${\mathrm{H}}_2$ (red is KS DFT using PBE [14]), and (c) XC free energy per particle at $r_S=1$ as a function of reduced temperature ($T_F$ is the Fermi temperature). Exact from Ref. [17] in (a), Ref. [18] in (c). Hartree atomic units used throughout. See supplemental materials for numerical values.

Figure 2

Pure blue electron approximation (purple curve), half pure blue approximation (orange curve), interpolation [Eq. (14), blue curve], and exact (black curve). (a) Normalized XC hole densities for the uniform gas at $r_s=1$ with exact parametrization from Ref. [26], (b) $U_c(R)$ from ${\mathrm{H}}_2$, using exact $n(r)$ and producing errors below 20%, (c) Hooke’s atom ${\tilde{v}}_S({\mathbf{r}}^{\prime }|\mathbf{r})$, with the external potential $r^{\prime 2}/8$ (black dashed), and (d) Hooke’s atom ${\tilde{n}}_{\mathbf{r}}(r^{\prime })$.

Figure 3

Percentage error of uniform gas potential XC free energy per electron for the $CP$ DFT calculations within KS (blue), TF (purple), and classical (green) approximations relative to the parametrization of Groth et al. [18].