Conditional probability density functional theory

We present conditional probability (CP) density functional theory (DFT) as a formally exact theory. In essence, CP-DFT determines the ground-state energy of a system by finding the CP density from a series of independent Kohn-Sham (KS) DFT calculations. By directly calculating CP densities, we bypass the need for an approximate XC energy functional. In this paper we discuss and derive several key properties of the CP density and corresponding CP-KS potential. Illustrative examples are used throughout to help guide the reader through the various concepts and theory presented. We explore a suitable CP-DFT approximation and discuss exact conditions, limitations, and results for selected examples.

Figure 1

Exact CP densities and potentials in 1D He (left) and 1D Be (right): ${\tilde{v}}_{\mathrm{S},y}\left(y^{\prime }\right)$ is the CP-KS potential with corresponding CP density ${\tilde{n}}_y\left(y^{\prime }\right)$, $v_{\mathrm{S}}\left(y^{\prime }\right)$ the KS potential with corresponding ground-state density $n\left(y^{\prime }\right)$. Quantities are plotted for reference positions $y=0$ and $y=0.8$.

Figure 2

Exact CP densities and potentials in 1D Li with $N_{\uparrow }=2,N_{\downarrow }=1$. The CP-KS potential is ${\tilde{v}}_{\mathrm{S},y}\left(y^{\prime }\right)$ with corresponding total CP density ${\tilde{n}}_y\left(y^{\prime }\right)$ (dashed), the spin-CP-KS potential is ${\tilde{v}}_{\mathrm{S},x}\left(y^{\prime }\right)$ with corresponding spin-CP density ${\tilde{n}}_x\left(y^{\prime }\right)$, $v_{\mathrm{S},\sigma }\left(y^{\prime }\right)$ is the KS spin potential with corresponding ground-state spin density $n_{\sigma }\left(y^{\prime }\right)$, and $n\left(y^{\prime }\right)$ is the total ground-state density. Quantities are plotted for reference space-spin positions $x=(0,\uparrow )$, $x=(0.8,\uparrow )$ (left) and $x=(0,\downarrow )$, $x=(0.8,\downarrow )$ (right).

Figure 3

CP densities and potentials in 1D He ($\lambda =1$): ${\tilde{v}}_{\mathrm{S},y}^{\mathrm{BEA}}\left(y^{\prime }\right)$ is the 1D BEA approximation to the CP-KS potential with corresponding CP density ${\tilde{n}}_y^{\mathrm{BEA}}\left(y^{\prime }\right)$, ${\tilde{v}}_{\mathrm{S},y}\left(y^{\prime }\right)$ is the exact CP-KS potential with corresponding exact CP density ${\tilde{n}}_y\left(y^{\prime }\right)$, $v_{\mathrm{S}}\left(y^{\prime }\right)$ the exact KS potential with corresponding exact ground-state density $n\left(y^{\prime }\right)$, and $v\left(y^{\prime }\right)$ the external potential for our 1D He atom. Quantities are plotted for reference position $y=0$ (left) and for $y=0.8$ (right).

Figure 4

CP densities and potentials in 1D Be ($\lambda =1$): ${\tilde{v}}_{\mathrm{S},y}^{\mathrm{BEA},\mathrm{LDA}}\left(y^{\prime }\right)$ is the 1D BEA approximation to the CP-KS potential using the 1D LDA [20] approximation, see Eq. (96). The corresponding CP density is ${\tilde{v}}_{\mathrm{S},y}^{\mathrm{BEA},\mathrm{LDA}}\left(y^{\prime }\right)$.

Figure 5

Spin-CP densities and potentials in 1D Li with $N_{\uparrow }=2,N_{\downarrow }=1$, and $\lambda =1$: ${\tilde{v}}_{\mathrm{S},x}^{\mathrm{BEA},\mathrm{LDA}}\left(y^{\prime }\right)$ is the 1D BEA approximation to the spin-CP-KS potential using the 1D LDA [20] approximation, see Eq. (96), with corresponding spin-CP density ${\tilde{v}}_{\mathrm{S},x}^{\mathrm{BEA},\mathrm{LDA}}\left(y^{\prime }\right)$. The exact CP-KS potential is ${\tilde{v}}_{\mathrm{S},y}\left(y^{\prime }\right)$ with corresponding exact CP density ${\tilde{n}}_y\left(y^{\prime }\right)$, $v_{\mathrm{S}}\left(y^{\prime }\right)$ the exact KS potential with corresponding exact ground-state density $n\left(y^{\prime }\right)$, and $v\left(y^{\prime }\right)$ the external potential for our 1D Li atom.

Figure 6

CP densities and potentials for (3D) He atom ($\lambda =1$): contour plots for the exact CP density ${\tilde{n}}_{\mathbf{r}}\left({\mathbf{r}}^{\prime }\right)$ (top left) and corresponding exact CP-KS potential ${\tilde{v}}_{\mathbf{r}}\left({\mathbf{r}}^{\prime }\right)$ (bottom left) are plotted within the $(x^{\prime },y^{\prime },z^{\prime }=0)$ plane for reference position $\mathbf{r}=(0.563,0,0)$ (displayed as red cross). The corresponding LBEA results are plotted on the right.