Nuclear cusps and singularities in the nonadditive kinetic potential bifunctional from analytical inversion

The nonadditive kinetic potential $v^{\text{NAD}}$ is a key quantity in density-functional theory (DFT) embedding methods, such as frozen density embedding theory and partition DFT. $v^{\text{NAD}}$ is a bifunctional of electron densities ${\rho }_{\mathrm{B}}$ and ${\rho }_{\mathrm{tot}}={\rho }_{\mathrm{A}}+{\rho }_{\mathrm{B}}$. It can be evaluated using approximate kinetic-energy functionals, but accurate approximations are challenging. The behavior of $v^{\text{NAD}}$ in the vicinity of the nuclei has long been questioned, and singularities were seen in some approximate calculations. In this article, the existence of singularities in $v^{\text{NAD}}$ is analyzed analytically for various choices of ${\rho }_{\mathrm{B}}$ and ${\rho }_{\mathrm{tot}}$, using the nuclear cusp conditions for the density and Kohn-Sham potential. It is shown that no singularities arise from smoothly partitioned ground-state Kohn-Sham densities. We confirm this result by numerical calculations on diatomic test systems HeHe, ${\mathrm{HeLi}}^{+}$, and ${\mathrm{H}}_2$, using analytical inversion to obtain a numerically exact $v^{\mathrm{NAD}}$ for the local density approximation. We examine features of $v^{\mathrm{NAD}}$ which can be used for development and testing of approximations to $v^{\mathrm{NAD}}[{\rho }_{\mathrm{B}},{\rho }_{\mathrm{tot}}]$ and kinetic-energy functionals.

Figure 1

Difference between the analytically inverted $v^{\mathrm{inv}}\left[{\rho }_1\right]\left(\mathbf{r}\right)$ and $v^{\mathrm{KS}}\left(\mathbf{r}\right)$ for HeHe, showing agreement to 1 part in ${10}^{10}$. Blue: $v^{\mathrm{KS}}\left(\mathbf{r}\right)$. Red: $\mathrm{\Delta }v=v^{\mathrm{KS}}\left(\mathbf{r}\right)-v^{\mathrm{inv}}\left[{\rho }_1\right]\left(\mathbf{r}\right)$.

Figure 2

Ground-state charge density distribution of HeHe in two dimensions. The vertical line indicates the cutoff plane where $z=z_0$, dividing the two localized densities which integrate to two electrons.

Figure 3

Nonadditive kinetic potential for HeHe, in a 1D representation. Purple dashed lines mark the location of the nuclei and a black solid line marks the cutoff $z_0$. (a) Total and partitioned densities. (b) Analytically inverted kinetic potential $v^{\mathrm{NAD}/\mathrm{INV}}[{\rho }_{\mathrm{B}},{\rho }_{\mathrm{tot}}]\left(\mathbf{r}\right)$ [from Eq. (24)], where the localized density ${\rho }_{\mathrm{B}}$ is on the left, compared to the same from von Weizsäcker theory [from Eq. (25)]. Blue: $v^{\mathrm{NAD}/\mathrm{INV}}$; Red curve: $\mathrm{\Delta }v=v^{\text{NAD/vW}}-v^{\mathrm{NAD}/\mathrm{INV}}$.

Figure 4

Nonadditive kinetic potential for HeHe in a 2D representation, comparing (a) $v^{\text{NAD/INV}}$ with (b) $v^{\text{NAD/vW}}$, both in Ry. The density is localized on the left nucleus. Black dots mark the nuclei and a black line marks the cutoff $z_0$.

Figure 5

Nonadditive kinetic potential for ${\mathrm{HeLi}}^{+}$ (with Li on the left side), in a 1D representation. Purple dashed lines mark the location of the nuclei and a black solid line marks the cutoff $z_0$. (a) Total and partitioned densities, with a taller peak for Li. (b) Analytically inverted kinetic potential $v^{\mathrm{NAD}/\mathrm{INV}}[{\rho }_{\mathrm{B}},{\rho }_{\mathrm{tot}}]\left(\mathbf{r}\right)$ [from Eq. (24)], which is localized on the Li side (blue), compared to the same from von Weizsäcker theory [from Eq. (25)], with $\mathrm{\Delta }v=v^{\text{NAD/vW}}-v^{\mathrm{NAD}/\mathrm{INV}}$ in red. Also shown is $v^{\mathrm{NAD}/\mathrm{INV}}[{\rho }_{\mathrm{A}},{\rho }_{\mathrm{tot}}]\left(\mathbf{r}\right)$ which is localized on the He side (purple).

Figure 6

Nonadditive kinetic potential for ${\mathrm{HeLi}}^{+}$ in a 2D representation, comparing (a) $v^{\text{NAD/INV}}[{\rho }_{\mathrm{B}},{\rho }_{\mathrm{tot}}]\left(\mathbf{r}\right)$ with (b) $v^{\text{NAD/vW}}[{\rho }_{\mathrm{B}},{\rho }_{\mathrm{tot}}]\left(\mathbf{r}\right)$, both in Ry. The density is localized on Li, on the left. Black dots mark the nuclei and a black line marks the cutoff $z_0$.

Figure 7

Nonadditive kinetic potential for ${\mathrm{H}}_2$, in a 1D representation, where one electron is localized. Purple dashed lines mark the location of the nuclei and a black solid line marks the cutoff $z_0$. (a) Total and partitioned densities. (b) Analytically inverted kinetic potential $v^{\mathrm{NAD}/\mathrm{INV}}[{\rho }_{\mathrm{B}},{\rho }_{\mathrm{tot}}]\left(\mathbf{r}\right)$ [from Eq. (24)], where the localized density ${\rho }_{\mathrm{B}}$ is on the left, compared to the same from von Weizsäcker theory [from Eq. (25)]. Blue: $v^{\mathrm{NAD}/\mathrm{INV}}$; Red curve: $\mathrm{\Delta }v=v^{\text{NAD/vW}}-v^{\mathrm{NAD}/\mathrm{INV}}$.

Figure 8

Nonadditive kinetic potential for ${\mathrm{H}}_2$ in a 2D representation, comparing (a) $v^{\text{NAD/INV}}$ with (b) $v^{\text{NAD/vW}}$, both in Ry. Black dots mark the nuclei and a black line marks the cutoff $z_0$.