Approximate forms of the pair-density-functional kinetic energy on the basis of a rigorous expression with coupling-constant integration

We propose approximate kinetic energy (KE) functionals of the pair-density (PD)-functional theory on the basis of the rigorous expression with the coupling-constant integration (RECCI) that has been recently derived [Phys. Rev. A 85, 062508 (2012)]. These approximate functionals consist of the noninteracting KE and correlation energy terms. It is found that the Thomas-Fermi-Weizsäcker functional is shown to be better as the noninteracting KE term than the Thomas-Fermi and Gaussian model functionals. It is also shown that the correlation energy term is also indispensable for the reduction of the KE error, i.e., reductions of both inappropriateness of the approximate functional and error of the resultant PD. Concerning the correlation energy term, we further propose an approximate functional in addition to using the existing familiar functionals. This functional satisfies the scaling property of the KE functional, and yields a reasonable PD in a sense that the KE, electron-electron interaction, and potentials energies tend to be improved with satisfying the virial theorem. The present results not only suggest the usefulness of the RECCI but also provide the guideline for the further improvement of the RECCI-based KE functional.

Figure 1

Profiles of electron densities of the Ne atom. These profiles are calculated by using the TF-PBE (chain line), GM-PBE (dashed line), and TFW-PBE functionals (solid line). As a reference, the electron density by the CI method [82] is also shown (dotted line).

Figure 2

Profiles of electron densities of the Ne atom. These profiles are calculated by using the TF (chain line), TF-LDA (dashed line), and TF-PBE functionals (solid line). As a reference, the electron density by the CI method [82] is also shown (dotted line).

Figure 3

Profiles of electron densities of the Ne atom. These profiles are calculated by using the GM (chain line), GM-LDA (dashed line), and GM-PBE functionals (solid line). As a reference, the electron density by the CI method [82] is also shown (dotted line).

Figure 4

Profiles of electron densities of the Ne atom. These profiles are calculated by using the TFW (chain line), TFW-LDA (dashed line), TFW-PBE (solid line), and TFW-$E_c^{\mathrm{scale}}$ functionals (gray bold line). As a reference, the electron density by the CI method [82] is also shown (dotted line).

Figure 5

Profiles of the xc holes $\mathrm{[}{n}_{\mathrm{xc}}(\mathbf{r},{\mathbf{r}}^{\prime })\text{]}$ of the Ne atom along $z$ axis. The reference electron is placed at $z=0.2$. These profiles are calculated by using the TF-PBE (chain line), GM-PBE (dashed line), and TFW-PBE functionals (solid line). As a reference, the xc hole by the CI method [83] is also shown (dotted line).

Figure 6

Profiles of the xc holes $\mathrm{[}{n}_{\mathrm{xc}}(\mathbf{r},{\mathbf{r}}^{\prime })\text{]}$ of the Ne atom along $z$ axis. The reference electron is placed at $z=0.2$. These profiles are calculated by using the TFW (chain line), TFW-LDA (dashed line), TFW-PBE (solid line) and TFW-$E_c^{\mathrm{scale}}$ functionals (gray bold line). As a reference, the xc hole by the CI method [83] is also shown (dotted line).

Figure 7

Profiles of the xc holes $\mathrm{[}{n}_{\mathrm{xc}}(\mathbf{r},{\mathbf{r}}^{\prime })\text{]}$ of the Ne atom along $z$ axis. The reference electron is placed at $z=0.2$. These profiles are calculated by using the TFW (chain line), TFW-LDA (dashed line), TFW-PBE (solid line), and TFW-$E_c^{\mathrm{scale}}$ functionals (gray bold line). As a reference, the xc hole by the CI method [83] is also shown (dotted line).

Figure 8

Profiles of the xc holes $\mathrm{[}{n}_{\mathrm{xc}}(\mathbf{r},{\mathbf{r}}^{\prime })\text{]}$ of the Ne atom along $z$ axis. The reference electron is placed at $z=0.4$. These profiles are calculated by using the TFW (chain line), TFW-LDA (dashed line), TFW-PBE (solid line), and TFW-$E_c^{\mathrm{scale}}$ functionals (gray bold line). As a reference, the xc hole by the CI method [83] is also shown (dotted line).

Figure 9

Dependence of $\beta$ on the total number of electrons $N$.