Polarons free from many-body self-interaction in density functional theory

We develop a unified theoretical framework encompassing one-body and many-body forms of self-interaction. We find an analytic expression for both the one-body and the many-body self-interaction energies, and quantitatively connect the two expressions through the dielectric constant. The two forms of self-interaction are found to coincide in the absence of electron screening. This analysis confers superiority to the notion of many-body self-interaction over the notion of one-body self-interaction. Next, we develop a semilocal density functional scheme that addresses the many-body self-interaction of polarons, thereby overcoming the limitations of standard density functional theory. Polaron localization is achieved through the addition of a weak local potential in the Kohn-Sham Hamiltonian that enforces the piece-wise linearity of the total energy upon partial electron occupation. Our method equivalently applies to electron and hole polarons and does not require any constraint on the wave functions during the self-consistent optimization. The implementation of this scheme does not produce any computational overhead compared to standard semilocal calculations and achieves fast convergence. This approach results in polaron properties, including the atomic geometry, the electron density, and the formation energy, which are close to those achieved with a hybrid functional that similarly satisfies the piece-wise linearity condition. This suggests that addressing the many-body self-interaction results in a polaron description that is robust with respect to the functional adopted. We illustrate our approach through applications to the electron polaron in ${\mathrm{BiVO}}_4$, the hole polaron in MgO, and the hole trapped at the Al impurity in $\alpha -{\mathrm{SiO}}_2$.

Figure 1

Schematics of (a) the total energy $E^{\alpha }$ and of (b) the polaron level ${\epsilon }_{\text{p}}^{\alpha }$ as a function of the electron occupation $f$ of the polaron state for different PBE0($\alpha$) functionals.

Figure 2

Polaron isodensity surface at 5% of its maximum calculated with PBE0(${\alpha }_{\text{k}}$) for the electron polaron in ${\mathrm{BiVO}}_4$, the hole polaron in MgO, and the hole trapped at the Al impurity in $\alpha -{\mathrm{SiO}}_2$ (Bi in orange, V in cyan, O in red, Mg in pink, Si in blue, Al in grey).

Figure 3

[(a)–(c)] Energy levels and [(d)–(f)] formation energies obtained with PBE0($\alpha$) as a function $\alpha$ for the electron polaron in ${\mathrm{BiVO}}_4$, the hole polaron in MgO, and the hole trapped at the Al impurity in $\alpha -{\mathrm{SiO}}_2$. The polaron levels are identified by their respective polaron charge. In all cases, the polaron geometry is obtained with PBE0(${\alpha }_{\text{k}}$).

Figure 4

Many-body self-interaction corrected [(a),(b)] total energy $E^0\left(q\right)$ and [(c),(d)] polaron level ${\epsilon }_{\text{p}}^0\left(q\right)$ as a function of the charge $q$, for the electron polaron in ${\mathrm{BiVO}}_4$ and for the trapped hole at the Al impurity in $\alpha -{\mathrm{SiO}}_2$. The localized polarons are obtained with the hybrid functional PBE0(${\alpha }_{\text{k}}$). The results for MgO are shown in Ref. [47].

Figure 5

[(a), (d)] Polaron density $n_{\text{p}}$ and [(b), (c), (e), (f)] variation of the valence-electron densities $n_{\sigma \text{val}}$ upon the occupation $f$ of the polaron level [cf. Eq. (40)] for the electron polaron in ${\mathrm{BiVO}}_4$, and the hole polaron in $\alpha -{\mathrm{SiO}}_2$, as obtained with the functional PBE0(${\alpha }_{\text{k}}$). The densities are integrated over the $z$ direction and plotted in the $xy$ plane. The results for the hole polaron in MgO are shown in Ref. [47].

Figure 6

One-body and many-body SI energy correction for a hole polaron level ${\epsilon }_{\text{p}}^{\alpha }(Q=+1)$, illustrating Eq. (46).

Figure 7

Polaron energy levels ${\epsilon }_{\text{p}}^{\gamma }\left(Q\right)$ and ${\epsilon }_{\text{p}}^{\gamma }\left(0\right)$ for the structure ${\mathbf{R}}_Q^{\gamma }$ as a function of $\gamma$ for the electron polaron in ${\mathrm{BiVO}}_4$, the hole polaron in MgO, and the hole trapped at the Al impurity in $\alpha -{\mathrm{SiO}}_2$. The polaron levels are identified by their respective polaron charge. The value ${\gamma }_{\text{k}}$ is found such that ${\epsilon }_{\text{p}}^{{\gamma }_{\text{k}}}\left(Q\right)={\epsilon }_{\text{p}}^{{\gamma }_{\text{k}}}\left(0\right)$.

Figure 8

[(a), (b)] Polaron densities for the electron polaron in ${\mathrm{BiVO}}_4$ and for the hole trapped at the Al impurity in $\alpha -{\mathrm{SiO}}_2$ as obtained with the hybrid functional PBE0(${\alpha }_{\mathrm{k}}$) and with the semilocal scheme introduced in this paper. The polaron densities are integrated over $xy$ planes. [(c),(d)] Electrostatic potential $V_{\text{elec}}=V_{\text{H}}\left[Q{n}_{\text{p}}^{{\gamma }_{\text{k}}}\right]$ and potential $V_{{\sigma }_{\text{p}}}^{{\gamma }_{\text{k}}}\left(Q\right)$ averaged over $xy$ planes ($Q=-1$ for electron polarons, $Q=+1$ for hole polarons). The results for the hole polaron in MgO are shown in Ref. [47].

Figure 9

Graphical representation related to Eq. (71) to show the robustness of polaron formation energies obtained with semilocal and hybrid functional approaches.