# The Influence of One-Electron Self-Interaction on d-Electrons

^{*}

## Abstract

**:**

## 1. Introduction

`DARSEC`[86,107,108]. As a consequence, we are restricted to computations of molecules with two atomic centers. We find that for the transition metal-containing molecules investigated in this work, local and global hybrids exhibit great similarities for the description of the DOS. In particular, the investigated local hybrid functionals do not show a greater sensitivity to localized d-states despite the formal fulfillment of Equation (1). Thus, the main conclusion of Reference [67] is further confirmed in this work. While transition metal atoms in diatomic molecules differ from atoms in the solid-state limit of typical d-electron metals, we see the present study as an indicator for what one can qualitatively expect from local hybrids for d-electron metals.

## 2. Counteracting Electronic Self-Interaction with Hybrid Functionals

## 3. Computational Details

`DARSEC`using a highly accurate real-space grid. In

`DARSEC`, the Kohn–Sham equations are solved self-consistently with an explicit consideration of all electrons. A local, multiplicative potential for orbital-dependent functionals (i.e., EXX and the global and local hybrid functionals) was obtained by using the KLI approximation [123] to the OEP. All calculations were performed in a non-relativistic way.

_{2}, CuCl, and Pd

_{2}. The former three molecules were evaluated based on their experimental ground-state bond lengths ${R}_{\mathrm{ZnO}}=3.2162\phantom{\rule{4pt}{0ex}}\mathrm{bohr}$, ${R}_{\mathrm{CuCl}}=3.8762\phantom{\rule{4pt}{0ex}}\mathrm{bohr}$, and ${R}_{{\mathrm{Cu}}_{2}}=4.1946\phantom{\rule{4pt}{0ex}}\mathrm{bohr}$ (see Reference [124]). The bond length of Pd

_{2}was determined as ${R}_{{\mathrm{Pd}}_{2}}=4.9063\phantom{\rule{4pt}{0ex}}\mathrm{bohr}$ based on a geometry optimization using the LSDA, since no experimental value is available.

## 4. Results and Discussion

_{2}, as shown in Figure 2 and Figure 3. It becomes evident that both PBEh and ISO with varying values of their parameters gradually change the energetic position of Kohn–Sham states that are affected by SI between the limiting cases of the semilocal PBE/LSDA and nonlocal, SI-free EXX. Interestingly, for some states—such as, for instance, the lowest state shown for Cu

_{2}—inclusion of small parts of EXX results in an upshift on the energy scale, while larger amounts of EXX reverse the direction of the shift. However, the local hybrid functional does not exhibit a special sensitivity to states affected by SI, as can be seen be comparing the DOS obtained with PBEh($a=0.25$) and ISO($c=0.5$). These functionals can be regarded as comparable in the sense that both use the parametrization that was determined as ideal for the description of thermochemical properties such as binding energies [86,111]. Yet, despite the fact that ISO in contrast to PBEh is formally free from SI, the spectra of PBEh($a=0.25$) and ISO($c=0.5$) are rather similar for the systems investigated here.

_{2}, a different scenario occurs. Figure 4 demonstrates that the DOS changes insignificantly when using EXX instead of semilocal functionals. This indicates that all states are affected by SI to the same extent [13,115]. In this case, counteracting SI by using a global hybrid functional with increasing amounts of EXX does not change the relative energetic positions of the Kohn–Sham states, as can be seen from Figure 4a. Similarly, the results obtained with ISO in Figure 4b exhibit a DOS that is nearly independent of the value of the functional parameter. Consequently, also for systems whose electronic structure is less influenced by SI, the local and global hybrid perform very similarly.

_{2}and the DOS of the heteronuclear dimer CuCl are greatly affected by SI. The relative position of the Kohn–Sham eigenvalues of the homonuclear diatomic molecule Pd

_{2}, on the other hand, is hardly influenced by SI. We conclude from this finding that it is difficult to predict trends in the DOS based on general arguments about formally being free from SI. For this reason, the results for the Pd

_{2}dimer also cannot be generalized to bulk Pd.

_{2}and Pd

_{2}. These graphs demonstrate that ISOII, even though it explicitly uses the detection function ${\tau}_{\mathbf{W}}\left(\mathbf{r}\right)/\tau \left(\mathbf{r}\right)$ also for spin-unpolarized systems, provides a Kohn–Sham DOS that is similar to the results of PBEh and ISO. For ZnO, CuCl, and Cu

_{2}, the energetic positions of the states affected by SI change gradually from the result of the LSDA to the one of pure EXX with increasing values of the functional parameter ${c}^{*}$, and good agreement with SI-free calculations is only achieved for large values of ${c}^{*}$. In the case of Pd

_{2}, again, changing the parametrization of ISOII has virtually no effect on the relative Kohn–Sham eigenvalue spectrum. In general, also for ISOII, the intrinsic amount of EXX appears to be the decisive factor for the outcome of the Kohn–Sham DOS.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Kohn–Sham density of states (DOS) of ZnO obtained with (

**a**) PBE (blue), exact-exchange (EXX, black), and PBEh in dependence on a (green); (

**b**) LDA (blue), EXX (black), and ISO in dependence on c (red). In this and all following figures, each panel contains the value of $-{\epsilon}_{\mathrm{ho}}$ in eV in order to indicate the absolute position of the DOS on the energy scale. Further, the relative shift of the d-states—which were identified in

`DARSEC`as the Kohn–Sham states with angular momentum quantum number $m=\pm 2$ (cf. Reference [107])—is highlighted by the dashed grey line.

**Figure 2.**Kohn–Sham DOS of CuCl obtained with (

**a**) PBE (blue), EXX (black), and PBEh in dependence on a (green); (

**b**) LDA (blue), EXX (black), and ISO in dependence on c (red).

**Figure 3.**Kohn–Sham DOS of Cu

_{2}obtained with (

**a**) PBE (blue), EXX (black), and PBEh in dependence on a (green); (

**b**) LDA (blue), EXX (black), and ISO in dependence on c (red).

**Figure 4.**Kohn–Sham DOS of Pd

_{2}obtained with (

**a**) PBE (blue), EXX (black), and PBEh in dependence on a (green); (

**b**) LDA (blue), EXX (black), and ISO in dependence on c (red).

**Figure 5.**Kohn–Sham DOS obtained with LDA (blue), pure EXX (black), and ISOII in dependence on ${c}^{*}$ (orange) for (

**a**) ZnO and (

**b**) CuCl.

**Figure 6.**Kohn–Sham DOS obtained with LDA (blue), pure EXX (black), and ISOII in dependence on ${c}^{*}$ (orange) for (

**a**) Cu

_{2}and (

**b**) Pd

_{2}.

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Schmidt, T.; Kümmel, S.
The Influence of One-Electron Self-Interaction on *d*-Electrons. *Computation* **2016**, *4*, 33.
https://doi.org/10.3390/computation4030033

**AMA Style**

Schmidt T, Kümmel S.
The Influence of One-Electron Self-Interaction on *d*-Electrons. *Computation*. 2016; 4(3):33.
https://doi.org/10.3390/computation4030033

**Chicago/Turabian Style**

Schmidt, Tobias, and Stephan Kümmel.
2016. "The Influence of One-Electron Self-Interaction on *d*-Electrons" *Computation* 4, no. 3: 33.
https://doi.org/10.3390/computation4030033