# 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

## References

- Hohenberg, P.; Kohn, W. Inhomogeneous electron gas. Phys. Rev.
**1964**, 136, B864–B871. [Google Scholar] [CrossRef] - Kohn, W.; Sham, L.J. Self-consistent equations including exchange and correlation effects. Phys. Rev.
**1965**, 140, A1133–A1138. [Google Scholar] [CrossRef] - Parr, R.G.; Yang, W. Density-Functional Theory of Atoms and Molecules; Oxford University Press: New York, NY, USA, 1989. [Google Scholar]
- Becke, A.D. Perspective: Fifty years of density-functional theory in chemical physics. J. Chem. Phys.
**2014**, 140, 18A301. [Google Scholar] [CrossRef] [PubMed] - Cramer, C.J.; Truhlar, D.G. Density functional theory for transition metals and transition metal chemistry. Phys. Chem. Chem. Phys.
**2009**, 11, 10757–10816. [Google Scholar] [CrossRef] [PubMed] - Szotek, Z.; Temmerman, W.M.; Winter, H. Application of the self-interaction correction to transition-metal oxides. Phys. Rev. B
**1993**, 47, 4029–4032. [Google Scholar] [CrossRef] - Schulthess, T.C.; Temmerman, W.M.; Szotek, Z.; Butler, W.H.; Stocks, G.M. Electronic structure and exchange coupling of Mn impurities in III–V semiconductors. Nat. Mater.
**2005**, 4, 838–844. [Google Scholar] [CrossRef] - Strange, P.; Svane, A.; Temmerman, W.M.; Szotek, Z.; Winter, H. Understanding the valency of rare earths from first-principles theory. Nature
**1999**, 399, 756–758. [Google Scholar] [CrossRef] - Perdew, J.P. Orbital functional for exchange and correlation: Self-interaction correction to the local density approximation. Chem. Phys. Lett.
**1979**, 64, 127–130. [Google Scholar] [CrossRef] - Perdew, J.P.; Zunger, A. Self-interaction correction to density-functional approximations for many-electron systems. Phys. Rev. B
**1981**, 23, 5048–5079. [Google Scholar] [CrossRef] - Kümmel, S.; Kronik, L. Orbital-dependent density functionals: Theory and applications. Rev. Mod. Phys.
**2008**, 80, 3–60. [Google Scholar] [CrossRef] - Tsuneda, T.; Hirao, K. Self-interaction corrections in density functional theory. J. Chem. Phys.
**2014**, 140, 18A513. [Google Scholar] [CrossRef] [PubMed] - Körzdörfer, T.; Kümmel, S.; Marom, N.; Kronik, L. When to trust photoelectron spectra from Kohn–Sham eigenvalues: The case of organic semiconductors. Phys. Rev. B
**2009**, 79, 201205(R). [Google Scholar] [CrossRef] - Körzdörfer, T.; Kümmel, S.; Marom, N.; Kronik, L. Erratum: When to trust photoelectron spectra from Kohn–Sham eigenvalues: The case of organic semiconductors [Phys. Rev. B
**79**, 201205 (2009)]. Phys. Rev. B**2010**, 82, 129903. [Google Scholar] [CrossRef] - Handy, N.C.; Cohen, A.J. Left-right correlation energy. Mol. Phys.
**2001**, 99, 403–412. [Google Scholar] [CrossRef] - Cremer, D. Density functional theory: Coverage of dynamic and non-dynamic electron correlation effects. Mol. Phys.
**2001**, 99, 1899–1940. [Google Scholar] [CrossRef] - Polo, V.; Kraka, E.; Cremer, D. Electron correlation and the self-interaction error of density functional theory. Mol. Phys.
**2002**, 100, 1771–1790. [Google Scholar] [CrossRef] - Fermi, E.; Amaldi, E. Le Orbite [infinito] s Ddegli Elementi; R. Accademia d’Italia: Roma, Italy, 1934; Volume 6, p. 119. [Google Scholar]
- Cortona, P. New self-interaction-corrected local-density approximation to tzhe density-functional theory. Phys. Rev. A
**1986**, 34, 769–776. [Google Scholar] [CrossRef] - Guo, Y.; Whitehead, M.A. An alternative self-interaction correction in the generalized exchange local-density functional theory. J. Comput. Chem.
**1991**, 12, 803–810. [Google Scholar] [CrossRef] - Lundin, U.; Eriksson, O. Novel method of self-interaction corrections in density functional calculations. Int. J. Quantum Chem.
**2001**, 81, 247–252. [Google Scholar] [CrossRef] - Unger, H.J. Self-interaction correction with an explicitly density-dependent functional. Phys. Lett. A
**2001**, 284, 124–129. [Google Scholar] [CrossRef] - Vydrov, O.A.; Scuseria, G.E. A simple method to selectively scale down the self-interaction correction. J. Chem. Phys.
**2006**, 124, 191101. [Google Scholar] [CrossRef] [PubMed] - Vieira, D.; Capelle, K. Investigation of self-interaction corrections for an exactly solvable model system: Orbital dependence and electron localization. J. Chem. Theory Comput.
**2010**, 6, 3319–3329. [Google Scholar] [CrossRef] [PubMed] - Constantin, L.A.; Fabiano, E.; Della Sala, F. Improving atomization energies of molecules and solids with a spin-dependent gradient correction from one-electron density analysis. Phys. Rev. B
**2011**, 84, 233103. [Google Scholar] [CrossRef] - Dinh, P.M.; Reinhard, P.G.; Suraud, E.; Vincendon, M. The two-set and average-density self-interaction corrections applied to small electronic systems. In Advances in Atomic, Molecules, and Optical Physics; Elsevier: Amsterdam, The Netherlands, 2015; Volume 64, pp. 87–103. [Google Scholar]
- Borghi, G.; Ferretti, A.; Nguyen, N.L.; Dabo, I.; Marzari, N. Koopmans-compliant functionals and their performance against reference molecular data. Phys. Rev. B
**2014**, 90, 075135. [Google Scholar] [CrossRef] - Nguyen, N.L.; Borghi, G.; Ferretti, A.; Dabo, I.; Marzari, N. First-principles photoemission spectroscopy and orbital tomography in molecules from koopmans-compliant functionals. Phys. Rev. Lett.
**2015**, 114, 166405. [Google Scholar] [CrossRef] [PubMed] - Pederson, M.R.; Heaton, R.A.; Lin, C.C. Local-density Hartree-Fock theory of electronic states of molecules with self-interaction correction. J. Chem. Phys.
**1984**, 80, 1972. [Google Scholar] [CrossRef] - Vydrov, O.A.; Scuseria, G.E. Effect of the Perdew-Zunger self-interaction correction on the thermochemical performance of approximate density functionals. J. Chem. Phys.
**2004**, 121, 8187–8193. [Google Scholar] [CrossRef] [PubMed] - Messud, J.; Dinh, P.M.; Reinhard, P.G.; Suraud, E. Time-dependent density-functional theory with a self-interaction correction. Phys. Rev. Lett.
**2008**, 101, 1–4. [Google Scholar] [CrossRef] [PubMed] - Ruzsinszky, A.; Perdew, J.P.; Csonka, G.I.; Scuseria, G.E.; Vydrov, O.A. Understanding and correcting the self-interaction error in the electrical response of hydrogen chains. Phys. Rev. A
**2008**, 77, 060502. [Google Scholar] [CrossRef] - Körzdörfer, T.; Mundt, M.; Kümmel, S. Electrical Response of Molecular Systems: The Power of Self-Interaction Corrected Kohn–Sham Theory. Phys. Rev. Lett.
**2008**, 100, 133004. [Google Scholar] [CrossRef] [PubMed] - Kümmel, S. Self-interaction correction as a Kohn–Sham scheme in ground-state and time-dependent density functional theory. In Advances in Atomic, Molecular, and Optical Physics; Elsevier: Amsterdam, The Netherlands, 2015; Volume 64, pp. 143–151. [Google Scholar]
- Chen, J.; Krieger, J.B.; Li, Y.; Iafrate, G.J. Kohn–Sham calculations with self-interaction-corrected local-spin-density exchange-correlation energy functional for atomic systems. Phys. Rev. A
**1996**, 54, 3939–3947. [Google Scholar] [CrossRef] [PubMed] - Garza, J.; Nichols, J.A.; Dixon, D.A. The optimized effective potential and the self-interaction correction in density functional theory: Application to molecules. J. Chem. Phys.
**2000**, 112, 7880. [Google Scholar] [CrossRef] - Patchkovskii, S.; Autschbach, J.; Ziegler, T. Curing difficult cases in magnetic properties prediction with self-interaction corrected density functional theory. J. Chem. Phys.
**2001**, 115, 26–42. [Google Scholar] [CrossRef] - Legrand, C.; Suraud, E.; Reinhard, P.G. Comparison of self-interaction-corrections for metal clusters. J. Phys. B Atomic Mol. Opt. Phys.
**2002**, 35, 1115–1128. [Google Scholar] [CrossRef] - Vieira, D.; Capelle, K.; Ullrich, C.A. Physical signatures of discontinuities of the time-dependent exchange-correlation potential. Phys. Chem. Chem. Phys.
**2009**, 11, 4647–4654. [Google Scholar] [CrossRef] [PubMed] - Pemmaraju, C.D.; Sanvito, S.; Burke, K. Polarizability of molecular chains: A self-interaction correction approach. Phys. Rev. B
**2008**, 77, 121204(R). [Google Scholar] [CrossRef] - Körzdörfer, T.; Kümmel, S.; Mundt, M. Self-interaction correction and the optimized effective potential. J. Chem. Phys.
**2008**, 129, 014110. [Google Scholar] [CrossRef] [PubMed] - Körzdörfer, T.; Kümmel, S. Self-interaction correction in the Kohn–Sham framework. In Theoretical Computational Developments in Modern Density Functional Theory; Roy, A.K., Ed.; Nova Science Publishers: New York, NY, USA, 2012. [Google Scholar]
- Svane, A.; Gunnarsson, O. Transition-metal oxides in the self-interaction-corrected density-functional formalism. Phys. Rev. Lett.
**1990**, 65, 1148–1151. [Google Scholar] [CrossRef] [PubMed] - Temmerman, W.M.; Szotek, Z.; Winter, H. Self-interaction-corrected electronic strucutre of La
_{2}CuO_{4}. Phys. Rev. B**1993**, 47, 11533–11536. [Google Scholar] [CrossRef] - Goedecker, S.; Umrigar, C.J. Critical assessment of the self-interaction-corrected-local-density-functional method and its algorithmic implementation. Phys. Rev. A
**1997**, 55, 1765–1771. [Google Scholar] [CrossRef] - Pederson, M.R.; Heaton, R.A.; Lin, C.C. Density-functional theory with self-interaction correction: Application to the lithium molecule. J. Chem. Phys.
**1985**, 82, 2688. [Google Scholar] [CrossRef] - Pederson, M.R.; Lin, C.C. Localized and canonical atomic orbitals in self-interaction corrected local density functional approximation. J. Chem. Phys.
**1988**, 88, 1807–1817. [Google Scholar] [CrossRef] - Klüpfel, S.; Klüpfel, P.; Jónsson, H. Importance of complex orbitals in calculating the self-interaction- corrected ground state of atoms. Phys. Rev. A
**2011**, 84, 050501. [Google Scholar] [CrossRef] - Hofmann, D.; Klüpfel, S.; Klüpfel, P.; Kümmel, S. Using complex degrees of freedom in the Kohn–Sham self-interaction correction. Phys. Rev. A
**2012**, 85, 062514. [Google Scholar] [CrossRef] - Klüpfel, S.; Klüpfel, P.; Jónsson, H. The effect of the Perdew-Zunger self-interaction correction to density functionals on the energetics of small molecules. J. Chem. Phys.
**2012**, 137, 124102. [Google Scholar] [CrossRef] [PubMed] - Hofmann, D.; Körzdörfer, T.; Kümmel, S. Kohn–Sham Self-Interaction Correction in Real Time. Phys. Rev. Lett.
**2012**, 108, 146401. [Google Scholar] [CrossRef] [PubMed] - Lehtola, S.; Jónsson, H. Variational, self-consistent implementation of the Perdew-Zunger self-interaction correction with complex optimal orbitals. J. Chem. Theory Comput.
**2014**, 10, 5324–5337. [Google Scholar] [CrossRef] [PubMed] - Gudmundsdóttir, H.; Jónsson, E.Ö.; Jónsson, H. Calculations of Al dopant in α-quartz using a variational implementation of the Perdew-Zunger self-interaction correction. New J. Phys.
**2015**, 17, 83006. [Google Scholar] [CrossRef] - Lehtola, S.; Jónsson, E.Ö.; Jónsson, H. Effect of complex-valued optimal orbitals on atomization energies with the Perdew–Zunger self-interaction correction to density functional theory. J. Chem. Theory Comput.
**2016**. [Google Scholar] [CrossRef] [PubMed] - Lehtola, S.; Head-Gordon, M.; Jónsson, H. Complex orbitals, multiple local minima and symmetry breaking in Perdew-Zunger self-interaction corrected density-functional theory calculations. J. Chem. Theory Comput.
**2016**, 12, 3195–3207. [Google Scholar] [CrossRef] [PubMed] - Pederson, M.R.; Ruzsinszky, A.; Perdew, J.P. Communication: Self-interaction correction with unitary invariance in density functional theory. J. Chem. Phys.
**2014**, 140, 121103. [Google Scholar] [CrossRef] [PubMed] - Pederson, M.R.; Baruah, T. Self-interaction corrections within the fermi-orbital-based formalism. In Advance in Atomic, Molecular, and Optical Physics; Elsevier: Amsterdam, The Netherlands, 2015; Volume 64, pp. 153–180. [Google Scholar]
- Hahn, T.; Liebing, S.; Kortus, J.; Pederson, M.R. Fermi orbital self-interaction corrected electronic structure of molecules beyond local density approximation. J. Chem. Phys.
**2015**, 143, 224104. [Google Scholar] [CrossRef] [PubMed] - Ruzsinszky, A.; Perdew, J.P.; Csonka, G.I.; Vydrov, O.A.; Scuseria, G.E. Spurious fractional charge on dissociated atoms: Pervasive and resilient self-interaction error of common density functionals. J. Chem. Phys.
**2006**, 125, 194112. [Google Scholar] [CrossRef] [PubMed] - Mori-Sánchez, P.; Cohen, A.J.; Yang, W. Many-electron self-interaction error in approximate density functionals. J. Chem. Phys.
**2006**, 125, 201102. [Google Scholar] [CrossRef] [PubMed] - Perdew, J.P.; Parr, R.G.; Levy, M.; Balduz, J.L. Density-Functional Theory for Fractional Particle Number: Derivative Discontinuities of the Energy. Phys. Rev. Lett.
**1982**, 49, 1691–1694. [Google Scholar] [CrossRef] - Cohen, A.J.; Mori-Sánchez, P.; Yang, W. Insights into current limitations of density functional theory. Science
**2008**, 321, 792–794. [Google Scholar] [CrossRef] [PubMed] - Sai, N.; Barbara, P.F.; Leung, K. Hole localization in molecular crystals from hybrid density functional theory. Phys. Rev. Lett.
**2011**, 106, 226403. [Google Scholar] [CrossRef] [PubMed] - Stein, T.; Autschbach, J.; Govind, N.; Kronik, L.; Baer, R. Curvature and frontier orbital energies in density functional theory. J. Phys. Chem. Lett.
**2012**, 3, 3740–3744. [Google Scholar] [CrossRef] [PubMed] - Li, C.; Zheng, X.; Cohen, A.J.; Mori-Sánchez, P.; Yang, W. Local scaling correction for reducing delocalization error in density functional approximations. Phys. Rev. Lett.
**2015**, 114, 053001. [Google Scholar] [CrossRef] [PubMed] - Dauth, M.; Caruso, F.; Kümmel, S.; Rinke, P. Piecewise linearity in the GW approximation for accurate quasiparticle energy predictions. Phys. Rev. B
**2016**, 93, 121115. [Google Scholar] [CrossRef] - Schmidt, T.; Kümmel, S. One- and many-electron self-interaction error in local and global hybrid functionals. Phys. Rev. B
**2016**, 93, 165120. [Google Scholar] [CrossRef] - Atalla, V.; Zhang, I.Y.; Hofmann, O.T.; Ren, X.; Rinke, P.; Scheffler, M. Enforcing the linear behavior of the total energy with hybrid functionals: Implications for charge transfer, interaction energies, and the random-phase approximation. Phys. Rev. B
**2016**, 94, 035140. [Google Scholar] [CrossRef] - Yanai, T.; Tew, D.P.; Handy, N.C. A new hybrid exchange-correlation functional using the Coulomb-attenuating method (CAM-B3LYP). Chem. Phys. Lett.
**2004**, 393, 51–57. [Google Scholar] [CrossRef] - Peach, M.J.G.; Helgaker, T.; Sałek, P.; Keal, T.W.; Lutnæs, O.B.; Tozer, D.J.; Handy, N.C. Assessment of a Coulomb-attenuated exchange-correlation energy functional. Phys. Chem. Chem. Phys.
**2006**, 8, 558–562. [Google Scholar] [CrossRef] [PubMed] - Iikura, H.; Tsuneda, T.; Yanai, T.; Hirao, K. A long-range correction scheme for generalized-gradient- approximation exchange functionals. J. Chem. Phys.
**2001**, 115, 3540. [Google Scholar] [CrossRef] - Vydrov, O.A.; Scuseria, G.E. Assessment of a long-range corrected hybrid functional. J. Chem. Phys.
**2006**, 125, 234109. [Google Scholar] [CrossRef] [PubMed] - Chai, J.D.; Head-Gordon, M. Systematic optimization of long-range corrected hybrid density functionals. J. Chem. Phys.
**2008**, 128, 084106. [Google Scholar] [CrossRef] [PubMed] - De Queiroz, T.B.; Kümmel, S. Tuned range separated hybrid functionals for solvated low bandgap oligomers. J. Chem. Phys.
**2015**, 143, 034101. [Google Scholar] [CrossRef] [PubMed] - Karolewski, A.; Kronik, L.; Kümmel, S. Using optimally tuned range separated hybrid functionals in ground-state calculations: Consequences and caveats. J. Chem. Phys.
**2013**, 138, 204115. [Google Scholar] [CrossRef] [PubMed] - Becke, A.D. A new mixing of Hartree-Fock and local density-functional theories. J. Chem. Phys.
**1993**, 98, 1372–1377. [Google Scholar] [CrossRef] - Becke, A.D. Density-functional thermochemistry. III. The role of exact exchange. J. Chem. Phys.
**1993**, 98, 5648–5652. [Google Scholar] [CrossRef] - Perdew, J.P.; Ernzerhof, M.; Burke, K. Rationale for mixing exact exchange with density functional approximations. J. Chem. Phys.
**1996**, 105, 9982–9985. [Google Scholar] [CrossRef] - Cruz, F.G.; Lam, K.C.; Burke, K. Exchange-correlation energy density from virial theorem. J. Phys. Chem. A
**1998**, 102, 4911–4917. [Google Scholar] [CrossRef] - Jaramillo, J.; Scuseria, G.E.; Ernzerhof, M. Local hybrid functionals. J. Chem. Phys.
**2003**, 118, 1068–1073. [Google Scholar] [CrossRef] - Arbuznikov, A.V.; Kaupp, M.; Bahmann, H. From local hybrid functionals to “localized local hybrid” potentials: Formalism and thermochemical tests. J. Chem. Phys.
**2006**, 124, 204102. [Google Scholar] [CrossRef] [PubMed] - Janesko, B.G.; Scuseria, G.E. Local hybrid functionals based on density matrix products. J. Chem. Phys.
**2007**, 127, 164117. [Google Scholar] [CrossRef] [PubMed] - Bahmann, H.; Rodenberg, A.; Arbuznikov, A.V.; Kaupp, M. A thermochemically competitive local hybrid functional without gradient corrections. J. Chem. Phys.
**2007**, 126, 011103. [Google Scholar] [CrossRef] [PubMed] - Kaupp, M.; Bahmann, H.; Arbuznikov, A.V. Local hybrid functionals: An assessment for thermochemical kinetics. J. Chem. Phys.
**2007**, 127, 194102. [Google Scholar] [CrossRef] [PubMed] - Perdew, J.P.; Staroverov, V.N.; Tao, J.; Scuseria, G.E. Density functional with full exact exchange, balanced nonlocality of correlation, and constraint satisfaction. Phys. Rev. A
**2008**, 78, 052513. [Google Scholar] [CrossRef] - Schmidt, T.; Kraisler, E.; Makmal, A.; Kronik, L.; Kümmel, S. A self-interaction-free local hybrid functional: Accurate binding energies vis-à-vis accurate ionization potentials from Kohn–Sham eigenvalues. J. Chem. Phys.
**2014**, 140, 18A510. [Google Scholar] [CrossRef] [PubMed] - de Silva, P.; Corminboeuf, C. Local hybrid functionals with orbital-free mixing functions and balanced elimination of self-interaction error. J. Chem. Phys.
**2015**, 142, 074112. [Google Scholar] [CrossRef] [PubMed] - Kümmel, S.; Perdew, J.P. Two avenues to self-interaction correction within Kohn–Sham theory: Unitary invariance is the shortcut. Mol. Phys.
**2003**, 101, 1363–1368. [Google Scholar] [CrossRef] - Duffy, P.; Chong, D.P.; Casida, M.E.; Salahub, D.R. Kohn–Sham density-functional orbitals as approximate Dyson orbitals scattering for the calculation. Phys. Rev. A
**1994**, 50, 4707–4728. [Google Scholar] [CrossRef] [PubMed] - Chong, D.P.; Gritsenko, O.V.; Baerends, E.J. Interpretation of the Kohn–Sham orbital energies as approximate vertical ionization potentials. J. Chem. Phys.
**2002**, 116, 1760–1772. [Google Scholar] [CrossRef][Green Version] - Kronik, L.; Kümmel, S. Gas-phase valence-electron photoemission spectroscopy using density functional theory. In First Principles Approaches to Spectroscopic Properties of Complex Materials; Topics in Current Chemistry; di Valentin, C., Botti, S., Coccoccioni, M., Eds.; Springer: Berlin, Germany, 2014. [Google Scholar]
- Akola, J.; Manninen, M.; Häkkinen, H.; Landman, U.; Li, X.; Wang, L.S. Aluminum cluster anions: Photoelectron spectroscopy and ab initio simulations. Phys. Rev. B
**2000**, 62, 13216. [Google Scholar] [CrossRef] - Khanna, S.N.; Beltran, M.; Jena, P. Relationship between photoelectron spectroscopy and the magnetic moment of Ni
_{7}clusters. Phys. Rev. B**2001**, 64, 235419. [Google Scholar] [CrossRef] - Kronik, L.; Fromherz, R.; Ko, E.; Ganteför, G.; Chelikowsky, J.R. Highest electron affinity as a predictor of cluster anion structures. Nat. Mater.
**2002**, 1, 49–53. [Google Scholar] [CrossRef] [PubMed] - Moseler, M.; Huber, B.; Häkkinen, H.; Landman, U.; Wrigge, G.; Hoffmann, M.A.; Issendorff, B.V. Thermal effects in the photoelectron spectra of Na-N clustres (N = 4–19). Phys. Rev. B
**2003**, 68, 165413. [Google Scholar] [CrossRef] - Häkkinen, H.; Moseler, M.; Kostko, O.; Morgner, N.; Hoffmann, M.A.; Issendorff, B.V. Symmetry and electronic structure of noble-metal nanoparticles and the role of relativity. Phys. Rev. Lett.
**2004**, 93, 093401. [Google Scholar] [CrossRef] [PubMed] - Mundt, M.; Kümmel, S.; Huber, B.; Moseler, M. Photoelectron spectra of sodium clusters: The problem of interpreting Kohn–Sham eigenvalues. Phys. Rev. B
**2006**, 73, 205407. [Google Scholar] [CrossRef] - Leppert, L.; Kümmel, S. The electronic structure of gold-platinum nanoparticles: Collecting clues for why they are special. J. Phys. Chem. C
**2011**, 115, 6694–6702. [Google Scholar] [CrossRef] - Leppert, L.; Albuquerque, R.Q.; Foster, A.S.; Kümmel, S. Interplay of electronic structure and atomic mobility in nanoalloys of Au and Pt. J. Phys. Chem. C
**2013**, 117, 17268–17273. [Google Scholar] [CrossRef] - Zamudio-Bayer, V.; Leppert, L.; Hirsch, K.; Langenberg, A.; Rittmann, J.; Kossick, M.; Vogel, M.; Richter, R.; Terasaki, A.; Möller, T.; et al. Coordination-driven magnetic-to-nonmagnetic transition in manganese-doped silicon clusters. Phys. Rev. B
**2013**, 88, 115425. [Google Scholar] [CrossRef] - Capelo, R.G.; Leppert, L.; Albuquerque, R.Q. The concept of localized atomic mobility: Unraveling properties of nanoparticles. J. Phys. Chem. C
**2014**, 118, 21647–21654. [Google Scholar] [CrossRef] - Leppert, L.; Kempe, R.; Kümmel, S. Hydrogen binding energies and electronic structure of Ni–Pd particles: A clue to their special catalytic properties. Phys. Chem. Chem. Phys.
**2015**, 17, 26140–26148. [Google Scholar] [CrossRef] [PubMed] - Cherepanov, P.V.; Melnyk, I.; Skorb, E.V.; Fratzl, P.; Zolotoyabko, E.; Dubrovinskaia, N.; Dubrovinsky, L.; Avadhut, Y.S.; Senker, J.; Leppert, L.; et al. The use of ultrasonic cavitation for near-surface structuring of robust and low-cost AlNi catalysts for hydrogen production. Green Chem.
**2015**, 17, 2745. [Google Scholar] [CrossRef] - Aslan, M.; Davis, J.B.A.; Johnston, R.L. Global optimization of small bimetallic Pd–Co binary nanoalloy clusters: A genetic algorithm approach at the DFT level. Phys. Chem. Chem. Phys.
**2016**, 18, 6676. [Google Scholar] [CrossRef] [PubMed] - Grabo, T.; Kreibich, T.; Gross, E.K.U. Optimized effective potential for atoms and molecules. Mol. Eng.
**1997**, 7, 27–50. [Google Scholar] [CrossRef] - Kümmel, S.; Perdew, J.P. Simple Iterative Construction of the Optimized Effective Potential for Orbital Functionals, Including Exact Exchange. Phys. Rev. Lett.
**2003**, 90, 043004. [Google Scholar] [CrossRef] [PubMed] - Makmal, A.; Kümmel, S.; Kronik, L. Fully Numerical All-Electron Solutions of the Optimized Effective Potential Equation for Diatomic Molecules. J. Chem. Theory Comput.
**2009**, 5, 1731–1740. [Google Scholar] [CrossRef] [PubMed] - Makmal, A.; Kümmel, S.; Kronik, L. Dissociation of diatomic molecules and the exact-exchange Kohn–Sham potential: The case of LiF. Phys. Rev. A
**2011**, 83, 062512. [Google Scholar] [CrossRef] - Burke, K.; Cruz, F.G.; Lam, K.C. Unambiguous exchange-correlation energy density. J. Chem. Phys.
**1998**, 109, 8161–8167. [Google Scholar] [CrossRef] - Arbuznikov, A.V.; Kaupp, M. Towards improved local hybrid functionals by calibration of exchange-energy densities. J. Chem. Phys.
**2014**, 141, 204101. [Google Scholar] [CrossRef] [PubMed] - Adamo, C.; Barone, V. Toward reliable density functional methods without adjustable parameters: The PBE0 model. J. Chem. Phys.
**1999**, 110, 6158–6170. [Google Scholar] [CrossRef] - Ernzerhof, M.; Scuseria, G.E. Assessment of the Perdew-Burke-Ernzerhof exchange-correlation functional. J. Chem. Phys.
**1999**, 110, 5029–5036. [Google Scholar] [CrossRef] - Perdew, J.P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett.
**1996**, 77, 3865–3868. [Google Scholar] [CrossRef] [PubMed] - Perdew, J.P.; Burke, K.; Ernzerhof, M. Erratum: Generalized Gradient Approximation Made Simple [Phys. Rev. Lett.
**77**, 3865 (1996)]. Phys. Rev. Lett.**1997**, 78, 1396. [Google Scholar] [CrossRef] - Körzdörfer, T.; Kümmel, S. Single-particle and quasiparticle interpretation of Kohn–Sham and generalized Kohn–Sham eigenvalues for hybrid functionals. Phys. Rev. B
**2010**, 82, 155206. [Google Scholar] [CrossRef] - Imamura, Y.; Kobayashi, R.; Nakai, H. Linearity condition for orbital energies in density functional theory (II): Application to global hybrid functionals. Chem. Phys. Lett.
**2011**, 513, 130–135. [Google Scholar] [CrossRef] - Atalla, V.; Yoon, M.; Caruso, F.; Rinke, P.; Scheffler, M. Hybrid density functional theory meets quasiparticle calculations: A consistent electronic structure approach. Phys. Rev. B
**2013**, 88, 165122. [Google Scholar] [CrossRef] - Ceperley, D.M.; Alder, B.J. Ground state of the electron gas by a stochastic method. Phys. Rev. Lett.
**1980**, 45, 566–569. [Google Scholar] [CrossRef] - Vosko, S.H.; Wilk, L.; Nusair, M. Accurate spin-dependent electron liquid correlation energies for local spin density calculations: A critical analysis. Can. J. Phys.
**1980**, 58, 1200–1211. [Google Scholar] [CrossRef] - Wang, Y.; Perdew, J.P. Accurate and simple analytic representation of the electron-gas correlation energy. Phys. Rev. B
**1992**, 45, 13244–13249. [Google Scholar] - Schmidt, T.; Kraisler, E.; Kronik, L.; Kümmel, S. One-electron self-interaction and the asymptotics of the Kohn–Sham potential: An impaired relation. Phys. Chem. Chem. Phys.
**2014**, 16, 14357–14367. [Google Scholar] [CrossRef] [PubMed] - Kurth, S.; Perdew, J.P.; Blaha, P. Molecular and solid-state tests of density functional approximations: LSD, GGAs, and meta-GGAs. Int. J. Quantum Chem.
**1999**, 75, 889–909. [Google Scholar] [CrossRef] - Krieger, J.B.; Li, Y.; Iafrate, G.J. Construction and application of an accurate local spin-polarized Kohn–Sham potential with integer discontinuity: Exchange-only theory. Phys. Rev. A
**1992**, 45, 101–126. [Google Scholar] [CrossRef] [PubMed] - Lide, D.R. (Ed.) CRC Handbook of Chemistry and Physics, 92nd ed.; CRC: London, UK, 2011.
- Gritsenko, O.V.; Mentel, L.M.; Baerends, E.J. On the errors of local density (LDA) and generalized gradient (GGA) approximations to the Kohn–Sham potential and orbital energies. J. Chem. Phys.
**2016**, 144, 204114. [Google Scholar] [CrossRef] [PubMed]

**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