^{1,a)}, Peter Klüpfel

^{1,2}and Hannes Jónsson

^{1,3}

### Abstract

Self-consistent calculations using the Perdew-Zunger self-interaction correction (PZ-SIC) to local density and gradient dependent energyfunctionals are presented for the binding energy and equilibrium geometry of small molecules as well as energy barriers of reactions. The effect of the correction is to reduce binding energy and bond lengths and increase activation energy barriers when bond breaking is involved. The accuracy of the corrected functionals varies strongly, the correction to the binding energy being too weak for the local density approximation but too strong for the gradient dependent functionals considered. For the Perdew, Burke, and Ernzerhof (PBE) functional, a scaling of the PZ-SIC by one half gives improved results on average for both binding energy and bond lengths. The PZ-SIC does not necessarily give more accurate total energy, but it can result in a better cancellation of errors. An essential aspect of these calculations is the use of complex orbitals. A restriction to real orbitals leads to less accurate results as was recently shown for atoms [S. Klüpfel, P. Klüpfel, and H. Jónsson, Phys. Rev. A84, 050501 (2011)10.1103/PhysRevA.84.050501]. The molecular geometry of radicals can be strongly affected by PZ-SIC. An incorrect, non-linear structure of the C_{2}H radical predicted by PBE is corrected by PZ-SIC. The CH_{3} radical is correctly predicted to be planar when complex orbitals are used, while it is non-planar when the PZ-SIC calculation is restricted to real orbitals.

This work was supported by the Graduate Fellowship Fund of The University of Iceland, the Icelandic Research Fund, the Marie Curie Research Training Network (MCRTN) “Hydrogen,” and the Nordic Energy Research network “Solar Fuel.” The calculations were carried out on the computer clusters “Sol,” funded by the Icelandic Research Fund, and “Gardar,” operated by the Nordic High Performance Computing (NHPC) project.

I. INTRODUCTION

II. COMPUTATIONAL METHOD

III. PZ-SIC ORBITALS

IV. ATOMIZATION ENERGY OF MOLECULES

V. EQUILIBRIUM GEOMETRY

VI. STRUCTURE OF MOLECULAR RADICALS

VII. REACTION BARRIERS

VIII. SUMMARY AND CONCLUSION

### Key Topics

- Chemical bonds
- 11.0
- Hydrogen energy
- 11.0
- Real functions
- 9.0
- Density functional theory
- 8.0
- Total energy calculations
- 6.0

## Figures

Complex (a) and real (b) energy minimizing valence orbitals of N_{2} calculated with PBE+SIC. Only one spin channel is shown, orbitals of the other spin have the same shape. The top three orbitals represent the triple bond, the bottom two represent lone pairs.

Complex (a) and real (b) energy minimizing valence orbitals of N_{2} calculated with PBE+SIC. Only one spin channel is shown, orbitals of the other spin have the same shape. The top three orbitals represent the triple bond, the bottom two represent lone pairs.

Mean error (ME, horizontal black lines) and mean absolute error (MAE, columns) of calculated atomization energy compared to experimental values (with zero point energy removed).^{48} For comparison, results obtained from calculations restricted to real orbitals are shown by striped columns. The best overall agreement is obtained with PBE+SIC/2, apart from the hybrid functionals, in particular B3LYP.

Mean error (ME, horizontal black lines) and mean absolute error (MAE, columns) of calculated atomization energy compared to experimental values (with zero point energy removed).^{48} For comparison, results obtained from calculations restricted to real orbitals are shown by striped columns. The best overall agreement is obtained with PBE+SIC/2, apart from the hybrid functionals, in particular B3LYP.

Optimized orbitals corresponding to the single bond in F_{2} calculated with (a) PBE+SIC/2 and (b) PBE+SIC. Both spin-up and spin-down orbitals are shown. For PBE+SIC/2, the total density is not spin polarized. For PBE+SIC, the orbitals are localized to some extent on one of the atoms and the electron density is spin-polarized.

Optimized orbitals corresponding to the single bond in F_{2} calculated with (a) PBE+SIC/2 and (b) PBE+SIC. Both spin-up and spin-down orbitals are shown. For PBE+SIC/2, the total density is not spin polarized. For PBE+SIC, the orbitals are localized to some extent on one of the atoms and the electron density is spin-polarized.

Errors per electron in the total energy of the molecules (x-axis) and of the constituent atoms (y-axis). The diagonal line indicates a cancellation of errors in atomization energy. Systems in the upper left area are over bound, in the lower right area binding energy is too small. Grey points indicate the hydrogen containing molecules.

Errors per electron in the total energy of the molecules (x-axis) and of the constituent atoms (y-axis). The diagonal line indicates a cancellation of errors in atomization energy. Systems in the upper left area are over bound, in the lower right area binding energy is too small. Grey points indicate the hydrogen containing molecules.

Mean error (ME, horizontal lines) and mean absolute error (MAE, bars) of calculated equilibrium bond lengths compared to experimental values.^{50} For SIC/2 and SIC, results obtained using real orbitals are indicated by striped columns. F_{2} was excluded for all functionals, as it is not bound with respect to the atoms for BLYP+SIC.

Mean error (ME, horizontal lines) and mean absolute error (MAE, bars) of calculated equilibrium bond lengths compared to experimental values.^{50} For SIC/2 and SIC, results obtained using real orbitals are indicated by striped columns. F_{2} was excluded for all functionals, as it is not bound with respect to the atoms for BLYP+SIC.

Bond angle deviations for H_{2}O, NH_{3}, and CH_{2}. The difference of calculated and experimental^{50} angles H-X-H are shown for the various functionals.

Bond angle deviations for H_{2}O, NH_{3}, and CH_{2}. The difference of calculated and experimental^{50} angles H-X-H are shown for the various functionals.

(a) Complex and (b) real PBE+SIC optimized valence orbitals of the planar CH_{3} radical. Isosurfaces of the spin-majority valence orbital densities are shown in side view (left) and top view (right). The orbital of the unpaired electron is colored. The complex orbitals have mirror symmetry with respect to the plane. The real orbital for the unpaired electron has *sp* ^{3} character and the C–H-binding orbitals are out of plane. The arrangement of the real orbitals is not favored and the ground state geometry is predicted to be pyramidal.^{51} Figure by Simon Klüpfel from ‘Implementation and reassessment of the Perdew-Zunger self-interaction correction’, ISBN: 978-9935-9053-8-3. Used under a Creative Commons Attribution license.

(a) Complex and (b) real PBE+SIC optimized valence orbitals of the planar CH_{3} radical. Isosurfaces of the spin-majority valence orbital densities are shown in side view (left) and top view (right). The orbital of the unpaired electron is colored. The complex orbitals have mirror symmetry with respect to the plane. The real orbital for the unpaired electron has *sp* ^{3} character and the C–H-binding orbitals are out of plane. The arrangement of the real orbitals is not favored and the ground state geometry is predicted to be pyramidal.^{51} Figure by Simon Klüpfel from ‘Implementation and reassessment of the Perdew-Zunger self-interaction correction’, ISBN: 978-9935-9053-8-3. Used under a Creative Commons Attribution license.

Energy of the bent ethynyl radical relative to the energy of the linear geometry. The bond lengths have been optimized for each value of the bond angle. The PBE ground state geometry is bent with an angle of ≈166° in agreement with previous calculations.^{52} PBE+SIC and PBE+SIC/2 both favor the linear geometry in agreement with experiment.^{50}

Energy of the bent ethynyl radical relative to the energy of the linear geometry. The bond lengths have been optimized for each value of the bond angle. The PBE ground state geometry is bent with an angle of ≈166° in agreement with previous calculations.^{52} PBE+SIC and PBE+SIC/2 both favor the linear geometry in agreement with experiment.^{50}

Energy barrier and bond lengths at the saddle points for the four reactions of Table V. The deviation from reference energy, in eV, and bond lengths, in Å, is shown for the various functionals. The rectangle, 1 pm by 0.1 eV, at the origin emphasizes the different energy and length scales of the different graphs. For HFH and NH_{3} the results based on real orbitals are depicted by grey symbols, for H_{4} and H_{3} real and complex orbitals give identical results.

Energy barrier and bond lengths at the saddle points for the four reactions of Table V. The deviation from reference energy, in eV, and bond lengths, in Å, is shown for the various functionals. The rectangle, 1 pm by 0.1 eV, at the origin emphasizes the different energy and length scales of the different graphs. For HFH and NH_{3} the results based on real orbitals are depicted by grey symbols, for H_{4} and H_{3} real and complex orbitals give identical results.

Errors in total energy for the PBE-type functionals. For each of the reactions, the errors per electron for the saddle point (S), reactants (R), and separated atoms (A) is shown. The vertical difference between neighboring points corresponds to errors in atomization energy (A-R) and barrier height (R-S).

Errors in total energy for the PBE-type functionals. For each of the reactions, the errors per electron for the saddle point (S), reactants (R), and separated atoms (A) is shown. The vertical difference between neighboring points corresponds to errors in atomization energy (A-R) and barrier height (R-S).

Energy along a path for the H + H_{2} → H_{2} + H reaction. The x-axis shows the distance between the two fragments. The H_{2} bond length has been relaxed for each separation. LSD+SIC(/2) predicts an energy barrier, but also reveals an intermediate configuration that is more stable than the separated reactants. Full SIC applied to PBE gives good agreement with the CI results and corrects the LSD result enough to avoid the formation of a stable hydrogen trimer.

Energy along a path for the H + H_{2} → H_{2} + H reaction. The x-axis shows the distance between the two fragments. The H_{2} bond length has been relaxed for each separation. LSD+SIC(/2) predicts an energy barrier, but also reveals an intermediate configuration that is more stable than the separated reactants. Full SIC applied to PBE gives good agreement with the CI results and corrects the LSD result enough to avoid the formation of a stable hydrogen trimer.

## Tables

Deviation (in eV) of calculated atomization energy *E* _{b} from experiment (with zero point energy removed^{48}). For H_{2} an accurate result was used as reference.^{49} The energy has been calculated for the respective equilibrium geometry.

Deviation (in eV) of calculated atomization energy *E* _{b} from experiment (with zero point energy removed^{48}). For H_{2} an accurate result was used as reference.^{49} The energy has been calculated for the respective equilibrium geometry.

Atomization energy and equilibrium bond length of F_{2}. In BLYP+SIC, the molecule is not stable. The binding energy decreases from the uncorrected functionals to SIC/2 to SIC. The equilibrium bond length, however, changes non-monotonously with the fraction of SIC for the GGA functionals.

Atomization energy and equilibrium bond length of F_{2}. In BLYP+SIC, the molecule is not stable. The binding energy decreases from the uncorrected functionals to SIC/2 to SIC. The equilibrium bond length, however, changes non-monotonously with the fraction of SIC for the GGA functionals.

Deviation (in pm) of calculated bond length *d* _{b} from experimentally determined geometry.^{50}

Deviation (in pm) of calculated bond length *d* _{b} from experimentally determined geometry.^{50}

Equilibrium structure of the CH_{3} radical. The “out of plane” angle, α, in degrees and energy difference between planar and pyramidal structure, Δ*E*, in meV is shown for the various SIC functionals for complex (c.) and real (r.) orbitals. The uncorrected functionals all predict the correct planar ground state.

Equilibrium structure of the CH_{3} radical. The “out of plane” angle, α, in degrees and energy difference between planar and pyramidal structure, Δ*E*, in meV is shown for the various SIC functionals for complex (c.) and real (r.) orbitals. The uncorrected functionals all predict the correct planar ground state.

Energy barrier for four reactions. For each saddle point, the point group, energy barrier with respect to the reactants, and bond-length are listed. The labels in bold face are used throughout the text for the saddle point.

Energy barrier for four reactions. For each saddle point, the point group, energy barrier with respect to the reactants, and bond-length are listed. The labels in bold face are used throughout the text for the saddle point.

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