^{1,a)}, Ting Zhou

^{1}and Weitao Yang

^{2,b)}

### Abstract

A nonempirical scaling correction (SC) approach has been developed for improving bandgap prediction in density functional theory[X. Zheng, A. J. Cohen, P. Mori-Sánchez, X. Hu, and W. Yang, Phys. Rev. Lett.107, 026403 (Year: 2011)10.1103/PhysRevLett.107.026403]. For finite systems such as atoms and molecules, the SC approach restores the Perdew–Parr–Levy–Balduz condition [Phys. Rev. Lett.49, 1691 (Year: 1982)10.1103/PhysRevLett.49.1691] that the total electronic energy should scale linearly with number of electrons between integers. Although the original SC approach is applicable to a variety of mainstream density functional approximations, it gives zero correction to the Hartree–Fock method. This is because the relaxation of orbitals with the change in electron number is completely neglected. In this work, with an iterative scheme for the evaluation of Fukui function, the orbital relaxation effects are accounted for explicitly via a perturbative treatment. In doing so, the SC approach is extended to density functionals involving substantial amount of Hartree–Fock exchange. Our new SC approach is demonstrated to improve systematically the predicted Kohn–Sham frontier orbital energies, and alleviate significantly the mismatch between fundamental and derivative gaps.

Support from the National Science Foundation of China (Grant Nos. 21103157 and 21233007) (X.Z.), the Fundamental Research Funds for the Central Universities of China (Grant Nos. 2340000034 and 2340000025) (X.Z.), the Office of Naval Research (ONR) (N00014-09-1-0576) (W.Y.), and the National Science Foundation (NSF) (CHE-09-11119) (W.Y.) is gratefully appreciated.

I. INTRODUCTION

II. METHODOLOGY

III. RESULTS AND DISCUSSIONS

IV. CONCLUDING REMARKS

### Key Topics

- Self consistent field methods
- 17.0
- Density functional theory
- 16.0
- Band gap
- 14.0
- Electron correlation calculations
- 5.0
- Laser Doppler velocimetry
- 4.0

## Figures

Deviation of (a) KS kinetic energy T s and (b) nuclear-electron attraction energy E ne versus n from linearity. Here, n is the fractional electron number, and n = 0 and n = 1 correspond to a carbon cation (C+) and a neutral carbon atom (C), respectively.

Deviation of (a) KS kinetic energy T s and (b) nuclear-electron attraction energy E ne versus n from linearity. Here, n is the fractional electron number, and n = 0 and n = 1 correspond to a carbon cation (C+) and a neutral carbon atom (C), respectively.

Absolute deviation of E v (N + n) from linearity for unscaled (original) and scaled HF methods. The n = 0 point represents (a) a carbon atom and (b) a water molecule. The green dashed line marks zero deviation as a reference. The HF curve is obtained through SCF calculation, while the scaled HF curve includes the post-SCF energy corrections of Eqs. (11) and (12) for positive and negative n, respectively. The inset of (b) shows the calculated E v (N + n) of a water molecule in unit of hartree.

Absolute deviation of E v (N + n) from linearity for unscaled (original) and scaled HF methods. The n = 0 point represents (a) a carbon atom and (b) a water molecule. The green dashed line marks zero deviation as a reference. The HF curve is obtained through SCF calculation, while the scaled HF curve includes the post-SCF energy corrections of Eqs. (11) and (12) for positive and negative n, respectively. The inset of (b) shows the calculated E v (N + n) of a water molecule in unit of hartree.

(a) Calculated εHOMO versus calculated −I for 70 molecules, and (b) εLUMO versus −A for 47 molecules of the G2–97 set. The green solid line indicates εHOMO = −I in (a) and εLUMO = −A in (b). εHOMO and εLUMO of the scaled HF method are calculated by using Eqs. (11) and (12) , and the vertical I and A are calculated by the ΔSCF method. The mean absolute deviations (MADs) between calculated εHOMO and −I are 0.32 and 1.60 eV for scaled and unscaled HF methods; and the MADs between εLUMO and −A are 0.38 and 0.90 eV with and without SC, respectively.

(a) Calculated εHOMO versus calculated −I for 70 molecules, and (b) εLUMO versus −A for 47 molecules of the G2–97 set. The green solid line indicates εHOMO = −I in (a) and εLUMO = −A in (b). εHOMO and εLUMO of the scaled HF method are calculated by using Eqs. (11) and (12) , and the vertical I and A are calculated by the ΔSCF method. The mean absolute deviations (MADs) between calculated εHOMO and −I are 0.32 and 1.60 eV for scaled and unscaled HF methods; and the MADs between εLUMO and −A are 0.38 and 0.90 eV with and without SC, respectively.

HF calculated versus experimentally measured −I for 18 atoms (H–Ar). The calculated εHOMO by using the HF method with and without SC are also depicted. The green solid line indicates perfect agreement with experimental data of −I. Taking the experimental −I as references, the MADs for the calculated −I, , and are 0.86, 0.47, and 1.13 eV, respectively. Whereas taking the calculated −I as references, the MADs for the calculated , and are 1.09 and 0.27 eV, respectively.

HF calculated versus experimentally measured −I for 18 atoms (H–Ar). The calculated εHOMO by using the HF method with and without SC are also depicted. The green solid line indicates perfect agreement with experimental data of −I. Taking the experimental −I as references, the MADs for the calculated −I, , and are 0.86, 0.47, and 1.13 eV, respectively. Whereas taking the calculated −I as references, the MADs for the calculated , and are 1.09 and 0.27 eV, respectively.

## Tables

The vertical ionization potentials and the HOMO energies of M2(hpp)4. The Is are calculated with the ΔSCF approach. All energies are in units of eV.

The vertical ionization potentials and the HOMO energies of M2(hpp)4. The Is are calculated with the ΔSCF approach. All energies are in units of eV.

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