Energy band gaps and lattice parameters evaluated with the Heyd-Scuseria-Ernzerhof screened hybrid functional

This work assesses the Heyd-Scuseria-Ernzerhof (HSE) screened Coulomb hybrid density functional for the prediction of lattice constants and band gaps using a set of 40 simple and binary semiconductors. An extensive analysis of both basis set and relativistic effects is given. Results are compared with established pure density functionals. For lattice constants, HSE outperforms local spin-density approximation (LSDA) with a mean absolute error (MAE) of 0.037 A for HSE vs 0.047 A for LSDA. For this specific test set, all pure functionals tested produce MAEs for band gaps of 1.0-1.3 eV, consistent with the very well-known fact that pure functionals severely underestimate this property. On the other hand, HSE yields a MAE smaller than 0.3 eV. Importantly, HSE correctly predicts semiconducting behavior in systems where pure functionals erroneously predict a metal, such as, for instance, Ge. The short-range nature of the exchange integrals involved in HSE calculations makes their computation notably faster than regular hybrid functionals. The current results, paired with earlier work, suggest that HSE is a fast and accurate alternative to established density functionals, especially for solid state calculations.     

Accurate solid-state band gaps via screened hybrid electronic structure calculations

The band energy differences of solids calculated with screened hybrid density functionals, such as the functional of Heyd-Scuseria-Ernzerhof (HSE), reproduce experimental band gaps with a high degree of accuracy. This unexpected result is here rationalized by observing that band energy differences obtained from generalized Kohn-Sham calculations with screened (short-range) Hartree-Fock-type exchange approach the excitation energies obtained via time-dependent density functional calculations with the corresponding unscreened functional. The latter are expected to be the accurate predictions of the experimental optical absorption spectra. While the optimum screening parameter (omega) is system dependent, the HSE standard value of omega=0.11 bohr(-1) represents a reasonable compromise across diverse systems.     

Analysis of the Heyd-Scuseria-Ernzerhof density functional parameter space

The Heyd-Scuseria-Ernzerhof (HSE) density functionals are popular for their ability to improve upon the accuracy of standard semilocal functionals such as Perdew-Burke-Ernzerhof (PBE), particularly for semiconductor band gaps. They also have a reduced computational cost compared to hybrid functionals, which results from the restriction of Fock exchange calculations to small inter-electron separations. These functionals are defined by an overall fraction of Fock exchange and a length scale for exchange screening. We systematically examine this two-parameter space to assess the performance of hybrid screened exchange (sX) functionals and to determine a balance between improving accuracy and reducing the screening length, which can further reduce computational costs. Three parameter choices emerge as useful: "sX-PBE" is an approximation to the sX-LDA screened exchange density functionals based on the local density approximation (LDA); "HSE12" minimizes the overall error over all tests performed; and "HSE12s" is a range-minimized functional that matches the overall accuracy of the existing HSE06 parameterization but reduces the Fock exchange length scale by half. Analysis of the error trends over parameter space produces useful guidance for future improvement of density functionals.     

Theoretical study of CeO2 and Ce2O3 using a screened hybrid density functional

The predicted structures and electronic properties of CeO(2) and Ce(2)O(3) have been studied using conventional and hybrid density functional theory. The lattice constant and bulk modulus for CeO(2) from local (LSDA) functionals are in good agreement with experiment, while the lattice parameter from a generalized gradient approximation (GGA) is too long. This situation is reversed for Ce(2)O(3), where the LSDA lattice constant is much too short, while the GGA result is in reasonable agreement with experiment. Significantly, the screened hybrid HSE functional gives excellent agreement with experimental lattice constants for both CeO(2) and Ce(2)O(3). All methods give insulating ground states for CeO(2) with gaps for the 4f band lying between 1.7 eV (LSDA) and 3.3 eV (HSE) and 6-8 eV for the conduction band. For Ce(2)O(3) the local and GGA functionals predict a semimetallic ground state with small (0-0.3 eV) band gap but weak ferromagnetic coupling between the Ce(+3) centers. By contrast, the HSE functional gives an insulating ground state with a band gap of 3.2 eV and antiferromagnetic coupling. Overall, the hybrid HSE functional gives a consistent picture of both the structural and electronic properties of CeO(2) and Ce(2)O(3) while treating the 4f band consistently in both oxides.     

Influence of the exchange screening parameter on the performance of screened hybrid functionals

This work reexamines the effect of the exchange screening parameter omega on the performance of the Heyd-Scuseria-Ernzerhof (HSE) screened hybrid functional. We show that variation of the screening parameter influences solid band gaps the most. Other properties such as molecular thermochemistry or lattice constants of solids change little with omega. We recommend a new version of HSE with the screening parameter omega=0.11 bohr(-1) for further use. Compared to the original implementation, the new parametrization yields better thermochemical results and preserves the good accuracy for band gaps and lattice constants in solids.     

Theoretical study of the electronic properties of narrow single-walled carbon nanotubes: beyond the local density approximation

In this work we present a systematic density functional theory study of the electronic properties of single-walled carbon nanotubes (SWNT) with diameters ranging from 3 to 5 A. In this work meta-generalized-gradient approximation, hybrid, and screened exchange hybrid functionals are utilized to compute energy band gaps in these narrow SWNT. Our calculations using hybrid functionals show that the only true exceptions to the zone folding predictions are the (4,0) and (5,0) SWNT. The remaining chiral SWNT are semiconducting with band gaps that can be as large as 1.7 eV. However, the calculated energy band gaps are significantly smaller than those predicted by the zone folding scheme. This difference is primarily attributed to the sigma-pi hybridization present in such narrow SWNT.     

Assessment and validation of a screened Coulomb hybrid density functional

This paper presents a revised and improved version of the Heyd-Scuseria-Ernzerhof screened Coulomb hybrid functional. The performance of this functional is assessed on a variety of molecules for the prediction of enthalpies of formation, geometries, and vibrational frequencies, yielding results as good as or better than the successful PBE0 hybrid functional. Results for ionization potentials and electron affinities are of slightly lower quality but are still acceptable. The comprehensive test results presented here validate our assumption that the screened, short-range Hartree-Fock (HF) exchange exhibits all physically relevant properties of the full HF exchange. Thus, hybrids can be constructed which neglect the computationally demanding long-range part of HF exchange while still retaining the superior accuracy of hybrid functionals, compared to pure density functionals.     

Comparison of DFT methods for molecular orbital eigenvalue calculations

We report how closely the Kohn-Sham highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) eigenvalues of 11 density functional theory (DFT) functionals, respectively, correspond to the negative ionization potentials (-IPs) and electron affinities (EAs) of a test set of molecules. We also report how accurately the HOMO-LUMO gaps of these methods predict the lowest excitation energies using both time-independent and time-dependent DFT (TD-DFT). The 11 DFT functionals include the local spin density approximation (LSDA), five generalized gradient approximation (GGA) functionals, three hybrid GGA functionals, one hybrid functional, and one hybrid meta GGA functional. We find that the HOMO eigenvalues predicted by KMLYP, BH&HLYP, B3LYP, PW91, PBE, and BLYP predict the -IPs with average absolute errors of 0.73, 1.48, 3.10, 4.27, 4.33, and 4.41 eV, respectively. The LUMOs of all functionals fail to accurately predict the EAs. Although the GGA functionals inaccurately predict both the HOMO and LUMO eigenvalues, they predict the HOMO-LUMO gap relatively accurately (approximately 0.73 eV). On the other hand, the LUMO eigenvalues of the hybrid functionals fail to predict the EA to the extent that they include HF exchange, although increasing HF exchange improves the correspondence between the HOMO eigenvalue and -IP so that the HOMO-LUMO gaps are inaccurately predicted by hybrid DFT functionals. We find that TD-DFT with all functionals accurately predicts the HOMO-LUMO gaps. A linear correlation between the calculated HOMO eigenvalue and the experimental -IP and calculated HOMO-LUMO gap and experimental lowest excitation energy enables us to derive a simple correction formula.     

Description of core excitations by time-dependent density functional theory with local density approximation, generalized gradient approximation, meta-generalized gradient approximation, and hybrid functionals

Time-dependent density functional theory (TDDFT) is employed to investigate exchange-correlation-functional dependence of the vertical core-excitation energies of several molecules including H, C, N, O, and F atoms. For the local density approximation (LDA), generalized gradient approximation (GGA), and meta-GGA, the calculated X1s-->pi* excitation energies (X = C, N, O, and F) are severely underestimated by more than 13 eV. On the other hand, time-dependent Hartree-Fock (TDHF) overestimates the excitation energies by more than 6 eV. The hybrid functionals perform better than pure TDDFT because HF exchange remedies the underestimation of pure TDDFT. Among these hybrid functionals, the Becke-Half-and-Half-Lee-Yang-Parr (BHHLYP) functional including 50% HF exchange provides the smallest error for core excitations. We have also discovered the systematic trend that the deviations of TDHF and TDDFT with the LDA, GGA, and meta-GGA functionals show a strong atom-dependence. Namely, their deviations become larger for heavier atoms, while the hybrid functionals are significantly less atom-dependent.     

Improved hybrid functional for solids: the HSEsol functional

We introduce the hybrid functional HSEsol. It is based on PBEsol, a revised Perdew-Burke-Ernzerhof functional, designed to yield accurate equilibrium properties for solids and their surfaces. We present lattice constants, bulk moduli, atomization energies, heats of formation, and band gaps for extended systems, as well as atomization energies for the molecular G2-1 test set. Compared to HSE, significant improvements are found for lattice constants and atomization energies of solids, but atomization energies of molecules are slightly worse than for HSE. Additionally, we present zero-point anharmonic expansion corrections to the lattice constants and bulk moduli, evaluated from ab initio phonon calculations.     

Interaction of atomic hydrogen with single-walled carbon nanotubes: a density functional theory study

We have studied the interaction of atomic hydrogen with (5,5) and (10,0) single-walled carbon nanotubes (SWNT) using density functional theory. These calculations use Gaussian orbitals and periodic boundary conditions. We compare results from the local spin density approximation, generalized gradient approximation (GGA), and hybrid density functionals. We have first kept the SWNT geometric structure fixed while a single H atom approaches the tube on top of a carbon atom. In that case, a weakly bound state with binding energies from -0.8 to -0.4 eV was found. Full geometry relaxation leads to a strong SWNT deformation, weakening the nearest C-C bonds and increasing the binding energy by about 1 eV. Full hydrogen coverage of the (5,5) SWNT converts this metallic nanotube into an insulator with a band gap of 3.4 eV for the GGA functional and 4.8 eV for the hybrid functional. Hybrid functionals perform similar to pure density functional theory functionals for the calculation of binding energies while band gaps critically depend on the functional choice.     

Communication: a new hybrid exchange correlation functional for band-gap calculations using a short-range Gaussian attenuation (Gaussian-Perdue-Burke-Ernzerhof)

We have developed a new hybrid functional [Gaussian-Perdue-Burke-Ernzerhof (Gau-PBE)] that is suitable for the calculation of solid state bandgaps using a periodic boundary condition. The characteristic of this functional is the use of a Gaussian attenuation scheme (Gau) to include a short-range Hartree-Fock (HF) exchange. This new functional can perform barrier height calculations with an accuracy comparable to the middle-range hybrid functional and bandgap calculations with an accuracy comparable to the Heyd-Scuseria-Ernzerhof (HSE) functional. However, the point is that the performance can be achieved using a Gaussian HF exchange, while the HISS functional calculates twice HF-exchange integrations using an error function to obtain both performances. In addition, Gau-PBE functional can decrease the time cost of bandgap calculations by an average of 40% compared to the HSE functional.     

The performance and relationship among range-separated schemes for density functional theory

The performance and relationship among different range-separated (RS) hybrid functional schemes are examined using the Coulomb-attenuating method (CAM) with different values for the fractions of exact Hartree-Fock (HF) exchange (α), long-range HF (β), and a range-separation parameter (μ), where the cases of α + β = 1 and α + β = 0 were designated as CA and CA0, respectively. Attenuated PBE exchange-correlation functionals with α = 0.20 and μ = 0.20 (CA-PBE) and α = 0.25 and μ = 0.11 (CA0-PBE) are closely related to the LRC-ωPBEh and HSE functionals, respectively. Time-dependent density functional theory calculations were carried out for a number of classes of molecules with varying degrees of charge-transfer (CT) character to provide an assessment of the accuracy of excitation energies from the CA functionals and a number of other functionals with different exchange hole models. Functionals that provided reasonable estimates for local and short-range CT transitions were found to give large errors for long-range CT excitations. In contrast, functionals that afforded accurate long-range CT excitation energies significantly overestimated energies for short-range CT and local transitions. The effects of exchange hole models and parameters developed for RS functionals for CT excitations were analyzed in detail. The comparative analysis across compound classes provides a useful benchmark for CT excitations.