HSE hybrid functional

B3LYP hybrid functional (Becke 1993), the PBEO hybrid functional (Adamo and Barone 1999), and the Heyd-Scuseria-Ernzerhof (HSE) hybrid functional (Heyd et al. 2003). 

We discuss here the main features of the HSE hybrid functional and its application in soUd-state LCAO calculations.The HSE functional for solids is much faster than regular hybrids and can be used in metals also. 

The latter studies conhrmed the efficiency of HF/DFT approach for solids and also use of the HSE hybrid functional. In [419] a set of 40 semiconductors was chosen based on the following criteria all considered systems were both closed shell and of simple or binary composition the majority of them have simple zincblende or rocksalt structures but also several systems with wurtzite structure were included. In addition, the availability of experimental data for lattice constants (and to a lesser extent bandgaps) was an important factor (the references to the experimental data can be found in [419]). These criteria led to the semiconductor/40 (SC/40) set of 40 soUds containing 13 group IIA-VI systems, 6 group IIB-VI systems, 17 group III-V systems, and 4 group IV systems. 

Hybrid functionals other than B3LYP have also proved successful in localised orbital methods. Yang and Dolg found that the hybrid functional B3PW including spin-orbit coupling reproduced the band gap of BiB30g well, while Prodan et u/." found that the HSE (Heyd, Scuseria and Emzerhof) screened coulomb hybrid potential described the oxides of uranium and plutonium well. The performance of hybrid functionals is discussed by Cora et Just as there is no universal... 

Concludig the discussion of hybrid functionals we stress that the HSE functional is universally apphcable and does not contain any system-dependent parameter. It yields excellent results, in molecules and solids, for many different properties. In contrast to other methods, HSE can be employed for both structural and electronic properties HSE provides a unique and powerful alternative for the study of large complex systems, such as chemisorption at surfaces and three-dimensional impurities in semiconductors. These fields of HSE functional applications await future study. 

The success of the hybrid functionals HSE and TPSSh in predicting the peak position of optical transitions has been attributed to unknown error cancellations (Kiimmel and Kronik 2008 Spataru et al. 2008). However, excitonic effects in metallic nanotubes are up to two orders of magnitude smaller than in semiconducting tubes (Deslippe et al. 2007) and remarkably, the same hybrid functionals that are able to describe the optical peaks in semiconducting tubes also produce excellent results in metallic tubes (Barone et al. 2005a). 

As a first example of the predictive capabilities of these functionals, we show in O Fig. 24-7 calculated first-order optical transitions, En as a function of the corresponding experimental values in a set of five semiconducting and five metallic chiral nanotubes. AH non-hybrid functionals employed here (LDA, PBE, and TPSS) underestimate En in metallic tubes by approximately 0.3 eV. This error is comparable to the error for En in semiconducting tubes. The best overall performance is achieved by the hybrids TPSSh and HSE, which yield comparable first-order transitions in the case of metallic SWNTs. 

As a second example, we compare in O Table 24-2 first-order transitions calculated using the hybrid TPSSh and HSE functionals (Barone et al. 2005a), and calculations considering GW plus electron-hole interactions (GW + e-h) (Spataru et al. 2008), with experimental values. Here, it is worth to point out the results obtained with hybrid functionals, that predict peak positions in agreement with more complex quasiparticle and excitonic effects approaches. An explanation for this behavior has been recently presented by Brothers et al. (2008). 

Another way to deal with the Coulomb self-interaction error is to use a hybrid functional that combines exact Hartree-Fock exchange with standard LDA/GGA. Recently, the hybrid HSE functional has been reported to describe successfully the localization of a single 4f electron in Cc203 (Da Silva et al., 2007). Even though the hybrid functional approach in some cases exhibits better results than the DFT -F U approach, DFT -F U can stiU compete well in terms of computational cost. Therefore, all the reported results in this chapter were obtained using DFT -F U calculations. 

Theoretical and experimental studies reveal that the optical peaks of AGNRs might be utilized as tools to determine the nature of their edges (Barone et al. 2006 Pimenta et al. 2007). The first calculations of the optical spectrum of GNRs was presented by Barone et al. (2006) by means of DFT using the screened-exchange hybrid HSE functional. As expected from an inter-band transitions framework, first optical excitations present the corresponding oscillations as a function of the width. Second-order transitions also exhibit these oscillations, as shown in O Fig- 24-8. 

---