The PBE functional

The non-empirical GGA functional of Perdew, Buike and Emzerhof (PBE) [28] can be considered as the most promising non-empirical functional. In particular, it was constructed to respect a number of physical constraints both in the correlation and in the exchange parts. A detailed discussion of the physical background of the PBE functional is given in references 33 and 34. Here we just recall that it obeys the following six conditions  

In this functional the correlation part is similar to the Perdew-Wang (PW) correlation functional [36], while the exchange contribution is  

Mean absolute errors (mae s, kJ/mol) and mayimiim errors for atomization energies of the original G2 set (55 molecules). The values have been computed using the MP2/6-31G(d) geometries of reference 39 and the 6-311+G(2df,2pd) basis set. 

These results give a flavor of the performances of the different functionals with respect to this set of covalently bonded molecules, and can be considered as a starting point for a deeper discussion about chemical applications. From these data, it is quite apparent that the PBE functional performs as well as more empirical DFT approaches, like the BLYP model (Becke 88 exhange [14] and Lee-Yang-Parr correlation [19]). 

In table 2 we report the deviations for the geometrical parameters and harmonic vibrational fiequencies of 32 molecules belonging to the G2 set. Here, the deviations of PBE are close to those provided by the BLYP functional, thus giving further support to the reliability of this model. It is clear, anyway, that these results are stiU far from the accuracy required for chemical applications (e.g. about 5 kJ/mol for atomization energies). Furthermore, the PBE functional suffers from other problems. For instance, the energy barriers for proton transfer reactions [22], as well as some chemisorption energies [31] are still significantly underestimated. 

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Having these severe approximations in mind, SCC-DFTB performs surprisingly well for many systems of interest, as discussed above. However, it has a lower overall accuracy than DFT or post HF methods. Therefore, applying it to new classes of systems should be only done after careful examination of its performance. This can be done e.g. by conducting reference calculations on smaller model systems with DFT or ab initio methods. A second source of errors is related to some intrinsic problems with the GGA functionals also used in popular DFT methods (SCC-DFTB uses the PBE functional), which are inherited in SCC-DFTB. This concerns the well known GGA problems in describing van der Waals interactions [32], extended conjugate n systems [45,46] or charge transfer excited states [47, 48],... 

Chemical Potential and Hardness Values (in eV) Determined Using the Different Approaches with the PBE Functional and the aug-cc-pVTZ Basis Set, Compared to Experimental Values Determined Using the Data in Ref. [50]... 

Note All calculated quantities were obtained using the PBE functional. All quantities are in eV. a Calculated using the aug-cc-pVTZ basis set. b Calculated using the cc-pVTZ basis set. 

Another revision to the PBE functional has been dubbed revPBE see Y. K. Zhang and W. T. Yang, Comment on Gradient Approximation Made Simple, Phys. Rev. Lett. 80 (1998), 890, and the reply from the PBE authors J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 80 (1998), 891. 

Ernzerhof and Scuseria53 analyzed the electron affinities and ionization potentials obtained using the LDA-, PBE-, and PBE1PBE functionals with the reference date taken from the G2-1 data set.99 Whereas the PBE functional leads to significantly more accurate results than does LDA, no noticeable improvement occurs at the meta-GGA level. 

This functional which is elfectively the PBE functional with 25% of exact Hartree-Fock exchange included, performed better than GGA + U. Alkauskas and Pasquarello found that PBEO increased the calculated band gap of a-quartz to 8.3 eV close to the experimental value of 9 eV and a great improvement on the value of 5.8 eV obtained with PBE. Blaha s group showed that the hybrid functionals PBEO and B3PW91 gave comparable or superior results to LDA + U for the transition metal monoxides MnO, FeO and CoO. 

They called this functional RPBE. So, while the revPBE functional deviates form the PBE functional in the value of one parameter (k) in the exchange enhancement factor Fx(s), the RPBE functional deviates from the PBE functional in the form of the functional itself. It must be pointed out that RPBE preserves all the correct features of the parent PBE model. This functional provides very good chemisorption energies, but has not yet been tested on molecular systems. The behavior of the revPBE and RPBE functionals, with to respect the LO limit is shown in figure 1. 

We have recently shown that the numerical performances of some of these models are comparable to those of current 3-parameter hybrids like B3LYP [47,48]. In particular, we have obtained the PBEO model, casting the PBE functional in equation (15) [49]. This model provides very good results, both for the termochemistry of molecules belonging to the G2 set (see table 1) and for the corresponding geometric parameters (see table 2). 

Our set of molecules allows also some analysis of N absolute shieldings. In particular, it is remarkable that already the PBE functional gives results closer to experiment than the B3LYP values, and a further improvement is obtained when going to the PBEO model. In contrast, an excessive shielding is obtained by the MP2 method and just the opposite occurs at the B3LYP level. 

Liu et al. used GGA calculations with the PBE functional to probe adsorption and reaction of 02 and CO on Au supported on defect-free TiO2(110).185 These calculations used a bilayer of Au forming a continuous strip across the support surface. 02 adsorption was found to be favorable only for adsorption on Au adjacent to the support surface. An analysis of the bonding character of these states indicated that the support enhances charge transfer from Au to 02 for Au atoms in close proximity to the support. CO oxidation via a reaction between adsorbed CO and adsorbed 02 at the Au/support interface was found to have a small activation barrier (0.1 eV). CO oxidation pathways involving adsorbed atomic oxygen were not examined, although adsorbed O is created by the pathway mentioned above. Also, dissociation of adsorbed 02 was found to occur with a barrier of 0.5 eV. 

As the basis sets developed for glycine produced excellent results for fhat amino acid (see Table 5.1 of Ref. [31] for details), we chose to use the same basis also for fhe presenf calculations [68]. Preliminary calculations were also carried out at the level of time-dependent density functional theory (TD-DFT) using the PBE functional [56]. However, the results differed by less fhan 1% from fhe RPA result and are thus not reported here. 

The agreement between the calculated zero-field sphtting parameters using the PBE functional and the experimental values can be considered excellent taking into account the magnitudes involved. Only the case of the C04 complexes shows considerable deviations from the experimental values. 

Tao-Perdew-Staroverov-Scuseria) exchange-correlation functional, on the other hand, is a non-empirical version that represents a further development of the PKZB (Perdew-Kurth-Zupan-Blaha) functional, and can be considered as the next improvement over the PBE functional. ... 

The t-HCTH functional has been augmented with exact exchange to produce the acronym r-HCTH-hybrid. The PBE functional has also been improved by addition... 

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