PBE functional

The second class of GGA exchange functionals use for F a rational function of the reduced density gradient. Prominent representatives are the early functionals by Becke, 1986 (B86) and Perdew, 1986 (P), the functional by Lacks and Gordon, 1993 (LG) or the recent implementation of Perdew, Burke, and Emzerhof, 1996 (PBE). As an example, we explicitly write down F of Perdew s 1986 exchange functional, which, just as for the more recent PBE functional, is free of semiempirical parameters ... 

Slightly better results have been obtained, however, with the recently developed mPW and PBE functionals as can be seen from the data compiled in Table 13-5. The improvements do not exceed 2 kcal/mol though. It is interesting to note that the application of smaller (6-31+G(d)) or larger basis sets (6-311++G(3df,3pd)) does not change the picture much, in contrast to the strong influence of the basis set quality we have noted before. 

We now compare the PM3-D method with previous uncorrected DFT calculations on the S22 complexes [130], For the dispersion-bonded complexes the errors in the interaction distances for the PBE, B3LYP and TPSS functionals are reported to be 0.63, 1.16 and 0.69 A which are reduced to 0.17, 0.00 and 0.02 A when appropriate dispersive corrections are included. We see in Table 5-9 that the PM3-D method is capable of predicting the structures of dispersion-bonded complexes with greater accuracy than some uncorrected DFT functionals and with an accuracy comparable to that for the dispersion corrected PBE functional [130],... 

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. 

Nonempirical GGA functionals satisfy the uniform density limit. In addition, they satisfy several known, exact properties of the exchange-correlation hole. Two widely used nonempirical functionals that satisfy these properties are the Perdew-Wang 91 (PW91) functional and the Perdew-Burke-Ernzerhof (PBE) functional. Because GGA functionals include more physical ingredients than the LDA functional, it is often assumed that nonempirical GGA functionals should be more accurate than the LDA. This is quite often true, but there are exceptions. One example is in the calculation of the surface energy of transition metals and oxides. 

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. 

PBEIPBE One-parameter hybrid PBE functional incorporating 25% HF exchange (sometimes alternatively called PBEO, PBEOPBE, or PBEl). Adamo, C., Cossi, M., and Barone, V. 1999. J. Mol. Struct. (Theochem), 493, 145. 

There is an interesting exception to the rule that pure functionals predict near-degenerate singlet and triplet states. The modified form of the pure PBE functional described by Hammer et al [14], called RPBE, has been reported [35] to predict a triplet-singlet energy difference for Fe(CO)4 of 4.0 kcal/mol, which is probably much closer to the correct value than that computed with any other pure functional. Whilst the RPBE functional has been very enthusiastically adopted by the periodic DFT community for calculations on surfaces - the area for which it was developed - very few applications have been reported for molecular TM complexes, so it is not yet clear whether the good result for Fe(CO)4 will be transposable to other systems. 

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. 

The mPBE functional39 is a modification of the nonempirical PBE functional into which one empirical parameter was reintroduced. Compared to PBE, mPBE leads to significantly worse interaction energy for the water dimer, whereas the interaction energies in the cases of hydrogen fluoride and hydrogen chloride dimers are only slightly improved. 

Table 8.1 Convergence of the bulk energy for M0O3, with respect to the k-point grid using a planewave basis ( < , = 520eV) and PBE functional within the VASP code.

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. 

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 ... 

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. 

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. 

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