Global hybrids

In (1.1), Ef is an approximate correlation density-functional and ao is a parameter, which can be determined by fitting some reference datasets or, possibly, by theoretieal considerations. The functionals described by (1.1) are the so-called one-parameter global hybrids, which are probably best represented by the PBEO functional [12, 13]. 

An even more general expression could be obtained if the HE exchange is replaced by a global hybrid in the long-range contribution. However, the resulting three-parameter hybrid would not have the correct asymptotic behavior. For this reason, we have not considered this latter possibility in our work. 

Fig. 1.1 MAEs in the barrier heights of the DBH24 dataset (in kcal/mol) of the global hybrids based on the S-RC, PBE-TCA, PBE-RevTCA, and ModPBE-RevTCA functionals. The results are given in function of the mixing parameter oq. Full line S-RC. Long-dashed line PBE-TCA. Dotted line PBE-RevTCA. Short-dashed line ModPBE-RevTCA...

Table 1.1 MAEs (in kcal/mol) in atomization energies, barrier heights, and noncovalent binding energies of the global hybrids based on various local or semi-local functionals. The value of the mixing parameter ao for each case is also reported. The MAEs of the original local or semi-local functionals are given in parenthesis...

As in the case of the global hybrids, the final co values for the various functionals are determined by the need of finding a compromise between the value which optimize the barriers heights and the noncovalent binding energies on one hand, and the atomization energies on the other hand. Once more, only one functional (the one based on PBE-TCA, in the present case) is optimized by approximately the same value of co for the three dataset. In Table 1.2, the co values we have chosen are reported, as well as the resulting MAEs of the four functionals for the three datasets. 

Finally, we consider the range-separated global hybrids. In this case, there are two parameters to be optimized, ao and co. The results we have obtained by the hybrid based on the PBE-TCA functional are shown in Figs. 1.7, 1.8, and 1.9. In... 

Fig. 1.7 MAEs in the atomization energies of the G2 dataset (in kcal/mol) of the range-separated global hybrids based on the PBE-TCA functional. The results for various ao values are given in function of the range-separation parameter Full line, ao = 0.0. Long-dashed hne ao = 0.2. Dashed-dotted line, ao = 0.25.

Recently, Heyd et al. [262] used the screening of 1/m to adapt global hybrid functionals to calculations of periodic systems with small band gaps. Such an adaptation is necessary because direct evaluation of exact exchange in periodic systems with metallic character is prohibitively expensive, rendering conventional hybrids utterly impractical for many solids. Heyd et al. start with rewriting the one-parameter global hybrid functional of Eq. (187) as... 

An independent line of thought also revives our interest in energy densities. The standard global hybrid functionals [2] mix a large constant fraction of a semilocal (LDA, GGA, or meta-GGA) exchange energy with a complementary... 

This section explains how to use the LCOE to global hybrid functionals and assesses their performance regarding orbital energies. Numerical assessments on ionization potentials (IPs) and concluding remarks are also given. 

We describe two procedures of how to construct the OS global hybrid functionals using Eq. (14.61) the determination of OS parameters and estimation of orbital energies. 

Figures 14.1 illustrate the 01s and Sis orbital energies with respect to FON electrons. A similar tendency is observed for the Ols and Sis orbitals As the number of electrons increases, orbital energies of HFx -F DFTc functional decrease and those of the conventional DFT XC functionals increase. In contrast to HOMO, the dependences of DFTc functionals for HFx -F DFTc are smaller than those for the conventional XC functionals. The curves of the OS global hybrid functionals with appropriate a,- determined through the LCOE are approximately flat for Ols and Sis. The OS parameters aois, sis are approximately 0.57,0.71, which are slightly larger than those of CV-B3LYP, determined by numerical assessment for core excitations [50-52].

In order to assess the performance of the OS global hybrid functionals from a different point of view, we also compare the orbital energies and IPs of valence and core orbitals for OCS molecule in a sense of Koopmans theorem. IPs obtained by the OS hybrid functionals are shown in Table 14.16. The deviations from experimental IPs [53] and values of a, are shown in parentheses and square brackets, respectively. For HOMO, the OS global hybrid functionals provide comparatively similar IPs 11.45,10.99, 11.18, and 11.17 eV for SVWN5, BLYP, PBE, and TPSS functionals, and the corresponding deviations are at most 0.25 eV. The OS hybrid functionals also reproduce Ols and Sis IPs within the deviation of 2.5 eV for the LDA, GGA, and meta-GGA functionals, though the accurate estimation of large IPs is rather difficult. 

Table 14.16 OCS IPs (eV) by OS global hybrid functionals of SVWN5, BLYP, PBE, and TPSS functionals and Oj determined for the OS in eV...

The above assessment reveals that the LCOE improves FON dependence and estimation of IPs significantly for all global hybrid functionals, which bases S VWN5, BLYP, PBE, and TPSS XC functionals and an added HFx term. Finally, let us compare the results of the OS functional based on LC-BLYP. For core orbitals, the global hybrid-based OS functionals basically perform slightly better than the OS functional of LC-BLYP does, although the obtained a, values are relatively different. For valence orbitals, all OS functionals provide MAEs less than 0.5 eV. The MAE of the conventional LC-BLYP is the smallest among all functionals, which is consistent with the previous reports [100, 101]. The overall MAEs of the OS functional of LC-BLYP are comparable to those of the LDA, GGA, and meta-GGA functionals. 

We have constructed and assessed the OS global hybrid functionals satisfying the LCOE for core and valence orbitals. As was reported for LC hybrid functionals [53], the LCOE drastically reduces the FON dependence and enables accurate estimates... 

The valence s OS HFx portions obtained for global hybrid functionals are significantly larger than those for LC hybrid functionals [53], although the core s ones are similar to those of LC hybrid functionals. The effect of HFx has been discussed theoretically and numerically from various points of view. 

For these complexes, we examined the following exchange-correlation (XC) functionals (i) GGA functional, BLYP [38, 39] (ii) four global hybrid functionals,... 

Mn(H20)6] , but we omitted anisotropic values because the anisotropic values are not observed with the EPR experiment for solutions [53]. For these complexes, the dependence of all H °values on XC functionals is similar all DFT and hybrid DFT underestimate values, while HF overestimates the 1 ° value. As for global hybrid DFT such as B3LYP and TPSSh, the results we obtained are similar to those showed in previous studies [21, 22], Even if we employed any range-separated... 

Meta-GGA functionals incorporate the Laplacian of the electron density and generally depend on the electron kinetic energy density. These features allow a systematic improvement of results for many quantum chemical calculations. We have used the exchange functional BR89 (Becke-Roussel 1989 represented in analytic form) [87,90] and the correlation functional B94 (Becke 1994) [90, 91]. Minnesota functionals tested are M05 [92], M05-2X [93], M06-L [94], M06-HF [95], M06 [96], M06-2X [96], and Ml 1 [97]. Another global hybrid studied is BMK [98]. A hybrid extension of the nonempirical exchange-correlation TPSS (Tao, Perdew, Staroverov, and Scuseria) [99] and functional TPSSh [100] is also examined. 

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