Hybrid GGA

The classes of functionals shown in Fig. 10.2 do not exhaust the kinds of functionals that can be constructed. Any functional that incorporates exact exchange is called a hybrid functional. The hypcr-GGA functionals listed above can therefore also be referred to as hybrid-GGA methods. Hybrid-meta-GGA functionals also exist—these combine exact exchange with meta-GGA functionals this group of functionals is not shown in Fig. 10.2. 

B3LYP Three-parameter hybrid GGA functional with 20% HF exchange... 

Exchange-correlation functionals, which determine the reliability of Kohn-Sham calculations, are compared in terms of the basic concepts in their development, and for their features and problems, in Chap. 5. This chapter uses as examples the major local density approximation (LDA) and generalized gradient approximation (GGA) exchange-correlation functionals and meta-GGA, hybrid GGA, and semi-empirical functionals to enhance the degree of approximation in terms of their concepts, applicabilities, and problems. 

Among the hybrid GGAs, B3PW91 and PBEO have MUEs of 0.20 eV or less for CEBEs, but these two along with all the other hybrid GGAs have somewhat higher MUEs of 0.25-0.30 eV for the chemical shifts. 

Hybrid exchange-correlation functionals are widely used. A hybrid functional mixes together the formula (16.60) for with GGA (or meta-GGA) and Ec formulas and was first proposed by Becke. For example, the popular B3LYP (or BeckeSLYP) hybrid GGA functional (where the 3 indicates a three-parameter functional) is defined by... 

As an improvement on the B3LYP, B3PW91, and B1B95 hybrid functionals, Becke [A. D. Becke, J. Chem. Phys., 107,8554 (1997) H. L. Schmider and Becke, J. Chem. Phys., 108, 9624 (1998)] proposed the B97 hybrid GGA functional... 

GGA, meta-GGA, hybrid-GGA, and hybrid-meta-GGA functionals give not only good equilibrium geometries, vibrational frequencies, and dipole moments, but also generally accurate molecular atomization energies. Eor example, B3LYP/6-31 H-G(2d,p) calculations on the G2 data set gave an MAE of 3.1 kcal/mol (Foresman and Frisch, Chapter 7). 

Grimme proposed improving DFT energy results by using the MP2 second-order energy-correction formula, as follows [S. Grimme, J. Chem. Phys., 124,034108 (2006)]. One first defines a hybrid-GGA DFT functional of the form... 

Hujo W, Grimme S (2011) Performance of the van der Waals density functional WIO and (hybrid) GGA variants for thermochemistry and noncovalent interactions. J Chem Theory Comput 7 3866-3871... 

Fourth-rung functionals include Hartree-Fock exchange and are known as hybrid-GGAs. A number of hybrid functionals have been intentionally parameterized using datasets that... 

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