Exchange-energy density functional: Reparametrization of Becke's formula and derivation of second-order gradient correction

Density functionals of chemical bonding

The behavior of electrons in general many-electronic systems throughout the density functionals of energy is reviewed. The basic physico-chemical concepts of density functional theory are employed to highlight the energy role in chemical structure while its extended influence in electronic localization function helps in chemical bonding understanding. In this context the energy functionals accompanied by electronic localization functions may provide a comprehensive description of the global-local levels electronic structures in general and of chemical bonds in special. Becke-Edgecombe and author's Markovian electronic localization functions are discussed at atomic, molecular and solid state levels. Then, the analytical survey of the main workable kinetic, exchange, and correlation density functionals within local and gradient density approximations is undertaken. The hierarchy of various energy functionals is formulated by employing both the parabolic and statistical correlation degree of them with the electronegativity and chemical hardness indices by means of quantitative structure-property relationship (QSPR) analysis for basic atomic and molecular systems.

The extended Perdew-Burke-Ernzerhof functional with improved accuracy for thermodynamic and electronic properties of molecular systems

Density functional theory (DFT) has become the method of choice for many applications of quantum mechanics to the study of the electronic properties of molecules and solids. Despite the enormous progress in improving the functionals, the current generation is inadequate for many important applications. As part of the quest of finding better functionals, we consider in this paper the Perdew-Burke-Ernzerhof (PBE) functional, which we believe to have the best theoretical foundation, but which leads to unacceptable errors in predicting thermochemical data (heats of formation) of molecular systems [mean absolute deviation (MAD)=16.9 kcal/mol against the extended G2 data set of 148 molecules]. Much improved thermochemistry is obtained with hybrid DFT methods that include part of the Hartree-Fock exchange [thus B3LYP (Becke's three parameter scheme combining Hartree-Fock exchange, Becke gradient corrected exchange functional and Lee-Yang-Parr correlational functional) with MAD=3.1 kcal/mol and PBE0 (Perdew's hybrid scheme using PBE exchange and correlation functionals) with MAD=4.8 kcal/mol]. However we wish to continue the quest for a pure density-based DFT. Thus we optimized the four free parameters (micro, kappa, alpha, and beta) in PBE theory against experimental atomic data and the van der Waals interaction properties of Ne(2), leading to the xPBE extended functional, which significantly outperforms PBE for thermochemical properties MAD reduced to 8.0 kcal/mol while being competitive or better than PBE for predictions of geometric parameters, ionization potentials, electron affinities, and proton affinities and for the description of van der Waals and hydrogen bond interactions. Thus xPBE significantly enlarges the field of applications available for pure DFT. The functional forms thus obtained for the exchange and correlational functionals may be useful for discovering new improved functionals or formalisms.