Coulomb potential derivatives, first-order

The physical interpretation of the electron-interaction component Wge(r) was originally proposed by Harbola and Sahni [9], and derived by them via Coulomb s law. It is based on the observation that the pair-correlation density g(r,r ) is not a static but rather a dynamic charge distribution whose structure changes as a function of electron position. The dynamic nature of this charge then must be accounted for in the description of the potential. Thus, in order to obtain the local potential in which the electron moves, the force field due to this charge distribution must first be determined. According to Coulomb s law this field is... 

Coulomb-like potential. The zeroth-order regular approximation (ZORA) avoids this disadvantage by expanding in /(2c — V) up to the first order so that the ZORA Hamiltonian is variationally stable. The ZORA Hamiltonian was first derived by Chang et al. in 1986 [16], and later rediscovered as an approximation to the FW transformation by van Lenthe et al. [ 17-19]. The ZORA Hamiltonian of one electron in the external potential V is given by... 

On the other hand, the orbital-dependent treatment of correlation represents a much more serious challenge than that of exchange The systematic derivation of such functionals via standard many-body theory leads to rather complicated expressions. Their rigorous application within the OPM not only requires the evaluation of Coulomb matrix elements between the complete set of KS states, but, in principle, also relies on the knowledge of higher order response functions. In practical calculations, these first-principles functionals necessarily turn out to be rather inefficient, even if they are only treated perturbatively. In addition, the potential resulting from a large class of such functionals is non-physical for finite systems. Both problems are related to the presence of unoccupied states in the functionals which seems inevitable as soon as some variant of standard many-body theory is used for the derivation. 

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