For localized systems

While the supercell approach works well for localized systems, it is typically necessary to consider a very large supercell. This results in a plane-wave basis replicating not only the relevant electronic states but also vacuum regions imposed by the supercell. A much more efficient method to implement for investigating the electronic structures of localized systems is to use real space methods such as the recursion methods [27] and the moments methods [28], These methods do not require symmetry and their cost grows linearly with the number of inequivalent atoms being considered. For these reasons, real space methods are very useful for a description of the electronic properties of complex systems, for which the usual k-space methods are either inapplicable or extremely costly. 

Inutility of the Softnesses and Discrepancy Between the Ensemble and the Kohn-Sham Fukui Functions for Localized Systems. .. 154... 

We shall refer to Eq. (72) as the BP relation. BP presumed their proof to be valid for all systems. It does not hold for localized systems at T = 0 when p = INo+1 and N0 < Jf < N0 + 1 because p(r) is then indeterminate at fixed p and the definition of Eq. (66) loses its meaning. However, an alternative proof has been constructed for gapless systems using the GCE along the lines of the arguments outlined in Sect. 4 [42,44]. If one uses the definition of Eq. (67) instead of Eq. (66), one can show that the BP relation holds for all systems at T = 0, provided that... 

What one has done here is to recapture the BP relation for localized systems at the cost of avoiding a proper definition of charge-transfer chemical reactivities... 

The analysis of [3] which leads to Eq. (68a) is also valid for localized systems for which it yields the result... 

The electron-transfer reactivities are defined as derivatives of the electron-density p(r) with respect to total electron number Jf, Ufr), or chemical potential p, s r). The treatment of JT as a continuous variable [8-12] is justified by reference to the ensemble formulation of density-functional theory [8,18] and, in consequence, of the Kohn-Sham theory. We show in Sect. 4, in previously unpublished work [42], that this ensemble formulation yields either vanishing or infinite local and global softnesses for localized systems with... 

It was pointed out in [2,3] that nuclear-configuration changes define chemical reactions so that nuclear reactivities should be defined and set on equal footing with the corresponding electronic reactivities. Thus nuclear Fukui functions , Eq. (59), and nuclear softnesses a Eq. (60), were defined in [2], and explicit Kohn-Sham expressions were found for them, Eqs. (61H64), as reviewed in Sect. 5. These are electron-transfer reactivities and are valid only for extended systems, leaving open the question of nuclear electron-transfer reactivities for localized systems and nuclear isoelectronic reactivities for all systems. 

Atomic Radio Clocks [35] the IEEE 1588 standard is designed for local systems requiring better accuracy than that provided by NTP. It is also designed for use where the cost of a GPS receiver in each communicating component is too high, or for where GPS signals are not reliable (or accessible)... 

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