Potentials asymptotic properties

Long-range asymptotic properties of KS orbitals were recalled in Sect. 4, as dictated by asymptotic properties of all potentials of KS equations. A new, simple and direct method was applied to obtain the asymptotic form of the exchange potential. For pure-state systems the known results were confirmed in Sects. 4.2 and 4.3, while for mixed-state systems a new, exact result was obtained in Sect. 5.6. 

In some cases when detailed ab initio calculations of potential surfaces have become available, they have confirmed the major qualitative features of the surfaces deduced from experimental data and from preliminary data on correlation diagrams and the asymptotic properties of reactant, product, and intermediate states. One such case is the C+-H2 system,451 which has aroused considerable interst. Angular and energy distributions were determined experimentally for the CH+ + H products from the reactions of the C+(2P) ground state and the C+(4P) excited state with H2 (Table I, Jones et al.9b and references cited therein, and Jones et al.326). Chemiluminescence was also observed from this hydrogen-atom transfer reaction (Table IV).442,443 In addition, the reaction... 

Rigorous properties of the optimized effective potential (OEP) are derived. We present a detailed analysis of the asymptotic form of the OEP, going beyond the leading term. Furthermore, the asymptotic properties of the approximate OEP scheme of Krieger, Li and Iafrate [Phys. Lett. A 146, 256 (1990)] are analysed, showing that the leading asymptotic behavior is preserved by this approximation. 

Asymptotic Properties of Radiae Distribution Functions and Potential Truncations... 

One clever approach to obtaining better convergence is to include asymptotic properties of the pair correlation functions (Lebowitz and Percus 1963). In particular, exact asymptotic expressions have been obtained by Attard and coworkers (Attard 1990 Attard et al. 1991), such as for dipolar fluids. Other work has extended simulation results for a system with a truncated potential to give those for the full potential (Lado 1964). The effects on pair distribution functions of potential truncations are important. 

The condition (118) is related to the asymptotic properties of the exact XC potential (for finite and neutral systems) ... 

Asymptotic properties of the EXX potential. Condition (119) have been shown by Della Sala and Gorling to be only valid only for spherical systems (e.g. atoms). For non-spherical systems (e.g. molecules) the investigation of the asymptotic properties of the exact XC potential on points T/ with a chosen direction f/, i.e., r/ = r/f/, is required, and it turned out that... 

The CEDA was originally derive for the exchange-only functional as the discussion about the asymptotic properties of the KLI exchange-only potential also holds for the CEDA potential, we obtain... 

Many important properties, such as critical dimensionality and critical exponents describing the divergence of correlation length and other quantities can thus be obtained from renormalization group analysis. It is worth noting that for X well below X only the quadratic terms of the potential contribute to the asymptotic properties of P, which reduces therefore to a multigaussian distribution in accordance with the central limit theorem [13]. There exists,however,a (frequently very narrow) vicinity of X ... 

This snbchapter presents the results of theoretical and experimental studies on the influence of solvent properties on the nature of the swelling kinetics for elastomeric polymer networks at finite strains of the polymer matrix. Emphasis is focused on the study of the relationship of the asymptotic properties of the swelling kinetic curves with the thermodynamic quality of a solvent. The chemical potential of a solvent in the sample under swelling is given by the following equation ... 

In this review we will give an overview of the properties (asymptotics, shell-structure, bond midpoint peaks) of exact Kohn-Sham potentials in atomic and molecular systems. Reproduction of these properties is a much more severe test for approximate density functionals than the reproduction of global quantities such as energies. Moreover, as the local properties of the exchange-correlation potential such as the atomic shell structure and the molecular bond midpoint peaks are closely related to the behavior of the exchange-correlation hole in these shell and bond midpoint regions, one might be able to construct... 

In this nonvariational approach for the first term represents the potential of the exchange-correlation hole which has long range — 1/r asymptotics. We recognize the previously introduced splitup into the screening and screening response part of Eq. (69). As discussed in the section on the atomic shell structure the correct properties of the atomic sheU structure in v arise from a steplike behavior of the functional derivative of the pair-correlation function. However the WDA pair-correlation function does not exhibit this step structure in atoms and decays too smoothly [94]. A related deficiency is that the intershell contributions to E c are overestimated. Both deficiencies arise from the fact that it is very difficult to represent the atomic shell structure in terms of the smooth function p. Substantial improvement can be obtained however from a WDA scheme dependent on atomic shell densities [92,93]. In this way the overestimated intershell contributions are much reduced. Although this orbital-depen-... 

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