Asymptotic behavior function

However, it is known that the direct correlation functions have an exact long-range asymptotic form, arising due to intramolecular correlations in clusters formed via the association mechanism. This asymptotics is not included in the Percus-Yevick approximation. Other common liquid state approximations also do not provide correct asymptotic behavior of Ca ir). 

The functional equations (1 ), (4), (7), (8), (2.22), which have been established earlier and proved in the present paper, not only summarize the recursion formulas for the numbers R, S, Q, R but allow also general inferences (e.g.. Sec. 60), in particular on the asymptotic behavior (in Chapter 4). 

In Section II.D(4c), it was pointed out that, in treating correlation effects in a molecular system, it is of essential importance that the improved wave function leads to an energy curve having correct asymptotic behavior for separated atoms. It has been shown (Frost, Braunstein, and Schwemer 1948) that this condition may be fulfilled by a convenient choice of a correlation factor g. Let us consider the H2 molecule and a wave function of the type... 

Following the discussion in connection with the expansion III. 127, we note that, for a molecular or a solid-state system, the wave function III. 129 will lead to a correct asymptotic behavior of the energy for separated atoms, provided that the factor g has been conveniently chosen so that it increases indefinitely when any one of the electrons is taken away from the others. A more detailed study of g may sometimes be necessary in order to ensure that no excessive accumulation of ions will occur when the system is separated into its constituents. 

A common problem for both methods lies in the use of potentials that do not possess the correct net attractiveness. This can have the consequence that continuum feamres appear shifted in energy. In particular, there is evidence that the LB94 exchange-correlation potential currently used for the B-spline calculations, although possessing the correct asymptotic behavior for ion plus electron, is too attractive, and near threshold features can then disappear below the ionization threshold. An empirical correction can be made, offsetting the energy scale, but this can mean that dynamics within a few electronvolts of threshold get an inadequate description or are lost. There is limited scope to tune the Xa potential, principally by adjustment of the assumed a parameter, but for the B-spline method a preferable alternative for the future may well be use of the SAOP functional that also has correct asymptotic behavior, but appears to be better calibrated for such problems [79]. 

In some cases we may not be able to obtain estimates of all the parameters by examining the asymptotic behavior of the model. However, it can still be used to obtain estimates of some of the parameters or even establish an approximate relationship between some of the parameters as functions of the rest. 

Becke, A. D., 1988b, Density-Functional Exchange-Energy Approximation With Correct Asymptotic Behavior , Phys. Rev. A, 38, 3098. 

Chermette, H., Lembarki, A., Razafinjanahary, H., Rogemond, 1998, Gradient-Corrected Exchange Potential Functional With the Correct Asymptotic Behavior , Adv. Quant. Chem., 33, 105. 

Lembarki, A., F. Regemont, and H. Chermette. 1995. Gradient-corrected exchange potential with the correct asymptotic behavior and the corresponding exchange-energy functional obtained from virial theorem. Phys. Rev. A 52, 3704. 

Glossman, M. D., L. C. Baibas, A. Rubio, and I. A. Alonso. 1994. Nonlocal Exchange and Kinetic Energy Density Functionals with Correct Asymptotic Behavior for Electronic Systems. Int. I. Q. Chem. 49, 171. 

The curves 1 in Figs. 4.6a and b show the functions Fr and FA calculated by formulae (4.3.35) and (4.3.38) for the case of normal molecular orientations (e Oz) and plotted versus the argument AQ/( +AQ). The dimensionless argument and functions of this kind normalized with respect to the sum of the resonance and the band widths were introduced so as to depict their behavior in both limiting cases, ACl rj and Af2 77. The deviation of the solid lines from the dotted ones indicates to which degree the one-parameter approximation defined by Eq. (4.3.38) differs from the realistic dispersion law. As seen, this approximation shows excellent adequacy, but for the region AQ r/, where the asymptotic behavior of the approximation (4.3.38) and Eq. (4.3.35) are as follows ... 

Glossman, M. D., Baibas, L. C., Rubio, A. and Alonso, J. A. Nonlocal exchange and kinetic energy density functionals with correct asymptotic behavior for electronic systems, Int.J. Quantum ( hem., 49 (1994), 171-184... 

In the case of finite star chains with very high functionality, the units are concentrated near and in the star core. Therefore, their theoretical behavior can approximately be described by a rigid sphere [2]. The form factor of a sphere presents a series of oscillations. The experimental data of stars with 128 arms [67] show a smooth function covering the first two oscillations of the sphere, followed by a peak coincident with the third oscillation and the asymptotic behavior for high q previously described for stars of lower functionalities. It seems that the chain resembles a soft spherical core with a peripheral region of considerably smaller density. 

Levy identified the unknown part of the exact universal D functional as the correlation energy Ed D] and investigated a number of properties of c[ D], including scaling, bounds, convexity, and asymptotic behavior [11]. He suggested approximate explicit forms for Ec[ D] for computational purposes as well. Redondo presented a density-matrix formulation of several ab initio methods [26]. His generalization of the HK theorem followed closely Levy s... 

Since momentum densities are unfamiliar to many. Section II outlines the connection between the position and momentum space representations of wavefunctions and reduced-density matrices, and the connections among one-electron density matrices, densities, and other functions such as the reciprocal form factor. General properties of momentum densities, including symmetry, expansion methods, asymptotic behavior, and moments, are described in... 

However, one should ask whether the ansatz Eq. (23) is a valid one, and exactly how good is the TF approximation. It is certain that for systems other than the PEG, the idempotency property in Eq. (9) satisfied by any idempotent DM1 will no longer be true for Eq. (23). Hence, the TF functional is actually not an approximation for the Ts functional, the KS idempotent KEDF. Further, Eq. (23) has the wrong asymptotic behavior for isolated finite systems as both r and r become large, where the exact DM1 goes like the product of the highest occupied molecular orbital (HOMO) of Eq. (10) at two different points rand... 

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