Uniform density limit approximation

We argue that the uniform density limit is an important theoretical constraint which should not be sacrificed in a functional that needs to be universal. The density functionals discussed here can be exact only for uniform densities. Approximations ought to be exact in those limits where they can be. Moreover, the unexpected success of LSD outside its formal domain of validity... 

The BLYP [6, 7] and B3LYP [18] functionals are widely and successfully used in quantum chemistry. But, as Tables I and II show, they fail seriously in the uniform density limit, where they underestimate the magnitude of the correlation for = 0 and even more for = 1 (where BLYP reduces to the exchange-only approximation). For uniform densities, B3LYP reduces to a peculiar combination of 81% LYP and 19% RPA (VWN). 

The correlation energy can in principle be resolved as a sum of contributions from tT> ii> ti correlations. Such a resolution even in the uniform density limit, is not really needed for the construction of density functional approximations, and no assumption about the spin resolution has been made in any of the functionals from our research group (which are all correct by construction in the uniform density limit). 

Langreth and Mehl [134] used the sharp cut-olf procedure in momentum space to eliminate spurious contributions to and an empirical exponential function to damp the gradient contribution to the energy. The Langreth-Mehl (LM) functional [134,160,161] has now mostly a historical significance. A few years later, Perdew [162] improved the LM functional by imposing two additional requirements that it recover the correct second-order DGE in the slowly varying density limit and reduce in the uniform density limit to LDA, not to the random-phase approximation (RPA), as the LM functional does. Perdew s 1986 correlation functional is... 

In contrast to the gradient expansion of the exchange hole of Perdew [135], the BR functional does not reduce to LSDA in the uniform density limit. To recover this limit approximately, Becke and Roussel multiplied the term t — t ) in Eq. (141) by an adjustment factor of 0.8. At the same time, the BR exchange energy density has the correct — p(r)/2r asymptotic behavior in the r — oo limit. 

These LDA correlation functionals have been used in various property calculations, especially in solid state calculations. However, we should recall that these LDA correlation functionals are not exact functionals but are inductively derived approximative functionals. Actually, even though the exact correlation energy of a uniform electron gas in a quanmm Monte Carlo calculation has (9(p) and density dependences at the high and low density limits, respectively... 

Several boundary conditions have been used to prescribe the outer limit of an individual rhizosphere, (/ = / /,). For low root densities, it has been assumed that each rhizosphere extends over an infinite volume of. soil in the model //, is. set sufficiently large that the soil concentration at r, is never altered by the activity in the rhizosphere. The majority of models assume that the outer limit is approximated by a fixed value that is calculated as a function of the maximum root density found in the simulation, under the assumption that the roots are uniformly distributed in the soil volume. Each root can then extract nutrients only from this finite. soil cylinder. Hoffland (31) recognized that the outer limit would vary as more roots were formed within the simulated soil volume and periodically recalculated / /, from the current root density. This recalculation thus resulted in existing roots having a reduced //,. New roots were assumed to be formed in soil with an initial solute concentration equal to the average concentration present in the cylindrical shells stripped away from the existing roots. The effective boundary equation for all such assumptions is the same ... 

If the main limitations of HF theory are overcome by the introduction of electron correlation, those of density functional theory are expanded by the use of more accurate functionals. These functionals, that improve the uniform gas description of the LDA approach, are labeled as non-local or Generalize Gradient Approximation (GGA). 

This approximation is equivalent to assuming that the differences in internal densities and, consequently, in solvent draining, between a branched chain and the homologous linear chain, when included in their corresponding mean sizes, can describe both the friction coefficient and the viscosity. Besides these theoretical considerations, an empirical correlation in terms of a log-log fit of h vs f was employed by Roovers et al. [51]. Kurata and Fukatsu [48] and Ptitsyn [82] performed a more general Kirwood evaluation of the friction coefficient for different types of ideal branched molecules (uniform and randomly distributed stars, combs and random-branched structures). Their results for different structures are included within the limits l[Pg.60]

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