Uniform density limit

Uniform Density Limit of Exchange-Correlation Energy Functionals... 

The uniform density limit for any of these functionals is easily evaluated Just set Vn = 0, = 0,... 

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... 

On the practical side, we note that nature provides a number of extended systems like solid metals [29, 30], metal clusters [31], and semiconductors [30, 32]. These systems have much in common with the uniform electron gas, and their ground-state properties (lattice constants [29, 30, 32], bulk moduli [29, 30, 32], cohesive energies [29], surface energies [30, 31], etc.) are typically described much better by functionals (including even LSD) which have the right uniform density limit than by those that do not. There is no sharp boundary between quantum chemistry and condensed matter physics. A good density functional should describe all the continuous gradations between localized and delocalized electron densities, and all the combinations of both (such as a molecule bound to a metal surface a situation important for catalysis). 

Recent work [16,19-21] suggests that functionals which respect the uniform density limit and other exact constraints can still achieve high accuracy for molecules. Just as the empirical electron-ion pseudopotentials of the 1960 s have been replaced by non-empirical ones, we expect tiiat the empirical density functionals of the 1990 s will be replaced by ones that are fully or largely non-empirical. 

What is Known About the Uniform Density Limit ... 

The uniform density limit has been well-studied by a combination of analytic and numerical methods. This section will review some (but not all) of what is known about it. 

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). 

A second observation about the LYP functional is that it predicts no correlation energy for a fully spin-polarized system of electrons. Yet, in the uniform-density limit, the correlation energy at full spin-polarization is about half that of the unpolarized system [3, 4, 57]. Even in the Ne atom, the parallel-spin contribution accounts for about 24% of the total correlation energy (Sect. 3.4). 

Nonempirical GGA functionals satisfy the uniform density limit. In addition, they satisfy several known, exact properties of the exchange-correlation hole. Two widely used nonempirical functionals that satisfy these properties are the Perdew-Wang 91 (PW91) functional and the Perdew-Burke-Ernzerhof (PBE) functional. Because GGA functionals include more physical ingredients than the LDA functional, it is often assumed that nonempirical GGA functionals should be more accurate than the LDA. This is quite often true, but there are exceptions. One example is in the calculation of the surface energy of transition metals and oxides. 

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... 

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