Correlation energy density limit

Notice that this expression for the correlation energy density functional is obtainable only in the high- and low-density limits, and one must... 

Like the electron density, it has to be computed only once, for one value of F. The correlation energy density has no exact scaling equality [25] and must be computed separately for each F. In the low-density limit (k 0), correlation scales like exchange, but in the high-density limit (k oo) correlation varies much more weakly with k. 

In Part 2 of their paper, Hohenberg and Kohn go on to investigate the form of the functional F[P(r)] in the special cases of certain limiting charge densities. They find that F[P(r)] can be expressed in terms of the correlation energy and electric polarizabilities. 

The correlation energy of a uniform electron gas has been determined by Monte Carlo methods for a number of different densities. In order to use these results in DFT calculations, it is desirable to have a suitable analytic interpolation formula. This has been constructed by Vosko, Wilk and Nusair (VWN) and is in general considered to be a very accurate fit. It interpolates between die unpolarized ( = 0) and spin polarized (C = 1) limits by the following functional. 

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

In the low-density limit (r, — ), correlation and exchange are of comparable strength, and are together independent of exc is then nearly equal to the electrostatic energy per electron of the Wigner crystal [33-36] ... 

The correlation energy is known analytically in the high-and low-density limits. For typical valence electron densities (1 < r, < 10) and lower densities (r, > 10), it is known numerically from release-node Diffusion Monte Carlo studies [33]. Various parametrizations have been developed to interpolate between the known limits while fitting the Monte Carlo data. The first, simplest and most transparent is that of Perdew and Zunger (PZ) [34] ... 

Spin Resolution of the Correlation Energy in the Uniform Density Limit... 

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). 

The exact high- and low-density limits can be found from arguments given in Ref. [57]. For = 0 in the high-density limit (where the random phase approximation becomes exact), the parallel-spin and anti-parallel-spin correlation energies are equal [57], so... 

Eq. (23) is at Erst surprising, since it implies that the parallel-spin correlation energy for = 0 is slightly positive [-0.02ec(/ = 0)] in the low-density limit. But this cannot be ruled out, since the total correlation energy is of course properly negative. Eqs. (22) and (23) are at least consistent with the increase with r, of the GSB ratio - 0)/e(r, = 0), as shown in Table... 

Because of these difficulties, great interest arose in the last decade in methods free of such limitations, based on the density functional theory (DFT). The DFT equations contain terms that evaluate—already at the SCF level—a significant amount (ca. 70%) of the correlation energy. On the other hand, very accurate DFT methods require calculation of much fewer integrals (n ) than ab initio, which is why they have been widely used in theoretical studies of large systems. The DFT [2] is based upon Hohenberg-Kohn (HK) theorems, which legitimize the use of electron density as a basis variable [22]. 

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). 

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