Universal functionals, density-functional

Part I of the paper develops an exact variational principle for the ground-state energy, in which the density (r) is the variable function (i.e. the one allowed to vary). The authors introduce a universal functional F[n(r)] which applies to all electronic systems in their ground states no matter what the external potential is. This functional is used to develop a variational principle. 

The existence of the first HK theorem is quite surprising since electron-electron repulsion is a two-electron phenomenon and the electron density depends only on one set of electronic coordinates. Unfortunately, the universal functional is unknown and a plethora of different forms have been suggested that have been inspired by model systems such as the uniform or weakly inhomogeneous electron gas, the helium atom, or simply in an ad hoc way. A recent review describes the major classes of presently used density functionals [10]. 

This, at first glance innocuous-looking functional FHK[p] is the holy grail of density functional theory. If it were known exactly we would have solved the Schrodinger equation, not approximately, but exactly. And, since it is a universal functional completely independent of the system at hand, it applies equally well to the hydrogen atom as to gigantic molecules such as, say, DNA FHK[p] contains the functional for the kinetic energy T[p] and that for the electron-electron interaction, Eee[p], The explicit form of both these functionals lies unfortunately completely in the dark. However, from the latter we can extract at least the classical Coulomb part J[p], since that is already well known (recall Section 2.3),... 

The universal function x(x) obtained by numerical integration and valid for all neutral atoms decreases monotonically. The electron density is similar for all atoms, except for a different length scale, which is determined by the quantity b and proportional to Z. The density is poorly determined at both small and large values of r. However, since most electrons in complex atoms are at intermediate distances from the nucleus the Thomas-Fermi model is useful for calculating quantities that depend on the average electron density, such as the total energy. The Thomas-Fermi model therefore cannot account for the periodic properties of atoms, but provides a good estimate of initial fields used in more elaborate calculations like those to be discussed in the next section. 

V. The second term is the classical Coulomb energy of a density distribution p. The quantity F p] is a universal functional of the density, which means that it is uniquely specified by the density p of the interacting electrons and does not depend on the particular external potential V acting on the electrons. The functional F contains whatever is necessary to make the energy in Eq. (6) equal to the expected value in Eq. (2). 

Two approaches to the excited-state problem have been the focus of this chapter. The nonvariational one, based on Kato s theorem, is pleasing in that it does not require a bifunctional, but it presumes that the excited-state density is known. On the other hand, the bifunctional approach is appealing in that it actually generates the desired excited-state density, which results in the generation of more known constraints on the universal functional for approximation purposes. 

If the external magnetic field B(r), and m(r) have only a nonvanishing Z-component, B(r) = (0,0, B(r)) and m(r) = (0,0, m(r)), the universal functional F[p, m] may then be considered as a functional of the spin densities ps(r) and p(r), F[ps(r), p(r)], because the spin density is proportional to the z-component of the magnetization m(r) = p-bPsW P-b is the electron Bohr magneton. It is of worth mentioning that it is possible to define two spin densities that are the diagonal elements of the density matrix introduced by von Barth and Hedin [3]. These correspond to the spin-up (alpha) electrons density pT(r), and the spin-down (beta) electrons density p (r). In terms of these quantities, the electron and spin densities can be written as... 

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... 

After the discovery of the relativistic wave equation for the electron by Dirac in 1928, it seems that all the problems in condensed-matter physics become a matter of mathematics. However, the theoretical calculations for surfaces were not practical until the discovery of the density-functional formalism by Hohenberg and Kohn (1964). Although it is already simpler than the Hartree-Fock formalism, the form of the exchange and correlation interactions in it is still too complicated for practical problems. Kohn and Sham (1965) then proposed the local density approximation, which assumes that the exchange and correlation interaction at a point is a universal function of the total electron density at the same point, and uses a semiempirical analytical formula to represent such universal interactions. The resulting equations, the Kohn-Sham equations, are much easier to handle, especially by using modern computers. This method has been the standard approach for first-principles calculations for solid surfaces. 

The exchange-correlation energy v,c(r) in Eq. (4.35) is still a complicated matter. In the great majority of all practical calculations, it is assumed that Vxc(r) is a universal function of the charge density at a point r. 

Density functional approaches to molecular electronic structure rely on the existence theorem [10] of a universal functional of the electron density. Since this theorem does not provide any direction as to how such a functional should be constructed, the functionals in existence are obtained by relying on various physical models, such as the uniform electron gas and others. In particular, the construction of an exchange-correlation potential that depends on the electron density only locally seems impossible without some approximations. Such approximate exchange-correlation potentials have been derived and applied with some success for the description of molecular electronic ground states and their properties. However, there is no credible evidence that such simple constructions can lead to either systematic approximate treatments, or an exact description of molecular electronic properties. The exact functional that seems to... 

The Hohenberg-Kohn theory of /V-clcctron ground states is based on consideration of the spin-indexed density function. Much earlier in the development of quantum mechanics, Thomas-Fermi theory [402, 108] (TFT) was formulated as exactly such a density-dependent formalism, justified as a semiclassical statistical theory [231, 232], Since Hohenberg-Kohn theory establishes the existence of an exact universal functional Fs [p] for ground states, it apparently implies the existence of an exact ground-state Thomas-Fermi theory. The variational theory that might support such a conclusion is considered here. 

Given the formal similarity between the Hamiltonians defined in Eqs.(2) and (5), it follows that the ground-state energy, E, is given in terms of a universal functional of the pair (or n-particle) density, n(x), which attains its minimum value for the exact pair density. Furthermore, within a Kohn-Sham scheme, the form of this functional is identical to the functional of ordinary DFT but is given in terms of the correlated pair density. The details of this derivation... 

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