Hohenberg-Kohn-Sham theorem

Theory (DFT). The basic ideas of Density Functional Theory are contained in the two original papers of Hohenberg, Kohn and Sham, [22, 23] and are referred to as the Hohenberg-Kohn-Sham theorem. This theory has had a tremendous impact on realistic calculations of the properties of molecules and solids, and its applications to different problems continue to expand. A measure of its importance and success is that its main developer, W. Kohn (a theoretical physicist) shared the 1998 Nobel prize for Chemistry with J.A. Pople (a computational chemist). We will review here the essential ideas behind Density Functional Theory. 

The local-scaling transformation version of density functional theory (LS-DFT), [1-12] is a constructive approach to DFT which, in contradistinction to the usual Hohenberg-Kohn-Sham version of this theory (HKS-DFT) [13-18], is not based on the IIohenberg-Kohn theorem [13]. Moreover, in the context of LS-DFT it is possible to generate explicit energy density functionals that satisfy the variational principle [8-12]. This is achieved through the use of local-scaling transformations. The latter are coordinate transformations that can be expressed as functions of the one-particle density [19]. 

The easier way to solve the problem of correlation is by using Density Functional Theory (DFT). DFT is less expensive than any of the correlated methods, and it is much more precise in many situations. It is our only possibility to include electron-correlation of large systems. DFT is based on the first Hohenberg-Kohn (1964) theorem, which establishes that properties in the ground state are functionals of the electron density (12.1). In 1965, Kohn-Sham demonstrated that the electron density of a molecular system of interacting electrons can be represented with the electron density of an equivalent system of non-interactive electrons subjected to an effective potential. Exact functionals for exchange and correlation are unknown and, thus, approximations found in the literature are needed to perform calculations using DFT. 

The Hohenberg-Kohn and Kohn-Sham theorems simply state that the total energy can be obtained by applying the variation principle to the total energy density functional and suggest that the one-electron equations obtained in this way also account for electronic correlation. HKS derive another theorem that the total energy is uniquely determined by the density. 

The Kohn-Sham orbitals are found as follows. The Hohenberg-Kohn variational theorem tells us that we can find the ground-state energy by varying p (subject to the constraint f p dr = n) so as to minimize the functional E [p]. Equivalently, instead of varying p, we can vary the KS orbitals dP, which determine p by (16.45). (In doing so, we must constrain the dP s to be orthonormal, since orthonormality was assumed when we evaluated T. )... 

Two core elements of DFT are the Hohenberg-Kohn (HK) theorems [328,331] and the Kohn-Sham equations [332]. The former is mainly conceptual, but via the second the most common implementations of DFT have been done. 

The fact that an exact density functional exists is known from a theorem proved by Hohenberg and Sham and Kohn. However, this is a non-constructive proof since it does not actually give the form of the exact functional. DFT theorists must try to approximate this functional as well as they can. 

According to a theorem by Hohenberg, Kohn and Sham [4], the total energy E of an electron gas can be written as a functional of the electronic density n(r) in the following form ... 

So the highest occupied Kohn-Sham orbital has a fractional occupation number Hohenberg-Kohn theorem applied to the non-interacting system. The proof of... 

In fact, the true form of the exchange-correlation functional whose existence is guaranteed by the Hohenberg-Kohn theorem is simply not known. Fortunately, there is one case where this functional can be derived exactly the uniform electron gas. In this situation, the electron density is constant at all points in space that is, n(r) = constant. This situation may appear to be of limited value in any real material since it is variations in electron density that define chemical bonds and generally make materials interesting. But the uniform electron gas provides a practical way to actually use the Kohn-Sham equations. To do this, we set the exchange-correlation potential at each position to be the known exchange-correlation potential from the uniform electron gas at the electron density observed at that position ... 

Hohenberg-Kohn theorems, but use the Kohn-Sham construction and local approximations to such non-local potentials and often lump together the exchange and the correlation energies into an exchange-correlation energy Exc[n], This yields a local exchange-correlation potential vxc(t) in the Kohn-Sham equations that determine the Kohn-Sham spin orbitals j, i.e. 

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