Electron correlation Hohenberg-Kohn theorem

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 ... 

The third and final approach to the electron correlation problem included briefly here is density functional theory (DFT), a review of which has been given by Kohn in his Nobel lecture [38]. The Hohenberg-Kohn theorem [39] states tiiat there is a one-to-one mapping between the potential V r) in which the electrons in a molecule move, the associated electron density p(r), and the ground state wave function 4>o. A consequence of this is that given tiie density p r), the potential and wave function Pq are functionals of that density. An additional theorem provided by Kohn and Sham [40] states that it is possible to construct an auxiliary reference system of non-interacting... 

In Chap. 4, the Kohn-Sham equation, which is the fundamental equation of DFT, and the Kohn-Sham method using this equation are described for the basic formalisms and application methods. This chapter first introduces the Thomas-Fermi method, which is conceptually the first DFT method. Then, the Hohenberg-Kohn theorem, which is the fundamental theorem of the Kohn-Sham method, is clarified in terms of its basics, problems, and solutions, including the constrained-search method. The Kohn-Sham method and its expansion to more general cases are explained on the basis of this theorem. This chapter also reviews the constrained-search-based method of exchange-correlation potentials from electron densities and... 

We begin with the most important issue. It is assumed that in principle it is possible to obtain a full description of many-electron systems by DFT if the exact (yet unknown) density-dependent exchange-correlation functional is employed. This seemingly undisputed tenet, based on the Hohenberg-Kohn theorem, was recently called into question [106, 107]. In his works Kaplan shows that the conventional Kohn-Sham equations are invariant with the respect to the total spin,... 

The resulting single-particle eigenvalue equations are the Kohn-Sham equations. The Hohenberg-Kohn theorems ensure that the exchange-correlation energy in Eq. (9) is a functional of the electron density. 

From what has been said already with respect to the variational collapse and the minimax principle, it is clear from the beginning that the standard derivation of the Hohenberg-Kohn theorems [386], which are the fundamental theorems of nonrelativistic DFT and establish a variational principle, must be modified compared to nonrelativistic theory [383-385]. Also, we already know that the electron density is only the zeroth component of the 4-current, and we anticipate that the relativistic, i.e., the fundamental, version of DFT should rest on the 4-current and that different variants may be derived afterwards. The main issue of nonrelativistic DFT for practical applications is the choice of the exchange-correlation energy functional [387], an issue of equal importance in relativistic DFT [388,389] but beyond the scope of this book. 

While considering the observability models in chemistry, the density functional theory has at its center the electronic density p, which helps in determining the atomic or molecular energy as a density functional as well as its properties. One of these properties is that for a state density coherently formulated in relation to an external V potential (Hohenberg-Kohn Theorem), the following succession of identities that define the chemical potential as the virtual flow of the driven electrons and therefore correlated to the electronegativity itself, % actually validate the reality of chemical reactivity... 

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]. 

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