The Hohenberg-Kohn Existence Theorem

let us assume that two different external potentials can each be consistent with the same nondegenerate ground-state density po- We will call these two potentials Va and wj, and the different Hamiltonian operators in which they appear and Ht,. With each Hamiltonian will be associated a ground-state wave function Pq and its associated eigenvalue Eq. The variational theorem of molecular orbital theory dictates that the expectation value of the Hamiltonian a over the wave function b must be higher than the ground-state energy of a, i.e.. 

Since die potentials u are one-electron operators, the integral in the last line of Eq. (8.9) can be written in terms of the ground-state density 

As we have made no distinction between a and b, v/e can interchange the indices in Eq. (8.10) to anive at the equally valid 

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The original Hohenberg-Kohn theorem was directly applicable to complete systems [14], The first adaptation of the Hohenberg-Kohn theorem to a part of a system involved special conditions the subsystem considered was a part of a finite and bounded entity regarded as a hypothetical system [21], The boundedness condition, in fact, the presence of a boundary beyond which the hypothetical system did not extend, was a feature not fully compatible with quantum mechanics, where no such boundaries can exist for any system of electron density, such as a molecular electron density. As a consequence of the Heisenberg uncertainty relation, molecular electron densities cannot have boundaries, and in a rigorous sense, no finite volume, however large, can contain a complete molecule. 

The preceding theorem falls well short of the Hohenberg-Kohn theorem because it is restricted to Coulombic external potentials. The theorem is not true for all external potentials. In fact, for any Coulombic system, there always exists a one-electron system, with external potential,... 

Within the Hohenberg-Kohn approach [17, 18], the possibility of transforming density functional theory into a theory fully equivalent to the Schrodinger equation hinges on whether the elusive universal energy functional can ever be found. Unfortunately, the Hohenberg-Kohn theorem, being just an existence theorem, does not provide any indication of how one should proceed in order to find this functional. Moreover, the contention that such a functional should exist - and that it should be the same for systems that have neither the same number of particles nor the same symmetries (for an atom, for example, those symmetries are defined by U, L, S, and the parity operator ft) -certainly opens the door to dubious speculation. 

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 ... 

It should not surprise you that the LDA is not the only functional that has been tried within DFT calculations. The development of functionals that more faithfully represent nature remains one of the most important areas of active research in the quantum chemistry community. We promised at the beginning of the chapter to pose a problem that could win you the Nobel prize. Here it is Develop a functional that accurately represents nature s exact functional and implement it in a mathematical form that can be efficiently solved for large numbers of atoms. (This advice is a little like the Hohenberg-Kohn theorem—it tells you that something exists without providing any clues how to find it.)... 

The Hohenberg-Kohn theorem does not go beyond this point it offers no guidance on the nature of the functionals that it shows must exist. 

As already stated in the preceding section, the PB equation neglects ion size effects and interparticle correlations. One route to improve the theory can be done on a density functional level. The PB equation can be derived via a variational principle out of a local density functional [25, 31]. This is also a convenient formulation to overcome its major deficiencies, namely the neglect of ion size effects and interparticle correlations. The Hohenberg-Kohn theorem gives an existence proof of a density functional that will produce the correct density profile upon variation. However, it does not specify its... 

The Hohenberg-Kohn theorem [120-123] states that there exists a one to one map between external potentials and the ground state electronic density ... 

It is important to realize that in the proofs of the Hohenberg-Kohn theorem and the Holographic Electron Density Theorem, some very natural properties of molecular electron densities have been assumed. Two of these assumptions are i) the very existence of a ground-state electron density function and ii) the assumption of continuity of this function in the space variable r. 

The Hohenberg-Kohn theorem implies the existence of a functional Ftp], but says nothing about its form nor how it can be constructed. Assuming that the problems of representability have been solved, the following is a modified presentation of the Kohn-Sham (KS) procedure [19] developed to solve the problem of the Schrodinger equation in general, and to help find the universal functional Ftp] in particular. 

An overview of relativistic density functional theory (RDFT) is presented with special emphasis on its field theoretical foundations and the construction of relativistic density functionals. A summary of quantum electrodynamics (QED) for bound states provides the background for the discussion of the relativistic generalization of the Hohenberg-Kohn theorem and the effective single-particle equations of RDFT. In particular, the renormalization procedure of bound state QED is reviewed in some detail. Knowledge of this renormalization scheme is pertinent for a careful derivation of the RDFT concept which necessarily has to reflect all the features of QED, such as transverse and vacuum corrections. This aspect not only shows up in the existence proof of RDFT, but also leads to an extended form of the single-particle equations which includes radiative corrections. The need for renormalization is also evident in the construction of explicit functionals. 

The Hohenberg-Kohn theorem assures only that the functional F[p exists, but the actual form of F[p] is unknown (except for the term 7[p]) and must be approximated. Once the number of electrons N is fixed, Hamiltonian operators for any two systems differ only by the external potential v(r). The functional F[p] is therefore universal. 

According to the actual formulation of the Hohenberg-Kohn theorem 44], there exists a universal variational functional F(fi) of trial electron densities p, such that if o is the ground-state energy and Po(r) is the ground-state electron density that belongs to the given external potential V(r) specified in Eq. 15, then. 

Such methods owe their modern origins to the Hohenberg-Kohn theorem, published in 1964, which demonstrated the existence of a unique functional which determines the ground state energy and density exactly. The theorem does not provide the form of this functional, however. 

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