Hohenberg-Kohn existence theorem

The first step of any DFT is the proof of a Hohenberg-Kohn type theorem [6]. In its traditional form, this theorem demonstrates that there exists a one-to-one correspondence between the external potential and the (one-body) density. The first implication is clear With the external potential it is alwaj possible (in principle) to solve the many-body Schrodinger equation to obtain the many-body wave-function. From the wave-function we can trivially obtain the density. The second implication, i.e. that the knowledge of the density is sufficient to obtain the external potential, is much harder to prove. In their seminal paper, Hohenberg and Kohn used the variational principle to obtain a proof by reductio ad ahsurdum. Unfortunately, their method cannot be easily generalized to arbitrary DFTs. The Hohenberg-Kohn theorem is a very strong statement From the density, a simple property of the quantum mechanical system, it is possible to obtain the external potential and therefore the many-body wave-function. The wave-function, in turn, determines every observable of the system. This implies that every observable can be written as a junctional of the density. 

The main problem relating to practical applications of the Hohenberg and Kohn theorems is obvious the theorems are existence theorems and do not give us any clues as to the calculation of the quantities involved. 

The First Hohenberg-Kohn Theorem Proof of Existence... 

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,... 

Like the first Hohenberg-Kohn theorem, the preceding theorems are existence theorems they say that the shape function is enough but they do not provide any guidance for evaluating properties based on the shape function alone. Once one knows that shape functionals exist, however, there are systematic ways to construct them using, for example, the moment expansion technique [48-51]. For atomic... 

As will be developed in more detail below, the paper by Hohenberg and Kohn (1964) [7], which proved the existence theorem that the ground state energy is a functional of n(r), but now without the approximations (valid for large N) in the explicit energy functional (1), formally completed the TFD theory. The work of Kohn and Sham (1965) [8] similarly gave the formal completion of Slater s 1951 proposal. 

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. 

The first theorem of Hohenberg and Kohn is an existence theorem. As such, it is provocative with potential, but altogether unhelpful in providing any indication of how to predict the density of a system. Just as with MO theory, we need a means to optimize our fundamental quantity. Hohenberg and Kohn showed in a second theorem that, also just as with MO theory, the density obeys a variational principle. 

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... 

---