Hohenberg-Kohn theorem ground-state electron density

As a consequence of the Hohenberg-Kohn theorem [14], a non-degenerate ground state electron density p(r) determines the Hamiltonian H of the system within an additive constant, implying that the electron density p(r) also determines all ground state and all excited state properties of the system. 

According to the Hohenberg-Kohn theorem of the density functional theory, the ground-state electron density determines all molecular properties. E. Bright Wilson [46] noticed that Kato s theorem [47,48] provides an explicit procedure for constructing the Hamiltonian of a Coulomb system from the electron density ... 

Density functional theory (DFT) uses the electron density p(r) as the basic source of information of an atomic or molecular system instead of the many-electron wave function T [1-7]. The theory is based on the Hohenberg-Kohn theorems, which establish the one-to-one correspondence between the ground state electron density of the system and the external potential v(r) (for an isolated system, this is the potential due to the nuclei) [6]. The electron density uniquely determines the number of electrons N of the system [6]. These theorems also provide a variational principle, stating that the exact ground state electron density minimizes the exact energy functional F[p(r)]. 

Density functional theory is originally based on the Hohenberg-Kohn theorem [105, 106]. In the case of a many-electron system, the Hohenberg-Kohn theorem establishes that the ground-state electronic density p(r), instead of the potential v(r), can be used as the fundamental variable to describe the physical properties of the system. In the case of a Hamiltonian given by... 

Density functional theory purists are apt to argue that the Hohenberg-Kohn theorem [1] ensures that the ground-state electron density p(r) determines all the properties of the ground state. In particular, the electron momenmm density n( ) is determined by the electron density. Although this is true in principle, there is no known direct route from p to IT. Thus, in practice, the electron density and momentum density offer complementary approaches to a qualitative understanding of electronic structure. 

The spirit of the Hohenberg-Kohn theorem is that the inverse statement is also true The external potential v(r) is uniquely determined by the ground-state electron density distribution, n(r). In other words, for two different external potentials vi(r) and V2(r) (except a trivial overall constant), the electron density distributions ni(r) and 2(r) must not be equal. Consequently, all aspects of the electronic structure of the system are functionals of n(r), that is, completely determined by the function (r). 

The so-called Hohenberg-Kohn theorem states that the ground-state electron density p(r) describing an N-electron system uniquely determines the potential V(r) in the Hamiltonian... 

The first Hohenberg-Kohn theorem, then, says that any ground state property of a molecule is a functional of the ground state electron density function, e.g. for the energy... 

Density functional theory is based on the two Hohenberg-Kohn theorems, which state that the ground-state properties of an atom or molecule are determined by its electron density function, and that a trial electron density must give an energy greater than or equal to the true energy. Actually, the latter theorem is true only if the exact functional (see below) is used with the approximate functionals in use... 

Density functional theory (DFT) is based on the Hohenberg-Kohn theorem, which states that there is a functional which gives the exact ground-state energy for the exact electron density. DFT models have become very popular because they are not more costly than Hartree-Fock models. The reason is that in the HF, Cl and MP models, a wavefunction for an N-electron system depends on 3N coordinates, whereas in the DFT approach, the electron density depends on only three coordinates, irrespective of the number of electrons. The problem is that the exact functional would be the Schrodinger equation itself Several approximate functionals have been developed by many authors (Becke, Parr, Perdew, and others) and different forms of the functional can yield slightly different results. Some of the most common DFT models are ... 

The density functional theory of Hohenberg, Kohn and Sham [173,205] has become the standard formalism for first-principles calculations of the electronic structure of extended systems. Kohn and Sham postulate a model state described by a singledeterminant wave function whose electronic density function is identical to the ground-state density of an interacting /V-clcctron system. DFT theory is based on Hohenberg-Kohn theorems, which show that the external potential function v(r) of an //-electron system is determined by its ground-state electron density. The theory can be extended to nonzero temperatures by considering a statistical electron density defined by Fermi-Dirac occupation numbers [241], The theory is also easily extended to the spin-indexed density characteristic of UHF theory and of the two-fluid model of spin-polarized metals [414],... 

The strength of Density Functional Theory (DFT) [120-122] is based on the fact that the electronic wavefunction Fo (ri,..., r r) of the electronic ground state of the system can be entirely described only by its electron density n°(r), as stated by the Hohenberg-Kohn theorem [123]. It is based on a minimization principle stating that the ground state electronic density minimizes an energy functional. Thus in principle we have to consider and manipulate a much simpler object, the electronic density, which is simply a function of R. ... 

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

The sweeping theorem of Hohenberg and Kohn is that, like the wave function, the ground state s electron density determines all the properties of an electronic system [1]. The result is proved in three steps. First, one recalls that the number of electrons is determined from the electron density using Eq. (14). Next, one demonstrates that the external potential can be determined from the ground-state electron density. From N and v(r), we may determine the electronic Hamiltonian and solve Schro-dinger s equation for the wave function, subsequently determining all observable properties of the system. 

Evidently, molecular informatics strongly relies on electron density. The central role of electron density in molecular informatics underlines the importance of the Hohenberg-Kohn theorem, referred to above the nondegenerate ground-state electron density of a molecule determines the molecular energy and, through the Hamiltonian, all other molecular properties [2]. 

The interpretation of the fundamental results of molecular informatics is rather straightforward if one considers the actual material making up the molecules. The atomic nuclei and the fuzzy, boundary-less electron density cloud are the only physical entities contained in molecules, where the electron density contains all information about the location and nature of the nuclei. Consequently, simply on information-theoretical grounds, the result of the Hohenberg-Kohn theorem [2], stating that the... 

Based on the tools employed in the Hohenberg-Kohn theorem, also in part on the result of Riess and Miinch, and on a four-dimensional version of the Alexandrov one-point compactification method of topology applied to the complete three-dimensional electron density, it was possible to prove recently that for nondegenerate ground-state electron densities, the Holographic Electron Density Theorem applies any nonzero volume part of the nondegenerate ground-state electron density cloud contains all information about the molecule [4,5]. 

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. 

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. 

These two auxiliary results, in fact, state all of the conclusions of the original Hohenberg-Kohn theorem the ground-state energy of a molecule, as well as the ground-state wavefunction T consequently, the expectation values of all spin-free observables are unique functionals of the ground-state electron density p(r). 

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