Hohenberg-Kohn theorems theory

Let us recall that the Hohenberg-Kohn theorems allow us to construct a rigorous many-body theory using the electron density as the fundamental quantity. We showed in the previous chapter that in this framework the ground state energy of an atomic or molecular system can be written as... 

Gorling, A., 1999, Density-Functional Theory Beyond the Hohenberg-Kohn Theorem , Phys. Rev. A, 59, 3359. 

The electron density of a non-degenerate ground state system determines essentially all physical properties of the system. This statement of the Hohenberg-Kohn theorem of Density Functional Theory plays an exceptionally important role among all the fundamental relations of Molecular Physics. 

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 ... 

This chapter has dealt with introducing the main concepts within a theory called MQS. It has discussed the different steps to be taken to evaluate and quantify a degree of similarity between molecules in some molecular set but also fragments in molecules. QSM provides a scheme that relieves the arbitrariness of molecular similarity by using the electron density function as the sole descriptor, in agreement with the Hohenberg-Kohn theorems. It also addressed the different pitfalls that are present, for example the dependence on proper molecular alignment. 

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)]. 

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. 

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. 

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 Levy construction [222] can be used to prove Hohenberg-Kohn theorems for the ground state of any such theory. It should be noted that any explicit model of the Hohenberg-Kohn functional F[p] implies a corresponding orbital functional theory. The relevant density function p(r) is that constructed from an OFT ground state. This has the orbital decomposition , as postulated by Kohn and Sham [205]. Unlike the density p,, for an exact A-electron wave function T, which cannot be determined for most systems of interest, the OFT ground-state density function is constructed from explicit solutions of the orbital Euler-Lagrange equations, and the theory is self-contained. 

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