Density functional theory Hohenberg-Kohn theorems

Key words Density functional theory -Hohenberg-Kohn theorems... 

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 ... 

Consequently, from the density the Hamiltonian can be readily obtained, and then every property of the system can be determined by solving the Schrodinger equation to obtain the wave function. One has to emphasize, however, that this argument holds only for Coulomb systems. By contrast, the density functional theory formulated by Hohenberg and Kohn is valid for any external potential. Kato s theorem is valid not only for the ground state but also for the excited states. Consequently, if the density n, of the f-th excited state is known, the Hamiltonian H is also known in principle and its eigenvalue problem ... 

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)]. 

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]. 

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... 

The entire field of density functional theory rests on two fundamental mathematical theorems proved by Kohn and Hohenberg and the derivation of a... 

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. 

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