Hohenberg-Kohn principle

The idea of the Kohn-Sham method is best understood as follows. Consider a generalized Hamiltonian of Eq. (2) in which the term 4e is scaled by an electron-electron coupling constant A. We are interested in values of A between 0 and 1. Each value of A corresponds to a distinct universal functional of the density. In Levy s constraint search formulation [36] of the Hohenberg-Kohn principle, this is explicitly stated as... 

The density depends only on three spatial coordinates instead of 3N, reducing the complexity of the task enormously. The Hohenberg-Kohn principles prove that the electron density is the most central quantity determining the electronic interactions and forms the basis of an exact expression of the electronic ground state. 

The Hohenberg-Kohn principles provide the theoretical basis of Density Functional Theory, specifically that the total energy of a quantum mechanical system is determined by the electron density through the Kohn-Sham functional. In order to make use of this very important theoretical finding, Kohn-Sham equations are derived, and these can be used to determine the electronic ground state of atomic systems. 

The Second Hohenberg-Kohn Theorem Variational Principle... 

In this section we introduce a different way of looking at the variational search connected to the Hohenberg-Kohn treatment. Recall the variational principle, equation (1-13) as introduced in Chapter 1... 

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)]. 

But there is a more basic difficulty in the Hohenberg-Kohn formulation [19-21], which has to do with the fact that the functional iV-representability condition on the energy is not properly incorporated. This condition arises when the many-body problem is presented in terms of the reduced second-order density matrix in that case it takes the form of the JV-representability problem for the reduced 2-matrix [19, 22-24] (a problem that has not yet been solved). When this condition is not met, an energy functional is not in one-to-one correspondence with either the Schrodinger equation or its equivalent variational principle therefore, it can lead to energy values lower than the experimental ones. 

P. W. Ayers, S. Golden, and M. Levy, Generalizations of the Hohenberg—Kohn theorem I. Legendre transform constructions of variational principles for density matrices and electron distribution functions. J. Chem. Phys. 124, 054101 (2006). 

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 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],... 

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 local-scaling transformation version of density functional theory (LS-DFT), [1-12] is a constructive approach to DFT which, in contradistinction to the usual Hohenberg-Kohn-Sham version of this theory (HKS-DFT) [13-18], is not based on the IIohenberg-Kohn theorem [13]. Moreover, in the context of LS-DFT it is possible to generate explicit energy density functionals that satisfy the variational principle [8-12]. This is achieved through the use of local-scaling transformations. The latter are coordinate transformations that can be expressed as functions of the one-particle density [19]. 

We begin with a survey of the hardness and softness quantities in the local resolution [3,21,22]. In this description, the equilibrium (ground-state) density satisfies the Hohenberg- Kohn (HK) variational principle ... 

We begin by giving here a generalized Hohenberg-Kohn theorem by giving the variational principles for equilibrium ensembles of quantum states. We consider a many-electron system with Hamiltonian... 

By functional we understand a function which depends on the form of another function—loosely, a function of a function. In the present context, a functional can be considered a recipe for extracting a single number from a function. For example, the variational principle involves a functional of the wavefunction, E = E -. A more elegant formulation of the first Hohenberg-Kohn theorem is the statement the wavefunction is a unique functional of the density. 

The second Hohenberg-Kohn theorem is a variational principle for the density functional, requiring that... 

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