Hohenberg-Kohn variational theorem

The Kohn-Sham orbitals are found as follows. The Hohenberg-Kohn variational theorem tells us that we can find the ground-state energy by varying p (subject to the constraint f p dr = n) so as to minimize the functional E [p]. Equivalently, instead of varying p, we can vary the KS orbitals dP, which determine p by (16.45). (In doing so, we must constrain the dP s to be orthonormal, since orthonormality was assumed when we evaluated T. )... 

For the GS, the two Hohenberg-Kohn (HK) theorems legitimize the density p(r) (a function of only 3 coordinates) as the basie variational variable henee, all terms in the GS eleetronie energy of a quantum system are frmetionals of the density ... 

This theorem means that the ground state electron density, as obtained from the Hohenberg-Kohn variational principle, uniquely determines the ground state properties of the system of interest. The electron density is obtained from the variational principle... 

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 Second Hohenberg-Kohn Theorem Variational Principle... 

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)]. 

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). 

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 ... 

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 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... 

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