Kohn-Hohenberg theorem

In 1964, the concept of the Thomas-Fermi method was revived by a theorem called the Hohenberg-Kohn theorem (Hohenberg and Kohn 1964). This theorem consists of the following two subsidiary theorems for nondegenerate ground electronic states  

External potentials, which correspond to the nuclear-electron interaction 

The energy variational principle is always established for any electron density. 

Since these theorems were also proven mathematically, establishing the validity of the concept (Kutzelnigg 2006), they can be interpreted as the basic theorems of a quantum theory based on electron density. 

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The first Hohenberg-Kohn theorem states that, for a nondegenerate ground state, there is a one-to-one mapping among p. V. and iq... 

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

The main difficulty with DFT is that the Hohenberg-Kohn theorem shows that the ground-state values of T,, V, etc. are all unique functionals of the ground-state p (i.e.. 

The First Hohenberg-Kohn Theorem Proof of Existence... 

The Second Hohenberg-Kohn Theorem Variational Principle... 

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. 

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. 

The original Hohenberg-Kohn theorem was directly applicable to complete systems [14], The first adaptation of the Hohenberg-Kohn theorem to a part of a system involved special conditions the subsystem considered was a part of a finite and bounded entity regarded as a hypothetical system [21], The boundedness condition, in fact, the presence of a boundary beyond which the hypothetical system did not extend, was a feature not fully compatible with quantum mechanics, where no such boundaries can exist for any system of electron density, such as a molecular electron density. As a consequence of the Heisenberg uncertainty relation, molecular electron densities cannot have boundaries, and in a rigorous sense, no finite volume, however large, can contain a complete molecule. 

According to Hohenberg-Kohn theorem, 8p(F) given in Eq (36) does never vanishes because pA(r) and pY(r) are determined by different external potentials [26], Moreover, 8p(r) represents the electronic polarization contribution due to the isoelectronic change under the influence of the external electrostatic field. 

The conclusion that it may be possible to formulate the quantum mechanics of many-electron systems solely in terms of the single-particle density was put on a firm foundation by the two Hohenberg-Kohn theorems (1964), which are stated below, without proof. 

Hohenberg-Kohn Theorems for the Ground State of a Many-Electron System [4,5,10-12]... 

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 ... 

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