Hohenberg and Kohn

Parr B 2000 webpage http //net.chem.unc.edu/facultv/rap/cfrap01. html Professor Parr was among the first to push the density functional theory of Hohenberg and Kohn to bring it into the mainstream of electronic structure theory. For a good overview, see the book ... 

Hohenberg and Kohn demonstrated that is determined entirely by the (is a functional of) the electron density. In practice, E is usually approximated as an integral involving only the spin densities and possibly their gradients ... 

Hohenberg and Kohn s 1964 paper was widely regarded by physicists, but its tme importance in chemistry has only become apparent during the last decade... 

The main problem relating to practical applications of the Hohenberg and Kohn theorems is obvious the theorems are existence theorems and do not give us any clues as to the calculation of the quantities involved. 

In Part 2 of their paper, Hohenberg and Kohn go on to investigate the form of the functional F[P(r)] in the special cases of certain limiting charge densities. They find that F[P(r)] can be expressed in terms of the correlation energy and electric polarizabilities. 

Instead of treating all electrons in the metal plus adsorbate system individually, one considers the electron density of the system. Hohenberg and Kohn (Kohn received the 1999 Nobel Prize in Chemistry for his work in this field) showed that the ground state Eq of a system is a unique functional of the electron density in its ground state Wq- Neglecting electron spin, the energy functional can be written as... 

Density-functional theory, developed 25 years ago (Hohenberg and Kohn, 1964 Kohn and Sham, 1965) has proven very successful for the study of a wide variety of problems in solid state physics (for a review, see Martin, 1985). Interactions (beyond the Hartree potential) between electrons are described with an exchange and correlation potential, which is expressed as a functional of the charge density. For practical purposes, this functional needs to be approximated. The local-density approximation (LDA), in which the exchange and correlation potential at a particular point is only a function of the charge density at that same point, has been extensively tested and found to provide a reliable description of a wide variety of solid-state properties. Choices of numerical cutoff parameters or integration schemes that have to be made at various points in the density-functional calculations are all amenable to explicit covergence tests. 

The electronic wave function of an n-electron molecule is defined in 3n-dimensional configuration space, consistent with any conceivable molecular geometry. If the only aim is to characterize a molecule of fixed Born-Oppenheimer geometry the amount of information contained in the molecular wave function is therefore quite excessive. It turns out that the three-dimensional electron density function contains adequate information to uniquely determine the ground-state electronic properties of the molecule, as first demonstrated by Hohenberg and Kohn [104]. The approach is equivalent to the Thomas-Fermi model of an atom applied to molecules. 

Beginning way back in the 20s, Thomas and Fermi had put forward a theory using just the diagonal element of the first-order density matrix, the electron density itself. This so-called statistical theory totally failed for chemistry because it could not account for the existence of molecules. Nevertheless, in 1968, after years of doing wonders with various free-electron-like descriptions of molecular electron distributions, the physicist John Platt wrote [2] We must find an equation for, or a way of computing directly, total electron density. [This was very soon after Hohenberg and Kohn, but Platt certainly was not aware of HK by that time he had left physics.]... 

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 ... 

The original premise of Hohenberg and Kohn has led to new methods of calculating energies, very useful for large systems [1], We shall not go into this important area, but instead concentrate on another aspect of DFT the new chemical concepts that have arisen, mainly from the work of Parr and his coworkers [2],... 

The advent of density functional theory (DFT) [1,2] has had a profound impact on quantum and computational chemistry. The ingenious proof, given in 1964 by Hohenberg and Kohn [1], that the wave function of a many-electron system... 

It is clear that the functional F[n], which plays such a prominent role in the theory of Hohenberg and Kohn, must be a quantity of unusual complexity. It is... 

As will be developed in more detail below, the paper by Hohenberg and Kohn (1964) [7], which proved the existence theorem that the ground state energy is a functional of n(r), but now without the approximations (valid for large N) in the explicit energy functional (1), formally completed the TFD theory. The work of Kohn and Sham (1965) [8] similarly gave the formal completion of Slater s 1951 proposal. 

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