Hohenberg-Kohn properties

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. 

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 ... 

Secondly, information is obtained on the nature of the nuclei in the molecule from the cusp condition [11]. Thirdly, the Hohenberg-Kohn theorem points out that, besides determining the number of electrons, the density also determines the external potential that is present in the molecular Hamiltonian [15]. Once the number of electrons is known from Equation 16.1 and the external potential is determined by the electron density, the Hamiltonian is completely determined. Once the electronic Hamiltonian is determined, one can solve Schrodinger s equation for the wave function, subsequently determining all observable properties of the system. In fact, one can replace the whole set of molecular descriptors by the electron density, because, according to quantum mechanics, all information offered by these descriptors is also available from the electron density. 

According to the Hohenberg-Kohn theorem [48], the properties of a system of electrons and nuclei in its ground state are determined entirely by p(r). Thus the total energy, for example, is a functional of p(r), E = 9[p(r)]. 

Like the first Hohenberg-Kohn theorem, the preceding theorems are existence theorems they say that the shape function is enough but they do not provide any guidance for evaluating properties based on the shape function alone. Once one knows that shape functionals exist, however, there are systematic ways to construct them using, for example, the moment expansion technique [48-51]. For atomic... 

The DFT approach (for an excellent introduction, see Parr and Yang 1989) is different and somewhat simpler. The electron density p(r) has been recognized to be a feature that uniquely determines all properties of the electronic ground state (1st Hohenberg-Kohn theorem). Instead of minimizing E with respect to coefficients of the wave function as in HF, E is minimized with respect to the electron density p... 

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... 

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 first Hohenberg-Kohn theorem, then, says that any ground state property of a molecule is a functional of the ground state electron density function, e.g. for the energy... 

Density functional theory is based on the two Hohenberg-Kohn theorems, which state that the ground-state properties of an atom or molecule are determined by its electron density function, and that a trial electron density must give an energy greater than or equal to the true energy. Actually, the latter theorem is true only if the exact functional (see below) is used with the approximate functionals in use... 

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