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 [Pg.529]

The ground-state electronic density p(r) is uniquely related to the external potential Vext(r) as stated by the fundamental theorems of DFT [1,2,8]. At zero field, the external potential of an atom is due to its nuclei and vext(r) = —Ze2/r where Z is the nuclear charge. It is shifted by the quantity V [Pg.335]

Since the ground-state electron density minimizes the energy, subject to the normalization constraint, Jp(r)dr — N = 0, theEuler-Lagrange equation (see Equation 4.23) becomes [Pg.48]

The non-degenerate ground state electron density p/(r) over any subset d of manifold S3, S3 zd d, where subset d has non-zero volume on S3, determines uniquely [Pg.67]

Since the complete ground state energy is a functional of the ground state electron density so must be its individual components and we can write (where we revert to the subscript Ne to specify the kind of external potential present in our case, which is fully defined by the attraction due to the nuclei) [Pg.51]

For a fixed external potential v(f), the ground-state electron density po satisfies the variational equation [Pg.161]

The so-called Hohenberg-Kohn theorem states that the ground-state electron density p(r) describing an N-electron system uniquely determines the potential V(r) in the Hamiltonian [Pg.374]

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 [Pg.122]

N. H. March, The local potential determining the square root of the ground-state electron-density of atoms and molecules from the schriklinger equation. Phys. Lett. A 113, 476 78 (1986). [Pg.481]

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. [Pg.67]

The important conclusion is that in principle, all ground-state molecular properties may be calculated from the ground-state electron density p(x, y, z). The challenge is to find the density and use it to calculate energies. A partial solution was found by Kohn and Sham [105]. [Pg.397]

This concludes the proof that there cannot be two different Vext that yield the same ground state electron density, or, in other words, that the ground state density uniquely specifies the external potential Vext. Using again the terminology of Section 1.2 we can simply add p0 as the property which contains the information about N, ZA, RA and summarize this as [Pg.51]

Wesolowski, T. A. and J. Weber. 1996. Kohn-Sham equations with constrained electron density an iterative evaluation of the ground-state electron density of interacting molecules. Chem. Phys. Lett. 248,71. [Pg.130]

Now, what if we abandon the orbital-by-orbital electron partitioning in favor of a description based on the stationary ground-state electron density p(r) Clearly, this will oblige us to redefine the coie-valence separation. In sharp contrast with what was done in orbital space, we need a partitioning in real space. Let us begin with isolated atoms. [Pg.18]

The implication of this theorem is that it gives a prescription for the variational determination of the ground-state electron density, since the latter minimizes the energy. [Pg.48]

These results, as most related results of density functional theory, have direct connections to the fundamental statement of the Hohenberg-Kohntheorem the nondegenerate ground state electron density p(r) of a molecule of n electrons in a local spin-independent external potential V, expressed in a spin-averaged form as [Pg.66]

The spirit of the Hohenberg-Kohn theorem is that the inverse statement is also true The external potential v(r) is uniquely determined by the ground-state electron density distribution, n(r). In other words, for two different external potentials vi(r) and V2(r) (except a trivial overall constant), the electron density distributions ni(r) and 2(r) must not be equal. Consequently, all aspects of the electronic structure of the system are functionals of n(r), that is, completely determined by the function (r). [Pg.113]

In Equation 9.21, T yields n and is orthogonal to the first i — 1 state of the Hamiltonian for which n0 is the ground-state density. Here, this Hamiltonian is the // in Equation 9.19. Note that instead of the ground-state electron density n0, we could use the external potential v or any ground-state Kohn-Sham orbital, etc. Thus we could use Ft[n, v]. The extension to degenerate states is studied in Section 9.4. [Pg.126]

The total energy E of the system is also a functional of the density distribution, E = [] (r)]. Therefore, if the form of this functional is known, the ground-state electron density distribution n t) can be determined by its Euler-Lagrange equation. However, except for the electron gas of almost constant density, the form of the functional [ (r)] cannot be determined a priori. [Pg.113]

On the other hand, when the solute is under the influence of the external perturbation, the effective energy functional is minimized by a new electron density p(r), which differs from the ground state electron density of the isolated solute p°(r) by an amount Sp(r). In other words, the effective energy functional E p may be written as [38] [Pg.109]

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