Electron density Kohn-Sham theory

The basic concepts of the one-electron Kohn-Sham theory have been presented and the structure, properties and approximations of the Kohn-Sham exchange-correlation potential have been overviewed. The discussion has been focused on the most recent developments in the theory, such as the construction of from the correlated densities, the methods to obtain total energy and energy differences from the potential, and the orbital dependent approximations to v. The recent achievements in analysis of the atomic shell and molecular bond midpoint structure of have been... 

This argument shows that the locality hypothesis fails for more than two electrons because the assumed Frechet derivative must be generalized to a Gateaux derivative, equivalent in the context of OEL equations to a linear operator that acts on orbital wave functions. The conclusion is that the use by Kohn and Sham of Schrodinger s operator t is variationally correct, but no equivalent Thomas-Fermi theory exists for more than two electrons. Empirical evidence (atomic shell structure, chemical binding) supports the Kohn-Sham choice of the nonlocal kinetic energy operator, in comparison with Thomas-Fermi theory [288]. A further implication is that if an explicit approximate local density functional Exc is postulated, as in the local-density approximation (LDA) [205], the resulting Kohn-Sham theory is variation-ally correct. Typically, for Exc = f exc(p)p d3r, the density functional derivative is a Frechet derivative, the local potential function vxc = exc + p dexc/dp. 

The electron-transfer reactivities are defined as derivatives of the electron-density p(r) with respect to total electron number Jf, Ufr), or chemical potential p, s r). The treatment of JT as a continuous variable [8-12] is justified by reference to the ensemble formulation of density-functional theory [8,18] and, in consequence, of the Kohn-Sham theory. We show in Sect. 4, in previously unpublished work [42], that this ensemble formulation yields either vanishing or infinite local and global softnesses for localized systems with... 

Symmetrised density-functionals, which have been proposed recently [88] as the correct solution of the symmetry dilemma in Kohn-Sham theory, also naturally lead to fractional occupations. The symmetry dilemma occurs because the density or spin-density of KS theory may exhibit lower symmetry them the external potential due to the nuclear conformation. This in turn leads to a KS Hamiltonian with broken symmetry, leading to electronic orbitals that cannot be assigned to an irreducible representation... 

The property associated [9] with the purely electron-interaction component of the Kohn-Sham theory many-body potential as well as the electron-interaction energy is the pair-correlation density g(r, r ). It is defined in terms of the pair-correlation operator... 

Within the context of Kohn-Sham theory, the assumption underlying the LDA is that each point of the nonuniform electron density is uniform but with a density corresponding to the local value. In the LDA for exchange, the wavefunction is therefore assumed to be a Slater determinant of plane waves at each electron position. The corresponding pair-correlation density g[ r, r p(r) is thus the expectation of Eq. (66) taken with res[ ct to this Slater determinant, with the resulting expression then assumed valid locally. (The superscript (0) indicates the result is derived from uniform electron gas theory.)... 

As we have already observed above, within the hardness (interaction) representation (see Tables 1, 3) the FF indices provide important weighting factors in combination formulas which express the global CS and potentials in terms of the local properties, relevant for the resolution in question. The FF expressions from the EE equations are invalid in the MO resolution, since no equalization of the orbital potentials can take place, due to obvious constraints on the MO occupations in the Hartree-Fock (HF) theory [61]. Moreover, standard chain-rule transformations of derivatives are not applicable in the MO resolution since some of the derivatives involved are not properly defined. Various approaches to the local FF, f(f), have been proposed e.g., those expressing f(f), in terms of the frontier orbital densities [11, 25], or the spin densities [38]. Also the finite difference estimates of the chemical potential (electronegativity) and hardness have been proposed in the MO and Kohn-Sham theories for various electron configurations [10, 11, 19, 52, 61b, 62, 63]. 

Within the local density formulation of the Kohn-Sham theory, - the ground state energy is given as a functional of the electron density p(r) in the presence of an external potential v(r) ... 

In this work, an alternative strategy for deriving the spin-density of an embedded molecule is used. Instead of applying Kohn-Sham theory to the whole system, the spin/electron densities of different subsystems are treated separately. The two subsystems correspond to a) the embedded molecule the radical b) the embedding molecule(s) the atoms or molecules forming the complex with the radical. For a given electron-density of the embedding molecules. 

As in the Kohn-Sham theory, the electron density pl is represented by means of one-electron functions... 

Time-dependent density-functional theory (TDDFT) extends the basic ideas of ground-state density-functional theory (DFT) to the treatment of excitations and of more general time-dependent phenomena. TDDFT can be viewed as an alternative formulation of time-dependent quantum mechanics but, in contrast to the normal approach that relies on wave-functions and on the many-body Schrodinger equation, its basic variable is the one-body electron density, n(r,t). The advantages are clear The many-body wave-function, a function in a 3A-dimensional space (where N is the number of electrons in the system), is a very complex mathematical object, while the density is a simple function that depends solely on the 3-dimensional vector r. The standard way to obtain n r,t) is with the help of a fictitious system of noninteracting electrons, the Kohn-Sham system. The final equations are simple to tackle numerically, and are routinely solved for systems with a large number of atoms. These electrons feel an effective potential, the time-dependent Kohn-Sham potential. The exact form of this potential is unknown, and has therefore to be approximated. 

The premise behind DFT is that the energy of a molecule can be determined from the electron density instead of a wave function. This theory originated with a theorem by Hoenburg and Kohn that stated this was possible. The original theorem applied only to finding the ground-state electronic energy of a molecule. A practical application of this theory was developed by Kohn and Sham who formulated a method similar in structure to the Hartree-Fock method. 

Kohn-Sham orbitals functions for describing the electron density in density functional theory calculations... 

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