Potentials Hartree potential

With these eharge densities defined, it is possible to define eorresponding potentials. The Coulomb or Hartree potential, V, is defined by... 

The environment affects the charge density of the QM zone in a self-consistent way by the addition of a point charge potential (VMM)to the Hartree potential as in ... 

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 third term of Eq (54) is the electronic Hartree potential, whereas the fourth one represents the exchange-correlation potential. This last term is usually obtained from a model exchange-correlation energy functional xc[pl To a first order approximation, the effective KS potential compatible with the electron density p f) given in Eq (51) may be written as ... 

While Dirac [3] chose to solve Eq. (4) as a quadratic equation for in terms of the Hartree potential yHC "), it was Slater in 1951 ([6] see also [4]) who chose an alternative, and more fruitful, route by regarding Eq. (4) as demonstrating that it could be viewed as a modified Hartree equation, with the Hartree potential Unfr) now supplemented by the exchange n -potential (the so-called Dirac-Slater (DS) exchange potential), to yield a total one-body potential energy... 

If you have a vague sense that there is something circular about our discussion of the Kohn-Sham equations you are exactly right. To solve the Kohn-Sham equations, we need to define the Hartree potential, and to define the Hartree potential we need to know the electron density. But to find the electron density, we must know the single-electron wave functions, and to know these wave functions we must solve the Kohn-Sham equations. To break this circle, the problem is usually treated in an iterative way as outlined in the following algorithm 1 2 3... 

The third term on the left-hand side is the same Hartree potential we saw in Eq. (1.5) ... 

We now have all the pieces in place to perform an HF calculation—a basis set in which the individual spin orbitals are expanded, the equations that the spin orbitals must satisfy, and a prescription for forming the final wave function once the spin orbitals are known. But there is one crucial complication left to deal with one that also appeared when we discussed the Kohn-Sham equations in Section 1.4. To find the spin orbitals we must solve the singleelectron equations. To define the Hartree potential in the single-electron equations, we must know the electron density. But to know the electron density, we must define the electron wave function, which is found using the individual spin orbitals To break this circle, an HF calculation is an iterative procedure that can be outlined as follows ... 

The KS (local) effective potential has three components the external potential, the Hartree potential, and the XC potential. 

In turn, veia is subdivided into the standard Hartree potential vCoui of electrostatic electron repulsion and the exchange-correlation (xc) potential... 

Density Functional Theory (DFT) has become a powerful tool for ab-initio electronic structure calculations of atoms, molecules and solids [1, 2, 3]. The success of DFT relies on the availability of accurate approximations for the exchange-correlation (xc) energy functional Exc or, equivalently, for the xc potential vxc. Though these quantities are not known exactly, a number of properties of the exact xc potential vxc(r) are well-known and may serve as valuable criteria for the investigation of approximate xc functionals. In this contribution, we want to focus on one particular property, namely the asymptotic behavior of the xc potential For finite systems, the exact xc potential vxc(r) is known to decrease like — 1/r as r —oo, reflecting also the proper cancellation of spurious self-interaction effects induced by the Hartree potential. 

For the details and derivation of the physical interpretation we refer the reader to the original literature14,15. Since the Coulomb self-energy component of the KS electron-interaction energy functional and its derivative, the Hartree potential, are known functionals of the density, we provide in Section HA the expressions governing the interpretation of the KS exchange-correlation energy... 

Hartree potential, (d) the Fock non-local exchange potential, (e-g) second-order contributions to the self-energy Xj(E) (e) direct (optical potential), (f) exchange and (g) Fermi sea correlation... 

Higher order corrections are then included according to Eqs. (42) and (43). The TDAE part of the atomic dielectric response including the effect of the Hartree potential of a spectator 4d hole is taken into account by our choice of basis set ... 

The classical electronic potential vn(r) in Eq. (17-2), referred to as Hartree potential, is defined by... 

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