Functionals of the electronic density

In our opinion, a major requirement for a succesfiiU exchange-correlation functional is its generality a good functional should treat with the same accuracy different chemical interactions and properties, avoiding any excessive specialization for a specific subset of interactions or properties. In the next paragraph we will discuss in some detail this last point. Of course we cannot be exhaustive and we refer the reader to published reviews and textbooks for a more complete analysis [1-3]. 

In the Kohn-Sham (KS) approach to DFT [1,5], the total energy can be expressed as  

V [p] is the potential energy in the field of the nuclei plus any external perturbation, T [p] is the kinetic energy of a set of n independent electrons, moving in an effective one-electron potential which leads to the density p(r), and J p] is the total Coulomb interaction [1]. [p] is the remainder, usually described as the exchange-correlation energy. This term represents the key-problem in DFT, since the exact E c is unknown, and approximations must be used. The simplest approach is the local spin density approximation (LSD), in which the functional for the uniform electron gas of density p is integrated over the whole space  

While e ipfj) in equation (2) is uniquely defined, there is no unique 

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Wang L-W and Teter M P 1992 Kinetic-energy functional of the electron density Phys. Rev. B 45 13 196-220... 

Lee C, W Yang and R G Parr 1988. Development of the Colle-Salvetti Correlation Energy Formula into a Functional of the Electron Density. Physical Review B37 785-789. 

DFT methods compute electron correlation via general functionals of the electron density (see Appendix A for details). DFT functionals partition the electronic energy into several components which are computed separately the kinetic energy, the electron-nuclear interaction, the Coulomb repulsion, and an exchange-correlation term accounting for the remainder of the electron-electron interaction (which is itself... 

Density functional theory-based methods ultimately derive from quantum mechanics research from the 1920 s, especially the Thomas-Fermi-Dirac model, and from Slater s fundamental work in quantum chemistry in the 1950 s. The DFT approach is based upon a strategy of modeling electron correlation via general functionals of the electron density. 

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 ... 

All three terms are again functionals of the electron density, and functionals defining the two components on the right side of Equation 57 are termed exchange functionals and correlation functionals, respectively. Both components can be of two distinct types local functionals depend on only the electron density p, while gradient-corrected functionals depend on both p and its gradient, Vp. ... 

C. Lee, W. Yang and R. G. Parr, Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density, Physical Review B, 37, 785 (1988). 

Thus v t) is a unique functional of the electron density since u(r) fixes the Hamiltonian we see that the full many-particle ground state is a unique functional of the electron density. 

Addition of these two inequalities gives Eq + Eo>Eq + Eo, showing that the assumption was wrong. In other words, for the ground state there is a one-to-one correspondence between the electron density and the nuclear potential, and thereby also with the Hamilton operator and tlie energy. In the language of Density Functional Theory, the energy is a unique functional of the electron density, [p]. 

If now a second, time varying, voltage is applied to the horizontal plates, this will cause a vertical deflection of the beam, and the result is a trace on the screen, which corresponds to the variation of this second voltage with time. The action of the beam on the screen causes a fluorescent trace to appear on the screen as the beam is deflected. The time for which this trace persists is a function of the electron density in the beam and the material with which the screen is coated. 

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... 

In DFT the total energy is expressed as a functional of the electron density p of the molecular system of interest. The derivation of the SCC-DFTB method starts by choosing a reference density po as a superposition of densities p of the neutral atoms a constituting the molecular system,... 

Several methods have been used for analyzing the electron density in more detail than we have done in this paper. These methods are based on different functions of the electron density and also the kinetic energy of the electrons but they are beyond the scope of this article. They include the Laplacian of the electron density ( L = - V2p) (Bader, 1990 Popelier, 2000), the electron localization function ELF (Becke Edgecombe, 1990), and the localized orbital locator LOL (Schinder Becke, 2000). These methods could usefully be presented in advanced undergraduate quantum chemistry courses and at the graduate level. They provide further understanding of the physical basis of the VSEPR model, and give a more quantitative picture of electron pair domains. 

One knows, however, that the simple density-functional theories cannot produce an oscillatory density profile. The energy obtained by Schmickler and Henderson55 is, of course, lower than that of Smith54 because of the extra parameters, but the oscillations in the profile found are smaller than the true Friedel oscillations. Further, the density-functional theories often give seriously inexact results. The problem is in the incorrect treatment of the electronic kinetic energy, which is, of course, a major contributor to the total electronic energy. The electronic kinetic energy is not a simple functional of the electron density like e(n) + c Vn 2/n, but a... 

Wesolowski and Warshel197 introduced a DFT based approach in which all short-range terms in the effective Hamiltonian (Eq. 4.25) were derived entirely from density functional theory and were involved in the construction of the Fock matrix195 196. In this approach, the H croEnv is expressed using explicit functionals of the electron density ... 

An alternative approach to conventional methods is the density functional theory (DFT). This theory is based on the fact that the ground state energy of a system can be expressed as a functional of the electron density of that system. This theory can be applied to chemical systems through the Kohn-Sham approximation, which is based, as the Hartree-Fock approximation, on an independent electron model. However, the electron correlation is included as a functional of the density. The exact form of this functional is not known, so that several functionals have been developed. 

The basic idea in DFT is to express the total energy as a functional of the electron density, i.e., Etot= E p(r). We are thereby able to (formally) reduce the many-body electronic problem to a dependence on three coordinates only (the value of the density at... 

Figure 3.5 The inverse Helmholtz capacity at the pzc as a function of the electronic density the latter is plotted in atomic units (a.u.), where 1 a.u. of density = 6.76 x 1024 cm-3. The dashed line is based on a model calculation of Schmickler and Henderson [3].

According to a theorem by Hohenberg, Kohn and Sham [4], the total energy E of an electron gas can be written as a functional of the electronic density n(r) in the following form ... 

As previously mentioned, this expression is conceptually clumsy because the number of electrons is a function of the electron density through Equation 19.2. As one cannot vary the number of electrons if the electron density is fixed, the naive expression for the total differential... 

Equation 24.17 shows that the energy gained by the system when a field E is applied is a function of the electronic density represented by p. According to the variational principle of DFT, the energy in the ground state (in the absence of a field) is minimum [1,2]. 

Colle-Salvetti Correlation-Energy Formula into a Functional of the Electron Density. 

Figure 2-11 compares the observed work function, 4>, with that calculated based on the jeUium model as a function of the electron density, n.,in metals here, n, is represented in terms of the Wigner-Seitz radius which is inversely proportional to the cube root of n.. The chemical potential term (p. = —1.5 to-2.5 eV) predominates in the work function of metals of low valence electron density, while on the contrary the surface term (- e x = -0-1 -5.0 eV) predominates for... 

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