Electronic density, Fermi distribution

Crystal can compute a number of properties, such as Mulliken population analysis, electron density, multipoles. X-ray structure factors, electrostatic potential, band structures, Fermi contact densities, hyperfine tensors, DOS, electron momentum distribution, and Compton profiles. 

The electron density distribution is determined by the electrostatic attraction between the nuclei and the electrons, the electrostatic repulsion between the electrons, the Fermi correlation between same spin electrons (due to the operation of the Pauli principle), and the Coulombic correlation (due to electrostatic repulsion). 

Under the conditions of maximum localization of the Fermi hole, one finds that the conditional pair density reduces to the electron density p. Under these conditions the Laplacian distribution of the conditional pair density reduces to the Laplacian of the electron density [48]. Thus the CCs of L(r) denote the number and preferred positions of the electron pairs for a fixed position of a reference pair, and the resulting patterns of localization recover the bonded and nonbonded pairs of the Lewis model. The topology of L(r) provides a mapping of the essential pairing information from six- to three-dimensional space and the mapping of the topology of L(r) on to the Lewis and VSEPR models is grounded in the physics of the pair density. 

Typically the contributions of the two bands to the current are of rather unequal magnitude, and one of them dominates the current. Unless the electronic densities of states of the two bands differ greatly, the major part of the current will come from the band that is closer to the Fermi level of the redox system (see Fig. 7.6). The relative magnitudes of the current densities at vanishing overpotential can be estimated from the explicit expressions for the distribution functions Wled and Wox ... 

Beginning way back in the 20s, Thomas and Fermi had put forward a theory using just the diagonal element of the first-order density matrix, the electron density itself. This so-called statistical theory totally failed for chemistry because it could not account for the existence of molecules. Nevertheless, in 1968, after years of doing wonders with various free-electron-like descriptions of molecular electron distributions, the physicist John Platt wrote [2] We must find an equation for, or a way of computing directly, total electron density. [This was very soon after Hohenberg and Kohn, but Platt certainly was not aware of HK by that time he had left physics.]... 

Fig. 1-3. Probability density of electron energy distribution, fli), state density, D(t), and occupied electron density. Die) fit), in an allowed energy band much higher than the Fermi level in solid semiconductors, where the Boltzmann function is applicable.

Electrons thermally excited from the valence band (VB) occupy successively the levels in the conduction band (CB) in accordance with the Fermi distribution function. Since the concentration of thermally excited electrons (10 to 10 cm" ) is much smaller than the state density of electrons (10 cm ) in the conduction band, the Fermi function may be approximated by the Boltzmann distribution function. The concentration of electrons in the conduction band is, then, given by the following integral [Blakemore, 1985 Sato, 1993] ... 

Figure 6.3 The Fermi distribution function (a) at absolute zero and (b) at a finite temperature, (c) The population density of electrons in a metal as a function of energy. From Z. Jastrzebski, The Nature and Properties of Engineering Materials, 2nd ed. Copyright 1976 by John Wiley Sons, Inc. This material is used by permission of John Wiley Sons, Inc.

The Fermi distribution law deals with the probability of occupancy by electrons in metals of states of a given energy. The density of states represents a number of states per unit volume having a given energy, (e) What, then, is an expression for the number of electrons per cubic centimeter having an energy between E and E + dE ... 

Modification of this model to get the potential function is obtained considering the Fermi-Dirac distribution function for the electron density and the Boltzmann distribution for the ionic density. This was done by Stewart and Pyatt [58] to get the energy levels and the spectroscopic properties of several atoms under various plasma conditions. Here the electron density was given by... 

Analogously to the formulations in (4.8), the total number of electrons per unit volume in the conduction band is found by integrating the density of states per energy interval multiplied by the Fermi distribution in (4.4) over the energy range of the conduction band ... 

It is worth remembering that we are still working with the one-electron picture, and that we have applied the Boltzmann relation in order to approximate Fermi and quasi-Fermi distribution functions, assuming the quasi-free electron and hole densities of states in the bands. 

The position of an edge denotes the ionization threshold of the absorbing atom. The inflection in the initial absorption rise marks the energy value of the onset of allowed energy levels for the ejected inner electron (216). For a metal this represents the transition of an inner electron into the first empty level of the Fermi distribution (242) and in case of a compound the transition of an inner electron to the first available unoccupied outer level of proper symmetry. Chemical shifts in the absorption-edge position due to chemical combination (reflecting the initial density of states) were first observed by Bergergren (27). 

The magnitudes of the hyperfine splitting parameters also yield information about the electron distribution in the molecule. The theory of the electron-nuclear coupling interaction was first worked out by Fermi, who showed that the constant a depended on the electron density at the nucleus. For a free hydrogen atom, a is given by the Fermi contact interaction in the forrn ... 

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