Homogeneous electron/energy

Metalloporphyrin Excited State Structural Dyuamics (Homogeneous Electron/Energy Trausfer)... 

Inserting equation (6-14) into equation (6-12) retrieves the p4/3 dependence of the exchange energy indicated in equation (3-5). This exchange functional is frequently called Slater exchange and is abbreviated by S. No such explicit expression is known for the correlation part, ec. However, highly accurate numerical quantum Monte-Carlo simulations of the homogeneous electron gas are available from the work of Ceperly and Alder, 1980. 

One obvious drawback of the LDA is that, when we replace unknown exchange-correlation energy by the known form of the exchange-correlation for a homogeneous electron gas in Equation (17), we have a problem in that cancelation of self-Coulomb... 

An exact expression for the correlation energy per particle ec( o) of a homogeneous electron gas does not exist, but good approximations to this nevertheless do exist.24 Also, nearly exact correlation energies have been obtained numerically for different densities25 and the results have been parametrized as useful functions ec n).26 The corresponding LDA correlation potential... 

In Eq. [47], epc ( ) and exc (n) are the exchange-correlation energy densities for the nonpolarized (paramagnetic) and fully polarized (ferromagnetic) homogeneous electron gas. The form of both exc(n) and exc(n) has been conveniently parameterized by von Barth and Hedin. Other interpolations have also been proposed24,33 for eKC(n, J ). The results for the homogeneous electron gas can be used to construct an LSDA... 

Similar to homogeneous electron-transfer processes, one can consider the observed electrochemical rate constant, k, , to be related to the electrochemical free energy of reorganization for the elementary electron-transfer step, AG, by... 

It should be noted that a Maxwellian form of fie) is a reasonable approximation to the actual distribution at low electron energies. This observation is indicated in Figure 5 i24). However, the first ionization potential of most atoms and molecules is above eV. Thus, many of the important homogeneous processes that occur in glow discharges, such as ionization, take place as a result of high energy electrons in the "tail" of the distribution. These electrons are precisely the ones that are not adequately described by a Maxwellian distribution function. 

Figure 4.5, Potential energy diagrams for the homogeneous electron transfer reaction between an aromatic radical-anion and a second aromatic with a frangible R-X bond, (a) The situation where back electron transfer and bond cleavage have similar free energy of activation, (b) The situation where the RX radical-anicm has high energy and the R-X bond has low dissociation ertergy.

Based on the first-principles study of helium adsorption on metals (Zaremba and Kohn, 1977), Esbjerg and Nprskov (1980) made an important observation. Because the He atom is very tight (with a radius about 1 A), the surface electron density of the sample does not vary much within the volume of the He atom. Therefore, the interaction energy should be determined by the electron density of the sample at the location of the He nucleus. A calculation of the interaction of a He atom with a homogeneous electron distribution results in an explicit relation between the He scattering potential V r) and the local electron density p(r). For He atoms with kinetic energy smaller than 0.1 eV, Esbjerg and Nprskov (1980) obtained... 

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