Free-Electron Potential Parameters

The electrochemical potential of single ionic species cannot be determined. In systems with charged components, all energy effects and all thermodynamic properties are associated not with ions of a single type but with combinations of different ions. Hence, the electrochemical potential of an individual ionic species is an experimentally undefined parameter, in contrast to the chemical potential of uncharged species. From the experimental data, only the combined values for electroneutral ensembles of ions can be found. Equally inaccessible to measurements is the electrochemical potential, of free electrons in metals, whereas the chemical potential, p, of the electrons coincides with the Fermi energy and can be calculated very approximately. 

Methanol-water mixtures were studied by Adamovic and Gordon who used the so called effective-fragment-potential method, an approach of the type discussed in section IID, for finite (Me0H/H20) clusters with n = 2,3,. .., 8. Some of their findings were tested by using accurate, parameter-free, electronic-structure methods. The main outcome is the organization of the H2O and MeOH molecules relative to each other, which then is expected to be representative for macroscopic mixtures of water and methanol. 

In Fenske and Hall s parameter-free SCF calculations (80-84), the He1t 1-electron operator is substituted by a model 1-electron operator that has a kinetic energy and potential energy term for each atomic center in the complex. This approach assumes that the electron density may be assigned to appropriate centers. The partitioning of electron density is done through Mulliken population analyses (163) until self-consistency is obtained. The Hamiltonian elements are evaluated numerically, and the energies of the MO s depend only on the choice of basis functions and the intemuclear distance. 

The equilibrium redox potential, the free energy change per mole electron for a given reduction, represents the oxidizing intensity of the couple at equilibrium. It is conveniently expressed for many applications in terms of the parameter, pE, as proposed by Jorgensen (8) and popularized by Sillen (14). This parameter is defined by the relation,... 

Eq. (1) is used to find the d-band width (6.5 eV) once the other parameters of the band shape are determined. Similarly, Eq. (2) is used to determine the s-band width (12.9 eV) of a free-electron density of states symmetric in energy about the middle of the band. The d-band density of states, Nj(E). rises sharply at the lower band edge to about 1.5 states/eV atom then falls off to 0.47 states/eV atom near the middle. With the general shape of Nj. (E) and Ns(E) given, the critical magnitude of Nd( q ), the chemical potential in d-orbital, is determined from the observed linear part of the low-temperature specific heat as follows ... 

In section 2 the theory of ensembles is reviewed. Section 3 summarizes the parameter-free theory of G par[ll]. The self-consistently determined ensemble a parameters of the ensemble Xa potential are presented. In section 4 spin-polarized calculations using several ground-state exchange-correlation potentials are discussed. In section 5 the w dependence of the ensemble a parameters is studied. It is emphasized that the excitation energy can not generally be calculated as a difference of the one-electron energies. The additional term should also be determined. Section 6 presents accurate... 

However, because of the pathologieal behaviour ofXa for some ( magnetic ) transition-metal systems (see below) and because the other potentials include correlation in a more explicit manner and are parameter-free (except for parameters needed to fit electron-gas results), there seems to be little reason to retain the Xa potential, except perhaps for the sake of compatibility with previous calculations. Since the VWN or the potential represents the local limit, it should be used as the standard of reference. If an LSD-VWN problem has been aceurately solved, any remaining errors may be attributed to non-local effects (Section II.B). 

Fig.3.1. Estimates of the logarithmic-derivative function for free s electrons compared with the exact result D(x) = x cot x - 1, x = S/E explained in Sect. 4.4. The curve labelled to is the second-order estimate (3.51), E(D) is the third-order estimate (3.50). while Lau is the Laurent expansion (3.30) valid to third order in (E - EV)S2. The potential parameters used in the three estimates are derived in Sect.4.4 and listed in Table 4.4. The two open circles in the figure refer to the points (EVS2,DV) and (EVS2,D ), where EVS2 is K2S2 of Table 4.4...

As an example of a set of standard parameters, Table 4.1 lists all the potential-dependent information needed to perform an energy-band calculation for (non-magnetic) chromium metal. In the following, chromium is used as an example when we discuss the physical significance of each of the four potential parameters (4.1). At the end of the chapter we derive free-electron potential parameters, give expressions for the volume derivatives of some se-... 

The potential parameters for the free-electron case are interesting for several reasons. First of all they are easy to calculate analytically from standard expressions for the spherical Bessel functions [4.2] and therefore useful in order-of-magnitude estimates. Secondly, they may be used in empty-lattice tests of the LMTO method in order to indicate the accuracy of that method in various applications. Such tests are also useful for programme debugging purposes. Thirdly, muffin-tin orbitals with free-electron parameters are used in Sect.6.9 to derive a correction to the atomic-sphere approxir mation. 

The local free-electron potential parameters may be found by differentiation with respect to energy of the logarithmic derivatives. To see this we multiply the Taylor expansion of D(e = E - E )... 

The free-electron potential parameters are particularly simple when D = - n - 1 or D = l In the former case they may be found in a straight-... 

Table 4.5. Standard potential parameters for free electrons. D = , v = 0,...

With this definition, xL is equal to the proper orbital in the interstitial region only, while inside the spheres it is derived from a constant (pseudo) potential v(r) = E - k. For that reason the integral over the sphere appearing in (6.44) is simply the LMTO overlap matrix (5.47) evaluated for the free-electron potential parameters from Sect.4.4 corresponding to = D j (kS). Hence, the contribution from the second term in (6.44) may be included in the LMTO equations (5.45) by subtracting... 

The basic differences between the FEMO and the HMO methods are (1) the FEMO method presumes the electron exists in a space (box) of continuous potential and (2) the FEMO method contains only one parameter, the neighboring distance D, rather than two parameters as in the HMO method. The meaning of the parameter D in terms of measurable quantities is defined more precisely than the integral parameters in the HMO method, and the FEMO model is closer to being an absolute theory with no adjustable parameters. When the existence of the free electron is restricted to the onedimensional carbon-bond skeleton of the molecule, one obtains the free-electron network model. Depending upon the conditions imposed at the branch points of the C skeleton, a very close relationship between the FEMO and HMO methods and results can be demonstrated. 

To derive the shift in the optical band gap, Tiedje et al (1984) used the conduction and valence densities of states corresponding to the model of free electrons and holes in a one-dimensional periodic potential shown in Fig. 5a. The parameters of the model were chosen as follows bulk amorphous silicon band gap 1.8 eV conduction- and valence-band-edge discontinuities at the a-Si H/a-SiN t H interfaces [4= 1.0 eV and [4 = 0.6 eV, respectively... 

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