Self-consistent calculations density

Modern theories of electronic structure at a metal surface, which have proved their accuracy for bare metal surfaces, have now been applied to the calculation of electron density profiles in the presence of adsorbed species or other external sources of potential. The spillover of the negative (electronic) charge density from the positive (ionic) background and the overlap of the former with the electrolyte are the crucial effects. Self-consistent calculations, in which the electronic kinetic energy is correctly taken into account, may have to replace the simpler density-functional treatments which have been used most often. The situation for liquid metals, for which the density profile for the positive (ionic) charge density is required, is not as satisfactory as for solid metals, for which the crystal structure is known. 

The adequacy of the spin-averaged approach has been confirmed in self-consistent spin-density-functional calculations for H in Si by Van de Walle et al. (1989). The deviation from the spin-averaged results is expected to be largest for H at the tetrahedral interstitial (T) site, where the crystal charge density reaches its lowest value. For neutral H at the T site, it was found that inclusion of spin polarization lowered the total energy of the defect only by 0.1 eV. The defect level was split into a spin-up and a spin-down level, which were separated by 0.4 eV. These results are consistent with spin-polarized linearized-muffin-tin-orbital (LMTO) Green s-function calculations (Beeler, 1986). 

Figure 6 shows the behavior of the reduced monomer density p z)Rp/Np at increasing anchoring density. The stretching of the chains with increasing surface coverage, which is due to the repulsion between monomers, is evident. This plot has to be compared with Fig. 3b, where the same type of rescaling has been used. However, note that at this point, direct and quantitative comparison is not possible, since it is a priori not clear which value of the interaction parameter /3 in the self-consistent calculation corresponds to which set of simulation parameters ct, N, pa. 

Appelbaum, J. A., and Hamann, D. R. (1973a). Surface potential, charge density, and ionization potential of Si(lll) - a self-consistent calculation. Phys. Rev. Lett. 32, 225-228. 

Fig. 3.6 Binding energy curves for the hydrogen molecule (lower panel). HF and HL are the Hartree-Fock and Heitler-London predictions, whereas LDA and LSDA are those for local density and local spin density approximations respectively. The upper panel gives the local magnetic moment within the LSDA self-consistent calculations. (After Gunnarsson and Lundquist (1976).)...

Fig. 3.12 The binding energies, equilibrium internuclear separations and vibrational frequencies across the first-row diatomic molecules. Note the good agreement between the self-consistent local density approximation calculations and experiment for R and coe but the larger systematic error of up to 2 eV for the binding energy. (After Gunnarsson et aL (1977).)...

In a supercooled liquid near the glass transition temperature, the self-consistent calculation is the only way to explain the anomalies in different dynamical quantities. As mentioned before, the first such self-consistent calculation was done by Geszti to explain the behavior of the viscosity near the glass transition temperature. He had argued that an increase in the viscosity slows down the structural relaxation and thus the relaxation of the density mode. This in turn increases the density mode contribution to the viscosity, t]spp... 

In this section, we illustrate the self-consistent calculation of these charge current densities in the plane-wave approximation, using plane waves in the X, Y, and Z directions. In general, the solution of the field equation (459) must be found numerically, and it is emphasized that the plane-wave approximation is a first approximation only. In the internal space, there is the real vector ... 

Xu et al. carried out self-consistent periodic density functional theory calculations (GGA-PW91) to study the adsorption of atomic oxygen and molecular oxygen, and the dissociation of 02 on the... 

Fig. 1 Asymptotic structure coefficients as(j8), c ks,xG ) crw03), a CS), and aKS gOS) as fimction of barrier height parameter )S =VW/eF, where W is the barrier height and eF the Fermi energy. Corresponding values of the Wigner-Seitz radius rs for jellium and structureless-pseudopotential models over the metallic range of densities are also given. The relationship between rs and ff is via self-consistent calculations in the local density approximation for exchange-correlation.

A plot of wfPp(z) for rs = 3.24 employing the orbitals of the finite-linear-potential moder is given in Fig. 6. The corresponding local density approximation (LDA) potential is also plotted. In the interior and about the surface of the metal the two potentials are equivalent. But outside the surface vxapp(z) improves upon the LDA significantly and approaches the exact structure asymptotically. We thus expect that properties such as the surface energy and work function obtained with Eapp[p] and v pp(r) to be superior to those of the LDA. Such self-consistent calculations are in progress. 

A self-consistent calculation of electron-density profiles at strongly charged jellium surfaces, similar to the approach of Halley and co-workers, was made by Gies and Gerhardts [143]. This work was applied by the Patey group... 

From the practical standpoint, the first attempt to solve the self-consistent TF equation for a diatomic molecule was made by Hund.82 Following this, the density method was applied to the benzene molecule and compared with both the molecular orbital prediction for the density and with relevant experiments.88 Various other early molecular calculations are discussed in ref. 16 we refer here to the recent studies of Dreizler and his co-workers.84 The importance of such self-consistent calculations will be emphasized below, even though we shall not use them in any detail in the ensuing discussion. 

For perfectly ordered crystals at absolute zero, solutions to the Schrodinger equation can be calculated on fast computers using density functional theory (DFT) based on the self-consistent local density approximation (LDA) simplifying procedures using different basis functions include augmented... 

The details of the modified electron-gas (MEG) ionic model method have been fully described by Gordon and Kim (1972). The fundamental assumptions of the method are (1) the total electron density at each point is simply the sum of the free-ion densities, with no rearangements or distortion taking place (2) ion-ion interactions are calculated using Coulomb s law, and the free-electron gas approximation is employed to evaluate the electronic kinetic, exchange, and correlation energies (3) the free ions are described by wave functions of Hartree-Fock accuracy. Note that this method does not iterate to a self-consistent electron density. 

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