Electronic self-consistent calculation

Inglesfield J E and Benesh G A 1988 Surface electronic structure embedded self-consistent calculations Phys. Rev. 

Raffenetti, R.C. Pre-processing two-electron integrals for efficient utilization in many-electron self-consistent field calculations. Chem. Phys. LeUera 20 335-338, 1973. 

Ab initio calculations are iterative procedures based on self-consistent field (SCF) methods. Normally, calculations are approached by the Hartree-Fock closed-shell approximation, which treats a single electron at a time interacting with an aggregate of all the other electrons. Self-consistency is achieved by a procedure in which a set of orbitals is assumed, and the electron-electron repulsion is calculated this energy is then used to calculate a new set of orbitals, which in turn are used to calculate a new repulsive energy. The process is continued until convergence occurs and self-consistency is achieved." ... 

Figure 6.32 Self-consistent calculation of the electronic structure of CO adsorbed on Al and Pt. The sharp 5

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. 

Thus far, I have mainly discussed neutral impurities. From the treatment of the electronic states, however, it should be clear that occupation of the defect level with exactly one electron is by no means required. In principle, zero, one, or two electrons can be accommodated. To alter the charge state, electrons are taken from or removed to a reservoir the Fermi level determines the energy of electrons in this reservoir. In a self-consistent calculation, the position of the defect levels in the band structure changes as a function of charge state. For H in Si, it was found that with H fixed at a particular site, the defect level shifted only by 0.1 eV as a function of charge state (Van de Walle et al., 1989). 

So far we have assumed that the electronic structure of the crystal consists of one band derived, in our approximation, from a single atomic state. In general, this will not be a realistic picture. The metals, for example, have a complicated system of overlapping bands derived, in our approximation, from several atomic states. This means that more than one atomic orbital has to be associated with each crystal atom. When this is done, it turns out that even the equations for the one-dimensional crystal cannot be solved directly. However, the mathematical technique developed by Baldock (2) and Koster and Slater (S) can be applied (8) and a formal solution obtained. Even so, the question of the existence of otherwise of surface states in real crystals is diflBcult to answer from theoretical considerations. For the simplest metals, i.e., the alkali metals, for which a one-band model is a fair approximation, the problem is still difficult. The nature of the difficulty can be seen within the framework of our simple model. In the first place, the effective one-electron Hamiltonian operator is really different for each electron. If we overlook this complication and use some sort of mean value for this operator, the operator still contains terms representing the interaction of the considered electron with all other electrons in the crystal. The Coulomb part of this interaction acts in such a way as to reduce the effect of the perturbation introduced by the existence of a free surface. A self-consistent calculation is therefore essential, and the various parameters in our theory would have to be chosen in conformity with the results of such a calculation. 

The primary particle involved in the screening process is the mobile electron. One has then the problem of a self-consistent calculation of the charge distribution in the neighborhood of a test charge. The Thomas-Fermi approach to this problem is the analog of the Debye-Huckel calculation wherein allowance has been made for the Pauli exclusion principle. From any standard text one can obtain the Poisson equation (19)... 

In the last configuration a particle-hole pair is considered in the system promoting an electron from the valence band (i = h) to a conduction band (i = e). For this reason the method is also called constrained DFT. The excitation energy of the many-electron system is the difference in total energy between two self-consistent calculations with the occupations described above, i.e. ... 

We adopt the same chemisorption model as in our previous work [3], which within the unrestricted Hartree-Fock approximation involves a self-consistent calculation of the electronic charge on the adatom. The basis elements needed for the calculation are the... 

Results of Self-Consistent Calculations LDA electronic structure at x =... 

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... 

This does not mean that the LCAO approach of the type we have used is incorrect or not useful. Recent applications of LCAO theory, based only upon electron orbitals that are occupied in the free atom, have been made to the study of simple metals (Smith and Gay, 1975), noble-metal surfaces (Gay, Smith, and Arlinghaus, 1977), and transition metals (Rath and Callaway, 1973). In fact, the LCAO approach seems a particularly effective way to obtain self-consistent calculations. The difficulty from the point of view taken in this book is that, as with many other band-calculational techniques, LCAO theory has not provided a means for the elementary calculations of properties emphasized here, but pseudo-potentials have. 

One of the earliest treatments of a metal surface was based upon a jellium model (Bardeen, 19.36). If the electron gas terminated abruptly at the surface of the jellium there would be no net potential to contain the electrons in the metal. Therefore the electron gas extends beyond the metal, giving a dipole layer, as illustrated in Fig. 17-5. Bardeen attempted the self-consistent calculation of the resulting potential. It should be mentioned that the Fermi-Thomas approximation is not adequate for this task and was not used by Bardeen it is not difficult to see that it would predict the Fermi energy to be at the vacuum level, corresponding to a vanishing work function. 

The surface relaxation on both SrTiOs (001) terminations has been calculated by various numerical approaches [45,189-193]. The surface rumpling is usually reasonably accounted for, but all calculations predict an inward relaxation for the Ti02 termination, in contradiction to experiments [194-196]. A particular attention has been focused on the energy of surface states, since the first study based on non-self-consistent calculations predicted that they were located deep in the gap [197,198]. All subsequent self-consistent calculations have contradicted this prediction [191,192,199,200], in agreement with photoemission and EELS results [201,202]. When calculated [191,192], the surface energy is rather low, an indication that no surface instability takes place, and there is no evidence of anomalous filling of electronic states. 

The CD and UV spectra of the compound with a twisted n-electron system can be calculated by the jc-electron Self-Consistent-Field Configuration-Interaction Dipole-Velocity Molecular Orbital method (the Tc-electron SCF-CI-DV MO me-thod).8-10 In the dipole velocity method, the rotational strength I ba and dipole strength Dba which govern the sign and intensity of a CD Cotton effect and the intensity of a UV absorption band, respectively are formulated as follows ... 

FIGURE 5.12 Dependence of radial probability densities on distance from the nucleus for Hartree orbitals in argon with n = 1, 2, 3. The results were obtained from self-consistent calculations using Hartree s method. Distance is plotted in the same dimensionless variable used in Figure 5.10 to facilitate comparison with the results for hydrogen. The fact that the radial probability density for all orbitals with the same value of n have maxima very near one another suggests that the electrons are arranged in "shells" described by these orbitals. 

In brief, the CNDO (the acronym stands for complete neglect of differential overlap) approach is an all valence electron, self-consistent field calculation in which multicenter integrals have been neglected and some of the two electron integrals parameterized using atomic data. Slater type atomic orbitals are used as the basis 2s, 2px, 2p, 2p for carbon and oxygen. In these calculations two-electron in egrafs are approximated as... 

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