Self-consistent band calculation

Cu is no longer in evidence and the corrected AS values range from —3 to —7 eV. This energy arises from extraatomic relaxation, from charge renormalization and from any accumulated errors such as the omission of Self-consistent band calculations, made for all the metals except Zn, provide estimates of the renormalization shifts, — 3 eV for Ti and Cu and —4 to — 5 eV for the other metals (21). These values suggest quite modest contributions to AS from extraatomic relaxation. 

In principle, the canonical number-of-states function n(P) contains all the structural information needed to perform self-consistent band calculations for a given crystal structure. No diagonalisations but only scalings... 

Figure 3 Flow diagram of the self-consistent band calculation.

The one-electron band model which has been used so successfully to interpret the photoelectron spectra of the valence bands of many metals clearly will not be satisfactory by itself to interpret the 4f spectra of Ce. Indeed, self-consistent band calculations for both y- and a-Ce yield a 4f band about 1 eV wide which straddles the Fermi level (Glotzel 1978, Pickett et aL 1981, Podloucky and Glotzel, 1983). The filled portion extends only 0.1eV below the Fermi level and contains approximately one electron. It is hybridized with other states and is not a pure 4f band. The simple one-electron picture of photoemission based on these bands would predict a narrow 4f-derived peak at the Fermi level in both phases of Ce, and this is not observed. To describe adequately the photoemission spectra, we must then move beyond the one-electron picture. As discussed above, photoelectron spectra dxt final state spectra which reflect the initial states to a greater or lesser extent depending on the localization of the states themselves. 

We calculate the total RHF energy of the ionized metallic final state, the second term on the right side of eq. (25), by the methods described in section 2.1. RHF computations are performed for the 4f" Sd"" 6s free ions, renormalized atom crystal potentials are constructed, and self-consistent band calculations are carried out. Normalization of the wave functions to the WS sphere ensures that the final state cell has charge -l-lle. The q = 0 component of the full crystal potential, which arises from the charge of the other WS cells, is not included in the total energy since our intent is to compare to the completely screened limit where no such term appears (each cell in that case being neutral). Multiplet theory is again employed to place the 4f electrons into their Hund-rule states.. 

Computation of the pressure P instead of the crystal energy somewhat simplifies calculations of the equilibrium volume Qq- This is because near 2o the relation P( 2) is close to linear and therefore, to determine 2o> it is sufficient to calculate P(Qj) for two or three volumes 2,-, whereas computation of ( 2) and double differentiation are much more tedious. However, in the case of refractory carbides and nitrides nonlinearity of the relation P( 2) near 2q markedly affects the results of the bulk modulus B calculations. As an example. Table 2.1 furnishes the values of (the equilibrium lattice constant) and B obtained from two self-consistent band calculations. Most of the tabulated values agree with experimental data, but when the calculated a and experimental Uq values differ by 0.2 au, the bulk modulus is overestimated by 30-40% (Zhukov, 1988). 

Jarlborg, T. (1977). Self-Consistent Band Calculations on A15 Compounds. Doctor of Philosophy. Thesis, Chalmers Univ. Technol., Goreborg. 

Fig. 6. Self-consistent band structure (48 valence and 5 conduction bands) for the hexagonal II arrangement of nanotubes, calculated along different high-symmetry directions in the Brillouin zone. The Fermi level is positioned at the degeneracy point appearing between K-H, indicating metallic behavior for this tubule array[17. ...

We have carried out impurity calculations for a zinc atom embedded in a copper matrix. We first perform self consistent band theory calculations on pure Cu and Zn on fee lattices with the lattice constant of pure Cu, 6.76 Bohr radii. This yields Fermi energies, self consistent potentials, scattering matrices, and wave functions for both metals. The Green s function for a system with a Zn atom embedded in a Cu matrix... 

Van der Woude and Miedema [335] have proposed a model for the interpretation of the isomer shift of Ru, lr, Pt, and Au in transition metal alloys. The proposed isomer shift is that derived from a change in boundary conditions for the atomic (Wigner-Seitz) cell and is correlated with the cell boundary electron density and with the electronegativity of the alloying partner element. It was also suggested that the electron density mismatch at the cell boundaries shared by dissimilar atoms is primarily compensated by s —> electron conversion, in agreement with results of self-consistent band structure calculations. 

Early band structure calculations for the actinide metals were made both with and without relativistic effects. As explained above, at least the mass velocity and Darwin shifts should be included to produce a relativistic band structure. For this reason we shall discuss only the relativistic calculations. There were some difficulties with the f-band structure in these studies caused by the f-asymptote problem , which have since been elegantly solved by linear methods . Nevertheless the non-self-consistent RAPW calculations for Th through Bk indicated some interesting trends that have also been found in more recent self-consistent calculations ... 

The electronic structure of bulk VO has been calculated by different band structure methods [110-114] and using correlated electron procedures [115]. This Mott-Hubbard metal, which forms a rocksadt type lattice, is the simplest of all single valence oxides of vanadium and has been treated theoretically already a long time ago. As an example, Neckel et al. [114] have published results from self-consistent APW calculations for the experimentally known lattice geometry... 

Calculations of the self-consistent band structures predict relatively large band gaps for the optimized lattices. For example, in the case of AN, the band gaps at the T(0,0,0) point for phases V, IV, III, and II have values between 3.37-3.51 eV while for ADN the band gap is about 3 eV. These results indicate that both these two materials are electrical insulators at ambient conditions. 

Another scheme known as LDA-t-U has been developed [121-125] to add aspects of the Hubbard model [126,127] to self-consistent band structure calculations. It introduces additional interactions which depend on the occupation of the individual orbitals, and in that way an extra symmetry... 

This volume proposes to describe one particular method by which the self-consistent electronic-structure problem may be solved in a highly efficient manner. Although the technique under consideration, the Linear Muffin-Tin Orbital (LMTO) method, is quite general, we shall restrict ourselves to the case of crystalline solids. That is, it will be shown how one may perform self-consistent band-structure calculations for infinite crystals, and apply the results to estimate ground-state properties of real materials. 

To avoid misunderstanding I mention that the above scaling cycle is used in the self-consistency procedure mainly to reduce the number of band-structure calculations needed. If one wants very accurate self-consistent bands one must include an energy-band calculation at the end of each self-consistent scaling cycle. However, the scaling procedure is so efficient that fully converged bands of most metals may be obtained with only one or two band calculations included in the complete self-consistency procedure. 

Krasovski E E, Starrost F and Schattke W1999 Augmented Fourier components method for constructing the crystal potential in self-consistent band-structure calculations Phys. Rev. B 59 10 504... 

In (a) are shown the energy bands of a-quartz obtained by Chelikowsky and Schliiter (1977) froi a self-consistent pseudopotential calculation. In part (b) are shown the bands of the simpler /1-cristobalite structure (Pantelides and Harrison, 1976) from an LCAO calculation and a Bond Orbital Approximation that calculation was restricted to the valence bands. The corresponding density of states is shown to the right. Because the symmetry of the structures is different, different symmetry points are indicated below each figure. [Part (a) after Chelikowsky and Schliiter, 1977 part (b) after Pantiledes and Harrison, 1976.]... 

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