Rectangular density-of-states

The trend in the f-pressure is almost parabolic with band filling and this is typical for a transition metal (with d replaced by f). The physical basis was given by the Friedel who assumed that a rectangular density of states was being filled monotonically and thus was able to reproduce the parabolic trend in transition metal cohesive energies analytically. Pettifor has shown that the pressure formula can similarly be integrated analytically. 

Rectangular Density of States Model for Electronic Entropy In this problem we imagine two competing structures, both characterized by electronic densities of states that are of the rectangular-band-type. In... 

In this problem, elaborate on the Einstein model for structural change given in the chapter by replacing the delta function densities of states by rectangular densities of states. Assume that the phase with the lower internal energy has the broader vibrational density of states. Compute the transformation temperature as a function of the difference in the widths of the two rectangular bands. 

We may therefore assume that the 5 f non-spin-polarized band splits into two subbands because of spin-polarization. Approximation of the two sub-bands, according to Friedel s model, by two rectangular ones, having densities of state N+(E) = N (E) = 7/ Wf, and occupation numbers n+ and n, leads to the following expression for the total Pspd pressure ... 

The heats of formation of equiatomic AB transition-metal alloys may be predicted by generalizing the rectangular d band model for the elements to the case of disordered binary systems, as illustrated in the lower panel of Fig. 7.13. Assuming that the A and transition elements are characterized by bands of width WA and WB, respectively, then they will mix together in the disordered AB alloy to create a common band with some new width, WAB. The alloy bandwidth, WAB may be related to the elemental bond integrals, hAA and , and the atomic energy level mismatch, AE — EB — EAt by evaluating the second moment of the total alloy density of states per atom ab( ), namely... 

The charge transfer, - Qde, which accompanies a given atomic energy-level separation, AE, in the binary AB alloy may be obtained by filling the local densities of states up to the Fermi level as shown in Fig. 7.13. For the skew rectangular local densities of states this gives... 

Hence, within the rectangular d band model for the AB alloy density of states, from eqs (7.33) the bond energy becomes... 

This expression is stationary with respect to small variations in AE for just that value of AE = A LCN, which results from filling up the skew rectangular partial densities of states and requiring local charge neutrality. Thus, this simple model is internally consistent. 

The rectangular band model posits that the density of states is constant over some range of energies and zero otherwise. The constant value associated with the density of states is chosen such that... 

Fig. 4.16. Comparison of exact and rectangular band densities of states for osmium (adapted from Harrison (1980)) (a) density of states for osmium as computed using density functional theory and (b) density of states constructed by superposiug a rectaugular baud model with a free-electrou-bke deusity of states. Rising Unes in the two curves are a measure of the integrated density of states.

Fig. 4.17. Band energy associated with rectangular band density of states. Energy is plotted in dimensionless units, scaled by the width of the band, W. Similarly, the band filling is plotted as a fraction of the full band.

Galanakis et al. [9] correlated the surface energy of some d-metals to the broken bond in the tight-binding approximation. The is the number of d-electrons. Ws and Wb are the bandwidths for the surface and the bulk density of states, which took in rectangular forms. 

Figure 7.29 The surface states for Si(lOO) in the 2x1 Feconstmction. Circles indicate experimental data and dashed curves show the calculated surface states. The bands indicated represent a range of energies covered by the surface component of the bulk band structure. Bulk band details are conventionally omitted in such diagrams to emphasize the surface states. Because the surface is reconstructed the surface Brillouin zone is rectangular. The results demonstrate why a maximum in the surface-state density of states occurs at an energy of -0.5 eV. After Surface Science 299/300, Himpsel F.J., Electronic structure of semiconductor surfaces and interfaces. , 525-540 (1994) with permission, copyright Elsevier 1994.

We shall finish with a small example that illustrates the difference between the multiconfigurational wave function approach described in this chapter and the commonly used density functional (DFT) theory. It concerns the quadratic cyclobutadiene system, which is a transition state between the two equivalent rectangular forms of the molecule (cf. Fig. 5-3). 

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