Muffin-tin constant

The interstitial region between these spheres is represented by a constant potential valve (the muffin-tin constant or muffin-tin zero). This is the region of unscattered electron propagation, in which simple plane-wave or spherical-wave behavior describes the electron wave-field. The plane-wave description is computationally advantageous when dealing with multiple scattering by periodic lattices, since such lattices diffract any plane wave into other well-defined plane waves. This approach is often useful even with non-periodic overlayers adsorbed on an otherwise periodic substrate. 

The muffin-tin constant Co is simply fixed to the average of the full potential in the space outside the potential sphere, but inside the unit cell the interstitial. If the muffin-tins overlap, one can get another equation for the muffin-tin constant... 

With the atomic scattering represented by a set of phase shifts 5 , we now assemble atoms into layers and calculate the latter s coherent elastic scattering, that is, diffraction. This is according to the so-called muffin-tin potential displayed in Figure 3.2.1.21a. In the example shown, the muffin-tin constant - equivalent to the real part of the inner potential Vor - consists of a contribution of a possible adlayer and that of the subsequent substrate. As we are dealing with a two-dimensionally periodic arrangement, we will get only intensities in discrete directions as described by the layer s reciprocal lattice vectors gy = gj, of the layer. Accordingly, this will create... 

Having recognized that the muffin-tin approximation, widely used in molecular calculations, is rather severe, they use the overlapping atomic spheres concept (60). This concept has been considered to lead to an improved description of ionisation potentials of molecules where a substantial fraction of the charge due to the valence electrons is distributed over the interatomic region of constant potential (60). The SCF Xa calculations on Nis—CO and Ni4—CO clusters yield three main peaks, one due to the 5-band and two which can be related to the n and 5relative energies of the peaks agree well with the experimental values. In contradiction to the initial tentative assumption of Eastman and Cushion and to the CNDO results of Blyholder, the first peak of the adsorbed CO is found to be due to the 1 n orbital and the second due to the 5a orbital. 

The parameter is listed in the Solid State Table for each transition clement. The theory used in Section 20-E does not give useful values for the three universal constants and but Muffin-Tin Orbital theory (Section 20-D) sug-... 

Imagine again that spheres are constructed around each atom of a transition metal. A muffin-tin potential, constant between spheres and spherically symmetric within, is assumed. In the context of this section, it will be best to let the spheres be nonoverlapping. 

The particular functional form of Equation 10 is certainly not the only conceivable dimensionally-consistent possibility, but was chosen on the basis of algebraic simplicity (12). Viewed in a semi-empirical spirit, it is found that Equation To, with suitable values of the constants b and c, fits exact atomic exchange energies throughout the periodic table with extremely high accuracy (i.e. roughly 0.2% error compared with 5-10% error for the exchange-only LDA). It should be noted in this connection that constants b and c are "universal" constants, and not adjustable from atom to atom as is the a parameter in so-called "muffin-tin" Xa schemes. 

Methfessel, M., Elastic constants and phonon frequencies of Si calculated by a fast full-potential linear-muffin-tin-orbital method, Phys. Rev. B, 38, 1537, 1988. 

According to the muffin-tin approximation the potential V(f) inside the muffin-tin spheres is replaced by its spherical mean V(r), and outside the atomic spheres V(f) is approximated by a constant Vq. The electron-electron interaction in this region is calculated from the constant charge density pg defined by ... 

Table 12. Charges inside the atomic spheres Qy and the ratios Qy VOy of the 1 partial charges for LiTl and NaTl. For both compounds the self consistent scalar relativistic APW calculations have been performed for both structures B2 and B32. The lattice constants are chosen aB32 = 2 agi where aB32(NaTl) = 7.47 A and aB2(Lin) = 3.42 A are the lattice constants found experimentally. The volumes of the muffin-tin sphere are Wu(LiTl) = w-nCLiTI) = 12.47 A and ti)nj(NaTl) = o)Ti(NaTl) = 16.84 A ...

The case of Cr(CO)g illustrates the extreme difficulty of obtaining a convincing ETS assignment using calculation alone. The MS-Xa method is of course limited by its use of a muffin-tin potential. For unoccupied orbitals a significant fraction of the electron density occurs in the interatomic region the least accurately treated part of the molecule. In general, diffuse orbitals tend to be overstabilized by the constant interatomic potential. 

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