Muffin tin potentials

The muffin-tin potential around each atom in the unit cell has been calculated in the framework of the Local-Spin-Density-Approximation using the ASW method. The ASW method uses the atomic sphere approximation (ASA), i.e. for each atom a sphere radius is chosen such that the sum of the volumes of all the overlapping spheres equals the unit cell volume. The calculation yields the expected ferromagnetic coupling between Cr and Ni. From the self-consistent spin polarized DOS, partial and total magnetic moment per formula unit can be computed. The calculated total magnetic moment is 5.2 pg in agreement with the experimental value (5.3 0.1 e calculations presented here have been performed... 

These approximations, and additional ones involving the muffin tin potential, have been described in recent reviews. (3 - ) In particular, the work reported here is entirely analogous to our earlier calculations on porphyrins,(5-10) where we discuss the method of calculation in some detail. 

Subsequent cellular methods, on which there is an enormous literature, will not be described here. We shall, however, need to introduce- certain ideas, particularly that of the pseudopotential. We begin by introducing the concept of the muffin-tin potential due to Ziman (1964a). This is illustrated in Fig. 1.9. The tight-binding approximation is appropriate for states with energies below the muffin-tin zero ( bound bands in Ziman s notation). If the energy is above the... 

Fig. 1.9 The muffin-tin potential for an electron in a solid. The potential differs appreciably from zero in the shaded region. From Ziman (1964b).

It should be emphasized that in metals the d-states, for which tight-binding functions may be used, lie above the zero of the muffin-tin potential The reason why the tight-binding method can still be used is the following. The radial part of the Schrodinger equation is... 

Classical anharmonic spring models with or without damping [9], and the corresponding quantum oscillator models seem well removed from the molecular problems of interest here. The quantum systems are frequently described in terms of coulombic or muffin tin potentials that are intrinsically anharmonic. We will demonstrate their correspondence after first discussing the quantum approach to the nonlinear polarizability problem. Since we are calculating the polarization of electrons in molecules in the presence of an external electric field, we will determine the polarized molecular wave functions expanded in the basis set of unperturbed molecular orbitals and, from them, the nonlinear polarizability. At the heart of this strategy is the assumption that perturbation theory is appropriate for treating these small effects (see below). This is appropriate if the polarized states differ in minor ways from the unpolarized states. The electric dipole operator defines the interaction between the electric field and the molecule. Because the polarization operator (eq lc) is proportional to the dipole operator, there is a direct link between perturbation theory corrections (stark effects) and electronic polarizability [6,11,12]. 

Gyorffy, B.L. (1972). Coherent-potential approximation for a nonoverlapping muffin-tin potential model of random substitutional alloys, Phys. Rev. B 5, 2382-2384. 

Results obtained by this method have been compared with the Xa scattered-wave results for FeCU anions.144 Results for ferrocene compare well with ab initio results and experiment.145 They appear to give better ionization potentials than the ab initio results of Veillard.83 This method has several clear advantages. The use of the LCAO form avoids the muffin-tin potentials and may make the method more easily visualizable for chemists. The numerical integration imposes no restriction on the form of the basic functions. STOs can be used as readily as CGTOs. The Ni problem is avoided since only N2 matrix elements need be stored. 

Band Structure for the Muffin-Tin Potential. The muffin-tin potential assumes... 

Augmented Plane Waves (APW). This method is a detailed application of the muffin-tin potential, as is the Korringa-Kohn-Rostocker Method (KKR) [30],... 

The well-known muffin-tin model of the electron atom interaction potential has in most cases proved to be an adequate compromise between accuracy and computational efficiency, at least for electron kinetic energies exceeding 20 eV./27,28,29/ The muffin-tin potential is spherical inside the muffin-tin spheres. This ion core usually provides the dominant contribution to electron scattering and emission, to be discussed shortly. 

In the case of photoexcitation from a bound state of a single atom a with a muffin-tin potential, one obtains/82/ the following amplitudes for out going spherical waves L = (l,m) ... 

Perhaps the most successful representation of the wave functions for band calculations for semiconductors has been the OPW method (orthogonalized plane-wave method), developed by Herring (1940). The success of the method has been due to the ease of obtaining and using realistic potentials in the calculation, in contrast to methods that utilize the muffin-tin approximation to the potential (discussed in Chapter 20). Only recently have difficulties with the application of muffin-tin potentials to semiconductors been overcome. (P or discussion and references see Johnson, Norman, and Connolly, 1973.) For any given potential, any of the accurate methods should give the same bands if the necessary effort is applied. 

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. 

Andersen s assumption of a muffin-tin potential equal to the energy of the d slate, which was used in the preceding section to obtain muffin-tin orbitals, is used here in two ways. (This analysis has only very recently been made see Harrison and Froyen, 1979.) First, it is used to obtain the matrix elements and second, it is used in writing F, - = in Eq. (20-35). We begin with... 

We again take the flat portion of the muffin-tin potential equal to the energy as we did in Fig. 20-13,b, but now it is desirable to take the muffin-tin radius smaller than r, such that the potential is continuous at Fig. 20-13 has been modified to do this, in Fig. 20-17. Taking this muffin-tin potential as the potential in the metal, we immediately see that the potential in the metal minus that in the free atom is simply... 

The atomic d state in chromium and the atomic potential, V (r), as in Fig, 20-13. A muffin-tin potential is constructed with r , chosen to make the potential continuous, and equal to Sj for r >. The difference... 

Eq. (20-33). That equation requires more scrutiny now that we have the form of the hybridization matrix elements. Let us look first at the energy at k = 0, = 0 and, since is proportional to <, the final term is zero also. The energy is simply <0 kE 0>. But we have measured energies from the flat portion of the muffin tin potential, which was taken at the r/-state energy, so it follows that... 

The quantum-mechanical description of minerals containing transition metals is at a less advanced stage. The accuracy of simple Hartree-Fock-Roothaan methods has not been fully determined for such systems. Local-density-functional methods have been successful for calculating the structural properties of high-symmetry materials, but excitation energies are still poorly reproduced. Local-density-functional cluster calculations have so far been restricted mostly to model potentials (e.g., muffin-tin potentials) so that their full power has not been utilized. We need to determine the accuracy and efficiency of Hartree-Fock-Roothaan (or Har-... 

A band-theoretical method employing density-functional theory, a composite basis set, and a muffin-tin potential... 

Scattered-Wave (SW) or Multiple Scattering (MS) (J 9) Muffin-tin potential Partial-wave expansion rapid good one-electron props., total energy unreliable... 

FIG. 4. A schematic overview of the muffin-tin potential in- a molecule with the division of the space in three regions as described in the text. 

FIG. 6. The valence molecular eigenvalues,MSM, in the muffin-tin potential for FeCl4 calculated using different basis sets as discussed in the text [78]. 

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