Muffin-tin model

The computational survey of electronic structure of metals by Moruzzi et al. [25 6] is a landmark example of the original KKR formalism, using the muffin-tin model and... 

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. 

For energies lower than 20 eV, the muffin-tin model is often inadequate outside the muffin-tin spheres. In the case of surfaces, this applies especially to low-energy studies of adsorbed molecules. For instance, HREELS is normally performed at kinetic energies around 5-10 eV (to benefit from a strong reflection... 

Electron-atom scattering is central to all techniques under discussion here. As we have mentioned, electron-atom scattering at surfaces has been treated almost exclusively by means of the muffin-tin model. 

The phase shifts <5, are calculated by standard partial-wave scattering theory. It involves the electron-atom interaction potential of the muffin-tin model. There are a variety of ways to obtain this potential, which consists of electrostatic and exchange parts (spin dependence may be included, especially when the spin polarization of the outgoing electrons is of interest). One usually starts from known atomic wave functions within one muffin-tin sphere and spherically averages contributions to the total charge density or potential from nearby... 

The investigations of Asada et al. and Christensen - were carried out with linear-muffin-tin orbitals within the atomic sphere approximation (LMTO- AS A) Within the muffin-tin model suitable s, p and d basis functions (muffin-tin orbitals, MTO) are chosen. In contrast to the APW procedure the radial wave functions chosen in the linear MTO approach are not exact solutions of the radial Schrodinger (or Dirac) equation. Furthermore, in the atomic sphere approximation (ASA) the radii of the atomic spheres are of the Wigner-Seitz type (for metals the spheres have the volume of the Wigner-Seitz cell) and therefore the atomic spheres overlap. The ASA procedure is less accurate than the APW method. However, the advantage of the ASA-LMTO method is the drastic reduction of computer time compared to the APW procedure. 

Most of the present implementations of the CPA on the ab-initio level, both for bulk and surface cases, assume a lattice occupied by atoms with equal radii of Wigner-Seitz (or muffin-tin) spheres. The effect of charge transfer which can seriously influence the alloy energetics is often neglected. Several methods were proposed to account for charge transfer effects in bulk alloys, e.g., the so-called correlated CPA , or the screened-impurity model . The application of these methods to alloy surfaces seems to be rather complicated. 

Theoretical calculations were performed with the linear muffin tin orbital (LMTO) method and the local density approximation for exchange and correlation. This method was used in combination with supercell models containing up to 16 atoms to calculate the DOS. The LMTO calculations are run self consistently and the DOS obtained are combined with the matrix elements for the transitions from initial to final states as described in detail elsewhere (Botton et al., 1996a) according to the method described by Vvedensky (1992). A comparison is also made between spectra calculated for some of the B2 compounds using the Korringa-Kohn-Rostoker (KKR) method. 

The CPA has proved to be an enormously successful tool in the study of alloys, and has been implemented within various frameworks, such as the TB, linear muffin-tin orbital and Korringa-Kohn-Rostoker (Kumar et al 1992, Turek et al 1996), and is still considered to be the most satisfactory single-site approximation. Efforts to do better than the single-site CPA have focused on multi-site (or cluster) CPA s (see, e.g., Gonis et al 1984, Turek et al 1996), in which a central site and its set of nearest neighbours are embedded in an effective medium. Still, for present purposes, the single-site version of the CPA suffices, and we derive the necessary equations here, within the framework of the TB model. 

In this paper we present preliminary results of an ab-initio study of quantum diffusion in the crystalline a-AlMnSi phase. The number of atoms in the unit cell (138) is sufficiently small to permit computation with the ab-initio Linearized Muffin Tin Orbitals (LMTO) method and provides us a good starting model. Within the Density Functional Theory (DFT) [15,16], this approach has still limitations due to the Local Density Approximation (LDA) for the exchange-correlation potential treatment of electron correlations and due to the approximation in the solution of the Schrodinger equation as explained in next section. However, we believe that this starting point is much better than simplified parametrized tight-binding like s-band models. 

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. 

Anderson and Woolley148 have proposed a linear combination of muffin-tin orbitals method for molecules based on the earlier cellular model.147 No applications to large molecules have been reported. 

PWPP refers to plane-wave pseudopotential results. bK-H Keating-Harrison model. Linear muffin-tin orbitals. dLinear combination of atomic orbitals (value at 300 K). 

In the one-electron transition model it is assumed that only one core electron is excited to an unfilled state present in the initial, unperturbed solid. The remaining electrons are assumed to be unaffected, remaining frozen in their original states. In essence the one-electron model describes the potential seen by the final-state electrons as nonoverlapping spherically symmetric spin-independent potentials (muffin-tin scatterers) centered around an atom from which the X-ray cross section from a deep core level of an atom to final states above the Fermi level can in principle be calculated for any energy above threshold. 

More recently, Boness et al. (1986) calculated the band structure of both e-Fe and 7-Fe using the linear muffin-tin-orbital method under conditions corresponding to pressures ranging from 4 to 980 GPa. The results were used to, in turn, calculate electronic thermodynamic quantities such as the electronic specific heat and the electronic Gruneisen parameter, useful quantities with which to constrain models of the Earth s core. 

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-... 

All the TB models discussed so far are based on the nearest neighbor interactions. Recently, Sapra et al. proposed [73] the sp d tight-binding model with cation-anion nearest neighbor and anion-anion next nearest neighbor (NNN) interactions for the A B semiconductor compounds with A = Zn, Cd, Hg and B = S, Se, Te. The model was chosen after a careful analysis of the bulk band structures of these compounds obtained from the linearized muffin tin orbital (LMTO) method as described earlier in this section. These calculations were car-... 

Furthermore, within the (R)APW method the so called muffin-tin approach is used for calculating V(f). According to this model the volume of the unit cell to is separated into the volume tOy of non overlapping and approximately touching atomic spheres (muffin-tin spheres, cf. Fig. 4) centred at the lattice sites y and the volume to between the spheres. In Table 5 the radii ry and the volumes o), which are used for the RAPW calculations of Zintl phases are given. Because of the arrangement of the atoms in the crystal shown in Sect. B, the volumes to a and Wigner-Seitz volumes are listed too. 

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