Muffin-tin approximation

Foulis, D.L., Pettifer, R.F., and Sherwood, P, 1995, The removal of the muffin-tin approximation and use of self-consistent-field electron densities for calculating the K-edge X-ray absorption nearedge structure of chlorine, Europhys. Lett. 29 647. 

Accurate LCAO-SCF-Xa calculations (no muffin-tin approximation) for a Ni5 CO cluster203,2041 give the following picture ... 

Loubriel205) uses a SCF-MSXct approach with muffin-tin approximation to study a Ni(CO)4 molecule, reaching the following conclusions ... 

A variant of the MsXa method has been proposed in which a is varied,256 and applications to N2 and Oa have been reported.257 For a = 0.70, results were in good agreement with experiment, mainly because of the removal of the muffin-tin approximations. 

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 disadvantage is the inadequacy of the muffin-tin approximation for systems with low symmetry or open structure, which renders the determination of orbital energies and total binding energies unreliable. In the following section we shall describe an improved method which relaxes the muffin-tin approximation and is potentially valuable in adsorbate structure determination. 

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. 

Due to the muffin-tin approximation of the local multi-centre potential, the applicability of the scattered-wave (SW) method is limited to relatively small and high-symmetry clusters and molecules. On the other hand, it is the prerequisite for generalizing the in-out integration scheme of atomic one-electron wavefiinctions to the multi-centre case. This approach furnishes the multi-centre electronic structure in a particular atom sphere in its single-centre representation. Thus atomic transition-matrix elements can be readily rescaled in terms of near-nucleus electron amplitudes. 

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 ... 

Figure 5.16 Top experimental normal emission (T-L) ARUPS spectrum for Cu (111), hv = 21.2 eV (Geriach etal. 1998). Bottom corresponding calculated ARUPS spectram using the muffin-tin approximation and the full potential version of relativistic multiple scattering theory. Reproduced with permission from Fluchtmann et aL (1998). Elsevier Science.

We will also use the so called muffin-tin approximation for the potential ... 

In the earliest implementation applied to molecular problems, K. Johnson [39] used scattered-plane waves as a basis and the exchange-correlation energy was represented by (13). This SW-Xa method employed in addition an (muffin-tin) approximation to the Coulomb potential of (17) in which Vc is replaced by a sum of spherical potentials around each atom. This approximation is well suited for solids for which the SW-Xa method originally was developed [40]. However, it is less appropriate in molecules where the potential around each atom might be far from spherical. The SW-Xa method is computationally expedient compared to standard ab initio techniques and has been used with considerable success [41] to elucidate the electronic structure in complexes and clusters of transition metals. However, the use of the muffin-tin approximation precludes accurate calculations of total energies. The method has for this reason not been successful in studies involving molecular structures and bond energies [42]. 

The first implementations of a self-consistent KS-scheme, without recourse to the muffin-tin approximations, are due to Ellis and Painter [43], Baerends et al. [44], Sambe and Felton [45], Dunlap et al. [46] as well as Gunnarson et al. [47]. The accurate representation of Vc is in general accomplished by fitting the molecular density to a set of one-center auxiliary functions [44, 46] fn r ) as... 

The extraction of the variation of the interatomic distances in unknown systems is complicated by the determination of = (ii/2 m(E — Eg -b V) because V, the average interstitial potential in the muffin tin approximation, is unknown. In order to overcome the problem of V, which cannot be determined experimentally, it has been pointed out that the energy separation of a multiple scattering resonance in the continuum from a bound state at threshold AE = (E,. — E ) can be used to determine the variation of the interatomic distance. 

The magnitude of Slater s exchange term in the Hamiltonian was questioned three years later by Caspar [10] who, through a different derivation, obtained the same form, but with a factor of 2/3. This new value was confirmed later by Kohn and Sham. The factor under debate, then named a, was used later on as an atom-dependent parameter [11] in the so-called Xa methodologies. The first among these was derived from the original muffin-tin approximation of Slater [8] and generalized by Johnson [12,... 

The linear methods devised by Andersen [1.19] are characterised by using fixed basis functions constructed from partial waves and their first energy derivatives obtained within the muffin-tin approximation to the potential. 

Fig.5.1. The muffin-tin approximation. a) the unit cell, the muffin-tin sphere of radius Sjvjj, and the escribed sphere of radius Sjr. 

Fig.6.1. The kinetic energy k of the interstitial region in (a) the muffin-tin approximation, and (b) the atomic-sphere approximation...

Johnson KH (1966) Multiple-scattering model for polyatomic molecules. J Chem Phys 45 3085-3095 Johnson KH (1973) Scattered-wave theory of the chemical bond. Adv Quantum Chem 7 143-185 Joly Y (2001) X-ray absorption near-edge stmcture calculations beyond the muffin-tin approximation. Phys Rev B63 125120-1-10... 

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