Linear combination of muffin-tin orbitals

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. 

The method they used to solve the density functional equations, the so-called linear combination of muffin tin orbitals (LCMTO), due originally to Andersen and Woolley,101-102 is described by them in detail and for these molecules we shall simply discuss the results. 

The seeds for the development of linear methods may be found in the 1971 paper by Andersen [1.21] which contains a definition of muffin-tin orbitals, an addition theorem for tails of partial waves, and the tail cancellation theorem. Soon after, these ideas were developed into a practicable band-calculation method, the linear combination of muffin-tin orbitals (LCMTO) method [1.22,... 

Instead of applying tail cancellation as in Sect.2.1 where we derived the KKR-ASA equations, one may use the linear combination of muffin-tin orbitals (5.27) directly in a variational procedure. This has the advantages that it leads to an eigenvalue problem and that it is possible to include non-muffin-tin perturbations to the potential. According to the Rayleigh-Ritz variational principle, one varies y to make the energy functional stationary, i.e. 

The calculated gaps of the band structures are 5.5 and 5.8 eV at the most stable conformation. These values are somewhat smaller than the one obtained by Kasowski et a/.80 using a linear combination of muffin tin orbitals in their LDA calculation. It should be mentioned that these values are still essentially smaller than the gap of 7.2 eV which we have obtained for our quasi particle band structure at the MP2 level.67 One should observe also that the lower limit of the conduction band is essentially higher (—1.5-2.0 eV), than in the HF + MP2 calculation ( — 5.7 eV67). Being, however, negative in both approximations (GKS and PZ, respectively) it still indicates a possibility of n-doping. 

Implementations have been realized using Gaussian functions (GTO s) ([38, 39] and Slater-type orbitals (STO s) [5, 40, 41], and numerical basis sets [42, 43, 44]. The auxiliary basis may be avoided by the use of a purely numerical representation of the potential on a grid (usually called DVM - Discrete Variational Method [45, 5]), by certain approximations for the potential (Multiple Scattering concept within the so-called mufl5n-tin approximation - [46]), the linear combination of muffin-tin orbitals [47, 3], and in connection with the pseudopotential concept the application of plane-wave basis expansions - see, e.g.. Ref. [112]. 

Linearized band structure methods were developed in the 1970s the linearized augmented plane wave (LAPW) method (36), the linear combination of muffin-tin orbitals (LMTO) method (37), the augmented spherical wave (ASW) method (38), and some others. In the LAPW method a warped muffin tin potential is frequently used, in which the real shape of the crystal potential in the interstitial region between the atomic spheres is taken into account. In the LMTO and ASW approaches the atomic sphere approximation (ASA) is frequently applied, in which— contrary to the muffin-tin approximation—overlapping atomic spheres are used. The crystal potential in the spheres is again assumed to be spherically symmetric. The sum of the atomic sphere volumes must be equal to the total volume of the unit cell. No interstitial space remains. 

Table 9. Calculated surface free energy y of metals for various orientations. The subscripts A and B refer to the two possible surface terminations of (1010) surfaces of hep crystals [910ve], where the termination with the smaller lattice spacing is denoted A [98Vit]. Calculations were performed for T = 0 K. The method of calculation is indicated FS empirical n-body Finnis-Sinclair potential, PSP total energy pseudopotential, EAM embedded atom method, DFT density functional theory, FPLAPW full potential linear combination of augmented waves, FPLMTO full potential linear combination of muffin tin orbitals.

We have used the basis set of the Linear-Muffin-Tin-Orbital (LMTO) method in the atomic sphere approximation (ASA). The LMTO-ASA is based on the work of Andersen and co-workers and the combined technique allows us to treat all phases on equal footing. To treat itinerant magnetism we have employed the Vosko-Wilk-Nusair parametrization for the exchange-correlation energy density and potential. In conjunction with this we have treated the alloying effects for random and partially ordered phases with a multisublattice generalization of the coherent potential approximation (CPA). 

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. 

A different approach was taken by Hao and Cooper (1994), who used a combination of the him linear muffin-tin orbital (LMTO) method and an ab initio molecular quantum cluster method, to investigate S02 adsorption on a Cu monolayer supported by 7—AI2O3. Emphasis here was on the geometry of adsorption sites, with the conclusion that the preferred adsorption site is the Al—Al bridging one. 

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). 

DOS = Density of states BO = Bloch orbital IBZ = Irreducible Brillouin zone BZ = Brillouin zone PZ = Primitive zone COOP = Crystal orbital overlap population CDW = Charge density wave MO = Molecular orbital DFT = Density functional theory HF = Hartree-Fock LAPW = Linear augmented plane wave LMTO = Linear muffin tin orbital LCAO = Linear combination of atomic orbitals. 

There are a number of band-structure methods that make varying approximations in the solution of the Kohn-Sham equations. They are described in detail by Godwal et al. (1983) and Srivastava and Weaire (1987), and we shall discuss them only briefly. For each method, one must eon-struct Bloch functions delocalized by symmetry over all the unit cells of the solid. The methods may be conveniently divided into (1) pesudopo-tential methods, (2) linear combination of atomic orbital (LCAO) methods (3) muffin-tin methods, and (4) linear band-structure methods. The pseudopotential method is described in detail by Yin and Cohen (1982) the linear muffin-tin orbital method (LMTO) is described by Skriver (1984) the most advanced of the linear methods, the full-potential linearized augmented-plane-wave (FLAPW) method, is described by Jansen... 

Hence, any linear combination of the orthogonal functions (r) and (r), e.g. J (r), is orthogonal to the core states. Therefore, the augmentation of the muffin-tin orbital by a (r) function (mentioned at the end of the previous section) is one particular way of orthogonalising the orbital to the core states of the neighbouring atoms. 

LA LCAO LDA LED LEED LMTO LO LPE LTLPE longitudinal acoustic linear combination of atomic orbitals local density approximation light emitting diode low energy electron diffraction linear muffin-tin orbital longitudinal optical liquid phase epitaxy low temperature liquid phase epitaxy... 

By using the tight-binding linearized muffin-tin orbital method combined with die coherent-potential approximation (TB-LMTO-CPA) the total energies, bulk moduli, equilibrium lattice parameters, magnetie moments, and hyperfine fields of bcc solid solution were ealeulated by [2000San], and are in qualitative agreement with experimental trends. 

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