LMTO

We now discuss the most important theoretical methods developed thus far the augmented plane wave (APW) and the Korringa-Kolm-Rostoker (KKR) methods, as well as the linear methods (linear APW (LAPW), the linear miiflfm-tin orbital [LMTO] and the projector-augmented wave [PAW]) methods. 

The LMTO method [58, 79] can be considered to be the linear version of the KKR teclmique. According to official LMTO historians, the method has now reached its third generation [79] the first starting with Andersen in 1975 [58], the second connnonly known as TB-LMTO. In the LMTO approach, the wavefimction is expanded in a basis of so-called muffin-tin orbitals. These orbitals are adapted to the potential by constmcting them from solutions of the radial Scln-ddinger equation so as to fomi a minimal basis set. Interstitial properties are represented by Hankel fiinctions, which means that, in contrast to the LAPW teclmique, the orbitals are localized in real space. The small basis set makes the method fast computationally, yet at the same time it restricts the accuracy. The localization of the basis fiinctions diminishes the quality of the description of the wavefimction in die interstitial region. 

In the connnonly used atomic sphere approximation (ASA) [79], the density and the potential of the crystal are approximated as spherically synnnetric within overlapping imifiBn-tin spheres. Additionally, all integrals, such as for the Coulomb potential, are perfonned only over the spheres. The limits on the accuracy of the method imposed by the ASA can be overcome with the fiill-potential version of the LMTO (FP-LMTO)... 

The LMTO method is the fastest among the all-electron methods mentioned here due to the small basis size. The accuracy of the general potential teclmique can be high, but LAPW results remain the gold standard . 

Tank R W and Arcangell C 2000 An Introduction to the third-generation LMTO method Status Solid B 217 89... 

Figure 3-18. LMTO correction factors 2-pass, cross-flow bol+i fluids unmixed. (From Gas Processors Suppliers Association, Engineering Data Book, 9th Edition.)...

We have used the multisublattice generalization of the coherent potential approximation (CPA) in conjunction with the Linear-MufRn-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 for the local spin density approximation (LSDA) the Vosko-Wilk-Nusair parameterization". 

In summary, we have demonstrated the possibility of calculating the phase stability of a magnetic random alloy from first principles by means of LMTO-CPA theory. Our calculated phase diagram is in good agreement with experiment and shows a transition from the partially ordered a phase to an hep random alloy at 85% Co concentration. 

We observe that for the Fe-Co system a sim le spin polarized canonical model is able to reproduce qualitatively the results obtained by LMTO-CPA calculations. Despite the simplicity of this model the structural properties of the Fe-Co alloy are explained from simple band-filling arguments. 

H.L. Skriver, The LMTO Method (Springer-Verlag, Berlin, 1984). 

Our results demonstrate that the augmented space recursion and the orbital peeling method in conjunction with the LMTO formalism, constitute a viable and computationally feasible approach to the calculation of phase stability in binary substitutionally disordered alloys. ... 

Second, using the fully relativistic version of the TB-LMTO-CPA method within the atomic sphere approximation (ASA) we have calculated the total energies for random alloys AiBi i at five concentrations, x — 0,0.25,0.5,0.75 and 1, and using the CW method modified for disordered alloys we have determined five interaction parameters Eq, D,V,T, and Q as before (superscript RA). Finally, the electronic structure of random alloys calculated by the TB-LMTO-CPA method served as an input of the GPM from which the pair interactions v(c) (superscript GPM) were determined. In order to eliminate the charge transfer effects in these calculations, the atomic radii were adjusted in such a way that atoms were charge neutral while preserving the total volume of the alloy. The quantity (c) used for comparisons is a sum of properly... 

Figure 1. The effective pair interactions as functions of alloy composition for the alloy system Al-Ni. The results of the CWIS based on the FP-LAPW calculations for 5 ordered structures (full line) and on the TB-LMTO-CPA for 5 disordered alloys (dash line) are compared with the the results of the GPM (dotted line).

We conclude that more work is need<. In particular it would be useful to repeat the TB-LMTO-CPA calculations using also other methods for description of charge transfer effects, e.g., the so-called correlated CPA, or the screened-impurity modeP. One may also cisk if a full treatment of relativistic effects is necessary. The answer is positive , at least for some alloys (Ni-Pt) that contain heavy elements. 

In a previous work we showed that we could reproduce qualitativlely the LMTO-CPA results for the Fe-Co system within a simple spin polarized canonical band model. The structural properties of the Fe-Co alloy can thus be explained from the filling of the d-band. In that work we presented the results in canonical units and we could of course not do any quantitative comparisons. To proceed that work we have here done calculations based on the virtual crystal approximation (VGA). In this approximation each atom in the alloy has the same surrounding neighbours, it is thus not possible to distinguish between random and ordered alloys, but one may analyze the energy difference between different crystal structures. 

To summarize we have reproduced the intricate structural properties of the Fe-Co, Fe-Ni and the Fe-Cu alloys by means of LMTO-ASA-CPA theory. We conclude that the phase diagram of especially the Fe-Ni alloys is heavily influenced by short range order effects. The general trend of a bcc-fcc phase transition at lower Fe concentrations is in accordance with simple band Ailing effects from canonical band theory. Due to this the structural stability of the Fe-Co alloys may be understood from VGA and canonical band calculations, since the common band model is appropriate below the Fermi energy for this system. However, for the Fe-Ni and the Fe-Cu system this simple picture breaks down. 

For the description of the random Hamiltonian we employ TB-LMTO formalism in the most tight binding representation . The Hamiltonian for the binary random alloy takes the form ... 

Figure 2. Total energies of ordered (LIq structure, squares), random (circles) and segregated (triangles) fee RhsoPdso alloys as a function of the number of neighboring shells included in the local interaction zone. Values obtained by the LSGF-CPA method are shown by filled symbols and full lines. The energies obtained by the reference calculations are shown by a dashed line (LMTO, ordered sample), a dotted line (LMTO-CPA, random sample), and a dot-dashed line (interface Green s function technique, segregated sample).

A different approach is adopted here. Within the LMTO-ASA method, it is possible to vary the atomic radii in such a way that the net charges are non-random while preserving the total volume of the system . The basic assumption of a single-site theory of electronic structure of disordered alloys, namely that the potential at any site R depends only on the occupation of this site by atom A or B, and is completely independent of the occupation of other sites, is fulfilled, if the net charges... 

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. 

Figure 1. Ni L23 edges in NiAl and Ni (fee) for a) EELS experiments and b) LMTO ealeulations.

Figure 2. Co L23 edges in CoAl shown for EELS experiments (dots), LMTO and KKR calculations (lines).

Figure 3. Al L23 edges in FeAI and TiAI (EELS experiments dots) and LMTO calculations for FeAI (line).

Figure 4. Co L23 edges in Ni 43 Co o Al 50 EELS (dots) and LMTO calculation of the Co L23 edge in the Ni7CoA18 supercell (full line).

Calculations were done with a full-potential version of the LMTO method with nonoverlapping spheres. The contributions from the interstitial region were accounted for by expanding the products of Hankel functions in a series of atom-ce- -ered Hankels of three different kinetic energies. The corrected tetrahedron method was used for Brillouin zone integration. Electronic exchange and correlation contributions to the total energy were obtained from the local-density functional calculated by Ceperley and Alder " and parametrized by Vosko, Wilk, and Nusair. ... 

Three LMTO envelopes were used with the tail energies -0.01 Ry, -1 Ry and -2.3 Ry. In the first two of them, s,p,d orbitals were included and in the last one only. s and p were used. It was necessary to treat the Ti 3p and 3-s states in the semi-core state, i.e. to do a so called 2-panel calculation. The basis set for the second panel consisted of 3-s, 3p, 3d orbitals on the Ti sites and 3-s, 3p orbitals on the Si sites. The same quality k-mesh was used in all calculations to ensure maximum cancellation of numerical errors and to obtain accurate energy differences. 

To be consistent, the minimum energies from the LMTO-program were used even though this underestimated the lattice constant. Murnaghan s equation of state was used to determine bulk moduli and equilibrium volumes. The energy calculations were converged within 1 mRy/atom. 

Composition Structure Vq ((a.u.) /atom) Bo (GPa) -AH (mRy/atom) LMTO Expt LMTO Expt LMTO Expt ... 

IMPROVED LMTO-ASA METHODS PART II TOTAL ENERGY... 

In this paper we will present a way to improve the evaluation of total energies in LMTO calculations. The Kohn Sham energy functional [1] can be written in the form... 

Consider the electrostatic terms. These are hard to evaluate because the output charge density is a complicated non spherical function in space. In traditional LMTO calculations the charge density is first spheridised before Ees is calculated. In this method p(r) is reduced to a sum of spherically symmetric balls of charge inside each ASA sphere [2]. 

We will refer to this as the one center expansion. The components pRx are easily calculated in LMTO and are known on a radial mesh. In the interstitial region we choose to expand the charge density in the SSW s xp) and their first two energy derivatives. 

To summarise, we have presented a way to improve an LMTO-ASA calculation of the electrostatic energy in a crystal. The method is stable and general in its formalism so that it should be applicable to a wide range of systems. In this talk we did not mention the exchange correlation energy. It is possible to make an expansion of the (xc(p(r)) in terms of the SSW s. Then the integral... 

---