Activities 19952002
MgB2
Resistivity Saturation
Alkalidoped Fullerenes
Clusters
Cuprates
GW Method
HighTemperature Superconductors
LMTObased methods
LmtART
Linear Response DFLMTO method
Quantum MonteCarlo
Stuttgart TBLMTOASA Program
Thirdgeneration MTO
The surprising discovery of a relatively high superconducting transition temperature (39K) in a rather common binary alloy, MgB2, has stimulated intense research in this and related materials. The superconductivity is driven by strong coupling of electrons in the sigmabands to boronboron bond stretching optical phonons. Although the coupling is mediated by classical electronphonon interaction, many researchers now believe that MgB2 shows the first clear manifestation of multiband superconductivity.
A substantial effort, centered around Olle Gunnarsson, has gone into work on the alkalidoped fullerenes. These systems are novel in the sense that the energy scales for electrons and phonons are comparable, and this leads to interesting issues. There are many competing effects, which make the physics rich and pose a challenge to the theoretical treatment and, as we have demonstrated, many properties can be calculated thanks to the clear separation of these solids into weaklycoupled C_{60}molecules. This causes the abovementioned common energy range for the electronic bandwidths and molecular vibrations to be distinct from that of the intramolecular electronic energies, as well as from that of the intermolecular phononic energies. The parameters of the basic Hamiltonian for the electronphonon system was obtained in the early nineties using DF calculations and subsequent adjustments to experiments. The electronic band structures are so simple that they can be given analytically [1].
One important problem is the superconductivity. It is determined both by the electronphonon coupling, , and the Coulomb pseudopotential,, which describes the effects of the Coulomb repulsion. In cooperation with the experimental group of W. Eberhardt in Jülich, the electronphonon coupling parameters were extracted from photoemission experiments for free C_{60 }molecules and it was found that [2]. For conventional superconductors, retardation effects are believed to reduce to a value of about 0.1, but for the alkalidoped fullerenes, A_{3}C_{60} (A=Rb or Cs), we found that these effects are relatively small due to the similarity of the energy scales for phonons and electrons and that, instead, screening effects reduce [3]. As a result, is substantially larger than for conventional superconductors, specifically 0.30.4. This agrees with recent experimental estimates and does not suffice to suppress the electronphonon mechanism for superconductivity.
Another important issue is why not all alkalidoped fullerenes are insulators, as one might expect from estimates of the Coulomb interaction, U, and the band width, W. We have studied the effects of the orbital degeneracy [4], the band filling, and the lattice structure [5]. We find that the orbital degeneracy substantially increases the ratio U/W for which a Mott transition occurs. The lattice structure is important for the difference between A_{3}C_{60} (metal) and A_{4}C_{60} (insulator). We also find that the electronphonon interaction plays a substantial role [5], with an interesting competition between the JahnTeller effect and the Hund's rule coupling.
The electrical resistivity of metallic A_{3}C_{60} is exceptionally large, corresponding to an apparent mean free path l ~ 12 Å at large temperatures T. Thus l is much shorter than the separation d =10 Å between two molecules. This contradicts an observation made for many systems, i.e. that the resistivity saturates when l ~ d, earlier considered as a universal behavior. We have performed essentially exact quantum MonteCarlo calculations for a model of A_{3}C_{60} , including the scattering against intramolecular phonons. We find that for large T indeed l <<d . This illustrates that saturation when l ~ d is not a universal phenomenon [6]. The reasons for the lack of saturation were analyzed, and it was found that the small electronic bandwidth and the coupling to intramolecular phonons are important [See also: FKF Annual Report 1999, p I100104].
A review of superconductivity in the fullerenes has been written [7].
The alkalidoped fullerenes raise some fundamental questions, e.g., whether we can have l <<d for the resistivity, which can be answered only by accurate manybody calculations without uncontrolled approximations. We have therefore developed the tools for doing quantum MonteCarlo calculations of groundstate properties at T = 0 [4] and dynamical correlation functions for T > 0 [6]. The first method uses a projection method approach and a fixed node approximation, while the second method uses a determinantal method. The T > 0 method gives essentially exact results for the electronphonon problem to which it was applied above, and we have demonstrated that the fixednode approximation is rather accurate for the problems where it was used.
The socalled sudden approximation is applied in almost all calculations of photoemission. We have studied its validity for the case of coupling to localized excitations. We found that the sudden approximation becomes valid for rather small photon energies, in contrast to the case of coupling to plasmons. The relevant energy scales for the validity of the sudden approximation were determined [8].
Experimentally, the core level photoemission spectra of cuprates have been found to show a clear dependence on the CuO network, in spite of the locality of the probe. We have calculated the spectrum in the Anderson impurity model, determining the parameters from ab initio DFLMTO calculations [9]. The model was solved using an expansion in the inverse degeneracy, developed in the group. Fig. 1 illustrates the good agreement with experiment. The leading peak results from a valence electron filling the Cu 3d level to screen the Cu core hole in the photoemission process. The core spectrum therefore depends on the valence electronic structure. In Bi_{2}CuO_{4}, Ba_{3}Cu_{2}O_{4}Cl_{2}and Li_{2}CuO_{2} there is little hopping between the Cu atoms because they belong to strings of edgesharing, and therefore uncoupled CuO_{4}squares. As a result, the screening electron comes from a narrow resonance, leading to a narrow peak. In Sr_{2}CuO_{3} and SrCuO_{2}, the CuO_{4}squares form cornersharing and therefore strongly coupled 1D networks. In Sr_{2}CuO_{2}Cl_{2}the CuO_{4}squares form cornersharing 2D networks, like in the superconducting cuprates, and in Ba_{2}Cu_{3}O_{4}Cl_{2} with extra, weakly coupled Cu atoms at the center of every second O_{4}square. LaCuO_{3} has a 3D CuO network. In all these systems additional screening channels enter, leading to more structured leading peaks.
Contrary to the case for the fullerenes, the mechanism behind the superconductivity in the cuprates remains a major puzzle. These hightemperature superconductors (HTSCs) are doped MottHubbard insulators in which superconductivity occurs below a critical temperature, T_{c}(p). The latter depends parabolically on the holedoping p with a maximum at socalled optimal doping. For different materials, max_{p}T_{c} varies substantially.
Research in our group was initially focussed on DF (LDA) calculations for the optimally doped, stoichiometric YBa_{2}Cu_{3}O_{7} of energy bands, optical spectra, phonon frequencies in the normal and superconducting phases, the electronphonon interaction, Raman spectra, and the Fermi surface. The LDA, however, completely fails to describe the electronic structure of the undoped, insulating compounds, and this lead Zaanen et al. to develop the LDA+U method for magnetically ordered MottHubbard insulators at the beginning of the nineties. In this method, the LDA is corrected by addition of the onsite Coulomb interactions, U for the correlated electrons in a static meanfield treatment. Using the LDA+U, Lichtenstein et al. now argued that orbital polarization should be treated on equal footing with spin polarization, and they demonstrated that orbital ordering is a necessary condition for obtaining the crystal structure and exchange couplings observed for the 3d MottHubbard insulator KCuF_{3}[10]. This early work fertilized research on orbital polarization in transitionmetal compounds. Another application of the LDA+U formalism, this time to the 4felectrons in REBa_{2}Cu_{3}O_{7} , enabled Lichtenstein and Mazin to explain why Prdoping suppresses superconductivity more efficiently in NdBa_{2}Cu_{3}O_{7} than in YBa_{2}Cu_{3}O_{7}[11]. Most recently, Dudarev et al. compared the charge density in NiO measured by highenergy electron diffraction with those calculated using various DF approximations. The best match to the experimental charge density was obtained by the generalized gradient approximation (GGA) and the LDA+U with U and J adjusted, but neither method sufficently suppresses the covalent charge density between Ni and O [12].
Whereas the LDA (+U ) gives a parameterfree and reasonable description of the highenergy properties of the HTSCs and produces all electronic bands in a broad energy range, almost any theory of the lowenergy properties uses a oneband, correlated model Hamiltonian. The rationale for this is that, since the superconductivity originates in the CuO_{2} layers and a single layer gives rise to merely one band at the Fermi level, a oneband model should suffice. We have developed a generally applicable method for computing, from a given singleelectron potential, the Wannierlike orbitals which form a basis for such lowenergy, fewband Hamiltonians [13(a),13(b)]. The LDA result for a HTSC is shown in Fig. 2. This orbital is centered on Cu, has d_{x}2_{y}2symmetry, and extends to the 2ndnearest neighbor oxygen atoms. As a consequence, its integrals for singleparticle hopping on the Cucentered square lattice, t, t', t'', ..., have rather long range. We have succeeded in computing this orbital and its hopping integrals for a series of HTSC materials with single layers and find that there is essentially only one parameter which depends on the material, namely the rangeparameter , which gives the ratio, t^{(n)}/t, of any hopping integral to the nearestneighbor one. Most importantly, our calculated rvalues shown in Fig. 3 correlate with the observed max T_{c}. To understand why, is a key to understanding the mechanism of HTSC and poses a challenge to manybody theorists, such as those who currently use a kspace renormalizationgroup method to search for the leading correlationdriven instabilities for various band shapes.
We have understood the much simpler issue of how r is controlled by structural and chemical factors, such as the distance between copper and apical oxygen [14, 15]. In order to explain this, we must first transform to a larger basis of orbitals which are more localized than the one shown on the front page. It is customary to use _{ }O_{x }p_{x , }Cu d_{x}2_{y}2 , and O_{y }p_{y}, in which case the nearestneighbor CuCu hopping (t) is described as: Cu d_{x}2_{y}2 – O_{y }p_{y} – Cu d_{x}2_{y}2 , while the 2nd and farther CuCu hoppings require various intermediate OO hoppings. We have found that by adding one further diffuse Cu 4slike orbital with axial symmetry, the resulting four orbitals are so localized that merely nearestneighbor hoppings (Cu and O) need to be included. Hence, the various OO hoppings of the 3orbital model, and thereby t', t'', ..., of the 1orbital model, all proceed via the axial orbital, and this leads to relations like: t' = 2t''. In the 4orbital model, the dispersion of the bare conduction band is simply:

(1) 
and the rangeparameter is essentially the Cu s character:

(2) 
The energy of the axial orbital is eV's above the Fermi level, , which itself is eV's above the O plevel, . Expansion of (12ru)^{1} yields the oneband form:
The axial orbital not only provides the dominant channel for the intralayer hopping beyond nearest Cuneighbors, but also provides the dominant channel for singleparticle hopping out of the layer, , which according to Eq. (1) depends on k like v^{2 }which has as a nodal line [14]. For materials like La_{2}CuO_{4}, Bi_{2}Sr_{2}CuO_{4}, and Tl_{2}Ba_{2}CuO_{6}, where the stacking of the CuO_{2}layers is bodycentered, the interlayer hopping has an additional nodal line along . This supresses the perpendicular transport significantly, and provides a strong argument against the socalled interlayer tunnelling mechanism for HTSC [15].
In multilayer HTSCs, the neighboring CuO_{2}layers are stacked on top of each other and the integral for hopping between the layers proceeds mainly from Cu s to Cu s[14]. Our predicted value of the interlayer exchange coupling in the bilayer insulator YBa_{2}Cu_{3}O_{6} was verified by Keimer et al. using neutron scattering. Also for metallic multilayer HTSCs, it is usually believed that the Coulomb repulsion localizes the electrons onto the individual layers. If that is not the case, there will be electronic subbands whose (bare) dispersions are given by Eqs.(1)(2), but with added to for a bilayer, and 0 for a triplelayer, and for an infinite layer. Hence, the subbands will have different rvalues and the lowest, most bonding subband will have the highest r. For bilayer YBa_{2}Cu_{3}O_{7} with maxT_{c} we find , for triplelayer HgBa_{2}Ca_{2}Cu_{3}O_{4} with maxT_{c}=135K we find and 0.37, and for infinitelayer CaCuO_{2} we find r ranging from 28 to 0.42. We thus have the unexpected result that the correlation between r and maxT_{c} observed for singlelayer HTSCs, extends to multilayer HTSCs with up to 3 layers, provided that most bonding subband is considered! This speaks against the vanHove scenario, because for YBa_{2}Cu_{3}O_{7} it is the saddlepoint of the antibonding band which is close to but it is consistent with our finding that for YBa_{2}Cu_{3}O_{6+y} it is the bonding subband which can support antiferromagnetic spinfluctuations [16].
In bilayer YBa_{2}Cu_{3}O_{7} the oxygens are dimpled out of the Cuplane by towards Y and away from Ba. This allows for a linear electronphonon coupling of the corresponding modes, of which the Ramanactive ones were studied intensively in Cardona's and our groups at the beginning of the nineties. We found these modes to have relatively strong electronphonon couplings, despite the smallness of the static dimple. The reason is that when the energy of the axial orbital is low, it pushes the like conductionband orbital down in energy so that it becomes nearly degenerate with the orbitals, whereby the hopping between them may be strong. This neardegeneracy term adds to the usual change of t_{pd} with CuO bondlength and angle. By adding the O_{x }p_{z}and O_{y }p_{z }orbitals to the abovementioned 4orbital model, the matrix elements for scattering of an electron from k to k'k+q by the buckling modes of wavevector q can be given analytically [17(a),17(b),17(c)]. For the O_{x}O_{y} mode, this scattering is maximal when k= k'= (p, 0) and it vanishes when k=(p, 0) and k' =(0 ,p). Since the observed d_{x}2_{y}2 wave pairing is presumably caused by a Udriven repulsive pairing interaction which vanishes at small q and peaks for q (p, p), this buckling mode provides a pairing interaction which, although being attractive, might enhance d_{x}2_{y}2wave pairing. The electronphonon coupling constants for both buckling modes, _{x²–y² }, turn out to be identical in the model, and equal to a positive number times the density of states on the Fermi surface weighted by v^{2}. When the Fermi level is at the van Hove singularity, _{x²–y²}=_{s}. This speculation about the buckling modes was confirmed by a full LDALMTO linearresponse calculation for holedoped CaCuO_{2} of the equilibrium structure, all phonon branches, and their electronphonon coupling constants in both s and d_{x}2_{y}2wave channels [17(a),17(b),17(c)]. This calculation showed that the mirror symmetry around a CuO_{2}plane is broken to develop a static O_{x}–O_{y}dimple, that_{x²–y²} 0.3 is contributed predominantly by the buckling modes, and that 0.4.
Over ten years ago, Zaanen and Gunnarsson pointed out that in HTSCs at low doping concentrations, the holes have a tendency to order along parallel, magnetic domain lines. Such striped phases have recently drawn a lot of attention. Magnetic stripes with Q= 2p were observed in La_{1.61/8}Nd_{0.4}Sr_{1/8}CuO_{4} where they are presumably pinned by the nonsuperconducting LTT phase. Recent experiments hint that the stripes might be dynamic and are the cause of the lowenergy incommensurate spin excitations observed in La_{2x}Sr_{x}CuO_{4} throughout the doping range for superconductivity. Whether such dynamical stripes are responsible for or decremental to HTSC remains to be seen. Fleck et al. treated the 2D Hubbard model with a dynamical meanfield scheme, using exact diagonalization, and did find stable [10]oriented striped phases for hole dopings between 0.05 and 0.2 [18(a), 18(b)]. Also most features of the photoemission spectra could be reproduced and interpreted with the subband Hamiltonian generated by the real part of the dynamical mean field. The energy cost for moving a hole perpendicular to the stripe was calculated as a function of the rangeparameter r and found to be negative for r 0.3.
In concluding on HTSCs, possible reasons for the observed correlation between maxT_{c} and the hopping range r may be listed: 1) A longer hopping range screens U better, and maybe even overscreens it locally. 2) t' supresses static stripe order. 3) A low energy of the axial orbital increases the tendency towards buckling and, maybe, formation of bipolarons.
During the last 6 years, several new LMTObased methods were developed. These methods treat d and felectrons as easily as s and pelectrons, and may do so with a small basis set, because the orbitals are constructed from the muffintin (MT) approximation to the potential in the solid. A MTpotential is spherically symmetric inside nonoverlapping spheres surrounding the atoms and constant in between. In the interstitial region, a linear muffintin orbital (LMTO) of the 1st generation is a spherical harmonics times the appropriate spherical Hankel or Neumann function, regular at infinity and with a fixed wavenumber k_{0 }. This envelope function is continued smoothly inside any sphere as the appropriate linear combination of spherical harmonics times radial Schrödingerequation solutions, , and their first energy derivatives, with the energy chosen at the centre of interest. It is intuitively obvious that the way in which a linear combination of LMTOs reproduces a wave function with energy of the solid, is by adding to the head of each LMTO that amount of tails, which makes the total become . Hence, the LMTO set spans the wave functions of the MTpotential to linear order in inside the spheres, but merely to 0th order in the interstitial. For closepacked solids and low energies, one often uses the atomicspheres approximation (ASA), which substitutes the MTspheres with slightly overlapping (spacefilling) atomic spheres and sets k_{0}=0, so that the envelope functions become simple multipole potentials, . In the ASA, the ingredients to the band structure are potential parameters, obtained from the radial functions at the sphere radii, s_{R} and a structure matrix, which specifies the sphericalharmonics expansion around site R' of an LMTOenvelope at site R. In open structures, also the potentials in the voids are taken to be spherically symmetric. The long range of the envelopes with k_{0}=0 were gotten rid of in the early eighties by screening with multipoles on the neighboring sites. This made it possible to generate the structure matrix in real space and to use tightbinding (TB) techniques in DF calculations, e.g. to treat imperfect crystals. The twocentre TBHamiltonian is simply the structure matrix pre and postmultiplied by potential parameters. The TBLMTO set is merely a similarity transformation of the bare set. Another similarity transformation generates an orthonormal set. Downfolding is a third feature of this 2ndgeneration method: If one wants to omit, say the Rlmorbital from a given basis, one must first transform to a representation in which the tails of the neighboring envelopes, when expanded around site R have an lmprojection with the proper radial logarithmic derivative at the sphere, . After this transformation, the Rlmorbital can be deleted from the set, which is now complete to 0th order in the deleted channel. Similarity transformations and downfolding made the 2ndgeneration method intelligible, fast, and useful beyond the realm of closepacked, crystalline transition metals.
Our Stuttgart TBLMTOASA program is in use world wide because it is extremely fast, userfriendly, hardwareindependent, and free of charge. Input is the crystal structure and output is the charge and spinselfconsistent band structure, the partial densities of states, the Fermi surface, plots of the full charge and spindensities, the total energy, and the partial pressures. In addition, the program delivers tools for analysing the electronic structure and chemical bonding such as: Orbitalprojected band structures, COHPs for describing bond strengths, and electron localization functions (ELFs). The division of space into potentialspheres is done automatically [19]. We maintain the programme, give handson courses, and follow up with consultations on specific projects (coauthorship is not a condition).
Hedin's GW approximation for calculating the electron selfenergy has been widely applied to spelectron systems and shown to be quite accurate for these weakly correlated systems. In the early nineties, Aryasetiawan and Gunnarsson developed a GW method based on LMTOs. This made it possible to apply the GW approximation to systems with localized states, e.g., transition metal compounds, and to test its applicability for such strongly correlated systems, in particular NiO [20]. Although the method showed deficiencies for some properties of NiO, it gave surprisingly accurate results for the band structure and magnetic moment. A review of the method was written [21].
For computing totalenergy differences associated with symmetrylowering displacements of the atoms, the ASA is too crude. Various fullpotential (FP) LMTO methods were developed in the late eighties and early nineties. In these more accurate methods, the LMTOs are defined with respect to nonoverlapping potentials, but since the interstitial region is now large, one must use double or triplek sets. A further blow up of the basis set is caused by core states which extend beyond the MTsphere. The charge density in the interstitial is expanded in Hankel functions or plane waves. Similarity transformations and downfolding are not attempted. Hence, the FPLMTO methods are numerically accurate, but not really fast and intelligible.
Savrasov developed an FPLMTO linearresponse method with which phonon dispersions and electronphonon interactions can be calculated efficiently throughout the Brillouin zone [22]. Compared with the supercell method used previously for YBa_{2}Cu_{3}O_{7 } this was a big step forward, although for reasons of computer storage only the simplest HTSC, idealized CaCuO_{2 }could be treated [17(a),17(b),17(c)]. Like the methods of Zein and Baroni, based on planewaves and pseudopotentials, Savrasov's method uses the Sternheimer approach and thereby avoids inverting a large dielectric matrix. To prove the method, Savrasov et al. computed the phonon dispersions and the electronphonon spectral functions for elemental Al, Pb, V, Nb, Ta, Mo, Cu, and Pd, and compared with experimental tunnelling data, temperaturedependent electrical and thermal resistivities, as well as the electronic specific heats. Only for Pd did their specificheat enhancements correctly leave room for a contribution from spinfluctuations. It was concluded that, for the materials considered, the electronphonon interaction is described with 10 per cent accuracy [22(a),22(b)].
Next, Savrasov generalized the method to treat spinfluctuations. Lacking an accurate approximation for the kernel describing dynamical exchangecorrelation effects in timedependent DF theory, the socalled adiabatic GGA was adopted. The efficiency of the approach was demonstrated by applications to the transverse spin fluctuations at T=0 in ferromagnetic Fe and Ni, as well as to the paramagnetic response in Cr and Pd. Some discrepancies between neutron scattering experiments and early semiempirical calculations, as well as a recent frozenmagnon calculation, were resolved [23][see also: FKF Annual Report 1998, I5759]. Savrasov's LmtARTprogram is free software which besides FPLMTO and linearresponse includes TBLMTOASA. A Windows surface is available.
LDA+U calculations of for CaCuO_{2} were initiated, but at that stage it seemed more fruitful that Savrasov go to Rutgers and join the effort initiated by G. Kotliar, A. Georges, and A. Lichtenstein to develop a method with which it should be possible to describe strongly correlated, metallic, real systems. This method is based on dynamical meanfield theory, which includes the energy but not the momentum dependence of the selfenergy through mapping of the Hubbard Hamiltonian obtained from DFLMTO calculations onto the Anderson impurity model, subject to a selfconsistency condition.
We recently succeeded in deriving 3rdgeneration MTOs which treat the energydependence in the interstitial region and in the downfolded channels on the same footing as in the active channels of the spherical regions, and yet keep the short range [24(a),24(b), 13(a), 13(b)]. The first step is to generate socalled screened spherical waves, . These are waveequation solutions in the interstitial with the following boundary conditions: equals at its own sphere and angular momentum, and it vanishes at any other sphere or angular momentum, except in the channels to be downfolded which have logarithmic derivatives equal to The screened spherical waves are localized as long as their energy is below the bottom of the hardsphere continuum. Next, to each solution, of Schrödinger's equation inside a sphere, we attach the corresponding screened spherical wave as a tail extending into the interstitial, and thus form the kinked partial wave . The set of kinked partial waves is characterized by a matrix whose element K_{R'l'm',Rlm}(e) is the kink of in the R'l'm'channel. This kink matrix is generated from the structure matrix and the potential parameters by inversion of positive definite matrices for small clusters. Now, the solutions of Schrödinger's equation at energy e may be expressed as those linear combinations, of kinked partial waves which are smooth, i.e. those for which all kinks vanish: S_{Rlm}K_{R'l'm',Rlm}(e)c_{Rlm,k} =0 for all R'l'm'. Hence, by substituting the partial waves, which in the ASA are truncated outside the atomic spheres, with kinked partial waves, we have recovered the ASA formalism without invoking this approximation.
It turns out that the kinked partial waves span the solutions of Schrödinger's equation for the MTpotential to leading order in the overlap of the spherically symmetric potential wells. As a consequence, the 3rdgeneration MTOs are so accurate, that only minimal sets and atomcentered MTwells are required. This has been confirmed in extensive test calculations for diamondstructured silicon. We find that an sp^{3}set suffices to describe the valence band and that only when the potentialoverlap exceeds 30 per cent in radius, must the overlap correction to the kinetic energy be included explicitly. This correction treats overlaps up to about 60 per cent accurately. Since MTspheres that large are like van der Waal's spheres, we believe that 3rdgeneration MTOs will be useful not only for bulk solids, but also for molecules. We have shown that the overlapping MTpotential should be the leastsquares fit to the full potential, weighted with the valencecharge density, and we have sought ways to solve the resulting coupled radial equations efficiently. The total energies calculated from the full charge density have been found to account well for the pressurevolume curves for the crystalline phases of silicon [24(a), 24(b)].
The kinkcancellation equation, is a screened version of the socalled KKR equation of multiplescattering theory. However, rather than energydependent kinked partial waves, we want a minimal set of energyindependent orbitals which span the solutions of Schrödinger's equation in some range of energies, say the range of the valence band in Si or the range of the conduction band in a HTSC. In the spirit of polynomial approximation, we specify a mesh of N+1 energies, e_{0}, ..., e_{N} and demand that the set of Nthorder MTOs (NMTOs) span the solutions of Schrödinger's equation to within an error proportional to . The set of 0thorder MTOs is obviously the set of kinked partial waves at the energy e_{0}, and the corresponding Hamiltonian and overlap matrices are respectively and . For N=1, one obtains a set of LMTOs expressed in terms of f(e_{0},r), f(e_{1},r), K(e_{0}) and K(e_{1}). Although it is more practical to use discrete energies, we may let e_{1} approach e_{0} and find that the LMTO becomes This is just like in the ASAformalism, except that is now a matrix. It turns out that the LMTO Hamiltonian and overlap matrices are expressed in terms of K(e_{0}), , K(e_{1}), and . We have derived and demonstrated the use of a formalism for MTOs of arbitrary order [13(a), 13(b)]. A summary may be found in the FKF Annual Report 1999 on pp I1623. Fig. 1 of this report shows 0th and 1storder MTOs for Si and demonstrates the point that, as N increases, each NMTO becomes more smooth and less localized. This is the way in which the set can cover a wider energy range without increasing its size. Fig. 3 demonstrates for GaAs how well the 18 valence and lowest conduction bands over a 35 eV range, including the Ga 3d semicore bands, are described with the minimal set of 9 Ga sp^{3}d^{5} and 16 As sp^{3}d^{5}f^{7}QMTOs. Fig. 4 finally demonstrates how well a single orbital, like the one on the front page, obtained by downfolding of all partial waves except Cu d_{x}2_{y}2 can pick out the conduction band of a HTSC.
In conclusion, the 3rdgeneration formalism solves the longstanding
problem of deriving useful minimal sets of shortranged orbitals from scattering
theory. Enter into a calculation: 1) The phase shifts of the (overlapping)
potential wells. 2) A choice of which orbitals to include in the set. 3)
For these, a choice of screening radii to control the orbital ranges. 4)
An energy mesh on which the set will provide exact solutions.
References