Scientific Activities

Activities 1995-2000
Alkali-doped Fullerenes
Clusters
Cuprates
GW Method
High-Temperature Superconductors
LMTO-based methods
LmtART
Linear Response DF-LMTO method
Quantum Monte-Carlo
Stuttgart TB-LMTO-ASA Program
Third-generation MTO
 
 

Activities 1995-2000

Our theory group attempts to improve the understanding of the properties of condensed matter by means of electronic structure calculations. This involves performing parameter-free density-functional (DF) calculations for specific systems of current interest, extending DF theory and going beyond, deriving electronically correlated model Hamiltonians, solving these Hamiltonians, and developing numerical tools for doing so. For most applications, it is crucial to use an intelligible, general, and numerically efficient single-electron representation, and this causes recurrent developments of the LMTO method.

A substantial effort, centered around Olle Gunnarsson, has gone into work on the alkali-doped 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 weakly-coupled C60-molecules. This causes the above-mentioned common energy range for the electronic bandwidths and molecular vibrations to be distinct from that of the intra-molecular electronic energies, as well as from that of the inter-molecular phononic energies. The parameters of the basic Hamiltonian for the electron-phonon 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 electron-phonon 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 electron-phonon coupling parameters were extracted from photoemission experiments for free C60 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 alkali-doped fullerenes, A3C60 (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 0.3-0.4. This agrees with recent experimental estimates and does not suffice to suppress the electron-phonon mechanism for superconductivity.

Another important issue is why not all alkali-doped 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 A3C60 (metal) and A4C60 (insulator). We also find that the electron-phonon interaction plays a substantial role [5], with an interesting competition between the Jahn-Teller effect and the Hund's rule coupling.

The electrical resistivity of metallic A3C60  is exceptionally large, corresponding to an apparent mean free path l ~ 1-2 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 Monte-Carlo calculations for a model of A3C60 , including the scattering against intra-molecular 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 intra-molecular phonons are important [See also: FKF Annual Report 1999, p I-100-104].

A review of superconductivity in the fullerenes has been written [7].

The alkali-doped fullerenes raise some fundamental questions, e.g., whether we can have l <<d for the resistivity, which can be answered only by accurate many-body calculations without uncontrolled approximations. We have therefore developed the tools for doing quantum Monte-Carlo calculations of ground-state 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 electron-phonon problem to which it was applied above, and we have demonstrated that the fixed-node approximation is rather accurate for the problems where it was used.

The so-called sudden approximation is applied in almost all calculations of photo-emission. 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 Cu-O 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 DF-LMTO 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 Bi2CuO4, Ba3Cu2O4Cl2and Li2CuO2 there is little hopping between the Cu atoms because they belong to strings of edge-sharing, and therefore uncoupled CuO4-squares. As a result, the screening electron comes from a narrow resonance, leading to a narrow peak. In Sr2CuO3 and SrCuO2, the CuO4-squares form corner-sharing -and therefore strongly coupled- 1D networks. In Sr2CuO2Cl2the CuO4-squares form corner-sharing 2D networks, like in the superconducting cuprates, and in Ba2Cu3O4Cl2 with extra, weakly coupled Cu atoms at the center of every second O4-square. LaCuO3 has a 3D Cu-O 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 high-temperature superconductors (HTSCs) are doped Mott-Hubbard insulators in which superconductivity occurs below a critical temperature, Tc(p). The latter depends parabolically on the hole-doping p with a maximum at so-called optimal doping. For different materials, maxpTc  varies substantially.

Research in our group was initially focussed on DF (LDA) calculations for the optimally doped, stoichiometric  YBa2Cu3O7  of energy bands, optical spectra, phonon frequencies in the normal and superconducting phases, the electron-phonon 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 Mott-Hubbard insulators at the beginning of the nineties. In this method, the LDA is corrected by addition of the on-site Coulomb interactions, U for the correlated electrons in a static mean-field 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 Mott-Hubbard insulator KCuF3[10]. This early work fertilized research on orbital polarization in transition-metal compounds. Another application of the LDA+U  formalism, this time to the 4f-electrons in REBa2Cu3O7 , enabled Lichtenstein and Mazin to explain why Pr-doping suppresses superconductivity more efficiently in NdBa2Cu3O7  than in YBa2Cu3O7[11]. Most recently, Dudarev et al. compared the charge density in NiO measured by high-energy 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 parameter-free and reasonable description of the high-energy properties of the HTSCs and produces all electronic bands in a broad energy range, almost any theory of the low-energy properties uses a one-band, correlated model Hamiltonian. The rationale for this is that, since the superconductivity originates in the CuO2 -layers and a single layer gives rise to merely one band at the Fermi level, a one-band model should suffice. We have developed a generally applicable method for computing, from a given single-electron potential, the Wannier-like orbitals which form a basis for such low-energy, few-band Hamiltonians [13(a),13(b)]. The LDA result for a HTSC is shown in Fig. 2. This orbital is centered on Cu, has dx2-y2symmetry, and extends to the 2nd-nearest neighbor oxygen atoms. As a consequence, its integrals for single-particle hopping on the Cu-centered 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 range-parameter , which gives the ratio,  t(n)/t, of any hopping integral to the nearest-neighbor one. Most importantly, our calculated r-values shown in Fig. 3 correlate with the observed max Tc. To understand why, is a key to understanding the mechanism of HTSC and poses a challenge to many-body theorists, such as those who currently use a k-space renormalization-group method to search for the leading correlation-driven 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  Ox px , Cu dx2-y2 ,  and  Oy py, in which case the nearest-neighbor Cu-Cu hopping  (t) is described as: Cu dx2-y2 –  Oy py – Cu dx2-y2 ,  while the 2nd and farther Cu-Cu hoppings require various intermediate O-O hoppings. We have found that by adding one further -diffuse Cu 4s-like- orbital with axial symmetry, the resulting four orbitals are so localized that merely nearest-neighbor hoppings (Cu and O) need to be included. Hence, the various O-O hoppings of the 3-orbital model, and thereby t', t'', ...,  of the 1-orbital model, all proceed via the axial orbital, and this leads to relations like:   t' = 2t''. In the 4-orbital model, the dispersion of the bare conduction band is simply:
(1)

and the range-parameter is essentially the Cu s character:
, with
(2)

The energy  of the axial orbital is eV's above the Fermi level, , which itself is eV's above the O p-level, . Expansion of  (1-2ru)-1  yields the one-band form:

e(k) = const. – 2t (cos kx+cos ky) + 4t' cos kx cos ky – 2t'' (cos 2kx+cos 2ky) + ...,
and, hence, the expressions for t, t',  t'', ..., in terms of  and r . In k-space, the constant-energy contours develop from being predominantly [11]-oriented, to being predominantly [10]-oriented, as r increases from 0 to 0.5. Finally, one must realize that the axial orbital is a hybrid between Cu 4s apical oxygen pz , and further orbitals on La, Bi, Tl, or Hg, and Cu 3d3z2–1  which couples to apical oxygen pz . We thus realize that the way to increase r and presumably max Tc  is to lower the energy, , of the axial orbital as much as possible by decreasing the coupling between Cu 4s  and apical oxygen pz . This can be done by moving apical oxygen away from Cu and by passivating apical oxygen pz by presenting it with a rival to Cu 4s. The lines in Fig. 3  give the structural effect of displacing apical oxygen, e.g. away from Cu and towards the rival. We see that in Tl2Ba2CuO6  and HgBa2CuO4  the Cu s character is saturated so that  is the energy of the pure Cu  4s orbital. We find some experimental support for our findings in measured 63Cu Knight shifts (isotropic B-term), but the strongest support is from the group of J.-P. Locquet, who produced ultra-thin strained La1.9Sr0.1CuO4 films by MBE and measured their local structures and physical properties in accord with the La-line in Fig. 3.

The axial orbital not only provides the dominant channel for the intra-layer hopping beyond nearest Cu-neighbors, but also provides the dominant channel for single-particle hopping out of the layer, , which according to Eq. (1)  depends on k like v2 which has  as a nodal line [14]. For materials like La2CuO4, Bi2Sr2CuO4, and  Tl2Ba2CuO6, where the stacking of the CuO2-layers is body-centered, the inter-layer hopping has an additional nodal line along . This supresses the perpendicular transport significantly, and provides a strong argument against the so-called inter-layer tunnelling mechanism for HTSC [15].

In multi-layer HTSCs, the neighboring CuO2-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 inter-layer exchange coupling in the bi-layer insulator YBa2Cu3O6  was verified by Keimer et al. using neutron scattering. Also for metallic multi-layer 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 bi-layer,  and 0 for a triple-layer, and for an infinite layer. Hence, the subbands will have different r-values and the lowest, most bonding subband will have the highest r. For bi-layer YBa2Cu3O7  with maxTc  we find , for triple-layer HgBa2Ca2Cu3O4 with maxTc=135K we find  and 0.37, and for infinite-layer CaCuO2 we find r ranging from 28 to 0.42. We thus have the unexpected result that the correlation between r and maxTc  observed for single-layer HTSCs, extends to multi-layer HTSCs with up to 3 layers, provided that most bonding subband is considered! This speaks against the van-Hove scenario, because for YBa2Cu3O7 it is the saddle-point of the anti-bonding band which is close to  but it is consistent with our finding that for YBa2Cu3O6+y it is the bonding subband which can support anti-ferromagnetic spin-fluctuations [16].

In bi-layer YBa2Cu3O7 the oxygens are dimpled out of the Cu-plane by  towards Y and away from Ba. This allows for a linear electron-phonon coupling of the corresponding modes, of which the Raman-active ones were studied intensively in Cardona's and our groups at the beginning of the nineties. We found these modes to have relatively strong electron-phonon 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 conduction-band orbital down in energy so that it becomes nearly degenerate with the -orbitals, whereby the hopping between them may be strong. This near-degeneracy term adds to the usual change of tpd with Cu-O bond-length and angle. By adding the Ox pzand Oy pz orbitals to the above-mentioned 4-orbital 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 Ox-Oy mode, this scattering is maximal when k= k'= (p, 0) and it vanishes when k=(p, 0) and k' =(0 ,p).  Since the observed dx2-y2 -wave pairing is presumably caused by a U-driven 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 dx2-y2-wave pairing. The electron-phonon 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 v2. When the Fermi level is at the van Hove singularity, x²–y²=s. This speculation about the buckling modes was confirmed by a full LDA-LMTO linear-response calculation for hole-doped CaCuO2 of the equilibrium structure, all phonon branches, and their electron-phonon coupling constants in both s- and dx2-y2-wave channels [17(a),17(b),17(c)]. This calculation showed that the mirror symmetry around a CuO2-plane is broken to develop a static Ox–Oy-dimple, thatx²–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 La1.6-1/8Nd0.4Sr1/8CuO4  where they are presumably pinned by the non-superconducting LTT phase. Recent experiments hint that the stripes might be dynamic and are the cause of the low-energy incommensurate spin excitations observed in La2-xSrxCuO4 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 mean-field 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 range-parameter r and found to be negative for r 0.3.

In concluding on HTSCs, possible reasons for the observed correlation between maxTc  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 LMTO-based methods were developed. These methods treat d- and f-electrons as easily as s- and p-electrons, and may do so with a small basis set, because the orbitals are constructed from the muffin-tin (MT) approximation to the potential in the solid. A MT-potential is spherically symmetric inside non-overlapping spheres surrounding the atoms and constant in between. In the interstitial region, a linear muffin-tin 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 k0 . This envelope function is continued smoothly inside any sphere as the appropriate linear combination of spherical harmonics times radial Schrödinger-equation 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 MT-potential to linear order in  inside the spheres, but merely to 0th order in the interstitial. For close-packed solids and low energies, one often uses the atomic-spheres approximation (ASA), which substitutes the MT-spheres with slightly overlapping (space-filling) atomic spheres and sets k0=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, sR and a structure matrix, which specifies the spherical-harmonics expansion around site R'  of an LMTO-envelope 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 k0=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 tight-binding (TB) techniques in DF calculations, e.g. to treat imperfect crystals. The two-centre TB-Hamiltonian is simply the structure matrix pre- and post-multiplied by potential parameters. The TB-LMTO set is merely a similarity transformation of the bare set. Another similarity transformation generates an orthonormal set. Downfolding is a third feature of this 2nd-generation method: If one wants to omit, say the Rlm-orbital 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 lm-projection with the proper radial logarithmic derivative at the sphere, . After this transformation, the Rlm-orbital can be deleted from the set, which is now complete to 0th order in the deleted channel. Similarity transformations and downfolding made the 2nd-generation method intelligible, fast, and useful beyond the realm of close-packed, crystalline transition metals. 

Our Stuttgart TB-LMTO-ASA program is in use world wide because it is extremely fast, user-friendly, hardware-independent, and free of charge. Input is the crystal structure and output is the charge- and spin-selfconsistent band structure, the partial densities of states, the Fermi surface, plots of the full charge and spin-densities, the total energy, and the partial pressures. In addition, the program delivers tools for analysing the electronic structure and chemical bonding such as: Orbital-projected band structures, COHPs for describing bond strengths, and electron localization functions (ELFs). The division of space into potential-spheres is done automatically [19]. We maintain the programme, give hands-on courses, and follow up with consultations on specific projects (coauthorship is not a condition).

Hedin's GW approximation for calculating the electron self-energy has been widely applied to sp-electron 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 total-energy differences associated with symmetry-lowering displacements of the atoms, the ASA is too crude. Various full-potential (FP) LMTO methods were developed in the late eighties and early nineties. In these more accurate methods, the LMTOs are defined with respect to non-overlapping potentials, but since the interstitial region is now large, one must use double- or triple-k sets. A further blow up of the basis set is caused by core states which extend beyond the MT-sphere. The charge density in the interstitial is expanded in Hankel functions or plane waves. Similarity transformations and downfolding are not attempted. Hence, the FP-LMTO methods are numerically accurate, but not really fast and intelligible. 

Savrasov developed an FP-LMTO linear-response method with which phonon dispersions and electron-phonon interactions can be calculated efficiently throughout the Brillouin zone [22]. Compared with the supercell method used previously for YBa2Cu3O7  this was a big step forward, although for reasons of computer storage only the simplest HTSC, idealized CaCuO2 could be treated [17(a),17(b),17(c)]. Like the methods of Zein and Baroni, based on plane-waves 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 electron-phonon spectral functions  for elemental Al, Pb, V, Nb, Ta, Mo, Cu, and Pd, and compared with experimental tunnelling data, temperature-dependent electrical and thermal resistivities, as well as the electronic specific heats. Only for Pd did their specific-heat enhancements -correctly- leave room for a contribution from spin-fluctuations. It was concluded that, for the materials considered, the electron-phonon interaction is described with 10 per cent accuracy [22(a),22(b)].

Next, Savrasov generalized the method to treat spin-fluctuations. Lacking an accurate approximation for the kernel  describing dynamical exchange-correlation effects in time-dependent DF theory, the so-called 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 frozen-magnon calculation, were resolved [23][see also: FKF Annual Report 1998, I-57-59]. Savrasov's LmtARTprogram is free software which besides FP-LMTO and linear-response includes TB-LMTO-ASA. A Windows surface is available.

LDA+U calculations of  for CaCuO2 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 mean-field theory, which includes the energy- but not the momentum- dependence of the self-energy through mapping of the Hubbard Hamiltonian -obtained from DF-LMTO calculations- onto the Anderson impurity model, subject to a selfconsistency condition. 

We recently succeeded in deriving 3rd-generation MTOs which treat the energy-dependence 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 so-called screened spherical waves, . These are wave-equation 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 hard-sphere 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 KR'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: SRlmKR'l'm',Rlm(e)cRlm,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 MT-potential to leading order in the overlap of the spherically symmetric potential wells. As a consequence, the 3rd-generation MTOs are so accurate, that only minimal sets and atom-centered MT-wells are required. This has been confirmed in extensive test calculations for diamond-structured silicon. We find that an sp3-set suffices to describe the valence band and that only when the potential-overlap 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 MT-spheres that large are like van der Waal's spheres, we believe that 3rd-generation MTOs will be useful not only for bulk solids, but also for molecules. We have shown that the overlapping MT-potential should be the least-squares fit to the full potential, weighted with the valence-charge 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 pressure-volume curves for the crystalline phases of silicon [24(a), 24(b)].

The kink-cancellation equation,  is a screened version of the so-called KKR equation of multiple-scattering theory. However, rather than energy-dependent kinked partial waves, we want a minimal set of energy-independent 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, e0, ..., eN and demand that the set of Nth-order MTOs (NMTOs) span the solutions of Schrödinger's equation to within an error proportional to . The set of 0th-order MTOs is obviously the set of kinked partial waves at the energy e0, and the corresponding Hamiltonian and overlap matrices are respectively  and . For N=1, one obtains a set of LMTOs expressed in terms of f(e0,r), f(e1,r), K(e0)  and K(e1).  Although it is more practical to use discrete energies, we may let e1 approach e0 and find that the LMTO becomes  This is just like in the ASA-formalism, except that  is now a matrix. It turns out that the LMTO Hamiltonian and overlap matrices are expressed in terms of K(e0), , K(e1), 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 I-16-23. Fig. 1 of this report shows 0th and 1st-order 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 semi-core bands, are described with the minimal set of 9 Ga sp3d5 and 16 As sp3d5f7QMTOs. 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 dx2-y2  can pick out the conduction band of a HTSC.

In conclusion, the 3rd-generation formalism solves the long-standing problem of deriving useful minimal sets of short-ranged 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.
 

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