Angular Momentum Expansion Research Articles

Calculation of the XAFS Debye-Waller Factor with Spherical-Wave Corrections

To calculate XAFS spherical-wave corrections and multiple scattering contributions have to be included improving the single scattering plane-wave approximation. A suitable method is described to include such corrections. The temperature dependence of the spectra is determined by a Debye-Waller factor in the plane-wave approximation. An additional temperature dependence arises in a spherical-wave description. An approximative expression is developed, which is valid for the exact angular momentum expansion and any spherical-wave approximation. Due to the thermal movement of the atoms the elements of the t-matrix tl have to be replaced by tlTD=tl+k2 σ102 \ ildetl where \ ildet is a modified t-matrix. σ102 is the average relative atomic displacement between the emitter and the scatterer. t and \ ildet have a different k-dependence. The consequence is as shift in the peaks of the absolut value of the fouriertransformed spectrum. For Ni and NiO we get a shift up to 0.1 atomic units for the nearest and next nearest neighbour.

Shortcomings of separable representations of the Green function in multiple-scattering theory for photoelectron diffraction and extended X-ray absorption fine structure

In most of the approximations for spherical-wave scattering, the angular momenta l and l′ in the two-centre angular momentum representation of the free electron Green function are decoupled, in order to reduce the mathematical complexity. In the present paper, it is shown that the errors of these so-called separable approximations (e.g. the Taylor series magnetic quantum number expansion of Barton and Shirley, the reduced angular momentum expansion of Fritzsche and Rennert, and the scattering matrix formulation of Rehr and Albers) increase considerably with the kinetic energy of the electrons, if the distance between source and scatterer is small (i.e. if they are nearest neighbours). In particular, low accuracy in the forward scattering direction is a great problem. The origin of such errors is the oscillatory nature of the coupling between the two angular momenta in the Green function, which cannot be described adequately by separable approximations.

Bloch-wave scattering by crystal defects

The Korringa, Kohn, and Rostoker (KKR) conditions for Bloch waves that describe the motion of electrons in a perfect crystal, and the scattering of these waves by crystal impurities, are formulated by the methods of time-independent scattering theory without recourse to angular-momentum expansions. Generalized Levinson theorems (Friedel sum rules) are proved, including the possibility of half-bound states.

Real-Space Multiple Scattering Theory Method: Formalism and Applications

ABSTRACTDifferent forms of the real-space multiple scattering theory (RS-MST) formalism are compared in order to understand its convergence behavior with respect to truncation in the angular momentum expansions. In particular the so-called “folding method” or (1,n) mode is considered, in which the renormalized t-matrix T of the semi-infinite system has the dimension of n (n > 1) repeating units and the self-consistent equation for T is constructed by adding and contracting one such unit. It has been demonstrated in previous studies of layered structures that the folding method converges rapidly in both angular momentum (L) and site (n) indices and yields accurate results. The convergence behavior is an important factor to be considered in applying the various forms of the RS-MST method to layered structures as well as other systems with extended defects.

Full-potential Korringa-Kohn-Rostoker band theory applied to the Mathieu potential.

The band theory of Korringa, Kohn, and Rostoker (KKR) based on the Green-function method is extended to space-filling potentials. A numerical test using the Mathieu potential shows good convergence for the bands up to 1.5 Ry with {ital l}{le}4 included in the angular-momentum expansion for the wave functions. Our results strongly support the applicability of the full-potential KKR to bulk electronic-structure problems.

Angle-resolved auger electron spectra for a Ni(110) surface: Calculations with the reduced angular momentum expansion

Results of multiple-scattering cluster calculations are presented for angle-resolved M 2,3VV Auger electron intensities from a clean Ni(110) surface, which are obtained using the reduced angular momentum expansion (RAME). This method goes beyond the popular plane-wave approximation and takes into account both the spherical character and the anisotropy of the electron waves. The positions of the maxima and minima in the calculated polar intensity profiles agree well with the experimentally determined structures.

Application of curved-wavefront approximations to multiple-scattering effects in EXAFS of ReO3

Multiple-scattering effects in EXAFS of the rhenium L3 edge are calculated with the reduced angular momentum expansion (RAME) and with the modified small-scattering-centre approximation (MSSCA). The multiple-scattering enhancement in the nearly collinear Re-O-Re bonds is very sensitive to changes in the bond angle. A comparison of the theoretical results with the pressure-dependent measurements of Alberding et al. (1986) confirms the relation between the pressure and the bond angle that follows from the structure model of Jorgensen et al. (1986) for the high-pressure phases in ReO3.

Phase shifts in the EXAFS of crystalline iron–experimental results and theoretical calculations

For crystalline iron, it has been shown that the approximation of the spherical photoelectron wave by a plane wave causes systematic errors. Shell-dependent energy shifts E 0 up to 10 eV are necessary to correct these errors of the plane-wave SSCA (small-scattering-centre approximation). An analytical expression E 0( R) is given which can be used as preinformation in the analysis of an experimental EXAFS spectrum. Consequently, the great number of free parameters, especially the “dummy” energy zeros for different coordination shells at a distance R, can be reduced drastically. Furthermore, two approximations within the curved-wave theory, MSSCA (modified small-scattering-centre approximation) and RAME (reduced angular momentum expansion), that partially correct the systematic E 0 errors are proposed.

Approximations for photoelectron scattering

The errors of several approximations in the theoretical approach of photoelectron scattering are systematically studied, in tungsten, for electron energies ranging from 10 to 1000 eV. The large inaccuracies of the plane-wave approximation (PWA) are substantially reduced by means of effective scattering amplitudes in the modified small-scattering-centre approximation (MSSCA). The reduced angular momentum expansion (RAME) is so accurate that it allows reliable calculations of multiple-scattering contributions for all the energies considered.

Local Spin Density Calculations for the High Tc Superconductors

In this paper we present local spin density (LSD) calculations for La2CuO4 and YBa2Cu3O6. We find both La2CuO4 and YBa2Cu3O6 to be anti-ferro-magnetic semi-metals with a very small of nearly non-existent moment. The value of this moment depends on the Brillouin zone integration sampling and also on the angular momentum expansion of the Cu and O sites in the CuO2 layers. Expanding the Wigner-Seitz spheres by 10% or including the breathing mode failed to make La2CuO4 an anti-ferromagnetic semiconductor.

A local spin-density study of antiferromagnetism in La2CuO4and YBa2Cu3O6

The authors present local spin-density (LSD) calculations for YBa2Cu3O6 and La2CuO4. They find YBa2Cu3O6 to be an antiferromagnetic semimetal with a very small or nearly nonexistent moment. The value of this moment is not only dependent on the Brillouin zone integration sampling, but is also sensitive to the angular momentum expansion of the Cu and O sites in the CuO2 layers. Expanding (or squeezing) the Wigner-Seitz spheres up to 10% (or down to 9%) or including the breathing mode failed to make La2CuO4 an antiferromagnetic semiconductor.

A complete basis set model chemistry. I. The total energies of closed-shell atoms and hydrides of the first-row elements

The major source of errror in most ab initio calculations of molecular energies is the truncation of the one-electron basis set. A complete basis set model chemistry is defined to include corrections for basis set truncation errors. This model uses double zeta plus polarization level atomic pair natural orbital basis sets to calculate molecular self-consistent-field (SCF) energies and correlation energies. The small corrections to give the complete basis set SCF energies are then estimated using the l−6 asymptotic convergence of the multicenter angular momentum expansion. The calculated correlation energies of the atoms He, Be, and Ne, and of the hydrides LiH, BH3, CH4, NH3, H2O, and HF, using the double zeta plus polarization basis sets vary from 83.0% to 91.2% of the experimental correlation energies. However, extrapolation of each of the pair energies and pair-coupling terms to the complete basis set values using the asymptotic convergence of pair natural orbital expansions retrieves from 99.5±0.7% to 101.1±0.6% of the experimental correlation energies for these atoms and molecules. With the exception of ammonia which gave 101.1%, the calculated correlation energies agree with the experimental values to within the error limits of the experiments for all these atoms and molecules with more than four electrons. The total extrapolated energies (ESCF+ECorrelation) are then in agreement with experiment to within ±0.0014 hartree (root mean square deviation) and represent the most accurate total energy calculations yet reported for the molecules. The largest discrepancies with experiment occur for methane, where we obtain ETotal =−40.5112 hartree compared to EExpt =−40.514±0.002 hartree, and ammonia, where we obtain ETotal =−56.5659 hartree compared to EExpt =−56.563±0.002 hartree.

Curved Wave and Multiple Scattering Corrections in Calculated EXAFS Spectra

AbstractThe curved wave theory RAME (reduced angular momentum expansion) is used to develope an expression for EXAFS. Systematic errors of the standard plane wave evaluation SSCA are found. Partially they can be corrected by a suitable choice of the energy zero E0 in the SSCA expression. But different values E0 have to be used for different neighbour shells at the distance R. An analytical expression E0(R) is given. Furthermore it is shown that there are other important multiple scattering contributions besides those due to the focussing effect for shadowed atoms. The comparision of RAME and the modified small scattering centre approximation MSSCA shows that MSSCA is sufficient accurate for EXAFS. In the XANES region below electron energies of 50 eV RAME is needed. For the first nearest neighbour RAME corrects MSSCA up to 100 eV.

Approximations for the Scattering of Excited Electrons in Solids

AbstractThe errors of several approximations for photoelectron scattering are systematically studied as a function of the electron energy in the range from 10 to 1000 eV. The accuracy of the approximations does not improve with rising energy, as it was commonly supposed. The errors increase considerably with the angular momentum of the incident wave. The large discrepancies of the small‐scattering‐centre approximation (SSCA) are substantially reduced by means of effective scattering amplitudes in the modified small‐scattering‐centre approximation (MSSCA). Above all the reduced angular momentum expansion (RAME) has such a high degree of accuracy that allows reliable calculations of multiple‐scattering contributions for all purposes. Both the MSSCA and the RAME are further simplified in order to facilitate applications.

Efficient computational method for interstitial Green functions

A computational scheme is developed for the numerical evaluation of Green functions expanded around interstitial positions. The Green function matrix elements, consisting of a Brillouin zone and a Fermi surface integral, are calculated at one energy. The explicit occurrence of free-electron singularities in the integrand of the Brillouin zone integral complicates the evaluation and requires special computational effort. It is shown how, within the framework of the tetrahedron method, the problem of singular integrands can be resolved efficiently and adequately. A comparison with other evaluation schemes, based on a Kramers-Kronig relation or an approximate angular momentum expansion, is made. Special attention is given to the octahedral position in a FCC lattice.

A reduced angular momentum expansion in the multiple‐scattering theory

AbstractIn order to improve the calculations of ARPES, ARAES, and EXAFS a new approximation in the multiple‐scattering theory for the excited electrons is introduced, which is called reduced angular momentum expansion (RAME). In the RAME the incoming waves are approximated in the range of the scattering potential by a limited set of spherical waves with low angular momenta. This is a much better description than the plane wave approximation in the often used small scattering‐centre approximation. The RAME possesses a high accuracy for energies down to 40 eV and needs a substantially lower numerical effort than the full angular momentum expansion. For higher energies it can be further simplified and it modified small‐scattering‐centre approximation is obtained, which takes into account the curvature of the waves by means of effective scattering amplitudes.

Two-electron wave functions in hyperspherical coordinates

In order to investigate the potential precision of quantitative calculations using hyperspherical coordinates, we have reexamined this method in detail. By developing an analytic solution of the angular equation in the form of a series expansion in the variable x = tan(..cap alpha../2), we are able to isolate the numerical errors in the solution from those caused by the truncation of the angular momentum expansion and by the exclusion of radial coupling terms, which we then examine separately. This new method enables us to extend the calculation of the potential curves and channel functions to the large-R region and to obtain asymptotic expansions for the potential curves and radial couplings. From these we determine general expressions for the boundary conditions of the radial equation. Finally, we present the first nonadiabatic result for the ground-state energy of helium.

Variational cellular method for polyatomic molecules: SiF4and SiCl4

Presents the new developments in the theory and applications of the variational cellular method. The most important advance in the theory was the introduction of a continuity condition for the wavefunctions and potential at the cell boundaries. The continuity condition gave a very remarkable stability to the results for changes in the size of the angular momentum expansions. The calculated molecules were SiF4 and SiCl4, for which the ionisation potentials are better than those of the multiple-scattering or existing ab initio methods.

Local density of states and EXAFS in the small scattering atom approximation

Multiple scattering theory in the small scattering atom approximation is used to evaluate the Green function at high energies. A rearrangement of the series expansion in terms of the individual t matrices is used where the leading term is the Green function of the free atom. The energy density matrix well above the Fermi energy can then be calculated from the wavefunctions of the central atom as well as from the position and scattering amplitudes of the other atoms. An angular momentum expansion appears only for the central atom. The sum over all closed loops can be restricted to the neighbouring atoms inside a cluster around the central atom due to the imaginary part of the optical potential. The EXAFS amplitude is calculated using this approach for an arbitrary angular momentum of the initial state. The result contains all multiple scattering contributions and reduces in the single scattering approximation to the well known results for the K and L edges.

Quantum theory of neutral-atom scattering at long range from solid cylinders

Qualitative arguments suggest that diffraction effects might be important for the interpretation of the experiments of Shih et al. We derive a tractable quantum theory to describe the scattering of neutral atoms at long range by van der Waals forces from solid cylinders. The theory is based on an angular-momentum expansion and evaluates the scattered intensity close to the cylinder in the geometrical shadow region. The formal quantum theory indicates that one is far from the conventional classical limit, in the sense that WKB and stationary-phase approximations are not valid. Yet an essentially exact evaluation appears to agree with the classical prediction, at least over the range of deflections that have been experimentally studied. This surprising classical-quantum correspondence is clarified by a Regge-pole analysis. Finally we also set limits on the extent of specular reflection from the cylinder surfaces. The discrepancy between theory and experiment remains unresolved.