Volker Heine

The Fitting of Pseudopotentials to Experimental Data and Their Subsequent Application

Publisher Summary This chapter summarizes results from the theory of pseudopotentials that relate to choosing suitable mathematical forms forfitting. The procedures used are described. It is reported that it is important to distinguish between what is exact and what is approximate, and to discuss where various approximations might be reasonably appropriate and where not. The chapter reviews all the experimentally fitted pseudopotentials. Primarily they come from Fermi surface studies in the case of metals and analysis of optical properties for semiconductors, but phonon spectra, resistivities of liquid metals, and other data have also been important. It is suggested that by making use of all the theory, all the fitted results by interpolating, extrapolating, cajoling, and cudgeling, one should be able to produce a useful pseudopotential for many elements in many situations.

Pseudopotential Theory of Cohesion and Structure

Publisher Summary This chapter discusses the pseudopotential theory of cohesion and structure. The pseudopotential theory of properties related to the total energy of simple (sp-bonded) metals is reviewed with emphasis on the structure dependence of the total energy. Some of the topics covered in addition to the explanation of the observed structures of simple metals are phonon dispersion, stacking faults, ordering in alloys and its relation to electronegativity, structures of liquid metals, and the various interpretations of covalent bonding effects. After an elementary introduction that seeks to put this theory in perspective by expounding its general features and relating it to prevailing ideas on metallic bonding derived from earlier theories, the details of the formalism are explored. The conditions for its validity or breakdown are examined. The results of various calculations of structural properties are quoted element by element and discussed. The theory is used to account for observed trends in structures in the periodic table.

The Pseudopotential Concept

Publisher Summary This chapter reviews the basic ideas about pseudopotentials. These ideas have evolved and intersected with other developments, therefore, the concept is developed it both from the point of view of an equivalent operator potential and from scattering theory. It is suggested that a correct pseudopotential is defined as one that gave the same energy levels as the real potential, a t least for a range of states of interest, with the emphasis on transforming the Schroedinger equation from. In the chapter, the augmented plane wave, orthogonalized plane wave, Ziman, and Abarenkov forms are discussed. It is reported that with the perspective of time, it also seems appropriate to review why the whole concept of a pseudopotential is necessary, and the cancellation theorem is treated more satisfactorily than previously.

Rigid-unit phonon modes and structural phase transitions in framework silicates

Abstract The rigid-unit mode model provides many new insights into the stability and physical properties of framework silicates. In this model the SiO4 and AlO4 tetrahedra are treated as very stiff, to a first approximation as completely rigid, in comparison with intertetrahedral forces. In this paper we apply the model to several important examples. The model is reviewed by a detailed study of quartz, and it is shown that the α-β phase transition involves a rigid-unit mode that preserves the Si-O-Si bond angle. The model is used to explain the phase transitions in cristobalite and the different feldspar, sodalite, and leucite structures. We also use the model to explain the nature of the high-temperature disordered phases of cristobalite and tridymite, to interpret the observations of streaks of diffuse scattering in electron diffraction patterns, to interpret the structures in the kalsilite-nepheline solid solution, to explain volume anomalies in the cubic leucite structures, and to explain qualitatively the negative linear thermal expansion in cordierite. The results for the highest symmetry sodalite structure show that there is a rigid-unit mode at every wave vector, a finding with significant implications for the understanding of the sorption and catalytic behavior of zeolites.

Electronic Structure from the Point of View of the Local Atomic Environment

Publisher Summary The chapter presents a study on electronic structure from the perspective of the local atomic environment. The chapter begins by discussing fundamental ideas, such as throwing out k space, the invariance theorem, and mathematics of the Green functions. Several aspects of solid state physics, where the model of perfect lattice periodicity is simply not appropriate are a part of the discussion. It must be emphasized that the density functional theory is purely concerned with the ground state, that the eigenfunctions Ψn are only a device for calculating the kinetic energy, and that the energy values En do not have any direct physical significance in contrast to the Landau quasi-particle energies, which are genuine excitation energies of the system. The chapter presents a summary on the basic results of the density functional formalism without proof and considers a system in its ground state of stationary nuclei with an electron density p(r). The chapter discusses the special and important type of perturbation or rather difference calculation, namely, switching on or off a coupling interaction between two subsystems, or between an adsorbed atom and the adsorbate surface. The chapter presents the calculation of the coupling energy.

Theory of lattice contraction at aluminium surfaces

The authors show how the elementary concepts which govern the bulk total energy and crystal structure of sp bonded metals are also relevant at the surface. A simple model explains the observed contraction in the interlayer spacing of the (110) surface layer of aluminium compared with the (100) and (111) surfaces.

The Determination of Rigid-Unit Modes as Potential Soft Modes for Displacive Phase Transitions in Framework Crystal Structures

This paper describes a computational method for the determination of all possible phonon modes in framework crystal structures that leave the fundamental structural units (tetrahedra and octahedra) undistorted. Such rigid-unit modes (RUMs) are prime candidates as soft modes for displacive phase transitions, such as in the perovskite structure, and this computational method can be used to rationalize the phase transitions in any framework structure. The method has been programmed for general use. The RUM approach is illustrated by consideration of the perovskite, quartz and cristobalite structures.

Rigid unit modes in framework silicates

Abstract We describe a model for framework silicates in which the SiO4 (and AlO4) tetrahedra are treated as perfectly rigid and freely jointed. From this model we are able to identify low-energy modes of distortion of the structure, which we call Rigid unit modes. These modes can act as soft modes to allow easy distortions at phase transition. We discuss three forces that will operate at a phase transition in conjunction with the candidate soft modes to determine which of the rigid unit modes will actually precipitate a phase transition, and illustrate these ideas by detailed discussions of the phase transitions in quartz, leucite and cristobalite. The model can also be used to estimate the transition temperature, and the theory highlights an important role for the stiffness of the tetrahedra.

Distribution of tetrahedral and octahedral A1 sites in gamma alumina

Abstract Experimental and computational studies of γ-Al 2 O 3 show that 70±2% of Al ions occupy octahedral interstitial sites in the fcc oxygen structure, and the rest tetrahedral sites. The experimental data come from 27 Al MAS NMR and the theoretical results from ab initio quantum mechanical energy calculations for nine superlattice structures, analysed to extract various parameters which were then fed into a Monte Carlo simulation of the γ-Al 2 O 3 structure. The simulation throws new light on the reasons for the structural disorder.

Inter-layer interactions and the origin of SiC polytypes

The authors investigate, up to the fourth-nearest neighbour, the interaction energies J1 to J4 between SiC double layers by calculating the total energies of five SiC polytypes with norm-conserving pseudopotentials. They find J1=1.89 mod J2 mod with J2 negative. This is very close to the multi-phase degeneracy point J1=2 mod J2 mod in the ANNNI model at T=0 where an infinite number of polytypes are degenerate. The third-(J3) and fourth-(J4) neighbour interactions turn out to be very small but they stabilise the phase (3) (in Zhdanov notation) which has the lowest energy of those considered. The splitting of the multi-phase degeneracy and the origin of polytypes as possible equilibrium phases are discussed.