The stopping and range of ions in matter is physically very complex, and there are few simple approximations which are accurate. However, if modern calculations are performed, the ion distributions can be calculated with good accuracy, typically better than 10%. This review will be in several sections: 

a)

A brief exposition of what can be determined by modern calculations.




b)

A review of existing widely-cited tables of ion stopping and ranges.




c)

A review of the calculation of accurate ion stopping powers.

The stopping and range of ions in solids

We construct a generalized gradient approximation (GGA) for the density ${\mathit{n}}_{\mathrm{xc}}$(r,r+u) at position r+u of the exchange-correlation hole surrounding an electron at r, or more precisely for its system and spherical average 〈${\mathit{n}}_{\mathrm{xc}}$(u)〉=(4\ensuremath{\pi}${)}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\int}d${\mathrm{\ensuremath{\Omega}}}_{\mathit{u}}$ ${\mathit{N}}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\int}${\mathit{d}}^{3}$r n(r)${\mathit{n}}_{\mathrm{xc}}$(r,r+u). Starting from the second-order density gradient expansion, which involves the local spin densities ${\mathit{n}}_{\mathrm{\ensuremath{\uparrow}}}$(r),${\mathit{n}}_{\mathrm{\ensuremath{\downarrow}}}$(r) and their gradients \ensuremath{\nabla}${\mathit{n}}_{\mathrm{\ensuremath{\uparrow}}}$(r),\ensuremath{\nabla}${\mathit{n}}_{\mathrm{\ensuremath{\downarrow}}}$(r), we cut off the spurious large-u contributions to restore those exact conditions on the hole that the local spin density (LSD) approximation respects. Our GGA hole recovers the Perdew-Wang 1991 and Perdew-Burke-Ernzerhof GGA's for the exchange-correlation energy, which therefore respect the same powerful hole constraints as LSD. When applied to real systems, our hole model provides a more detailed test of these energy functionals, and also predicts the observable electron-electron structure factor. \textcopyright{} 1996 The American Physical Society.

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Generalized gradient approximation for the exchange-correlation hole of a many-electron system

A scheme that reduces the calculations of ground-state properties of systems of interacting particles exactly to the solution of single-particle Hartree-type equations has obvious advantages. It is not surprising, then, that the density functional formalism, which provides a way of doing this, has received much attention in the past two decades. The quality of the energy surfaces calculated using a simple local-density approximation for exchange and correlation exceeds by far the original expectations. In this work, the authors survey the formalism and some of its applications (in particular to atoms and small molecules) and discuss the reasons for the successes and failures of the local-density approximation and some of its modifications.

The density functional formalism, its applications and prospects

Preface Acknowledgements Notation Part I. Overview and Background Topics: 1. Introduction 2. Overview 3. Theoretical background 4. Periodic solids and electron bands 5. Uniform electron gas and simple metals Part II. Density Functional Theory: 6. Density functional theory: foundations 7. The Kohn-Sham ansatz 8. Functionals for exchange and correlation 9. Solving the Kohn-Sham equations Part III. Important Preliminaries on Atoms: 10. Electronic structure of atoms 11. Pseudopotentials Part IV. Determination of Electronic Structure, The Three Basic Methods: 12. Plane waves and grids: basics 13. Plane waves and grids: full calculations 14. Localized orbitals: tight binding 15. Localized orbitals: full calculations 16. Augmented functions: APW, KKR, MTO 17. Augmented functions: linear methods Part V. Predicting Properties of Matter from Electronic Structure - Recent Developments: 18. Quantum molecular dynamics (QMD) 19. Response functions: photons, magnons ... 20. Excitation spectra and optical properties 21. Wannier functions 22. Polarization, localization and Berry's phases 23. Locality and linear scaling O (N) methods 24. Where to find more Appendixes References Index.

Electronic Structure: Basic Theory and Practical Methods

Density functional theory (DFT) is a (in principle exact) theory of electronic structure, based on the electron density distribution n(r), instead of the many-electron wave function Ψ(r1,r2,r3,...). Having been widely used for over 30 years by physicists working on the electronic structure of solids, surfaces, defects, etc., it has more recently also become popular with theoretical and computational chemists. The present article is directed at the chemical community. It aims to convey the basic concepts and breadth of applications:  the current status and trends of approximation methods (local density and generalized gradient approximations, hybrid methods) and the new light which DFT has been shedding on important concepts like electronegativity, hardness, and chemical reactivity index.

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Density Functional Theory of Electronic Structure

Calculated electronic properties of metals

We present a conceptual model and calculational procedure for the study of the electronic structure of metallic compounds. The model consists of spherical atoms compressed into finite volumes appropriate to the solid. The model involves no adjustable or experimentally derived parameters. All contributions to the total energy (other than the Madelung energy) are obtained from independent compressed-atom calculations. Interatomic interactions enter the calculations through the electronic configuration (the distribution of the valence charge among $s$, $p$, $d$, etc., states) and boundary conditions which give the atomic valence levels a finite width. These environmental constraints, which specify the state of the compressed atoms, are obtained from energy-band calculations. For the latter we introduce a new method, which we call the augmented-spherical-wave (ASW) method to stress its conceptual similarity to Slater's augmented-plane-wave (APW) method. The ASW method is a direct descendant of the linear-muffin-tin-orbitals technique introduced by Andersen; when applied to pure metals, it yields results which closely approximate those of the much more elaborate Korringa-Kohn-Rostoker calculations of Moruzzi, Williams, and Janak. The combined ASW compressed-atom procedure is tested on (i) the empty lattice, (ii) the pure metals Na, Al, Cu, and Mo, and (iii) the ordered stoichiometric compounds NaCl, NiAl, and CuZn. Finally, we demonstrate the utility of the procedure by using it to study the anomalous tendency of Ni and Pd (as compared to their Periodic Table neighbors Co, Cu, Rh, and Ag) to form hydride phases. We have calculated the total energies of the six pure metals and their monohydrides. The total energy differences exhibit the anomaly and an analysis of quantities internal to the calculation reveals its origin.

Cohesive properties of metallic compounds: Augmented-spherical-wave calculations

A new theory of cluster expansions has been derived, which allows one, for the first time, to estimate the energy of a disordered system from first principles. The cluster variables are derived from a series of density-functional calculations on ordered compounds. The disordering temperatures calculated with this theory show the correct trends for binary alloys of $4d$ transition metals, and are in excellent agreement with the experimental phase diagrams in most cases.

Density-functional theory applied to phase transformations in transition-metal alloys

The electronic structure of the insulating antiferromagnetic transition-metal compounds MnO, FeO, CoO, and NiO, which have been regarded as the prototypes of the concept of a Mott insulator, is discussed with use of energy-band theory based on the local-spin-density treatment of exchange and correlation. It is shown that the band structure is very sensitive to the magnetic ordering and that the ground-state magnetic ordering is special in the sense that it makes the ${e}_{g} ({x}^{2}\ensuremath{-}{y}^{2},3{z}^{2}\ensuremath{-}{r}^{2})$ band particularly narrow, which is crucial to the insulating nature of NiO. A detailed analysis is made of this particular aspect of the ground-state magnetic ordering. As for FeO and CoO, it is suggested that the population imbalance among the ${t}_{2g} (xy, yz, zx)$ orbitals induced by the intra-atomic exchange interaction may cause a gap to open at the Fermi level.

Band theory of insulating transition-metal monoxides: Band-structure calculations

We use the atom-jellium model of chemisorption to study the properties of an atom bonded to the surface of a simple metal. Beyond the jellium simulation of the metal substrate and the Kohn-Sham local-density description of exchange and correlation, we make no significant approximations: our solutions are parameter free, wave mechanical, and fully self-consistent. We solve this model for a set of adatoms exhibiting a variety of chemical behaviors. Properties studied include the electron-density distribution, the state density, the dipole moment, and the heat of adsorption. We discuss also the reintroduction of the discrete lattice structure of the substrate using perturbation theory.

Arthur Robert Williams

Papers

Calculated electronic properties of metals

Cohesive properties of metallic compounds: Augmented-spherical-wave calculations

Density-functional theory applied to phase transformations in transition-metal alloys

Band theory of insulating transition-metal monoxides: Band-structure calculations

Theory of atomic chemisorption on simple metals