Table of contents

The topic of Atomistic Simulation has been raising its profile in diverse fields of physics, chemistry, materials science and biology. So while it is clear that a comprehensive condensed matter physics journal should pay attention to this field, and a Special issue devoted to it does just that, some readers will be aghast to find their favourite atomistic simulation methods are not represented in this issue, while others will find no mention of what they consider to be the most important applications. So let me say a few words about the motivation for this selection of papers. Knowledge and insight about the behaviour of materials and large molecules have been won over the past fifty years or so by making molecular dynamic or static calculations using empirically based interatomic potentials. Their functional forms have varying degrees of complexity, and with the simplest models one can nowadays do calculations on systems of hundreds of millions of atoms. The price to be paid for such simple models is always some uncertainty about their reliability for calculating total energies outside the regime in which their parameters were fitted. At the other extreme, using configuration interaction or quantum Monte Carlo techniques, the many electron problem can be solved accurately in simple systems, whatever the arrangement of atoms, but with a formidable price now for computation. Such high quality models are not yet state of the art for handling most of problems that today's atomistic simulators wish to solve. The middle way is based on the versatile and very widely used density functional theory of Hohenberg, Kohn and Sham, with some extensions, and this is the focus for the present issue. Readers unfamiliar with the density functional approach will find a good introduction in the first paper. The papers generally will illustrate how we can make the most of this theory to do useful atomistic simulations. There are a number of codes such as CASTEP, VASP or SIESTA, which are very widely used tools for atomistic simulation within a state of the art, first-principles density functional framework, and examples are described in these pages. To extend the range of atoms that can be treated within this framework from hundreds to thousands, so called `order N' approaches have been developed and implemented in the codes SIESTA and CONQUEST, and these methods are described for the first time here in considerable detail. Nevertheless, many applications are still only possible when the equations of density functional theory are simplified, usually with the introduction of some semi-empirical parameters. The effective medium method and the tight binding method which are represented here are different approaches to such a simplification; they belong to the class of methods for which Volker Heine, without disrespect, coined the phrase `quick and dirty'. The useful strategy of carefully parameterising a model, using first principles calculations to provide data on the energies of different arrangements of atoms, is also illustrated here at its most developed. Recognizing that the length and timescales of even the largest possible atomic simulations are tiny on the scale of microstructural evolution in materials, or biological processes, many workers are attempting to do `coarse graining', or find ways of connecting a hierarchy of models, each suitable for different length and time scales. Nieminen's paper in particular addresses the issue and some approaches for dealing with it. A different kind of approach to building a bridge from the atomistic length scale to macroscopic properties is also in the paper of Benedek {\it et al}. The front line of progress on the one hand is into novel systems, with methods which are by now standard, or almost so; Segall's review of applications in biology gives the flavour of what can and has been achieved already, and what might be done in the future. On the other hand methodological developments are continuing in order to enable simulations of systems and processes which were hitherto intractable. Magnetic systems are a case in point, for which Fang and many others are applying extensions to density functional theory, which include explicit orbitals in the functionals. The final papers in this issue describe new approaches that are needed when the dynamics of the electrons must be treated, for example to calculate current-induced forces or various non-adiabatic processes. Time dependent density functional theory in various guises offer a methodology for tackling transport problems on the molecular scale in all the disciplines mentioned before. Finally, this field owes much of its progress to teamwork. Very many person years of experience and cooperative effort are embodied in the hundreds of thousands of lines of some of the larger codes. Multi-author papers are the norm. We are all grateful to the various funding agencies for supporting networks such as the CCPs (EPSRC) and PSIK (EC and ESF) and CECAM in Europe, which have contributed to much of the progress reported here.

First-principles simulation, meaning density-functional theory calculations with plane waves and pseudopotentials, has become a prized technique in condensed-matter theory. Here I look at the basics of the suject, give a brief review of the theory, examining the strengths and weaknesses of its implementation, and illustrating some of the ways simulators approach problems through a small case study. I also discuss why and how modern software design methods have been used in writing a completely new modular version of the CASTEP code.

We have developed and implemented a selfconsistent density functional method using standard norm-conserving pseudopotentials and a flexible, numerical linear combination of atomic orbitals basis set, which includes multiple-zeta and polarization orbitals. Exchange and correlation are treated with the local spin density or generalized gradient approximations. The basis functions and the electron density are projected on a real-space grid, in order to calculate the Hartree and exchange-correlation potentials and matrix elements, with a number of operations that scales linearly with the size of the system. We use a modified energy functional, whose minimization produces orthogonal wavefunctions and the same energy and density as the Kohn-Sham energy functional, without the need for an explicit orthogonalization. Additionally, using localized Wannier-like electron wavefunctions allows the computation time and memory required to minimize the energy to also scale linearly with the size of the system. Forces and stresses are also calculated efficiently and accurately, thus allowing structural relaxation and molecular dynamics simulations.

We describe recent progress in developing linear scaling ab initio electronic structure methods, referring in particular to our highly parallel code CONQUEST. After reviewing the state of the field, we present the basic ideas underlying almost all linear scaling methods, and discuss specific practical details of the implementation. We also note the connection between linear scaling methods and embedding techniques.

We review some recent conceptual improvements of the Korringa–Kohn–Rostoker (KKR) Green function method for electronic structure calculations. After an introduction into the KKR–Green function method we present an extension of this method into an accurate full-potential scheme, which allows calculation of forces and lattice relaxations. The additional numerical effort compared to the atomic sphere approximation scales only linear with the number of atoms. In addition, we discuss the recently developed screened KKR method which represents a reformulation of the multiple scattering theory with exponentially decreasing structure constants. This method, which has the same accuracy as the standard KKR method, exhibits strong advantages for two-dimensional systems like multilayers or surfaces, since the numerical effort scales linearly with the number of layers. The strength of both methods is illustrated in typical applications.

We present an overview of recent work on quantum-based atomistic simulation of materials properties in transition metals performed in the Metals and Alloys Group at Lawrence Livermore National Laboratory. Central to much of this effort has been the development, from fundamental quantum mechanics, of robust many-body interatomic potentials for bcc transition metals via model generalized pseudopotential theory (MGPT), providing close linkage between ab initio electronic-structure calculations and large-scale static and dynamic atomistic simulations. In the case of tantalum (Ta), accurate MGPT potentials have been so obtained that are applicable to structural, thermodynamic, defect, and mechanical properties over wide ranges of pressure and temperature. Successful application areas discussed include structural phase stability, equation of state, melting, rapid resolidification, high-pressure elastic moduli, ideal shear strength, vacancy and self-interstitial formation and migration, grain-boundary atomic structure, and dislocation core structure and mobility. A number of the simulated properties allow detailed validation of the Ta potentials through comparisons with experiment and/or parallel electronic-structure calculations. Elastic and dislocation properties provide direct input into higher-length-scale multiscale simulations of plasticity and strength. Corresponding effort has also been initiated on the multiscale materials modelling of fracture and failure. Here large-scale atomistic simulations and novel real-time characterization techniques are being used to study void nucleation, growth, interaction, and coalescence in series-end fcc transition metals. We have so investigated the microscopic mechanisms of void nucleation in polycrystalline copper (Cu), and void growth in single-crystal and polycrystalline Cu, undergoing triaxial expansion at a large, constant strain rate - a process central to the initial phase of dynamic fracture. The influence of pre-existing microstructure on the void growth has been characterized both for nucleation and for growth, and these processes are found to be in agreement with the general features of void distributions observed in experiment. We have also examined some of the microscopic mechanisms of plasticity associated with void growth.

This paper discusses some current trends in computational materials science, especially the striving to forge links between modelling activities at various length and timescales. At the atomistic scale, methods based on quantum mechanical, especially density-functional, theories for electronic properties link to atomic/molecular dynamics and kinetic Monte Carlo simulations. Coarse graining leads to lattice-gas and cellular automata, and eventually to continuum equations solved by finite-element and finite-difference techniques. As examples of hierarchical modelling of materials, the paper describes recent work on anisotropic chemical etching of silicon, irradiation processing of fullerenes, oxygen clustering in silicon and self-diffusion in the compound semiconductor GaSb.

Most previous atomistic simulations of heterophase interfaces have neglected misfit, the discrepancy between the interatomic length scales parallel to the interface of the two phases. The obstacles to quantitative calculations of interface energies in the presence of misfit are assessed. The most straightforward approach is to perform simulations for a supercell whose size is of the order of the cube of the smallest common periodic length scale (essentially the coincidence-site-lattice periodicity), which varies inversely with the misfit parameter. Such supercells typically contain at least thousands of atoms. First-principles simulations are highly accurate, but are feasible only for a few selected heterophase interfaces with large misfit. Classical interatomic potentials, on the other hand, are efficient numerically, but their accuracy has not been demonstrated in the context of heterophase interface calculations. An approximate formulation of the interface energy is presented here which can be evaluated numerically by first-principles calculations for supercells of only moderate size. This formulation explores the relationship between the interface energies for coherent and semi-coherent interfaces. A numerical application to an interface between tetragonal TiAl and perovskite Ti₃AlC is presented.

The environment-dependent interaction potential is a transferable empirical potential for carbon which is well suited for studying disordered systems. Ab initio data are used to motivate and parametrize the functional form, which includes environment-dependence in the pair and triple terms, and a generalized aspherical coordination describing dihedral rotation and non-bonded π-repulsion. Simulations of liquid carbon compare very favourably with Car-Parrinello calculations, while amorphous networks generated by liquid quench have properties superior to Tersoff, Brenner and orthogonal tight-binding calculations. The efficiency of the method enables the first simulations of tetrahedral amorphous carbon by deposition, and a new model for the formation of diamond-like bonding is presented.

We discuss atomistic simulations of dislocation processes in copper based on effective medium theory interatomic potentials. Results on screw dislocation structures and processes are reviewed with particular focus on point defect mobilities and processes involving cross slip. For example, the stability of screw dislocation dipoles is discussed. We show that the presence of jogs will strongly influence cross slip barriers and dipole stability.

We furthermore present some new results on jogged edge dislocations and edge dislocation dipoles. The jogs are found to be extended, and simulations of vacancy controlled climb show the jogs to climb easily in their extended form. The stability of small vacancy dipoles is discussed and it is seen that the introduction of jogs may lead to the formation of  Z-type faulted vacancy dipoles.

Biological systems provide a particularly challenging set of problems for the application of ab initio quantum mechanical simulations. Despite this, these methods are providing insights into biological structures and processes at an atomistic level.

This paper outlines current methods for first-principles modelling of biological systems. Example applications to the cytochrome P450 family of metabolic enzymes, photoreactive rhodopsin proteins and the calculation of NMR chemical shifts are described. Finally, trends in the development of new algorithms and the biological problems to which they will be applied are discussed.

The calculation of total energy from electronic structure is now well established, and recent interest has moved to evaluation of free energies and equations of state. This paper discusses various methods for evaluating free energies, for equilibrium phases, for reaction pathways and for phase transformations.

A characteristic feature of the physics in transition-metal oxides is that the charge, spin, and lattice degrees of freedom are strongly coupled. The key to understanding these strong mutual couplings is the orbital degree of freedom (ODF), which plays a crucial role in controlling the phases and various physical properties. We have been working on TMO extensively in recent years. Examples are FeO and MnO, La_1-xSr_xMnO₃, Ca_2-xSr_xRuO₄, Sr₂FeMoO₆, and SrTiO₃, not only for the bulk but also for the surfaces. A review will be given in this article, with concentration on the strong coupling between the structural distortion and the magnetism mediated by ODF. Most of the studies were conducted by our STATE (simulational tool for atom technology) code, which is particularly designed for the transition-metal systems. Some particular aspects of STATE code, such as LDA + U method and virtual crystal approximation, will be also discussed.

The present status of development of the density-functional-based tight-binding (DFTB) method is reviewed. As a two-centre approach to density-functional theory (DFT), it combines computational efficiency with reliability and transferability. Utilizing a minimal-basis representation of Kohn-Sham eigenstates and a superposition of optimized neutral-atom potentials and related charge densities for constructing the effective many-atom potential, all integrals are calculated within DFT. Self-consistency is included at the level of Mulliken charges rather than by self-consistently iterating electronic spin densities and effective potentials. Excited-state properties are accessible within the linear response approach to time-dependent (TD) DFT. The coupling of electronic and ionic degrees of freedom further allows us to follow the non-adiabatic structure evolution via coupled electron-ion molecular dynamics in energetic particle collisions and in the presence of ultrashort intense laser pulses. We either briefly outline or give references describing examples of applications to ground-state and excited-state properties. Addressing the scaling problems in size and time generally and for biomolecular systems in particular, we describe the implementation of the parallel `divide-and-conquer' order-N method with DFTB and the coupling of the DFTB approach as a quantum method with molecular mechanics force fields.

The tight-binding (TB) approach to the modelling of electrical conduction in small structures is introduced. Different equivalent forms of the TB expression for the electrical current in a nanoscale junction are derived. The use of the formalism to calculate the current density and local potential is illustrated by model examples. A first-principles time-dependent TB formalism for calculating current-induced forces and the dynamical response of atoms is presented. An earlier expression for current-induced forces under steady-state conditions is generalized beyond local charge neutrality and beyond orthogonal TB. Future directions in the modelling of power dissipation and local heating in nanoscale conductors are discussed.