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Modeling Materials

Material properties emerge from phenomena on scales ranging from angstroms to mil-limeters, and only a multiscale treatment can provide a complete understanding. Materialsresearchers must therefore understand fundamental concepts and techniques from differentfields, and these are presented in a comprehensive and integrated fashion for the first timein this book.

Incorporating continuum mechanics, quantum mechanics, statistical mechanics, atom-istic simulations, and multiscale techniques, the book explains many of the key theoreticalideas behind multiscale modeling. Classical topics are blended with new techniques todemonstrate the connections between different fields and highlight current research trends.Example applications drawn from modern research on the thermomechanical properties ofcrystalline solids are used as a unifying focus throughout the text.

Together with its companion book, Continuum Mechanics and Thermodynamics(Cambridge University Press, 2012), this work presents the complete fundamentals ofmaterials modeling for graduate students and researchers in physics, materials science,chemistry, and engineering.

Ellad B. Tadmor is Professor of Aerospace Engineering and Mechanics, University ofMinnesota. His research focuses on the development of multiscale theories and com-putational methods for predicting the behavior of materials directly from the interaction ofthe atoms.

Ronald E. Miller is Professor of Mechanical and Aerospace Engineering, Carleton University.He has worked in the area of multiscale materials modeling for over 15 years and haspublished more than 40 scientific articles in the area.

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Cambridge University Press978-0-521-85698-0 - Modeling Materials: Continuum, Atomistic and Multiscale TechniquesEllad B. Tadmor and Ronald E. MillerFrontmatterMore information

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Modeling MaterialsContinuum, Atomistic and Multiscale Techniques

ELLAD B. TADMORUniversity of Minnesota, USA

RONALD E. MILLERCarleton University, Canada






C A M B R I D G E U N I V E R S I T Y P R E S S

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Library of Congress Cataloguing in Publication dataTadmor, Ellad B., 1965–

Modeling materials : continuum, atomistic, and multiscale techniques /Ellad B. Tadmor, Ronald E. Miller.

p. cm.Includes bibliographical references and index.

ISBN 978-0-521-85698-0 (hardback)1. Materials – Mathematical models. I. Miller, Ronald E. II. Title.

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Contents

Preface page xiiiAcknowledgments xviNotation xxi

1 Introduction 11.1 Multiple scales in crystalline materials 1

1.1.1 Orowan’s pocket watch 11.1.2 Mechanisms of plasticity 31.1.3 Perfect crystals 41.1.4 Planar defects: surfaces 71.1.5 Planar defects: grain boundaries 101.1.6 Line defects: dislocations 121.1.7 Point defects 151.1.8 Large-scale defects: cracks, voids and inclusions 16

1.2 Materials scales: taking stock 17Further reading 18

Part I Continuum mechanics and thermodynamics 19

2 Essential continuum mechanics and thermodynamics 212.1 Scalars, vectors, and tensors 22

2.1.1 Tensor notation 222.1.2 Vectors and higher-order tensors 262.1.3 Tensor operations 332.1.4 Properties of second-order tensors 372.1.5 Tensor fields 39

2.2 Kinematics of deformation 422.2.1 The continuum particle 422.2.2 The deformation mapping 432.2.3 Material and spatial descriptions 442.2.4 Description of local deformation 462.2.5 Kinematic rates 49

2.3 Mechanical conservation and balance laws 512.3.1 Conservation of mass 512.3.2 Balance of linear momentum 532.3.3 Balance of angular momentum 582.3.4 Material form of the momentum balance equations 59

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2.4 Thermodynamics 612.4.1 Macroscopic observables, thermodynamic equilibrium and state variables 612.4.2 Thermal equilibrium and the zeroth law of thermodynamics 652.4.3 Energy and the first law of thermodynamics 672.4.4 Thermodynamic processes 712.4.5 The second law of thermodynamics and the direction of time 722.4.6 Continuum thermodynamics 83

2.5 Constitutive relations 902.5.1 Constraints on constitutive relations 912.5.2 Local action and the second law of thermodynamics 922.5.3 Material frame-indifference 972.5.4 Material symmetry 992.5.5 Linearized constitutive relations for anisotropic hyperelastic solids 101

2.6 Boundary-value problems and the principle of minimum potential energy 105Further reading 108Exercises 109

Part II Atomistics 113

3 Lattices and crystal structures 1153.1 Crystal history: continuum or corpuscular? 1153.2 The structure of ideal crystals 1193.3 Lattices 119

3.3.1 Primitive lattice vectors and primitive unit cells 1203.3.2 Voronoi tessellation and the Wigner–Seitz cell 1223.3.3 Conventional unit cells 1233.3.4 Crystal directions 124

3.4 Crystal systems 1253.4.1 Point symmetry operations 1253.4.2 The seven crystal systems 129

3.5 Bravais lattices 1343.5.1 Centering in the cubic system 1343.5.2 Centering in the triclinic system 1373.5.3 Centering in the monoclinic system 1373.5.4 Centering in the orthorhombic and tetragonal systems 1383.5.5 Centering in the hexagonal and trigonal systems 1383.5.6 Summary of the fourteen Bravais lattices 139

3.6 Crystal structure 1393.6.1 Essential and nonessential descriptions of crystals 1423.6.2 Crystal structures of some common crystals 142

3.7 Some additional lattice concepts 1463.7.1 Fourier series and the reciprocal lattice 1463.7.2 The first Brillouin zone 1483.7.3 Miller indices 149

Further reading 151Exercises 151






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4 Quantum mechanics of materials 1534.1 Introduction 1534.2 A brief and selective history of quantum mechanics 154

4.2.1 The Hamiltonian formulation 1574.3 The quantum theory of bonding 160

4.3.1 Dirac notation 1604.3.2 Electron wave functions 1634.3.3 Schrodinger’s equation 1684.3.4 The time-independent Schrodinger equation 1714.3.5 The hydrogen atom 1724.3.6 The hydrogen molecule 1794.3.7 Summary of the quantum mechanics of bonding 187

4.4 Density functional theory (DFT) 1884.4.1 Exact formulation 1884.4.2 Approximations necessary for computational progress 1964.4.3 The choice of basis functions 1994.4.4 Electrons in periodic systems 2004.4.5 The essential machinery of a plane-wave DFT code 2104.4.6 Energy minimization and dynamics: forces in DFT 221

4.5 Semi-empirical quantum mechanics: tight-binding (TB) methods 2234.5.1 LCAO 2234.5.2 The Hamiltonian and overlap matrices 2244.5.3 Slater–Koster parameters for two-center integrals 2274.5.4 Summary of the TB formulation 2284.5.5 TB molecular dynamics 2284.5.6 From TB to empirical atomistic models 229


5 Empirical atomistic models of materials 2375.1 Consequences of the Born–Oppenheimer approximation (BOA) 2385.2 Treating atoms as classical particles 2405.3 Sensible functional forms 241

5.3.1 Interatomic distances 2425.3.2 Requirement of translational, rotational and parity invariance 2435.3.3 The cutoff radius 245

5.4 Cluster potentials 2465.4.1 Formally exact cluster potentials 2475.4.2 Pair potentials 2515.4.3 Modeling ionic crystals: the Born–Mayer potential 2565.4.4 Three- and four-body potentials 2575.4.5 Modeling organic molecules: CHARMM and AMBER 2595.4.6 Limitations of cluster potentials and the need for interatomic functionals 261

5.5 Pair functionals 2625.5.1 The generic pair functional form: the glue–EAM–EMT–FS model 2635.5.2 Physical interpretations of the pair functional 2645.5.3 Fitting the pair functional model 2655.5.4 Comparing pair functionals to cluster potentials 266






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5.6 Cluster functionals 2685.6.1 Introduction to the bond order: the Tersoff potential 2685.6.2 Bond energy and bond order in TB 2715.6.3 ReaxFF 2745.6.4 The modified embedded atom method 276

5.7 Atomistic models: what can they do? 2795.7.1 Speed and scaling: how many atoms over how much time? 2795.7.2 Transferability: predicting behavior outside the fit 2825.7.3 Classes of materials and our ability to model them 285

5.8 Interatomic forces in empirical atomistic models 2885.8.1 Weak and strong laws of action and reaction 2885.8.2 Forces in conservative systems 2915.8.3 Atomic forces for some specific interatomic models 2945.8.4 Bond stiffnesses for some specific interatomic models 2975.8.5 The cutoff radius and interatomic forces 298


6 Molecular statics 3046.1 The potential energy landscape 3046.2 Energy minimization 306

6.2.1 Solving nonlinear problems: initial guesses 3066.2.2 The generic nonlinear minimization algorithm 3076.2.3 The steepest descent (SD) method 3086.2.4 Line minimization 3106.2.5 The conjugate gradient (CG) method 3116.2.6 The condition number 3126.2.7 The Newton–Raphson (NR) method 313

6.3 Methods for finding saddle points and transition paths 3156.3.1 The nudged elastic band (NEB) method 316

6.4 Implementing molecular statics 3216.4.1 Neighbor lists 3216.4.2 Periodic boundary conditions (PBCs) 3256.4.3 Applying stress and pressure boundary conditions 3286.4.4 Boundary conditions on atoms 330

6.5 Application to crystals and crystalline defects 3316.5.1 Cohesive energy of an infinite crystal 3326.5.2 The universal binding energy relation (UBER) 3346.5.3 Crystal defects: vacancies 3386.5.4 Crystal defects: surfaces and interfaces 3396.5.5 Crystal defects: dislocations 3476.5.6 The γ-surface 3576.5.7 The Peierls–Nabarro model of a dislocation 360

6.6 Dealing with temperature and dynamics 371Further reading 371Exercises 372






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Part III Atomistic foundations of continuum concepts 375

7 Classical equilibrium statistical mechanics 3777.1 Phase space: dynamics of a system of atoms 378

7.1.1 Hamilton’s equations 3787.1.2 Macroscopic translation and rotation 3797.1.3 Center of mass coordinates 3807.1.4 Phase space coordinates 3817.1.5 Trajectories through phase space 3827.1.6 Liouville’s theorem 384

7.2 Predicting macroscopic observables 3877.2.1 Time averages 3877.2.2 The ensemble viewpoint and distribution functions 3897.2.3 Why does the ensemble approach work? 392

7.3 The microcanonical (NVE) ensemble 4037.3.1 The hypersurface and volume of an isolated Hamiltonian system 4037.3.2 The microcanonical distribution function 4067.3.3 Systems in weak interaction 4097.3.4 Internal energy, temperature and entropy 4127.3.5 Derivation of the ideal gas law 4187.3.6 Equipartition and virial theorems: microcanonical derivation 420

7.4 The canonical (NVT) ensemble 4237.4.1 The canonical distribution function 4247.4.2 Internal energy and fluctuations 4287.4.3 Helmholtz free energy 4297.4.4 Equipartition theorem: canonical derivation 4317.4.5 Helmholtz free energy in the thermodynamic limit 432


8 Microscopic expressions for continuum fields 4408.1 Stress and elasticity in a system in thermodynamic equilibrium 442

8.1.1 Canonical transformations 4428.1.2 Microscopic stress tensor in a finite system at zero temperature 4478.1.3 Microscopic stress tensor at finite temperature: the virial stress 4508.1.4 Microscopic elasticity tensor 460

8.2 Continuum fields as expectation values: nonequilibrium systems 4658.2.1 Rate of change of expectation values 4668.2.2 Definition of pointwise continuum fields 4678.2.3 Continuity equation 4698.2.4 Momentum balance and the pointwise stress tensor 4698.2.5 Spatial averaging and macroscopic fields 475

8.3 Practical methods: the stress tensor 4798.3.1 The Hardy stress 4808.3.2 The virial stress tensor and atomic-level stresses 4818.3.3 The Tsai traction: a planar definition for stress 4828.3.4 Uniqueness of the stress tensor 4878.3.5 Hardy, virial and Tsai stress expressions: numerical considerations 488

Exercises 489






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9 Molecular dynamics 4929.1 Brief historical introduction 4929.2 The essential MD algorithm 4959.3 The NVE ensemble: constant energy and constant strain 497

9.3.1 Integrating the NVE ensemble: the velocity-Verlet (VV) algorithm 4979.3.2 Quenched dynamics 5049.3.3 Temperature initialization 5049.3.4 Equilibration time 507

9.4 The NVT ensemble: constant temperature and constant strain 5079.4.1 Velocity rescaling 5089.4.2 Gauss’ principle of least constraint and the isokinetic thermostat 5099.4.3 The Langevin thermostat 5119.4.4 The Nose–Hoover (NH) thermostat 5139.4.5 Liouville’s equation for non-Hamiltonian systems 5169.4.6 An alternative derivation of the NH thermostat 5179.4.7 Integrating the NVT ensemble 518

9.5 The finite strain NσE ensemble: applying stress 5209.5.1 A canonical transformation of variables 5219.5.2 The hydrostatic stress state 5279.5.3 The Parrinello–Rahman (PR) approximation 5289.5.4 The zero-temperature limit: applying stress in molecular statics 5309.5.5 The kinetic energy of the cell 533

9.6 The NσT ensemble: applying stress at a constant temperature 533Further reading 534Exercises 534

Part IV Multiscale methods 537

10 What is multiscale modeling? 53910.1 Multiscale modeling: what is in a name? 53910.2 Sequential multiscale models 54110.3 Concurrent multiscale models 543

10.3.1 Hierarchical methods 54410.3.2 Partitioned-domain methods 546

10.4 Spanning time scales 547Further reading 549

11 Atomistic constitutive relations for multilattice crystals 55011.1 Statistical mechanics of systems in metastable equilibrium 554

11.1.1 Restricted ensembles 55411.1.2 Properties of a metastable state from a restricted canonical ensemble 556

11.2 Relating mean positions to applied deformation: the Cauchy–Born rule 55811.2.1 Multilattice crystals and mean positions 55811.2.2 Cauchy–Born kinematics 55911.2.3 Centrosymmetric crystals and the Cauchy–Born rule 56111.2.4 Extensions and failures of the Cauchy–Born rule 562






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11.3 Finite temperature constitutive relations for multilattice crystals 56311.3.1 Periodic supercell of a multilattice crystal 56311.3.2 Helmholtz free energy density of a multilattice crystal 56611.3.3 Determination of the reference configuration 56711.3.4 Uniform deformation and the macroscopic stress tensor 57011.3.5 Elasticity tensor 575

11.4 Quasiharmonic approximation 57811.4.1 Quasiharmonic Helmholtz free energy 57811.4.2 Determination of the quasiharmonic reference configuration 58211.4.3 Quasiharmonic stress and elasticity tensors 58611.4.4 Strict harmonic approximation 590

11.5 Zero-temperature constitutive relations 59211.5.1 General expressions for the stress and elasticity tensors 59211.5.2 Stress and elasticity tensors for some specific interatomic models 59311.5.3 Crystal symmetries and the Cauchy relations 595


12 Atomistic–continuum coupling: static methods 60112.1 Finite elements and the Cauchy–Born rule 60112.2 The essential components of a coupled model 60412.3 Energy-based formulations 608

12.3.1 Total energy functional 60812.3.2 The quasi-continuum (QC) method 61012.3.3 The coupling of length scales (CLS) method 61312.3.4 The bridging domain (BD) method 61412.3.5 The bridging scale method (BSM) 61612.3.6 CACM: iterative minimization of two energy functionals 61712.3.7 Cluster-based quasicontinuum (CQC-E) 618

12.4 Ghost forces in energy-based methods 62012.4.1 A one-dimensional Lennard-Jones chain of atoms 62212.4.2 A continuum constitutive law for the Lennard-Jones chain 62312.4.3 Ghost forces in a generic energy-based model of the chain 62312.4.4 Ghost forces in the cluster-based quasicontinuum (CQC-E) 62712.4.5 Ghost force correction methods 630

12.5 Force-based formulations 63112.5.1 Forces without an energy functional 63112.5.2 FEAt and CADD 63312.5.3 The hybrid simulation method (HSM) 63412.5.4 The atomistic-to-continuum (AtC) method 63412.5.5 Cluster-based quasicontinuum (CQC-F) 63612.5.6 Spurious forces in force-based methods 636

12.6 Implementation and use of the static QC method 63812.6.1 A simple example: shearing a twin boundary 63812.6.2 Setting up the model 64012.6.3 Solution procedure 64212.6.4 Twin boundary migration 64412.6.5 Automatic model adaption 645






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12.7 Quantitative comparison between the methods 64712.7.1 The test problem 64812.7.2 Comparing the accuracy of multiscale methods 65012.7.3 Quantifying the speed of multiscale methods 65412.7.4 Summary of the relative accuracy and speed of multiscale methods 655

Exercises 656

13 Atomistic–continuum coupling: finite temperature and dynamics 65813.1 Dynamic finite elements 65913.2 Equilibrium finite temperature multiscale methods 661

13.2.1 Effective Hamiltonian for the atomistic region 66213.2.2 Finite temperature QC framework 66713.2.3 Hot-QC-static: atomistic dynamics embedded in a static continuum 67013.2.4 Hot-QC-dynamic: atomistic and continuum dynamics 67213.2.5 Demonstrative examples: thermal expansion and nanoindentation 675

13.3 Nonequilibrium multiscale methods 67713.3.1 A naıve starting point 67813.3.2 Wave reflections 67813.3.3 Generalized Langevin equations 68313.3.4 Damping bands 687

13.4 Concluding remarks 689Exercises 689

Appendix A Mathematical representation of interatomic potentials 690A.1 Interatomic distances and invariance 691A.2 Distance geometry: constraints between interatomic distances 693A.3 Continuously differentiable extensions of V int(s) 696A.4 Alternative potential energy extensions and the effect on atomic forces 698

References 702Index 746






Preface

Studying materials can mean studying almost anything, since all of the physical, tangibleworld is necessarily made of something. Normally, we think of studying materials in thesense of materials science and engineering – an endeavor to understand the properties ofnatural and man-made materials and to improve or exploit them in some way – but eventhis includes broad and disparate goals. One can spend a lifetime studying the strengthand toughness of steel, for example, and never once concern oneself with its magneticor electric properties. At the same time, modeling in science can mean many things tomany people, ranging from computer simulation to analytical effective theories to abstractmathematics. To combine these two terms “modeling materials” as the title of a singlebook, then, is surely to invite disaster. How could it be possible to cover all the topics thatthe product modeling × materials implies? Although this book remains true to its title, itwill be necessary to pick and choose our topics so as to have a manageable scope. To startwith, then, we have to decide: what models and what materials do we want to discuss?

As far as modeling goes, we must first recognize the fact that materials exhibit phe-nomena on a broad range of spatial and temporal scales that combine together to dictatethe response of a material. These phenomena range from the bonding of individual atomsgoverned by quantum mechanics to macroscopic deformation processes described by con-tinuum mechanics. Various aspects of materials behavior and modeling, which tends tofocus on specific phenomena at a given scale, have traditionally been treated by differentdisciplines in science and engineering. The great disparity in scales and the interdisciplinarynature of the field are what makes modeling materials both challenging and exciting. It isunlikely that any one researcher has sufficient background in engineering, physics, mate-rials science and mathematics to understand materials modeling at every length and timescale. Furthermore, there is increased awareness that materials must be understood, notonly by rigorous treatment of phenomena at each of these scales alone, but rather throughconsideration of the interactions between these scales. This is the paradigm of multiscalemodeling that will also be a persistent theme throughout the book.

Recognizing the need to integrate models from different disciplines creates problemsof nomenclature, notation and jargon. While we all strive to make our research specialtiesclear and accessible, it is a necessary part of scientific discourse to create and use spe-cific terms and notation. An unintended consequence of this is the creation of barriers tointerdisciplinary understanding. One of our goals is to try to facilitate this understandingby providing a unified presentation of the fundamentals, using a common nomenclature,notation and language that will be accessible to people across disciplines. The result is thisbook on Modeling Materials (MM) and our companion book on Continuum Mechanicsand Thermodynamics (CMT) [TME12].

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xiv Preface�

The subject matter in MM is divided into four parts. Part I covers continuum mechanicsand thermodynamics concepts that serve as the basis for the rest of the book. The descrip-tion of continuum mechanics and thermodynamics is brief and is only meant to make MMa stand-alone book. The reader is referred to CMT for a far deeper view of these subjectsconsistent with the rest of MM. Part II covers atomistics, discussing the basic structure andsymmetries of crystalline materials, quantum mechanics and more approximate empiricalmodels for describing bonding in materials, and molecular statics – a computational ap-proach for studying static properties of materials at the atomic scale. Part III focuses on theatomistic foundations of continuum concepts. Here, using ideas from statistical mechanics,connections are forged between the discrete world of atoms – described by atomic positions,velocities and forces – and continuum concepts such as fields of stress and temperature.Finally, the subject of molecular dynamics (MD) is presented. We treat MD as a computa-tional method for studying dynamical properties of materials at the atomic scale subject tocontinuum-level constraints, so it is yet another unique connection between the atomisticand continuum views. Part IV on multiscale methods describes a class of computationalmethods that attempt to model material response by simultaneously describing its behavioron multiple spatial and temporal scales. This final part of the book draws together andunifies many of the concepts presented earlier and shows how these can be integrated intoa single modeling paradigm.

By bringing together this unusual combination of topics, we provide a treatment that isuniquely different from other books in the field. First, our focus is on a critical analysis andunderstanding of the fundamental assumptions that underpin these methods and that areoften taken for granted in other treatments. We believe that this focus on fundamentals isessential for anyone seeking to combine different theories in a multiscale setting. Secondly,some of the topics herein are often treated from the perspective of the gaseous or liquid states.Here, our emphasis is on solids, and this changes the presentation in important ways. Forexample, in statistical mechanics we comprehensively discuss the subject of the stress tensor(not just pressure) and the concept of a restricted ensemble for metastable systems. Similarly,we talk at length about constant stress simulations in MD and how to correctly interpretthem in a setting of finite deformation beyond that of simple hydrostatic compression. Third,while covering this broad range of topics we strive to regularly make connections betweenthe atomistic, statistical and continuum worldviews. Finally, we have tried to create a healthybalance between fundamental theory and practical “how to.” For example, we present, atlength, the practical implementation of such topics as density functional theory, empiricalatomistic potentials, molecular statics and dynamics, and multiscale partitioned-domainmethods. It is our hope that someone with basic computer programming skills will be ableto use this book to implement any of these methods, or at least to better understand animplementation in a pre-existing code.

Although the modeling methods we describe are, in principle, applicable to any material,we focus our scope of materials and properties on those that we, the authors, know best.The answer to “What materials?” then is crystalline solids and their thermomechanical(as opposed to electrical, optical or chemical) properties. For the most part, these serveas examples to illustrate the application and usefulness of the modeling methods that we






xv Preface�

describe, but we hope that the reader will also learn something new regarding the materialsthemselves along the way.

Even starting from this narrow mandate we have already failed, to some degree, in ourgoal of putting all the fundamentals in one place. This is because the binding of thesesubjects into a single volume becomes unwieldy if we wish to maintain the level of detailthat we feel is necessary. To make room in this book, we sacrificed coverage of continuummechanics, leaving only a concise summary in Chapter 2 of the key results needed to makecontact with the rest of the topics. CMT [TME12], the companion volume to this one,provides the full details of the continuum mechanics and thermodynamics that we believeto be fundamental to materials modeling.

Both books, MM and CMT, are addressed to graduate students and researchers in chem-istry, engineering, materials science, mathematics, mechanics and physics. The interdisci-plinary nature of materials modeling means that researchers from all of these fields havecontributed to and continue to be engaged in this field. The motivation for these bookscame from our own frustration, and that of ours students, as we tried to acquire the breadthof knowledge necessary to do research in this highly interdisciplinary field. We have madeevery effort to eliminate this frustration in the future by making our writing accessible toall readers with an undergraduate education in an engineering or scientific discipline. Thewriting is self-contained, introducing all of the necessary basic concepts and building upfrom there. Of course, by necessity that means that our coverage of the different topics islimited and skewed to our primary focus on materials modeling. At the end of each chap-ter, we recommend sources for further reading for readers interested in expanding theirunderstanding in a particular direction.






Acknowledgments

One of our favorite teachers when we were graduate students at Brown University, BenFreund, has said that writing a book is like giving birth. The process is long and painful, itinvolves a lot of screaming, but in the end something has to come out. We find this analogyso apt that we feel compelled to extend it: in some cases, you are blessed with twins.1 Aswe initially conceived it, our goal was to have everything in a single volume. But as timewent on, and what we were “carrying” grew bigger and bigger, it became clear that it reallyneeded to be two separate books.

Since the book has been split in two, we choose to express our gratitude twice, once ineach book, to everyone who has helped us with the project as a whole. Surely, thankingeveryone twice is the least we can do. Some people helped in multiple ways, and so theirnames appear even more often. Our greatest debt goes to our wives, Jennifer and Granda,and to our children: Maya, Lea, Persephone and Max. They have suffered more than anyoneduring the long course of this project, as their preoccupied husbands and fathers stole toomuch time from so many other things. They need to be thanked for such a long list ofreasons that we would likely have to split these two books into three if we were thoroughwith the details. Thanks, all of you, for your patience and support. We must also thank ourown parents Zehev and Ciporah and Don and Linda for giving us the impression – perhapsmistaken – that everybody will appreciate what we have to say as much as they do.

The writing of a book as diverse as this one is really a collaborative effort with somany people whose names do not appear on the cover. These include students in courses,colleagues in the corridors and offices of our universities and unlucky friends cornered atconferences. The list of people that offered a little piece of advice here, a correction there,or a word of encouragement somewhere else is indeed too long to include, but there are afew people in particular that deserve special mention.

Some colleagues generously did calculations for us, verified results, or provided othercontributions from their own work. We thank Quiying Chen at the NRC Institute forAerospace Research in Ottawa for his time in calculating UBER curves with DFT. TsvetaSendova, a postdoctoral fellow at the University of Minnesota, coded and ran the simulationsfor the two-dimensional NEB example we present. Another postdoctoral fellow at theUniversity of Minnesota, Woo Kyun Kim, performed the indentation and thermal expansionsimulations used to illustrate the hot-QC method. We thank Yuri Mishin (George Mason

1 This analogy is made with the utmost respect for our wives, and anyone else who actually has given birth.Assuming labor has units of power, then we feel that the integral of this power over the very different timescalesof the two processes should yield quantities of work that are on the same order of magnitude. Our wives disagree,no doubt in part because some of the power consumed by book-writing indirectly comes from them, whereasour contribution to childbirth amounts mainly to sending around e-mail photos of the newborn children.

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xvii Acknowledgments�

University) for providing figures, and Christoph Ortner (Oxford University) for providingmany insights into the problem of full versus sequential minimization of multivariatefunctions, including the example we provide in the book. The hot-QC project has greatlybenefited from the work of Laurent Dupuy (SEA Saclay) and Frederic Legoll (EcoleNationale des Ponts et Chaussees). Their help in preparing a journal paper on the subjecthas also proven extremely useful in preparing the chapter on dynamic multiscale methods.Furio Ercolessi must be thanked in general for his fantastic web-based notes on so manyimportant subjects discussed herein, and specifically for providing us with his moleculardynamics code as a teaching tool to provide with this book.

Other colleagues patiently taught us the many subjects in this book about which weare decidedly not experts. Dong Qian at the University of Cincinnati and Michael Parksat Sandia National Laboratories very patiently and repeatedly explained the nuances ofvarious multiscale methods to us. Similarly, we would like to thank Catalin Picu at theRensselaer Polytechnic Institute for explaining CACM, and Leo Shilkrot for his frank con-versations about CADD and the BSM. Noam Bernstein at the Navy Research Laboratorieswas invaluable in explaining DFT in a way that an engineer could understand, and PeterWatson at Carleton University was instrumental in our eventual understanding of quantummechanics. Roger Fosdick University of Minnesota discussed, at length, many topics relatedto continuum mechanics including tensor notation, material frame-indifference, Reynold’stransport theorem and the principle of action and reaction. He also took the time to readand comment on our take on material frame-indifference.

We are especially indebted to those colleagues that were willing to take the time tocarefully read and comment on drafts of various sections of the book – a thankless anddelicate task. James Sethna (Cornell University) and Dionisios Margetis (University ofMaryland) read and commented on the statistical mechanics chapter. Noam Bernstein(Navy Research Laboratories) must be thanked more than once for reading and commentingon both the quantum mechanics chapter and the sections on cluster expansions. NikhilAdmal, a graduate student working with Ellad at the University of Minnesota, contributedsignificantly to our understanding of stress and read and commented on the continuummechanics chapter, Marcel Arndt helped by translating an important paper on stress byWalter Noll from German to English and worked with Ellad on developing algorithmsfor lattice calculations, while Gang Lu at the California State University (Northridge) setus straight on several points about density functional theory. Ryan Elliott, our coauthorof the companion book to this one, must also be thanked countless times for his carefulreading of quantum mechanics and his many helpful suggestions and discussions. Otherpatient readers to whom we say “thank you” include Mitch Luskin from the University ofMinnesota (numerical analysis of multiscale methods and quantum mechanics), Bill Curtinfrom Brown University (static multiscale methods), Dick James from the University ofMinnesota (restricted ensembles and the definition of stress) and Leonid Berlyand fromPennsylvania State University (thermodynamics).

There are a great many colleagues that were willing to talk to us at length about varioussubjects in this book. We hope that we did not overstay our welcome in their offices toooften, and that they do not sigh too deeply anymore when they see a message from usin their inbox. Most importantly, we thank them very much for their time. In addition to






xviii Acknowledgments�

those already mentioned above, we thank David Rodney (Institut National PolytechniqueGrenoble), Perry Leo and Tom Shield (University of Minnesota), Miles Rubin and EliAltus (Technion), Joel Lebowitz, Sheldon Goldstein and Michael Kiessling (Rutgers),2 andAndy Ruina (Cornell University). We would also be remiss if we did not take the time tothank Art Voter (Los Alamos National Laboratory), John Moriarty (Lawrence LivermoreNational Laboratory) and Mike Baskes (Sandia National Laboratory) for many insightfuldiscussions and suggestions of valuable references.

There are some things in these books that are so far outside our area of expertise that wehave even had to look beyond the offices of professors and researchers. Elissa Gutterman,an expert in linguistics, provided phonetic pronunciation of French and German names.As neither of us is an experimentalist, our brief foray into pocket watch “testing” wouldnot have been very successful without the help of Steve Truttman and Stan Conley in thestructures laboratories at Carleton University. The story of our cover images involves somany people, it deserves its own paragraph.

As the reader will see in the introduction to both books, we are fond of the symbolicconnection between pocket watches and the topics we discuss herein. There are manybeautiful images of pocket watches out there, but obtaining one of sufficient resolution,and getting permission to use it, is surprisingly difficult. As such, we owe a great debtto Mr. Hans Holzach, a watchmaker and amateur photographer at Beyer ChronometrieAG in Zurich. Not only did he generously agree to let us use his images, he took overthe entire enterprise of retaking the photographs when we found out that his images didnot have sufficient resolution! This required Hans to coordinate with many people that wealso thank for helping make the strikingly beautiful cover images possible. These includethe photographer, Dany Schulthess (www.fotos.ch), Mr. Rene Beyer, the owner of BeyerChronometrie AG in Zurich, who compensated the photographer and allowed photographsto be taken at his shop, and also to Dr. Randall E. Morris, the owner of the pocket watch,who escorted it from California to Switzerland (!) in time for the photo shoot. The fact thattotal strangers would go to such lengths in response to an unsolicited e-mail contact is atestament to their kind spirits and, no doubt, to their proud love of the beauty of pocketwatches.

We cannot forget our students. Many continue to teach us things every day just bybringing us their questions and ideas. Others were directly used as guinea pigs with earlydrafts of parts of this book.3 Ellad would like to thank his graduate students and post-doctoral fellows over the last five years who have been fighting with this book for attention;specifically Nikhil Admal, Yera Hakobian, Hezi Hizkiahu, Dan Karls, Woo Kyun Kim,Leonid Kucherov, Amit Singh, Tsvetanka Sendova, Valeriu Smiricinschi, Slava Sorkin andSteve Whalen. Ron would likewise like to thank Ishraq Shabib, Behrouz Shiari and DenisSaraev, whose work helped shape his ideas about atomistic modeling. Harley Johnsonand his 2008–2009 graduate class at the University of Illinois (Urbana-Champaign) usedthe book extensively and provided great feedback to improve the manuscript, as did Bill

2 Ellad would particularly like to thank the Rutgers trio for letting him join them on one of their lunches to discussthe foundations of statistical mechanics – a topic which is apparently standard lunch fare for them along withthe foundations of quantum mechanics.

3 Test subjects were always treated humanely and no students were harmed during the preparation of this book.






xix Acknowledgments�

Curtin’s class at Brown University in 2009–2010. The 2009 and 2010 classes of Ron’s“Microstructure and properties of engineering materials” class caught many initial errorsin the chapters on crystal structures and molecular statics and dynamics. Some students ofEllad’s continuum mechanics course are especially noted for their significant contributions:Yilmaz Bayazit (2008), Pietro Ferrero (2009), Zhuang Houlong (2008), Jenny Hwang(2009), Karl Johnson (2008), Dan Karls (2008), Minsu Kim (2009), Nathan Nasgovitz(2008), Yintao Song (2008) and Chonglin Zhang (2008).

Of course, we should also thank our own teachers. Course notes from Michael Ortiz,Janet Blume, Jerry Weiner and Tom Shield were invaluable to us in preparing our ownnotes and this book. Thanks also to our former advisors at Brown University, Michael Ortizand Rob Phillips (both currently at Caltech), whose irresistible enthusiasm, curiosity andencouragement pulled us down this most rewarding of scientific paths.

We note that many figures in this book were prepared with the drawing package Asymp-tote (see http://asymptote.sourceforge.net/), an open-source effort that we think deservesto be promoted here. Finally, we thank our editor Simon Capelin and the entire team atCambridge, for their advice, assistance and truly astounding patience.






Notation

In a book covering such a broad range of topics, notation is a nightmare. We have attempted,as much as possible, to use the most common and familiar notation from within each field aslong as this did not lead to confusion. However, this does mean that the occasional symbolwill serve multiple purposes, as the tables below will help to clarify. To keep the amountof notation to a minimum, we generally prefer to append qualifiers to symbols rather thanintroducing new symbols. For example, f is force, which if relevant can be divided intointernal, f int , and external, f ext , parts.

We use the following general conventions:

• Descriptive qualifiers generally appear as superscripts and are typeset using a Roman (asopposed to Greek) nonitalic font.

• The weight and style of the font used to render a variable indicates its type. Scalarvariables are denoted using an italic font. For example, T is temperature. Array variablesare denoted using a sans serif font, such as A for the matrix A. Vectors and tensors (inthe technical sense of the word) are rendered in a boldface font. For example, σ is thestress tensor.

• Variables often have subscript and superscript indices. Indices referring to the compo-nents of a matrix, vector or tensor appear as subscripts in italic Roman font. For example,vi is the ith component of the velocity vector. Superscripts are used as counters of vari-ables. For example, F e is the deformation gradient in element e. Superscripts referringto atoms are distinguished by using a Greek letter. For example, the velocity of atom α

is denoted vα . Iteration counters appear in parentheses, for example f (i) is the force initeration i.

• The Einstein summation convention is followed on repeated indices (e.g. vivi = v21 +

v22 + v2

3 ), unless otherwise clear from the context. (See Section 2.1.1 for more details.)• One special type of superscript concerns the denotation of Bravais lattices and crystals.

For example, the position vector R[�λ ] denotes the λth basis atom associated with Bravaislattice site �. (See Section 3.6 for details.)

• A subscript is used to refer to multiple equations on a single line, for example“Eqn. (2.66)2” refers to the second equation in Eqn. (2.66) (“ai(x, t) ≡ . . . ”).

• Important equations are emphasized by shading.

Below, we describe the main notation and symbols used in the book, and indicate the pageon which each is first defined. We also include a handy list of fundamental constants andunit conversions at the end of this section.

xxi






xxii Notation�

Mathematical notation

Notation Description Page

≡ equal to by definition 28:= variable on the left is assigned the value on the right 24∀ for all 28∈ contained in 28iff if and only if 28O(f) terms proportional to order f 188O(n) orthogonal group of degree n 31SO(n) proper orthogonal (special orthogonal) group of degree n 31R set of all real numbers 26Rn real coordinate space (n-tuples of real numbers) 27| • | absolute value of a real number 28‖•‖ norm of a vector 28〈•, •〉 inner product of two vectors 28〈•|•〉 inner product of two vectors (bra-ket notation) 161〈•| • |•〉 bra-operator-ket inner product 163[uvw] direction in a crystal (ua + vb + wc) 125〈uvw〉 family of crystal directions 125(hkl) Miller indices denoting a crystallographic plane 150{hkl} family of crystallographic planes 151{M | C} a set of members M such that conditions C are satisfied 245• time average of a quantity 388〈•〉 phase average of a quantity 391〈•; f〉 phase average of a quantity relative to distribution function f 391Pr(O) probability of outcome O 395Var(A) variance of A: Var(A) =

⟨A2

⟩− (〈A〉)2 396

Cov(A,B) covariance of A and B: Cov(A,B) = 〈AB〉 − 〈A〉〈B〉 461Covχ(A,B) covariance of A and B in a restricted ensemble 577

f(k) Fourier transform of f(x) 166f˜(s) Laplace transform of a f(t) 684

•∗ complex conjugate 161AT transpose of a matrix or second-order tensor: [AT ]ij = Aji 25A−T transpose of the inverse ofA: A−T ≡ (A−1)T 35a · b dot product (vectors): a · b = aibi 28a× b cross product (vectors): [a× b]k = εijkaibj 30a⊗ b tensor product (vectors): [a⊗ b]ij = aibj 33A : B contraction (second-order tensors): A : B = AijBij 36A · ·B transposed contraction (second-order tensors): A · ·B = AijBji 36λA

α , ΛAα αth eigenvalue and eigenvector of the second-order tensor A 38

Ik (A) kth principal invariant of the second-order tensor A 38det A determinant of a matrix or a second-order tensor 26trA trace of a matrix or a second-order tensor: trA = Aii 25






xxiii Notation�

∇•, grad • gradient of a tensor (deformed configuration) 40∇0•, Grad • gradient of a tensor (reference configuration) 45curl • curl of a tensor (deformed configuration) 41Curl • curl of a tensor (reference configuration) 45div • divergence of a tensor (deformed configuration) 41Div • divergence of a tensor (reference configuration) 45∇2• Laplacian of a tensor (deformed configuration) 41d inexact differential 81

rαβ position vector to closest periodic image of β to atom α 326�α,

#α unit cell and sublattice of atom α in a multilattice crystal 564

General symbols – Greek

Symbol Description Page

Γ phase space 382Γ, Γi set of extensive kinematic state variables 63Γi wave vector of the ith DFT plane wave basis function 211γ, γi set of intensive state variables work conjugate with Γ 78γ damping coefficient 511γs surface energy 340γGB grain boundary energy 346γSF stacking fault energy 357δij Kronecker delta 25ε energy of an electron 164ε, εij small strain tensor 49εijk permutation symbol 26ζαi fractional coordinates of basis atom α 142

καβγδ scalar atomistic stiffness term relating bonds α–β and γ–δ 297Λ de Broglie thermal wavelength 241Λi projection operator 162λ Lame constant 105λ plane wave wavelength 164μ shear modulus (solid) 105μ(m) mth moment of a function 230ν Poisson’s ratio 105νe number of atoms associated with element e in QC 612Π total potential energy of a system and the applied loads 107Πα , Πα

i pull-back momentum of atom α 452ρ mass density (deformed configuration) 52ρ electron density 188ρ0 mass density (reference configuration) 52ρpt pointwise (microscopic) mass density field 468






xxiv Notation�

ρα total electron density at atom α in a pair functional 263Σ(E;ΔE) hypershell in phase space with energy E and thickness ΔE 404σ, σij Cauchy stress tensor 56σinst , σinst

ij instantaneous atomic-level stress 457σpt , σpt

ij pointwise (microscopic) Cauchy stress tensor 470

σpt,K , σpt,Kij kinetic part of the pointwise (microscopic) Cauchy stress 471

σpt,V , σpt,Vij potential part of the pointwise (microscopic) Cauchy stress 471

ϕ, ϕi deformation mapping 43φ(r) pair potential as a function of distance r 251ϕ electron wave basis function 173ϕαβ scalar magnitude of force on atom α due to presence of atom β 291χ general, time-dependent electronic wave function 163χ characteristic function in restricted ensemble 554ψ specific Helmholtz free energy 95ψ general, time-independent electronic wave function 165ψsp single-particle, time-independent electronic wave function 194Ω volume of a periodic simulation cell in a DFT simulation 210Ω0 nonprimitive unit cell volume in reference configuration 124Ω volume of the first Brillouin zone 208Ω0 primitive unit cell volume in reference configuration 122Ω(E;ΔE) volume of hypershell Σ(E;ΔE) in phase space 404ω plane wave frequency 164

General symbols – Roman

Symbol Description Page

A macroscopic observable associated with phase function A(q,p) 387A(q,p) phase function associated with macroscopic observable A 387A1 , A2 , A3 reference nonprimitive lattice vectors 123A1 , A2 , A3 reference primitive lattice vectors 120a, ai acceleration vector 50a1 ,a2 ,a3 nonprimitive lattice vector (deformed configuration) 561B the first Brillouin zone 212B bulk modulus 112B(x;u,v) bond function at x due to the spatially averaged bond u–v 479B1 , B2 , B3 reciprocal reference lattice vectors 147B, Bij left Cauchy–Green deformation tensor 47BO bond order 272b, bi body force (spatial description) 55b, bi Burgers vector 351bpt , bpt

i pointwise (microscopic) body force field 470C the DFT simulation cell 210






xxv Notation�

Cv molar heat capacity at constant volume 69C, CIJ right Cauchy–Green deformation tensor 47C, CIJ K L referential elasticity tensor 101cv specific heat capacity at constant volume 70cI , cI j , cα

iI Ith eigenvector solution and its components (associated withplane wave j or orbital i on atom α)

176

c, cijkl spatial (or small strain) elasticity tensor 102c, cmn elasticity matrix (in Voigt notation) 104D(E) density of states (statistical mechanics) 405D(ε) electronic density of states 230Dα

i electronic density of states for orbital i on atom α 230D, DiJ kL mixed elasticity tensor 102d, dij rate of deformation tensor 50E total energy of a thermodynamic system 68E Young’s modulus 105Efree(Z) energy of a free (isolated) atom with atomic number Z 247Ecoh , E0

coh cohesive energy and equilibrium cohesive energy 332E, EIJ Lagrangian strain tensor 48ei orthonormal basis vectors 27F ext total external force acting on a system 54F , FiJ deformation gradient 46f occupancy of an electronic orbital 208f(q,p; t) distribution function at point (q,p) in phase space at time t 391fmc(q,p;E) microcanonical (NVE) distribution function 407fc(q,p;T ) canonical (NVT) distribution function 427fα , fα

i force on atom α 54fαβ , fαβ

i force on atom α due to the presence of atom β 289f int,α , f int,α

i internal force on atom α 289f ext,α , f ext,α

i external force on atom α 289f column matrix of finite element nodal forces 603Gα , Gα

i stochastic force on atom α 511g specific Gibbs free energy 96g(r) electron density function in a pair functional 264H Hamiltonian of a system 159H , Hi angular momentum 58H0 matrix of periodic cell vectors (reference configuration) 326H matrix of periodic cell vectors (deformed configuration) 326H matrix of reference primitive lattice vectors 120I identity tensor 34I identity matrix 25J Jacobian of the deformation gradient 46K macroscopic (continuum) kinetic energy 68K stiffness matrix or Hessian 312k wave vector and Fourier space variable 146






xxvi Notation�

L Lagrangian function 158L, Li linear momentum 54L, Li vectors defining a periodic simulation cell (reference) 325l, li vectors defining a periodic simulation cell (deformed) 563l, lij spatial gradient of the velocity field 50M total mass of a system of particles 380Mcell mass of a unit cell 567M ext total external moment acting on a system 58M finite element mass matrix 660m,mα mass, mass of atom α 54N α set of atoms forming the neighbor list to atom α 324N number of particles/atoms 54NB number of basis atoms 142NB number of basis atoms in the primitive unit cell 140Nlat number of lattice sites 563nd dimensionality of space 22Pdef deformation power 86Pext external power 85P , PiJ first Piola–Kirchhoff stress tensor 59P

α , Pαi reference momentum of atom α 452

p pressure (or hydrostatic stress) 57pα , pα

i momentum of atom α 54pα

rel center-of-mass momentum of atom α 380p, pi momentum of an electron or atom 164p, pi generalized momenta in statistical mechanics 382ΔQ heat transferred to a system during a process 68Q,Qαi orthogonal transformation matrix 31q, qi spatial heat flux vector 87q0 , q0I reference heat flux vector 88q, qi generalized positions in statistical mechanics 382q, qi generalized mean positions in restricted ensemble 556R rate of heat transfer 85R, RiJ finite rotation (polar decomposition) 47R[�λ ] reference position of the λth basis atom of lattice site � 141R, Ri center of mass of a system of particles 380Rα , Rα

i reference position of atom α 242r spatial strength of a distributed heat source 87r0 reference strength of a distributed heat source 88rα , rα

i spatial position of atom α 54rα , rα

i mean position of atom α in restricted ensemble 556rα

rel center-of-mass coordinates of atom α 380S electronic orbital overlap 225S entropy 73Sλ set of all atoms belonging to sublattice λ in a multilattice 564






xxvii Notation�

SI shape function for finite element node I 602SE hypersurface of constant energy E in phase space 382S, SIJ second Piola–Kirchhoff stress tensor 60s specific entropy 88s, sijkl spatial (or small strain) compliance tensor 103sλ , sλ

i shift vector of basis atom λ 560T instantaneous microscopic kinetic energy 158T vib microscopic (vibrational) kinetic energy 379T el instantaneous kinetic energy of the electrons 190T s instantaneous kinetic energy of the noninteracting electrons 192T temperature 65T , Ti nominal traction (stress vector) 60t, ti true traction (stress vector) 56t, ti true external traction (stress vector) 55U internal energy 68U potential energy of a quantum mechanical system 169U(ρ) embedding energy term in a pair functional 263U(z) unit step function (Heaviside function) 404U , UIJ right stretch tensor 47u spatial specific internal energy 85u0 reference specific internal energy 88u, ui displacement vector 48u, ui finite element approximation to the displacement field 602u column matrix of finite element nodal displacements 601V potential energy of a classical system of particles 158V int internal (interatomic) part of the potential energy 240Vext total external part of the potential energy 240Vext

fld , Vextcon potential energy due to external fields and external contact 240

V0 volume (reference configuration) 46V volume (deformed configuration) 46V α

0 volume of atom α (reference configuration) 457V α volume of atom α (deformed configuration) 457VR volume of region R in phase space 384V , Vij left stretch tensor 47v, vi velocity vector 50vpt , vpt

i pointwise (microscopic) velocity field 468vα , vα

i velocity of atom α 54vα

rel , vαrel,i velocity of atom α relative to center of mass 471

v column matrix of finite element nodal velocities 660ΔW work performed on a system during a process 67W virial of a system of particles 422W strain energy density function 96w(r), w(r) spatial averaging weighting function (general and spherical) 476w, wij spin tensor 50






xxviii Notation�

wα , wαi displacement of atom α relative to its mean position 556

X , XI position of a point in a continuum (reference configuration) 43X column matrix of finite element nodal coordinates 601x, xi position of a point in a continuum (deformed configuration) 43x, xi position of an electron 156Z atomic number 176Z partition function 426ZK ,ZV kinetic and potential parts of the partition function 427Zα position of basis atom α relative to the Bravais site 141z valence of an atom (or charge on an ion) 198

Fundamental constants

Avogadro’s constant (NA ) 6.0221× 1023 mol−1

Bohr radius (r0) 0.52918 ABoltzmann’s constant (kB ) 1.3807× 10−23 J/K

8.6173× 10−5 eV/Kcharge of an electron (e) 1.6022× 10−19 Ccharge-squared per Coulomb constant(e2/4πε0 ≡ e2) 14.4 eV · Amass of an electron (mel) 9.1094× 10−31 kgpermittivity of free space (ε0) 8.8542× 10−12 C2/(J ·m)Planck’s constant (h) 6.6261× 10−34 J · s

4.1357× 10−15 eV · sPlanck’s constant, reduced (� = h/2π) 1.0546× 10−34 J · s

6.5821× 10−16 eV · suniversal gas constant (Rg ) 8.3145 J/(K ·mol)

Unit conversion

1 fs = 10−15 s (femto)1 ps = 10−12 s (pico)1 ns = 10−9 s (nano)1 μs = 10−6 s (micro)1 ms = 10−3 s (milli)1 A = 10−10 m = 0.1 nm (angstrom)

1 eV = 1.60212× 10−19 J1 eV/A = 1.60212× 10−9 N = 1.60212 nN

1 eV/A2 = 16.0212 J/m2 = 16.0212 N/m

1 eV/A2.5 = 1.60212× 106 N/m1.5 = 1.60212 MPa ·

√m

1 eV/A3 = 1.60212× 1011 N/m2 = 160.212 GPa

1 amu = 1.66054× 10−27 kg = 1.03646× 10−4eV · ps2/A2






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