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Coupled Quantum–
Atomistic and Quantum–Continuum Mechanics Methods in Materials Research
Ashwin Ramasubramaniam and Emily A. Carter
Abstract
The interface of quantum mechanics methods with classical atomistic simulation techniques, such as molecular dynamics and Monte Carlo, continues to be an area of considerable promise and interest. Such coupled quantum–atomistic approaches have been developed and employed, for example, to gain a comprehensive understanding of the energetics, kinetics, and

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MRS BULLETIN ã VOLUME 32 ã NOVEMBER 2007 ãwww/mrs.org/bulletin913
C
oupled Quantum–Atomistic andQuantum–ContinuumMechanics Methodsin MaterialsResearch
Ashwin Ramasubramaniam andEmily A. Carter
Abstract
The interface of quantum mechanics methods with classical atomistic simulationtechniques, such as molecular dynamics and Monte Carlo, continues to be an area ofconsiderable promise and interest. Such coupled quantum–atomistic approaches havebeen developed and employed, for example, to gain a comprehensive understanding ofthe energetics, kinetics, and dynamics of chemical processes involving surfaces andinterfaces of hard materials. More recently, it has become possible to directly couplefirst-principles electronic structure techniques to continuum solid mechanics, either onthe fly with feedback between length scales or by information passing between lengthscales. We discuss, with tutorial examples, the merging of quantum mechanics withmolecular dynamics and Monte Carlo simulations, as well as quantum–continuumcoupled techniques. We illustrate the opportunities offered by incorporation ofinformation from quantum mechanics (reducing assumptions in higher length-scalemodels) and outline the challenges associated with achieving full predictive capabilityfor the behavior of materials.
uum mechanics methods—conceptually,the molecular or continuum mechanicssimulation region provides correct bound-ary conditions for the quantum mechanicsregion. While this coupling is motivated inpart by necessity, the realization that it isunnecessary to use quantum mechanics totreat atoms that behave similarly providesfurther impetus for developing coupledmethods.This viewpoint presented may bethought of as a bottom-up approach tocoupled models. One may also adopt thetop-down point of view wherein the breakdown of continuum models at smalllength scales necessitates the inclusion of additional physics from these smallerscales—this could be done by introducingmicroscopic parameters in the continuummodel or, in cases where the couplingacross scales is strong, by concurrent modeling at both scales. Whicheverthepoint of view, coupled methods pro-vide a powerful approach for
accurate
modeling of
realistic
situations with finiteresources.The building blocks of coupled schemesexist in well-honed numerical techniquesat each length scale (Figure 1). At thequantum mechanics level, quantumchemistry approaches (e.g., configurationinteraction or quantum Monte Carlo) provide the most accurate description of electronic structure. Unfortunately, theircomputational cost is still too high to beviable for quantum mechanics/molecularmechanics (QM/MM) coupling in thematerials context. A more practicalapproach is density functional theory(DFT),
1,2
which reduces the task of deter-mining the many-electron wave functionto a less-demanding problem of optimiz-ing the electronic density. DFT is themethod of choice for large systems, sinceit usually provides sufficient accuracy at alower computational cost per atom.Traditionally, empirical potentials areused for molecular mechanics modeling.These potentials are typically fit to equi-librium properties and hence are less reliable far from equilibrium.
Ab initio
data are used increasingly to improveoverall reliability, as will be discussedlater. Nevertheless, processes such as bond formation, bond breaking, andcharge transfer are best handled withquantum mechanics, leaving molecularmechanics to handle near-equilibrium sit-uations. Finally, at the continuum level,the finite element method (FEM)
3
contin-ues to be the technique of choice, mostlydue to theease of modeling arbitrarygeometrieswith wide-ranging boundaryconditions. Defect kinetics (e.g., disloca-tion dynamics
4–6
) also can be includedphenomenologically within FEM to pro-duce mesoscopic-scale models.This list of methods is by no meansexhaustive. Computational materials sci-ence is an intersection of many disciplines,each of which has developed specializedtools at differing scales of interest. Thechallenge for coupling schemes lies inintegrating these tools while making con-trollable approximations and withoutintroducing spurious physics through
adhoc
assumptions.Coupled methods are broadly classifi-able as either multiscale models seekingtocouple two or more spatial and/or
Introduction
Over the past few decades, theory/computation has firmly established itself as a partner to experiment in unravelingfundamental principles behind materials behavior. The ever-improving perform-ance of computers and the development of accurate and efficient algorithms progres-sively bring predictive quantum mechan-ics models of materials within reach.Nevertheless, full quantum mechanicstreatments remain all but intractable atpresent for more than a few hundredatoms. Several schemes have been devisedto couple quantum mechanics with less-expensive molecular mechanics or contin-
914MRS BULLETIN ã VOLUME 32 ã NOVEMBER 2007 ãwww/mrs.org/bulletin
Coupled Quantum–Atomistic and Quantum–Continuum Mechanics Methods
temporal scales; or multiphysics modelswhere different physical descriptions, notnecessarily at different scales, are coupled.In this article, we will focus attention onissues related to quantum-based multiscalemethods. Our aim is not to provide a com-prehensive review, which may be found inseveral excellent articles and proceed-ings,
7–9
but rather to provide examples thataddress the twin issues of methodologyand capability.
Improved Continuum Modelingthrough Quantum MechanicsMethods
Continuum theories have a well- established tradition within the engineeringcommunity. The limitations of continuumapproaches in materials modeling arealsowell recognized—notable examples,among others, include the lack of a funda-mental failure criterion and the inability topredict structural and dynamic aspects of defects (dislocations, grain boundaries,etc.). The solution to these problems typi-cally involves the inclusion of additionalphysics, often in an
adhoc
manner (e.g.,cohesive zone models for fracture,
10–12
regu-larization for dislocation cores,
13,14
etc.).
Informed Continuum Models
The simplest remedy for the deficienciesof continuum models is to construct“informed” models (sequential multiscalemodels) where the material response is calibrated from a more fundamentalmodeland propagated to higher lengthscales. Tadmor etal.’s
15
characterizationofthe electromechanical response of apiezoelectric crystal (PbTiO
3
) is illustra-tive.They built a quantum mechanicalHamiltonian—in essence, a constitutivemodel—for a finite element calculation. Inthis case, the Hamiltonian spanned theneighborhood of phases of interest and the barriers between these phases, which theydeemed an adequate description of theenergy landscape for their purposes. Themodel yielded qualitatively correct resultsfor the highly nonlinear, hysteretic behaviorof these materials and provided insight intothe underlying microscopic mechanisms.Another example of a continuummodel informed by quantum mechanics isthe work of Serebrinsky etal.
16
on crackpropagation in hydrogen-embrittledsteels (Figure 2). Fracture is modeledwithin continuum approaches through anempirical “cohesive law” that allows forthe smooth decohesion of atomic planes atthe crack tip, thereby enabling crack propagation. Hydrogen preferentiallysegregates to crack faces and the incipi-entfracture zone. The first-principlesdecohesion data
17,18
and a renormalizationprocedure
19,20
produced a realistic hydrogen-dependent cohesive law. Theirfindings on crack initiation times anddependence on applied stress intensityand yield strength were in agreement withexperimental trends, as were their obser-vations on intermittent crack growth. Themain outcome was to demonstrate thathydrogen-induced decohesion could be adominant fracture mechanism of steels inaqueous environments.The primary criticism of informed con-tinuum models is the
apriori
assumptionof “important” physics that invariablygoes into the upscaling procedure. Intheeffective Hamiltonian approach,thisoccurs by restricting the functionaldependence to certain parameters and to aparticular region of the overall energylandscape. In the hydrogen embrittlementscenario, competing mechanisms (e.g.,interaction of dislocations with crack tips,hydrogen trapping at dislocations) otherthan coverage-dependent decohesion areeither ignored or treated approximately at best. Nevertheless, such models do have arole to play in imparting qualitative firstinsights into microscopic mechanismsunderlying macroscopic observations. Weare working on ways to incorporate mech-anisms other than coverage-dependentdecohesion to provide more sophisticatedmodels of embrittlement.
Direct
Ab Initio
–Continuum Coupling
A more rigorous and conceptuallyappealing procedure to remedy the limita-tions of continuum models is to explicitlymodel the material at various requisitelength scales with feedback between thesescales. Relevant pioneering work by Ortiz,Phillips, and co-workers (see Reference 21and references therein) focused on cou-pling FEM and atomistic calculations.Their technique, the quasi-continuummethod, has found several applicationsand has undergone much refinement overthe last decade or so. The srcinal quasi-continuum method is based on con-strained minimization of an atomisticenergy, obtained from an empirical poten-tial, with the ability to adapt the computa-tional mesh to the deformation field. Thelatter feature enables a seamless transitionfrom a fully atomically resolved materialto a discretized continuum, as long as theempirical potential is short-ranged. Werefer the reader to the companion article in
Figure 1. Schematic illustration of the coupling strategies discussed in this article.Representative length and time scales for each modeling strategy are indicated along theaxes. Spatial modeling strategies are indicated in green boxes, time-stepping methods inpurple boxes, direct coupling in gray ellipses, and sequential coupling (information-passing)in blue ellipses. Terminal points for time-stepping methods indicate the maximum timescales that are accessible in practice. Abbreviations used in this figure: AIMD,
ab initio
molecular dynamics; DFT, density functional theory; EAM, embedded atom method; KMC,kinetic Monte Carlo; MD, molecular dynamics; MM, molecular mechanics; PES, potentialenergy surface; QC, quasi-continuum; QM, quantum mechanics.
this
MRS Bulletin
issue by Tadmor andMiller for more details.Another approach developed byKaxiras, Abraham, and co-workers,
22,23
dubbed “macroscopic atomistic
abinitio
dynamics” (MAAD), couples FEM withan atomistic region, which in turn isdescribed by a combination of empiricalpotentials and quantum mechanics atthelevel of tight-binding. Differentregions in the simulation interact witheach other through interfaces or “hand-shake” regions. The construction of thesehandshake regions is not unique andrequires special care to avoid discontinu-ities and wave reflections. MAAD wasused successfully to simulate brittle crackpropagation in silicon. The tight-bindingregion adaptively tracked the crack tipwhere increased accuracy, beyond thescope of empirical potentials, is requiredto describe bond breaking. At the timeMAAD was presented, adaptive refine-ment of the continuum region or defectpropagation between the atomistic andcontinuum region was not possible. Also,the use of tight-binding at the crack tip(mostly appropriate for localized elec-tronic densities) makes the extension tometallic systems problematic; recent work by Lu etal.
24
attempts to rectify this situa-tion by replacing tight-binding with DFT(albeit within a quasi-continuum frame-work, rather than MAAD). On the otherhand, the quasi-continuum method suf-fers from the lack of transferability of empirical potentials, especially in far-from-equilibrium situations. These short-comings motivate direct coupling of quantum mechanics to FEM.Nanoindentation tests are routinelyused to evaluate the response of thin filmsand the onset of plastic flow in small vol-umes.
25,26
Simulations can facilitate adeeper interpretation of these tests throughanalysis of individual events occurringduring indentation. Computational chal-lenges involve modeling realistic indenterand sample sizes, large deformations of thematerial, nucleation and interaction of dis-locations below the indenter, and thermaleffects, among others. Recently, Fago etal.
27
and Hayes etal.
28,29
extended the quasi-continuum method to study nano-indentation of aluminum, and aluminummagnesium alloys (Al
3
Mg), wherein theatomistic regions, normally described inthe quasi-continuum method by empiricalembedded-atom method (EAM) poten-tials, were replaced by quantum mechanicsregions. To render the quantum mechani-cal calculations computationally afford-able, they used orbital-free densityfunctional theory (OFDFT),
30
which scaleslinearly with size, as opposed to traditionalKohn–Sham density functional theory(KSDFT), which scales cubically. For met-als, OFDFT is 3–5 orders of magnitudefaster than KSDFT. Their findings on theonset of dislocation emission—reported interms of the nucleation site, indenter dis-placement, and load—were at significantvariance with quasi-continuum calcula-tions using EAM potentials (Figure 3). Thisis not surprising, since empirical potentialssuch as EAM potentials that are fitted toequilibrium properties cannot be expectedto correctly capture behavior at large
Coupled Quantum–Atomistic and Quantum–Continuum Mechanics Methods
MRS BULLETIN ã VOLUME 32 ã NOVEMBER 2007 ãwww/mrs.org/bulletin915
H
2
DiffusionIncorporationEmbrittlementIntrinsic cohesive law(zero coverage)Fully embrittledcohesive lawNormalized OpeningDisplacement
N o r m a l i z e d S t r e s s
abcd
Figure 2. Schematic illustration of hydrogen embrittlement in metals. (a) Some individual processes involved in embrittlement: a preexistingcrack is attacked by hydrogen from an external source, e.g., hydrogen molecules dissociate and adsorb on the crack flanks; some of theadsorbed atoms are absorbed in the bulk and can diffuse under stress gradients; accumulation of hydrogen near the crack tip lowers cohesionamong host atoms near the crack tip and eventually leads to failure. (b) Finite element mesh with high resolution at crack tip. (c) Cohesiveelements surrounding the crack tip. (d) The coverage-dependent traction–separation law that governs the behavior of the cohesive elements.The intrinsic cohesive law (no hydrogen) and the fully embrittled cohesive law (complete hydrogen coverage) are indicated; intermediate levelsof coverage fall within the shaded region. (After Serebrinsky etal.
16
)
deformations. However, the DFT-basedquasi-continuum method is not yet devel-oped to the point where the EAM-basedapproach is obsolete. Chief among thepresent limitations is the restriction to peri-odic OFDFT calculations. Consequently,the finite element mesh cannot be resolveddown to individual atoms, which wouldthen need to be handled as nonperiodicclusters. Also, at present, only metals arewell described by OFDFT, and transferabil-ity toother materials (through improvedpseudopotentials
31,32
and kinetic energyfunctionals
33
) remains a matter of currentresearch.Finally, we mention briefly the work of Woodward and co-workers,
34–36
whohaveemployed Green’s functions in conjunction with quantum mechanicsmethods to simulate isolated dislocationsin bulk metals and alloys. While there isno explicit coupling of continuum andquantum mechanics regions, the Green’sfunctions in effect mimic the continuum by providing appropriate boundary con-ditions tothe quantum mechanics region.This approach eliminates the overhead of a finite element calculation. Reference 36provides an interesting application of thismethod to solid-solution softening in bccmetals.
Coupled Quantum Mechanics/ Molecular Mechanics Approaches:Extending Length and Time Scales
Quantum mechanics provides highaccuracy in materials models, albeit at ahigh computational price per atom. Thisnaturally leads to a limit on systemsizesamenable to quantum mechanicstreatment. A more serious restrictionarisesindynamics simulations (e.g., theCar–Parrinello
37
or Born–Oppenheimer
38–40
methods) from the inherently small timesteps required to stably integrate the equa-tions of motion, not to mention the expenseof frequent quantum mechanics force eval-uations. Hence, from a practical perspec-tive,
abinitio
molecular dynamics (AIMD)can be used only for studying short-time-scale phenomena (e.g., picoseconds). Also,it is still beyond the reach of AIMD togather reliable statistics, which requirehundreds to thousands of trajectories.Accordingly, AIMD trajectories should betreated as anecdotal rather than definitive.Nevertheless, AIMD, with its accuracy andpredictive capability, has come to be animportant addition to the arsenal. In partic-ular, DFT-based AIMD enjoys greaterapplicability, due to its lower cost, and has been applied to a wide spectrum of prob-lems such as surface and interface chem-istry, catalysis, chemical reactions, reactionsin solution, solvation of proteins, and proton transfer in biomolecules. A recentcomprehensive review on the theory andapplications of AIMD may be found inReference 41. The interested reader mightalso wish to peruse reviews on surfacechemistry
42
and applications to mole-cules.
43
Here, we focusonly on examplesthat seek to extendthe length and timescales of
abinitio
methods.
Ab Initio
–Derived Molecular Dynamics
We first consider the approach of
ab-initio
–derived potentials. This is the atom-istic analogue of an informed continuummodel. The basic idea is to produce ananalytical expression for the potentialenergy surface, whose parameters are fitto data obtained from
abinitio
calcula-tions, in distinct contrast to the empiricalapproach of using experimental materialproperties. Experimental observables rep-resent a thermal sampling of a limited setof configurations within the experimen-tally accessible phase space. Hence,empirical potentials typically suffer from alack of transferability; that is, they canonly be relied upon to accurately repro-duce regions of the potential energy sur-face to which they were fit. Situationsoften arise where experimental data arenot available or cannot be produced (e.g.,for oxides in the Earth’s mantle or radioac-tive waste);
abinitio
calculations providethe only viable route for designing poten-tials under these circumstances. The mainchallenge lies in designing a potentialthatis accurate, transferable, and, if possi- ble, computationally inexpensive—fittinga good potential is as much an art as itisascience. Thereafter, the well-honedmachinery of molecular dynamics can be brought to bear upon the problem.An early example of
abinitio
potentialsused with molecular dynamics examinedfluorine reactions with silicon surfaces.
44,45
The parameters of the classical Stillinger–Weber potential were refit for
abinitio
dataon Si-F interactions. The new potentialaccurately captured the initial stages of etching silicon. Additionally, new phe-nomena were predicted, such as local-heating-induced disorder at the siliconsurface caused by the reaction’s highexothermicity, which was subsequently
Coupled Quantum–Atomistic and Quantum–Continuum Mechanics Methods
916MRS BULLETIN ã VOLUME 32 ã NOVEMBER 2007 ãwww/mrs.org/bulletin
ab
[111][1 1 2][111][1 1 2]
Figure 3. Nanoindentation along the [111] direction of a 2
µ
m
×
1
µ
m
×
1
µ
m Al sample using both (a) OFDFT- and (b) EAM-based quasi-continuum methods. The OFDFT calculation predicts nucleation of a dislocation at 50 nm indentation at a distance of 0.15
µ
m below theindenter, whereas EAM predicts nucleation at 35 nm indentation and at 0.04
µ
m below the indenter. The magenta and black arrows in theinsets indicate the slip plane normal and Burgers vectors, respectively—both EAM and OFDFT predict the formation of [01–1] dislocations,butthe slipplanes are (111) for the former and (1–11) for the latter.
27

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