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MRS BULLETIN VOLUME 32 NOVEMBER 2007 www/mrs.org/bulletin 913

Coupled QuantumAtomistic andQuantumContinuumMechanics Methodsin Materials

ResearchAshwin Ramasubramaniam and

Emily A. Carter

AbstractThe 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 quantumatomistic approaches have

been developed and employed, for example, to gain a comprehensive understanding of

the energetics, kinetics, and dynamics of chemical processes involving surfaces andinterfaces of hard materials. More recently, it has become possible to directly couple

first-principles electronic structure techniques to continuum solid mechanics, either on

the fly with feedback between length scales or by information passing between length

scales. We discuss, with tutorial examples, the merging of quantum mechanics with

molecular dynamics and Monte Carlo simulations, as well as quantumcontinuum

coupled techniques. We illustrate the opportunities offered by incorporation of

information from quantum mechanics (reducing assumptions in higher length-scale

models) and outline the challenges associated with achieving full predictive capability

for the behavior of materials.

uum mechanics methodsconceptually,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 small

length scales necessitates the inclusion ofadditional physics from these smallerscalesthis could be done by introducingmicroscopic parameters in the continuummodel or, in cases where the couplingacross scales is strong, by concurrentmodeling at both scales. Whicheverthe point of view, coupled methods pro-vide a powerful approach for accuratemodeling ofrealistic 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 ofelectronic structure. Unfortunately, theircomputational cost is still too high to beviable for quantum mechanics/molecular

mechanics (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 lessreliable 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 the ease of modeling arbitrarygeometries with wide-ranging boundaryconditions. Defect kinetics (e.g., disloca-tion dynamics46) also can be included

phenomenologically within FEM to pro-duce mesoscopic-scale models.This list of methods is by no means

exhaustive. 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 throughad hoc assumptions.

Coupled methods are broadly classifi-able as either multiscale models seekingto couple two or more spatial and/or

Introduction

Over the past few decades, theory/computation has firmly established itselfas a partner to experiment in unravelingfundamental principles behind materials

behavior. The ever-improving perform-ance of computers and the development ofaccurate 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-


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914 MRS BULLETIN VOLUME 32 NOVEMBER 2007 www/mrs.org/bulletin

Coupled QuantumAtomistic and QuantumContinuum 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,79but rather to provide examples thataddress the twin issues of methodologyand capability.

Improved Continuum Modelingthrough Quantum Mechanics

MethodsContinuum theories have a well-established tradition within the engineeringcommunity. The limitations of continuumapproaches in materials modeling arealso well recognizednotable examples,among others, include the lack of a funda-mental failure criterion and the inability topredict structural and dynamic aspects ofdefects (dislocations, grain boundaries,etc.). The solution to these problems typi-cally involves the inclusion of additionalphysics, often in an ad hoc manner (e.g.,cohesive zone models for fracture,1012 regu-larization for dislocation cores,13,14 etc.).

Informed Continuum ModelsThe simplest remedy for the deficiencies

of continuum models is to constructinformed models (sequential multiscalemodels) where the material response iscalibrated from a more fundamentalmodel and propagated to higher lengthscales. Tadmor et al.s15 characterizationof the electromechanical response of apiezoelectric crystal (PbTiO3) is illustra-tive. They built a quantum mechanicalHamiltonianin essence, a constitutivemodelfor a finite element calculation. Inthis case, the Hamiltonian spanned the

neighborhood of phases of interest and thebarriers 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 et al.16 on crackpropagation in hydrogen-embrittledsteels (Figure 2). Fracture is modeledwithin continuum approaches through anempirical cohesive law that allows for

the smooth decohesion of atomic planes atthe crack tip, thereby enabling crackpropagation. Hydrogen preferentiallysegregates to crack faces and the incipi-ent fracture zone. The first-principlesdecohesion data17,18 and a renormalizationprocedure19,20 produced a realistichydrogen-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 a priori assumptionof important physics that invariablygoes into the upscaling procedure. In

the effective Hamiltonian approach,this occurs 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.

DirectAb InitioContinuumCoupling

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 original 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 initiomolecular 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.


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this MRS Bulletin issue by Tadmor andMiller for more details.

Another approach developed byKaxiras, Abraham, and co-workers,22,23

dubbed macroscopic atomistic ab initiodynamics (MAAD), couples FEM withan atomistic region, which in turn isdescribed by a combination of empiricalpotentials and quantum mechanics atthe level of tight-binding. Differentregions in the simulation interact with

each 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 et al.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 ofempirical potentials, especially in far-

from-equilibrium situations. These short-comings motivate direct coupling ofquantum 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 et al.27

and Hayes et al.28,29 extended the quasi-continuum method to study nano-indentation of aluminum, and aluminummagnesium alloys (Al3Mg), 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 density

functional theory (OFDFT),30

which scaleslinearly with size, as opposed to traditionalKohnSham density functional theory(KSDFT), which scales cubically. For met-als, OFDFT is 35 orders of magnitudefaster than KSDFT. Their findings on theonset of dislocation emissionreported interms of the nucleation site, indenter dis-placement, and loadwere 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



H2

Diffusion

Incorporation

Embrittlement

Intrinsic cohesive law(zero coverage)

Fully embrittled

cohesive law

Normalized OpeningDisplacement

NormalizedStress

a

b c

d

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 tractionseparation 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 et al.16)


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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 to other materials (through improvedpseudopotentials31,32 and kinetic energyfunctionals33) remains a matter of currentresearch.

Finally, we mention briefly the work ofWoodward and co-workers,3436 whohave employed Greens functions inconjunction with quantum mechanicsmethods to simulate isolated dislocationsin bulk metals and alloys. While there isno explicit coupling of continuum andquantum mechanics regions, the Greensfunctions in effect mimic the continuum

by providing appropriate boundary con-

ditions to the quantum mechanics region.This approach eliminates the overhead ofa 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 systemsizes amenable to quantum mechanics

treatment. A more serious restrictionarises in dynamics simulations (e.g., theCarParrinello37 or BornOppenheimer3840

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, ab initio 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 surfacechemistry42 and applications to mole-cules.43 Here, we focus only on examplesthat seek to extend the length and timescales ofab initio methods.

Ab InitioDerived MolecularDynamics

We first consider the approach of ab-initioderived potentials. This is the atom-istic analogue of an informed continuum

model. The basic idea is to produce ananalytical expression for the potentialenergy surface, whose parameters are fitto data obtained from ab initio 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 a

lack 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 Earths mantle or radioac-tive waste); ab initio calculations providethe only viable route for designing poten-tials under these circumstances. The mainchallenge lies in designing a potentialthat is accurate, transferable, and, if possi-

ble, computationally inexpensivefittinga good potential is as much an art as it

is a science. Thereafter, the well-honedmachinery of molecular dynamics can bebrought to bear upon the problem.

An early example ofab initio potentialsused with molecular dynamics examinedfluorine reactions with silicon surfaces.44,45

The parameters of the classical StillingerWeber potential were refit for ab initio dataon Si-F interactions. The new potentialaccurately captured the initial stages ofetching silicon. Additionally, new phe-nomena were predicted, such as local-heating-induced disorder at the siliconsurface caused by the reactions highexothermicity, which was subsequently



a b[111]

[1 1 2]

[111]

[1 1 2]

Figure 3. Nanoindentation along the [111] direction of a 2 m 1 m 1 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 m below theindenter, whereas EAM predicts nucleation at 35 nm indentation and at 0.04 m below the indenter. The magenta and black arrows in the

insets indicate the slip plane normal and Burgers vectors, respectivelyboth EAM and OFDFT predict the formation of [011] dislocations,but the slip planes are (111) for the former and (1

11) for the latter.27


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verified experimentally. The key point ofthis example is that even refitting anempirical potential to accurate ab initiodata can lead to a much improved poten-tial energy surface and enhanced model-ing capability.

Other instances exist of increasinglysophisticated interatomic potentialsderived from ab initio methods. For exam-ple, Madden and co-workers46 developedtechniques for ionic materials. Their poten-tials account for many-body effects viamultipole expansions and coordination-dependent parameters such as ionic radii.Potential fitting is achieved via moleculardynamics and quantum mechanics calcu-lations across a range of states and fittingparameters to minimize discrepancies

between the two methods, thus ensuringtransferability. The availability of quantummechanical information for various states

allows one to increase or decrease thecomplexity of a potential and therebyascertain the importance of differentphysical effects. For covalently bondedmaterials, Goddard and co-workers47,48

developed reactive potentials (dubbedReaxFF for reactive force field) fit todata sets derived from ab initio calculationson clusters and condensed phases. Thekey ingredient is the notion of a bond-order parameter updated continuouslyfrom instantaneous interatomic posi-tions, which allows for bond creation anddissociation. The ability to handle bondformation and breaking, albeit more

approximately than true ab initio methods,greatly enhances an interatomic potentialscapability to simulate surface reactions,interfaces in materials, and proteins insolution, among others. An application todynamic fracture is found in Buehleret al.s work.49

Finally, in the context of potential fit-ting, Csnyi et al.50,51 recently presented alearn-on-the-fly strategy. The idea is touse ab initio methods as a black boxfrom which information can be gleanedduring the course of a simulation. Thealgorithm starts by using a simple empiri-

cal potential to evolve a short moleculardynamics trajectory (predictor phase).The validity of the potential is then tested,particularly in defective regions of thecrystal, with respect to a more accuratemethod (tight-binding or DFT), and theparameters are refitted locally to minimizeerrors in the forces. The molecular dynam-ics trajectory is then rerun (correctorphase) with an interpolation of thepotentials between the initial and new val-ues. The approachs efficacy has beendemonstrated thus far for defects in sili-con; localized covalent bonds in siliconallow the use of small cluster models in

the ab initio calculations. The extension tometals, which has not yet been done, ismore delicate, since the delocalized elec-tron distribution would necessitate a moreelaborate embedding scheme.52 Also,varying the interatomic potential amountsto using a time-varying Hamiltonian thatdoes not conserve energy. Hence, simula-tions must be run in the canonical (con-stant NVT, where N is the number ofparticles, V is their volume, and T is theirtemperature) ensemble with a thermostat.Nevertheless, the methods advantage liesin eliminating the need for patchingthrough handshake regions or, alterna-tively, the need for elaborate interatomicpotentials that will be less accurate thanthe parent ab initio method. On the otherhand, the on-the-fly potential, as with allanalytical potentials, is forever limited bychoice of functional form, which will con-

strain behavior that can be observed.

Quantum-Based Kinetic MonteCarlo Methods

Kinetic Monte Carlo (KMC) meth-ods53,54 are a stochastic way of extendingtime scales in simulations. When com-

bined with a spatial coarse-graining pro-cedure, this additionally leads to amethod of bridging length scales.Examples of applications of KMC to mate-rials modeling aboundone that encom-passes several modeling techniques is therecent work of Reuter and co-workers,55,56

who used KMC to study oxidation of CO

to CO2 on RuO2. The relevant processes,such as molecular adsorption and recom-

binative desorption, are thermally acti-vated rare events, not easily accessible todirect simulation. DFT was used in combi-nation with transition-state theory toobtain process rates. Surface binding siteswere modeled as a simplified lattice, sincethese were clearly identifiable, and KMCwas performed. A similar approach,employing configuration interactioninstead of DFT to obtain rates, was usedearlier57 for studying hydrogen desorp-tion from silicon. The key outcome of the

DFT KMC work was the ability to predictphase diagrams and pressure-dependentturnover frequencies (a measure of cat-alytic performance), which agreed withexperiments over wide ranges of temper-ature and pressure.

The main drawback of KMC is the lackof true time evolution, in contrast to molec-ular dynamics. Accordingly, interestabounds in developing accelerated meth-ods that extend molecular dynamics tolong timesthe review by Voter et al.58

provides seminal details. Also, restrictingmyriad processes to predetermined eventsis fraught with danger, as important non-

intuitive processes may be overlooked.Feibelman59 provided the cautionary tale:the commonly assumed surface diffusionmechanism for metals (direct hops) isincorrect; instead, diffusion occurs by con-certed motion of two atoms. The mostreliable way to overcome this serious defi-ciency is to employ on-the-fly KMC meth-ods that only utilize information aboutthe current state and then exhaustivelydetermine connected transition states.60,61

The high computational cost of on-the-flyKMC, however, makes it impractical foruse with ab initio methods. The use ofab-initioderived potentials, possibly inconjunction with a self-learning proce-dure,62 may render such methods veryeffective.

Concluding RemarksIn this short review, we attempted to give

a flavor of quantum-based multiscale/multiphysics methods. It is by no meanscomprehensive; most glaringly, we omitteddiscussion of patched quantum mechanicsand molecular mechanics (QM/MM)methods.6365 We also limited our attentionto hard materials, which is a small subset ofmaterials research.

In conclusion, much work remainsto extend quantum-based multiscale/multiphysics methods to full realismand accuracy. We believe the solutions lienot just in greater computing power(always at a premium), but primarily in

theoretical and algorithmic work thatspans disciplines.

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