A POSTERIORI ERROR ESTIMATE AND ADAPTIVITY FOR QMMM MODELS OF CRYSTALLINE DEFECTS YANGSHUAI WANG JAMES R. KERMODE CHRISTOPH ORTNER AND LEI ZHANG

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A POSTERIORI ERROR ESTIMATE AND ADAPTIVITY

FOR QM/MM MODELS OF CRYSTALLINE DEFECTS

YANGSHUAI WANG, JAMES R. KERMODE, CHRISTOPH ORTNER, AND LEI ZHANG

Abstract. Hybrid quantum/molecular mechanics (QM/MM) models play a pivotal role in molec-

ular simulations. These models provide a balance between accuracy, surpassing pure MM models,

and computational eﬃciency, oﬀering advantages over pure QM models. Adaptive approaches have

been developed to further improve this balance by allowing on-the-ﬂy selection of the QM and MM

subsystems as necessary. We propose a novel and robust adaptive QM/MM method for practical

material defect simulations. To ensure mathematical consistency with the QM reference model, we

employ machine-learning interatomic potentials (MLIPs) as the MM models [13, 31]. Our adap-

tive QM/MM method utilizes a residual-based error estimator that provides both upper and lower

bounds for the approximation error, thus indicating its reliability and eﬃciency. Furthermore,

we introduce a novel adaptive algorithm capable of anisotropically updating the QM/MM parti-

tions. This update is based on the proposed residual-based error estimator and involves solving a

free interface motion problem, which is eﬃciently achieved using the fast marching method. We

demonstrate the robustness of our approach via numerical tests on a wide range of crystalline

defects.

1. Introduction

Quantum mechanics and molecular mechanics (QM/MM) coupling methods have emerged as

indispensable tools for simulating large molecular systems in both materials science and biology [4,

16, 28, 35, 39, 56]. The essence of this methodology lies in the partition of the computational

domain into QM and MM regions, where the region of primary interest is described by a QM

model embedded in an ambient environment represented by a MM model. The goal of QM/MM

coupling methods is to achieve (near-)QM accuracy at (near-)MM computational cost, to make

quantitative large-scale atomistic simulations viable.

An important question in the ﬁeld of QM/MM coupling methods concerns the optimal assign-

ment of atoms to either QM or MM subsystems in order to balance accuracy and computational

cost. To address this issue, adaptive QM/MM methods have been developed by providing an

on-the-ﬂy partition of QM/MM subsystems based on error estimators during simulations.

Adaptive QM/MM methods have been proposed for various applications such as the analysis

of molecular fragments in macromolecules, monitoring of molecules entering/leaving binding sites

and tracking proton transfer through the Grotthuss mechanism (as discussed in [20] and references

therein). However, the majority of adaptive QM/MM methods developed thus far are for solute-

solvent systems and rely on heuristic criteria. Further information can be found in [5, 30, 33, 34, 50,

52, 55]. In the context of materials featuring defects, adaptive computations are often guided by

the distance to these defects [16, 35]. Similar concepts can also be observed in the quasi-continuum

methods for density functional theory [29, 44].

Adaptive QM/MM methods normally rely on empirical error estimators. By contrast, Chen et

al. [7] ﬁrst introduced mathematically rigorous a posteriori error estimators for the QM/MM model

residual inspired by classical adaptive ﬁnite element methods [18, 49]. Their approach employed

a weighted ℓ2-norm on the residual forces, providing an upper bound for the approximation error.

Date: February 20, 2024.

arXiv:2210.04856v2 [physics.comp-ph] 17 Feb 2024

2 YANGSHUAI WANG, JAMES R. KERMODE, CHRISTOPH ORTNER, AND LEI ZHANG

However, it fell short in providing a lower bound, which is essential for ensuring eﬃciency of

adaptive schemes. To overcome this limitation, Wang et al. [51] proposed a reliable and eﬃcient

a posteriori error estimator by connecting the natural dual norm of the residual with solving

an auxiliary Poisson equation. They developed an inner-outer adaptive strategy with an outer

adaptive algorithm for selecting QM and MM regions and an inner algorithm for computing the

estimators with the desired accuracy. However, the inner algorithm necessitated a ﬁnite element

mesh, which introduced additional errors requiring careful handling, not to mention additional

algorithmic complexity. More importantly, this work was primarily centered on energy-mixing

schemes for simple point defects and the adaptive algorithm only adjusts the radius of QM and

MM sub-regions, limiting the ability to capture signiﬁcant anisotropy in the defect core, elastic

ﬁeld, or defect nucleation observed in practical material simulations.

The purpose of the present work is to develop a more practical adaptive QM/MM method

for material defect simulations, while maintaining the rigourous approach of [7, 51]. To ensure

consistency of the QM/MM scheme and improve computational eﬃciency, we employ state-of-art

machine-learning interatomic potentials (MLIPs) as the MM models [13, 31]. Next, we propose a

practical and ﬂexible approach to obtain the error estimator, essentially replacing the PDE operator

from [51] with a generalization of the graph-Laplacian [42]. Algorithmically, this approach ﬁts much

better into the setting of atomistic modeling. A practical error estimator is further developed by

(i) truncating to a ﬁnite computational domain and (ii) facilitating the QM force constant to give

a linear approximation of the residual force in the MM region. Moreover, to evolve the QM/MM

partitions anisotropically rather than only adjusting the radius, a free interface motion problem

(i.e., Eikonal equation [57]) is formulated and solved using the fast marching method [14, 57], where

the practical error estimator is regarded as the extending speed. We develop a novel strategy

to assign atoms to QM or MM subsystems based on the solution of the corresponding Eikonal

equation.

We test our algorithm by performing adaptive computations for three common defect types:

edge dislocations, in-plane cracks, and di-interstitials. Our ﬁndings reveal that the practical error

estimator we introduce attains convergence rates comparable to those of the approximation error,

oﬀering substantial computational cost reductions when employing a realistic electronic structure

model. The adaptive algorithm showcases robustness by eliminating the need for user a priori

input, thereby aligning with our objective of achieving a fully adaptive QM/MM scheme. The

analysis and adaptive algorithm presented in this paper demonstrates a considerable degree of

independence from the underlying approximation scheme, thereby rendering the proposed frame-

work widely applicable to various coarse-graining or multiscale methods. As a proof of concept,

we will focus solely on geometry equilibration problems (statics).

Outline. This paper is organized as follows: Section 2 introduces the variational formulation

for defect equilibration and the QM/MM coupling methods we use. Section 3 outlines the con-

struction of our novel a posteriori error estimator, which provides upper and lower bounds for the

approximation error, along with practical approximations to enhance its implementation. Section 4

presents our adaptive QM/MM algorithm, incorporating a free interface problem to dynamically

update QM/MM partitions using the practical error estimator. We showcase our ﬁndings with

numerical examples, validating the eﬃcacy of our adaptive algorithm. Section 5 concludes our key

ﬁndings and outlines future research directions. Appendices provide supplementary information

for interested readers.

Notation. We use the symbol ⟨·,·⟩ to denote the duality pairing between a Banach space and

its dual space. The symbol |·|normally denotes the Euclidean or Frobenius norm, while ∥·∥

ADAPTIVITY FOR QM/MM MODELS 3

denotes an operator norm. For the sake of brevity of notation, we will denote A\{a}by A\a, and

{b−a|b∈A}by A−a. For E∈C2(X), the ﬁrst and second variations are denoted by ⟨δE(u), v⟩

and ⟨δ2E(u)v, w⟩for u, v, w ∈X. For a ﬁnite set A, we will use #Ato denote the cardinality of

A. The closed ball with radius rand center xis denoted by Br(x), or Brif the center is the origin.

We use the standard deﬁnitions and notations Lp,Wk,p,Hkfor Lebesgue and Sobolev spaces. In

addition we deﬁne the homogeneous Sobolev spaces ˙

Hk(Ω) := f∈Hk

loc(Ω) |∇kf∈L2(Ω).

2. QM/MM Coupling for Crystalline Defects

2.1. Variational model for crystalline defects. A rigorous framework for modelling the geo-

metric equilibration of crystalline defects has been developed in [9, 21]. These works formulate the

equilibration of a single crystalline defect as a variational problem in a discrete energy space and

establish qualitatively sharp far-ﬁeld decay estimates for the corresponding equilibrium conﬁgura-

tion. This analytical foundation will serve as the basis for our a posteriori error estimates and the

corresponding adaptive algorithm.

2.1.1. Displacement space. Given d∈ {2,3}, a homogeneous crystal reference conﬁguration is

given by the Bravais lattice Λh=AZd, for some non-singular matrix A∈Rd×d. A reference lattice

with a single defect is denoted by Λ ⊂Rd. For the sake of simplicity we admit only single-species

Bravais lattices. There are no conceptual obstacles to generalising this work to multi-lattices [40],

however, the notational and technical details become more involved.

A deformed conﬁguration of the inﬁnite lattice Λ is a map y: Λ →Rd. We can decompose the

conﬁguration yinto

y(ℓ) = ℓ+u0(ℓ) + u(ℓ) = y0(ℓ) + u(ℓ),(2.1)

where u0: Λ →Rdis a far-ﬁeld predictor solving a continuum linearised elasticity (CLE) equa-

tion [21] enforcing the presence of defect and u: Λ →Rdis a core corrector. For point defects

we simply take u0(ℓ)=0∀ℓ∈Λ. The derivation of u0for straight dislocations and cracks are

reviewed in the Appendix B.

The set of admissible atomic conﬁgurations is

Adm0(Λ) := [

m>0

Admm(Λ) with

Admm(Λ) := y: Λ →Rd,|y(ℓ)−y(m)|>m|ℓ−m| ∀ ℓ, m ∈Λ,

where the parameter m>0 prevents the accumulation of atoms.

For ℓ∈Λ and ρ∈Λ−ℓ, we deﬁne the ﬁnite diﬀerence Dρu(ℓ) := u(ℓ+ρ)−u(ℓ). For a

subset R ⊂ Λ−ℓ, we deﬁne DRu(ℓ) := (Dρu(ℓ))ρ∈R, and we consider Du(ℓ) := DΛ−ℓu(ℓ) to be a

ﬁnite-diﬀerence stencil with inﬁnite range. For a stencil Du(ℓ), we deﬁne the stencil norms

Du(ℓ)N:= X

ρ∈N (ℓ)−ℓDρu(ℓ)21/2

and ∥Du∥ℓ2

N:= X

ℓ∈Λ|Du(ℓ)|2

N1/2

,(2.2)

where N(ℓ) is the set containing nearest neighbours of site ℓ,

N(ℓ) := nm∈Λ\ℓ∃a∈Rds.t. |a−ℓ|=|a−m| ≤ |a−k| ∀k∈Λo.(2.3)

We can then deﬁne the corresponding functional space of ﬁnite-energy displacements

U1,2(Λ) := u: Λ →Rd∥Du∥ℓ2

N(Λ) <∞

(2.4)

4 YANGSHUAI WANG, JAMES R. KERMODE, CHRISTOPH ORTNER, AND LEI ZHANG

with the associated semi-norm ∥u∥U1,2:= ∥Du∥ℓ2

N. Then the associated class of admissible dis-

placements is given by

A(Λ) := u∈U1,2(Λ) : y0+u∈Adm0(Λ).

2.1.2. The QM site potential. We consider the site potential to be a collection of mappings Vℓ:

(Rd)Λ−ℓ→R, which represent the energy distributed to each atomic site. For technical reasons

we make the following assumptions on the regularity and locality of the site potentials, which has

been justiﬁed for some basic quantum mechanic models [8, 10, 12, 41], but we emphasize that it

is not a universally valid assumption.

(RL) Regularity and locality: For all ℓ∈Λ, VℓDu(ℓ)possesses partial derivatives up to the

third order. For j= 1,2,3, there exist constants Cjand ηjsuch that

Vℓ,ρDu(ℓ)≤Cjexp −ηj

l=1 |ρl|

(2.5)

for all ℓ∈Λ and ρ∈(Λ −ℓ)j.

We refer to [9, §2.3 and §4] for a justiﬁcation and discussion of this assumption.

If the reference conﬁguration Λ is a homogeneous lattice, Λ = Λh, then the site potential does

not depend on site ℓ∈Λhdue to the translation invariance. In this case, we will denote the

site potential by Vh: (Rd)Λh\0→R. for the homogeneous lattice. Although the site potentials

are deﬁned on inﬁnite stencils (Rd)Λ−ℓ, the setting also applies to ﬁnite systems or to ﬁnite range

interactions. In particular, we denote by VΩ

ℓthe site potential of a ﬁnite system with the reference

conﬁguration lying in Λ ∩Ω.

2.1.3. Equilibration of crystalline defects. Let the site potential Vℓsatisfy the assumptions (RL).

The energy-diﬀerence functional is then given by

E(u) := X

ℓ∈ΛVℓDu0(ℓ) + Du(ℓ)−VℓDu0(ℓ).(2.6)

It was shown in [9, Theorem 2.1] that after an elementary renormalisation, (2.6) is well-deﬁned

on the admissible displacements set A(Λ), and that it is (n−1)-times continuously diﬀerentiable

with respect to the ∥·∥U1,2norm.

Instead of the energy minimization problem [13, 51], we will focus on the (formally equivalent)

force equilibrium formulation, that is,

Find ¯u∈A(Λ),s.t.Fℓ(¯u) = 0,∀ℓ∈Λ,(2.7)

where Fℓ:= −∇ℓE(u) represents the force on the atomic site ℓ. For uh∈A(Λh) on a homogeneous

lattice Λh, the force on each atomic site ℓsatisﬁes Fℓ(uh) = Fhuh(·−ℓ)with some homogeneous

force Fhthat does not depend on site ℓ.

2.2. QM/MM Coupling. In this section, we describe the QM/MM models utilized in our study.

Our approach is inspired by [11, 13], tailored to better align with the speciﬁc context of the present

work.

ADAPTIVITY FOR QM/MM MODELS 5

2.2.1. Domain decomposition. We ﬁrst decompose the reference conﬁguration Λ into three disjoint

sets, Λ = ΛQM ∪ΛMM ∪ΛFF, where ΛQM and ΛMM denote the QM and MM regions, respectively, and

ΛFF represents the far-ﬁeld region where atom positions are frozen. This yields the approximated

admissible set

AH(Λ) := u∈U1,2(Λ) : y0+u∈Adm0(Λ) and u= 0 in ΛFF.(2.8)

Moreover, we deﬁne a buﬀer region ΛBUF ⊂ΛMM surrounding ΛQM such that all atoms in ΛBUF ∪

ΛQM are involved in the evaluation of the site energies deﬁned on ΛQM.

In contrast to the method proposed in [7, 51], which employed approximately spherical regions

centered at the defect core for domain partitioning, we introduce a more versatile domain decompo-

sition scheme in our study. Rather than relying on region radii as model parameters, we utilize the

number of atoms within the QM, buﬀer, and MM regions, denoted respectively by NQM := #ΛQM,

NBUF := #ΛBUF, and NMM := #ΛMM. Figure 1 presents a typical QM/MM decomposition for a

(001)[100] edge dislocation in Tungsten (W) with non-spherical subdomains.

Figure 1. Decomposition of (001)[100] edge dislocation in W into QM, MM, buﬀer

(BUF), and far-ﬁeld (FF) regions.

2.2.2. Speciﬁcation of MM model. To represent interatomic interactions at a distance from the

defect core in the MM region, it is desirable to construct the MM site energy (force) that is com-

putationally eﬃcient, accurately captures the underlying physical behavior, and is “compatible”

with the QM model. These requirements have led to the adoption of machine-learned interatomic

potentials (MLIPs) as the MM models [13, 31].

The concrete MLIPs ansatz we employ is the atomic cluster expansion (ACE) method [19, 1, 37],

although we emphasise the workﬂow is readily transferable to other MLIPs approaches. The ACE

model stands out for its ability to achieve high accuracy, despite being a linear model [37]. The

linear ACE model provides a parameterised site potential, for g

g∈RdΛh\0,

VACE(g

g;{cB}B∈B

B) = X

B∈B

cBB(g

g),(2.9)

where Bare the basis functions and cBthe parameters that we will estimate by minimizing a least

square loss (cf. (2.12)). The resulting ACE forces are denoted by FACE

ℓ(g

g).

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APOSTERIORIERRORESTIMATEANDADAPTIVITYFORQM/MMMODELSOFCRYSTALLINEDEFECTSYANGSHUAIWANG,JAMESR.KERMODE,CHRISTOPHORTNER,ANDLEIZHANGAbstract.Hybridquantum/molecularmechanics(QM/MM)modelsplayapivotalroleinmolec-ularsimulations.Thesemodelsprovideabalancebetweenaccuracy,surpassingpureMMmodels,andcomputation...

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A POSTERIORI ERROR ESTIMATE AND ADAPTIVITY FOR QMMM MODELS OF CRYSTALLINE DEFECTS YANGSHUAI WANG JAMES R. KERMODE CHRISTOPH ORTNER AND LEI ZHANG

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