Grand-canonical Monte-Carlo simulation methods for charge-decorated cluster expansions Fengyu XiePeichen Zhong Luis Barroso-Luque Bin Ouyang and Gerbrand Cedery

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Grand-canonical Monte-Carlo simulation methods for charge-decorated cluster

expansions

Fengyu Xie,∗Peichen Zhong, Luis Barroso-Luque, Bin Ouyang, and Gerbrand Ceder†

Department of Materials Science and Engineering,

University of California, Berkeley, California 94720, United States and

Materials Sciences Division, Lawrence Berkeley National Laboratory, California 94720, United States

Monte-Carlo sampling of lattice model Hamiltonians is a well-established technique in statistical

mechanics for studying the conﬁgurational entropy of crystalline materials. When species to be

distributed on the lattice model carry charge, the charge balance constraint on the overall system

prohibits single-site Metropolis exchanges in MC. In this article, we propose two methods to perform

MC sampling in the grand-canonical ensemble in the presence of a charge-balance constraint. The

table-exchange method (TE) constructs small charge-conserving excitations, and the square-charge

bias method (SCB) allows the system to temporarily drift away from charge neutrality. We illustrate

the eﬀect of internal hyper-parameters on the eﬃciency of these algorithms and suggest practical

strategies on how to apply these algorithms to real applications.

INTRODUCTION

Conﬁgurational disorder is particularly important for

understanding the thermodynamic properties of materi-

als at non-zero temperatures, especially in systems com-

posed of multiple components. The cluster-expansion

(CE) method has been a successful approach to study

the statistical mechanics of conﬁgurational disorder in

solids[1–4], and has been used to calculate phase di-

agrams in alloys[5–8] and ionic solids [9–12], predict

the short-range order related properties under ﬁnite

temperatures[13–16], ﬁnd the ground-state ordering in

alloys[17–22], and even compute voltage proﬁle of bat-

tery electrode materials[23–27].

The CE model can be understood as a generalization of

the Ising model. The micro-states in a solid solution are

represented as a series of occupancy variables σ, which

denote the chemical species occupying each lattice site.

The energy of a micro-state is described as a function

of occupancy and is expanded as a sum of many-body

interactions:

E(σ) = X

mβJβhΦα(σ)iα∈β,(1)

where Φα’s are a set of cluster basis functions that take

as input the occupancy values of diﬀerent clusters of mul-

tiple sites. The cluster basis functions are then grouped

and averaged over lattice symmetry orbits βto gener-

ate the correlation functions hΦαiα∈β; and mβis the

multiplicity of orbit βper crystallographic unit cell. The

linear-expansion coeﬃcients Jβare called eﬀective cluster

interactions (ECI). In a typical approach, ECIs are ﬁtted

to the ﬁrst-principles calculated energy of a large num-

ber of ordered super-cells, through a variety of suggested

procedures [28–36]. Thermodynamic quantities can be

obtained by sampling the CE energy with Monte-Carlo

simulations (CE-MC) [6, 37–39]. This workﬂow allows

fast statistical mechanics computation of conﬁgurational

disorder, using only a relatively small number of ﬁrst-

principles calculations. More detailed descriptions of the

CE-MC method can be found in various review papers

[35, 40–44].

CE-MC can be performed in a canonical ensemble or

in a grand-canonical ensemble. In a canonical ensemble,

the conﬁguration states are sampled with a ﬁxed com-

position of each species. Using the Metropolis-Hastings

algorithm [45, 46], a typical Metropolis step involves the

swapping of the species occupying two randomly chosen

sites (canonical swap). In a grand-canonical ensemble,

the states are sampled under ﬁxed chemical potentials

allowing the relative amounts of each species to vary.

A Metropolis step in the grand-canonical ensemble usu-

ally replaces the occupying species on one randomly cho-

sen site with another species (single-species exchange).

Grand canonical simulations are the preferred approach

for studying phase transition in solids, as the simulation

cell is always in a single-phase state, and phase transi-

tions are relatively easy to observe. In contrast, multiple

phases can coexist in canonical simulations, giving a dis-

proportionate inﬂuence to the interfacial energy between

phases.

Single-species exchanges can be applied without issue

when the species are all charge-neutral atoms. How-

ever, in an ionic system in which all the species carry

charge, net zero charge needs to be maintained, essen-

tially coupling allowed species exchanges. For simulating

ionic liquids, various methods have been proposed such

as: inserting and removing only charge-neutral combi-

nations of ions[47]; performing single insertion or dele-

tion while controlling the statistical average of the net

charge equal to be zero[48, 49]; or using an expanded

grand-canonical ensemble[50]. However, charge balance

in lattice-model CE-MC with arbitrary complexity has

not been addressed in the literature yet.

In this study, we introduce two CE-MC sampling meth-

ods to handle the charge-balance constraint in the grand-

canonical CE-MC for ionic systems with charge deco-

arXiv:2210.01165v1 [cond-mat.mtrl-sci] 3 Oct 2022

ration. The ﬁrst is the table-exchange (TE) method,

in which MC samples are kept charge-neutral by using

charge conserving multi-species exchanges. The second

is the square-charge bias (SCB) method, which combines

single-species exchanges with a penalty on the net charge

to drive the system towards zero charge. We benchmark

the computational eﬃciency over hyper-parameters in a

complex rocksalt system with conﬁgurational disorder,

and demonstrate proper usage strategies of these sam-

pling methods.

METHODS

For simplicity, the formalism in the following discus-

sion is limited to materials with a single sub-lattice. How-

ever, the methodology can be easily extended to multiple

sub-lattices. We also limit our investigation to the ap-

plication of a charge-balance constraint; although more

generic integral constraints on the composition (e.g., ﬁx-

ing the atomic ratio between particular components to

follow a speciﬁc hyper-plane in the composition space)

can be addressed in a similar manner.

Table-exchange method

In the grand-canonical ensemble with species carrying

charge, every possible occupancy state must satisfy the

following constraints:

s=1

Csns= 0,

s=1

ns=N,

ns∈N,∀s∈ {1,2,· · · , S}

(2)

where sis the label of a species, nsis the amount of

species sin conﬁguration σ, and Nis the total number

of sites in the system. The ﬁrst equation is a charge-

balance constraint, where Csis the charge of species s.

The second equation requires the number of species to

be equal to the number of sites. Equation 2 is a system

of linear Diophantine equations with natural number so-

lutions. All integral solutions n= (n1,· · · , nS) to these

Diophantine equations can be represented as a bounded

fraction of a (S−2)-dimensional integer grid in NS[51],

speciﬁed as follows:

n=n0+

S−2

i=1

xivi,

s.t. xi∈Z,vi∈ZS

ns∈N, ns≤N

(3)

where n0is a base integer solution to Equation 2, the

vi’s are S−2 linearly independent basis vectors, and the

xi’s are integer coordinates on the grid.

Any vector u=n0−npointing from one solution (n)

on the integer grid to another solution (n0) is called an ex-

change direction. An exchange direction physically rep-

resents a composition transfer under the charge-balance

constraint. A selected set Vamong all possible uis

called an exchange table. Based on the exchange table

V, we can deﬁne a random walk process between charge-

balanced compositions as follows:

(1) Using the current composition n, select one direc-

tion ufrom all feasible directions in the predeﬁned ex-

change table V. The feasibility of a direction uis deﬁned

with the requirement, that for all us<0 (i.e. species sis

being removed), we have ns>−us, ensuring a move to-

wards direction uwould not result in a negative amount

of any species.

(2) Perform the operation to the occupancy conﬁgu-

ration according to the selected exchange u, such that

the composition nchanges to n+u. Given u=

(u1, u2,· · · , uS), one such operation can be achieved by

removing −usof species sfrom the occupancy for all

us<0; then inserting usof species sinto the empty

sites, for all us>0. Such an operation is called a ta-

ble exchange. It results in a simultaneous exchange of

species on multiple sites and is always charge conserv-

ing. The number of sites Uto be exchanged is called

the exchange size in direction u. Because any exchange

should conserve the site number, Psus= 0. Therefore,

U=Pus>0us=Pus<0−us.

A complete exchange table should have ergodicity,

which means an MC simulation should be able to reach

any charge-balanced composition from an arbitrary start-

ing conﬁguration. Once ergodicity is satisﬁed, the num-

ber of sites involved in the exchange directions should

be minimal, as exchanging a large number of sites in a

Metropolis step can lead to low acceptance ratio and thus

ineﬃcient sampling of the conﬁguration space. It is not

necessary, nor practical, to include all possible directions

uin the table. Usually, as a minimal setup, one can

choose S−2 linearly independent basis vectors ({vi})

with minimal exchange size as well as their inverse vec-

tors ({−vi}). The ergodicity of a table can be checked

by enumerating charge balanced compositions in a spe-

ciﬁc super-cell size as vertices of a graph, and checking

graph connectivity between the compositions using vec-

tors in the table as the edges of the graph. If ergodicity

is not satisﬁed with the minimal setup, and the unreach-

able compositions are of interest, vectors linking the dis-

connected composition to other compositions should be

added to the table, until the ergodicity is guaranteed.

According to the statements above, given an exchange

table V, one can propose grand-canonical Metropo-

lis steps using the following procedure, as illustrated

schematically in Figure 1:

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Grand-canonicalMonte-Carlosimulationmethodsforcharge-decoratedclusterexpansionsFengyuXie,PeichenZhong,LuisBarroso-Luque,BinOuyang,andGerbrandCederyDepartmentofMaterialsScienceandEngineering,UniversityofCalifornia,Berkeley,California94720,UnitedStatesandMaterialsSciencesDivision,LawrenceBerkeleyNati...

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Grand-canonical Monte-Carlo simulation methods for charge-decorated cluster expansions Fengyu XiePeichen Zhong Luis Barroso-Luque Bin Ouyang and Gerbrand Cedery

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