A Methodology to Generate Crystal-based Molecular Structures for Atomistic Simulations Christian F. A. Negre1 Andrew Alvarado23 Himanshu Singh1

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A Methodology to Generate Crystal-based

Molecular Structures for Atomistic Simulations

Christian F. A. Negre1, Andrew Alvarado2,3, Himanshu Singh1,

Joshua Finkelstein1, Enrique Martinez3,4, and Romain Perriot1

1 Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545,

USA

2 Advances System Development, Los Alamos National Laboratory, Los Alamos, NM

87545, USA

3 Department of Mechanical Engineering, Clemson University, Clemson, SC 29623,

USA

4 Department of Materials Science and Engineering, Clemson University, Clemson,

SC 29623, USA

E-mail: cnegre@lanl.gov

12 October 2022

Abstract. We propose a systematic method to construct crystal-based molecular

structures often needed as input for computational chemistry studies. These structures

include crystal “slabs” with periodic boundary conditions (PBCs) and non-periodic

solids such as Wulﬀ structures. We also introduce a method to build crystal slabs with

orthogonal PBC vectors. These methods are integrated into our code, Los Alamos

Crystal Cut (LCC), which is open source and thus fully available to the community.

Examples showing the use of these methods are given throughout the manuscript.

Keywords: Quantum Chemistry, Extended Structures, Crystal Structures, Unit cells,

Miller indices

arXiv:2210.01358v2 [cond-mat.mtrl-sci] 11 Oct 2022

1. Introduction

Surface science is essential to understand and

predict many physical phenomena including

heterogeneous catalysis [1, 2], photo-catalysis

[3, 4], material interfaces [5, 6], and optical

properties [7, 8]. Surface science is also

crucial to study the shape and properties

of nanocrystals [9], which are essential to

quantum dot applications [10, 11], and 2D

materials, exhibiting exceptional electrical,

optical and mechanical properties [12, 13].

Beyond physical chemistry, surface science

also plays a crucial role in biomedical [14]

and bioengineering [15] applications, and

dictates crystal growth [16], which is known

to aﬀect, for instance, the performance of

high explosives [17, 18]. Moreover, in all

applications where the material exhibits a

high surface to volume ratio, the properties

of the surface (exposed crystal faces) largely

determine the properties of the material.

Despite the progress of characterization

techniques, simulations remain a fundamen-

tal part of surface science, either to comple-

ment [19] or fully predict [20, 21] the proper-

ties of surfaces, interfaces, and nanoparticles.

Electronic transport, optical properties, and

even surface reconstructions can be a signiﬁ-

cant challenge for empirical models and, in or-

der to perform these simulations, ab initio level

of theory is often required due to the complex-

ity of the phenomena involved [22]. In most

cases, the ﬁrst step involved in these calcula-

tions will involve building a model crystal slab,

which should obey the following constraints:

the system must give us access to the surfaces

of interest to the particular problem, it must

be periodic in all other directions, and, in order

to improve computational eﬃciency, it must be

as small as possible (electronic structure cal-

culations are usually performed on hundreds

to a few thousands atoms, at most). This

results in a crystal-based parallelepiped with

planes that are not necessarily orthogonal to

each other since, in the general case, the unit

cell is triclinic (i.e. the lattice vectors are non-

orthogonal to each other with diﬀering lengths

and angles to one another).

In order to study crystal surfaces with

quantum chemistry methods, it is often

necessary to have a crystal slab cut through

planes that expose the face one wants to study

and that also satisﬁes the periodic boundary

conditions (PBCs) imposed by the crystal

unit cell. A typical minimalistic system is

depicted in Figure 1. This is a z−axis view

of a 3 ×3×3 = 27 monoclinical unit cell

system showing the slab PBC vectors. In

this case the p2vector was enlarged in order

to have some vacuum that could expose the

surface of interest. At ﬁrst, this system

may not seem complicated as it can be

easily built with an ad-hoc procedure using

an oﬀ-the-shelf molecular visualization tool.

There are however, many cases in which

building such a system in this way could

turn into a complicated and time consuming

endeavor. Triclinic unit cells exposing some

crystal face with large Miller indices fall

within this category. Moreover, a lot of time

and eﬀort is consumed when errors in the

simulations arise due to an ill-chosen system

slab. How do we then proceed to construct

any desired crystallographic system by just

knowing the basic crystallographic data? In

this article, we explain a method based on

purely algebraic/geometrical transformations

that leads to a suﬃciently small crystal slab

exposing the desired crystal faces. We would

like to oﬀer a detailed and simple step-by-step

procedure that the reader could fully code up

on their own. Moreover, the method developed

in this paper can also be used to construct

Wulﬀ type of structures provided that the

exposed (hkl) planes are known.

For many applications, it is preferable

to build perfectly orthogonal faces to the

exposed surface, i.e. orthorhombic systems.

For instance, in shock simulations, a piston

hits the back surface of the sample (or vice

versa) and the shock propagates through the

material oriented in a speciﬁc way [23, 24];

thermodynamic quantities are then estimated

within slices of the material perpendicular to

the shock direction, thus the use of orthogonal

planes makes data processing a lot simpler.

In addition, certain simulation codes explicitly

require orthorhombic simulation boxes.

Previously, the authors of Ref. [25] pro-

posed a method where the vectors deﬁning the

slab are tentatively constrained to satisfy both

the orthogonality and periodicity conditions;

however, in the general case, fulﬁlling these

two conditions is not always possible. There is

thus a balance between two eﬀects: the larger

the cell, the higher the probability of fulﬁll-

ing PBCs, although resulting in an increase of

the computational cost when the slab is used

as an input for a quantum chemistry appli-

cation code. On the other hand, a system

that is not periodic will induce possibly large

strain and stress, compromising some thermo-

dynamic properties such as volume or pressure

and yielding artiﬁcial responses. In Ref. [25],

the algorithm usually produces a cell that is

periodic but not exactly orthogonal, with small

deviations in the lattice angles allowed to pre-

serve this condition. This occasionally results

in cells that do not have the exact requested

orientation. In this paper we propose an eﬃ-

cient algorithm to build perfectly orthorhom-

bic cells where the lattice periodicity mismatch

is used to assess the validity of the slab.

The following sections are organized as

follows: We ﬁrst introduce some basic crys-

p2Surface of interest

Figure 1. Representation of a typical system slab

needed to study a particular physical chemistry surface

property. The system slab is composed of lattice points

that are illustrated as bright magenta spheres together

with the slab PBC vectors p1and p2.

tallographic concepts in order to keep consis-

tent notation throughout the manuscript. In

Section 3 we introduce our method to cut a

crystal lattice, and in Section 3.2 we develop

the techniques to determine the PBC vectors.

Section 4 is dedicated to explaining how the

method can be used to construct non-periodic

solids using Wulﬀ structures as an example.

Finally, in Section 5 we explain a method to

construct crystal slabs with orthogonal PBC

vectors. Sections in the Appendices are used

for support and clariﬁcation throughout the

text. Units of length and angles used in all the

examples are in Angstroms (˚

A) and degrees (◦)

respectively.

2. Background

A crystal lattice is a set of points L ⊆ R3that

is fully determined by the primitive unit cell

described by the lattice vectors a1,a2, and

a3. For any point rbelonging to L, there exist

three integers, n1,n2,n3such that

r=n1a1+n2a2+n3a3.(1)

Formally, L={r∈R3|n1, n2, n3∈

Z}. A conventional unit cell (such as the

cubical systems by Bravais), is just a more

elaborate cell in which symmetry is increased.

This increase of symmetry in some cases will,

for instance, render lattice vectors that are

orthogonal to each other; a highly desirable

property for many applications. Regardless

of which type of cell we have, the convention

in crystallography is to report the so-called

lattice parameters a,b,c,α,β, and γ; where

a,b, and care the lengths of lattice vectors

a1,a2, and a3, respectively, and α,β,γ

are, respectively, the angle between vectors

a2and a3,a3and a1, and a1and a2[26].

This reduces the arbitrariness of having to

choose a lattice orientation given by the lattice

vectors. Note that if the lattice is rotated,

our lattice vectors will need to be rotated as

well, whereas the lattice parameters will stay

the same. Finally, a full representation of the

system needs a “basis,” which is the minimal

molecular fragment contained by each unit cell.

It is common to express the coordinates of

the basis in fractions of the lattice vectors.

By choosing this coordinate system we make

the orientation of the basis invariant to lattice

rotations.

Although working with lattice parameters

has some advantages, it is convenient to

compute the lattice vectors in order to do all

the necessary transformations to build a PBC

slab. In order to compute the lattice vectors

from the lattice parameters one needs to apply

the following transformations:

a1x=a

a1y= 0

a1z= 0

(2)

a2x=bcos(γ)

a2y=bsin(γ)

a2z= 0

(3)

a3x=ccos(β)

a3y=c(cos(α)−cos(γ) cos(β))

sin(γ)

a3z=qc2−a2

3x−a2

3y

(4)

where we have arbitrarily set a1to be aligned

with the x-axis, or in more formal terms,

the ﬁrst canonical vector e1= (1,0,0) in

the canonical basis for R3. An equivalent

reverse transformation is used to compute the

parameters given the lattice vectors:

a=q(a2

1x+a2

1y+a2

1z)

b=q(a2

2x+a2

2y+a2

2z)

c=q(a2

3x+a2

3y+a2

3z)

(5)

γ=360

2πarccos ((a1·a2)/(ab))

β=360

2πarccos ((a1·a3)/(ac))

α=360

2πarccos ((a2·a3)/(bc))

(6)

3. Building extended systems

Using the lattice vectors, a crystal slab can be

built simply by adding lattice points according

to Eq. (1) for a ﬁnite number of ni’s. The

resulting slab would expose the (100), (010)

and (001) crystalline faces as well as the

respective opposite faces given by (100), (010)

and (001). Note that this slab will form a

parallelepiped whose edge directions are not

necessarily orthogonal to one another in E ≡

{e1,e2,e3}, the standard basis for R3. We

shall call this slab the “canonical slab.” An

example canonical slab of the monoclinic phase

of benzene is shown in Figure 2. At this

point, a natural question emerges. What if

now we need to expose other crystalline faces

to perform speciﬁc computational physico-

chemical studies? In this case, the periodicity

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AMethodologytoGenerateCrystal-basedMolecularStructuresforAtomisticSimulationsChristianF.A.Negre1,AndrewAlvarado2;3,HimanshuSingh1,JoshuaFinkelstein1,EnriqueMartinez3;4,andRomainPerriot11TheoreticalDivision,LosAlamosNationalLaboratory,LosAlamos,NM87545,USA2AdvancesSystemDevelopment,LosAlamosNationalL...

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A Methodology to Generate Crystal-based Molecular Structures for Atomistic Simulations Christian F. A. Negre1 Andrew Alvarado23 Himanshu Singh1

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