A High-Throughput Framework for Materials Research and Space Group Determination Algorithm

A High-Throughput Framework for Materials

Research and Space Group Determination

Algorithm

Richard Taylor

Department of Mechanical Engineering and Materials Science Duke University

Date: Approved:

Stefano Curtarolo, Supervisor

Teh Tan

Nico Hotz

Thesis submitted in partial fulﬁllment of the requirements for the degree of Master of Science in the Department of Mechanical Engineering and Materials Science in the Graduate School of Duke University 2012 Abstract A High-Throughput Framework for Materials Research and Space Group Determination Algorithm

Richard Taylor

Department of Mechanical Engineering and Materials Science Duke University

Date: Approved:

Stefano Curtarolo, Supervisor

Teh Tan

Nico Hotz

An abstract of a thesis submitted in partial fulﬁllment of the requirements for the degree of Master of Science in the Department of Mechanical Engineering and Materials Science in the Graduate School of Duke University 2012 Copyright c 2012 by Richard Taylor All rights reserved except the rights granted by the Creative Commons Attribution-Noncommercial License Abstract

Effective computational materials search, categorization, and design necessitates a high-throughput (HT) approach. System by system analyses lack the scope and speed needed to uncover large portions of the materials landscape. By performing broad searches over structural or chemical classes of materials and guided by fundamental physical principles, materials with specific desired properties can be systematically found. Furthermore, the HT approach is an effective general tool for materials classification. Depending on the application, various properties can be computed leading to powerful classification schemes. To implement HT materials studies, however, a versatile and robust framework must first be developed. In this paper, the HT framework AFLOW that has been developed and used successfully over the last decade is presented. Specifically, attention is given to an origin-specific symmetry algorithm. The algorithm is designed to determine the relevant symmetry properties of an arbitrary crystal structure (e.g., point group, space group, etc.).

iv Contents

Abstract iv

List of Tables vii

List of Figures viii

List of Abbreviations and Symbols xi

Acknowledgements xii

1 Introduction1

2 Symmetry in Crystallography: Crystal System, Point Group, and Space Group5

2.1 Crystal class or point group...... 5

2.2 Lattice system and crystal system...... 6

2.2.1 Lattice system...... 6

2.2.2 Crystal system...... 7

2.3 Space group...... 7

3 AFLOW: Structure analysis tools and library 13

3.0.1 POSCAR ﬁle template...... 15

3.0.2 WYCCAR ﬁle template...... 16

3.0.3 ABCCAR ﬁle template...... 16

3.1 AFLOWLIB.ORG...... 17

v 4 Space group determination algorithm and related functions 21

4.1 Overview of the algorithm...... 21

4.2 Origin-referenced symmetry operations...... 22

4.3 Unit cell conventionalizing...... 24

4.3.1 Cubic...... 28

4.3.2 Hexagonal...... 28

4.3.3 Rhombohedral...... 28

4.3.4 Tetragonal...... 29

4.3.5 Orthorhombic...... 30

4.3.6 Monoclinic...... 30

4.3.7 Triclinic and the reduced cell...... 31

4.4 Crystal point group and the standard crystallographic origin..... 33

4.5 Symmetry classes and comparison routine...... 39

4.5.1 Improper rotations and the RotoInversion class...... 40

4.5.2 Glide class...... 42

4.5.3 Screw class...... 44

4.5.4 Operator overloading in the symmetry classes...... 44

A Stereographic projections of the 32 point groups including crystal system and the Bravais lattices 46

Bibliography 58

vi List of Tables

2.1 Conditions on the symmetry of crystals in the crystal systems.....8

2.2 The 230 space groups with point group and crystal system...... 9

4.1 The standard origins of the 230 space groups with point group and crystal system...... 38

vii List of Figures

2.1 The seven lattice systems in 3 dimensions with unfolded faces..... 11

2.2 Classiﬁcation of crystal structures. Based on Fig. 8.2.1.1 in ITC (Hahn(2002))...... 12

3.1 A typical view of the online aconvasp interface. Note the input ﬁle is in the POSCAR format described in 3.0.1...... 14

3.2 An example of a parent lattice (a) and a derived superstructure (b). The superstructure is constructed by a relabeling of a subset of the lattice points. The lattice vectors of the derived structure become multiples of the parent lattice...... 18

3.3 The opening view of the online binary materials database interface. Lattice-based structures such as fcc and bcc can be selected to reﬁne the search...... 19

3.4 Using hundreds of ﬁrst-principles calculations, the T 0 K phase diagram can be constructed, and the low temperature stability of the binary system can be investigated. The diagram represented here is the phase diagram of the Au-Mg (gold-magnesium) system...... 20

4.1 Flow chart describing the basic design of the space group algorithm in four thematically distinct parts...... 23

4.2 The geometric objects related to the crystallographic symmetry operations in three dimensions: equilateral triangle, isosceles right triangle, 120-30-30 triangle, plane, and line. Small solid points indicate the intersection of an axis of symmetry with the plane deﬁned by three points...... 24

viii 4.3 Cartoon diagram illustrating the method used to find point group operations in an origin-specific manner. a) shows a mirror reflection requiring selection of two atoms. The plane normal is defined by the line between the two points and intersects at the midpoint. b) shows a three-fold rotation about an axes. The axes is uniquely determined by the normal to the plane intersecting at the center of mass. This is extended to other crystallographic point operations (n 1, 2, 3, 4, 6) by reduction of co-planer atoms...... 25

4.4 Three choices of reduced basis for a square lattice. Each has the same, minimal volume...... 26

4.5 Ambiguity in the choice of unit cell is shown. A rhombohedral unit cell (solid lattice vectors) or the larger hexagonal cell (dashed lattice vectors) generate the same lattice...... 27

4.6 The Bravais lattices for the cubic lattice system...... 28

4.7 The Bravais lattices for the hexagonal (hP) and rhombohedral (hR) lattice systems...... 29

4.8 The Bravais lattices for the tetragonal lattice system...... 29

4.9 The Bravais lattices for the orthorhombic lattice system...... 30

4.10 The Bravais lattices for the monoclinic lattice system...... 31

4.11 The Bravais lattice for the triclinic lattice system...... 33

4.12 Threefold, fourfold, and sixfold rotoinversions are illustrated. The geometric shapes deﬁned by stereographic projections along possible axes are used to identify possible rotoinversion axes...... 36

4.13 Additional geometric objects comprising the full set of symmetry operations. (a) Screw axes (nf ) create triangles with non-simple (but determinable) angle and side relations. The possible geometries can be derived using the projected object (dashed triangle) geometry and the screw displacement (h and 2h). Because the screw axes are parallel to lattice directions, a finite set of plane normal-screw axis angles α exists. (b) Glide planes exist for a finite set of translations (n,d,a,b,c) defined in terms of the lattice vectors. Applying these translations and checking if an atom exists along the normal direction reveals the glide planes...... 37

4.14 Cartoon illustrating a rotation-reﬂection/rotation-inversion operation. 42

ix 4.15 Cartoon illustrating a glide operation...... 43

4.16 Cartoon illustrating a screw operation...... 44

x List of Abbreviations and Symbols

Symbols used in the stereographic projections

Stereograms for the 32 crystallographic point groups are listed in AppendixA. The symbols used are deﬁned below.

Twofold rotation axis 2

Threefold rotation axis 3

Fourfold rotation axis 4

Sixfold rotation axis 6

_ Inversion center 1

_ Threefold rotoinversion 3

_ Fourfold rotoinversion 4

_ Sixfold rotoinversion 6

Twofold rotation with center 2/m

Fourfold rotation with center 4/m

Sixfold rotation with center 6/m

xi Acknowledgements

The author thanks Stefano Curtarolo for his mentoring and Shidong Wang, Kesong Yang, and Junkai Xue for fruitful discussions. This work was supported by ONR (N00014-11-1-0136, N00014-10-1-0436, N00014-09-1-0921), NSF (DMR-0639822, DMR- 0908753), and the Department of Homeland Security Domes tic Nuclear Detection Ofce.

xii 1

Introduction

A number of theoretical approaches in materials science have been developed into powerful computational tools. From the standpoint of materials engineering, these tempt an intriguing vision of the future of materials design. Computational materials scientist often speak of “materials by design.” This is a compelling phrase, but the tools used by materials scientists have until recently lacked the unifying framework needed to facilitate the realization of their full potential. A system-specific view is often taken by computational materials scientists. That is, a particular compound or system of interest is selected and is studied by the directed application of computational tools. These include density functional theory (DFT), simulated annealing, and a host of others. The goal is to understand, for instance, the electronic structure and properties, or thermodynamic characteristics of one (or at most several) compound(s) to the extent possible. The system-specific approach has resulted in hundreds of scientific papers and a pittance of novel materials predictions. This highlights a fundamental limitation of the system-specific approach. In many respects, it is the natural extension of experimental materials science. In the real (in contrast to the computational) laboratory,

1 one seldom studies more than one or a few compounds at a time. The investigation is time-consuming and the object is thoroughness and accuracy. On the other hand, a unified computational framework founded on a high-throughput approach to materials design and discovery is a unique paradigm: it has no clear anal- ogy in the real laboratory. It is precisely this approach that offers enormous potential to the future of materials design. In contrast to the system-specific approach where the emphasis is on accuracy and detail, high-throughput computational searches leverage fundamental physical insights with the capability to do broad searches. A greater portion of the vast materials landscape can be accessed in this way. In con- ceptual terms, the landscape’s broad contours are “discovered” in a high-throughput fashion. This can then be followed by a more detailed analysis provided by the system-specific approach. In the context of structure discovery and optimization, the high-throughput (HT) approach is the unification of two recent theoretical advancements: ab initio quantum mechanical computation (e.g., DFT) and informatics data mining and evolutionary screening algorithms. This methodology has already been used on a number of occa- sions as a tool for materials discovery and development (Xiang et al.(1995); Chiang et al.(1998); Jóhannessonet al.(2002); Stucke and Crespi(2003); Curtarolo et al. (2003)). A sample of its applications—primarily focused on novel stable compound discovery—can be found in various references (Levy et al.(2010b,a); Taylor et al. (2011a)). The HT approach to materials discovery relies on an automatic optimization and search scheme to screen a large library of candidate compounds (provided by, for example, the Pauling File (P. Villars, M. Berndt, K. Brandenburg, K. Cenzual, J. Daams, F. Hulliger, T. Massalski, H. Okamoto, K. Osaki, A. Prince, H. Putz, and S. Iwata(2004)) and/or the ICSD Database (Mighell and Karen(1993a); Belsky et al. (2002)). To be effective, the framework should be capable of automatically directing

2 refinements and handling the failure of a computation. Common causes of failure that must be addressed include insufficient hardware resources and failure of the ab initio calculation itself. Specifically, the memory requirements of a computational task must be accurately estimated to facilitate the use of multi-node computer systems, and ab initio run-time errors should be addressed by input parameter refinement in a manner requiring minimal user intervention.

In this thesis I describe AFLOW, a HT framework following the prescription above, as well as an algorithm I have developed for the determination of crystallographic symmetry (i.e., crystal lattice, point group, space group). AFLOW has been under development for the last decade and is fully functional and available at aflowlib.org/aflow.html. It comprises over 150, 000 lines of C++ code written for UNIX systems with the GNU suite of compilers. Although it is designed to work in tandem with any program providing structure energy, it is, at the present, optimized

for the Vienna Ab Initio Simulation Package (VASP)(Kresse and Hafner(1993a)). Although I will give a general overview of its capabilities, I will focus primarily on my contributions to this project. In particular, this is the development of a space group determination algorithm and associated developed classes for the necessary symmetry operations. The symmetry algorithm described in Chapter 4 makes use of certain geometric objects underlying the possible crystallographic symmetry operations in order to find symmetry properties. An origin-specific search routine allows for a standardized structure to be constructed in two thematically distinct steps (conventional cell, standard origin). The importance of standardization is especially clear in HT applications such as crystal classification and structure library generation. In such cases, the ambiguity of crystal representations must be eliminated. This is typically done by the application of selected rules. The procedure described here uses symmetry as the guiding principal behind classification (as opposed to metric considerations

3 found in reduction schemes)—this is done primarily because it is consistent with the format used in the ITC (Hahn(2002)) and because useful symmetry information is obtained during the process. Specifically, the routine involves conventionalizing the unit cell followed by a high site-symmetry choice of origin. Using a comparison routine with C++ libraries containing the symmetry operations of the equivalently defined 230 space groups listed in the ITC, the space group can be determined. Finding the symmetry operations of an arbitrary crystal structure is a non-trivial task. All possible symmetry types are implemented in C++ classes with overloaded operators for ease of use and portability. The possible symmetry operations are pure rotations (2-fold, 3-fold, 4-fold, and 6-fold), reflections, inversions, rotation-inversions (rotoinversions), screw axes, and glide planes. Conveniently, the 230 space groups’ standard origins (ITC standard) make use of high site-symmetry points which depend only on the point group of the crystal (although this sometimes still necessiates determining higher-order operations such as glide planes). In 11 of the 32 point groups the existence of inversion symmetry makes this classification scheme especially useful. When it exists, an inversion point is chosen as the standard origin. Also, by this choice we can eliminate the need to find the non point group operations in the symmorphic space groups. For the others, screw axes and glide planes must be found. A method for this is described in Chapter 4. Due to the character of rotoinversions (all but one can be reduced to equivalent pure rotations/reflections), we need only directly search for fourfold rotoinversions.

4 2

Symmetry in Crystallography: Crystal System, Point Group, and Space Group

The symmetry properties of a crystal are the foundational characteristics for the categorization and study of materials. These properties can be unambiguously divided by several classification schemes including crystal system (or crystal family), lattice system, point group, and space group. Various physical properties are inti- mately connected with these symmetry descriptors. For example, crystals belonging to a centrosymmetric space group may not exhibit the piezoelectric effect due to the existence of an inversion center. Everything from electronic structure and op- tical properties to lattice vibrations are profoundly related to a crystal’s symmetry groups. In an effort to study the properties of materials, it is therefore necessary to have tools for the accurate measure of the symmetry relations.

2.1 Crystal class or point group

There are 32 crystal classes or crystallographic point groups. These comprise the possible groups of point operations that leave crystals invariant. For general objects,

5 there are infinitely many point groups. However, in crystals these are reduced by the requirement of translational symmetry which, for example, does not permit five-fold rotation symmetry. Each of the 32 crystal classes can be uniquely represented by a geometric, space-filling object called a crystal form. The notation used throughout this paper is the so-called Hermann-Mauguin notation which is the most commonly used in crystallography. Rotation axes are denoted by their rotation order (i.e., 1, 2, 3, 4, 6 corresponding to rotations of 2π, π, π{2, and so forth). A ‘¯’ is used to indicate an improper rotation (e.g., 2¯ is a two-fold rotation-inversion). Mirror planes—which are equivalently described as a two-fold rotation-inversion—are denoted m. When a rotation axis is parallel to the normal of a mirror plane, both are indicated by a fraction (e.g., 2{m). The Hermann-Mauiguin notation for point groups does not explicitly list the complete set of point operations. Instead, only those symmetries along special directions are given. These are the primary, secondary, and tertiary directions and are defined within a crystal system.

2.2 Lattice system and crystal system

2.2.1 Lattice system

The lattice system of a crystal is a classification scheme defined by the possible point groups of crystallographic lattices. In three dimensions there are seven lattice systems: cubic, tetragonal, hexagonal, rhombohedral, orthorhombic, monoclinic, triclinic (Fig. 2.1). It is important to note that the point group of the crystal—the set of point operations of the crystal—is not necessarily equivalent or even inclusive of the point group of the underlying lattice. However, it is possible to determine the lattice system from the full point group of the crystal by determining the Bravais class of the space group. The point group of the crystal is a subgroup of the Bravais lattice groups. When this condition is satisfied by more than one Bravais lattice

6 group, the one with the smallest possible order is selected. This uniquely assigns the crystal to a Bravais class and therefore to a lattice system.

2.2.2 Crystal system

The crystal systems are nominally similar to the lattice systems: There are seven crystal systems, five of which are identical to their lattice-system counterparts. The discrepancy arises in the trigonal and hexagonal crystal systems (the trigonal crystal system includes space groups belonging to both the rhombohedral and hexagonal lattice systems), and is due to underlying differences inherent in these modes of classification. The crystal system is defined by the point group of the crystal but not by its lattice. For example, crystals containing a 3-fold symmetry axis belong to the trigonal crystal system. However, such crystals may have either a rhombohedral or hexagonal lattice. Some of the confusion this causes may be averted by considering instead six crystal families and the equivalent six lattice families. In these schemes, the trigonal and hexagonal crystal systems are grouped into one hexagonal crystal family, and the rhombohedral and hexagonal lattice systems into one hexagonal lattice family (Fig. 4.5). The crystal system of a space group is derived from its crystal class by identifying key symmetry operations described in Table 2.1. The overlap between the hexagonal and rhombohedral lattice systems occurs for those space groups having 3,

31, 32 or 3¯ as principal axis. These comprise the trigonal crystal system but may be in either the hexagonal or rhombohedral lattice system.

2.3 Space group

There are 230 unique groupings of the crystallographic point operations, lattice translation symmetries, and compound symmetry operations (glide, screw, etc). Combin- ing the 14 Bravais lattices with the point operations gives the 73 simple space groups.

7 Table 2.1: Conditions on the symmetry properties of crystals in each of the seven crystal systems. When “only” is written, it should be taken to indicate that no other symmetry operations exist other than a possible inversion symmetry. The seven lattice systems correspond directly to the crystal systems in all cases except the trigonal system. In this case, the lattice may be either hexagonal or rhombohedral.

Crystal system Symmetry Cubic (a) four 3-fold axes. Tetragonal (a) one 4-fold axis; (b) only one 2-fold; (c) three 2-fold and two mirror planes; (d) only four mirror planes. Orthorhombic (a) only one 2-fold axis and two mirror planes; (b) only three 2-fold; (c) three 2-fold and three mirror planes. Trigonal (a) only one 3-fold axis; (b) three 2-fold, one 3-fold axes and zero or three mirror planes. Hexagonal (a) one 6-fold axis; (b) one 3-fold axis and one or 4 mirror planes. Monoclinic (a) only one mirror plane; (b) only one 2-fold; (b) one 2-fold and one mirror plane. Triclinic (a) no symmetry other than a 1-fold axis and possible inversion.

The simple space groups are also called the arithmetic crystal classes. The addition of glide planes and screw axes brings the total number of affine space groups to 219. The 219 space groups are classified by their underlying abstract groups: two space groups are equivalent if an affine transformation exists relating one to the other. In practice, we want to distinguish between right and left handed screw operations as well as preserving the handedness of the coordinate system. These conditions expand the space group types to the 230 crystallographic space groups. As a final note, the holosymmetric space groups are those constructed by only applying the possible lattice translations to a point. In these cases, the point group of the space group is equal to the point group of the lattice. The seven holohedral point groups are 1,¯ 2{m, mmm, 4{mmm, 3¯m, 6{mmm, and m3¯m. The relation between these symmetry categories is given in Figure 2.2.

8 Table 2.2: The 230 space groups with point groups and crystal systems. The space groups and point groups are given using Hermann-Mauguin notation. The Laue groups indicating the centrosymmetric point groups are boxed.

Crystal System Point Group Space Groups 1 P 1 Triclinic (1-2) 1¯ P 1¯ 2 P 2, P 21, C2 Monoclinic (3-15) m P m, P c, Cm, Cc

2{m P 2{m, P 21{m, C2{m, P 2{c, P 21{c, C2{c

222 P 222, P 221, P 21212, P 212121, C2221, C222, F 222, I222, I212121 mm2 P mm2, P mc21, P cc2, P ma2, P ca21, P nc2, P mn21, P ba2, Orthorhombic (16-74) P na21, P nn2, Cmm2, Cmc21, Ccc2, Amm2, Aem2, Ama2, Aea2, F mm2, F dd2, Imm2, Iba2, Ima2 mmm P mmm, P nnn, P ccm, P ban, P mma, P nna, P mna, P cca, P bam, P ccn, P bcm, P nnm, P mmn, P bcn, P bca, P nma, Cmcm, Cmce, Cmmm, Cccm, Cmme, Ccce, F mmm, F ddd, Immm, Ibam, Ibca, Imma 4 P 4, P 41, P 42, P 43, I4, I41 4¯ P 4,¯ I4¯

4{m P 4{m, P 42{m, P 4{n, P 42{n, I4{m, I41{a

422 P 422, P 4212, P 4122, P 41212, P 4222, P 42212, P 4322, P 43212, I422, I4122 4mm P 4mm, P 4bm, P 42cm, P 42nm, P 4cc, P 4nc, Tetragonal (75-142) P 42mc, P 42bc, I4mm, I4cm, I41md, I41cd 42¯ m P 42¯ m, P 42¯ c, P 42¯ 1m, P 42¯ 1c, P 4¯m2, P 4¯c2, P 4¯b2, P 4¯n2, I4¯m2, I4¯c2, I42¯ m, I42¯ d 4{mmm P 4{mmm, P 4{mcc, P 4{nbm, P 4{nnc, P 4{mbm, P 4{mnc, P 4{nmm, P 4{ncc, P 42{mmc,P 42{mcm, P 42{nbc, P 42{nnm, P 42{mbc, P 42{mnm, P 42{nmc, P 42{ncm, I4{mmm, I4{mcm, I41{amd, I41{acd 3 P 3, P 31, P 32, R3 3¯ P 3,¯ R3¯ Trigonal (143-167)

9 32 P 312, P 321, P 3112, P 3121, P 3212, P 3221, R32 3m P 3m1, P 31m, P 3c1, P 31c, R3m, R3c 3¯m P 31¯ m, P 31¯ c, P 3¯m1, P 3¯c1, R3¯m, R3¯c 6 P 6, P 61, P 65, P 62, P 64, P 63 6¯ P 6¯

6{m P 6{m, P 63{m Hexagonal (168-194) 622 P 622, P 6122, P 6522, P 6222, P 6422, P 6322 6mm P 6mm, P 6cc, P 63cm, P 63mc 6¯m2 P 6¯m2, P 6¯c2, P 62¯ m, P 62¯ c

6{mmm P 6{mmm, P 6{mcc, P 63{mcm, P 63{mmc

23 P 23, F 23, I23, P 213, I213 m3¯ P m3,¯ P n3,¯ F m3,¯ F d3,¯ Im3,¯ P a3,¯ Ia3¯ Cubic (195-230 432 P 432, P 4232, F 432, F 4132, I432, P 4332, P 4132, I4132 43¯ m P 43¯ m, F 43¯ m, I43¯ m, P 43¯ n, F 43¯ c, I43¯ d m3¯m P m3¯m, P n3¯n, P m3¯n, P n3¯m, F m3¯m, F m3¯c, F d3¯m, F d3¯c, Im3¯m, Ia3¯d

10 Cubic Tetragonal

Hexagonal

Rhombohedral

Orthorhombic

Monoclinic

Triclinic

Figure 2.1: The seven lattice systems in 3 dimensions with unfolded faces.

11 6 crystal/lattice families

7 lattice systems 7 crystal systems

14 Bravais lattice types 32 crystal classes

73 simple space groups Add glides and screws 219 afﬁne space groups Add chirality 230 crystallographic space groups

∞ space groups

Figure 2.2: Classiﬁcation of crystal structures. Based on Fig. 8.2.1.1 in ITC (Hahn (2002))

12 3

AFLOW: Structure analysis tools and library

AFLOW provides a suite of tools that are useful for structure analysis and manipulation. While these are built into the HT calculational framework and database

creation of AFLOW, they can be used separately (e.g., for system-speciﬁc studies). The tools available for structure analysis and manipulation are easily accessed via the command line version of AFLOW called ACONVASP; additionally, a variety of the functions are available online at aflowlib.org/awrapper.html. A typical view is shown in Fig 3.1. A few of these are listed below.

• Normal Primitive: Generates a maximally compact primitive unit cell. This is not always unique.

• Standard Primitive: Generates a primitive cell for which the Wigner-Seitz cell of reciprocal space coincides with one of the 24 Brillouin zones (Burns and Glazer(1990); Bradley and Cracknell(1972); Setyawan and Curtarolo(2010)).

• Standard Conventional: Generates a conventional unit cell by choosing maximally orthogonal basis vectors. This does not always coincide with the

13 Figure 3.1: A typical view of the online aconvasp interface. Note the input ﬁle is in the POSCAR format described in 3.0.1

conventional cell of the International Table of Crystallography (Hahn(2002); Wondratschek and M uller(2004)).

• Minkowski lattice reduction: Generates a maximally compact unit cell. Does not convert to a primitive cell.

• Niggli Standardized form: Generates a cell conforming with the Niggli standard (Niggli(1928)).

• WYCKOFF-CAR/ABCCAR to POSCAR: Converts a structure file containing the lattice information and Wyckoff positions to an explicit form (the POSCAR specifies the lattice and atomic basis.)

• POSCAR to ABCCAR: Converts the standard POSCAR containing lattice vectors to one containing lattice parameters (vector magnitudes and angles).

• Bring atoms in the cell: Calculates atomic coordinates modulo the lattice

14 vectors (e.g., an atom located at p1.9, ¡0.4, 0.5q with respect to the lattice becomes p0.9, 0.6, 0.5q.)

• Cartesian coordinates: The structure ﬁle is altered to provide atomic locations in terms of Cartesian coordinates rather than in terms of the lattice.

• Fractional coordinates: The structure ﬁle is altered to provide fractional atomic locations (i.e., in terms of the lattice), in contrast to Cartesian coordinates.

The typical form of the structure ﬁle is the so-called “POSCAR” and contains

(implicitly) the entirety of the structure’s symmetry properties. AFLOW, however, is equally capable of using alternative data forms: these include the “WYCCAR” and “ABCCAR” files. Template forms of these three files are given below. The WYCCAR file contains the wyckoff positions of the crystal’s space group that defines the structure, while the ABBCAR uses an angle-magnitude representation of the lattice vectors.

3.0.1 POSCAR ﬁle template

[Structure title] [Atomic volume (specified by a prefix "-") or lattice parmeter] [Lattice vector 1] [Lattice vector 2] [Lattice vector 3] [# of atom type 1] [# of atom type 2] ... [# of atom type n] [Representation: Options are "DIRECT" or "CARTESIAN"] [Atom 1] [Label (e.g., Cu)] [Atom 2] [Label (e.g., Cu)] ..

15 .. .. [Atom N] [Label (e.g., Cu)]

3.0.2 WYCCAR ﬁle template

[Structure title] [Atomic volume (specified by a prefix ‘-’) or lattice parmeter] [A] [B] [C] [ALPHA] [BETA] [GAMMA] [SG#] [SG OPTION# if necessary] [# of atom type 1] [# of atom type 2] ... [# of atom type n] [Representation: Options are "DIRECT" or "CARTESIAN"] [Wyckoff position 1] [Label (e.g., Cu)] [Wyckoff position 2] [Label (e.g., Cu)] ...... [Wyckoff position N] [Label (e.g., Cu)]

3.0.3 ABCCAR ﬁle template

[Structure title] [Atomic volume (specified by a prefix ‘-’) or lattice parmeter] [A] [B] [C] [ALPHA] [BETA] [GAMMA] [# of atom type 1] [# of atom type 2] ... [# of atom type n] [Representation: Options are "DIRECT" or "CARTESIAN"] [Atom 1] [Label (e.g., Cu)] [Atom 2] [Label (e.g., Cu)] .. ..

16 .. [Atom N] [Label (e.g., Cu)]

3.1 AFLOWLIB.ORG

In addition to providing a powerful tool for structure manipulation and analysis, an integral component of any materials HT framework is the structure library and associated created databases. The use of AFLOW over the last decade has resulted in a large (and continually expanding) repository of crystal structures, phase diagrams, and materials properties such as electronic structure, magnetic, thermoelectric, and vibrational properties. A primary component of the repository is a large collection of thermodynamic energies of compounds spanning the entire composition range and many of the possible binary systems. The database currently contains entries (over 150, 000) from 650 binary systems. Each entry comprises the ground state energy, magnetic moment per atom, space group and prototype (listed according to its Struckturbericht designation, if available). The structure database itself comprises roughly 400 experimental prototypes largely provided by the Pauling File (P. Villars, M. Berndt, K. Brandenburg, K. Cenzual, J. Daams, F. Hulliger, T. Massalski, H. Okamoto, K. Osaki, A. Prince, H. Putz, and S. Iwata(2004)) with additional structures taken from the Navy Crystal (nav(2011)) and ICSD (Mighell and Karen(1993a,b)) databases. The structure database also includes bcc- and fcc-derived superstructures (see Fig. 3.2) containing up to 24 atoms per cell (a few million in total). These were enumerated using the algorithms described in Hart and Forcade(2008a,b). The utility of derived superstructures is in the construction of a cluster expansion or by direct incorporation into a HT thermodynamic ground state search. In accordance with the HT paradigm, many materials properties and correlations can be extracted (discovered) a posteriori by appropriate searches through the com-

17 piled database. For example, by calculating thermodynamic energies of many known structures irrespective of supposed stability, the ground state phase diagram can be extracted.

(a) (b)

Figure 3.2: An example of a parent lattice (a) and a derived superstructure (b). The superstructure is constructed by a relabeling of a subset of the lattice points. The lattice vectors of the derived structure become multiples of the parent lattice.

Much of the information contained in the materials database constructed by

AFLOW is also available online at aflowlib.org/awrapper.html. The opening view is shown in Fig 3.3. From here it is possible to access much of the structure library and the phase diagrams constructed using the computed free energies for many of the binary systems. It is possible to search directly for the occurance and energies of specific lattice types (bcc, fcc, hcp) within a binary system in addition to those generated superlattices by the selection bar at the top of the view. Upon selection of a lattice(s) initial and relaxed structure files are provided for the available system. Selecting “phase-diagram” leads to a diagram after, for example, figure 3.4. This figure displays the phase diagram for the Au-Mg system presented (along with many other binary magnesium alloy systems) in Taylor et al.(2011b). This revealing

18 study of binary magnesium systems was conducted entirely within the AFLOW high-throughput framework.

Figure 3.3: The opening view of the online binary materials database interface. Lattice-based structures such as fcc and bcc can be selected to reﬁne the search.

19 Figure 3.4: Using hundreds of ﬁrst-principles calculations, the T 0 K phase diagram can be constructed, and the low temperature stability of the binary system can be investigated. The diagram represented here is the phase diagram of the Au-Mg (gold-magnesium) system.

20 4

Space group determination algorithm and related functions

4.1 Overview of the algorithm

In the most general situation, the only information explicitly known about a structure under investigation is contained in the structure ﬁle (e.g., POSCAR (Kresse and Hafner(1993b)), WYCCAR (S. Curtarolo, W. Setyawan, G. L. W. Hart, M. Jah- natek, R. V. Chepulskii, R. H. Taylor, S. Wang, J. Xue, K. Yang, O. Levy, M. Mehl, H. T. Stokes, D. O. Demchenko, and D. Morgan(2012))), etc.). Although this data is implicitly complete—that is, it implicitly contains all the symmetry information of the structure (crystal class, point group, space group and all other subgroups and derivatives)—little is explicitly provided. The explicitly known information comprises only lattice vectors (although the basis may not be reduced), the atomic basis vectors and their atomic types (speciﬁc atomic types are irrelevant to symmetry considerations). Hence, in the context of high-throughput (HT) computa- tions there is a need for extraction of this information via some algorithmic process. In a case-by-case scenario, it is feasible, however tedious, to explicitly determine the

21 symmetry relations by inspection. This is an entirely unsatisfactory method when hundreds or thousands, or even just several structures must be analyzed or when the process is part of an automated optimization, search or comparison scheme. The algorithm presented here relies principally on the data provided in the ﬁfth edition of the International Tables of Crystallography (ITC) (Hahn(2002)). The overarching methodology is simple in concept although non-trivial to implement. It assumes that only a standard POSCAR format will be provided as input (see Chap.3). The scheme may be understood as the sequential implementation of four thematically distinct parts (Fig. 4.1): a) Find the point group of the crystal by a symmetry search; b) Find the lattice type and conventionalize the unit cell according to the standard set in ITC; c) determine the minimal set of space group operations in an origin-aware manner; c) shift the origin of the ITC-conventionalized unit cell according to the prescription described in ITC (this requires choosing high site-symmetry points as origins); d) search space groups restricted by point group by systematic application of space group symmetry operations provided in standard origin format in the ITC.

4.2 Origin-referenced symmetry operations

To make use of the symmetry of a crystal structure and classify by lattice type, point group, and space group, we employ an approach which reveals the symmetry operations with reference to the origin. This is particularly useful for both the conventionalizing of the unit cell and the appropriate shift of origin, to facilitate comparison with the space group tables in the ITC, both of which make reference to specific symmetry directions. The use of specific symmetry directions of the lattice in defining a conventional cell is described in Section 4.3 and requires the point group of the lattice. For the determination of the standard crystallographic origin, the full crystal point group is necessary.

22 structure data

ﬁnd point group operations with respect to original origin

ﬁnd lattice and conventionalize unit cell (ITC)

shift origin according to ITC

search space groups using ITC-deﬁned symmetry operations

Figure 4.1: Flow chart describing the basic design of the space group algorithm in four thematically distinct parts.

In general, the method underlying the choice of origin can be divided into centrosymmetric and noncentrosymmetric space groups. The former defines an inversion point as the origin. The inversion point may, however, intersect other planes or axes of symmetry. The highest site-symmetry point defines a natural origin for the noncentrosymmetric space groups. In either case, we wish to know the location of symmetry axes, planes, and inversion points with respect to the initial origin so that the appropriate shift can be computed. The algorithm for computing origin-referenced symmetry directions makes use of the geometric properties underlying the crystallographic symmetries. Due to the requirement of translation symmetry in traditional crystals, the order of rotations considered are limited to n 1, 2, 3, 4, 6. Rotation symmetries of other orders exist only in quasi-crystalline structures. In addition to the rotation isomotries, there exist inversions, plane reflections and improper rotations (an n-fold rotation followed by inversion about a point). Improper rotations are termed “rotoinversions” or

23 “rotoreﬂections” and are indicated by their order with macron (e.g., a threefold rotoinversion would be 3).¯ The geometric structures related to the possible point symmetries of crystals are line, plane, equilateral triangle, isosceles right triangle and 120-30-30 triangle in connection with n 1, 2, 3, 4, 6 respectively.

Figure 4.2: The geometric objects related to the crystallographic symmetry operations in three dimensions: equilateral triangle, isosceles right triangle, 120-30-30 triangle, plane, and line. Small solid points indicate the intersection of an axis of symmetry with the plane deﬁned by three points.

We make use of the related geometric structures by noting the location of possible symmetry axes. The fact that at most three sites are necessary for the identiﬁcation of any symmetry operation is crucial to the characterization of large unit cells.

4.3 Unit cell conventionalizing

In order to compute the crystallographic origin in short time, performing an appropriate standardization of the unit cell is critical. Use of an arbitrary unit cell can result in ignoring symmetries of the crystal unless a large supercell is constructed, over which the symmetry search must be performed. The choice of unit cell is not unique and any number of criteria may be applied to select a mathematically equivalent lattice basis. A commonly sited example of lattice basis ambiguity (and frequent

24 →n → + + n

(a) (b)

Figure 4.3: Cartoon diagram illustrating the method used to find point group operations in an origin-specific manner. a) shows a mirror reflection requiring selection of two atoms. The plane normal is defined by the line between the two points and intersects at the midpoint. b) shows a three-fold rotation about an axes. The axes is uniquely determined by the normal to the plane intersecting at the center of mass. This is extended to other crystallographic point operations (n 1, 2, 3, 4, 6) by reduction of co-planer atoms. source of confusion) is found in the rhombohedral lattice system. A rhombohedral lattice can be viewed equivalently as an hexagonal lattice (see Fig. 4.5). When confronted with the problem of lattice unit cell choice, lattice reduction is often the chosen method. A reduced lattice basis refers to one with the smallest possible volume. The minimal volume condition does not imply a unique cell, however (see Fig 4.4. While there are a number of difficulties to the reduced-lattice approach (including implementing a stable algorithm) (Wübben et al.(2011); Grosse-Kunstleve et al.(2004)), the reduced cell is important in a number of crystallographic applications (e.g, crystallographic classification and categorization). In the 1920s, Niggli,

25 drawing on the work of Eisenstein, deﬁned algebraic conditions leading to a unique, reduced lattice basis (Niggli(1928)). The so-called Niggli reduced cell is frequently used; a Niggli reduction algorithm is implemented in AFLOW (Chap.3).

Figure 4.4: Three choices of reduced basis for a square lattice. Each has the same, minimal volume.

Reduction procedures such as the Niggli approach cited above rely on metric relations (i.e., relations restricting angles between lattice vectors and their lengths) to determine the unit cell. In contrast to a metric approach, it is also possible to use the symmetry properties of the lattice to determine the unit cell. Use of symmetry in this way leads to the conventional unit cell. This has the advantage of being consistent with the overall goal of the symmetry algorithm—determination of point group and space group. The method for conventionalizing used here conforms with that used in the ITC (Hahn(2002)) so that standard cells for comparison with the ITC are generated. This standard takes advantage of the symmetry properties of the underlying lattice,

26 together with a set of standardization rules. The unique choice of lattice basis deﬁned in this way is termed the conventional crystallographic basis and the generated cell is called the conventional crystallographic cell.

Figure 4.5: Ambiguity in the choice of unit cell is shown. A rhombohedral unit cell (solid lattice vectors) or the larger hexagonal cell (dashed lattice vectors) generate the same lattice.

The symmetry properties that are used to define the conventional unit cell comprise the seven holohedries in three dimensions (the point groups of the lattices) and the translation groups. One approach is to find the reduced lattice before proceed- ing to compute symmetry relations. This is due to the fact that mathematically equivalent representations in terms of a superlattice may be given. Once the reduction is performed, the lattice symmetry can be found. However, this procedure may result in the lattice having higher symmetry than the point group of the crystal. The classification in the ITC (Hahn(2002)) is based on so-called Bravais flocks (i.e., Bravais lattice types) that are determined by examining the point and translation group of the crystal. These are uniquely assigned to Bravais classes (and hence to the seven lattice systems) by finding the smallest possible compatible supergroup from the Bravais classes. To find the conventional cell, the basis vectors are chosen along high-symmetry directions with consideration given to metric properties in some cases

27 (i.e., in the monoclinic and triclinic classes), according to rules deﬁned by Donnay (1943). The rules are summarized by the seven lattice systems. In each case, the conventional form of the associated Bravais lattice(s) are given (Fig 4.6-4.11).

4.3.1 Cubic

The basis vectors are chosen parallel to three equivalent fourfold axes.

cP cI cF

Figure 4.6: The Bravais lattices for the cubic lattice system.

4.3.2 Hexagonal

In addition to the use of high-symmetry directions to determine the lattice basis, hexagonal lattices are restrained by metric conditions: The two axes perpendicular to the sixfold axis must be the shortest lattice vectors that span an angle of 1200. The direction labeled c is chosen parallel to the sixfold axes.

4.3.3 Rhombohedral

In the edition of the ITC used here (Hahn(2002)), rhombohedral lattices are deﬁned in two ways, using so-called hexagonal axes and rhombohedral axes. In the hexagonal version, the direction labeled c is taken along the threefold axis, and the additional two lattice basis vectors are taken along the equivalent twofold axes. The two non-c basis vectors span an angle of 1200 and are oriented using the obverse setting; that

28 hP hR

Figure 4.7: The Bravais lattices for the hexagonal (hP) and rhombohedral (hR) lattice systems. is, lattice points will occur at p2{3, 1{3, 1{3q and p1{3, 2{3, 2{3q. In the rhombohedral version, a primitive cell is selected with the following condition: The three lattice basis vectors are the shortest symmetrically equivalent (with respect to the threefold axis), non co-planar lattice vectors.

4.3.4 Tetragonal

The tetragonal lattice is determined by conditions analogous to hexagonal lattices. The vector labeled c is taken along the fourfold axis. The additional two lattice basis vectors are the shortest lattice vectors along perpendicular twofold axes.

tP tI

Figure 4.8: The Bravais lattices for the tetragonal lattice system.

29 4.3.5 Orthorhombic

Orthorhombic lattices contain three twofold axes. These are parallel to the three lattice basis vectors.

oP oI oC oF

Figure 4.9: The Bravais lattices for the orthorhombic lattice system.

4.3.6 Monoclinic

In the ITC two settings are given for monoclinic lattices. In the ﬁrst, the singular symmetry direction is labeled b. The basis vectors labeled a and c are chosen in the plane perpendicular to b and so the following metric condition is satisﬁed:

0 ¤ ¡2a ¤ c ¤ minpa2, c2q. (4.1)

In words, this means that a and c are the shortest lattice vectors in their plane and span a non-acute angle. The second setting is analogous. The symmetry direction is now labeled c. The vectors a and b satisfy the same condition given for a and c above. Additional consideration may be given to the selection of a and c (a and b ). Three cell choices are given in the ITC. If the basis vectors are chosen so c = e, a = f (e and f refer to the glide vector and projection of the centering vector in the monoclinic plane, respectively) with b normal to the plane deﬁned by c and a, the unit cell corresponds to the ﬁrst cell choice. However, ignoring such restrictions on the labeling other than the selection of shortest vectors in the plane normal to the

30 symmetry direction will result in one of the listed cell choices; the choice may simply not be the ﬁrst listed.

mP mC

Figure 4.10: The Bravais lattices for the monoclinic lattice system.

4.3.7 Triclinic and the reduced cell

The triclinic lattice is described using a reduced cell (Niggli(1928); Eisenstein(1851); Santoro and Mighell(1970); Hahn(2002)) due to the absence of any relevant symmetry characteristics. The metric conditions related to the determination of the reduced cell are as follows.

Right-handedness

The cell must be chosen to be right-handed: a ¢ b c.

Type

The type of the cell is determined by the sign of

T pa ¤ bqpb ¤ cqpc ¤ aq. (4.2)

Positive T corresponds to type I. If T ¤ 0, the cell is type II. Type-I cell conditions Main conditions:

1 1 a ¤ b ¤ c ¤ c; |b ¤ c| ¤ b ¤ b; |a ¤ c| ¤ a ¤ a (4.3) 2 2 31 b ¤ c ¡ 0 (4.4)

Special conditions: If a ¤ a b ¤ b then b ¤ c ¤ a ¤ c (4.5)

If b ¤ b c ¤ c then a ¤ c ¤ a ¤ b (4.6)

1 If b ¤ c b ¤ b then a ¤ b ¤ 2a ¤ c (4.7) 2 1 If a ¤ c a ¤ a then a ¤ b ¤ 2b ¤ c (4.8) 2 1 If a ¤ b a ¤ a then a ¤ c ¤ 2b ¤ c (4.9) 2 Type-II cell conditions Main conditions:

1 1 a ¤ b ¤ c ¤ c; |b ¤ c| ¤ b ¤ b; |a ¤ c| ¤ a ¤ a (4.10) 2 2

1 p|b ¤ c| |a ¤ c| |a ¤ b|q ¤ pa ¤ a b ¤ bq (4.11) 2

b ¤ c ¤ 0; a ¤ c ¤ 0; a ¤ b ¤ 0 (4.12)

Special conditions: If a ¤ a b ¤ b then |b ¤ c| ¤ |a ¤ c| (4.13)

If b ¤ b c ¤ c then |b ¤ c| ¤ |a ¤ b| (4.14)

1 If |b ¤ c| b ¤ b then a ¤ b 0 (4.15) 2 1 If |a ¤ c| a ¤ a then a ¤ b 0 (4.16) 2 1 If |a ¤ b| a ¤ a then a ¤ c 0 (4.17) 2 1 If p|b ¤ c| |a ¤ c| |a ¤ b| pa ¤ a b ¤ bq then a ¤ a ¤ 2|a ¤ c| a ¤ b (4.18) 2

32 aP

Figure 4.11: The Bravais lattice for the triclinic lattice system.

4.4 Crystal point group and the standard crystallographic origin

The standard for origin selection as set out in the ITC is based on the choice of high site-symmetry points or inversion centers as origin (Table 4.1). For an arbitrary crystallographic structure, the symmetry search will reveal the locations of point group operations. Once this is known for symmorphic space groups, determination of the highest site-symmetry point or inversion point is relatively straightforward. For the centrosymmetric space groups, at least one origin choice reported in the ITC has the inversion point as the origin. The appropriate transformation in the centrosymmetric space groups is, thus, simple. For noncentrosymmetric space groups, ﬁnding the highest site-symmetry point is done by calculating intersections of rotation axes and planes. The point of highest degeneracy is then selected. Thus, the site symmetry of the selected origin is either equal to the point group of the crystal or one of its subgroups. For nonsymmorphic space groups, additional searches must be done to identify the point symmetry-related space group operations.

Proper rotations, inversions, and reﬂections

The conventional unit cell does not impose a choice of origin. Therefore, to construct a cell amenable to comparison with the ITC, the appropriate origin shift must be determined. To do so requires the full point group of the crystal, and in some cases, additional space group operations. These are computed in much the same way the

33 point group of the lattice is computed: A search for the possible crystallographic symmetry-associated geometric structures over co-planar atoms (sets of three) is conducted for point operations and projected sets of atoms are used for space group operations. As examples, consider a search for mirror planes and threefold rotation axes (Fig. 4.3). Mirror planes must exist at the midpoint of two atoms with normal defined by the connecting line. Therefore, a search over all unique pairs of atoms with the appropriately defined candidate mirror planes will yield the plane symmetries. Similarly, a threefold rotation axes must exist at the center point of three atoms forming an equilateral triangle. The rotation axes must be normal to the plane defined by the three atoms. Searching all unique sets of three atoms, checking for an equidistant center of mass and then applying the candidate rotations to the unit cell will yield the threefold symmetries. The additional crystallographic point symmetries can be found using the same method with minor changes: Inversion points must exist at a midpoint between atoms. For fourfold symmetry, the triangle with atoms as vertices must be right- isosceles; the rotation axis exists at the midpoint of the base normal to the plane of atoms. Sixfold symmetry requires the existence of a 120 ¡ 30 ¡ 30 triangle with a rotation axes located along the line between the midpoint of the base and the obtuse angle, displaced by twice the height (Fig. 4.2). The direction is again perpendicular to the plane of atoms. Application of these rules provides the inversion points, axes of rotation, and planes of reflection with respect to the current origin.

Rotation-inversion symmetry

Rotation-inversion (rotoinversion) symmetries are compound symmetry operations that combine proper rotations with inversion about a point on the rotation axis. The possible rotoinversions in crystallography are one-fold, twofold, threefold, fourfold, and sixfold. In reference to symmetry determination of arbitrary crystallographic

34 configurations, it is convenient that all but one of the possible rotoinversion symmetries can be described by combinations of proper rotations, reflections, etc. One-fold rotoinversion is simply inversion. Twofold rotoinversion is equivalent to a reflection symmetry with mirror plane orthogonal to the rotation axis. Threefold rotoinversion occurs in crystals with threefold rotation and inversion (the inversion is used as the standard origin). Sixfold rotoinversion can be identified by a threefold rotation and a mirror symmetry orthogonal to the rotation axis. Fourfold rotoinversion cannot be equivalently described by simpler operations. In order to find the fourfold rotoinversion axes, similar to the other operations, a geometric approach be taken. First, however, note that the point groups that admit fourfold rotoinversion symmetry are in either the tetragonal or cubic crystal system. The primary axes are thus orthogonal, which restricts the possible rotoinversion axes to r100s, r010s, or r001s. Stereographic projections along these axes will reveal potential rotoinversion axes. In particular, the existence of four equidistant points, specify the location of the candidate axis (at the center point). Although unnecessary for the scheme described here, this approach can also be used to directly find threefold and sixfold rotoinversion axes (Fig. 4.12).

Full Space Group Symmetry

Although the crystal class uniquely deﬁnes the standard origin, this cannot be known a priori without some space group related information. This unfortunate fact is due to the way in which crystals are categorized by class which in turn is used in the selection of origin. Glide planes and screw axes along primary directions are associated to the point operations derived by excluding the translation part. For example, a c glide along 0, y, z becomes a mirror plane along 0, y, z. There are several ways to identify space groups through the determination of the full space group operations (e.g., glide, screw etc). To ﬁnd the additional operations, we extend the

35 _ + _ + 3 4

_ + 6

Figure 4.12: Threefold, fourfold, and sixfold rotoinversions are illustrated. The geometric shapes deﬁned by stereographic projections along possible axes are used to identify possible rotoinversion axes. symmetry determination routine to identify space group operations. This is done by checking for an expanded set of geometric objects (see Fig. 4.13) that now includes triangles deﬁned by the 11 possible screw operations (21, 31, 41, 42, 61, 62, 63, 32,

43, 64, 65) and glide planes (a, b, c, d, n), or by searching for relations among projected atoms along primary directions. In both these cases, the lattice vectors play an important role in uncovering the possible geometries, as the limited possible glide and screw translation vectors are given with reference to the principal lattice directions. Of course, it is advantageous to avoid the additional complexity of this approach unless knowledge of the full symmetry is needed. For space groups in the

36 noncentrosymmetric point groups 1,2, m, 4,¯ 6, 6,¯ 6mm, 6¯m2, 23, 43¯ m all or all but one of the space groups (with the exception of m for which a centering diﬀerence allows diﬀerentiation) contain only point operations as site symmetry of the origin. Nevertheless, some of these space groups may not actually contain the point group symmetry operations because of the conversion from glide planes and screw axes to point operations described above.

nf _f 2 n n,d,a,b,c

α _f n

(a) (b) ?

Figure 4.13: Additional geometric objects comprising the full set of symmetry operations. (a) Screw axes (nf ) create triangles with non-simple (but determinable) angle and side relations. The possible geometries can be derived using the projected object (dashed triangle) geometry and the screw displacement (h and 2h). Because the screw axes are parallel to lattice directions, a finite set of plane normal-screw axis angles α exists. (b) Glide planes exist for a finite set of translations (n,d,a,b,c) defined in terms of the lattice vectors. Applying these translations and checking if an atom exists along the normal direction reveals the glide planes.

37 Table 4.1: The standard origins of the 230 space groups with point group and crystal system from ITC (Hahn(2002)). The notation used borrows elements from the Hermann-Mauguin scheme: Rotations, rotoinversions, reflections, and screw rotations are defined equivalently (e.g., for screw rotation, 41 denotes a fourfold axis followed by a translation of 1{4 of the lattice vector parallel to the axis.) The symmetry symbols listed for each space group comprise those sufficient for defining the origin. Clusters of symmetry operations (e.g., 222) indicate the intersection of the constituent operations. In some cases more than are necessary for specification of the origin are listed for completeness (e.g., two twofold axes unambiguously define an origin but three may exist). Space groups with (*) are those for which the point group operations are not used alone to determine the standard origin. Boxed point group symbols are the centrosymmetric groups—in these cases, an inversion center is always used as the origin.

Cryst. Syst. Pnt. Group Standard Origin (SG number) 1 arbitrary(1) Triclinic (1-2) 1¯ 1¯(2) 2 2(3), 21(4*), 2(5) Monoclinic (3-15) m m(6), g(7*), m(8), g(9*,MB) 2{m 1¯(10-15)

222 222(16), 212(17*), 2p21 ¢ 21q(18*), Orthorhombic (16-74) Mp21, 21, 21q(19*), 221(20*), 222(21-23), Mp2, 2, 2q(24) mm2 mm2(25), mg21(26*), gg2(27*), g2(28*), g2(30*), mg(31*), 2(32), 21(33*), 2(34), mm2(35), mg21(36*), gg2(37*), mm2(38), gg2(39*), g2(40*), g2(41*), mm2(42), 2(43), mm2(44), gg2(45*), gg2(46*) mmm 1¯(47-74) 4 4(75), 41(76*), 2(77), 43(78*), 4(79), 2(80) 4¯ 4¯(81-82) 4{m 1¯(83-88)

422 422(89), 222(90), 241(91*), 221(92*), 222(93), 222(94), 243(95*), 221(96*), Tetragonal (75-142) 422(97), 222(98) 4mm 4mm(99), 4pk 2qg(100*), 2mm(101- 102),4pk 2qgg(103*), 4pk 2qg(104*), 2mm(105), 2(106), 4mm(107), 4pk 2qgg(108), 2mm(109), 2g(110) 42¯ m 42¯ m(111), 4¯g(112*-114*),4¯m2(115), 4¯g(116), 42¯ 1(117*), 4¯(118), 4¯m2(119), 4¯g21(120*), 42¯ m(121), 4¯(122) 4{mmm 1¯(123-142)

38 3 3(143), 31(144*), 33(145*), 3(146) 3¯ 1¯(147-148) Trigonal (143-167) 32 32(149), 32(150), 231(151*-152*), 232(153*-154*), 32(155) 3m 3m(156-157), 3g(158*-159*), 3m(160), 3g(161*) 3¯m 1¯(162-167) 6 6(168), 61(169), 65(170*), 2(171-172), 3(173) 6¯ 6¯(174) Hexagonal (168-194) 6{m 1¯(175-176)

622 622(177), 261(178*), 265(179*), 222(180- 181), 32(182) 6mm 6mm(183), 6gg(184*), 3m(185-186) 6¯m2 6¯m2(187), 32(188), 62¯ m(189), 32(190), 6{mmm 1¯(191-194)

23 23(195-197), 3Mp21, 21, 21q(198*), 3Mp2, 2, 2q(199) m3¯ 1¯(200-206) 432 432(207), 23(208), 432(209), 23(210), Cubic (195-230) 432(211), 3Mp43, 43, 43q (212*), 3Mp41, 41, 41q (213*-214*) 43¯ m 43¯ m (215-217), 23 (218), 23 (219), 3Mp4¯, 4¯, 4¯q (220) m3¯m 1¯ (221-230)

4.5 Symmetry classes and comparison routine

The search for point symmetries as well as comparison with the ITC space group tables is greatly facilitated by the construction of symmetry classes. The comparison routine itself makes use of a library of symmetry operations containing the information from the 230 space groups. Due to the large number of operations and space groups in question, the library does not contain explicit representations of the symmetry operations. The ITC notation is converted into Seitz matrices via member

39 functions of the various symmetry classes. An arbitrary space group, once converted into conventional form with standard origin, can be compared to the space group library through application of constituent symmetry operations. The crystal system and point group restrict the search to at most 28 space groups (this occurs in the orthorhombic crystal system with point group mmm). The symmetry operations of candidate space groups are applied to the atom positions of the crystal being studied. A crystal with n atoms will result in at most n groups of equivalent atoms generated by application of the space group symmetry.

i) Select candidate space group ii) Select atom from atomic basis set (this may be done without preference) iii) Apply candidate space group symmetry IF extraneous atom is created go to (i) ELSE compare generated set with complete basis set and discard matches. iv) Check basis set size. If nonzero go to (ii). If zero, end.

4.5.1 Improper rotations and the RotoInversion class

Rotoinversion (rotoreflection1) is the most general of the point operations (Fig. 4.14). Pure rotations, reflections, and inversions can be thought of as subsets of a rotoinversion operator. Thus, the RotoInversion class extends to all the point group operations. Rotation angle, rotation axis, and inversion point uniqely define the rotoinversion. In the RotoInversion class, these are implemented as private member functions.

1 Rotation about an axis followed by inversion about a point on the axes is equivalent to rotation about the axis followed by reﬂection in a plane perpendicular to the axis.

40 The user fills the class variables via the void get_roto_inversion(string s) function. The string s is a standard format for rotoinversions (similar to that used in the ITC). For example, a threefold counterclockwise rotation about an axes defined parametrically, x, x, 0, and an inversion point triple, p0, 0, 0q, is described by 3 x x 0; 0 0 0. The operation is applied using the matrix representations of the rotation and inversion operations. The rotation matrix is uniquely defined by the specification of the angle α and the rotation axis. The axis is defined by a direction vector pu, v, wq and a point pa, b, cq on the axis. Using this notation, the matrix is,

u2 x cospαq uvp1 ¡ cospαqq ¡ w sinpαq uvp1 ¡ cospαqq w sinpαq v2 y cospαq (4.19) uwp1 ¡ cospαqq ¡ v sinpαq vwp1 ¡ cospαqq u sinpαq 0 0

uwp1 ¡ cospαqq v sinpαq pax ¡ upbv cwqqp1 ¡ cospαqq pbw ¡ cvq sinpαq vwp1 ¡ cospαqq ¡ u sinpαq pby ¡ vpau cwqqp1 ¡ cospαqq pcu ¡ awq sinpαq w2 z cospαq pcz ¡ wpau bvqqp1 ¡ cospαqq pav ¡ buq sinpαq 0 1

To slightly simplify the unwieldy form, a few substitutions have been made:

v2 w2 x (4.20)

u2 w2 y (4.21)

u2 v2 z (4.22)

Following application of the 4x4 rotation matrix—a general rotation about a line is an aﬃne transformation and so requires an additional row and column to account

41 for the “translation part”—the inversion about a point px0, y0, z0q is given by, ¡1 0 0 2x0 0 ¡1 0 2y0 (4.23) 0 0 ¡1 2z0 0 0 0 1

In each of these cases, the operand is a vector with four components. The ﬁrst three are the standard coordinates of a point and the fourth component is always unity.

Rotation Reﬂection

Figure 4.14: Cartoon illustrating a rotation-reﬂection/rotation-inversion operation.

4.5.2 Glide class

The glide operation is defined as a reflection followed by a translation (or equivalently, the reverse). The Glide class in filled using the void get_glide(string str) function. The input string is in a form similar to that used in the ITC, with abbreviated notation for axial, diagonal, and diamond glide operations given by a, b, c, n, d. Because the reflection plane may not pass through the origin, the operation includes reflection and translation parts. The reflection component uses the plane

42 Glide plane

Reﬂection

Translation

Glide vector

Figure 4.15: Cartoon illustrating a glide operation.

normal (n1, n2, n3) and is represented by the 3x3 matrix, 2 2 ¡ ¡ n2 n3 n1n2 n1n3 ¡ 2 2 ¡ n1n2 n1 n3 n3n2 . (4.24) ¡ ¡ 2 2 n3n1 n3n2 n1 n2

The full operator is AP~ d~n |~n ¤ P~ ¡ d|~n T,~ (4.25) where d is deﬁned by the standard form of the plane, ax by cz d 0 (the triple pa, b, cq is parallel to the plane normal). Care must be given to the orientation of the selected normal. The normal should direct away from the operand, P . The Glide class corrects an incorrectly selected normal. T is the glide translation.

43 4.5.3 Screw class

A screw axis (Fig. 4.16) is a compound symmetry operations consisting of a rotation followed by a translation parallel to the rotation axis. Screw axes are implemented in the symmetry algorithm code with the Screw class and the member function void get_screw(string str). The input string follows the standard in the ITC. For example, a twofold screw axis about x, x, 1{4 with translation p1{2, 1{2, 0q is described by the string 2 p1{2 1{2 0q x x 1{4. The rotation matrix is exactly that described for the rotation-inversion class (4.19).

Rotation Screw axis

Translation Screw vector

Figure 4.16: Cartoon illustrating a screw operation.

4.5.4 Operator overloading in the symmetry classes

The ease and clarity with which the various symmetry classes can be used in the developed routines and algorithms greatly depends on the simplicity of their application to point triplets. Although the use of the classes here is solely for the determination the crystallographic space group (via a particular algorithmic approach), their

44 usefulness extends beyond this. They can, for example, be used for more general symmetry applications or for alternative space group determination algorithms (e.g., by complete determination of the space group symmetries). To facilitate their use and portability the symmetry types exist as distinct classes and are applied to a point by overloading the * operator. This allows the user to work with the (often complex) symmetry classes in a truly object-oriented fashion with limited concern for the nature of internal data types and functions (AFLOW matrix class, C++ vector functions, etc).

45 Appendix A

Stereographic projections of the 32 point groups including crystal system and the Bravais lattices

Triclinic System c

a 1 b

46 Monoclinic System c α = β = 90° c

a a b b 2

47 ... mono. 2/m cont.

Orthorhombic System α = β = γ = 90° c c c c

a b a b a b a b 222

48 ...ortho. mm2 cont.

mmm

49 ...ortho. cont. 4/m

422

4mm

50 Tetragonal System c c α = β = γ = 90° |a| = |b|

a b a b

51 ...tetr. 42m cont.

4m2

4/mmm

52 Trigonal System α β γ (Hexagonal Axes) = = 90°, = 120° c c |a| = |b|

a b a b 3

53 ...trig. cont. 321

3m1

54 Hexagonal System c

α = β = 90°, γ = 120° |a| = |b| a b

...hex. cont. 6/m

622

55 ...hex. 6mm cont.

6m2

6/mmm

56 Cubic System α = β = γ = 90° c c |a| = |b| = |c|c

a b a b a b 23

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