 1. Introduction
 2. Crystallographic databases and programs hosted at the Bilbao Crystallographic Server
 3. Mercury and the CSD
 4. Threedimensional structure visualization with VESTA
 5. Webbased visualization of crystal structure and symmetry using Jmol
 6. Application of the four tools for the structural analysis of thiourea at different temperatures and pressures
 7. Conclusion
 Supporting information
 References
 1. Introduction
 2. Crystallographic databases and programs hosted at the Bilbao Crystallographic Server
 3. Mercury and the CSD
 4. Threedimensional structure visualization with VESTA
 5. Webbased visualization of crystal structure and symmetry using Jmol
 6. Application of the four tools for the structural analysis of thiourea at different temperatures and pressures
 7. Conclusion
 Supporting information
 References
teaching and education
Free tools for
handling and visualization^{a}Institute of Applied Geosciences, Karlsruhe Institute of Technology, Karlsruhe, Germany, ^{b}Departamento de Física, Universidad del País Vasco UPV/EHU, Spain, ^{c}Cambridge Crystallographic Data Centre, United Kingdom, ^{d}National Museum of Nature and Science, Tokyo, Japan, ^{e}Department of Chemistry, St Olaf College, Northfield, Minnesota, USA, and ^{f}CryssmatLab/DETEMA, Facultad de Química, Universidad de la República, Montevideo, Uruguay
^{*}Correspondence email: gemma.delaflor@kit.edu
This article is part of a collection of articles from the IUCr 2023 Congress in Melbourne, Australia, and commemorates the 75th anniversary of the IUCr.
Online courses and innovative teaching methods have triggered a trend in education, where the integration of multimedia, online resources and interactive tools is reshaping the view of both virtual and traditional classrooms. The use of interactive tools extends beyond the boundaries of the physical classroom, offering students the flexibility to access materials at their own speed and convenience and enhancing their learning experience. In the field of crystallography, there are a wide variety of free online resources such as web pages, interactive applets, databases and programs that can be implemented in fundamental crystallography courses for different academic levels and curricula. This paper discusses a variety of resources that can be helpful for VESTA and Jmol. The utility of these resources is explained and shown by several illustrative examples.
handling and visualization, discussing four specific resources in detail: the Bilbao Crystallographic Server, the Cambridge Structural Database,Keywords: symmetry visualization; VESTA; Jmol/JSmol; Cambridge Structural Database; Bilbao Crystallographic Server.
1. Introduction
The pandemic of 2020 caused a sudden change in every aspect of our lives, transforming how we interact and socialize with others, as well as revolutionizing our way of working. The field of education was not exempt from this change, as educators were forced to transition their classes from inperson to remote learning. This radical change was a great challenge at all academic levels, since lecturers had to find ways to not only keep the classes running during lockdown but also still inspire and engage their students. One of the problems that many lecturers had to face was to adapt the traditional resources used in physical classrooms to the virtual learning environment. Many of the tools and materials necessary to perform this transition were already developed, but lecturers were either not aware of these resources or not familiar with them. In the field of fundamental crystallography, for example, there exist a wide variety of free online resources such as web pages, interactive applets, databases and programs that can be used in virtual classrooms as well as inperson ones. Many of these are useful for
handling and visualization.Basic fundamental crystallography courses usually start by introducing the concept of symmetry in crystallography. Students not only have to gain insight into the different types of symmetry exhibited by macroscopic crystals but also need to acquire the skills to identify them and determine their corresponding crystallographic interactive PDF files with embedded 3D models (Arribas et al., 2014), inspired by classical wooden crystallographic models, visualize for each of the 32 crystallographic point groups interactive threedimensional polyhedra (used to represent idealized crystals), their corresponding symmetry elements and the This material is freely available online (https://github.com/LluisCasas/GSP) and the resource can be very helpful for students to gain a solid grasp of the concepts of symmetry and point groups. With a more atombased approach, Jmol (Hanson, 2013) allows the visualization of threedimensional representations of molecules and crystal structures that can also be used in a classroom to analyse the symmetry and determine the of a given molecule (https://chemapps.stolaf.edu/jmol/jsmol/jpge/). In addition, the program offers the capability of visually demonstrating the action of the symmetry operations in the molecule, helping develop students' understanding of these concepts.
Traditionally, these competences were acquired by analysing the symmetry of classical crystallographic wooden models or molecular models. In an online classroom, however, it is necessary to replace the physical models used in a traditional classroom by virtual ones. TheOnce students have a solid understanding of point groups, the next step is to introduce the concept of translational symmetry followed by the definition of space groups. This knowledge is essential to describe the atomic structure of threedimensional crystals. Besides Jmol, the freely available program VESTA (https://jpminerals.org/vesta/en/; Momma & Izumi, 2011) can also be used for visualizing threedimensional crystal structures. Apart from crystal structural models, this program can be applied to visualize lattice planes and crystal morphologies. The programs Jmol and VESTA go beyond the simple visualization of crystal structures from uploaded crystallographic files. They allow the user to construct crystal structural models step by step, first building the by specifying the and lattice parameters, and then adding the atoms into the empty This feature is valuable for students since it provides a comprehensive learning experience.
The use and understanding of the crystallographic data on space groups in International Tables for Crystallography (2016), Vol. A, SpaceGroup Symmetry (henceforth abbreviated as ITA), is also an important part of the curriculum for graduate students. A significant number of students, however, lack access to either the printed or online editions of ITA. Fortunately, there are free resources on the web that contain essential information about space groups, such as the Bilbao Crystallographic Server (https://www.cryst.ehu.es/; Aroyo et al., 2011; Tasci et al., 2012; hereafter referred to as BCS) and the Hypertext Book of Crystallographic Space Group Diagrams and Tables (http://img.chem.ucl.ac.uk/sgp/mainmenu.htm). In addition to granting access to their extensive database, the BCS offers more complex programs to study problems in crystallography, such as the comparison of structures, data transformation or phase transitions – topics that are often part of the graduatelevel curriculum. Recent developments in Jmol allow the visualization of spacegroup symmetry (https://djohnston66.gitlab.io/sgexplorer/). The availability of such databases and programs can enhance these students' learning experience and capabilities.
Crystal structure databases also play an important role in the teaching of crystallography, providing realworld examples that allow students to apply theoretical concepts in a practical context. This practical approach can improve comprehension and reinforce their grasp of crystallographic principles. By exploring actual crystal structures, students gain insights into symmetry, atomic arrangements and other key principles. This allows a visual understanding of the concepts introduced in their studies. Depending on the discipline, there are many online freeofcharge databases, including the Protein Data Bank (https://www.rcsb.org/), Crystallography Open Database (Gražulis et al., 2009; https://www.crystallography.net/cod/), AFLOW Encyclopaedia of Prototypes (Mehl et al., 2017; https://aflowlib.org/prototypeencyclopedia/), Magnetic Structure Database (Gallego et al. 2016a,b; https://www.cryst.ehu.es/magndata/), Bilbao Incommensurate Database (https://www.cryst.ehu.eus/bincstrdb/), and the joint Cambridge Crystallographic Data Centre (CCDC) and FIZ Karlsruhe Access Structures Service (https://www.ccdc.cam.ac.uk/structures) which provides access to data sets in the Cambridge Structural Database (CSD; Groom et al., 2016) and the Inorganic Database (ICSD; Zagorac et al., 2019). The CCDC also offers the Mercury visualization tool (Macrae et al., 2020), allowing for the building of structural models based on symmetryrelated units, drawing symmetry elements within the and click calculation of interatomic distances and angles.
In this paper we describe four freely available tools that can be used in fundamental crystallography courses for bachelor's and master's curricula focused on symmetry handling and visualization: the BCS, the CSD and its associated Mercury program, VESTA, and Jmol. The utility of these tools will be shown using practical examples.
2. Crystallographic databases and programs hosted at the Bilbao Crystallographic Server
Operating since 1998, the BCS is a free web server that grants access to crystallographic databases and programs to resolve different types of problems related to crystallography, crystal chemistry, solidstate physics and materials science. The programs available on the server do not require a local installation and their use is free of charge. The server is built on a core of databases that includes data from ITA, International Tables for Crystallography, Vol. A1, Symmetry Relations Between Space Groups (2010a) (henceforth abbreviated as ITA1) and International Tables for Crystallography, Vol. E, Subperiodic Groups (2010b). Therefore, the server gives access to crystallographic data on space, subperiodic, plane and point groups. In addition to this, there is access to the normalizers of space groups database which contains data on the Euclidean, chiralitypreserving Euclidean and affine normalizers of the space groups. A kvector database with Brillouinzone figures and classification tables of all the wavevectors for all 230 space groups (Aroyo et al., 2014) and 80 layer groups (de la Flor et al., 2021) is also available, together with the magnetic (PerezMato et al., 2015) and doublespacegroup databases (Elcoro et al., 2017). The BCS also hosts the magnetic and incommensurate structure databases.
The main aim of the BCS is to bring the potential of group theory to those users who are not experts in this matter but nevertheless want to use it in their research. In addition, the server is a valuable resource for teaching crystallography to students across different academic levels, from bachelor's to PhD. This dual functionality highlights its versatility in both research and educational domains.
2.1. Crystallographic spacegroup databases
Most of the spacegroup data compiled in ITA, such as the generators, general positions, Wyckoff positions, geometric interpretation of symmetry operations, and normalizers, are available in the Spacegroup symmetry section of the BCS. These data can usually be retrieved by specifying the spacegroup number; if the number is unknown, the can be selected from a table with the Hermann–Mauguin symbols. The databases and programs of the BCS use specific settings of space groups, termed standard or default settings. These are settings that coincide with the conventional ones.^{1} For space groups with more than one conventional description in ITA, the following settings are chosen as standard: (i) unique axis b setting, cell choice 1 for monoclinic groups, (ii) hexagonal axes setting for rhombohedral groups, and (iii) origin choice 2 description for the centrosymmetric groups listed with respect to two origins in ITA (i.e. when the origin is at a centre of inversion). The BCS hosts data for all 230 space groups in their default settings, and in addition the crystallographic data of the settings for the monoclinic and orthorhombic space groups listed in Table 1.5.4.4 of ITA (henceforth abbreviated as ITA settings^{2}). Some of the programs available in the BCS and described in the following sections can also be found in the Symmetry Database (de la Flor et al., 2023; https://symmdb.iucr.org/), available only to subscribers of the online version of International Tables for Crystallography (https://it.iucr.org/).
2.1.1. Generators and general position of space groups
The generators and general position of space groups are shown by the program GENPOS. The generators and general position entries are specified by their coordinate triplets, the matrix–column representations of the corresponding symmetry operations and their geometric interpretations.
(i) The list of coordinate triplets (x, y, z) reproduces the data of the General position blocks of space groups found in ITA. The coordinate triplets may also be interpreted as shorthand descriptions of the matrix presentation of the corresponding symmetry operations.
(ii) The matrix–column presentation of symmetry operations is defined as follows. With reference to a coordinate system consisting of an origin O and a basis (a_{1}, a_{2}, a_{3}), the symmetry operations of space groups are described by (3 × 4) matrix–column pairs (W, w).
(iii) The geometric interpretation of symmetry operations is given (a) following the conventions in ITA [including the symbol of the its glide or screw component (if relevant), and the location of the related geometric element] and (b) using the Seitz notation [see Glazer et al. (2014)].
For space groups with a shows the general position for the (No. 82) in the standard setting. In this particular example, there are two blocks of general position, one for the `(0,0,0)+ set' and the other for the `(1/2,1/2,1/2)+ set'.
the general position is shown in blocks. The number of blocks is equal to the multiplicity of the centred cell. Fig. 1The program GENPOS lists the generators and general position entries of space groups in the standard setting (option Standard/Default Setting) as well as in one of the ITA settings. On clicking on the option ITA Settings, a list of the ITA settings of the chosen spacegroup settings is shown; the corresponding data can be calculated with respect to one of these settings by choosing it directly from this list. The general position of the selected setting is listed in the same way as in Fig. 1, together with the corresponding listing for the standard setting of the space group.
In addition, the program can produce the data in any other nonstandard setting via the option UserDefined Setting, provided the transformation relating the origin and basis of the nonstandard setting to those of the standard setting is specified. The coordinate transformation is described by a matrix–column pair (P, p) and consists of two parts: (i) a linear part P defined by a (3 × 3) matrix, which describes the change in direction and/or length of the basis vectors (a_{1}, a_{2}, a_{3})_{nonstand} = (a_{1}, a_{2}, a_{3})_{stand}P [where (a_{1}, a_{2}, a_{3})_{nonstand} is the basis of the nonstandard setting and (a_{1}, a_{2}, a_{3})_{stand} is the basis of the standard setting], and (ii) an origin shift p = (p_{1}, p_{2}, p_{3})^{T} defined as a (3 × 1) column matrix, whose coefficients describe the position of the nonstandard origin with respect to the standard one. The data of the matrix–column pair
are often written in the following concise form:
The program GENPOS allows the data to be transformed to any setting.
2.1.2. Wyckoff positions of space groups
The program WYCKPOS provides a list of Wyckoff positions, in the standard setting (option Standard/Default Setting) as well as in different settings, for a designated The listing follows that of ITA (cf. Fig. 2): the Wyckoffposition block starts with the generalposition data at the top, followed downwards by various special Wyckoff positions with decreasing multiplicity and increasing The data for each include (i) the multiplicity, i.e. the number of equivalent positions in the conventional (ii) the Wyckoff letter, which is an alphabetical label; (iii) the group; and (iv) a set of coordinate triplets of the equivalent points in the shown under the column `Coordinates'. For space groups with a the centring translations are listed above the coordinate triplets. The isomorphic to the sitesymmetry group is indicated by an oriented symbol (column three of Fig. 2), which is a variation of the Hermann–Mauguin pointgroup symbol that provides information about the orientation of the symmetry elements (for more details, see Section 2.1.3.12 of ITA). An explicit listing of the symmetry operations of the sitesymmetry group of a point is obtained by clicking directly on its coordinate triplet. Optionally, the symmetry operations of the sitesymmetry groups of an arbitrary point (specified by its coordinates but not necessarily within the unit cell) can be calculated using the auxiliary tool below the table.
Apart from the Standard/Default setting option, the program is also able to calculate the Wyckoff positions in different spacegroup settings, either by specifying the coordinate transformation (P, p) to a new basis (option UserDefined setting) or by selecting directly one of the ITA settings of the corresponding (option ITA Settings), thus enhancing and extending the data in ITA.
2.1.3. Geometric interpretation of matrix–column representations (W, w) of symmetry operations
Analysing the rotational part W and the translational part w of the matrix–column representation (W, w) of symmetry operations, it is possible to calculate their corresponding geometric interpretation if the coordinate system to which (W, w) refers is known [see Section 1.2.2.4. of ITA, and Stróż (2007)]. The geometric interpretation of symmetry operations can be independently calculated by the program SYMMETRY OPERATIONS of the BCS.
The input of this program consists of the spacegroup number or the Standard/Default Setting the program returns a table with the described by its coordinate triplet, matrix–column pair presentation, and geometric interpretation following the conventions of ITA and using the Seitz notation. As an example, Fig. 3 shows the geometric interpretation of the described with respect to the standard setting and belonging to the cubic calculated by the program SYMMETRY OPERATIONS. The program also allows these calculations in ITA settings, using the option ITA Settings, and in any other nonstandard setting by using the option UserDefined Setting.
and the matrix–column representation of the which can be given by its coordinate triplet or in matrix form. With the option  Figure 3 belonging to the cubic as determined by the program 
2.2. Group–subgroup relations of space groups
The data compiled in ITA1, such as the maximal of space groups, the minimal supergroups of space groups, and the relationships between the Wyckoff positions of space groups and their subgroups, are available in the sections Spacegroup symmetry and Group–subgroup relations of space groups of the BCS. The data in ITA1 on maximal subgroups of space groups of indices 2, 3 and 4 are extended to include the series of all isomorphic subgroups for indices up to 27 (125 for some cubic groups). In contrast to ITA1, where only spacegroup types of minimal supergroups are indicated, the database of the BCS contains individual information for each minimal including the transformation matrix that relates the conventional bases of the group and the minimal supergroup.
2.2.1. Maximal subgroups of space groups
The program MAXSUB gives access to the database of maximal subgroups of space groups. For a designated MAXSUB lists the maximal subgroups of indices up to index 9. This program first returns a table with the maximal types of the selected (lefthand table in Fig. 4). Each type is specified by its spacegroup number, Hermann–Mauguin symbol, index and type (t for translationengleiche and k for klassengleiche or isomorphic). The complete list of subgroups and their distribution in classes of conjugate subgroups is obtained by clicking on the link show.. . For example, the Pbca (No. 61) has three maximal translationengleiche subgroups of the type P2_{1}/c (No. 14) of index 2 (righthand table in Fig. 4). The transformation matrix–column pairs (P, p) that relate the standard bases of the and the group are also provided by the program. For certain applications, it is necessary to transform the generalposition representatives of the by the corresponding matrix–column pair (P, p)^{−1} to the coordinate system of the group by the option ChBasis. The relations between the Wyckoff positions of the and those of its maximal subgroups can also be calculated by MAXSUB, provided the option Show WP Splittings is selected in the input page of the program.
Every p, p^{2} or p^{3} (where p is a prime), and these are distributed into several series of maximal isomorphic subgroups. In most of the series, the Hermann–Mauguin symbol for each isomorphic is the same. However, if the belongs to one of the 11 pairs of enantiomorphic spacegroup types, the Hermann–Mauguin symbol of an isomorphic is either that of the group or that of its enantiomorphic pair. The program SERIES provides access to the database of maximal isomorphic subgroups. Apart from parametric descriptions of the series, the program provides individual listings for all maximal isomorphic subgroups. For each series, the Hermann–Mauguin symbol of the the restrictions on the parameters describing the series, and the transformation matrix relating the group and the are listed. There is also an option in the program that permits the online generation of maximal isomorphic subgroups of any allowed index. The format and content of the data are similar to those of the program MAXSUB.
has an infinite number of maximal isomorphic subgroups of indices of2.2.2. Splitting of the Wyckoff positions of space groups
The program WYCKSPLIT (Kroumova et al., 1998), available in the section Group–subgroup relations of space groups of the BCS, is designed to compute the relationships between the Wyckoff positions of a given and one of its subgroups. These relationships may involve splittings of the Wyckoff positions, reductions in the site symmetries or both. The program requires as input (i) the group and spacegroup numbers, (ii) the transformation matrix–column pair (P, p) that relates the basis of the group to that of the and (iii) the Wyckoff positions of the group to be split. As output it returns the relationships between the Wyckoff positions of the group and those of the these relationships being specified by their respective multiplicities and Wyckoff letters. The program WYCKSPLIT provides further information on the relations of the Wyckoff positions that are not listed in ITA1, namely the relations between the unitcell representatives of the orbit of the group and the corresponding representatives of the suborbits of the For example, the lefthand table in Fig. 5 shows the Wyckoffposition splitting schemes for the group–subgroup pair P6/mmm (No. 191) > (No. 187), (P, p) = (I, o) [here I is the threedimensional unit matrix and o is a (3 × 1) column matrix containing zeros as coefficients]. The 2c of P6/mmm splits into two independent positions of with no sitesymmetry reduction,
The relations between the coordinate triplets of the Wyckoff positions are displayed when clicking on Relations. These relations are presented in a table showing the Wyckoffposition coordinate triplets with respect to the standard group basis and the corresponding triplets referred to the basis of the (righthand table in Fig. 5).
2.3. Structure Utilities
There is a section in the BCS entitled Structure Utilities which contains interesting programs for coordinate transformations, comparison of structures and the evaluation of structure relationships between group–subgrouprelated structures. As input, these programs accept both standard crystallographic files in (CIF) format and the server's default structure definition format called BCS format; comments can be included by inserting a hash `#' sign at the beginning of a line:
The program SETSTRU performs coordinate transformations between ITA settings, and the program TRANSTRU can transform structural data to any spacegroup setting and can also be used to transform the and coordinates of a structure to those of a [Note that there are other programs such as VESTA (see Section 4), Mercury (see Section 3.7) and JANA (Petříček et al., 2023) (https://jana.fzu.cz/) that can perform coordinate transformations and transitions into subgroups. Recently, Jmol has also been adapted to be able to calculate and visualize group–subgroup relationships.] Practical examples of using the programs SETSTRU and TRANSTRU are available in the videos `Example SETSTRU' and `Example TRANSTRU', respectively (files numbered 7 and 8 in the supporting information).
The program COMPSTRU (de la Flor et al., 2016) is used to quantify the similarity of two structural models of the same phase or structures with different compositions which are isopointal, i.e. structures belonging to the same spacegroup type (or space groups that form an enantiomorphic pair), described in their standard settings, with the same or different chemical compositions and having the same sequence of occupied Wyckoff positions. The program STRUCTURE RELATIONS, however, is able to find the relation between two crystal structures having the same chemical composition and whose space groups are group–subgroup related. This program is used for the analysis and characterization of crystallinestate phase transitions with no change in the chemical composition. It calculates the index of the symmetry reduction and the transformation matrix–column pair (P, p) relating the coordinate system of the group to that of the More details about the use of the programs COMPSTRU and STRUCTURE RELATIONS are given in Section 6 and in the videos `Example COMPSTRU' and `Example STRUCTURE RELATIONS' (files numbered 9 and 5 in the supporting information).
3. Mercury and the CSD
3.1. Introduction to visualization software Mercury
Symmetry elements in structures in the CSD (Groom et al., 2016) or any structure stored in format can be explored using the CCDC visualization program Mercury (Macrae et al., 2020). Both experimentally determined and computationally generated structural models can be read directly into the free version of Mercury and symmetry can be visualized. Mercury runs on Microsoft Windows, macOS and Linux and the supported platforms are kept up to date on an FAQ (https://www.ccdc.cam.ac.uk/supportandresources/support/case/?caseid=f8eb6b5e0e5448678b01b21e84299e81). The installers for each of these three platforms can be downloaded from the CCDC website after signing in or registering for a free CCDC account. Once downloaded, the file needs to be double clicked and the instructions in the installer dialogue followed. Once installed, free Mercury will need to be activated by using the CCDC Activator tool and selecting the CSDCommunity option.
The Mercury software is available to download for free, with more advanced features available with a CSD licence (https://www.ccdc.cam.ac.uk/discover/blog/thedifferencebetweenfreemercuryandfulllicencemercury/). The free version of Mercury is designed to allow the exploration and visualization of 3D structures and enables users to generate highquality graphics and movies for scientific communication. Functionality free Mercury includes a comprehensive range of visualization features, the ability to explore crystallographic structures through their longerrange interactions such as hydrogen bonds, short contacts and polymer expansion, the generation of highresolution images and files to 3Dprint molecular and crystallographic structures, and the ability to simulate powder patterns and display the symmetry within a structure. Mercury functionality referenced in this section is all contained in the free version of Mercury, which has an accompanying instruction manual (https://www.ccdc.cam.ac.uk/supportandresources/documentationandresources/?category=All%20Categories&product=Mercury&type=User%20Guide) and training resources (https://www.ccdc.cam.ac.uk/community/trainingandlearning/).
3.2. Retrieving symmetry information from the CSD
For small organic, metal–organic and inorganic substances, the joint CCDC and FIZ Karlsruhe Access Structures website (https://www.ccdc.cam.ac.uk/structures/) is a helpful and comprehensive source of published experimental structural data. Access Structures enables users to view and retrieve any individual deposited data set in the CSD or ICSD, which together contain over 1.5 million crystal structures. Each entry in the CSD is assigned a database identifier known as a CSD Refcode (https://registry.identifiers.org/registry/csd) to enable individual data sets and structures of the same substance to be retrieved more easily. More advanced searches of the CSD can be performed using the CCDC desktop software ConQuest (Bruno et al., 2002) and web platform WebCSD (https://www.ccdc.cam.ac.uk/solutions/software/webcsd/). In these applications users can click on the spacegroup number to link to ITA to explore the spacegroup symmetry in more detail.
3.3. Educational resources available alongside the CSD
Alongside the CSD, the CCDC collates a collection of structures specifically chosen to help educators demonstrate key chemical concepts and teach others about chemistry and crystallography. This collection is called the CSD Teaching Subset (Battle et al., 2011) and this curated resource contains over 800 structures classified by the various concepts they can be used to demonstrate. Structures annotated as being in the symmetry category of this subset contain molecules that exhibit the most common pointgroup symmetries. The CSD Teaching Subset can be accessed in full and visualized in three dimensions in Mercury from the CSDCommunity menu (CSDCommunity > Open Teaching Database). Each individual structure can also be viewed and retrieved from the Access Structures website, with the individual categories linked from the CSD Teaching Subset website (https://www.ccdc.cam.ac.uk/community/educationandoutreach/education/teachingsubset/).
To complement the CSD Teaching Subset there is a collection of teaching modules (https://www.ccdc.cam.ac.uk/community/educationandoutreach/education/teachingmodules/), realized in collaboration with educators in the crystallographic community, which can be used as standalone educational packages for a broad range of chemistry science topics, including symmetry. To aid learning further there is also a collection of educational videos that are designed to teach students about symmetry and these are available through a Symmetry Operations and Symmetry Elements playlist (https://www.youtube.com/playlist?list=PLEtBZ08SGIScECmQvgvhZ0qXP3WWlBTnv).
3.4. Displaying symmetry in Mercury
Mercury allows the user to visualize the geometric elements of the symmetry operations of a in three dimensions, overlaid on the (Fig. 6). This feature can be activated from the Display menu by selecting Symmetry Elements and, in order to see how these elements are applied in the full structure, crystal packing can be displayed from the Calculate menu by selecting Packing and Slicing. In the Mercury software geometric elements are often referred to as symmetry elements in order to differentiate them from functionality used to explore intramolecular geometries and molecular conformations. The elements of the symmetry operations will be displayed on one by default, they are represented as follows:
(i) Inversion points are shown as orange spheres.
(ii) Rotation axes are shown as lines and screw axes as lines with halfarrow heads, coloured according to the order of rotation.
(iii) Glide and mirror planes are represented by planes, in magenta for glides and blue for mirrors.
The Symmetry Elements wizard interface will allow the user to personalize the display of the elements, as well as toggle them on and off.
It is also possible to learn more about the symmetry operations associated with each element. Clicking on the More Info button and selecting Symmetry Operators List will bring up a table with the list of symmetry operators, and for each a detailed description and further information, including a colour key matching the default colour of the elements displayed.
To complement further the visualization of symmetry in crystal structures, Mercury allows the user to colour molecules and atoms according to the symmetry relationship they have to the (Fig. 7). This can be done from the Colour dropdown menu, by selecting by Symmetry operation. An additional colouring option of interest is Colour: by Symmetry equivalence. This colour scheme assigns the same colour to different molecules or ions that are crystallographically identical. The video `Symmetry Features in Mercury' in the supporting information (file number 3) gives a detailed explanation of how to retrieve symmetry information from the CCDC, and an extended version of this video is also available on YouTube (How to View Symmetry Elements and Operations in Mercury, https://www.youtube.com/watch?v=umxCcRFDdss?si=twYSuwOJhYXx73_k).
3.5. Additional symmetry features
If the user is interested in exploring the contacts in a structure, more can be learned about it and the atoms involved in each contact pair from the More Info button by selecting Contacts List. The information displayed in the table will include the associated with both atoms.
It is also possible to visualize a network of symmetrygenerated molecules. This works by changing the Picking Mode in Mercury to Reveal SymmetryGenerated Molecules from the dropdown menu. Clicking on any point in the visualizer will bring up the closest symmetrygenerated molecule. From the More Info button the user can also select Structure to see more information about the entry, including the From here the user can click on the spacegroup number to link to ITA. The of the structure is also displayed alongside the CSD Refcode in the Structure Navigator toolbar.
Another feature is the possibility of using Mercury to identify stereocentres (Fig. 8) simply by using labels. Mercury allows the user to select the atoms they wish to show labels for, in this case Stereocentres, and which atom properties they wish to display in the label, for example Stereochemistry. As a result, once the user clicks on Show Labels, all stereocentres will be labelled with their atom label (element + number) and associated stereochemistry letter.
3.6. Highimpact visualization of structural symmetry
To communicate research outcomes effectively or to teach symmetryrelated concepts, once the user has explored the symmetry in a Mercury one can generate highquality publicationready images using POVRay (POVRay – The Persistence of Vision Ray Tracer, https://www.povray.org/) [Fig. 9(b)]. This functionality is accessed from the File menu by selecting POVRay Image. The POVRay Image interface will allow the user to customize the settings, such as the image size in pixels and the background colour, which includes a transparent option as well. If the user wishes to generate an animation, such as a GIF or a video, showing a structure rotating around an axis, they should click Generate Animation Frames and Mercury will generate frames for the rotation (as many as are assigned in Number of Frames); these frames can then be assembled into a GIF or video, for example, using thirdparty software.
they may want to create images or animations. InMercury additionally allows the creation of files for 3Dprinting structures or molecules [Fig. 9(a)]. This functionality is also accessed from the File menu, by selecting Print in 3D. The settings can be personalized from the 3D Printing interface, including the file format, the scale and whether to include a support framework.
3.7. Advanced functionality for structural comparisons
In Mercury it is possible to compare and overlay structures and molecules, as well as to change the spacegroup settings and reduce the symmetry. This functionality can be particularly powerful when comparing two related crystal structures, such as polymorphic structures or structures determined at different temperatures and pressures. In the Calculate menu of Mercury there is functionality to overlay pairs of structures or overlay molecules, and multiple structures can be displayed and manipulated using the Multiple Structures dialogue under the Structure Navigator. In the Edit menu there is functionality to transform, invert or translate molecules within a structure, change the spacegroup setting, invert crystal structures, remove lattice centring, reduce the symmetry, transform the structure to a and change the to a This advanced functionality is available in the licenced/full version of the software and can be a powerful tool for structure comparisons and analysis.
4. Threedimensional structure visualization with VESTA
VESTA is a crossplatform program for 3D visualization and investigation of data, volumetric (voxel) data and crystal morphology data. It runs on three major operating systems, Microsoft Windows (Version 7 or newer), macOS (Version 10.9 or newer) and Linux. For each platform, it is distributed as a compressed archive file (https://jpminerals.org/vesta/en/download.html). Once it has been uncompressed, the executable file can be directly executed without requiring any additional installation process.
When Text Area: spacegroup information, unitcell parameters and volume, a series of crystallographic sites and their site multiplicities, Wyckoff letters, and sitesymmetry symbols. On selection of atoms and bonds, further information is output to the Text Area:
data are loaded, the following basic information on the input data is output to the(i) Symmetry operations and translation vectors, by which the selected atoms are generated from the input coordinates.
(ii) Internally used orthogonal coordinates (Cartesian coordinates).
(iii) Principal axes (in both Cartesian and fractional coordinate systems) and their meansquare displacements of anisotropic displacement parameters (ADPs) when atoms are rendered as ADP ellipsoids.
(iv) Interatomic distances (l_{i}), bond angles and dihedral angles with their (s.u.) if those of the unitcell parameters and fractional coordinates are supplied.
Selecting Coordination polyhedra gives the following physical quantities in addition to information on each site and bonds:
(v) Polyhedral volumes.
(vi) Distortion indices (Baur, 1974).
(vii) Quadratic elongations (Robinson et al., 1971).
(viii) Bondangle variances (Robinson et al., 1971).
(ix) Effective ; Hoppe et al., 1989).
(Hoppe, 1979(x) Charge distribution (Hoppe et al., 1989; Nespolo et al., 1999).
(xi) Bondvalence sums (Brown & Altermatt, 1985).
(xii) Bond lengths expected from bondvalence parameters.
Volumetric data such as electron and nuclear densities, Patterson functions and wavefunctions are displayed as isosurfaces, twodimensional sections with contours or bird'seye views. Among a number of features of VESTA, we further describe some examples of advanced usage of handling.
4.1. Conversion of spacegroup symmetry
VESTA allows us to convert the of a to its or Before changing it would be safer first to convert a to P1 (No. 1) by clicking the Remove symmetry button in the Edit Data dialogue box. All the atomic positions in the will then be generated as independent sites. Although this process is not always necessary, it eliminates some restrictions on symmetry relations between the original and target crystal structures, making it easier to convert a properly. When the original and target space groups have different choices of origin, or the conversion requires changes to the principal axes, a userdefined transformation matrix (P, p) (Section 2.1.1) must be input. The user then presses the Remove symmetry button again (it will reset the transformation matrix to the identity matrix) and finally sets the target When the is changed to a highersymmetry group, or when the is transformed to a smaller one, the same atomic position may result from two or more sites, or atomic positions closely contacting each other around a special position may be generated. In such a case, the redundant atoms can be removed by clicking the Remove duplicate atoms… button. Atoms of the same element name closer than a threshold value, which is specified by the user, are merged onto a single site at their averaged position. The example in Section 6.3 shows how to use this feature of VESTA in a practical example.
4.2. Creation of superstructures and average structures
The userdefined transformation matrix is used to convert a standard spacegroup setting to a nonstandard one. The matrix can also be used for transformations between primitive and vice versa. When the determinant is larger than 1, the unitcell volume becomes larger than the original one. In this case VESTA can automatically locate additional atoms in the either as distinct sites or as additional equivalent positions. When the determinant is smaller than 1, the unitcell volume becomes smaller than the original one. By clicking the Remove duplicate atoms… button, average structures can then be generated from superstructures. Fig. 10 shows an example of a comparison between two silica polymorphs, moganite (I2/a, No. 15) and the righthanded αquartz (P3_{2}21, No. 154) with its converted to a monoclinic C2 (No. 5) by a transformation of (P, p) = (, ).
and for the creation of superstructures and average structures. The matrix transforms both the atomic positions and general equivalent positions (symmetry operations). If the determinant of the matrix is negative, the coordinate system is transformed from right to lefthanded, andIn addition to transformation of symmetry operations by the transformation matrix (P, p), symmetry operations may be manually edited. In that case, the symmetry operations do not necessarily form a closed group.
4.3. Comparison of multiple structures
VESTA allows the user to visualize and compare two or more crystal structures in the same 3D space. This feature facilitates the visualization and analysis of, for example, (i) small differences between similar crystal structures, (ii) interactions between crystal surfaces and noncrystalline molecules or clusters, and (iii) interface structures between two crystals, twins or layer structures with stacking faults between the two.
Data for second and subsequent phases are input from the Phase tab in the Edit Data dialogue box. Data can be entered manually, imported from existing files or copied from previously entered data. For each set of phase data, the position and orientation are then set as relative to another phase or the internal Cartesian coordinate system. Fig. 11 shows an example of a comparison between α and βquartz, the low and hightemperature polymorphs of SiO_{2}. The origin of the righthanded αquartz, P3_{2}21 (No. 154), is shifted to (0, 0, 2/3) to adjust its position to the standard setting of the righthanded βquartz, P6_{2}22 (No. 180). Fig. 12 shows another example of a comparison of multiple structures actually used to solve a real problem. In this example, the of the orthorhombic polytype of Ca_{2}B_{2}O_{5}, shimazakiite4O (P2_{1}2_{1}2_{1}, No. 19), was solved first, but a monoclinic polytype was also observed (Kusachi et al., 2013). However, the monoclinic structure was difficult to solve because of polysynthetic Starting from the layer structure of the 4O polytype, possible stacking structures of the 4M polytype were considered using VESTA. As a result, one type of stacking sequence was found to be most probable because the oxygen positions exactly matched at the interface of the adjacent layers. A monoclinic structure with P2_{1}/c (No. 14) was derived by this type of stacking sequence, and the same structure was later confirmed experimentally by singlecrystal Xray diffraction experiments.
5. Webbased visualization of and symmetry using Jmol
5.1. Jmol and JSmol
Jmol (project repositories SourceForge, https://sourceforge.net/projects/jmol, and GitHub, https://github.com/BobHanson/JmolSwingJS) is a versatile tool that consists of a standalone Java program which, when compiled, is simultaneously produced as an equivalent JavaScriptbased web application (sometimes referred to as JSmol) using a unique technology (https://github.com/BobHanson/java2script). Jmol runs in Java on Windows, MacOS and Linux, as well as in JavaScript in all standard browsers. The Java program is available either with 32bit floating precision (Jmol.jar) or 64bit double precision (JmolD.jar). (JavaScript is inherently double precision.) In both contexts, Jmol offers two modes: a `headless' commandline mode and an interactive graphical user interface (GUI) mode. Either mode, in Java or JavaScript, can run independently or as a library component of another headless or GUIbased program. As a standalone Java program, the headless mode (JmolData.jar or JmolDataD.jar) allows for fully scripted model construction, structure and symmetry calculation, and image production driven by Python or operating system batch scripts.
As JavaScript, JSmol can be integrated into web pages in one of three ways. In a headless `library' mode, JSmol can support other JavaScript functionality behind the scenes but never be displayed itself. Alternatively, it can be rendered in the form of an independent frame that `floats' on the web page and exactly duplicates its Java implementation. Finally, and most commonly, JSmol can be embedded directly into a web application or standard web page surrounded by text and JavaScript components that can drive it and `listen' to it interactively.
In particular, when JSmol is embedded within a web page as an interactive component, it becomes part of a larger context. It becomes a window through which the molecular and crystallographic world can be visualized and explored in ways that are customized to that context. For example, we might find JSmol being used in a biochemical context to explore protein secondary structure, validation issues or binding sites. We might find it involved in the description of atomic or molecular orbitals in small molecules or periodic systems. It is quite possible, in fact, that a `user' of Jmol may not even know that they are using it. All they know is that they are at a dynamic interactive website that lets them do interesting things with atoms and surfaces. The structures involved may be from a database, a study or a publication, or they might be ones the user has `dropped' into the website's page, allowing the user to explore their own structure within the designed context.
In all cases, whether 32bit or 64bit precision, Java, JavaScript, headless, framed or embedded, Jmol's scriptability is its common element. Scripting involves the giving of commands that effect changes in the visualization, that query the application for information about the models, that respond to the way the user is interacting with the models, and that initiate file reading or writing. It is this scriptability that we focus on in this discussion.
The scripting capability of Jmol is extensive. The Jmol/JSmol Interactive Scripting Documentation webpage (https://chemapps.stolaf.edu/jmol/docs) lists approximately 80 script commands, 100 script programming functions, 200 properties and over 400 command parameters. In this section, various snippets of Jmol scripting will be highlighted as a way of introducing the reader to some of this functionality. What we will describe here is only a small representative fraction of that capability, with a focus on symmetry, visualization and building. The goal is to give the reader an idea of what is generally available in Java or JavaScript.
5.2. Reading crystallographic data into Jmol
Jmol supports the reading of files of various flavours, including traditional CIF2, magnetic modulated structures (both magnetic and nonmagnetic) and topological (all defined in the dictionaries at https://www.iucr.org/resources/cif/dictionaries), mmCIF (Protein Data Bank in Europe mMCIF Information, https://www.ebi.ac.uk/pdbe/docs/documentation/mmcif.html), and BinaryCIF (Sehnal et al., 2020). In addition to reading files, Jmol can read over 20 additional experimental and computational crystallographic formats. Additional readers can be added with minimal effort. Jmol's file format `resolver' allows loading files (including dragdropping into any JSmol web application or into the Jmol Java application) without any filenamespecific requirements.
As for sourcing specific structural data, Jmol's load command includes a number of shortcuts relevant to this discussion. These direct Jmol to specific repositories of interest to users. Two of these, PubChem (https://pubchem.ncbi.nlm.nih.gov) and the National Cancer Institute Chemical Identifier Resolver (https://cactus.nci.nih.gov), allow quick loading of small molecules that can be viewed on their own or incorporated into models (see Section 6.1). Other databases are focused more on crystallography – most notably the Crystallographic Open Database (load =COD/ id), the American Mineralogist Society Database (load =AMS/<mineral name>) and a recent addition, the AFLOW Encyclopaedia of Crystallographic Prototypes (load =AFLOWLIB/nnn.m). These three databases represent three very different resource types. The COD contains over 500 000 curated structures deposited primarily by experimentalists (some structures are computational). If a COD structure ID is known, it can be used to load the structure directly into Jmol, which is also used on the COD website for previews. In Fig. 13, we see a Jmol model on the COD website, retrievable in Jmol using load =cod/2312394. Note that in this example, because of Jmol's inherent scriptability, a website user is not necessarily limited to what the website itself was designed to deliver. In this case, the user has enhanced the model by reloading with the centroid option in order to see better the packing of full molecules in the crystal structure.
Structures from the AMS database also can be accessed by ID, but more interesting is that a mineral name such as quartz or hematite or halite can be used, and in that case the database delivers a collection of structures. Over 1000 minerals are represented, many with several structures, for a total of nearly 6000 structures. For example, load =ams/quartz retrieves 45 models published between 1926 and 2008, with the following distribution, obtained using the Jmol script load =ams/quartz; print getProperty("fileInfo.models").select("(_journal_year)").pivot.format("JSON");: {"1926": 1, "1935": 1, "1939": 1, "1962": 2, "1980": 6, "1988": 1, "1989": 6, "1990": 17, "1992": 4, "2000": 1, "2005": 1, "2007": 3, "2008": 1}.
In contrast, the AFLOW Encyclopaedia of Crystallographic Prototypes allows for loading one or more example structures from every Pnma, No. 62). An example is shown in Fig. 14 from the Jmol application.
This database is perfect for use in a teaching or personal exploratory context, where we just want an example of one or more structures for a specific Every is represented by from one to 95 structures (Some of the more powerful features of Jmol involve how it loads files. Jmol can load crystallographic files with a number of options, all of which are scriptable. There are far too many possibilities to discuss here. Some of the more interesting options, though, are described in Table 1. Additional options allow the overriding of and/or origin for the data in the file as a way of investigating differences in symmetry of related subgroups and supergroups.

5.3. The Jmol crystallographic model kit
It is also possible to create crystal structures easily from scratch using Jmol. This is accomplished using simple scripting involving modelkit and related commands. Table 2 lists some of the many modelkit command options that can be used to build crystal structures. The basic idea is (i) to select a (ii) to define the and (iii) to add atoms. Over 600 settings of the 230 crystallographic space groups are available. Unit cells are defined using an array, indicating [a b c α β γ], or in relation to the current using the transformation notation as described in Section 2.1.1, such as .

Atoms can be added to a a–z or A) or G (for the general position). In the case of a that is not a single point, Jmol will place the atom on the on the basis of an internal default; the user can then move it to a more suitable location within that if desired. In all cases, Jmol will populate all equivalent atom positions. If a specific position is desired, fractional coordinates are recommended, i.e. `0.333' is not 1/3.
using decimal numbers for Cartesian coordinates in ångströms, such as {1.243, 2.336, 5.731}, or fractional coordinates, indicated by at least one value being expressed as a fraction using a forward slash `/', for example {0 0 1/2} or {0.123, 0.233, 0.500/1}. In addition, atoms can be placed using labels (So, for example for the model depicted in Fig. 15, the commands zap; modelkit spacegroup "P2/m"; modelkit add C wyckoff G packed created a model of P2/m (No. 10) which was rotated a bit by hand. Four atoms were added on the general position 4o, (x, y, z) (−x, y, −z) (−x, −y, −z) (x, y, z). The command draw spacegroup was used to visualize the twofold axis, mirror plane and inversion centre that characterize this and draw spacegroup ALL completed the diagram. Finally, after issuing set picking dragatom, one of the atoms was clicked and dragged, resulting in all four atoms moving to new symmetryequivalent positions.
A relatively new capability of the Jmol Crystallographic Model Kit (Jmol Version 16.2) is the ability to calculate and depict the relationships between super and subgroups. The top row of Fig. 16 illustrates P2_{1}2_{1}2 (No. 18) as a traditional general position diagram, as a 3D interactive model and as a 3D model showing various symmetry operations relating pairs of general position coordinates. The symmetry in this case involves three generators – two perpendicular twofold screw axes offset from the origin by 1/4 along the x axis, and a twofold rotation axis through the origin in the c direction. In the second row we see the relationships between the group and two settings of one of its maximal subgroups, P2_{1} (No. 4). Using the Jmol command color property site, we can see how the four general positions of P2_{1}2_{1}2 are split in each case by the loss of one or the other of the two perpendicular screw axes. The four 3D models were saved as PNGJ files, allowing the images to be dragged back into Jmol and investigated or modified further in three dimensions.
5.4. Jmol mathematical functions
Along with being scriptable, Jmol provides an extensive set of scriptable functions that can be used to obtain information about one or more of the currently loaded models, as well as general information about space groups. Two powerful functions are spacegroup() and symop(). The Jmol/JSmol interactive documentation includes extensive information about these functions. Table 3 provides some examples of their use to answer common questions about a model. Though in question format, it is to be understood that these queries would be done, at least for JSmol, principally by code on a web page as part of an integrated web application.

5.5. Jmol in the wild
As mentioned already, what sets Jmol apart is its versatility. On its own it may seem like just another program, but its true strength lies in its ability to be contextualized to suit any narrative or area of interest. Table 4 lists just a few of the JSmol implementations relating to crystallography and symmetry that are currently in service on the web. Readers interested in using Jmol are encouraged to join the Jmol Users List (https://sourceforge.net/p/jmol/mailman/jmolusers), where there are several `power users' who are always willing to assist in the design and implementation of JSmol applications and answer even the simplest of questions.
‡https://qs.pwmat.com. §https://chemapps.stolaf.edu/jmol/jsmol/jcse. ¶https://spacegroups.symotter.org. ∥https://chemapps.stolaf.edu/jmol/jsmol/jpge. #https://chemapps.stolaf.edu/jmol/jsmol/iucrdemo. 
6. Application of the four tools for the structural analysis of thiourea at different temperatures and pressures
6.1. Symmetry of thiourea molecule
Thiourea (CH_{4}N_{2}S) is a synthetic organic compound, an analogue of urea where O is replaced by S.^{3} The molecule is planar and displays mm2 symmetry (C_{2v} in Schönflies notation) where the two NH_{2} groups bonded to C are placed on opposite sides of the C=S molecular axis. In particular, molecules with mm2 symmetry can show a permanent along the twofold axis, in this case caused by a small negative charge on S and positive charge on H produced by polar C=S and N—H bonds. Jmol is capable of determining and visualizing the of a molecule, reporting detailed information about the symmetry elements in Hermann–Mauguin or Schönflies notation. In this case thiourea is present in the PubChem database so it can be input into JSmol just by opening the console and typing load :thiourea. The symmetry of the molecule can be determined using calculate pointgroup, returning C_{2v} (or mm2). The symmetry operations of the can be drawn through the molecule by typing draw pointgroup. The results of these three commands are shown in Fig. 17.
6.2. Crystal structures of thiourea
A search for thiourea in the CSD currently (as of March 2024) yields 33 entries (CSD Refcodes THIOUR and THIOUR01–32), 30 of them including refined atomic coordinates (two of them corresponding to deuterated thiourea, CD_{4}N_{2}S), with the first complete structural made by Truter (1967) (CSD Refcode THIOUR).^{4} In the group of 28 relevant entries, eight correspond to structure determinations at room pressure and temperature, seven using Xrays and one using neutrons. Ten entries correspond to lowtemperature determinations and 11 to highpressure studies. The video `Downloading cifs from Access Structures' in the supporting information (file 4) gives a detailed explanation of how to retrieve the structures of thiourea from the CCDC. The files used in this example are also available in the supporting information (file 1).
Commercial thiourea can be easily recrystallized from ethanol to yield large colourless rhombic crystals. At room temperature and pressure the Pnma (No. 62), with a = 7.655 (7), b = 8.537 (7) and c = 5.520 (7) Å, V = 360.74 Å^{3} and V/Z = 90.18 Å^{3}, as reported by Elcombe & Taylor (1968) (CSD Refcode THIOUR01) using singlecrystal neutron diffraction data. This of polymorph V of thiourea containing four molecules per (Z = 4, Z′ = 0.5) is shown in Fig. 18(a). The mm2 symmetric thiourea molecule crystallizes with one of its mirror planes coinciding with the crystallographic mirror plane of the normal to b at y = 1/4, making the of the crystal half a molecule, as shown in Fig. 18(b). This also implies that, in the solid state, the symmetry of the electron density of thiourea is m(σ). The difference in symmetry is determined by asymmetric interactions with neighbouring molecules (van der Waals forces in general, hydrogen bonds in this particular case) as discussed by Elcombe & Taylor (1968).
is orthorhombic,6.3. Comparison of the room and lowtemperature structures of thiourea at ambient pressure
Before the first ). One lowtemperature form of thiourea called polymorph I that was first reported by Elcombe & Taylor (1968) is also orthorhombic, P2_{1}ma^{5} (No. 26) with a = 7.516 (7), b = 8.519 (10) and c = 5.494 (5) Å, V = 351.77 Å^{3} and V/Z = 87.94 Å^{3} at 123 K (CSD Refcode THIOUR02), maintaining Z = 4 but with Z′ = 1 [see Fig. 19(a)]. Since the Pnma and P2_{1}ma unit cells have very similar cell parameters and share Z = 4 but P2_{1}ma is an order 2 of Pnma,^{6} the of the P2_{1}ma form is twice the volume of the Pnma form, leading to a content of twohalves of a thiourea molecule (Z′ = 2 × 1/2). In polymorph V, equivalent thiourea molecules are related by an inversion centre, and therefore the electric dipoles that accompany the molecules are antiparallel in the crystal, leaving a zero net dipolar moment. At low temperature inequivalent thiourea molecules are no longer antiparallel and the electric dipoles do not cancel each other, leaving a net polarization along the polar axis (a in the P2_{1}ma setting).
determination, it had been reported that roomtemperature paralectric crystals of thiourea convert to a ferroelectric phase below 200 K (Solomon, 1956The program STRUCTURE RELATIONS in the BCS (see Section 2.3) was used to compare the closely related structures of the room and lowtemperature forms of thiourea and quantify the differences. The files obtained from the CSD were used after removing by hand equivalent atoms added by the CSD [atoms N1G, H1G and H2G in THIOUR01.cif and N1B, H1B, H2B, N2B, H3B and H4B in THIOUR02.cif as given in the supporting information (file 1)]. THIOUR01.cif was input as a highsymmetry structure, THIOUR02.cif as a lowsymmetry structure, default tolerance values were maintained, and the box informing that one or both of the crystal structures were on a nonstandard setting (since P2_{1}ma is an ITA setting of Pmc2_{1}) was selected. On input, the BCS transforms P2_{1}ma into Pmc2_{1} and uses the standard setting for the rest of the analysis. After checking the correctness of the BCS listing of both crystal structures, the program determines the transformation matrix needed to represent the highsymmetry structure in the standard setting of the of the lowsymmetry one. In this case the transformation matrix (P, p) = (b, c, a; 0.01870, 1/4, 3/4) needs to be applied to go from Pnma to Pmc2_{1}, implying that a change in the names of the axes and a translation of the unitcell origin are required to make the two structures coincide. Using this transformation, the program calculates all the structural parameters of the first structure into the same setting as the second one, or represents the roomtemperature form of thiourea in Pmc2_{1}. This allows an atombyatom comparison of position, given in the Matching Atoms table, and calculation of the distance between each pair of equivalent atoms, given in the Differences in Atomic Positions table, such as the 0.2995 Å displacement of atom S12 relevant to ferroelectric behaviour. Finally, the evaluation of the global distortion of the lowtemperature structure with respect to the roomtemperature one is calculated in the form of spontaneous strain (S = 0.0065), maximum and average atomic displacements (d_{max} = 0.3975 Å for H1, d_{av} = 0.2497 Å) and the measure of similarity parameter (δ = 0.247 in this case). The video `Example STRUCTURE RELATIONS' in the supporting information (file 5) gives a detailed explanation of how to perform these calculations.
Once the transformation of the Pnma structure to the Pmc2_{1} is completed, the new description can be saved to input into the program COMPSTRU, which allows a much more detailed mathematical and graphical comparison of the two structures. COMPSTRU utilizes JSmol to visualize the two models represented in the same settings, highlighting the differences and similarities between the compared structures.
The transformation matrix found by STRUCTURE RELATIONS can also be used to convert the Pnma model to P2_{1}ma in VESTA to transform one model into the other `by hand' (more details about the procedure are given in Section 4.1). THIOUR01.cif is input into VESTA and the menu option Edit/Edit Data/Unit cell is selected where the Remove Symmetry button can be pressed. The Pnma model is converted into P1. Note that the number of atoms (in the Structure parameters tab) changes from five (one C, one S, one N and two H) to 32 (four C, four S, eight N and 16 H). At this point back in the Unit Cell tab the Transform button can be pressed and a menu will pop up asking for a transformation matrix. The matrix found by STRUCTURE RELATIONS can be input to transform the and coordinates of the Pnma model to the most similar Pmc2_{1} model. After applying the transformation the unitcell axes change their name and the positions of thiourea molecules in the also change. Note that C and S atoms that were located at y = 1/4 (the mirror plane m  y at x, 1/4, z, the coordinates of the plane where S and C atoms of thiourea are located) in the Pnma model are now located at x = 0 (the mirror at the yz plane) in Pmc2_{1}. Now the Pmc2_{1} symmetry needs to be added. This is done by changing the triclinic P1 to the orthorhombic Pmc2_{1} (No. 26), setting number 1, Pmc2_{1} (a, b, c), but the transformation matrix in the Transform window must be initialized first.
After the spacegroup symmetry has been converted to Pmc2_{1}, duplicate atoms must be removed in the Structure Parameters tab using the Remove duplicate atoms… button at the bottom. A threshold of maximum deviation between equivalent atoms can be selected, although the default is fine for exactly matching symmetryrelated atoms. Now the atom list changes again from 32 to ten, the new number of atoms in the of the Pmc2_{1} model of thiourea.
This model in Pmc2_{1} can be exported as a file and compared with the experimental structure that needs to be transformed in Pmc2_{1} for comparison. The P2_{1}ma setting can be obtained simply by selecting setting number 3, P2_{1}ma (c, a, b), and now the and atomic positions of the atoms become very similar to those in the THIOUR02.cif file. Note they are not identical, since the experimental structure corresponds to an experimental model at 123 K, while the structure obtained by the process just outlined here is a transformation of the roomtemperature structure into the same spacegroup setting.
Note that clicking one atom in a structure in VESTA provides the atomic coordinates of that atom and the required to obtain this atom from the one given in the atom list, which is assumed to be the general position x, y, z. Comparing the coordinates of N atoms in the same molecule in the Pnma and P2_{1}ma models of the high and lowtemperature forms of thiourea, respectively, allows us to visualize that, in both structures, there is a mirror symmetry relation between the N atoms. But two N atoms in nearby molecules in the of the cell are not symmetry related in P2_{1}ma. The video `THIOUR01 to 02 transformation VESTA' in the supporting information (file 6) gives a detailed explanation of how to perform these calculations.
6.4. Comparison of ambient and highpressure forms of thiourea at room temperature
Thiourea crystals have also been studied under applied isostatic pressure. Asahi et al. (2000) determined that, above 0.38 GPa, crystals of polymorph V convert into a new polymorph VI, orthorhombic, Pbnm (No. 62), with a = 5.503 (3), b = 7.138 (4) and c = 24.788 (4) Å, V = 973.68 Å^{3} and V/Z = 81.14 Å^{3}, as reported at 0.97 GPa (THIOUR19 in the CSD). Note that Pbnm is an ITA setting of Pnma that can be transformed to the standard one using the transformation matrix (P, p) = (b, c, a; 0, 0, 0).^{7} Fig. 20 shows the packing of thiourea in polymorphs V and VI side by side. Visual inspection of the unit cells of both forms indicates that the highpressure form of thiourea shows a tripled cell with respect to the ambientpressure form. The tripling of the cell is caused by the rotation of twothirds of the thiourea molecules in the crystal by a small angle along the molecular axis of symmetry, breaking the mirror symmetry of the arrangement and reconstructing the hydrogenbond network that holds the crystal together. The new (of the same type or isomorphic but with a tripled cell) is a of index 3 of the original one, with only one in every three of the translations along the elongated axis remaining,^{8} and its is also tripled with respect to the ambientpressure form containing one and a half thiourea molecules.
In this particular case, visual inspection of the unitcell parameters and atomic positions allows one to describe the transformation. The same analysis can be performed using STRUCTURE RELATIONS and COMPSTRU in the BCS. In this case the transformation from the small Pnma to the tripled Pbnm one requires the transformation matrix (P, p) = (c, a, 3b; 0, 0, 0).
The three crystalline forms of thiourea described so far are not the only ones published in the literature. Thiourea forms two incommensurate structures (polymorphs II and IV) on cooling between 160 and 205 K. Only polymorph II has a model included in the CSD (entry THIOUR06). The THIOUR05 entry, referred to as polymporph II′ (see Fig. 21), contains a periodic phase found in a narrow 2 K range around 179 K and shows a commensurate modulation vector very close to that in polymorph II. The structure of this polymorph II′ was refined by Tanisaki et al. (1988) and found to have an orthorhombic Pbnm a = 5.467 (1), b = 7.545 (1) and c = 76.867 (11) Å, V = 3170.65 Å^{3} and V/Z = 88.07 Å^{3}. This corresponds to an order 9 of the polymorph V structure (note that 8.537 × 9 = 76.833) that contains four thiourea halfmolecules per and can also be analysed using the tools described above to determine the relation between polymorphs V and VI.
All three graphical tools described in this paper have options to overlap different crystal structures. This can be done by overlapping unit cells with all of their contents (VESTA), molecules and packings (Mercury in the full version of the program) or average structures (JSmol in COMPSTRU). These overlap models can help the structure comparison and description of distortions in a visual way that complements mathematical calculations that could quantify them.
7. Conclusion
This work collects a set of tools that crystallographers use daily in their research which, among other features, allow the user to analyse, visualize and handle Mercury within the Cambridge Structural Database, VESTA and Jmol/JSmol are described in detail. The application of all the tools described to the comparison of different closely related polymorphic forms of thiourea is also included in the final part of the paper for the reader to reproduce the analyses and use them in the classroom.
These resources can also be used for educational purposes, offering valuable tools for teaching crystallography at all educational levels. The characteristics and utilities regarding symmetry analysis of the Bilbao Crystallographic Server,The authors hope that this publication, first discussed during a workshop at the IUCr 2023 Congress (Melbourne, Australia), will allow crystallography lecturers and practitioners to feel more confident with nonautomated analysis of
in their classes or everyday practice, using some of the best tools freely available for the task.Supporting information
https://doi.org/10.1107/S1600576724007659/dv5017sup1.zip
files for the examples in Section 6. DOI:Webbased visualization of https://doi.org/10.1107/S1600576724007659/dv5017sup2.pdf
and symmetry using Jmol. DOI:Video Symmetry features in Mercury. DOI: https://doi.org/10.1107/S1600576724007659/dv5017sup3.mp4
Video Downloading CIFs from Access Structures: related to the example in section 6 that shows how to retrieve the thiourea https://doi.org/10.1107/S1600576724007659/dv5017sup4.mp4
files from the CCDC. DOI:Video Example STRUCTURE RELATIONS: related to the example in section 6 that shows how to use the program STRUCTURE RELATIONS. DOI: https://doi.org/10.1107/S1600576724007659/dv5017sup5.mp4
Video THIOUR01 to 02 transformation VESTA: related to the example in section 6 that shows how to transform the structures using VESTA. DOI: https://doi.org/10.1107/S1600576724007659/dv5017sup6.mp4
Video Example SETSTRU: showing how to transform structures using the program SETSTRU with the examples from Section 6. DOI: https://doi.org/10.1107/S1600576724007659/dv5017sup7.mp4
Video Example TRANSTRU: showing how to transform structures using the program TRANSTRU with the examples from Section 6. DOI: https://doi.org/10.1107/S1600576724007659/dv5017sup8.mp4
Video Example COMPSTRU: showing how to compare structures using the program COMPSTRU with the examples from Section 6. DOI: https://doi.org/10.1107/S1600576724007659/dv5017sup9.mp4
Footnotes
^{1}The conventional settings of space groups are those fully described in ITA. This means that the corresponding data, including diagrams, asymmetric units, generators, and general and special Wyckoff positions, can be found in ITA.
^{2}In the BCS we refer to all those settings that are listed in the synoptic tables of plane and space groups found in ch. 1.5 of ITA as ITA settings. For us the Pnma setting is the conventional (and standard) setting of the Pnma (No. 62), while Pmnb, Pbnm, Pcmn, Pmcn and Pnam are just ITA settings of Pnma.
^{3}We have selected this small organic molecule for many reasons. The described in this section is accompanied by a simple to understand structure–property correlation. The molecule is small enough and the symmetry simple enough (without being obvious) to allow performing calculations by hand, and it is a great first step to enter into more complex structures (e.g. incommensurate phases). Finally, it is a cheap to purchase chemical that is very easy to recrystallize and shows its phase transitions in a temperature range accessible to most inhouse laboratories, allowing the transformation of the analysis in this paper into a practical session.
^{4}The four thiourea entries from the CSD discussed in this paper are available in the free Teaching Subset of the CSD.
^{5}P2_{1}ma is an ITA setting of Pmc2_{1} (No. 26) obtained by renaming the abc axes to cab (see Table 1.5.1.1 of ITA for transformation matrices). The P2_{1}ma setting of thiourea can be converted to the standard Pmc2_{1} with VESTA using the Edit/Edit Data/Unit Cell tab, selecting setting number 1 Pmc2_{1} (a, b, c) and clicking the Apply button.
^{6}Pmc2_{1} is a maximal of index 2 of Pnma, as could be confirmed using the MAXSUB program in the BCS (see Section 2.2.1). In particular it is a translationengleiche since translations are kept but the order of the is reduced.
^{7}Conversion from Pbnm to Pnma can be performed in VESTA using the Edit/Edit Data/Unit Cell tab, selecting setting number 1, Pnma (a, b, c), and clicking the Apply button; in Jmol it can be done with the command modelkit spacegroup "Pnma".
^{8}The tripled cell with Pnma symmetry is a klassengleiche of the original ambientpressure cell with the same spacegroup symmetry.
Acknowledgements
R. M. Hanson wishes to thank the University of the Basque Country and St Olaf College for sabbatical funding for collaboration visits to Bilbao in 2019, 2022 and 2023. L. Suescun thanks PEDECIBAQuímica for its constant support in educational and dissemination activities in crystallography. Open access funding enabled and organized by Projekt DEAL.
Funding information
M. I. Aroyo and G. de la Flor thank the Government of the Basque Country (Eusko Jaurlaritza) (Project No. IT145822).
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