# Point groups (property) This page documents an [OPTIMADE](https://www.optimade.org/) [Property Definition](https://schemas.optimade.org/#definitions). See [https://schemas.optimade.org/](https://schemas.optimade.org/) for more information. **ID: [`https://schemas.anyterial.se/defs/v0.1/datasets/pointgroups`](https://schemas.anyterial.se/defs/v0.1/datasets/pointgroups.md)** **Definition name:** `pointgroups` **Property name:** Point groups **Description:** Ordered table of crystallographic point-group records. Each item contains point-group classification, finite point-group operations, conjugacy classes, and real and complex character tables generated from cctbx and spgrep. **Type:** list **Requirements/Conventions**: - The table covers the 32 crystallographic point groups. - The point-group records are independent of any particular space-group setting. - Character-table rows use Mulliken labels and include formatted label variants when available. **Examples:** - `[{"hm_symbol": "2/m", "order": 4}]` **Formats:** [[JSON](pointgroups.json)] [[MD](pointgroups.md)] **JSON definition:** ``` json { "$id": "https://schemas.anyterial.se/defs/v0.1/datasets/pointgroups", "$schema": "https://schemas.optimade.org/meta/v1.3/optimade/property_definition.json", "title": "Point groups", "$comment": "Dataset of pointgroup entries.", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "pointgroups", "label": "pointgroups_pointgroups" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Ordered table of crystallographic point-group records.\nEach item contains point-group classification, finite point-group operations, conjugacy classes, and real and complex character tables generated from cctbx and spgrep.\n\n**Requirements/Conventions**:\n\n- The table covers the 32 crystallographic point groups.\n- The point-group records are independent of any particular space-group setting.\n- Character-table rows use Mulliken labels and include formatted label variants when available.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/entrytypes/pointgroups", "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "title": "Anyterial point group symmetry fields", "x-optimade-definition": { "kind": "entrytype", "format": "1.3", "version": "0.1.0", "name": "pointgroups", "label": "pointgroups_entrytype_anyterial" }, "properties": { "id": { "$id": "https://schemas.optimade.org/defs/v1.2/properties/core/id", "x-optimade-requirements": { "support": "must", "sortable": true, "query-support": "all mandatory", "response-level": "always" }, "title": "ID", "x-optimade-type": "string", "x-optimade-definition": { "label": "id_core", "kind": "property", "version": "1.2.0", "format": "1.2", "name": "id" }, "type": [ "string" ], "description": "A unique string referencing a specific entry in the database.\n\n**Requirements/Conventions:**\n\n- Taken together, the ID and entry type MUST uniquely identify the entry.\n- Reasonably short IDs are encouraged and SHOULD NOT be longer than 255 characters.\n- IDs MAY change over time.", "examples": [ "db/1234567", "cod/2000000", "cod/2000000@1234567", "nomad/L1234567890", "42" ], "x-optimade-unit": "inapplicable" }, "type": { "$id": "https://schemas.optimade.org/defs/v1.2/properties/core/type", "x-optimade-requirements": { "support": "must", "sortable": true, "query-support": "all mandatory", "response-level": "always" }, "title": "type", "x-optimade-type": "string", "x-optimade-definition": { "label": "type_core", "kind": "property", "version": "1.2.0", "format": "1.2", "name": "type" }, "type": [ "string" ], "description": "The name of the type of an entry.\n\n**Requirements/Conventions:**\n\n- MUST be an existing entry type.\n- The entry of type and ID MUST be returned in response to a request for // under the versioned or unversioned base URL serving the API.", "examples": [ "structures" ], "x-optimade-unit": "inapplicable" }, "immutable_id": { "$id": "https://schemas.optimade.org/defs/v1.2/properties/core/immutable_id", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "all mandatory", "response-level": "may" }, "title": "immutable ID", "x-optimade-type": "string", "x-optimade-definition": { "label": "immutable_id_core", "kind": "property", "version": "1.2.0", "format": "1.2", "name": "immutable_id" }, "type": [ "string", "null" ], "description": "The entry's immutable ID (e.g., a UUID).\n\n**Requirements/Conventions:**\n\n- This is important for databases having preferred IDs that point to \"the latest version\" of a record, but still offer access to older variants.\n- This ID maps to the version-specific record, in case it changes in the future.", "examples": [ "8bd3e750-b477-41a0-9b11-3a799f21b44f", "fjeiwoj,54;@=%<>#32" ], "x-optimade-unit": "inapplicable" }, "last_modified": { "$id": "https://schemas.optimade.org/defs/v1.2/properties/core/last_modified", "x-optimade-requirements": { "support": "should", "sortable": false, "query-support": "all mandatory", "response-level": "must" }, "title": "last modified", "x-optimade-type": "timestamp", "x-optimade-definition": { "label": "last_modified_core", "kind": "property", "version": "1.2.0", "format": "1.2", "name": "last_modified" }, "type": [ "string", "null" ], "format": "date-time", "description": "Date and time representing when the entry was last modified.", "examples": [ "2007-04-05T14:30:20Z" ], "x-optimade-unit": "inapplicable" }, "character_table_complex": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/pointgroups/character_table_complex", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Complex character table", "$comment": "Generated from data-generators JSON-LD fields without external definition URLs.", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "character_table_complex", "label": "character_table_complex_pointgroups" }, "type": [ "array", "null" ], "description": "Complex irreducible character table of the crystallographic point group.\n\nRows correspond to complex irreducible representations and columns follow the order of `conjugacy_classes`.\nCharacters are exact complex rational values stored as dictionaries with `re` and `im` fraction strings, so that they round-trip without floating-point loss.", "x-optimade-unit": "inapplicable", "items": { "x-optimade-type": "dictionary", "type": [ "object", "null" ], "description": "One row of a point-group character table.", "properties": { "label": { "x-optimade-type": "string", "type": [ "string", "null" ], "description": "Irreducible-representation label.", "x-optimade-unit": "inapplicable" }, "label_markup": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/string_markups", "title": "String markups", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "string_markups", "label": "string_markups_core" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Strings with alternate markup and/or encoding for display rendering.\n\nThe object is intended for display-oriented variants only, a sibling property should be used for canonical plain string value.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **html**: OPTIONAL; String.\n HTML rendering of the sibling string, using inline HTML elements where needed for typographic structure such as subscripts, superscripts, overlines, fractions, and line breaks.\n\n - **latex**: OPTIONAL; String.\n LaTeX rendering of the sibling string, suitable for use with a LaTeX or MathJax-like renderer.\n\n - **unicode**: OPTIONAL; String.\n Unicode rendering of the sibling string, using Unicode code points for display features where practical.", "properties": { "html": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "HTML rendering of the sibling string." }, "latex": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "LaTeX rendering of the sibling string." }, "unicode": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Unicode rendering of the sibling string." } }, "examples": [ { "html": "P 21/c", "latex": "\\mathit{P}\\,2_{1}/c", "unicode": "P2\u2081/c" } ] }, "dimension": { "x-optimade-type": "integer", "type": [ "integer", "null" ], "description": "Dimension of the irreducible representation.", "x-optimade-unit": "inapplicable" }, "characters": { "x-optimade-type": "list", "type": [ "array", "null" ], "description": "Characters for the conjugacy classes in table order.", "items": { "x-optimade-type": "dictionary", "type": [ "object", "null" ], "description": "One complex character value represented as exact real and imaginary parts.", "x-optimade-unit": "inapplicable", "properties": { "re": { "x-optimade-type": "string", "type": [ "string", "null" ], "description": "Exact real part represented as a string.", "x-optimade-unit": "inapplicable" }, "im": { "x-optimade-type": "string", "type": [ "string", "null" ], "description": "Exact imaginary part represented as a string.", "x-optimade-unit": "inapplicable" } } }, "x-optimade-unit": "inapplicable" }, "frobenius_schur_indicator": { "x-optimade-type": "integer", "type": [ "integer", "null" ], "description": "Frobenius-Schur indicator of the representation.", "x-optimade-unit": "inapplicable" }, "basis_linear": { "x-optimade-type": "list", "type": [ "array", "null" ], "description": "Linear basis functions transforming as this representation.", "items": { "x-optimade-type": "string", "type": [ "string", "null" ], "description": "One basis function.", "x-optimade-unit": "inapplicable" }, "x-optimade-unit": "inapplicable" }, "basis_rotation": { "x-optimade-type": "list", "type": [ "array", "null" ], "description": "Rotational basis functions transforming as this representation.", "items": { "x-optimade-type": "string", "type": [ "string", "null" ], "description": "One rotational basis function.", "x-optimade-unit": "inapplicable" }, "x-optimade-unit": "inapplicable" }, "basis_quadratic": { "x-optimade-type": "list", "type": [ "array", "null" ], "description": "Quadratic basis functions transforming as this representation.", "items": { "x-optimade-type": "string", "type": [ "string", "null" ], "description": "One quadratic basis function.", "x-optimade-unit": "inapplicable" }, "x-optimade-unit": "inapplicable" } }, "x-optimade-unit": "inapplicable" }, "examples": [ [ { "label": "A", "dimension": 1, "characters": [ 1 ], "frobenius_schur_indicator": 1, "label_markup": { "latex": "A", "unicode": "A" } } ], [ { "label": "Ag", "dimension": 1, "characters": [ 1, 1 ], "frobenius_schur_indicator": 1, "label_markup": { "latex": "A_{g}", "unicode": "Ag" } }, { "label": "Au", "dimension": 1, "characters": [ 1, -1 ], "frobenius_schur_indicator": 1, "label_markup": { "latex": "A_{u}", "unicode": "Au" } } ] ] }, "character_table_real": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/pointgroups/character_table_real", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Real character table", "$comment": "Generated from data-generators JSON-LD fields without external definition URLs.", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "character_table_real", "label": "character_table_real_pointgroups" }, "type": [ "array", "null" ], "description": "Real irreducible character table of the crystallographic point group.\n\nRows correspond to real irreducible representations and columns follow the order of `conjugacy_classes`.\nCharacters of the real table are exact integers and are therefore stored as plain integers, unlike the `re`/`im` fraction-string pairs of `character_table_complex`.", "x-optimade-unit": "inapplicable", "items": { "x-optimade-type": "dictionary", "type": [ "object", "null" ], "description": "One row of a point-group character table.", "properties": { "label": { "x-optimade-type": "string", "type": [ "string", "null" ], "description": "Irreducible-representation label.", "x-optimade-unit": "inapplicable" }, "label_markup": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/string_markups", "title": "String markups", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "string_markups", "label": "string_markups_core" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Strings with alternate markup and/or encoding for display rendering.\n\nThe object is intended for display-oriented variants only, a sibling property should be used for canonical plain string value.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **html**: OPTIONAL; String.\n HTML rendering of the sibling string, using inline HTML elements where needed for typographic structure such as subscripts, superscripts, overlines, fractions, and line breaks.\n\n - **latex**: OPTIONAL; String.\n LaTeX rendering of the sibling string, suitable for use with a LaTeX or MathJax-like renderer.\n\n - **unicode**: OPTIONAL; String.\n Unicode rendering of the sibling string, using Unicode code points for display features where practical.", "properties": { "html": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "HTML rendering of the sibling string." }, "latex": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "LaTeX rendering of the sibling string." }, "unicode": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Unicode rendering of the sibling string." } }, "examples": [ { "html": "P 21/c", "latex": "\\mathit{P}\\,2_{1}/c", "unicode": "P2\u2081/c" } ] }, "dimension": { "x-optimade-type": "integer", "type": [ "integer", "null" ], "description": "Dimension of the irreducible representation.", "x-optimade-unit": "inapplicable" }, "characters": { "x-optimade-type": "list", "type": [ "array", "null" ], "description": "Characters for the conjugacy classes in table order.", "items": { "x-optimade-type": "integer", "type": [ "integer", "null" ], "description": "One real character value represented as an exact integer.", "x-optimade-unit": "inapplicable" }, "x-optimade-unit": "inapplicable" }, "frobenius_schur_indicator": { "x-optimade-type": "integer", "type": [ "integer", "null" ], "description": "Frobenius-Schur indicator of the representation.", "x-optimade-unit": "inapplicable" }, "basis_linear": { "x-optimade-type": "list", "type": [ "array", "null" ], "description": "Linear basis functions transforming as this representation.", "items": { "x-optimade-type": "string", "type": [ "string", "null" ], "description": "One basis function.", "x-optimade-unit": "inapplicable" }, "x-optimade-unit": "inapplicable" }, "basis_rotation": { "x-optimade-type": "list", "type": [ "array", "null" ], "description": "Rotational basis functions transforming as this representation.", "items": { "x-optimade-type": "string", "type": [ "string", "null" ], "description": "One rotational basis function.", "x-optimade-unit": "inapplicable" }, "x-optimade-unit": "inapplicable" }, "basis_quadratic": { "x-optimade-type": "list", "type": [ "array", "null" ], "description": "Quadratic basis functions transforming as this representation.", "items": { "x-optimade-type": "string", "type": [ "string", "null" ], "description": "One quadratic basis function.", "x-optimade-unit": "inapplicable" }, "x-optimade-unit": "inapplicable" } }, "x-optimade-unit": "inapplicable" }, "examples": [ [ { "label": "A", "dimension": 1, "characters": [ 1 ], "basis_linear": [ "x", "y" ], "basis_rotation": [ "Rx", "Ry" ], "basis_quadratic": [ "x^2", "y^2" ], "label_markup": { "latex": "A", "unicode": "A" } } ], [ { "label": "Ag", "dimension": 1, "characters": [ 1, 1 ], "basis_rotation": [ "Rx", "Ry" ], "basis_quadratic": [ "x^2", "y^2" ], "label_markup": { "latex": "A_{g}", "unicode": "Ag" } }, { "label": "Au", "dimension": 1, "characters": [ 1, -1 ], "basis_linear": [ "x", "y" ], "label_markup": { "latex": "A_{u}", "unicode": "Au" } } ] ] }, "conjugacy_classes": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/pointgroups/conjugacy_classes", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Conjugacy classes", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "conjugacy_classes", "label": "conjugacy_classes_pointgroups" }, "type": [ "array", "null" ], "description": "Conjugacy classes of a crystallographic point group.\n\nEach class lists its member operation indices, a representative operation, a conventional class label, and operation-type metadata used by the character tables.\nOperation indices refer to positions in the `symops` list of the same point-group record, starting at 0.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries, one per conjugacy class, ordered consistently with the character tables.\n- **size** MUST equal the length of **members**.\n- **op\\_type** is the signed integer rotation-type code of the representative operation: `1`, `2`, `3`, `4`, `6` for proper rotations, and the negated value for the corresponding rotoinversions, with `-2` denoting a mirror plane.", "x-optimade-unit": "inapplicable", "items": { "x-optimade-type": "dictionary", "type": [ "object", "null" ], "description": "One conjugacy class of a crystallographic point group.", "properties": { "label": { "x-optimade-type": "dictionary", "type": [ "object", "null" ], "description": "Rendered class label.", "properties": { "ascii": { "x-optimade-type": "string", "type": [ "string", "null" ], "description": "Plain-text rendering intended for logs, command-line output, or compact display.", "x-optimade-unit": "inapplicable" }, "unicode": { "x-optimade-type": "string", "type": [ "string", "null" ], "description": "Unicode rendering of the same label.", "x-optimade-unit": "inapplicable" }, "latex": { "x-optimade-type": "string", "type": [ "string", "null" ], "description": "LaTeX rendering of the same label.", "x-optimade-unit": "inapplicable" } }, "x-optimade-unit": "inapplicable" }, "size": { "x-optimade-type": "integer", "type": [ "integer", "null" ], "description": "Number of point-group operations in the class. It MUST equal the length of `members`.", "x-optimade-unit": "inapplicable" }, "members": { "x-optimade-type": "list", "type": [ "array", "null" ], "description": "Indices into the `symops` list of the operations belonging to this class.", "items": { "x-optimade-type": "integer", "type": [ "integer", "null" ], "description": "Operation index into `symops`.", "x-optimade-unit": "inapplicable" }, "x-optimade-unit": "inapplicable" }, "representative": { "x-optimade-type": "integer", "type": [ "integer", "null" ], "description": "Index into `symops` of a representative operation for the class.", "x-optimade-unit": "inapplicable" }, "op_type": { "x-optimade-type": "integer", "type": [ "integer", "null" ], "description": "Signed integer rotation-type code of the representative operation; negative values denote rotoinversions, with `-2` denoting a mirror plane.", "enum": [ 1, -1, 2, -2, 3, -3, 4, -4, 6, -6 ], "x-optimade-unit": "inapplicable" }, "op_axis": { "x-optimade-type": "list", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Integer-vector axis or invariant direction of the representative operation; `[0, 0, 0]` when no axis is applicable.", "items": { "x-optimade-type": "integer", "type": [ "integer", "null" ], "description": "One integer component of the axis vector.", "x-optimade-unit": "inapplicable" }, "x-optimade-unit": "inapplicable" } }, "x-optimade-unit": "inapplicable" }, "examples": [ [ { "label": { "ascii": "E", "unicode": "E", "latex": "E" }, "size": 1, "members": [ 0 ], "representative": 0, "op_type": 1, "op_axis": [ 0, 0, 0 ] }, { "label": { "ascii": "i", "unicode": "i", "latex": "i" }, "size": 1, "members": [ 1 ], "representative": 1, "op_type": -1, "op_axis": [ 0, 0, 0 ] } ] ] }, "crystal_system": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/pointgroups/crystal_system", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Crystal system", "x-optimade-type": "string", "x-compatibility": [ "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group.crystal_system.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "crystal_system", "label": "crystal_system_pointgroups" }, "type": [ "string", "null" ], "description": "The crystal system of the space group or point group.\n\nValues use the conventional crystallographic system names.", "x-optimade-unit": "inapplicable", "enum": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ], "examples": [ "triclinic", "monoclinic" ] }, "hm_symbol": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/pointgroups/hm_symbol", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Hermann-Mauguin symbol", "x-optimade-type": "string", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "hm_symbol", "label": "hm_symbol_pointgroups" }, "type": [ "string", "null" ], "description": "Hermann-Mauguin point-group symbol used as the key and display symbol for a point-group record.\n\nThe value is one of the 32 crystallographic point-group symbols in Hermann-Mauguin notation, written in ASCII with `-` denoting rotoinversion.", "x-optimade-unit": "inapplicable", "enum": [ "1", "-1", "2", "m", "2/m", "222", "mm2", "mmm", "4", "-4", "4/m", "422", "4mm", "-42m", "4/mmm", "3", "-3", "32", "3m", "-3m", "6", "-6", "6/m", "622", "6mm", "-62m", "6/mmm", "23", "m-3", "432", "-43m", "m-3m" ], "examples": [ "1", "-1", "m-3m" ] }, "is_centrosymmetric": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/pointgroups/is_centrosymmetric", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "is centrosymmetric", "$comment": "Generated from data-generators JSON-LD fields without external definition URLs.", "x-optimade-type": "boolean", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "is_centrosymmetric", "label": "is_centrosymmetric_pointgroups" }, "type": [ "boolean", "null" ], "description": "Boolean flag indicating whether the point group contains inversion symmetry.", "x-optimade-unit": "inapplicable", "examples": [ false, true ] }, "laue_class": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/pointgroups/laue_class", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Laue class", "x-optimade-type": "string", "x-compatibility": [ "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group.Laue_class.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "laue_class", "label": "laue_class_pointgroups" }, "type": [ "string", "null" ], "description": "The Laue class associated with the space group or point group.\n\nThe Laue class groups point groups that become equivalent when inversion symmetry is included.", "x-optimade-unit": "inapplicable", "enum": [ "-1", "2/m", "mmm", "4/m", "4/mmm", "-3", "-3m", "6/m", "6/mmm", "m-3", "m-3m" ], "examples": [ "-1", "2/m" ] }, "n_conjugacy_classes": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/pointgroups/n_conjugacy_classes", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Number of conjugacy classes", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_conjugacy_classes", "label": "n_conjugacy_classes_pointgroups" }, "type": [ "integer", "null" ], "description": "Number of conjugacy classes in the crystallographic point group.\n\nThis value MUST equal the length of the `conjugacy_classes` list of the point-group entry.", "x-optimade-unit": "inapplicable", "examples": [ 1, 2 ] }, "order": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/pointgroups/order", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Order of the point group", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "order", "label": "order_pointgroups" }, "type": [ "integer", "null" ], "description": "Order of the point group, i.e. the number of operations in the finite point group.\n\nThis value MUST equal the length of the `symops` list of the point-group entry.", "x-optimade-unit": "inapplicable", "examples": [ 1, 2 ] }, "schoenflies": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/pointgroups/schoenflies", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Schoenflies symbol", "x-optimade-type": "string", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "schoenflies", "label": "schoenflies_pointgroups" }, "type": [ "string", "null" ], "description": "The Schoenflies symbol for the crystallographic point group.\n\nThe value is one of the 32 crystallographic point-group symbols in Schoenflies notation, written in ASCII without subscript formatting.\nThe symbol `S6` is used for the point group also known as `C3i`.", "x-optimade-unit": "inapplicable", "enum": [ "C1", "Ci", "C2", "Cs", "C2h", "D2", "C2v", "D2h", "C4", "S4", "C4h", "D4", "C4v", "D2d", "D4h", "C3", "S6", "D3", "C3v", "D3d", "C6", "C3h", "C6h", "D6", "C6v", "D3h", "D6h", "T", "Th", "O", "Td", "Oh" ], "examples": [ "C1", "C2v", "Oh" ] }, "symops": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/pointgroups/symops", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Symmetry operations", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "symops", "label": "symops_pointgroups" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Full list of symmetry-operation descriptors for a point group.\nEach list member is an `op` object as defined by `/defs/v0.1/properties/symmetry/op`.\nPoint-group operations have a zero translation part, so the `screw_glide` and `origin_shift` classification fields are omitted.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST follow the schema inherited from `/defs/v0.1/properties/symmetry/op`.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op", "title": "Operation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op", "label": "op_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Information related to a crystallographic operation acting within one coordinate setting.\n\nRepresents an affine_transformation that is a crystallographic operation within one setting.\nThe affine map itself is stored in the embedded `affine_transformation` field.\nThe remaining fields classify the operation crystallographically, for example by rotation type, axis, sense, and screw or glide component.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **affine\\_transformation**: REQUIRED; Dictionary.\n Exact affine map for the operation.\n It MUST follow `/defs/v0.1/properties/symmetry/affine_transformation`.\n\n - **rot\\_type**: OPTIONAL; String.\n Crystallographic operation-type label for the linear part.\n\n - **axis**: OPTIONAL; List of 3 Integers.\n Operation axis or invariant direction using the integer-vector convention returned by the generator.\n\n - **sense**: OPTIONAL; Integer.\n Rotation sense/sign convention returned by the generator; `0` is used when no handed rotation sense is applicable.\n\n - **screw\\_glide**: OPTIONAL; List of 3 Fractions (String).\n Screw-axis or glide-plane component associated with a space-group affine operation.\n\n - **origin\\_shift**: OPTIONAL; List of 3 Fractions (String).\n Origin shift associated with the screw/glide decomposition of a space-group affine operation.\n\n- Whether the operation is proper follows from the sign of the `det` field of `affine_transformation`; no separate properness flag is stored.", "properties": { "affine_transformation": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/affine_transformation", "title": "Affine transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_transformation", "label": "affine_transformation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "An affine transformation acting on fractional crystallographic coordinates.\n\nAn affine transformation is a geometric transformation preserving points, straight lines, and parallelism (collinearity), but may not preserve Euclidean distances and angles.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nThe transformation may, for example, represent an operation within one setting, a setting transform, a subgroup embedding, a normalizer representative, or a parametric coordinate map for a Wyckoff-position orbit representative.\nWhen used as a parametric coordinate map, the matrix may be singular because special Wyckoff positions can constrain or identify parameters.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **matrix**: REQUIRED; Exact 3x3 matrix.\n Matrix part of the affine transformation.\n It MUST be represented as a list of three row lists, each containing three exact rational entries represented as strings.\n\n - **vector**: REQUIRED; List of 3 Fractions (String).\n Translation or origin-shift vector of the affine transformation in fractional coordinates.\n\n - **xyz**: OPTIONAL; String.\n Coordinate expression for the affine transformation in `x,y,z` notation when available.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether `matrix` is orthogonal in the exact representation used by the generator.", "properties": { "matrix": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_lattice" ], "sizes": [ 3, 3 ] }, "type": [ "array", "null" ], "description": "Exact 3 by 3 matrix part of the affine transformation.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One row of the exact 3 by 3 matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "vector": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Exact fractional-coordinate vector part of the affine transformation.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "xyz": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op_xyz", "title": "Operation xyz", "x-optimade-type": "string", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz", "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op_xyz", "label": "op_xyz_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.", "x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$", "examples": [ "-x,-y,z", "x,1/2-y,1/2+z" ] }, "det": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Determinant of the matrix part when emitted by the generator." }, "is_orthogonal": { "x-optimade-type": "boolean", "x-optimade-unit": "inapplicable", "type": [ "boolean", "null" ], "description": "Whether the matrix part is orthogonal." } }, "examples": [ { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } ] }, "rot_type": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Symbolic crystallographic operation-type label for the linear part.", "enum": [ "1", "-1", "2", "m", "3", "-3", "4", "-4", "6", "-6" ] }, "axis": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Integer-vector axis or invariant-direction descriptor for the operation.", "items": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "One integer component of the axis vector." } }, "sense": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Rotation sense/sign convention returned by the generator." }, "screw_glide": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Screw-axis or glide-plane component represented exactly as a list of fraction strings.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "origin_shift": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Origin-shift descriptor represented exactly as a list of fraction strings.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "examples": [ { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true }, "rot_type": "2", "sense": 0, "axis": [ 0, 0, 1 ], "screw_glide": [ "0", "0", "0" ], "origin_shift": [ "0", "0", "0" ] } ] }, "examples": [ [ { "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "x,y,z", "det": 1, "is_orthogonal": true }, "rot_type": "1", "sense": 0, "axis": [ 0, 0, 0 ] } ] ] }, "schoenflies_markup": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/pointgroups/schoenflies_markup", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Schoenflies symbol markups", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "schoenflies_markup", "label": "schoenflies_markup_pointgroups" }, "description": "Display-oriented renderings of the Schoenflies symbol in `schoenflies`.\nThe plain string value is stored in the corresponding unsuffixed property; this object only provides alternate markup forms for display.", "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "properties": { "html": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "HTML rendering of the sibling string." }, "latex": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "LaTeX rendering of the sibling string." }, "unicode": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Unicode rendering of the sibling string." } }, "examples": [ { "html": "P 21/c", "latex": "\\mathit{P}\\,2_{1}/c", "unicode": "P2\u2081/c" } ] } } }, "examples": [ [ { "hm_symbol": "2/m", "order": 4 } ] ] } ```