httk property definitionsProperty definitions using the OPTIMADE property definition format, hosted by httk
This page documents an OPTIMADE Property Definition. See https://schemas.optimade.org/ for more information.
ID: https://schemas.httk.org/defs/v0.1/properties/pointgroups/conjugacy_classes
Definition name: conjugacy_classes
Property name: Conjugacy classes
Description: Conjugacy classes of a crystallographic point group.
Type: list
Each class lists its member operation indices, a representative operation, a conventional class label, and operation-type metadata used by the character tables.
Operation indices refer to positions in the symops list of the same point-group record, starting at 0.
Requirements/Conventions:
1, 2, 3, 4, 6 for proper rotations, and the negated value for the corresponding rotoinversions, with -2 denoting a mirror plane.Two operations g and h belong to the same class precisely when h = a*g*a^-1 for an operation a of this point group.
The members lists MUST partition the indices of symops, and representative MUST be a member of its class.
Class sizes therefore sum to order.
The class order defines character-table columns and must not be changed without applying the same permutation to every character row.
Class labels use Schoenflies-style rotation-reflection notation: S_n combines a rotation through 2*pi/n with reflection perpendicular to its axis.
For odd n the order of S_n is 2*n, so its inverse is S_n^(2*n-1); these powers must not be confused with crystallographic rotoinversion type codes in op_type.
The reported op_axis follows the same direct-lattice direction convention as symops[representative].axis; labels are display aids, while the operation matrix fixes the action.
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]}]JSON definition:
{
"$id": "https://schemas.httk.org/defs/v0.1/properties/pointgroups/conjugacy_classes",
"$schema": "https://schemas.optimade.org/meta/v1.3/optimade/property_definition.json",
"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.\n\nTwo operations g and h belong to the same class precisely when `h = a*g*a^-1` for an operation a of this point group.\nThe `members` lists MUST partition the indices of `symops`, and `representative` MUST be a member of its class.\nClass sizes therefore sum to `order`.\nThe class order defines character-table columns and must not be changed without applying the same permutation to every character row.\nClass labels use Schoenflies-style rotation-reflection notation: `S_n` combines a rotation through `2*pi/n` with reflection perpendicular to its axis.\nFor odd n the order of `S_n` is `2*n`, so its inverse is `S_n^(2*n-1)`; these powers must not be confused with crystallographic rotoinversion type codes in `op_type`.\nThe reported `op_axis` follows the same direct-lattice direction convention as `symops[representative].axis`; labels are display aids, while the operation matrix fixes the action.",
"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,
null
],
"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
]
}
]
]
}