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/transformations/wyckoff_splitting
Definition name: wyckoff_splitting
Property name: Wyckoff splitting
Description: Wyckoff-position splitting data associated with a subgroup or same-space-group transform.
Type: list
Each list item gives the split of one parent Wyckoff position.
The parent Wyckoff letter is stored in the parent field rather than as a JSON dictionary key.
Requirements/Conventions:
parent, the Wyckoff letter in the parent setting.splits, an ordered list of subgroup Wyckoff-position assignments, each carrying its target representative expression and exact affine map.For an embedding x_G = P*x_H + p, the split records partition the parent orbit, expressed in the subgroup cell, into distinct subgroup orbits.
For each piece let A be the first three columns of affine and b its last column.
Then q_H = A*q_G + b, where q_G is a fractional coordinate on the parent's published first_orbit branch and q_H is on the child's published representative branch.
The input is the actual parent representative coordinate, not its free-parameter vector; first evaluate the parent's orbit[0] map when starting from parameters.
The piece's xyz names the child representative branch and MUST equal that child's first_orbit; it is not a rendering of the piece's affine map on parent coordinates.
Repeated child letters are meaningful: they identify different child orbits with the same Wyckoff type and MUST NOT be deduplicated by letter.
For generic parent parameters, expanding all split pieces under the subgroup gives disjoint orbits whose union is the transformed parent orbit in the subgroup cell.
Their multiplicities sum to abs(det(P)) * parent_multiplicity; this factor accounts for the cells and is not generally the full subgroup index.
Keep the exact affine offsets when evaluating the maps; wrapping parent coordinates before applying a non-unimodular map can select a different child orbit.
Examples:
[{"parent": "a", "splits": [{"letter": "a", "xyz": "x,y,z", "affine": [["1", "0", "0", "0"], ["0", "1", "0", "0"], ["0", "0", "1/2", "0"]]}, {"letter": "a", "xyz": "x,y,z", "affine": [["1", "0", "0", "0"], ["0", "1", "0", "0"], ["0", "0", "1/2", "1/2"]]}]}]JSON definition:
{
"$id": "https://schemas.httk.org/defs/v0.1/properties/transformations/wyckoff_splitting",
"$schema": "https://schemas.optimade.org/meta/v1.3/optimade/property_definition.json",
"title": "Wyckoff splitting",
"x-optimade-type": "list",
"x-optimade-definition": {
"kind": "property",
"version": "0.1.0",
"format": "1.3",
"name": "wyckoff_splitting",
"label": "wyckoff_splitting_transformations"
},
"x-optimade-unit": "inapplicable",
"type": [
"array",
"null"
],
"description": "Wyckoff-position splitting data associated with a subgroup or same-space-group transform.\n\nEach list item gives the split of one parent Wyckoff position.\nThe parent Wyckoff letter is stored in the `parent` field rather than as a JSON dictionary key.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain `parent`, the Wyckoff letter in the parent setting.\n- Each dictionary MUST contain `splits`, an ordered list of subgroup Wyckoff-position assignments, each carrying its target representative expression and exact affine map.\n\nFor an embedding `x_G = P*x_H + p`, the split records partition the parent orbit, expressed in the subgroup cell, into distinct subgroup orbits.\nFor each piece let `A` be the first three columns of `affine` and `b` its last column.\nThen `q_H = A*q_G + b`, where `q_G` is a fractional coordinate on the parent's published `first_orbit` branch and `q_H` is on the child's published representative branch.\nThe input is the actual parent representative coordinate, not its free-parameter vector; first evaluate the parent's `orbit[0]` map when starting from parameters.\nThe piece's `xyz` names the child representative branch and MUST equal that child's `first_orbit`; it is not a rendering of the piece's `affine` map on parent coordinates.\nRepeated child letters are meaningful: they identify different child orbits with the same Wyckoff type and MUST NOT be deduplicated by letter.\nFor generic parent parameters, expanding all split pieces under the subgroup gives disjoint orbits whose union is the transformed parent orbit in the subgroup cell.\nTheir multiplicities sum to `abs(det(P)) * parent_multiplicity`; this factor accounts for the cells and is not generally the full subgroup index.\nKeep the exact affine offsets when evaluating the maps; wrapping parent coordinates before applying a non-unimodular map can select a different child orbit.",
"items": {
"x-optimade-type": "dictionary",
"x-optimade-unit": "inapplicable",
"type": [
"object"
],
"description": "Splitting data for one parent Wyckoff position.",
"required": [
"parent",
"splits"
],
"properties": {
"parent": {
"x-optimade-type": "string",
"x-optimade-unit": "inapplicable",
"type": [
"string"
],
"description": "Parent Wyckoff letter."
},
"splits": {
"x-optimade-type": "list",
"x-optimade-unit": "inapplicable",
"type": [
"array"
],
"description": "Ordered split records for this parent Wyckoff letter.",
"items": {
"x-optimade-type": "dictionary",
"x-optimade-unit": "inapplicable",
"type": [
"object"
],
"description": "One Wyckoff split record.",
"required": [
"letter",
"xyz",
"affine"
],
"properties": {
"letter": {
"x-optimade-type": "string",
"x-optimade-unit": "inapplicable",
"type": [
"string"
],
"description": "Subgroup Wyckoff letter assigned by this split branch."
},
"xyz": {
"x-optimade-type": "string",
"x-optimade-unit": "inapplicable",
"type": [
"string"
],
"description": "Coordinate expression for the split branch."
},
"affine": {
"x-optimade-type": "list",
"x-optimade-unit": "inapplicable",
"x-optimade-dimensions": {
"names": [
"dim_lattice",
"dim_affine"
],
"sizes": [
3,
4
]
},
"type": [
"array"
],
"description": "Exact affine representation for the split branch as a 3 by 4 augmented matrix.\nEach row holds the three linear coefficients followed by the translation component, all as exact fraction strings.",
"items": {
"x-optimade-type": "list",
"x-optimade-unit": "inapplicable",
"x-optimade-dimensions": {
"names": [
"dim_affine"
],
"sizes": [
4
]
},
"type": [
"array"
],
"description": "One row of the augmented affine matrix.",
"items": {
"$id": "https://schemas.httk.org/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": [
[
{
"parent": "a",
"splits": [
{
"letter": "a",
"xyz": "x,y,z",
"affine": [
[
"1",
"0",
"0",
"0"
],
[
"0",
"1",
"0",
"0"
],
[
"0",
"0",
"1/2",
"0"
]
]
},
{
"letter": "a",
"xyz": "x,y,z",
"affine": [
[
"1",
"0",
"0",
"0"
],
[
"0",
"1",
"0",
"0"
],
[
"0",
"0",
"1/2",
"1/2"
]
]
}
]
}
]
]
}