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/character_table_real
Definition name: character_table_real
Property name: Real character table
Description: Real irreducible character table of the crystallographic point group.
Type: list
Rows correspond to real irreducible representations and columns follow the order of conjugacy_classes.
Characters of the real table are exact integers and are therefore stored as plain integers, unlike the symbolic re/im string pairs of character_table_complex.
Irreducibility here is over the real numbers.
A pair of inequivalent complex-conjugate irreps can combine into one real irrep whose dimension and character are the sums of the constituent dimensions and characters.
Consequently the number of real rows can be smaller than n_conjugacy_classes; such a paired row has character norm two under the class-size-weighted complex character inner product, not one.
Every character list MUST follow conjugacy_classes and have that list's length, and the identity character MUST equal dimension.
Polynomial bases span each isotypic component, meaning all copies of the indicated real irrep in the chosen polynomial space.
The number of listed polynomials can therefore exceed dimension, but must be an integer multiple of it.
Linear and axial spaces each have dimension three; the homogeneous quadratic space has dimension six, including the scalar trace x^2+y^2+z^2.
Basis lists are independent spanning sets, not promised to be orthonormal or partitioned into conventional multiplets; the generator omits a basis field when that irrep is absent from that polynomial space.
The coordinates in these polynomial strings are Cartesian variables, unlike the fractional-coordinate variables in symops.
For a Cartesian operation Q, polar coordinates transform by Q and axial coordinates by det(Q)*Q; quadratic polynomials transform by substitution.
The final example displays selected A1 and B1 rows of 4mm rather than a complete table; full emitted tables contain every real irrep.
Examples:
[{"label": "A", "dimension": 1, "characters": [1], "basis_linear": ["x", "y", "z"], "basis_rotation": ["Rx", "Ry", "Rz"], "basis_quadratic": ["x^2", "y^2", "z^2", "xy", "xz", "yz"], "label_markup": {"latex": "A", "unicode": "A"}}][{"label": "Ag", "dimension": 1, "characters": [1, 1], "basis_rotation": ["Rx", "Ry", "Rz"], "basis_quadratic": ["x^2", "y^2", "z^2", "xy", "xz", "yz"], "label_markup": {"latex": "A_{g}", "unicode": "Ag"}}, {"label": "Au", "dimension": 1, "characters": [1, -1], "basis_linear": ["x", "y", "z"], "label_markup": {"latex": "A_{u}", "unicode": "Au"}}][{"label": "A1", "dimension": 1, "characters": [1, 1, 1, 1, 1], "basis_linear": ["z"], "basis_quadratic": ["x^2+y^2", "z^2"], "label_markup": {"latex": "A_{1}", "unicode": "A\u2081"}}, {"label": "B1", "dimension": 1, "characters": [1, 1, -1, 1, -1], "basis_quadratic": ["x^2-y^2"], "label_markup": {"latex": "B_{1}", "unicode": "B\u2081"}}]JSON definition:
{
"$id": "https://schemas.httk.org/defs/v0.1/properties/pointgroups/character_table_real",
"$schema": "https://schemas.optimade.org/meta/v1.3/optimade/property_definition.json",
"title": "Real character table",
"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 symbolic `re`/`im` string pairs of `character_table_complex`.\n\nIrreducibility here is over the real numbers.\nA pair of inequivalent complex-conjugate irreps can combine into one real irrep whose dimension and character are the sums of the constituent dimensions and characters.\nConsequently the number of real rows can be smaller than `n_conjugacy_classes`; such a paired row has character norm two under the class-size-weighted complex character inner product, not one.\nEvery character list MUST follow `conjugacy_classes` and have that list's length, and the identity character MUST equal `dimension`.\nPolynomial bases span each isotypic component, meaning all copies of the indicated real irrep in the chosen polynomial space.\nThe number of listed polynomials can therefore exceed `dimension`, but must be an integer multiple of it.\nLinear and axial spaces each have dimension three; the homogeneous quadratic space has dimension six, including the scalar trace `x^2+y^2+z^2`.\nBasis lists are independent spanning sets, not promised to be orthonormal or partitioned into conventional multiplets; the generator omits a basis field when that irrep is absent from that polynomial space.\nThe coordinates in these polynomial strings are Cartesian variables, unlike the fractional-coordinate variables in `symops`.\nFor a Cartesian operation Q, polar coordinates transform by Q and axial coordinates by `det(Q)*Q`; quadratic polynomials transform by substitution.\nThe final example displays selected A1 and B1 rows of 4mm rather than a complete table; full emitted tables contain every real irrep.",
"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.httk.org/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": "<i>P</i> 2<sub>1</sub>/<i>c</i>",
"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": "Optional Frobenius-Schur annotation; the current generator does not emit this field in the real table.\nThe indicator convention is defined for complex irreducible rows in `character_table_complex`; do not assume a real row formed from a conjugate pair is itself complex-irreducible or has indicator one.",
"x-optimade-unit": "inapplicable"
},
"basis_linear": {
"x-optimade-type": "list",
"type": [
"array",
"null"
],
"description": "Linear basis polynomials spanning the isotypic component of this representation in the space of linear functions.\nPolynomials use an orthonormal Cartesian frame (x along a, z along c for hexagonal axes) with exact rational coefficients, e.g. `x`, `x+y`.\nThe listed polynomials are linearly independent; repeated copies of a representation are not separated into conventional multiplets.",
"items": {
"x-optimade-type": "string",
"type": [
"string",
"null"
],
"description": "One basis polynomial in Cartesian x, y, z with rational coefficients.",
"x-optimade-unit": "inapplicable"
},
"x-optimade-unit": "inapplicable"
},
"basis_rotation": {
"x-optimade-type": "list",
"type": [
"array",
"null"
],
"description": "Axial-vector (rotation) basis polynomials spanning the isotypic component of this representation, in the same Cartesian frame as `basis_linear`, e.g. `Rz`, `Rx+Ry`.",
"items": {
"x-optimade-type": "string",
"type": [
"string",
"null"
],
"description": "One basis polynomial in Rx, Ry, Rz with rational coefficients.",
"x-optimade-unit": "inapplicable"
},
"x-optimade-unit": "inapplicable"
},
"basis_quadratic": {
"x-optimade-type": "list",
"type": [
"array",
"null"
],
"description": "Quadratic basis polynomials spanning the isotypic component of this representation in the space of homogeneous quadratic functions, in the same Cartesian frame as `basis_linear`, e.g. `x^2+y^2`, `x^2-y^2`, `xz`.",
"items": {
"x-optimade-type": "string",
"type": [
"string",
"null"
],
"description": "One quadratic basis polynomial in x^2, y^2, z^2, xy, xz, yz with rational coefficients.",
"x-optimade-unit": "inapplicable"
},
"x-optimade-unit": "inapplicable"
}
},
"x-optimade-unit": "inapplicable"
},
"examples": [
[
{
"label": "A",
"dimension": 1,
"characters": [
1
],
"basis_linear": [
"x",
"y",
"z"
],
"basis_rotation": [
"Rx",
"Ry",
"Rz"
],
"basis_quadratic": [
"x^2",
"y^2",
"z^2",
"xy",
"xz",
"yz"
],
"label_markup": {
"latex": "A",
"unicode": "A"
}
}
],
[
{
"label": "Ag",
"dimension": 1,
"characters": [
1,
1
],
"basis_rotation": [
"Rx",
"Ry",
"Rz"
],
"basis_quadratic": [
"x^2",
"y^2",
"z^2",
"xy",
"xz",
"yz"
],
"label_markup": {
"latex": "A_{g}",
"unicode": "Ag"
}
},
{
"label": "Au",
"dimension": 1,
"characters": [
1,
-1
],
"basis_linear": [
"x",
"y",
"z"
],
"label_markup": {
"latex": "A_{u}",
"unicode": "Au"
}
}
],
[
{
"label": "A1",
"dimension": 1,
"characters": [
1,
1,
1,
1,
1
],
"basis_linear": [
"z"
],
"basis_quadratic": [
"x^2+y^2",
"z^2"
],
"label_markup": {
"latex": "A_{1}",
"unicode": "A\u2081"
}
},
{
"label": "B1",
"dimension": 1,
"characters": [
1,
1,
-1,
1,
-1
],
"basis_quadratic": [
"x^2-y^2"
],
"label_markup": {
"latex": "B_{1}",
"unicode": "B\u2081"
}
}
]
]
}