Basis Set

Description

A description of a cluster expansion basis set.

Project files

This format is used for the following standard CASM project files:

basis.json: This file is generated by CASM when a basis set is constructed using the casm bset --update method.
eci.json: This file, with eci values included, is used to input coefficients. It may be generated using the casm learn --select method after using casm learn to fit coefficients, or it can be created manually by copying basis.json and adding eci values.

JSON Attributes List

Basis Set attributes:

Name	Description	Format
`bspecs`	Basis set specifications	Basis Set Specs
`orbits`	Cluster orbits and basis set functions	array of Cluster Basis Set Orbit
`prim`	Primitive crystal structure and DoF	Prim
`site_functions`	Description of site functions at each sublattice	array of Site Functions

Cluster Basis Set Orbit attributes:

Name	Description	Format
`cluster_functions`	Basis functions of the prototype cluster DoF values	array of Cluster Function
`linear_orbit_index`	Linear cluster orbit index	int
`mult`	Number of equivalent clusters in the orbit	int
`prototype`	Representation of the prototype cluster	Prototype Cluster

Cluster Function attributes:

Name	Description	Format
`\\Phi_{<linear_function_index>}`	Basis function formula, formatted for Latex	string
`linear_function_index`	Linear basis function index	int
`eci`	Effective cluster interaction value	number

Prototype Cluster attributes:

Name	Description	Format
`invariant_group`	Indices of prim factor group operations that leave the prototype cluster invariant	array of int
`invariant_group_descriptions`	Prototype cluster invariant group operation descriptions	array of string
`max_length`	Maximum distance between sites in the cluster.	number
`min_length`	Minimum distance between sites in the cluster.	number
`sites`	Cluster site coordinates	array of integral site coordinates

Site Functions attributes:

Name	Description	Format
`asym_unit`	Asymmetric unit index for the sublattice	int
`occ`	Occupation site basis functions	dict
`<site_dof_key>`	Continuous DoF site basis functions	dict
`sublat`	Sublattice index	int

JSON Attributes Description

Basis Set JSON object

bspecs: Basis Function Specs

A copy of the basis set specifications (bspecs.json) used to generate the basis set.
orbits: array of Cluster Basis Set Orbit.

An array of cluster basis set orbit objects, one for each orbit generated, including the null cluster orbit.
prim: Prim

A copy of the prim used to generate the basis set.
site_functions: array of Site Functions, shape=(n_sublattice,)

The site_functions array contains one site functions object for each sublattice in the prim.

Cluster Basis Set Orbit JSON object

cluster_functions: array of Cluster Function

An array describing the cluster functions associated with the prototype cluster.
linear_orbit_index: int

Linear cluster orbit index
mult: int

Cluster multiplicity, the number of symmetrically equivalent clusters in the orbit.
prototype: Prototype Cluster

Information about one cluster (the “prototype”) in the orbit of symmetrically equivalent clusters.

Cluster Function JSON object

\\Phi_{<linear_function_index>}: string

Basis function formula, formatted for Latex. This is the formula for the basis function evaluated from site DoF values on the prototype cluster sites and global DoF values. The complete basis set also includes symmetrically the equivalent basis functions for each cluster in the orbit.
Example:
```
"\\Phi_{24}" : "\\sqrt{1/2}(e_1\\phi_{0,0}(s_{0})\\phi_{0,0}(s_{1})+e_3\\phi_{0,0}(s_{0})\\phi_{0,0}(s_{1}))"
```
In this example:
- \\phi_{b,i} are site basis functions, where b is a sublattice index and i is a function index at that site
- s_{0}, s_{1} are the occupation values at sites in the clusters, where <site_index> is an index into the sites in the cluster.
- e_1, e_3 are strain DoF component values
- In general, other site and global DoF component values may also be included in the formulas using the standard basis names or user-specified axis_names as defined in the prim.
linear_function_index: int

Linear basis function index.
eci: number (optional)

Basis function coefficient value. Not printed in basis.json. This is an optional value in the eci.json file. Not including the eci attribute indicates the coefficient value is zero and the corresponding basis functions do not need to be evaluated for property prediction.

Site functions JSON object

asym_unit: int

Asymmetric unit index. Sublattices with the same asym_unit value are equivalent under prim factor group operations.

occ: dict

Specifies the value of occupation site basis functions as a function of which occupant is at the site. The discrete site basis functions are named \\phi_{b,i}, where b is the sublattice index and i is an index over site basis functions on this sublattice.

Example: Discrete site basis functions, for a ternary system, using the "occupation" basis

"occ" : {
  "basis" : {
    "\\phi_{0,0}" : {
      "A" : 0.000000000000,
      "B" : 1.000000000000,
      "C" : 0.000000000000
    },
    "\\phi_{0,1}" : {
      "A" : 0.000000000000,
      "B" : 0.000000000000,
      "C" : 1.000000000000
    }
  }
}

Example: Discrete site basis functions, for a ternary system, using the "chebychev" basis

"occ" : {
  "basis" : {
    "\\phi_{0,0}" : {
      "A" : -1.224744871392,
      "B" : 0.000000000000,
      "C" : 1.224744871392
    },
    "\\phi_{0,1}" : {
      "A" : -0.707106781187,
      "B" : 1.414213562373,
      "C" : -0.707106781187
    }
  }
}

<site_dof_key>: dict

Specifies the value of continuous site DoF basis functions as a function of the vector components, using the standard or user-specified axis names. The <site_dof_key> is the site DoF type as specified in the [prim] dofs dict for this sublattice. The continuous site basis functions are named var_{b}_{i}, where b is the sublattice index and i is an index over continuous site basis functions of this type on this sublattice.
Example: Atomic displacement site basis functions, using standard basis
```
"disp" : {
  "basis" : {
    "var_{0}_{0}" : "dx_{0}",
    "var_{0}_{1}" : "dy_{0}",
    "var_{0}_{2}" : "dz_{0}"
  }
}
```
Example: Atomic displacement site basis functions, using user-specified basis in a 2d subspace
```
"disp" : {
  "basis" : {
    "var_{0}_{0}" : "d1_{0}",
    "var_{0}_{1}" : "d2_{0}"
  }
}
```
sublat: int

Sublattice index.