Cluster space

Setting up a cluster space

The cluster space serves as a basis for cluster expansions. Setting up a clusterspace object is commonly done via

>>> from icet import ClusterSpace
>>> from ase.build import bulk
>>> primitive_structure = bulk('Si')
>>> cs = ClusterSpace(primitive_structure, cutoffs=[7.0, 5.0], chemical_symbols=['Si', 'Ge'])

The cutoffs are set up to include pairs with an interatomic distance smaller than 7 Å and triplets for which all pair-wise interatomic distances are smaller than 5 Å. Here, Si and Ge are allowed to occupy the reference lattice.

Sublattices

A cluster space can also be constructed with multiple sublattices. Consider for example a rocksalt lattice, on which we want to allow mixing of Na and Li on the Na sublattice and Cl and F on the Cl sublattices. This can be achived by

>>> from ase.build import bulk
>>> atoms = bulk('NaCl', 'rocksalt', a=4.0)
>>> chemical_symbols = [['Na', 'Li'], ['Cl', 'F']]
>>> cs = ClusterSpace(atoms, [7.0, 5.0], chemical_symbols)

where chemical_symbols now specifies which species are allowed for each lattice site in atoms.

Inactive sites

The sublattice functionality also allows one to have inactive sites. For example, if we consider the system above but would like to keep the Cl lattice fixed it can be achived via

>>> chemical_symbols = [['Na', 'Li'], ['Cl']]
>>> cs = ClusterSpace(atoms, [7.0, 5.0], chemical_symbols)

2D systems

icet requires input structures to have periodic boundary conditions (PBCs). In order to treat two-dimensional systems, or more generally without PBCs in at least one direction, one has to surround the prototype structure with vacuum and then apply PBCs in all directions. This can easily be achived with the ASE functions

>>> from ase.build import surface
>>> atoms = surface('Cu', (1,1,1), layers=7)

which creates ase atoms object without PBC in the z-direction. The structure can be modified to have PBC in the z-direction with vacuum via

>>> atoms.pbc = [True, True, True]
>>> atoms.center(vacuum=20, axis=2)

which can now be used to create a cluster space.

Interface

class icet.ClusterSpace(structure, cutoffs, chemical_symbols, symprec=1e-05, position_tolerance=None)[source]

This class provides functionality for generating and maintaining cluster spaces.

Note: In icet all ase.Atoms objects must have periodic boundary conditions. When carrying out cluster expansions for surfaces and nanoparticles it is therefore recommended to surround the structure with vacuum and use periodic boundary conditions. This can be done using e.g., ase.Atoms.center().

Parameters
  • structure (ase.Atoms) – atomic structure

  • cutoffs (list(float)) –

    cutoff radii per order that define the cluster space

    Cutoffs are specified in units of Angstrom and refer to the longest distance between two atoms in the cluster. The first element refers to pairs, the second to triplets, the third to quadruplets, and so on. cutoffs=[7.0, 4.5] thus implies that all pairs distanced 7 A or less will be included, as well as all triplets among which the longest distance is no longer than 4.5 A.

  • chemical_symbols (list(str) or list(list(str))) –

    list of chemical symbols, each of which must map to an element of the periodic table

    If a list of chemical symbols is provided, all sites on the lattice will have the same allowed occupations as the input list.

    If a list of list of chemical symbols is provided then the outer list must be the same length as the structure object and chemical_symbols[i] will correspond to the allowed species on lattice site i.

  • symprec (float) – tolerance imposed when analyzing the symmetry using spglib

  • position_tolerance (float) – tolerance applied when comparing positions in Cartesian coordinates

Examples

The following snippets illustrate several common situations:

>>> from ase.build import bulk
>>> from ase.io import read
>>> from icet import ClusterSpace

>>> # AgPd alloy with pairs up to 7.0 A and triplets up to 4.5 A
>>> prim = bulk('Ag')
>>> cs = ClusterSpace(structure=prim, cutoffs=[7.0, 4.5],
...                   chemical_symbols=[['Ag', 'Pd']])
>>> print(cs)

>>> # (Mg,Zn)O alloy on rocksalt lattice with pairs up to 8.0 A
>>> prim = bulk('MgO', crystalstructure='rocksalt', a=6.0)
>>> cs = ClusterSpace(structure=prim, cutoffs=[8.0],
...                   chemical_symbols=[['Mg', 'Zn'], ['O']])
>>> print(cs)

>>> # (Ga,Al)(As,Sb) alloy with pairs, triplets, and quadruplets
>>> prim = bulk('GaAs', crystalstructure='zincblende', a=6.5)
>>> cs = ClusterSpace(structure=prim, cutoffs=[7.0, 6.0, 5.0],
...                   chemical_symbols=[['Ga', 'Al'], ['As', 'Sb']])
>>> print(cs)

>>> # PdCuAu alloy with pairs and triplets
>>> prim = bulk('Pd')
>>> cs = ClusterSpace(structure=prim, cutoffs=[7.0, 5.0],
...                   chemical_symbols=[['Au', 'Cu', 'Pd']])
>>> print(cs)
assert_structure_compatibility(structure, vol_tol=1e-05)[source]

Raises error if structure is not compatible with ClusterSpace.

Todo

Add check for if structure is relaxed.

Parameters

structure (Atoms) – structure to check if compatible with ClusterSpace

Return type

None

property chemical_symbols

Species identified by their chemical symbols

Return type

List[List[str]]

copy()[source]

Returns copy of ClusterSpace instance.

property cutoffs

Cutoffs for different n-body clusters. The cutoff radius (in Angstroms) defines the largest interatomic distance in a cluster.

Return type

List[float]

evaluate_cluster_function()

Evaluates value of a cluster function.

property fractional_position_tolerance

tolerance applied when comparing positions in fractional coordinates

Return type

float

get_chemical_symbols()

Returns list of species associated with cluster space as chemical symbols.

get_cluster_vector(structure)[source]

Returns the cluster vector for a structure.

Parameters

structure (Atoms) – atomic configuration

Returns

Return type

the cluster vector

get_coordinates_of_representative_cluster(orbit_index)[source]

Returns the positions of atoms in the selected orbit

Parameters

orbit_index (int) – index of the orbit from which to calculate the positions of the atoms

Returns

Return type

list of positions of atoms in the selected orbit

get_cutoffs()
get_multi_component_vector_permutations()
get_multi_component_vectors_by_orbit()
get_number_of_allowed_species_by_site()
get_number_of_orbits_by_order()[source]

Returns the number of orbits by order.

Return type

OrderedDict

Returns

  • an ordered dictionary where keys and values represent order and number

  • of orbits, respectively

get_orbit()
get_possible_orbit_occupations(orbit_index)[source]

Returns possible occupation of the orbit.

Parameters

orbit_index (int) –

Return type

List[List[str]]

get_sublattices(structure)[source]

Returns the sublattices of the input structure.

Parameters

structure (Atoms) – structure the sublattices are based on

Return type

Sublattices

is_supercell_self_interacting(structure)[source]

Checks whether an structure has self-interactions via periodic boundary conditions.

Parameters

structure (Atoms) – structure to be tested

Returns

If True, the structure contains self-interactions via periodic boundary conditions, otherwise False.

Return type

bool

property orbit_data

list of orbits with information regarding order, radius, multiplicity etc

Return type

List[dict]

property orbit_list

Orbit list that defines the cluster in the cluster space

property position_tolerance

tolerance applied when comparing positions in Cartesian coordinates

Return type

float

property primitive_structure

Primitive structure on which cluster space is based

Return type

Atoms

print_overview(print_threshold=None, print_minimum=10)[source]

Print an overview of the cluster space in terms of the orbits (order, radius, multiplicity etc).

Parameters
  • print_threshold (Optional[int]) – if the number of orbits exceeds this number print dots

  • print_minimum (int) – number of lines printed from the top and the bottom of the orbit list if print_threshold is exceeded

Return type

None

static read(filename)[source]

Reads cluster space from filename.

Parameters

filename (str) – name of file from which to read cluster space

property space_group

space group of the primitive structure in international notion (via spglib)

Return type

str

property species_maps
property symprec

tolerance imposed when analyzing the symmetry using spglib

Return type

float

write(filename)[source]

Saves cluster space to a file.

Parameters

filename (str) – name of file to which to write

Return type

None