amd.calculate module¶

Functions for calculating the average minimum distance (AMD) and point-wise distance distribution (PDD) isometric invariants of periodic crystals and finite sets.

amd.calculate.AMD(pset: amd.periodicset.PeriodicSet, k: int) → numpy.ndarray[Any, numpy.dtype[numpy.floating]]¶

Return the average minimum distance (AMD) of a periodic set (crystal).

The AMD of a periodic set is a geometry based descriptor independent of choice of motif and unit cell. It is a vector, the (weighted) average of the PDD matrix, which has one row for each (unique) point in the unit cell containing distances to k nearest neighbors.

Parameters

pset (amd.PeriodicSet) – A periodic set (crystal) consisting of a unit cell and motif of points.
k (int) – The number of neighboring points (atoms) considered for each point in the unit cell.

Returns

A numpy.ndarray shape (k, ), the AMD of pset up to k.

Return type

numpy.ndarray

Examples

Make list of AMDs with k = 100 for crystals in data.cif:

amds = []
for periodic_set in amd.CifReader('data.cif'):
    amds.append(amd.AMD(periodic_set, 100))

Make list of AMDs with k = 10 for crystals in these CSD refcode families:

amds = []
for periodic_set in amd.CSDReader(['HXACAN', 'ACSALA'], families=True):
    amds.append(amd.AMD(periodic_set, 10))

Manually create a periodic set as a tuple (motif, cell):

# simple cubic lattice
motif = np.array([[0,0,0]])
cell = np.array([[1,0,0], [0,1,0], [0,0,1]])
periodic_set = amd.PeriodicSet(motif, cell)
cubic_amd = amd.AMD(periodic_set, 100)

amd.calculate.PDD(pset: amd.periodicset.PeriodicSet, k: int, lexsort: bool = True, collapse: bool = True, collapse_tol: float = 0.0001, return_row_groups: bool = False) → Union[numpy.ndarray[Any, numpy.dtype[numpy.floating]], Tuple[numpy.ndarray[Any, numpy.dtype[numpy.floating]], list]]¶

Return the point-wise distance distribution (PDD) of a periodic set (crystal).

The PDD of a periodic set is a geometry based descriptor independent of choice of motif and unit cell. It is a matrix where each row corresponds to a point in the motif, containing a weight followed by distances to the k nearest neighbors of the point.

Parameters

pset (amd.PeriodicSet) – A periodic set (crystal) consisting of a unit cell and motif of points.
k (int) – The number of neighbors considered for each atom (point) in the unit cell. The returned matrix has k + 1 columns, the first column for weights of rows.
lexsort (bool, default True) – Lexicographically order the rows.
collapse (bool, default True) – Collapse repeated rows (within tolerance collapse_tol).
collapse_tol (float, default 1e-4) – If two rows have all elements closer than collapse_tol, they are merged and weights are given to rows in proportion to the number of times they appeared.
return_row_groups (bool, default False) – If True, return a tuple (pdd, groups) where groups contains information about which rows in pdd correspond to which points. If pset.asym_unit is None, then groups[i] contains indices of points in pset.motif corresponding to pdd[i]. Otherwise, PDD rows correspond to points in the asymmetric unit, and groups[i] contains indices of points in pset.motif[pset.asym_unit].

Returns

pdd – A numpy.ndarray with k+1 columns, the PDD of pset up to k. The first column contains the weights of rows. If return_row_groups is True, returns a tuple with types (numpy.ndarray, list).

Return type

numpy.ndarray

Examples

Make list of PDDs with k=100 for crystals in data.cif:

pdds = []
for periodic_set in amd.CifReader('data.cif'):
    # do not lexicographically order rows
    pdds.append(amd.PDD(periodic_set, 100, lexsort=False))

Make list of PDDs with k=10 for crystals in these CSD refcode families:

pdds = []
for periodic_set in amd.CSDReader(['HXACAN', 'ACSALA'], families=True):
    # do not collapse rows
    pdds.append(amd.PDD(periodic_set, 10, collapse=False))

Manually create a periodic set as a tuple (motif, cell):

# simple cubic lattice
motif = np.array([[0,0,0]])
cell = np.array([[1,0,0], [0,1,0], [0,0,1]])
periodic_set = amd.PeriodicSet(motif, cell)
cubic_amd = amd.PDD(periodic_set, 100)

amd.calculate.PDD_to_AMD(pdd: numpy.ndarray[Any, numpy.dtype[numpy.floating]]) → numpy.ndarray[Any, numpy.dtype[numpy.floating]]¶

Calculate an AMD from a PDD. Faster than computing both from scratch.

Parameters: pdd (numpy.ndarray) – The PDD of a periodic set, so amd.PDD_to_AMD(amd.PDD(pset)) equals amd.AMD(pset).
Returns: The AMD of the periodic set.
Return type: numpy.ndarray

amd.calculate.AMD_finite(motif: numpy.ndarray[Any, numpy.dtype[numpy.floating]]) → numpy.ndarray[Any, numpy.dtype[numpy.floating]]¶

Return the AMD of a finite m-point set up to k = m - 1.

Parameters: motif (numpy.ndarray) – Collection of points.
Returns: A vector shape (motif.shape[0] - 1, ), the AMD of motif.
Return type: numpy.ndarray

Examples

The (L-infinity) AMD distance between finite trapezium and kite point sets:

trapezium = np.array([[0,0],[1,1],[3,1],[4,0]])
kite      = np.array([[0,0],[1,1],[1,-1],[4,0]])

trap_amd = amd.AMD_finite(trapezium)
kite_amd = amd.AMD_finite(kite)

l_inf_dist = np.amax(np.abs(trap_amd - kite_amd))

amd.calculate.PDD_finite(motif: numpy.ndarray[Any, numpy.dtype[numpy.floating]], lexsort: bool = True, collapse: bool = True, collapse_tol: float = 0.0001, return_row_groups: bool = False) → Union[numpy.ndarray[Any, numpy.dtype[numpy.floating]], Tuple[numpy.ndarray[Any, numpy.dtype[numpy.floating]], list]]¶

Return the PDD of a finite m-point set up to k = m - 1.

Parameters

motif (numpy.ndarray) – Coordinates of a set of points.
lexsort (bool, default True) – Whether or not to lexicographically order the rows.
collapse (bool, default True) – Whether or not to collapse repeated rows (within tolerance collapse_tol).
collapse_tol (float, default 1e-4) – If two rows have all elements closer than collapse_tol, they are merged and weights are given to rows in proportion to the number of times they appeared.
return_row_groups (bool, default False) – If True, return a tuple (pdd, groups) where groups[i] contains indices of points in motif corresponding to pdd[i].

Returns

pdd – A numpy.ndarray with m columns (where m is the number of points), the PDD of motif. The first column contains the weights of rows.

Return type

numpy.ndarray

Examples

Find PDD distance between finite trapezium and kite point sets:

trapezium = np.array([[0,0],[1,1],[3,1],[4,0]])
kite      = np.array([[0,0],[1,1],[1,-1],[4,0]])

trap_pdd = amd.PDD_finite(trapezium)
kite_pdd = amd.PDD_finite(kite)

dist = amd.EMD(trap_pdd, kite_pdd)

amd.calculate.PDD_reconstructable(pset: amd.periodicset.PeriodicSet, lexsort: bool = True) → numpy.ndarray[Any, numpy.dtype[numpy.floating]]¶

Return the PDD of a periodic set with k (number of columns) large enough such that the periodic set can be reconstructed from the PDD.

Parameters

pset (amd.PeriodicSet) – A periodic set (crystal) consisting of a unit cell and motif of points.
lexsort (bool, default True) – Whether or not to lexicographically order the rows.

Returns

pdd – The PDD of pset with enough columns to reconstruct pset using amd.reconstruct.reconstruct().

Return type

numpy.ndarray

amd.calculate.PPC(pset: amd.periodicset.PeriodicSet) → float¶

Return the point packing coefficient (PPC) of pset.

The PPC is a constant of any periodic set determining the asymptotic behaviour of its AMD and PDD. As \(k \rightarrow \infty\), the ratio \(\text{AMD}_k / \sqrt[n]{k}\) converges to the PPC, as does any row of its PDD.

For a unit cell \(U\) and \(m\) motif points in \(n\) dimensions,

\[\text{PPC} = \sqrt[n]{\frac{\text{Vol}[U]}{m V_n}}\]

where \(V_n\) is the volume of a unit sphere in \(n\) dimensions.

Parameters: pset (amd.PeriodicSet) – A periodic set (crystal) consisting of a unit cell and motif of points.
Returns: ppc – The PPC of pset.
Return type: float

amd.calculate.AMD_estimate(pset: amd.periodicset.PeriodicSet, k: int) → numpy.ndarray[Any, numpy.dtype[numpy.floating]]¶

Calculate an estimate of AMD based on the PPC.

Parameters: pset (amd.PeriodicSet) – A periodic set (crystal) consisting of a unit cell and motif of points.
Returns: amd_est – An array shape (k, ), where amd_est[i] \(= \text{PPC} \sqrt[n]{k}\) in n dimensions.
Return type: numpy.ndarray