SparseOrbitalBZSpin¶

class sisl.physics.SparseOrbitalBZSpin(geometry, dim=1, dtype=None, nnzpr=None, **kwargs)[source]¶

Sparse object containing the orbital connections in a Brillouin zone with possible spin-components

It contains an intrinsic sparse matrix of the physical elements.

Assigning or changing elements is as easy as with standard numpy assignments:

>>> S = SparseOrbitalBZSpin(...) 
>>> S[1,2] = 0.1 

which assigns 0.1 as the element between orbital 2 and 3. (remember that Python is 0-based elements).

Parameters:

geometry : Geometry: parent geometry to create a sparse matrix from. The matrix will have size equivalent to the number of orbitals in the geometry
dim : int or Spin, optional: number of components per element, may be a Spin object
dtype : np.dtype, optional: data type contained in the matrix. See details of Spin for default values.
nnzpr : int, optional: number of initially allocated memory per orbital in the matrix. For increased performance this should be larger than the actual number of entries per orbital.
spin : Spin, optional: equivalent to dim argument. This keyword-only argument has precedence over dim.
orthogonal : bool, optional: whether the matrix corresponds to a non-orthogonal basis. In this case the dimensionality of the matrix is one more than dim. This is a keyword-only argument.

Attributes

`S`
`dim`	Number of components per element
`dkind`	Data type of sparse elements (in str)
`dtype`	Data type of sparse elements
`finalized`	Whether the contained data is finalized and non-used elements have been removed
`geom`	Associated geometry
`geometry`	Associated geometry
`nnz`	Number of non-zero elements
`non_orthogonal`	True if the object is using a non-orthogonal basis
`orthogonal`	True if the object is using an orthogonal basis
`shape`	Shape of sparse matrix
`spin`	Associated spin class

Methods

`Rij`([what, dtype])	Create a sparse matrix with the vectors between atoms/orbitals
`Sk`([k, dtype, gauge, format])	Setup the overlap matrix for a given k-point
`__init__`(geometry[, dim, dtype, nnzpr])	Create sparse object with element between orbitals
`construct`(func[, na_iR, method, eta])	Automatically construct the sparse model based on a function that does the setting up of the elements
`copy`([dtype])	A copy of this object
`create_construct`(R, param)	Create a simple function for passing to the `construct` function.
`cut`(seps, axis, args, *kwargs)	Cuts the sparse orbital model into different parts.
`dSk`([k, dtype, gauge, format])	Setup the \(k\)-derivatie of the overlap matrix for a given k-point
`ddSk`([k, dtype, gauge, format])	Setup the double \(k\)-derivatie of the overlap matrix for a given k-point
`edges`([atom, exclude, orbital])	Retrieve edges (connections) of a given atom or list of atom’s
`eig`([k, gauge, eigvals_only])	Returns the eigenvalues of the physical quantity (using the non-Hermitian solver)
`eigh`([k, gauge, eigvals_only])	Returns the eigenvalues of the physical quantity
`eigsh`([k, n, gauge, eigvals_only])	Calculates a subset of eigenvalues of the physical quantity (default 10)
`eliminate_zeros`([atol])	Removes all zero elements from the sparse matrix
`empty`([keep_nnz])	See `empty` for details
`finalize`()	Finalizes the model
`fromsp`(geometry, P[, S])	Read and return the object with possible overlap
`iter`([local])	Iterations of the orbital space in the geometry, two indices from loop
`iter_nnz`([atom, orbital])	Iterations of the non-zero elements
`make_hermitian`()	Ensures the matrix is Hermitian by doing an in-place symmetrization
`nonzero`([atom, only_col])	Indices row and column indices where non-zero elements exists
`remove`(atom[, orb_index])	Remove a subset of this sparse matrix by only retaining the atoms corresponding to atom and optionally a subset of the atom orbitals
`repeat`(reps, axis)	Create a repeated sparse orbital object, equivalent to Geometry.repeat
`reset`([dim, dtype, nnzpr])	The sparsity pattern has all elements removed and everything is reset.
`rij`([what, dtype])	Create a sparse matrix with the distance between atoms/orbitals
`set_nsc`(args, *kwargs)	Reset the number of allowed supercells in the sparse orbital
`spalign`(other)	See `align` for details
`spsame`(other)	Compare two sparse objects and check whether they have the same entries.
`sub`(atom[, orb_index])	Create a subset of this sparse matrix by only retaining the atoms corresponding to atom
`swap`(a, b)	Swaps atoms in the sparse geometry to obtain a new order of atoms
`tile`(reps, axis)	Create a tiled sparse orbital object, equivalent to Geometry.tile
`toSparseAtom`([dtype])	Convert the sparse object (without data) to a new sparse object with equivalent but reduced sparse pattern
`tocsr`(index[, isc])	Return a `scipy.sparse.csr_matrix` of the specified index

Rij(what='orbital', dtype=<class 'numpy.float64'>)¶

Create a sparse matrix with the vectors between atoms/orbitals

Parameters:	what : {‘orbital’, ‘atom’} which kind of sparse vector matrix to return, either an atomic vector matrix or an orbital vector matrix. The orbital matrix is equivalent to the atomic one with the same vectors repeated for the same atomic orbitals. The default is the same type as the parent class. dtype : numpy.dtype, optional the data-type of the sparse matrix.

Notes

The returned sparse matrix with vectors are taken from the current sparse pattern. I.e. a subsequent addition of sparse elements will make them inequivalent. It is thus important to only create the sparse vector matrix when the sparse structure is completed.

S¶

Sk(k=(0, 0, 0), dtype=None, gauge='R', format='csr', *args, **kwargs)¶

Setup the overlap matrix for a given k-point

Creation and return of the overlap matrix for a given k-point (default to Gamma).

Parameters:

k : array_like, optional: the k-point to setup the overlap at (default Gamma point)
dtype : numpy.dtype, optional: the data type of the returned matrix. Do NOT request non-complex data-type for non-Gamma k. The default data-type is numpy.complex128
gauge : {‘R’, ‘r’}: the chosen gauge, R for cell vector gauge, and r for orbital distance gauge.
format : {‘csr’, ‘array’, ‘matrix’, ‘coo’, …}: the returned format of the matrix, defaulting to the scipy.sparse.csr_matrix, however if one always requires operations on dense matrices, one can always return in numpy.ndarray (‘array’) or numpy.matrix (‘matrix’).

Returns:

object : the overlap matrix for the \(k\)-point, format determines the object type.

See also

dSk: Overlap matrix derivative with respect to k
ddSk: Overlap matrix double derivative with respect to k

Notes

Currently the implemented gauge for the k-point is the cell vector gauge:

\[\mathbf S(k) = \mathbf S_{\nu\mu} e^{i k R}\]

where \(R\) is an integer times the cell vector and \(\nu\), \(\mu\) are orbital indices.

Another possible gauge is the orbital distance which can be written as

\[\mathbf S(k) = \mathbf S_{\nu\mu} e^{i k r}\]

where \(r\) is the distance between the orbitals.

construct(func, na_iR=1000, method='rand', eta=False)¶

Automatically construct the sparse model based on a function that does the setting up of the elements

This may be called in two variants.

Pass a function (func), see e.g. create_construct which does the setting up.
Pass a tuple/list in func which consists of two elements, one is R the radii parameters for the corresponding parameters. The second is the parameters corresponding to the R[i] elements. In this second case all atoms must only have one orbital.

Parameters:

func: callable or array_like

this function must take 4 arguments. 1. Is this object (self) 2. Is the currently examined atom (ia) 3. Is the currently bounded indices (idxs) 4. Is the currently bounded indices atomic coordinates (idxs_xyz) An example func could be:

>>> def func(self, ia, idxs, idxs_xyz=None): 
...     idx = self.geometry.close(ia, R=[0.1, 1.44], idx=idxs, idx_xyz=idxs_xyz) 
...     self[ia, idx[0]] = 0 
...     self[ia, idx[1]] = -2.7 

na_iR : int, optional

number of atoms within the sphere for speeding up the iter_block loop.

method : {‘rand’, str}

method used in Geometry.iter_block, see there for details

eta: bool, optional

whether an ETA will be printed

See also

create_construct: a generic function used to create a generic function which this routine requires
tile: tiling after construct is much faster for very large systems
repeat: repeating after construct is much faster for very large systems

copy(dtype=None)¶

A copy of this object

Parameters:	dtype : numpy.dtype, optional it is possible to convert the data to a different data-type If not specified, it will use `self.dtype`

create_construct(R, param)¶

Create a simple function for passing to the construct function.

This is simply to leviate the creation of simplistic functions needed for setting up the sparse elements.

Basically this returns a function:

>>> def func(self, ia, idxs, idxs_xyz=None): 
...     idx = self.geometry.close(ia, R=R, idx=idxs) 
...     for ix, p in zip(idx, param): 
...         self[ia, ix] = p 

Parameters:	R : array_like radii parameters for different shells. Must have same length as `param` or one less. If one less it will be extended with `R[0]/100` param : array_like coupling constants corresponding to the `R` ranges. `param[0,:]` are the elements for the all atoms within `R[0]` of each atom.

See also

construct: routine to create the sparse matrix from a generic function (as returned from create_construct)

Notes

This function only works for geometry sparse matrices (i.e. one element per atom). If you have more than one element per atom you have to implement the function your-self.

cut(seps, axis, *args, **kwargs)¶

Cuts the sparse orbital model into different parts.

Recreates a new sparse orbital object with only the cutted atoms in the structure.

Cutting is the opposite of tiling.

Parameters:	seps : int number of times the structure will be cut axis : int the axis that will be cut

dSk(k=(0, 0, 0), dtype=None, gauge='R', format='csr', *args, **kwargs)¶

Setup the \(k\)-derivatie of the overlap matrix for a given k-point

Creation and return of the derivative of the overlap matrix for a given k-point (default to Gamma).

Parameters:

k : array_like, optional: the k-point to setup the overlap at (default Gamma point)
dtype : numpy.dtype, optional: the data type of the returned matrix. Do NOT request non-complex data-type for non-Gamma k. The default data-type is numpy.complex128
gauge : {‘R’, ‘r’}: the chosen gauge, R for cell vector gauge, and r for orbital distance gauge.
format : {‘csr’, ‘array’, ‘matrix’, ‘coo’, …}: the returned format of the matrix, defaulting to the scipy.sparse.csr_matrix, however if one always requires operations on dense matrices, one can always return in numpy.ndarray (‘array’) or numpy.matrix (‘matrix’).

Returns:

tuple : for each of the Cartesian directions a \(\partial \mathbf S(k)/\partial k\) is returned.

See also

Sk: Overlap matrix at k
ddSk: Overlap matrix double derivative at k

Notes

Currently the implemented gauge for the k-point is the cell vector gauge:

\[\nabla_k \mathbf S_\alpha(k) = i R_\alpha \mathbf S_{\nu\mu} e^{i k R}\]

where \(R\) is an integer times the cell vector and \(\nu\), \(\mu\) are orbital indices. And \(\alpha\) is one of the Cartesian directions.

Another possible gauge is the orbital distance which can be written as

\[\nabla_k \mathbf S_\alpha(k) = i r_\alpha \mathbf S_{ij} e^{i k r}\]

where \(r\) is the distance between the orbitals.

ddSk(k=(0, 0, 0), dtype=None, gauge='R', format='csr', *args, **kwargs)¶

Setup the double \(k\)-derivatie of the overlap matrix for a given k-point

Creation and return of the double derivative of the overlap matrix for a given k-point (default to Gamma).

Parameters:

k : array_like, optional: the k-point to setup the overlap at (default Gamma point)
dtype : numpy.dtype, optional: the data type of the returned matrix. Do NOT request non-complex data-type for non-Gamma k. The default data-type is numpy.complex128
gauge : {‘R’, ‘r’}: the chosen gauge, R for cell vector gauge, and r for orbital distance gauge.
format : {‘csr’, ‘array’, ‘matrix’, ‘coo’, …}: the returned format of the matrix, defaulting to the scipy.sparse.csr_matrix, however if one always requires operations on dense matrices, one can always return in numpy.ndarray (‘array’) or numpy.matrix (‘matrix’).

Returns:

tuple of tuples : for each of the Cartesian directions

See also

Sk: Overlap matrix at k
dSk: Overlap matrix derivative at k

Notes

Currently the implemented gauge for the k-point is the cell vector gauge:

\[\nabla_k^2 \mathbf S_{\alpha\beta}(k) = - R_\alpha R_\beta \mathbf S_{\nu\mu} e^{i k R}\]

where \(R\) is an integer times the cell vector and \(\nu\), \(\mu\) are orbital indices. And \(\alpha\) and \(\beta\) are one of the Cartesian directions.

Another possible gauge is the orbital distance which can be written as

\[\nabla_k^2 \mathbf S_{\alpha\beta}(k) = - r_\alpha r_\beta \mathbf S_{ij} e^{i k r}\]

where \(r\) is the distance between the orbitals.

dim¶: Number of components per element

dkind¶: Data type of sparse elements (in str)

dtype¶: Data type of sparse elements

edges(atom=None, exclude=None, orbital=None)¶

Retrieve edges (connections) of a given atom or list of atom’s

The returned edges are unique and sorted (see numpy.unique) and are returned in supercell indices (i.e. 0 <= edge < self.geometry.no_s).

Parameters:	atom : int or list of int the edges are returned only for the given atom (but by using all orbitals of the requested atom). The returned edges are also atoms. exclude : int or list of int, optional remove edges which are in the exclude list. Default to atom. orbital : int or list of int the edges are returned only for the given orbital. The returned edges are orbitals.

See also

SparseCSR.edges: the underlying routine used for extracting the edges

eig(k=(0, 0, 0), gauge='R', eigvals_only=True, **kwargs)[source]¶

Returns the eigenvalues of the physical quantity (using the non-Hermitian solver)

Setup the system and overlap matrix with respect to the given k-point and calculate the eigenvalues.

All subsequent arguments gets passed directly to scipy.linalg.eig

Parameters:	spin : int, optional the spin-component to calculate the eigenvalue spectrum of, note that this parameter is only valid for `Spin.POLARIZED` matrices.

eigh(k=(0, 0, 0), gauge='R', eigvals_only=True, **kwargs)[source]¶

Returns the eigenvalues of the physical quantity

Setup the system and overlap matrix with respect to the given k-point and calculate the eigenvalues.

All subsequent arguments gets passed directly to scipy.linalg.eigh

Parameters:	spin : int, optional the spin-component to calculate the eigenvalue spectrum of, note that this parameter is only valid for `Spin.POLARIZED` matrices.

eigsh(k=(0, 0, 0), n=10, gauge='R', eigvals_only=True, **kwargs)[source]¶

Calculates a subset of eigenvalues of the physical quantity (default 10)

Setup the quantity and overlap matrix with respect to the given k-point and calculate a subset of the eigenvalues using the sparse algorithms.

All subsequent arguments gets passed directly to scipy.linalg.eigsh

Parameters:	spin : int, optional the spin-component to calculate the eigenvalue spectrum of, note that this parameter is only valid for `Spin.POLARIZED` matrices.

eliminate_zeros(atol=0.0)¶

Removes all zero elements from the sparse matrix

This is an in-place operation.

Parameters:	atol : float, optional absolute tolerance below this value will be considered 0.

empty(keep_nnz=False)¶: See empty for details

finalize()¶

Finalizes the model

Finalizes the model so that all non-used elements are removed. I.e. this simply reduces the memory requirement for the sparse matrix.

Note that adding more elements to the sparse matrix is more time-consuming than for a non-finalized sparse matrix due to the internal data-representation.

finalized¶: Whether the contained data is finalized and non-used elements have been removed

classmethod fromsp(geometry, P, S=None)¶: Read and return the object with possible overlap

geom¶: Associated geometry

geometry¶: Associated geometry

iter(local=False)¶

Iterations of the orbital space in the geometry, two indices from loop

An iterator returning the current atomic index and the corresponding orbital index.

>>> for ia, io in self: 

In the above case io always belongs to atom ia and ia may be repeated according to the number of orbitals associated with the atom ia.

Parameters:	local : bool, optional whether the orbital index is the global index, or the local index relative to the atom it resides on.

iter_nnz(atom=None, orbital=None)¶

Iterations of the non-zero elements

An iterator on the sparse matrix with, row and column

Parameters:	atom : int or array_like only loop on the non-zero elements coinciding with the orbitals on these atoms (not compatible with the `orbital` keyword) orbital : int or array_like only loop on the non-zero elements coinciding with the orbital (not compatible with the `atom` keyword)

Examples

>>> for i, j in self.iter_nnz(): 
...    self[i, j] # is then the non-zero value 

make_hermitian()¶: Ensures the matrix is Hermitian by doing an in-place symmetrization

nnz¶: Number of non-zero elements

non_orthogonal¶: True if the object is using a non-orthogonal basis

nonzero(atom=None, only_col=False)¶

Indices row and column indices where non-zero elements exists

Parameters:	atom : int or array_like of int, optional only return the tuples for the requested atoms, default is all atoms But for all orbitals. only_col : bool, optional only return then non-zero columns

See also

SparseCSR.nonzero: the equivalent function call

orthogonal¶: True if the object is using an orthogonal basis

remove(atom, orb_index=None)¶

Remove a subset of this sparse matrix by only retaining the atoms corresponding to atom and optionally a subset of the atom orbitals

See sub for details regarding atom and orb_index arguments.

Parameters:	atom : array_like of int or Atom indices of removed atoms or Atom for direct removal of all atoms orb_index : array_like of int, optional if atom is an instance of Atom, this variable correspond to the orbital indices for the atom to remove.

See also

Geometry.remove: equivalent to the resulting Geometry from this routine
Geometry.sub: the negative of Geometry.remove
sub: the opposite of remove, i.e. retain a subset of atoms

repeat(reps, axis)¶

Create a repeated sparse orbital object, equivalent to Geometry.repeat

The already existing sparse elements are extrapolated to the new supercell by repeating them in blocks like the coordinates.

Parameters:	reps : int number of repetitions along cell-vector axis axis : int 0, 1, 2 according to the cell-direction

See also

Geometry.repeat: the same ordering as the final geometry
Geometry.tile: a different ordering of the final geometry
tile: a different ordering of the final geometry

reset(dim=None, dtype=<class 'numpy.float64'>, nnzpr=None)¶

The sparsity pattern has all elements removed and everything is reset.

The object will be the same as if it had been initialized with the same geometry as it were created with.

Parameters:	dim: int, optional number of dimensions per element, default to the current number of elements per matrix element. dtype: numpy.dtype, optional the datatype of the sparse elements nnzpr: int, optional number of non-zero elements per row

rij(what='orbital', dtype=<class 'numpy.float64'>)¶

Create a sparse matrix with the distance between atoms/orbitals

Parameters:	what : {‘orbital’, ‘atom’} which kind of sparse distance matrix to return, either an atomic distance matrix or an orbital distance matrix. The orbital matrix is equivalent to the atomic one with the same distance repeated for the same atomic orbitals. The default is the same type as the parent class. dtype : numpy.dtype, optional the data-type of the sparse matrix.

Notes

The returned sparse matrix with distances are taken from the current sparse pattern. I.e. a subsequent addition of sparse elements will make them inequivalent. It is thus important to only create the sparse distance when the sparse structure is completed.

set_nsc(*args, **kwargs)¶

Reset the number of allowed supercells in the sparse orbital

If one reduces the number of supercells any sparse element that references the supercell will be deleted.

See SuperCell.set_nsc for allowed parameters.

See also

SuperCell.set_nsc: the underlying called method

shape¶: Shape of sparse matrix

spalign(other)¶: See align for details

spin¶: Associated spin class

spsame(other)¶

Compare two sparse objects and check whether they have the same entries.

This does not necessarily mean that the elements are the same

sub(atom, orb_index=None)¶

Create a subset of this sparse matrix by only retaining the atoms corresponding to atom

Indices passed MUST be unique.

Negative indices are wrapped and thus works.

Parameters:	atom : array_like of int or Atom indices of retained atoms or Atom for retaining only that atom orb_index : array_like of int, optional if atom is an instance of Atom, this variable correspond to the orbital indices for the atom to retain.

See also

Geometry.remove: the negative of Geometry.sub
Geometry.sub: equivalent to the resulting Geometry from this routine
remove: the negative of sub, i.e. remove a subset of atoms

Examples

>>> obj = SparseOrbital(...) 
>>> obj.sub(1) # only retain the second atom in the SparseGeometry
>>> obj.sub(obj.atoms.atom[0]) # retain all atoms which is equivalent to
>>>                            # the first atomic specie
>>> obj.sub(obj.atoms.atom[0], [1, 2]) # remove all but the 2nd and 3rd
>>>                                    # from the first atomic specie
>>>                                    # All other atomic species retain their orbitals.

swap(a, b)¶

Swaps atoms in the sparse geometry to obtain a new order of atoms

This can be used to reorder elements of a geometry.

Parameters:	a : array_like the first list of atomic coordinates b : array_like the second list of atomic coordinates

tile(reps, axis)¶

Create a tiled sparse orbital object, equivalent to Geometry.tile

The already existing sparse elements are extrapolated to the new supercell by repeating them in blocks like the coordinates.

Parameters:	reps : int number of repetitions along cell-vector axis axis : int 0, 1, 2 according to the cell-direction

See also

Geometry.tile: the same ordering as the final geometry
Geometry.repeat: a different ordering of the final geometry
repeat: a different ordering of the final geometry

toSparseAtom(dtype=None)¶

Convert the sparse object (without data) to a new sparse object with equivalent but reduced sparse pattern

This converts the orbital sparse pattern to an atomic sparse pattern.

Parameters:	dtype: numpy.dtype, optional the data-container for the sparse object. Defaults to the same.

tocsr(index, isc=None, **kwargs)¶

Return a scipy.sparse.csr_matrix of the specified index

Parameters:	index : int the index in the sparse matrix (for non-orthogonal cases the last dimension is the overlap matrix) isc : int, optional the supercell index, or all (if `isc=None`)