sisl.physics.DensityMatrix

class sisl.physics.DensityMatrix(geometry, dim=1, dtype=None, nnzpr=None, **kwargs)

Bases: _densitymatrix

Sparse density matrix object

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

>>> DM = DensityMatrix(...)
>>> DM.D[1,2] = 0.1

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

For spin matrices the elements are defined with an extra dimension.

For a polarized matrix:

>>> M = DensityMatrix(..., spin="polarized")
>>> M[0, 0, 0] = # onsite spin up
>>> M[0, 0, 1] = # onsite spin down

For non-colinear the indices are a bit more tricky:

>>> M = DensityMatrix(..., spin="non-colinear")
>>> M[0, 0, M.M11] = # Re(up-up)
>>> M[0, 0, M.M22] = # Re(down-down)
>>> M[0, 0, M.M12r] = # Re(up-down)
>>> M[0, 0, M.M12i] = # Im(up-down)

For spin-orbit it looks like this:

>>> M = DensityMatrix(..., spin="spin-orbit")
>>> M[0, 0, M.M11r] = # Re(up-up)
>>> M[0, 0, M.M11i] = # Im(up-up)
>>> M[0, 0, M.M22r] = # Re(down-down)
>>> M[0, 0, M.M22i] = # Im(down-down)
>>> M[0, 0, M.M12r] = # Re(up-down)
>>> M[0, 0, M.M12i] = # Im(up-down)
>>> M[0, 0, M.M21r] = # Re(down-up)
>>> M[0, 0, M.M21i] = # Im(down-up)

Thus the number of orbitals is unchanged but a sub-block exists for the spin-block.

When transferring the matrix to a k-point the spin-box is local to each orbital, meaning that the spin-box for orbital i will be:

>>> Dk = M.Dk()
>>> Dk[i*2:(i+1)*2, i*2:(i+1)*2]

Parameters:

geometry (Geometry) – parent geometry to create a density matrix from. The density 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 density matrix. See details of Spin for default values.
nnzpr (int, optional) – number of initially allocated memory per orbital in the density 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 density matrix corresponds to a non-orthogonal basis. In this case the dimensionality of the density matrix is one more than dim. This is a keyword-only argument.

Plotting

`plot`	Plotting functions for the SparseOrbital class.
`plot.atomicmatrix`([dim, isc, ...])	Builds a `AtomicMatrixPlot` by setting the value of "matrix" to the current object.

Methods

`Dk`([k, dtype, gauge, format])	Setup the density matrix for a given k-point
`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
`add`(other[, axis, offset])	Add two sparse matrices by adding the parameters to one set.
`append`(other, axis[, atol, scale])	Append other along axis to construct a new connected sparse matrix
`bond_order`([method, projection])	Bond-order calculation using various methods
`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.
`dDk`([k, dtype, gauge, format])	Setup the density matrix derivative for a given k-point
`dSk`([k, dtype, gauge, format])	Setup the \(\mathbf k\)-derivatie of the overlap matrix for a given k-point
`ddDk`([k, dtype, gauge, format])	Setup the density matrix double derivative for a given k-point
`ddSk`([k, dtype, gauge, format])	Setup the double \(\mathbf k\)-derivatie of the overlap matrix for a given k-point
`density`(grid[, spinor, atol, eta, method])	Expand the density matrix to the charge density on a grid
`edges`([atoms, exclude, orbitals])	Retrieve edges (connections) for all atoms
`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 using sparse matrices
`eliminate_zeros`(args, *kwargs)	Removes all zero elements from the sparse matrix
`empty`([keep_nnz])	See `empty` for details
`finalize`()	Finalizes the model
`fromsp`(geometry, P[, S])	Create a sparse model from a preset Geometry and a list of sparse matrices
`iter_nnz`([atoms, orbitals])	Iterations of the non-zero elements
`iter_orbitals`([atoms, local])	Iterations of the orbital space in the geometry, two indices from loop
`mulliken`([projection])	Calculate Mulliken charges from the density matrix
`nonzero`([atoms, only_cols])	Indices row and column indices where non-zero elements exists
`orbital_momentum`([projection, method])	Calculate orbital angular momentum on either atoms or orbitals
`prepend`(other, axis[, atol, scale])	See `append` for details
`read`(sile, args, *kwargs)	Reads density matrix from Sile using read_density_matrix.
`remove`(atoms)	Create a subset of this sparse matrix by removing the atoms corresponding to atoms
`remove_orbital`(atoms, orbitals)	Remove a subset of orbitals on atoms according to orbitals
`repeat`(reps, axis)	Create a repeated sparse orbital object, equivalent to Geometry.repeat
`replace`(atoms, other[, other_atoms, atol, scale])	Replace atoms in self with other_atoms in other and retain couplings between them
`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
`spin_align`(vec[, atoms])	Aligns all spin along the vector vec
`spin_rotate`(angles[, rad])	Rotates spin-boxes by fixed angles around the \(x\), \(y\) and \(z\) axis, respectively.
`spsame`(other)	Compare two sparse objects and check whether they have the same entries.
`sub`(atoms)	Create a subset of this sparse matrix by only retaining the atoms corresponding to atoms
`sub_orbital`(atoms, orbitals)	Retain only a subset of the orbitals on atoms according to orbitals
`swap`(atoms_a, atoms_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`([dim, dtype])	Convert the sparse object (without data) to a new sparse object with equivalent but reduced sparse pattern
`tocsr`([dim, isc])	Return a `csr_matrix` for the specified dimension
`transform`([matrix, dtype, spin, orthogonal])	Transform the matrix by either a matrix or new spin configuration
`translate2uc`([atoms, axes])	Translates all primary atoms to the unit cell.
`transpose`([hermitian, spin, sort])	A transpose copy of this object, possibly apply the Hermitian conjugate as well
`trs`()	Create a new matrix with applied time-reversal-symmetry
`unrepeat`(reps, axis[, segment, sym])	Unrepeats the sparse model into different parts (retaining couplings)
`untile`(reps, axis[, segment, sym])	Untiles the sparse model into different parts (retaining couplings)
`write`(sile, args, *kwargs)	Writes a density matrix to the Sile as implemented in the `Sile.write_density_matrix` method

Attributes

`D`	Access the density matrix elements
`S`	Access the overlap elements associated with the sparse matrix
`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
`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

Dk(k=(0, 0, 0), dtype=None, gauge: sisl.typing.GaugeType = 'cell', format='csr', *args, **kwargs)[source]

Setup the density matrix for a given k-point

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

Notes

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

\[\mathbf D(\mathbf k) = \mathbf D_{ij} e^{i \mathbf k\cdot\mathbf R}\]

where \(\mathbf R\) is an integer times the cell vector and \(i\), \(j\) are orbital indices.

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

\[\mathbf D(\mathbf k) = \mathbf D_{ij} e^{i \mathbf k\cdot\mathb r}\]

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

Parameters:

k (array_like) – the k-point to setup the density matrix at
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 – the chosen gauge, cell for cell vector gauge, and atom for atomic distance gauge.
format ({'csr', 'array', 'dense', '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’/’dense’/’matrix’). Prefixing with ‘sc:’, or simply ‘sc’ returns the matrix in supercell format with phases.
spin (int, optional) – if the density matrix is a spin polarized one can extract the specific spin direction matrix by passing an integer (0 or 1). If the density matrix is not Spin.POLARIZED this keyword is ignored.

See also

dDk: Density matrix derivative with respect to k
ddDk: Density matrix double derivative with respect to k

Returns:: matrix (numpy.ndarray or scipy.sparse.*_matrix) – the density matrix at \(\mathbf k\). The returned object depends on format.

Rij(what: str = 'orbital', dtype=np.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.

Sk(k: sisl.typing.KPoint = (0, 0, 0), dtype=None, gauge: sisl.typing.GaugeType = 'cell', format: str = '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).

Notes

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

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

where \(\mathbf R\) is an integer times the cell vector and \(i\), \(j\) are orbital indices.

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

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

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

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 – the chosen gauge, cell for cell vector gauge, and atom for atomic 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”/”dense”/”matrix”). Prefixing with “sc:”, or simply “sc” returns the matrix in supercell format with phases. This is useful for e.g. bond-current calculations where individual hopping + phases are required.

See also

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

Returns:: matrix (numpy.ndarray or scipy.sparse.*_matrix) – the overlap matrix at \(\mathbf k\). The returned object depends on format.

add(other, axis: int | None = None, offset: sisl.typing.Coord = (0, 0, 0))

Add two sparse matrices by adding the parameters to one set. The final matrix will have no couplings between self and other

The final sparse matrix will not have any couplings between self and other. Not even if they have commensurate overlapping regions. If you want to create couplings you have to use append but that requires the structures are commensurate in the coupling region.

Parameters:

other (SparseGeometry) – the other sparse matrix to be added, all atoms will be appended
axis – whether a specific axis of the cell will be added to the final geometry. For None the final cell will be that of self, otherwise the lattice vector corresponding to axis will be appended.
offset – offset in geometry of other when adding the atoms.

See also

append: append two matrices by also adding overlap couplings
prepend: see append

append(other, axis: int, atol: float = 0.005, scale: sisl.typing.SeqOrScalarFloat = 1)

Append other along axis to construct a new connected sparse matrix

This method tries to append two sparse geometry objects together by the following these steps:

Create the new extended geometry
Use neighbor cell couplings from self as the couplings to other This may cause problems if the coupling atoms are not exactly equi-positioned. If the coupling coordinates and the coordinates in other differ by more than 0.01 Ang, a warning will be issued. If this difference is above atol the couplings will be removed.

When appending sparse matrices made up of atoms, this method assumes that the orbitals on the overlapping atoms have the same orbitals, as well as the same orbital ordering.

Examples

>>> sporb = SparseOrbital(....)
>>> sporb2 = sporb.append(sporb, 0)
>>> sporbt = sporb.tile(2, 0)
>>> sporb2.spsame(sporbt)
True

To retain couplings only from the left sparse matrix, do:

>>> sporb = left.append(right, 0, scale=(2, 0))
>>> sporb = (sporb + sporb.transpose()) / 2

To retain couplings only from the right sparse matrix, do:

>>> sporb = left.append(right, 0, scale=(0, 2.))
>>> sporb = (sporb + sporb.transpose()) / 2

Notes

The current implementation does not preserve the hermiticity of the matrix. If you want to preserve hermiticity of the matrix you have to do the following:

>>> sm = (sm + sm.transpose()) / 2

Parameters:

other (object) – must be an object of the same type as self
axis – axis to append the two sparse geometries along
atol – tolerance that all coordinates must be within to allow an append. It is important that this value is smaller than half the distance between the two closests atoms such that there is no ambiguity in selecting equivalent atoms. An internal stricter tolerance is used as a baseline, see above.
scale (float or array_like, optional) – the scale used for the overlapping region. For scalar values it corresponds to passing: (scale, scale). For array-like input scale[0] refers to the scale of the matrix elements coupling from self, while scale[1] is the scale of the matrix elements in other.

See also

prepend: equivalent scheme as this method
add: merge two matrices without considering overlap or commensurability
transpose: ensure hermiticity by using this routine
replace: replace a sub-set of atoms with another sparse matrix

Geometry.append, Geometry.prepend

SparseCSR.scale_columns: method used to scale the two matrix elements values

Raises:: ValueError – if the two geometries are not compatible for either coordinate, orbital or supercell errors
Returns:: object – a new instance with two sparse matrices joined and appended together

bond_order(method: str = 'mayer', projection: Literal['atom', 'orbital'] = 'atom')

Bond-order calculation using various methods

For method='wiberg', the bond-order is calculated as:

\[\mathbf B_{ij}^{\mathrm{Wiberg}} &= \mathbf D_{ij}^2\]

For method='mayer', the bond-order is calculated as:

\[\mathbf B_{ij}^{\mathrm{Mayer}} &= (\mathbf D\mathbf S)_{ij} (\mathbf D\mathbf S)_{ji}\]

For method='mulliken', the bond-order is calculated as:

\[\mathbf B_{ij}^{\mathrm{Mulliken}} &= 2\mathbf D_{ij}\mathbf S_{ij}\]

The bond order will then be using the above notation for the summation for atoms:

\[\mathbf B_{IJ}^{\langle\rangle} &= \sum_{i\in I}\sum_{j\in J} \mathbf B^{\langle\rangle}_{ij}\]

The Mulliken bond-order is closely related to the COOP interpretation. The COOP is generally an energy resolved Mulliken bond-order. So if the density matrix represents a particular eigen-state, it would yield the COOP value for the energy of the eigenstate. Generally the density matrix is the sum over all occupied eigen states, and hence represents the full picture.

For all options one can do the bond-order calculation for the spin components. Albeit, their meaning may be more doubtful. Simply add ':spin' to the method argument, and the returned quantity will be spin-resolved with \(x\), \(y\) and \(z\) components.

Note

It is unclear what the spin-density bond-order really means.

Parameters:

method ({mayer, wiberg, mulliken}[:spin]) – which method to calculate the bond-order with
projection – whether the returned matrix is in orbital form, or in atom form. If orbital is used, then the above formulas will be changed

Returns:

SparseAtom (with the bond-order between any two atoms, in a supercell matrix.) – Returned only if projection is atom.
SparseOrbital (with the bond-order between any two orbitals, in a supercell matrix.) – Returned only if projection is orbital.

construct(func, na_iR: int = 1000, method: str = 'rand', eta=None)

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, atoms, atoms_xyz=None):
...     idx = self.geometry.close(ia, R=[0.1, 1.44], atoms=atoms, atoms_xyz=atoms_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) → _SparseGeometry

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 to relieve the creation of simplistic functions needed for setting up sparse elements.

For simple matrices this returns a function:

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

In the non-colinear case the matrix element \(\mathbf M_{ij}\) will be set to input values param if \(i \le j\) and the Hermitian conjugated values for \(j < i\).

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.

This method issues warnings if the on-site terms are not Hermitian for spin-orbit systems. Do note that it still creates the matrices based on the input.

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)

dDk(k=(0, 0, 0), dtype=None, gauge: sisl.typing.GaugeType = 'cell', format='csr', *args, **kwargs)[source]

Setup the density matrix derivative for a given k-point

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

Notes

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

\[\nabla_k \mathbf D_\alpha(\mathbf k) = i \mathbf R_\alpha \mathbf D_{ij} e^{i \mathbf k\cdot\mathbf R}\]

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

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

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

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

Parameters:

k (array_like) – the k-point to setup the density matrix at
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 – the chosen gauge, cell for cell vector gauge, and atom for atomic distance gauge.
format ({'csr', 'array', 'dense', '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’/’dense’/’matrix’).
spin (int, optional) – if the density matrix is a spin polarized one can extract the specific spin direction matrix by passing an integer (0 or 1). If the density matrix is not Spin.POLARIZED this keyword is ignored.

See also

Dk: Density matrix with respect to k
ddDk: Density matrix double derivative with respect to k

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

dSk(k: sisl.typing.KPoint = (0, 0, 0), dtype=None, gauge: sisl.typing.GaugeType = 'cell', format: str = 'csr', *args, **kwargs)

Setup the \(\mathbf k\)-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).

Notes

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

\[\nabla_{\mathbf k} \mathbf S_\alpha(\mathbf k) = i \mathbf R_\alpha \mathbf S_{ij} e^{i\mathbf k\cdot\mathbf R}\]

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

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

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

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

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 – the chosen gauge, cell for cell vector gauge, and atom for atomic 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”/”dense”/”matrix”).

See also

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

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

ddDk(k=(0, 0, 0), dtype=None, gauge: sisl.typing.GaugeType = 'cell', format='csr', *args, **kwargs)[source]

Setup the density matrix double derivative for a given k-point

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

Notes

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

\[\nabla_k^2 \mathbf D^{(alpha\beta)}(\mathbf k) = - \mathbf R_\alpha \mathbf R_\beta \mathbf D_{ij} e^{i \mathbf k\cdot\mathbf R}\]

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

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

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

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

Parameters:

k (array_like) – the k-point to setup the density matrix at
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 – the chosen gauge, cell for cell vector gauge, and atom for atomic distance gauge.
format ({'csr', 'array', 'dense', '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’/’dense’/’matrix’).
spin (int, optional) – if the density matrix is a spin polarized one can extract the specific spin direction matrix by passing an integer (0 or 1). If the density matrix is not Spin.POLARIZED this keyword is ignored.

See also

Dk: Density matrix with respect to k
dDk: Density matrix derivative with respect to k

Returns:: list of matrices – for each of the Cartesian directions (in Voigt representation); xx, yy, zz, zy, xz, xy

ddSk(k: sisl.typing.KPoint = (0, 0, 0), dtype=None, gauge: sisl.typing.GaugeType = 'cell', format: str = 'csr', *args, **kwargs)

Setup the double \(\mathbf k\)-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).

Notes

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

\[\nabla_{\mathbf k^2} \mathbf S_{\alpha\beta}(\mathbf k) = - \mathbf R_\alpha \mathbf R_\beta \mathbf S_{ij} e^{i\mathbf k\cdot\mathbf R}\]

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

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

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

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

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 – the chosen gauge, cell for cell vector gauge, and atom for atomic 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”/”dense”/”matrix”).

See also

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

Returns:: list of matrices – for each of the Cartesian directions (in Voigt representation); xx, yy, zz, zy, xz, xy

density(grid: Grid, spinor=None, atol: float = 1e-7, eta: bool | None = False, method: Literal['pre-compute', 'direct'] = 'pre-compute', **kwargs)

Expand the density matrix to the charge density on a grid

This routine calculates the real-space density components on a specified grid.

This is an in-place operation that adds to the current values in the grid.

Note: To calculate \(\boldsymbol\rho(\mathbf r)\) in a unit-cell different from the originating geometry, simply pass a grid with a unit-cell different than the originating supercell.

The real-space density is calculated as:

\[\boldsymbol\rho(\mathbf r) = \sum_{ij}\phi_i(\mathbf r)\phi_j(\mathbf r) \mathbf D_{ij}\]

While for non-collinear/spin-orbit calculations the density is determined from the spinor component (spinor) by

\[\boldsymbol\rho_{\boldsymbol\sigma}(\mathbf r) = \sum_{ij}\phi_i(\mathbf r)\phi_j(\mathbf r) \sum_\alpha [\boldsymbol\sigma \boldsymbol \rho_{ij}]_{\alpha\alpha}\]

Here \(\boldsymbol\sigma\) corresponds to a spinor operator to extract relevant quantities. By passing the identity matrix the total charge is added. By using the Pauli matrix \(\boldsymbol\sigma_x\) only the \(x\) component of the density is added to the grid (see Spin.X).

Parameters:

grid – the grid on which to add the density (the density is in e/Ang^3)
spinor ((2,) or (2, 2), optional) – the spinor matrix to obtain the diagonal components of the density. For un-polarized density matrices this keyword has no influence. For spin-polarized it has to be either 1 integer or a vector of length 2 (defaults to total density). For non-collinear/spin-orbit density matrices it has to be a 2x2 matrix (defaults to total density).
atol – DM tolerance for accepted values. For all density matrix elements with absolute values below the tolerance, they will be treated as strictly zeros.
eta – show a progressbar on stdout
method – It determines if the orbital values are computed on the fly (direct) or they are all pre-computed on the grid at the beginning (pre-compute). Pre computing orbitals results in a faster computation, but it requires more memory.

Notes

The method argument may change at will since this is an experimental feature.

edges(atoms: sisl.typing.AtomsIndex = None, exclude: sisl.typing.AtomsIndex = None, orbitals=None)

Retrieve edges (connections) for all atoms

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:

atoms – 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 – remove edges which are in the exclude list, this list refers to orbitals.
orbitals (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: sisl.typing.KPoint = (0, 0, 0), gauge: sisl.typing.GaugeType = 'cell', eigvals_only: bool = True, **kwargs)

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: sisl.typing.KPoint = (0, 0, 0), gauge: sisl.typing.GaugeType = 'cell', eigvals_only: bool = True, **kwargs)

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: sisl.typing.KPoint = (0, 0, 0), n: int = 1, gauge: sisl.typing.GaugeType = 'cell', eigvals_only: bool = True, **kwargs)

Calculates a subset of eigenvalues of the physical quantity using sparse matrices

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.sparse.linalg.eigsh.

Parameters:

n – number of eigenvalues to calculate Defaults to the n smallest magnitude eigevalues.
spin (int, optional) – the spin-component to calculate the eigenvalue spectrum of, note that this parameter is only valid for Spin.POLARIZED matrices.
**kwargs – arguments passed directly to scipy.sparse.linalg.eigsh.

Notes

The performance and accuracy of this method depends heavily on kwargs. Playing around with a small test example before doing large scale calculations is adviced!

eliminate_zeros(*args, **kwargs)

Removes all zero elements from the sparse matrix

This is an in-place operation.

See also

SparseCSR.eliminate_zeros: method called, see there for parameters

empty(keep_nnz: bool = 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.

classmethod fromsp(geometry: Geometry, P, S=None, **kwargs)

Create a sparse model from a preset Geometry and a list of sparse matrices

The passed sparse matrices are in one of scipy.sparse formats.

Parameters:

geometry (Geometry) – geometry to describe the new sparse geometry
P (list of scipy.sparse or scipy.sparse) – the new sparse matrices that are to be populated in the sparse matrix
S (scipy.sparse, optional) – if provided this refers to the overlap matrix and will force the returned sparse matrix to be non-orthogonal
**kwargs (optional) – any arguments that are directly passed to the __init__ method of the class.

Returns:

SparseGeometry – a new sparse matrix that holds the passed geometry and the elements of P and optionally being non-orthogonal if S is not none

iter_nnz(atoms: sisl.typing.AtomsIndex = None, orbitals=None)

Iterations of the non-zero elements

An iterator on the sparse matrix with, row and column

Examples

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

Parameters:

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

iter_orbitals(atoms: sisl.typing.AtomsIndex = None, local: bool = 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.iter_orbitals():

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:

atoms (int or array_like, optional) – only loop on the given atoms, default to all atoms
local (bool, optional) – whether the orbital index is the global index, or the local index relative to the atom it resides on.

Yields:

ia – atomic index
io – orbital index

See also

Geometry.iter_orbitals: method used to iterate orbitals

mulliken(projection: Literal['orbital', 'atom'] = 'orbital')

Calculate Mulliken charges from the density matrix

See here for details on the mathematical notation. Matrices \(\boldsymbol\rho\) and \(\mathbf S\) are density and overlap matrices, respectively.

For polarized calculations the Mulliken charges are calculated as (for each spin-channel)

\[\begin{split}\mathbf m_i &= \sum_{i} [\boldsymbol\rho_{ij} \mathbf S_{ij}] \\ \mathbf m_I &= \sum_{i\in I} \mathbf m_i\end{split}\]

For non-colinear calculations (including spin-orbit) they are calculated as above but using the spin-box per orbital (\(\sigma\) is spin)

\[\begin{split}\mathbf m_i &= \sum_\sigma\sum_j [\boldsymbol\rho_{ij,\sigma\sigma} \mathbf S_{ij,\sigma\sigma}] \\ \mathbf m^{S^x}_i &= \sum_j \Re [\boldsymbol\rho_{ij,\uparrow\downarrow} \mathbf S_{ij,\uparrow\downarrow}] + \Re [\boldsymbol\rho_{ij,\downarrow\uparrow} \mathbf S_{ij,\downarrow\uparrow}] \\ \mathbf m^{S^y}_i &= \sum_j \Im [\boldsymbol\rho_{ij,\uparrow\downarrow} \mathbf S_{ij,\uparrow\downarrow}] - \Im [\boldsymbol\rho_{ij,\downarrow\uparrow} \mathbf S_{ij,\downarrow\uparrow}] \\ \mathbf m^{S^z}_i &= \sum_{j} \Re [\boldsymbol\rho_{ij,\uparrow\uparrow} \mathbf S_{ij,\uparrow\uparrow}] - \Re [\boldsymbol\rho_{ij,\downarrow\downarrow} \mathbf S_{ij,\downarrow\downarrow}]\end{split}\]

Parameters:: projection – how the Mulliken charges are returned. Can be atom-resolved, orbital-resolved or the charge matrix (off-diagonal elements)
Returns:: numpy.ndarray – if projection does not contain matrix, otherwise [spin, no], for polarized spin is [T, Sz] and for non-colinear spin is [T, Sx, Sy, Sz]

nonzero(atoms: sisl.typing.AtomsIndex = None, only_cols: bool = False)

Indices row and column indices where non-zero elements exists

Parameters:

atoms – only return the tuples for the requested atoms, default is all atoms But for all orbitals.
only_cols – only return then non-zero columns

See also

SparseCSR.nonzero: the equivalent function call

orbital_momentum(projection: Literal['orbital', 'atom'] = 'orbital', method: Literal['onsite'] = 'onsite')[source]

Calculate orbital angular momentum on either atoms or orbitals

Currently this implementation equals the Siesta implementation in that the on-site approximation is enforced thus limiting the calculated quantities to obey the following conditions:

Same atom
\(l>0\)
\(l_i \equiv l_j\)
\(m_i \neq m_j\)
\(\zeta_i \equiv \zeta_j\)

This allows one to sum the orbital angular moments on a per atom site.

Parameters:

projection – whether the angular momentum is resolved per atom, or per orbital
method – method used to calculate the angular momentum

Returns:

numpy.ndarray – orbital angular momentum with the last dimension equalling the \(L_x\), \(L_y\) and \(L_z\) components

plot.atomicmatrix(dim: int = 0, isc: int | None = None, fill_value: float | None = None, geometry: Geometry | None = None, atom_lines: bool | dict = False, orbital_lines: bool | dict = False, sc_lines: bool | dict = False, color_pixels: bool = True, colorscale: Colorscale | None = 'RdBu', crange: tuple[float, float] | None = None, cmid: float | None = None, text: str | None = None, textfont: dict | None = {}, set_labels: bool = False, constrain_axes: bool = True, arrows: list[dict] = [], backend: str = 'plotly') → AtomicMatrixPlot

Builds a AtomicMatrixPlot by setting the value of “matrix” to the current object.

Plots a (possibly sparse) matrix where rows and columns are either orbitals or atoms.

Parameters:

dim – If the matrix has a third dimension (e.g. spin), which index to plot in that third dimension.
isc – If the matrix contains data for an auxiliary supercell, the index of the cell to plot. If None, the whole matrix is plotted.
fill_value – If the matrix is sparse, the value to use for the missing entries.
geometry – Only needed if the matrix does not contain a geometry (e.g. it is a numpy array) and separator lines or labels are requested.
atom_lines – If a boolean, whether to draw lines separating atom blocks, using default styles. If a dict, draws the lines with the specified plotly line styles.
orbital_lines – If a boolean, whether to draw lines separating blocks of orbital sets, using default styles. If a dict, draws the lines with the specified plotly line styles.
sc_lines – If a boolean, whether to draw lines separating the supercells, using default styles. If a dict, draws the lines with the specified plotly line styles.
color_pixels – Whether to color the pixels of the matrix according to the colorscale.
colorscale – The colorscale to use to color the pixels.
crange – The minimum and maximum values of the colorscale.
cmid – The midpoint of the colorscale. If crange is provided, this is ignored.

If None and crange is also None, the midpoint is set to 0 if the data contains both positive and negative values.
text – If provided, show text of pixel value with the specified format. E.g. text=”.3f” shows the value with three decimal places.
textfont – The font to use for the text. This is a dictionary that may contain the keys “family”, “size”, “color”.
set_labels – Whether to set the axes labels to the atom/orbital that each row and column corresponds to. For orbitals the labels will be of the form “Atom: (l, m)”, where Atom is the index of the atom and l and m are the quantum numbers of the orbital.
constrain_axes – Whether to set the ranges of the axes to exactly fit the matrix.
backend – The backend to use for plotting.

prepend(other, axis: int, atol: float = 0.005, scale: sisl.typing.SeqOrScalarFloat = 1)

See append for details

This is currently equivalent to:

>>> other.append(self, axis, atol, scale)

static read(sile, *args, **kwargs)[source]

Reads density matrix from Sile using read_density_matrix.

Parameters:

sile (Sile, str or pathlib.Path) – a Sile object which will be used to read the density matrix and the overlap matrix (if any) if it is a string it will create a new sile using get_sile.
* (args passed directly to read_density_matrix(,**))

remove(atoms: sisl.typing.AtomsIndex) → _SparseGeometry

Create a subset of this sparse matrix by removing the atoms corresponding to atoms

Negative indices are wrapped and thus works.

Parameters:: atoms – indices of removed atoms

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

remove_orbital(atoms: sisl.typing.AtomsIndex, orbitals)

Remove a subset of orbitals on atoms according to orbitals

For more detailed examples, please see the equivalent (but opposite) method sub_orbital.

Parameters:

atoms – indices of atoms or Atom that will be reduced in size according to orbitals
orbitals (array_like of int or Orbital) – indices of the orbitals on atoms that are removed from the sparse matrix.

See also

sub_orbital: retaining a set of orbitals (see here for examples)

repeat(reps: int, axis: int) → SparseOrbital

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 – number of repetitions along cell-vector axis
axis – 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
SparseOrbital.tile: a different ordering of the final geometry

replace(atoms: sisl.typing.AtomsIndex, other, other_atoms: sisl.typing.AtomsIndex = None, atol: float = 0.005, scale: sisl.typing.SeqOrScalarFloat = 1.0)

Replace atoms in self with other_atoms in other and retain couplings between them

This method replaces a subset of atoms in self with another sparse geometry retaining any couplings between them. The algorithm checks whether the coupling atoms have the same number of orbitals. Meaning that atoms in the overlapping region should have the same connections and number of orbitals per atom. It will _not_ check whether the orbitals or atoms _are_ the same, nor the order of the orbitals.

The replacement algorithm takes the couplings from self -> other on atoms belonging to self and other -> self from other. This will in some cases mean that the matrix becomes non-symmetric. See in Notes for details on symmetrizing the matrices.

Examples

>>> minimal = SparseOrbital(....)
>>> big = minimal.tile(2, 0)
>>> big2 = big.replace(np.arange(big.na), minimal)
>>> big.spsame(big2)
True

To ensure hermiticity and using the average of the couplings from big and minimal one can do:

>>> big2 = big.replace(np.arange(big.na), minimal)
>>> big2 = (big2 + big2.transpose()) / 2

To retain couplings only from the big sparse matrix, one should do the following (note the subsequent transposing which ensures hermiticy and is effectively copying couplings from big to the replaced region.

>>> big2 = big.replace(np.arange(big.na), minimal, scale=(2, 0))
>>> big2 = (big2 + big2.transpose()) / 2

To only retain couplings from the minimal sparse matrix:

>>> big2 = big.replace(np.arange(big.na), minimal, scale=(0, 2))
>>> big2 = (big2 + big2.transpose()) / 2

Notes

The current implementation does not preserve the hermiticity of the matrix. If you want to preserve hermiticity of the matrix you have to do the following:

>>> sm = (sm + sm.transpose()) / 2

Also note that the ordering of the atoms will be range(atoms.min()), range(len(other_atoms)), <rest>.

Algorithms that utilizes atomic indices should be careful.

When the tolerance atol is high, the elements may be more prone to differences in the symmetry elements. A good idea would be to check the difference between the couplings. The below variable diff will contain the difference (self -> other) - (other -> self)

>>> diff = sm - sm.transpose()

Parameters:

atoms – which atoms in self that are removed and replaced with other.sub(other_atoms)
other (object) – must be an object of the same type as self, a subset is taken from this sparse matrix and combined with self to create a new sparse matrix
other_atoms – to select a subset of atoms in other that are taken out. Defaults to all atoms in other.
atol – coordinate tolerance for allowing replacement. It is important that this value is at least smaller than half the distance between the two closests atoms such that there is no ambiguity in selecting equivalent atoms.
scale – the scale used for the overlapping region. For scalar values it corresponds to passing: (scale, scale). For array-like input scale[0] refers to the scale of the matrix elements coupling from self, while scale[1] is the scale of the matrix elements in other.

See also

prepend: prepend two sparse matrices, see append for details
add: merge two matrices without considering overlap or commensurability
transpose: may be used to ensure hermiticity (symmetrization of the matrix elements)
append: append two sparse matrices

Geometry.append, Geometry.prepend

SparseCSR.scale_columns: method used to scale the two matrix elements values

Raises:

ValueError – if the two geometries are not compatible for either coordinate, orbital or supercell errors
AssertionError – if the two geometries are not compatible for either coordinate, orbital or supercell errors

Warns:

SislWarning – in case the overlapping atoms are not comprising the same atomic specie. In some cases this may not be a problem. However, care must be taken by the user if this warning is issued.

Returns:

object – a new instance with two sparse matrices merged together by replacing some atoms

reset(dim: int | None = None, dtype=np.float64, nnzpr: int | None = 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 – 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 – number of non-zero elements per row

rij(what: str = 'orbital', dtype=np.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 Lattice.set_nsc for allowed parameters.

See also

Lattice.set_nsc: the underlying called method

spalign(other): See align for details

spin_align(vec: sisl.typing.SeqFloat, atoms: sisl.typing.AtomsIndex = None)

Aligns all spin along the vector vec

In case the matrix is polarized and vec is not aligned at the z-axis, the returned matrix will be a non-collinear spin configuration.

Parameters:

vec ((3,)) – vector to align the spin boxes against
atoms (AtomsIndex, optional) – only perform alignment for matrix elements on atoms. If multiple atoms are specified, the off-diagonal elements between the atoms will also be aligned. To only align atomic on-site values, one would have to do a loop.

See also

spin_rotate: rotate spin-boxes by a fixed amount (does not align spins)

Returns:: object – a new object with aligned spins

spin_rotate(angles: sisl.typing.SeqFloat, rad: bool = False)

Rotates spin-boxes by fixed angles around the \(x\), \(y\) and \(z\) axis, respectively.

The angles are with respect to each spin-boxes initial angle. One should use spin_align to fix all angles along a specific direction.

Notes

For a polarized matrix: The returned matrix will be in non-collinear spin-configuration in case the angles does not reflect a pure flip of spin in the \(z\)-axis.

Parameters:

angles ((3,)) – angle to rotate spin boxes around the Cartesian directions \(x\), \(y\) and \(z\), respectively (Euler angles).
rad (bool, optional) – Determines the unit of angles, for true it is in radians

See also

spin_align: align all spin-boxes along a specific direction

Returns:: object – a new object with rotated spins

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(atoms: sisl.typing.AtomsIndex) → SparseOrbital

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

Negative indices are wrapped and thus works, supercell atoms are also wrapped to the unit-cell.

Parameters:: atoms – indices of retained atoms or Atom for retaining only that atom

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

See also

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

sub_orbital(atoms: sisl.typing.AtomsIndex, orbitals)

Retain only a subset of the orbitals on atoms according to orbitals

This allows one to retain only a given subset of the sparse matrix elements.

Parameters:

atoms – indices of atoms or Atom that will be reduced in size according to orbitals
orbitals (array_like of int or Orbital) – indices of the orbitals on atoms that are retained in the sparse matrix, the list of orbitals will be sorted. One cannot re-arrange matrix elements currently.

Notes

Future implementations may allow one to re-arange orbitals using this method.

When using this method the internal species list will be populated by another species that is named after the orbitals removed. This is to distinguish different atoms.

Examples

>>> # a Carbon atom with 2 orbitals
>>> C = sisl.Atom('C', [1., 2.])
>>> # an oxygen atom with 3 orbitals
>>> O = sisl.Atom('O', [1., 2., 2.4])
>>> geometry = sisl.Geometry([[0] * 3, [1] * 3]], 2, [C, O])
>>> obj = SparseOrbital(geometry).tile(3, 0)
>>> # fill in obj data...

Now obj is a sparse geometry with 2 different species and 6 atoms (3 of each). They are ordered [C, O, C, O, C, O]. In the following we will note species that are different from the original by a ' in the list.

Retain 2nd orbital on the 2nd atom: [C, O', C, O, C, O]

>>> new_obj = obj.sub_orbital(1, 1)

Retain 2nd orbital on 1st and 2nd atom: [C', O', C, O, C, O]

>>> new_obj = obj.sub_orbital([0, 1], 1)

Retain 2nd orbital on the 1st atom and 3rd orbital on 4th atom: [C', O, C, O', C, O]

>>> new_obj = obj.sub_orbital(0, 1).sub_orbital(3, 2)

Retain 2nd orbital on all atoms equivalent to the first atom: [C', O, C', O, C', O]

>>> new_obj = obj.sub_orbital(obj.geometry.atoms[0], 1)

Retain 1st orbital on 1st atom, and 2nd orbital on 3rd and 5th atom: [C', O, C'', O, C'', O]

>>> new_obj = obj.sub_orbital(0, 0).sub_orbital([2, 4], 1)

See also

remove_orbital: removing a set of orbitals (opposite of this)

swap(atoms_a: sisl.typing.AtomsIndex, atoms_b: sisl.typing.AtomsIndex) → _SparseGeometry

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

This can be used to reorder elements of a geometry.

Parameters:

atoms_a – the first list of atomic coordinates
atoms_b – the second list of atomic coordinates

tile(reps: int, axis: int) → SparseOrbital

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 – number of repetitions along cell-vector axis
axis – 0, 1, 2 according to the cell-direction

See also

SparseOrbital.repeat: a different ordering of the final geometry
SparseOrbital.untile: opposite of this method
Geometry.tile: the same ordering as the final geometry
Geometry.repeat: a different ordering of the final geometry

toSparseAtom(dim: int = None, 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:

dim – number of dimensions allocated in the SparseAtom object, default to the same
dtype (numpy.dtype, optional) – used data-type for the sparse object. Defaults to the same.

tocsr(dim: int = 0, isc=None, **kwargs)

Return a csr_matrix for the specified dimension

Parameters:

dim – the dimension 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)

transform(matrix=None, dtype=None, spin=None, orthogonal=None)

Transform the matrix by either a matrix or new spin configuration

1. General transformation: * If matrix is provided, a linear transformation \(\mathbf R^n \rightarrow \mathbf R^m\) is applied to the \(n\)-dimensional elements of the original sparse matrix. The spin and orthogonal flags are optional but need to be consistent with the creation of an m-dimensional matrix.

This method will copy over the overlap matrix in case the matrix argument only acts on the non-overlap matrix elements and both input and output matrices are non-orthogonal.

2. Spin conversion: If spin is provided (without matrix), the spin class is changed according to the following conversions:

Upscaling * unpolarized -> (polarized, non-colinear, spinorbit): Copy unpolarized value to both up and down components * polarized -> (non-colinear, spinorbit): Copy up and down components * non-colinear -> spinorbit: Copy first four spin components * all other new spin components are set to zero

Downscaling * (polarized, non-colinear, spinorbit) -> unpolarized: Set unpolarized value to a mix 0.5*up + 0.5*down * (non-colinear, spinorbit) -> polarized: Keep up and down spin components * spinorbit -> non-colinear: Keep first four spin components * all other spin components are dropped

3. Orthogonality: If the orthogonal flag is provided, the overlap matrix is either dropped or explicitly introduced as the identity matrix.

Notes

The transformation matrix does not act on the rows and columns, only on the final dimension of the matrix.

Parameters:

matrix (array_like, optional) – transformation matrix of shape \(m \times n\). Default is no transformation.
dtype (numpy.dtype, optional) – data type contained in the matrix. Defaults to the input type.
spin (str, sisl.Spin, optional) – spin class of created matrix. Defaults to the input type.
orthogonal (bool, optional) – flag to control if the new matrix includes overlaps. Defaults to the input type.

translate2uc(atoms: sisl.typing.AtomsIndex = None, axes: sisl.typing.CellAxes | None = None)

Translates all primary atoms to the unit cell.

With this, the coordinates of the geometry are translated to the unit cell and the supercell connections in the matrix are updated accordingly.

Parameters:

atoms – only translate the specified atoms. If not specified, all atoms will be translated.
axes – only translate certain lattice directions, None specifies only the periodic directions

Returns:

SparseOrbital or SparseAtom – A new sparse matrix with the updated connections and a new associated geometry.

transpose(hermitian: bool = False, spin: bool = True, sort: bool = True)

A transpose copy of this object, possibly apply the Hermitian conjugate as well

Parameters:

hermitian – if true, also apply a spin-box Hermitian operator to ensure TRS, otherwise only return the transpose values.
spin – whether the spin-box is also transposed if this is false, and hermitian is true, then only imaginary values will change sign.
sort – the returned columns for the transposed structure will be sorted if this is true, default

trs()

Create a new matrix with applied time-reversal-symmetry

Time reversal symmetry is applied using the following equality:

\[2\mathbf M^{\mathrm{TRS}} = \mathbf M + \boldsymbol\sigma_y \mathbf M^* \boldsymbol\sigma_y\]

where \(*\) is the conjugation operator.

unrepeat(reps: int, axis: int, segment: int = 0, *args, sym: bool = True, **kwargs)

Unrepeats the sparse model into different parts (retaining couplings)

Please see untile for details, the algorithm and arguments are the same however, this is the opposite of repeat.

untile(reps: int, axis: int, segment: int = 0, *args, sym: bool = True, **kwargs)

Untiles the sparse model into different parts (retaining couplings)

Recreates a new sparse object with only the cutted atoms in the structure. This will preserve matrix elements in the supercell.

Parameters:

reps – number of repetitions the tiling function created (opposite meaning as in untile)
axis – which axis to untile along
segment – which segment to retain. Generally each segment should be equivalent, however requesting individiual segments can help uncover inconsistencies in the sparse matrix
*args – arguments passed directly to Geometry.untile
sym – if True, the algorithm will ensure the returned matrix is symmetrized (i.e. return (M + M.transpose())/2, else return data as is. False should generally only be used for debugging precision of the matrix elements, or if one wishes to check the warnings.
**kwargs – keyword arguments passed directly to Geometry.untile

Notes

Untiling structures with nsc == 1 along axis are assumed to have periodic boundary conditions.

When untiling structures with nsc == 1 along axis it is important to untile as much as possible. This is because otherwise the algorithm cannot determine the correct couplings. Therefore to create a geometry of 3 times a unit-cell, one should untile to the unit-cell, and subsequently tile 3 times.

Consider for example a system of 4 atoms, each atom connects to its 2 neighbors. Due to the PBC atom 0 will connect to 1 and 3. Untiling this structure in 2 will group couplings of atoms 0 and 1. As it will only see one coupling to the right it will halve the coupling and use the same coupling to the left, which is clearly wrong.

In the following the latter is the correct way to do it.

>>> SPM.untile(2, 0) != SPM.untile(4, 0).tile(2, 0)

Raises:: ValueError : – in case the matrix elements are not conseuctive when determining the new supercell structure. This may often happen if untiling a matrix too few times, and then untiling it again.

See also

tile: opposite of this method
Geometry.untile: same as this method, see details about parameters here

write(sile: sisl.typing.SileLike, *args, **kwargs) → None: Writes a density matrix to the Sile as implemented in the Sile.write_density_matrix method

property D: Access the density matrix elements

property S: Access the overlap elements associated with the sparse matrix

property dim: Number of components per element

property dkind: Data type of sparse elements (in str)

property dtype: Data type of sparse elements

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

property geometry: Associated geometry

property nnz: Number of non-zero elements

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

property orthogonal: True if the object is using an orthogonal basis

plot: Plotting functions for the SparseOrbital class.

property shape: Shape of sparse matrix

property spin: Associated spin class