sisl.physics.EnergyDensityMatrix
- class sisl.physics.EnergyDensityMatrix(geometry, dim=1, dtype=None, nnzpr=None, **kwargs)
Bases:
_densitymatrix
Sparse energy density matrix object
Assigning or changing elements is as easy as with standard
numpy
assignments:>>> EDM = EnergyDensityMatrix(...) >>> EDM.E[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 = EnergyDensityMatrix(..., 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 = EnergyDensityMatrix(..., 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 = EnergyDensityMatrix(..., 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:
>>> Ek = M.Ek() >>> Ek[i*2:(i+1)*2, i*2:(i+1)*2]
- Parameters
geometry (Geometry) – parent geometry to create a energy density matrix from. The energy 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
objectdtype (np.dtype, optional) – data type contained in the energy density matrix. See details of
Spin
for default values.nnzpr (int, optional) – number of initially allocated memory per orbital in the energy 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 overdim
.orthogonal (bool, optional) – whether the energy density matrix corresponds to a non-orthogonal basis. In this case the dimensionality of the energy density matrix is one more than
dim
. This is a keyword-only argument.
Methods
Ek
([k, dtype, gauge, format])Setup the energy 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[, eps, scale])Append other along axis to construct a new connected sparse matrix
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.dEk
([k, dtype, gauge, format])Setup the energy density matrix derivative for a given k-point
dSk
([k, dtype, gauge, format])Setup the \(k\)-derivatie of the overlap matrix for a given k-point
ddEk
([k, dtype, gauge, format])Setup the energy density matrix double derivative for a given k-point
ddSk
([k, dtype, gauge, format])Setup the double \(k\)-derivatie of the overlap matrix for a given k-point
density
(grid[, spinor, tol, eta])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 (default 10)
eliminate_zeros
(*args, **kwargs)Removes all zero elements from the sparse matrix
empty
([keep_nnz])See
empty
for detailsfinalize
()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_col])Indices row and column indices where non-zero elements exists
prepend
(other, axis[, eps, scale])See
append
for detailsread
(sile, *args, **kwargs)Reads density matrix from Sile using read_energy_density_matrix.
remove
(atoms)Remove a subset of this sparse matrix by only retaining 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, eps, 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
shift
(E, DM)Shift the energy density matrix to a common energy by using a reference density matrix
spalign
(other)See
align
for detailsspin_align
(vec)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
(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
([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 dimensiontransform
([matrix, dtype, spin, orthogonal])Transform the matrix by either a matrix or new spin configuration
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_energy_density_matrix
methodAccess the energy density matrix elements
Access the overlap elements associated with the sparse matrix
Number of components per element
Data type of sparse elements (in str)
Data type of sparse elements
Whether the contained data is finalized and non-used elements have been removed
Associated geometry
Number of non-zero elements
True if the object is using a non-orthogonal basis
True if the object is using an orthogonal basis
Shape of sparse matrix
Associated spin class
- property E
Access the energy density matrix elements
- Ek(k=(0, 0, 0), dtype=None, gauge='R', format='csr', *args, **kwargs)[source]
Setup the energy density matrix for a given k-point
Creation and return of the energy density matrix for a given k-point (default to Gamma).
Notes
Currently the implemented gauge for the k-point is the cell vector gauge:
\[\mathbf E(k) = \mathbf E_{\nu\mu} e^{i k R}\]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
\[\mathbf E(k) = \mathbf E_{\nu\mu} e^{i k r}\]where \(r\) is the distance between the orbitals.
- Parameters
k (array_like) – the k-point to setup the energy 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 ({'R', 'r'}) – the chosen gauge, R for cell vector gauge, and r for orbital 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 innumpy.ndarray
(‘array’/’dense’/’matrix’). Prefixing with ‘sc:’, or simply ‘sc’ returns the matrix in supercell format with phases.spin (int, optional) – if the energy density matrix is a spin polarized one can extract the specific spin direction matrix by passing an integer (0 or 1). If the energy density matrix is not
Spin.POLARIZED
this keyword is ignored.
See also
- Returns
matrix – the energy density matrix at \(k\). The returned object depends on format.
- Return type
numpy.ndarray or scipy.sparse.*_matrix
- 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.
- property S
Access the overlap elements associated with the sparse matrix
- 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).
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.
- 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 innumpy.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
- Returns
matrix – the overlap matrix at \(k\). The returned object depends on format.
- Return type
numpy.ndarray or scipy.sparse.*_matrix
- __init__(geometry, dim=1, dtype=None, nnzpr=None, **kwargs)[source]
Create sparse object with element between orbitals
- add(other, axis=None, offset=(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 (int or None, optional) – 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 ((3,), optional) – offset in geometry of other when adding the atoms.
- append(other, axis, eps=0.005, scale=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 eps 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 (int) – axis to append the two sparse geometries along
eps (float, optional) – 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 eps 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 inputscale[0]
refers to the scale of the matrix elements coupling from self, whilescale[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
a new instance with two sparse matrices joined and appended together
- Return type
- construct(func, na_iR=1000, method='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 theR[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)
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 \(M_{ij}\) 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 withinR[0]
of each atom.
See also
construct
routine to create the sparse matrix from a generic function (as returned from
create_construct
)
- dEk(k=(0, 0, 0), dtype=None, gauge='R', format='csr', *args, **kwargs)[source]
Setup the energy density matrix derivative for a given k-point
Creation and return of the energy 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:
\[\nabla_k \mathbf E_\alpha(k) = i R_\alpha \mathbf E_{\nu\mu} e^{i k R}\]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
\[\nabla_k \mathbf E_\alpha(k) = i r_\alpha \mathbf E_{\nu\mu} e^{i k r}\]where \(r\) is the distance between the orbitals.
- Parameters
k (array_like) – the k-point to setup the energy 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 ({'R', 'r'}) – the chosen gauge, R for cell vector gauge, and r for orbital 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 innumpy.ndarray
(‘array’/’dense’/’matrix’).spin (int, optional) – if the energy density matrix is a spin polarized one can extract the specific spin direction matrix by passing an integer (0 or 1). If the energy density matrix is not
Spin.POLARIZED
this keyword is ignored.
See also
- Returns
for each of the Cartesian directions a \(\partial \mathbf E(k)/\partial k\) is returned.
- Return type
- 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).
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.
- 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 innumpy.ndarray
(“array”/”dense”/”matrix”).
- Returns
for each of the Cartesian directions a \(\partial \mathbf S(k)/\partial k\) is returned.
- Return type
- ddEk(k=(0, 0, 0), dtype=None, gauge='R', format='csr', *args, **kwargs)[source]
Setup the energy density matrix double derivative for a given k-point
Creation and return of the energy 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:
\[\nabla_k^2 \mathbf E_{\alpha\beta}(k) = - R_\alpha R_\beta \mathbf E_{\nu\mu} e^{i k R}\]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
\[\nabla_k^2 \mathbf E_{\alpha\beta}(k) = - r_\alpha r_\beta \mathbf E_{\nu\mu} e^{i k r}\]where \(r\) is the distance between the orbitals.
- Parameters
k (array_like) – the k-point to setup the energy 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 ({'R', 'r'}) – the chosen gauge, R for cell vector gauge, and r for orbital 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 innumpy.ndarray
(‘array’/’dense’/’matrix’).spin (int, optional) – if the energy density matrix is a spin polarized one can extract the specific spin direction matrix by passing an integer (0 or 1). If the energy density matrix is not
Spin.POLARIZED
this keyword is ignored.
See also
- Returns
for each of the Cartesian directions (in Voigt representation); xx, yy, zz, zy, xz, xy
- Return type
list of matrices
- 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).
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.
- 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 innumpy.ndarray
(“array”/”dense”/”matrix”).
- Returns
for each of the Cartesian directions (in Voigt representation); xx, yy, zz, zy, xz, xy
- Return type
list of matrices
- density(grid, spinor=None, tol=1e-07, eta=None)
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 \(\rho(\mathbf r)\) 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:
\[\rho(\mathbf r) = \sum_{\nu\mu}\phi_\nu(\mathbf r)\phi_\mu(\mathbf r) D_{\nu\mu}\]While for non-collinear/spin-orbit calculations the density is determined from the spinor component (spinor) by
\[\rho_{\boldsymbol\sigma}(\mathbf r) = \sum_{\nu\mu}\phi_\nu(\mathbf r)\phi_\mu(\mathbf r) \sum_\alpha [\boldsymbol\sigma \mathbf \rho_{\nu\mu}]_{\alpha\alpha}\]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 (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).
tol (float, optional) – DM tolerance for accepted values. For all density matrix elements with absolute values below the tolerance, they will be treated as strictly zeros.
eta (bool, optional) – show a progressbar on stdout
- property dim
Number of components per element
- property dkind
Data type of sparse elements (in str)
- property dtype
Data type of sparse elements
- edges(atoms=None, exclude=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 (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 or None, optional) – 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=(0, 0, 0), gauge='R', eigvals_only=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=(0, 0, 0), gauge='R', eigvals_only=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=(0, 0, 0), n=10, gauge='R', eigvals_only=True, **kwargs)
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(*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=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.
- property finalized
Whether the contained data is finalized and non-used elements have been removed
- classmethod fromsp(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
a new sparse matrix that holds the passed geometry and the elements of P and optionally being non-orthogonal if
S
is not none- Return type
SparseGeometry
- property geometry
Associated geometry
- iter_nnz(atoms=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
- iter_orbitals(atoms=None, 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.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
- Yields
ia – atomic index
io – orbital index
See also
Geometry.iter_orbitals
method used to iterate orbitals
- mulliken(projection='orbital')
Calculate Mulliken charges from the density matrix
In the following \(\nu\) and \(\mu\) are orbital indices. Atomic indices noted by \(\alpha\), \(\beta\). 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)
\[\begin{split}M_{\nu} &= \sum_{\mu} [\boldsymbol\rho_{\nu\mu} \mathbf S_{\nu\mu}] \\ M_{\alpha} &= \sum_{\nu\in\alpha} M_{\nu}\end{split}\]For non-colinear calculations (including spin-orbit) they are calculated as above but using the spin-box per orbital (\(\sigma\) is spin)
\[\begin{split}M_{\nu} &= \sum_\sigma\sum_{\mu} [\boldsymbol\rho_{\nu\mu,\sigma\sigma} \mathbf S_{\nu\mu,\sigma\sigma}] \\ S_{\nu}^x &= \sum_{\mu} \Re [\boldsymbol\rho_{\nu\mu,\uparrow\downarrow} \mathbf S_{\nu\mu,\uparrow\downarrow}] + \Re [\boldsymbol\rho_{\nu\mu,\downarrow\uparrow} \mathbf S_{\nu\mu,\downarrow\uparrow}] \\ S_{\nu}^y &= \sum_{\mu} \Im [\boldsymbol\rho_{\nu\mu,\uparrow\downarrow} \mathbf S_{\nu\mu,\uparrow\downarrow}] - \Im [\boldsymbol\rho_{\nu\mu,\downarrow\uparrow} \mathbf S_{\nu\mu,\downarrow\uparrow}] \\ S_{\nu}^z &= \sum_{\mu} \Re [\boldsymbol\rho_{\nu\mu,\uparrow\uparrow} \mathbf S_{\nu\mu,\uparrow\uparrow}] - \Re [\boldsymbol\rho_{\nu\mu,\downarrow\downarrow} \mathbf S_{\nu\mu,\downarrow\downarrow}]\end{split}\]- Parameters
projection ({'orbital', 'atom'}) – how the Mulliken charges are returned. Can be atom-resolved, orbital-resolved or the charge matrix (off-diagonal elements)
- Returns
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]- Return type
- property nnz
Number of non-zero elements
- property non_orthogonal
True if the object is using a non-orthogonal basis
- nonzero(atoms=None, only_col=False)
Indices row and column indices where non-zero elements exists
- Parameters
See also
SparseCSR.nonzero
the equivalent function call
- property orthogonal
True if the object is using an orthogonal basis
- prepend(other, axis, eps=0.005, scale=1)
See
append
for detailsThis is currently equivalent to:
>>> other.append(self, axis, eps, scale)
- static read(sile, *args, **kwargs)[source]
Reads density matrix from Sile using read_energy_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_energy_density_matrix(,**)
) –
- remove(atoms)
Remove a subset of this sparse matrix by only retaining the atoms corresponding to atoms
- remove_orbital(atoms, 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
See also
sub_orbital
retaining a set of orbitals (see here for examples)
- 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
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
- replace(atoms, other, other_atoms=None, eps=0.005, scale=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 toself
andother -> self
fromother
. 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
andminimal
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 frombig
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 eps 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 (array_like) – 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 (array_like, optional) – to select a subset of atoms in other that are taken out. Defaults to all atoms in other.
eps (float, optional) – 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 (float or array_like, optional) – the scale used for the overlapping region. For scalar values it corresponds to passing:
(scale, scale)
. For array-like inputscale[0]
refers to the scale of the matrix elements coupling from self, whilescale[1]
is the scale of the matrix elements in other.
See also
prepend
prepend two sparse matrices, see
append
for detailsadd
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
a new instance with two sparse matrices merged together by replacing some atoms
- Return type
- 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
- property shape
Shape of sparse matrix
- shift(E, DM)[source]
Shift the energy density matrix to a common energy by using a reference density matrix
This is equal to performing this operation:
\[\mathfrak E_\sigma = \mathfrak E_\sigma + E \boldsymbol \rho_\sigma\]where \(\mathfrak E_\sigma\) correspond to the spin diagonal components of the energy density matrix and \(\boldsymbol \rho_\sigma\) is the spin diagonal components of the corresponding density matrix.
- Parameters
E (float or (2,)) – the energy (in eV) to shift the energy density matrix, if two values are passed the two first spin-components get shifted individually.
DM (DensityMatrix) – density matrix corresponding to the same geometry
- spalign(other)
See
align
for details
- property spin
Associated spin class
- spin_align(vec)
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
See also
spin_rotate
rotate spin-boxes by a fixed amount (does not align spins)
- Returns
a new object with aligned spins
- Return type
- spin_rotate(angles, rad=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, \(x\), \(y\) and :math`z`, respectively
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
a new object with rotated spins
- Return type
- 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)
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 (array_like of int or Atom) – 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
- sub_orbital(atoms, 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 (array_like of int or Atom) – 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 specie 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(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.
- toSparseAtom(dim=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 (int, optional) – 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=0, isc=None, **kwargs)
Return a
csr_matrix
for the specified dimension
- 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 \(R^n \rightarrow R^m\) is applied to the \(n\)-dimensional elements of the original sparse matrix. The
spin
andorthogonal
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.
- transpose(hermitian=False, spin=True, sort=True)
A transpose copy of this object, possibly apply the Hermitian conjugate as well
- Parameters
hermitian (bool, optional) – if true, also emply a spin-box Hermitian operator to ensure TRS, otherwise only return the transpose values.
spin (bool, optional) – whether the spin-box is also transposed if this is false, and hermitian is true, then only imaginary values will change sign.
sort (bool, optional) – 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, axis, segment=0, *args, sym=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 ofrepeat
.
- untile(reps, axis, segment=0, *args, sym=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 (int) – number of repetitions the tiling function created (opposite meaning as in
untile
)axis (int) – which axis to untile along
segment (int, optional) – 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 (bool, optional) – 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 neighbours. 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