def cut(self, seps, axis, *args, **kwargs):
""" Cuts the tight-binding model into different parts.
Creates a tight-binding model by retaining the parameters
for the cut-out region, possibly creating a super-cell.
Parameters
----------
seps : integer, optional
number of times the structure will be cut.
axis : integer
the axis that will be cut
"""
new_w = None
# Create new geometry
with warnings.catch_warnings(record=True) as w:
# Cause all warnings to always be triggered.
warnings.simplefilter("always")
# Create new cut geometry
geom = self.geom.cut(seps, axis, *args, **kwargs)
# Check whether the warning exists
if len(w) > 0:
if issubclass(w[-1].category, UserWarning):
new_w = str(w[-1].message)
new_w += (
"\n---\n"
"The tight-binding model cannot be cut as the structure "
"cannot be tiled accordingly. ANY use of the model has been "
"relieved from sisl.")
if new_w:
warnings.warn(new_w, UserWarning)
# Now we need to re-create the tight-binding model
H = self.tocsr(0)
has_S = self.S_idx > 0
if has_S:
S = self.tocsr(self.S_idx)
# they are created similarly, hence the following
# should keep their order
# First we need to figure out how long the interaction range is
# in the cut-direction
# We initialize to be the same as the parent direction
nsc = np.copy(self.nsc) // 2
nsc[axis] = 0 # we count the new direction
isc = np.zeros([3], np.int32)
isc[axis] -= 1
out = False
while not out:
# Get supercell index
isc[axis] += 1
try:
idx = self.sc_index(isc)
except:
break
# Figure out if the Hamiltonian has interactions
# to 'isc'
sub = H[0:geom.no, idx * self.no:(idx + 1) * self.no].indices[:]
if has_S:
sub = np.unique(
np.concatenate(
(sub, S[0:geom.no,
idx * self.no:(idx + 1) * self.no].indices[:]),
axis=0))
if len(sub) == 0:
break
c_max = np.amax(sub)
# Count the number of cells it interacts with
i = (c_max % self.no) // geom.no
ic = idx * self.no
for j in range(i):
idx = ic + geom.no * j
# We need to ensure that every "in between" index exists
# if it does not we discard those indices
if len(
np.where(
np.logical_and(idx <= sub,
sub < idx + geom.no))[0]) == 0:
i = j - 1
out = True
break
nsc[axis] = isc[axis] * seps + i
if out:
warnings.warn(
'Cut the connection at nsc={0} in direction {1}.'.format(
nsc[axis], axis), UserWarning)
# Update number of super-cells
nsc[:] = nsc[:] * 2 + 1
geom.sc.set_nsc(nsc)
# Now we have a correct geometry, and
# we are now ready to create the sparsity pattern
# Reduce the sparsity pattern, first create the new one
ham = self.__class__(geom,
nnzpr=np.amax(self._data.ncol),
spin=self.spin,
orthogonal=self.orthogonal)
def sco2sco(M, o, m, seps, axis):
# Converts an o from M to m
isc = np.copy(M.o2isc(o))
isc[axis] = isc[axis] * seps
# Correct for cell-offset
isc[axis] = isc[axis] + (o % M.no) // m.no
# find the equivalent cell in m
try:
# If a fail happens it is due to a discarded
# interaction across a non-interacting region
return (o % m.no, m.sc_index(isc) * m.no,
m.sc_index(-isc) * m.no)
except:
return None, None, None
# Copy elements
if has_S:
for jo in range(geom.no):
# make smaller cut
sH = H[jo, :]
sS = S[jo, :]
for io, iH in zip(sH.indices, sH.data):
# Get the equivalent orbital in the smaller cell
o, ofp, ofm = sco2sco(self.geom, io, ham.geom, seps, axis)
if o is None:
continue
ham.H[jo, o + ofp] = iH
ham.S[jo, o + ofp] = S[jo, io]
ham.H[o, jo + ofm] = iH
ham.S[o, jo + ofm] = S[jo, io]
if np.any(sH.indices != sS.indices):
# Ensure that S is also cut
for io, iS in zip(sS.indices, sS.data):
# Get the equivalent orbital in the smaller cell
o, ofp, ofm = sco2sco(self.geom, io, ham.geom, seps,
axis)
if o is None:
continue
ham.H[jo, o + ofp] = H[jo, io]
ham.S[jo, o + ofp] = iS
ham.H[o, jo + ofm] = H[jo, io]
ham.S[o, jo + ofm] = iS
else:
for jo in range(geom.no):
sH = H[jo, :]
for io, iH in zip(sH.indices, sH.data):
# Get the equivalent orbital in the smaller cell
o, ofp, ofm = sco2sco(self.geom, io, ham.geom, seps, axis)
if o is None:
continue
ham[jo, o + ofp] = iH
ham[o, jo + ofm] = iH
return ham
def density(self, grid, spinor=None, tol=1e-7, eta=False):
r""" 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 :math:`\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:
.. math::
\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
.. math::
\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 :math:`\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 :math:`\boldsymbol\sigma_x`
only the :math:`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
"""
try:
# Once unique has the axis keyword, we know we can safely
# use it in this routine
# Otherwise we raise an ImportError
unique([[0, 1], [2, 3]], axis=0)
except:
raise NotImplementedError(
self.__class__.__name__ +
'.density requires numpy >= 1.13, either update '
'numpy or do not use this function!')
geometry = self.geometry
# Check that the atomic coordinates, really are all within the intrinsic supercell.
# If not, it may mean that the DM does not conform to the primary unit-cell paradigm
# of matrix elements. It complicates things.
fxyz = geometry.fxyz
f_min = fxyz.min()
f_max = fxyz.max()
if f_min < 0 or 1. < f_max:
warn(
self.__class__.__name__ +
'.density has been passed a geometry where some coordinates are '
'outside the primary unit-cell. This may potentially lead to problems! '
'Double check the charge density!')
del fxyz, f_min, f_max
# Extract sub variables used throughout the loop
shape = _a.asarrayi(grid.shape)
dcell = grid.dcell
# Sparse matrix data
csr = self._csr
# In the following we don't care about division
# So 1) save error state, 2) turn off divide by 0, 3) calculate, 4) turn on old error state
old_err = np.seterr(divide='ignore', invalid='ignore')
# Placeholder for the resulting coefficients
DM = None
if self.spin.kind > Spin.POLARIZED:
if spinor is None:
# Default to the total density
spinor = np.identity(2, dtype=np.complex128)
else:
spinor = _a.arrayz(spinor)
if spinor.size != 4 or spinor.ndim != 2:
raise ValueError(
self.__class__.__name__ +
'.density with NC/SO spin, requires a 2x2 matrix.')
DM = _a.emptyz([self.nnz, 2, 2])
idx = array_arange(csr.ptr[:-1], n=csr.ncol)
if self.spin.kind == Spin.NONCOLINEAR:
# non-collinear
DM[:, 0, 0] = csr._D[idx, 0]
DM[:, 1, 1] = csr._D[idx, 1]
DM[:, 1,
0] = csr._D[idx,
2] - 1j * csr._D[idx, 3] #TODO check sign here!
DM[:, 0, 1] = np.conj(DM[:, 1, 0])
else:
# spin-orbit
DM[:, 0, 0] = csr._D[idx, 0] + 1j * csr._D[idx, 4]
DM[:, 1, 1] = csr._D[idx, 1] + 1j * csr._D[idx, 5]
DM[:, 1,
0] = csr._D[idx,
2] - 1j * csr._D[idx, 3] #TODO check sign here!
DM[:, 0, 1] = csr._D[idx, 6] + 1j * csr._D[idx, 7]
# Perform dot-product with spinor, and take out the diagonal real part
DM = dot(DM, spinor.T)[:, [0, 1], [0, 1]].sum(1).real
elif self.spin.kind == Spin.POLARIZED:
if spinor is None:
spinor = _a.onesd(2)
elif isinstance(spinor, Integral):
# extract the provided spin-polarization
s = _a.zerosd(2)
s[spinor] = 1.
spinor = s
else:
spinor = _a.arrayd(spinor)
if spinor.size != 2 or spinor.ndim != 1:
raise ValueError(
self.__class__.__name__ +
'.density with polarized spin, requires spinor '
'argument as an integer, or a vector of length 2')
idx = array_arange(csr.ptr[:-1], n=csr.ncol)
DM = csr._D[idx, 0] * spinor[0] + csr._D[idx, 1] * spinor[1]
else:
idx = array_arange(csr.ptr[:-1], n=csr.ncol)
DM = csr._D[idx, 0]
# Create the DM csr matrix.
csrDM = csr_matrix(
(DM, csr.col[idx], np.insert(np.cumsum(csr.ncol), 0, 0)),
shape=(self.shape[:2]),
dtype=DM.dtype)
# Clean-up
del idx, DM
# To heavily speed up the construction of the density we can recreate
# the sparse csrDM matrix by summing the lower and upper triangular part.
# This means we only traverse the sparse UPPER part of the DM matrix
# I.e.:
# psi_i * DM_{ij} * psi_j + psi_j * DM_{ji} * psi_i
# is equal to:
# psi_i * (DM_{ij} + DM_{ji}) * psi_j
# Secondly, to ease the loops we extract the main diagonal (on-site terms)
# and store this for separate usage
csr_sum = [None] * geometry.n_s
no = geometry.no
primary_i_s = geometry.sc_index([0, 0, 0])
for i_s in range(geometry.n_s):
# Extract the csr matrix
o_start, o_end = i_s * no, (i_s + 1) * no
csr = csrDM[:, o_start:o_end]
if i_s == primary_i_s:
csr_sum[i_s] = triu(csr) + tril(csr, -1).transpose()
else:
csr_sum[i_s] = csr
# Recreate the column-stacked csr matrix
csrDM = ss_hstack(csr_sum, format='csr')
del csr, csr_sum
# Remove all zero elements (note we use the tolerance here!)
csrDM.data = np.where(np.fabs(csrDM.data) > tol, csrDM.data, 0.)
# Eliminate zeros and sort indices etc.
csrDM.eliminate_zeros()
csrDM.sort_indices()
csrDM.prune()
# 1. Ensure the grid has a geometry associated with it
sc = grid.sc.copy()
if grid.geometry is None:
# Create the actual geometry that encompass the grid
ia, xyz, _ = geometry.within_inf(sc)
if len(ia) > 0:
grid.set_geometry(Geometry(xyz, geometry.atom[ia], sc=sc))
# Instead of looping all atoms in the supercell we find the exact atoms
# and their supercell indices.
add_R = _a.zerosd(3) + geometry.maxR()
# Calculate the required additional vectors required to increase the fictitious
# supercell by add_R in each direction.
# For extremely skewed lattices this will be way too much, hence we make
# them square.
o = sc.toCuboid(True)
sc = SuperCell(o._v, origo=o.origo) + np.diag(2 * add_R)
sc.origo -= add_R
# Retrieve all atoms within the grid supercell
# (and the neighbours that connect into the cell)
IA, XYZ, ISC = geometry.within_inf(sc)
# Retrieve progressbar
eta = tqdm_eta(len(IA), self.__class__.__name__ + '.density', 'atom',
eta)
cell = geometry.cell
atom = geometry.atom
axyz = geometry.axyz
a2o = geometry.a2o
def xyz2spherical(xyz, offset):
""" Calculate the spherical coordinates from indices """
rx = xyz[:, 0] - offset[0]
ry = xyz[:, 1] - offset[1]
rz = xyz[:, 2] - offset[2]
# Calculate radius ** 2
xyz_to_spherical_cos_phi(rx, ry, rz)
return rx, ry, rz
def xyz2sphericalR(xyz, offset, R):
""" Calculate the spherical coordinates from indices """
rx = xyz[:, 0] - offset[0]
idx = indices_fabs_le(rx, R)
ry = xyz[idx, 1] - offset[1]
ix = indices_fabs_le(ry, R)
ry = ry[ix]
idx = idx[ix]
rz = xyz[idx, 2] - offset[2]
ix = indices_fabs_le(rz, R)
ry = ry[ix]
rz = rz[ix]
idx = idx[ix]
if len(idx) == 0:
return [], [], [], []
rx = rx[idx]
# Calculate radius ** 2
ix = indices_le(rx**2 + ry**2 + rz**2, R**2)
idx = idx[ix]
if len(idx) == 0:
return [], [], [], []
rx = rx[ix]
ry = ry[ix]
rz = rz[ix]
xyz_to_spherical_cos_phi(rx, ry, rz)
return idx, rx, ry, rz
# Looping atoms in the sparse pattern is better since we can pre-calculate
# the radial parts and then add them.
# First create a SparseOrbital matrix, then convert to SparseAtom
spO = SparseOrbital(geometry, dtype=np.int16)
spO._csr = SparseCSR(csrDM)
spA = spO.toSparseAtom(dtype=np.int16)
del spO
na = geometry.na
# Remove the diagonal part of the sparse atom matrix
off = na * primary_i_s
for ia in range(na):
del spA[ia, off + ia]
# Get pointers and delete the atomic sparse pattern
# The below complexity is because we are not finalizing spA
csr = spA._csr
a_ptr = np.insert(_a.cumsumi(csr.ncol), 0, 0)
a_col = csr.col[array_arange(csr.ptr, n=csr.ncol)]
del spA, csr
# Get offset in supercell in orbitals
off = geometry.no * primary_i_s
origo = grid.origo
# TODO sum the non-origo atoms to the csrDM matrix
# this would further decrease the loops required.
# Loop over all atoms in the grid-cell
for ia, ia_xyz, isc in zip(IA, XYZ - origo.reshape(1, 3), ISC):
# Get current atom
ia_atom = atom[ia]
IO = a2o(ia)
IO_range = range(ia_atom.no)
cell_offset = (cell * isc.reshape(3, 1)).sum(0) - origo
# Extract maximum R
R = ia_atom.maxR()
if R <= 0.:
warn("Atom '{}' does not have a wave-function, skipping atom.".
format(ia_atom))
eta.update()
continue
# Retrieve indices of the grid for the atomic shape
idx = grid.index(ia_atom.toSphere(ia_xyz))
# Now we have the indices for the largest orbital on the atom
# Subsequently we have to loop the orbitals and the
# connecting orbitals
# Then we find the indices that overlap with these indices
# First reduce indices to inside the grid-cell
idx[idx[:, 0] < 0, 0] = 0
idx[shape[0] <= idx[:, 0], 0] = shape[0] - 1
idx[idx[:, 1] < 0, 1] = 0
idx[shape[1] <= idx[:, 1], 1] = shape[1] - 1
idx[idx[:, 2] < 0, 2] = 0
idx[shape[2] <= idx[:, 2], 2] = shape[2] - 1
# Remove duplicates, requires numpy >= 1.13
idx = unique(idx, axis=0)
if len(idx) == 0:
eta.update()
continue
# Get real-space coordinates for the current atom
# as well as the radial parts
grid_xyz = dot(idx, dcell)
# Perform loop on connection atoms
# Allocate the DM_pj arrays
# This will have a size equal to number of elements times number of
# orbitals on this atom
# In this way we do not have to calculate the psi_j multiple times
DM_io = csrDM[IO:IO + ia_atom.no, :].tolil()
DM_pj = _a.zerosd([ia_atom.no, grid_xyz.shape[0]])
# Now we perform the loop on the connections for this atom
# Remark that we have removed the diagonal atom (it-self)
# As that will be calculated in the end
for ja in a_col[a_ptr[ia]:a_ptr[ia + 1]]:
# Retrieve atom (which contains the orbitals)
ja_atom = atom[ja % na]
JO = a2o(ja)
jR = ja_atom.maxR()
# Get actual coordinate of the atom
ja_xyz = axyz(ja) + cell_offset
# Reduce the ia'th grid points to those that connects to the ja'th atom
ja_idx, ja_r, ja_theta, ja_cos_phi = xyz2sphericalR(
grid_xyz, ja_xyz, jR)
if len(ja_idx) == 0:
# Quick step
continue
# Loop on orbitals on this atom
for jo in range(ja_atom.no):
o = ja_atom.orbital[jo]
oR = o.R
# Downsize to the correct indices
if jR - oR < 1e-6:
ja_idx1 = ja_idx.view()
ja_r1 = ja_r.view()
ja_theta1 = ja_theta.view()
ja_cos_phi1 = ja_cos_phi.view()
else:
ja_idx1 = indices_le(ja_r, oR)
if len(ja_idx1) == 0:
# Quick step
continue
# Reduce arrays
ja_r1 = ja_r[ja_idx1]
ja_theta1 = ja_theta[ja_idx1]
ja_cos_phi1 = ja_cos_phi[ja_idx1]
ja_idx1 = ja_idx[ja_idx1]
# Calculate the psi_j component
psi = o.psi_spher(ja_r1,
ja_theta1,
ja_cos_phi1,
cos_phi=True)
# Now add this orbital to all components
for io in IO_range:
DM_pj[io, ja_idx1] += DM_io[io, JO + jo] * psi
# Temporary clean up
del ja_idx, ja_r, ja_theta, ja_cos_phi
del ja_idx1, ja_r1, ja_theta1, ja_cos_phi1, psi
# Now we have all components for all orbitals connection to all orbitals on atom
# ia. We simply need to add the diagonal components
# Loop on the orbitals on this atom
ia_r, ia_theta, ia_cos_phi = xyz2spherical(grid_xyz, ia_xyz)
del grid_xyz
for io in IO_range:
# Only loop halve the range.
# This is because: triu + tril(-1).transpose()
# removes the lower half of the on-site matrix.
for jo in range(io + 1, ia_atom.no):
DM = DM_io[io, off + IO + jo]
oj = ia_atom.orbital[jo]
ojR = oj.R
# Downsize to the correct indices
if R - ojR < 1e-6:
ja_idx1 = slice(None)
ja_r1 = ia_r.view()
ja_theta1 = ia_theta.view()
ja_cos_phi1 = ia_cos_phi.view()
else:
ja_idx1 = indices_le(ia_r, ojR)
if len(ja_idx1) == 0:
# Quick step
continue
# Reduce arrays
ja_r1 = ia_r[ja_idx1]
ja_theta1 = ia_theta[ja_idx1]
ja_cos_phi1 = ia_cos_phi[ja_idx1]
# Calculate the psi_j component
DM_pj[io, ja_idx1] += DM * oj.psi_spher(
ja_r1, ja_theta1, ja_cos_phi1, cos_phi=True)
# Calculate the psi_i component
# Note that this one *also* zeroes points outside the shell
# I.e. this step is important because it "nullifies" all but points where
# orbital io is defined.
psi = ia_atom.orbital[io].psi_spher(ia_r,
ia_theta,
ia_cos_phi,
cos_phi=True)
DM_pj[io, :] += DM_io[io, off + IO + io] * psi
DM_pj[io, :] *= psi
# Temporary clean up
ja_idx1 = ja_r1 = ja_theta1 = ja_cos_phi1 = None
del ia_r, ia_theta, ia_cos_phi, psi, DM_io
# Now add the density
grid.grid[idx[:, 0], idx[:, 1], idx[:, 2]] += DM_pj.sum(0)
# Clean-up
del DM_pj, idx
eta.update()
eta.close()
# Reset the error code for division
np.seterr(**old_err)
def func(self, ia, idxs, idxs_xyz=None):
idx = self.geom.close(ia, dR=dR, idx=idxs, idx_xyz=idxs_xyz)
for ix, p in zip(idx, param):
self.H[ia, ix] = p[:-1]
self.S[ia, ix] = p[-1]