def _parse_kwargs(self, wrap, eta=None, eta_key=""):
""" Parse kwargs """
bz = self._obj
parent = bz.parent
if wrap is None:
# we always return a wrap
def wrap(v, parent=None, k=None, weight=None):
return v
else:
wrap = allow_kwargs("parent", "k", "weight")(wrap)
eta = progressbar(len(bz), f"{bz.__class__.__name__}.{eta_key}", "k", eta)
return bz, parent, wrap, eta
def density(self, grid, spinor=None, tol=1e-7, eta=None):
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 Exception:
raise NotImplementedError(
f"{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()
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(
f"{self.__class__.__name__}.density with NC/SO spin, requires a 2x2 matrix."
)
DM = _a.emptyz([self.nnz, 2, 2])
idx = _a.array_arange(csr.ptr[:-1], n=csr.ncol)
if self.spin.kind == Spin.NONCOLINEAR:
# non-collinear
DM[:, 0, 0] = csr._D[idx, 0]
DM[:, 0, 1] = csr._D[idx, 2] + 1j * csr._D[idx, 3]
DM[:, 1, 0] = np.conj(DM[:, 0, 1])
DM[:, 1, 1] = csr._D[idx, 1]
else:
# spin-orbit
DM[:, 0, 0] = csr._D[idx, 0] + 1j * csr._D[idx, 4]
DM[:, 0, 1] = csr._D[idx, 2] + 1j * csr._D[idx, 3]
DM[:, 1, 0] = csr._D[idx, 6] + 1j * csr._D[idx, 7]
DM[:, 1, 1] = csr._D[idx, 1] + 1j * csr._D[idx, 5]
# 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(
f"{self.__class__.__name__}.density with polarized spin, requires spinor "
"argument as an integer, or a vector of length 2")
idx = _a.array_arange(csr.ptr[:-1], n=csr.ncol)
DM = csr._D[idx, 0] * spinor[0] + csr._D[idx, 1] * spinor[1]
else:
idx = _a.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], _ncol_to_indptr(csr.ncol)),
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()
# Find the periodic directions
pbc = [
bc == grid.PERIODIC or geometry.nsc[i] > 1
for i, bc in enumerate(grid.bc[:, 0])
]
if grid.geometry is None:
# Create the actual geometry that encompass the grid
ia, xyz, _ = geometry.within_inf(sc, periodic=pbc)
if len(ia) > 0:
grid.set_geometry(Geometry(xyz, geometry.atoms[ia], sc=sc))
# Instead of looping all atoms in the supercell we find the exact atoms
# and their supercell indices.
add_R = _a.fulld(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 + np.diag(2 * add_R), origin=o.origin - add_R)
# Retrieve all atoms within the grid supercell
# (and the neighbours that connect into the cell)
IA, XYZ, ISC = geometry.within_inf(sc, periodic=pbc)
XYZ -= grid.sc.origin.reshape(1, 3)
# Retrieve progressbar
eta = progressbar(len(IA), f"{self.__class__.__name__}.density",
"atom", eta)
cell = geometry.cell
atoms = geometry.atoms
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 = _ncol_to_indptr(csr.ncol)
a_col = csr.col[_a.array_arange(csr.ptr, n=csr.ncol)]
del spA, csr
# Get offset in supercell in orbitals
off = geometry.no * primary_i_s
origin = grid.origin
# TODO sum the non-origin 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, ISC):
# Get current atom
ia_atom = atoms[ia]
IO = a2o(ia)
IO_range = range(ia_atom.no)
cell_offset = (cell * isc.reshape(3, 1)).sum(0) - origin
# Extract maximum R
R = ia_atom.maxR()
if R <= 0.:
warn(
f"Atom '{ia_atom}' does not have a wave-function, skipping 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 = atoms[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.orbitals[jo]
oR = o.R
# Downsize to the correct indices
if jR - oR < 1e-6:
ja_idx1 = ja_idx
ja_r1 = ja_r
ja_theta1 = ja_theta
ja_cos_phi1 = ja_cos_phi
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.orbitals[jo]
ojR = oj.R
# Downsize to the correct indices
if R - ojR < 1e-6:
ja_idx1 = slice(None)
ja_r1 = ia_r
ja_theta1 = ia_theta
ja_cos_phi1 = ia_cos_phi
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.orbitals[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 test_progressbar_false():
eta = sm.progressbar(2, 'Hello', 'unit', False)
eta.update()
eta.update()
eta.close()
def test_progressbar_true():
eta = sm.progressbar(2, 'Hello', 'unit', True)
eta.update()
eta.update()
eta.close()