def __init__(self, parent, nkpt, displacement=None, size=None, trs=True): super(MonkhorstPack, self).__init__(parent) if isinstance(nkpt, Integral): nkpt = np.diag([nkpt] * 3) elif isinstance(nkpt[0], Integral): nkpt = np.diag(nkpt) # Now we have a matrix of k-points if np.any(nkpt - np.diag(np.diag(nkpt)) != 0): raise NotImplementedError( self.__class__.__name__ + " with off-diagonal components is not implemented yet") if displacement is None: displacement = np.zeros(3, np.float64) elif isinstance(displacement, Real): displacement = np.zeros(3, np.float64) + displacement if size is None: size = _a.onesd(3) elif isinstance(size, Real): size = _a.zerosd(3) + size # Retrieve the diagonal number of values Dn = np.diag(nkpt).astype(np.int32) if np.any(Dn) == 0: raise ValueError(self.__class__.__name__ + ' *must* be initialized with ' 'diagonal elements different from 0.') i_trs = -1 if trs: # Figure out which direction to TRS nmax = 0 for i in [0, 1, 2]: if displacement[i] == 0. and Dn[i] > nmax: nmax = Dn[i] i_trs = i # Calculate k-points and weights along all directions kw = [ self.grid(Dn[i], displacement[i], size[i], i == i_trs) for i in (0, 1, 2) ] self._k = _a.emptyd((kw[0][0].size, kw[1][0].size, kw[2][0].size, 3)) self._w = _a.onesd(self._k.shape[:-1]) for i in (0, 1, 2): k = kw[i][0].reshape(-1, 1, 1) w = kw[i][1].reshape(-1, 1, 1) self._k[..., i] = np.rollaxis(k, 0, i + 1) self._w[...] *= np.rollaxis(w, 0, i + 1) del kw self._k.shape = (-1, 3) self._k = np.where(self._k > .5, self._k - 1, self._k) self._w.shape = (-1, )
def grid(n, displ=0., size=1., trs=False): r""" Create a grid of `n` points with an offset of `displ` and sampling `size` around `displ` The :math:`k`-points are :math:`\Gamma` centered. Parameters ---------- n : int number of points in the grid. If `trs` is ``True`` this may be smaller than `n` displ : float, optional the displacement of the grid size : float, optional the total size of the Brillouin zone to sample trs : bool, optional whether time-reversal-symmetry is applied Returns ------- k : np.ndarray the list of k-points in the Brillouin zone to be sampled w : np.ndarray weights for the k-points """ # First ensure that displ is in the Brillouin displ = displ % 1. if displ > 0.5: displ -= 1. if trs and displ == 0.: n_half = n // 2 if n % 2 == 1: # Odd case, we have Gamma and remove all negative values k = _a.aranged(n_half + 1) * size / n + displ # Weights are all twice (except Gamma) w = _a.onesd(len(k)) / n * size w[1:] *= 2 else: # Even case, we do not have Gamma, but we shift to Gamma # All points except Gamma and edge have weights doubled k = _a.aranged(n_half + 1) * size / n + displ # Weights are all twice (except Gamma and band-edge) w = _a.onesd(len(k)) / n * size w[1:-1] *= 2 else: # Not TRS if n % 2 == 0: k = (_a.aranged(n) * 2 - n) * size / (2 * n) + displ else: k = (_a.aranged(n) * 2 - n + 1) * size / (2 * n) + displ w = _a.onesd(n) * size / n # Return values return k, w
def _setup(self, *args, **kwargs): """ Setup the special object for data containing """ self._data = dict() if self._access > 0: # Fake double calls access = self._access self._access = 0 # There are certain elements which should # be minimal on memory but allow for # fast access by the object. for d in ['cell', 'xa', 'lasto', 'E']: self._data[d] = self._value(d) # tbtrans does not store the k-points and weights # if the Gamma-point is used. try: self._data['kpt'] = self._value('kpt') except: self._data['kpt'] = _a.zerosd([3]) try: self._data['wkpt'] = self._value('wkpt') except: self._data['wkpt'] = _a.onesd([1]) # Create the geometry in the data file self._data['_geom'] = self.read_geometry() # Reset the access pattern self._access = access
def __init__(self, parent, k=None, weight=None): self.set_parent(parent) # Gamma point if k is None: self._k = _a.zerosd([1, 3]) self._w = _a.onesd(1) else: self._k = _a.arrayd(k).reshape(-1, 3) if weight is None: n = self._k.shape[0] self._w = _a.onesd(n) / n else: self._w = _a.arrayd(weight).ravel() if len(self.k) != len(self.weight): raise ValueError(self.__class__.__name__ + '.__init__ requires input k-points and weights to be of equal length.') # Instantiate the array call self.asarray()
def __init__(self, parent): try: # It probably has the supercell attached parent.cell parent.rcell self.parent = parent except: self.parent = SuperCell(parent) # Gamma point self._k = _a.zerosd([1, 3]) self._w = _a.onesd(1) # Instantiate the array call self.asarray()
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 __init__(self, parent, nkpt, displacement=None, size=None, centered=True, trs=True): super(MonkhorstPack, self).__init__(parent) if isinstance(nkpt, Integral): nkpt = np.diag([nkpt] * 3) elif isinstance(nkpt[0], Integral): nkpt = np.diag(nkpt) # Now we have a matrix of k-points if np.any(nkpt - np.diag(np.diag(nkpt)) != 0): raise NotImplementedError(self.__class__.__name__ + " with off-diagonal components is not implemented yet") if displacement is None: displacement = np.zeros(3, np.float64) elif isinstance(displacement, Real): displacement = np.zeros(3, np.float64) + displacement if size is None: size = _a.onesd(3) elif isinstance(size, Real): size = _a.zerosd(3) + size else: size = _a.arrayd(size) # Retrieve the diagonal number of values Dn = np.diag(nkpt).astype(np.int32) if np.any(Dn) == 0: raise ValueError(self.__class__.__name__ + ' *must* be initialized with ' 'diagonal elements different from 0.') i_trs = -1 if trs: # Figure out which direction to TRS nmax = 0 for i in [0, 1, 2]: if displacement[i] in [0., 0.5] and Dn[i] > nmax: nmax = Dn[i] i_trs = i if nmax == 1: i_trs = -1 if i_trs == -1: # If we still haven't decided (say for weird displacements) # simply take the one with the maximum number of k-points. i_trs = np.argmax(Dn) # Calculate k-points and weights along all directions kw = [self.grid(Dn[i], displacement[i], size[i], centered, i == i_trs) for i in (0, 1, 2)] self._k = _a.emptyd((kw[0][0].size, kw[1][0].size, kw[2][0].size, 3)) self._w = _a.onesd(self._k.shape[:-1]) for i in (0, 1, 2): k = kw[i][0].reshape(-1, 1, 1) w = kw[i][1].reshape(-1, 1, 1) self._k[..., i] = np.rollaxis(k, 0, i + 1) self._w[...] *= np.rollaxis(w, 0, i + 1) del kw self._k.shape = (-1, 3) self._k = np.where(self._k > .5, self._k - 1, self._k) self._w.shape = (-1,) # Store information regarding size and diagonal elements # This information is basically only necessary when # we want to replace special k-points self._diag = Dn # vector self._displ = displacement # vector self._size = size # vector self._centered = centered self._trs = i_trs
def grid(cls, n, displ=0., size=1., centered=True, trs=False): r""" Create a grid of `n` points with an offset of `displ` and sampling `size` around `displ` The :math:`k`-points are :math:`\Gamma` centered. Parameters ---------- n : int number of points in the grid. If `trs` is ``True`` this may be smaller than `n` displ : float, optional the displacement of the grid size : float, optional the total size of the Brillouin zone to sample centered : bool, optional if the points are centered trs : bool, optional whether time-reversal-symmetry is applied Returns ------- k : np.ndarray the list of k-points in the Brillouin zone to be sampled w : np.ndarray weights for the k-points """ # First ensure that displ is in the Brillouin displ = displ % 1. if displ > 0.5: displ -= 1. if displ < -0.5: displ += 1. # Centered _only_ has effect IFF # displ == 0. and size == 1 # Otherwise we resort to other schemes if displ != 0. or size != 1.: centered = False # We create the full grid, then afterwards we figure out TRS n_half = n // 2 if n % 2 == 1: k = _a.aranged(-n_half, n_half + 1) * size / n + displ else: k = _a.aranged(-n_half, n_half) * size / n + displ if not centered: # Shift everything by halve the size each occupies k += size / (2 * n) # Move k to the primitive cell and generate weights k = cls.in_primitive(k) w = _a.onesd(n) * size / n # Check for TRS points if trs and np.any(k < 0.): # Make all positive to remove the double conting terms k_pos = np.abs(k) # Sort k-points and weights idx = np.argsort(k_pos) # Re-arange according to k value k_pos = k_pos[idx] w = w[idx] # Find indices of all equivalent k-points (tolerance of 1e-10 in reciprocal units) # 1e-10 ~ 1e10 k-points (no body will do this!) idx_same = (np.diff(k_pos) < 1e-10).nonzero()[0] # The above algorithm should never create more than two duplicates. # Hence we can simply remove all idx_same and double the weight for all # idx_same + 1. w[idx_same + 1] *= 2 # Delete the duplicated k-points (they are already sorted) k = np.delete(k_pos, idx_same, axis=0) w = np.delete(w, idx_same) else: # Sort them, because it makes more visual sense idx = np.argsort(k) k = k[idx] w = w[idx] # Return values return k, w