mirror of
https://github.com/triqs/dft_tools
synced 2024-11-13 09:33:53 +01:00
8dc42b08ae
* restructuring * added user reference * started working on user guide * added schematic to structure
533 lines
16 KiB
Python
533 lines
16 KiB
Python
from math import sqrt
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from scipy.misc import factorial as fact
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from itertools import product
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import numpy as np
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# The interaction matrix in desired basis
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# U^{spherical}_{m1 m2 m3 m4} = \sum_{k=0}^{2l} F_k angular_matrix_element(l, k, m1, m2, m3, m4)
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def U_matrix(l, radial_integrals=None, U_int=None, J_hund=None, basis='spherical', T=None):
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r"""
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Calculate the full four-index U matrix being given either radial_integrals or U_int and J_hund.
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Parameters
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----------
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l : integer
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Angular momentum of shell being treated (l=2 for d shell, l=3 for f shell).
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radial_integrals : list, optional
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Slater integrals [F0,F2,F4,..].
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Must be provided if U_int and J_hund are not given.
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Preferentially used to compute the U_matrix if provided alongside U_int and J_hund.
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U_int : scalar, optional
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Value of the screened Hubbard interaction.
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Must be provided if radial_integrals are not given.
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J_hund : scalar, optional
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Value of the Hund's coupling.
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Must be provided if radial_integrals are not given.
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basis : string, optional
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The basis in which the interaction matrix should be computed.
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Takes the values
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- 'spherical': spherical harmonics,
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- 'cubic': cubic harmonics,
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- 'other': other basis type as given by the transformation matrix T.
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T : real/complex numpy array, optional
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Transformation matrix for basis change.
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Must be provided if basis='other'.
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Returns
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-------
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U_matrix : float numpy array
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The four-index interaction matrix in the chosen basis.
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"""
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# Check all necessary information is present and consistent
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if radial_integrals is None and (U_int is None and J_hund is None):
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raise ValueError("U_matrix: provide either the radial_integrals or U_int and J_hund.")
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if radial_integrals is None and (U_int is not None and J_hund is not None):
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radial_integrals = U_J_to_radial_integrals(l, U_int, J_hund)
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if radial_integrals is not None and (U_int is not None and J_hund is not None):
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if len(radial_integrals)-1 != l:
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raise ValueError("U_matrix: inconsistency in l and number of radial_integrals provided.")
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if (radial_integrals - U_J_to_radial_integrals(l, U_int, J_hund)).any() != 0.0:
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print "Warning: U_matrix: radial_integrals provided do not match U_int and J_hund. Using radial_integrals to calculate U_matrix."
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# Full interaction matrix
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# Basis of spherical harmonics Y_{-2}, Y_{-1}, Y_{0}, Y_{1}, Y_{2}
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# U^{spherical}_{m1 m2 m3 m4} = \sum_{k=0}^{2l} F_k angular_matrix_element(l, k, m1, m2, m3, m4)
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U_matrix = np.zeros((2*l+1,2*l+1,2*l+1,2*l+1),dtype=float)
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m_range = range(-l,l+1)
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for n, F in enumerate(radial_integrals):
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k = 2*n
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for m1, m2, m3, m4 in product(m_range,m_range,m_range,m_range):
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U_matrix[m1+l,m2+l,m3+l,m4+l] += F * angular_matrix_element(l,k,m1,m2,m3,m4)
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# Transform from spherical basis if needed
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if basis == "cubic": T = spherical_to_cubic(l)
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if basis == "other" and T is None:
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raise ValueError("U_matrix: provide T for other bases.")
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if T is not None: U_matrix = transform_U_matrix(U_matrix, T)
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return U_matrix
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# Convert full 4-index U matrix to 2-index density-density form
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def reduce_4index_to_2index(U_4index):
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r"""
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Reduces the four-index matrix to two-index matrices for parallel and anti-parallel spins.
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Parameters
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----------
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U_4index : float numpy array
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The four-index interaction matrix.
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Returns
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-------
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U : float numpy array
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The two-index interaction matrix for parallel spins.
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Uprime : float numpy array
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The two-index interaction matrix for anti-parallel spins.
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"""
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size = len(U_4index) # 2l+1
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U = np.zeros((size,size),dtype=float) # matrix for same spin
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Uprime = np.zeros((size,size),dtype=float) # matrix for opposite spin
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m_range = range(size)
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for m,mp in product(m_range,m_range):
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U[m,mp] = U_4index[m,mp,m,mp].real - U_4index[m,mp,mp,m].real
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Uprime[m,mp] = U_4index[m,mp,m,mp].real
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return U, Uprime
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# Construct the 2-index matrices for the density-density form
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def U_matrix_kanamori(n_orb, U_int, J_hund):
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r"""
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Calculate the Kanamori U and Uprime matrices.
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Parameters
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----------
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n_orb : integer
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Number of orbitals in basis.
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U_int : scalar
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Value of the screened Hubbard interaction.
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J_hund : scalar
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Value of the Hund's coupling.
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Returns
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-------
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U : float numpy array
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The two-index interaction matrix for parallel spins.
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Uprime : float numpy array
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The two-index interaction matrix for anti-parallel spins.
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"""
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U = np.zeros((n_orb,n_orb),dtype=float) # matrix for same spin
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Uprime = np.zeros((n_orb,n_orb),dtype=float) # matrix for opposite spin
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m_range = range(n_orb)
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for m,mp in product(m_range,m_range):
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if m == mp:
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Uprime[m,mp] = U_int
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else:
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U[m,mp] = U_int - 3.0*J_hund
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Uprime[m,mp] = U_int - 2.0*J_hund
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return U, Uprime
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# Get t2g or eg components
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def t2g_submatrix(U, convention=''):
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r"""
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Extract the t2g submatrix of the full d-manifold two- or four-index U matrix.
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Parameters
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----------
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U : float numpy array
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Two- or four-index interaction matrix.
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convention : string, optional
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The basis convention.
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Takes the values
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- '': basis ordered as ("xy","yz","z^2","xz","x^2-y^2"),
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- 'wien2k': basis ordered as ("z^2","x^2-y^2","xy","yz","xz").
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Returns
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-------
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U_t2g : float numpy array
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The t2g component of the interaction matrix.
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"""
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if convention == 'wien2k':
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return subarray(U, len(U.shape)*[(2,3,4)])
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else:
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return subarray(U, len(U.shape)*[(0,1,3)])
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def eg_submatrix(U, convention=''):
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r"""
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Extract the eg submatrix of the full d-manifold two- or four-index U matrix.
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Parameters
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----------
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U : float numpy array
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Two- or four-index interaction matrix.
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convention : string, optional
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The basis convention.
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Takes the values
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- '': basis ordered as ("xy","yz","z^2","xz","x^2-y^2"),
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- 'wien2k': basis ordered as ("z^2","x^2-y^2","xy","yz","xz").
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Returns
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-------
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U_eg : float numpy array
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The eg component of the interaction matrix.
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"""
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if convention == 'wien2k':
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return subarray(U, len(U.shape)*[(0,1)])
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else:
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return subarray(U, len(U.shape)*[(2,4)])
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# Transform the interaction matrix into another basis
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def transform_U_matrix(U_matrix, T):
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r"""
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Transform a four-index interaction matrix into another basis.
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Parameters
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----------
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U_matrix : float numpy array
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The four-index interaction matrix in the original basis.
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T : real/complex numpy array, optional
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Transformation matrix for basis change.
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Must be provided if basis='other'.
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Returns
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-------
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U_matrix : float numpy array
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The four-index interaction matrix in the new basis.
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"""
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return np.einsum("ij,kl,jlmo,mn,op",np.conj(T),np.conj(T),U_matrix,np.transpose(T),np.transpose(T))
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# Rotation matrices: complex harmonics to cubic harmonics
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# Complex harmonics basis: ..., Y_{-2}, Y_{-1}, Y_{0}, Y_{1}, Y_{2}, ...
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def spherical_to_cubic(l, convention=''):
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r"""
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Get the spherical harmonics to cubic harmonics transformation matrix.
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Parameters
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----------
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l : integer
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Angular momentum of shell being treated (l=2 for d shell, l=3 for f shell).
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convention : string, optional
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The basis convention.
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Takes the values
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- '': basis ordered as ("xy","yz","z^2","xz","x^2-y^2"),
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- 'wien2k': basis ordered as ("z^2","x^2-y^2","xy","yz","xz").
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Returns
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-------
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T : real/complex numpy array
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Transformation matrix for basis change.
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"""
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size = 2*l+1
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T = np.zeros((size,size),dtype=complex)
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if convention != 'wien2k' and l != 2:
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raise ValueError("spherical_to_cubic: wien2k convention only implemented only for l=2")
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if l == 0:
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cubic_names = ("s")
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elif l == 1:
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cubic_names = ("x","y","z")
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T[0,0] = 1.0/sqrt(2); T[0,2] = -1.0/sqrt(2)
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T[1,0] = 1j/sqrt(2); T[1,2] = 1j/sqrt(2)
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T[2,1] = 1.0
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elif l == 2:
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if convention == 'wien2k':
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cubic_names = ("z^2","x^2-y^2","xy","yz","xz")
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T[0,2] = 1.0
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T[1,0] = 1.0/sqrt(2); T[1,4] = 1.0/sqrt(2)
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T[2,0] = -1j/sqrt(2); T[2,4] = 1j/sqrt(2)
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T[3,1] = 1j/sqrt(2); T[3,3] = -1j/sqrt(2)
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T[4,1] = 1.0/sqrt(2); T[4,3] = 1.0/sqrt(2)
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else:
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cubic_names = ("xy","yz","z^2","xz","x^2-y^2")
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T[0,0] = 1j/sqrt(2); T[0,4] = -1j/sqrt(2)
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T[1,1] = 1j/sqrt(2); T[1,3] = 1j/sqrt(2)
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T[2,2] = 1.0
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T[3,1] = 1.0/sqrt(2); T[3,3] = -1.0/sqrt(2)
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T[4,0] = 1.0/sqrt(2); T[4,4] = 1.0/sqrt(2)
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elif l == 3:
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cubic_names = ("x(x^2-3y^2)","z(x^2-y^2)","xz^2","z^3","yz^2","xyz","y(3x^2-y^2)")
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T[0,0] = 1.0/sqrt(2); T[0,6] = -1.0/sqrt(2)
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T[1,1] = 1.0/sqrt(2); T[1,5] = 1.0/sqrt(2)
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T[2,2] = 1.0/sqrt(2); T[2,4] = -1.0/sqrt(2)
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T[3,5] = 1.0
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T[4,2] = 1j/sqrt(2); T[4,4] = 1j/sqrt(2)
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T[5,1] = 1j/sqrt(2); T[5,5] = -1j/sqrt(2)
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T[6,0] = 1j/sqrt(2); T[6,6] = 1j/sqrt(2)
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else: raise ValueError("spherical_to_cubic: implemented only for l=0,1,2,3")
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return T
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# Names of cubic harmonics
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def cubic_names(l):
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r"""
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Get the names of the cubic harmonics.
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Parameters
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----------
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l : integer or string
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Angular momentum of shell being treated.
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Also takes 't2g' and 'eg' as arguments.
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Returns
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-------
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cubic_names : tuple of strings
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Names of the orbitals.
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"""
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if l == 0 or l == 's':
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return ("s")
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elif l == 1 or l == 'p':
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return ("x","y","z")
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elif l == 2 or l == 'd':
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return ("xy","yz","z^2","xz","x^2-y^2")
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elif l == 't2g':
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return ("xy","yz","xz")
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elif l == 'eg':
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return ("z^2","x^2-y^2")
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elif l == 3 or l == 'f':
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return ("x(x^2-3y^2)","z(x^2-y^2)","xz^2","z^3","yz^2","xyz","y(3x^2-y^2)")
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else: raise ValueError("cubic_names: implemented only for l=0,1,2,3")
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# Convert U,J -> radial integrals F_k
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def U_J_to_radial_integrals(l, U_int, J_hund):
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r"""
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Determine the radial integrals F_k from U_int and J_hund.
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Parameters
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----------
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l : integer
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Angular momentum of shell being treated (l=2 for d shell, l=3 for f shell).
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U_int : scalar
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Value of the screened Hubbard interaction.
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J_hund : scalar
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Value of the Hund's coupling.
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Returns
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-------
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radial_integrals : list
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Slater integrals [F0,F2,F4,..].
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"""
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F = np.zeros((l+1),dtype=float)
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if l == 2:
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F[0] = U_int
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F[1] = J_hund * 14.0 / (1.0 + 0.63)
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F[2] = 0.630 * F[1]
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elif l == 3:
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F[0] = U_int
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F[1] = 6435.0 * J_hund / (286.0 + 195.0 * 451.0 / 675.0 + 250.0 * 1001.0 / 2025.0)
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F[2] = 451.0 * F[1] / 675.0
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F[3] = 1001.0 * F[1] / 2025.0
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else: raise ValueError("U_J_to_radial_integrals: implemented only for l=2,3")
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return F
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# Convert radial integrals F_k -> U,J
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def radial_integrals_to_U_J(l, F):
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r"""
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Determine U_int and J_hund from the radial integrals.
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Parameters
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----------
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l : integer
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Angular momentum of shell being treated (l=2 for d shell, l=3 for f shell).
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F : list
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Slater integrals [F0,F2,F4,..].
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Returns
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-------
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U_int : scalar
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Value of the screened Hubbard interaction.
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J_hund : scalar
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Value of the Hund's coupling.
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"""
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if l == 2:
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U_int = F[0]
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J_hund = F[1] * (1.0 + 0.63) / 14.0
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elif l == 3:
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U_int = F[0]
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J_hund = F[1] * (286.0 + 195.0 * 451.0 / 675.0 + 250.0 * 1001.0 / 2025.0) / 6435.0
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else: raise ValueError("radial_integrals_to_U_J: implemented only for l=2,3")
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return U_int,J_hund
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# Angular matrix elements of particle-particle interaction
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# (2l+1)^2 ((l 0) (k 0) (l 0))^2 \sum_{q=-k}^{k} (-1)^{m1+m2+q} ((l -m1) (k q) (l m3)) ((l -m2) (k -q) (l m4))
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def angular_matrix_element(l, k, m1, m2, m3, m4):
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r"""
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Calculate the angular matrix element
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.. math::
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(2l+1)^2
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\begin{pmatrix}
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l & k & l \\
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0 & 0 & 0
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\end{pmatrix}^2
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\sum_{q=-k}^k (-1)^{m_1+m_2+q}
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\begin{pmatrix}
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l & k & l \\
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-m_1 & q & m_3
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\end{pmatrix}
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\begin{pmatrix}
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l & k & l \\
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-m_2 & -q & m_4
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\end{pmatrix}.
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Parameters
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----------
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l : integer
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k : integer
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m1 : integer
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m2 : integer
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m3 : integer
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m4 : integer
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Returns
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-------
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ang_mat_ele : scalar
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Angular matrix element.
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"""
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ang_mat_ele = 0
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for q in range(-k,k+1):
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ang_mat_ele += three_j_symbol((l,-m1),(k,q),(l,m3))*three_j_symbol((l,-m2),(k,-q),(l,m4))*(-1.0 if (m1+q+m2) % 2 else 1.0)
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ang_mat_ele *= (2*l+1)**2 * (three_j_symbol((l,0),(k,0),(l,0))**2)
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return ang_mat_ele
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# Wigner 3-j symbols
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# ((j1 m1) (j2 m2) (j3 m3))
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def three_j_symbol(jm1, jm2, jm3):
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r"""
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Calculate the three-j symbol
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.. math::
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\begin{pmatrix}
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l_1 & l_2 & l_3\\
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m_1 & m_2 & m_3
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\end{pmatrix}.
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Parameters
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----------
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jm1 : tuple of integers
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(j_1 m_1)
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jm2 : tuple of integers
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(j_2 m_2)
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jm3 : tuple of integers
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(j_3 m_3)
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Returns
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-------
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three_j_sym : scalar
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Three-j symbol.
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"""
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j1, m1 = jm1
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j2, m2 = jm2
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j3, m3 = jm3
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if (m1+m2+m3 != 0 or
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m1 < -j1 or m1 > j1 or
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m2 < -j2 or m2 > j2 or
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m3 < -j3 or m3 > j3 or
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j3 > j1 + j2 or
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j3 < abs(j1-j2)):
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return .0
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three_j_sym = -1.0 if (j1-j2-m3) % 2 else 1.0
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three_j_sym *= sqrt(fact(j1+j2-j3)*fact(j1-j2+j3)*fact(-j1+j2+j3)/fact(j1+j2+j3+1))
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three_j_sym *= sqrt(fact(j1-m1)*fact(j1+m1)*fact(j2-m2)*fact(j2+m2)*fact(j3-m3)*fact(j3+m3))
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t_min = max(j2-j3-m1,j1-j3+m2,0)
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t_max = min(j1-m1,j2+m2,j1+j2-j3)
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|
|
|
t_sum = 0
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|
for t in range(t_min,t_max+1):
|
|
t_sum += (-1.0 if t % 2 else 1.0)/(fact(t)*fact(j3-j2+m1+t)*fact(j3-j1-m2+t)*fact(j1+j2-j3-t)*fact(j1-m1-t)*fact(j2+m2-t))
|
|
|
|
three_j_sym *= t_sum
|
|
return three_j_sym
|
|
|
|
# Clebsch-Gordan coefficients
|
|
# < j1 m1 j2 m2 | j3 m3 > = (-1)^{j1-j2+m3} \sqrt{2j3+1} ((j1 m1) (j2 m2) (j3 -m3))
|
|
def clebsch_gordan(jm1, jm2, jm3):
|
|
r"""
|
|
Calculate the Clebsh-Gordan coefficient
|
|
|
|
.. math::
|
|
\langle j_1 m_1 j_2 m_2 | j_3 m_3 \rangle = (-1)^{j_1-j_2+m_3} \sqrt{2 j_3 + 1}
|
|
\begin{pmatrix}
|
|
j_1 & j_2 & j_3\\
|
|
m_1 & m_2 & -m_3
|
|
\end{pmatrix}.
|
|
|
|
Parameters
|
|
----------
|
|
jm1 : tuple of integers
|
|
(j_1 m_1)
|
|
jm2 : tuple of integers
|
|
(j_2 m_2)
|
|
jm3 : tuple of integers
|
|
(j_3 m_3)
|
|
|
|
Returns
|
|
-------
|
|
cgcoeff : scalar
|
|
Clebsh-Gordan coefficient.
|
|
|
|
"""
|
|
norm = sqrt(2*jm3[0]+1)*(-1 if jm1[0]-jm2[0]+jm3[1] % 2 else 1)
|
|
return norm*three_j_symbol(jm1,jm2,(jm3[0],-jm3[1]))
|
|
|
|
# Create subarray containing columns in idxlist
|
|
# e.g. idxlist = [(0),(2,3),(0,1,2,3)] gives
|
|
# column 0 for 1st dim,
|
|
# columns 2 and 3 for 2nd dim,
|
|
# columns 0,1,2 and 3 for 3rd dim.
|
|
def subarray(a,idxlist,n=None) :
|
|
r"""
|
|
Extract a subarray from a matrix-like object.
|
|
|
|
Parameters
|
|
----------
|
|
a : matrix or array
|
|
idxlist : list of tuples
|
|
Columns that need to be extracted for each dimension.
|
|
|
|
Returns
|
|
-------
|
|
subarray : matrix or array
|
|
|
|
Examples
|
|
--------
|
|
idxlist = [(0),(2,3),(0,1,2,3)] gives
|
|
|
|
- column 0 for 1st dim,
|
|
- columns 2 and 3 for 2nd dim,
|
|
- columns 0, 1, 2 and 3 for 3rd dim.
|
|
|
|
"""
|
|
if n is None: n = len(a.shape)-1
|
|
sa = a[tuple(slice(x) for x in a.shape[:n]) + (idxlist[n],)]
|
|
return subarray(sa,idxlist, n-1) if n > 0 else sa
|