def cholesky_mos(mo_coeff):
'''
Calculates localized orbitals through a pivoted Cholesky factorization
of the density matrix.
Args:
mo_coeff: block of MO coefficients to be localized
Returns:
the localized MOs
'''
assert(mo_coeff.ndim == 2)
nao, nmo = mo_coeff.shape
# Factorization of a density matrix-like quantity.
D = np.dot(mo_coeff, mo_coeff.T)
L, piv, rank = pivoted_cholesky(D, lower=True)
if rank < nmo:
raise RuntimeError('rank of matrix lower than the number of orbitals')
# Permute L back to the original order of the AOs.
# Superfluous columns are cropped out.
P = np.zeros((nao, nao))
P[piv, np.arange(nao)] = 1
mo_loc = np.dot(P, L[:, :nmo])
return mo_loc
def partial_cholesky_orth_(S, canthr=1e-7, cholthr=1e-9):
'''Partial Cholesky orthogonalization for curing overcompleteness.
References:
Susi Lehtola, Curing basis set overcompleteness with pivoted
Cholesky decompositions, J. Chem. Phys. 151, 241102 (2019),
doi:10.1063/1.5139948.
Susi Lehtola, Accurate reproduction of strongly repulsive
interatomic potentials, Phys. Rev. A 101, 032504 (2020),
doi:10.1103/PhysRevA.101.032504.
'''
# Ensure the basis functions are normalized
normlz = numpy.power(numpy.diag(S), -0.5)
Snorm = numpy.dot(numpy.diag(normlz), numpy.dot(S, numpy.diag(normlz)))
# Sort the basis functions according to the Gershgorin circle
# theorem so that the Cholesky routine is well-initialized
odS = numpy.abs(Snorm)
numpy.fill_diagonal(odS, 0.0)
odSs = numpy.sum(odS, axis=0)
sortidx = numpy.argsort(odSs)
# Run the pivoted Cholesky decomposition
Ssort = Snorm[numpy.ix_(sortidx, sortidx)].copy()
c, piv, r_c = pivoted_cholesky(Ssort, tol=cholthr)
# The functions we're going to use are given by the pivot as
idx = sortidx[piv[:r_c]]
# Get the (un-normalized) sub-basis
Ssub = S[numpy.ix_(idx, idx)].copy()
# Orthogonalize sub-basis
Xsub = canonical_orth_(Ssub, thr=canthr)
# Full X
X = numpy.zeros((S.shape[0], Xsub.shape[1]))
X[idx, :] = Xsub
return X