def _nao_sub(mol, pre_occ, pre_nao, s=None):
if s is None:
s = mol.intor_symmetric('cint1e_ovlp_sph')
core_lst, val_lst, rydbg_lst = _core_val_ryd_list(mol)
nbf = mol.nao_nr()
cnao = numpy.empty((nbf,nbf))
if core_lst:
c = pre_nao[:,core_lst].copy()
s1 = reduce(lib.dot, (c.T, s, c))
cnao[:,core_lst] = c1 = lib.dot(c, orth.lowdin(s1))
c = pre_nao[:,val_lst].copy()
c -= reduce(lib.dot, (c1, c1.T, s, c))
else:
c = pre_nao[:,val_lst]
if val_lst:
s1 = reduce(lib.dot, (c.T, s, c))
wt = pre_occ[val_lst]
cnao[:,val_lst] = lib.dot(c, orth.weight_orth(s1, wt))
if rydbg_lst:
cvlst = core_lst + val_lst
c1 = cnao[:,cvlst].copy()
c = pre_nao[:,rydbg_lst].copy()
c -= reduce(lib.dot, (c1, c1.T, s, c))
s1 = reduce(lib.dot, (c.T, s, c))
cnao[:,rydbg_lst] = lib.dot(c, orth.lowdin(s1))
snorm = numpy.linalg.norm(reduce(lib.dot, (cnao.T, s, cnao)) - numpy.eye(nbf))
if snorm > 1e-9:
logger.warn(mol, 'Weak orthogonality for localized orbitals %s', snorm)
return cnao
def _nao_sub(mol, pre_occ, pre_nao):
core_lst, val_lst, rydbg_lst = _core_val_ryd_list(mol)
s = mol.intor_symmetric('cint1e_ovlp_sph')
nao = mol.nao_nr()
cnao = numpy.zeros((nao,nao))
if core_lst:
c = pre_nao[:,core_lst]
s1 = reduce(numpy.dot, (c.T, s, c))
cnao[:,core_lst] = numpy.dot(c, orth.lowdin(s1))
c1 = cnao[:,core_lst]
rm_core = numpy.eye(nao) - reduce(numpy.dot, (c1, c1.T, s))
c = numpy.dot(rm_core, pre_nao[:,val_lst])
s1 = reduce(numpy.dot, (c.T.conj(), s, c))
wt = pre_occ[val_lst]
cnao[:,val_lst] = numpy.dot(c, orth.weight_orth(s1, wt))
if rydbg_lst:
c1 = cnao[:,val_lst]
rm_val = numpy.eye(nao)-reduce(numpy.dot, (c1, c1.T, s))
c = reduce(numpy.dot, (rm_core, rm_val, pre_nao[:,rydbg_lst]))
s1 = reduce(numpy.dot, (c.T.conj(), s, c))
cnao[:,rydbg_lst] = numpy.dot(c, orth.lowdin(s1))
return cnao
def _nao_sub(mol, pre_occ, pre_nao, s=None):
if s is None:
s = mol.intor_symmetric('cint1e_ovlp_sph')
core_lst, val_lst, rydbg_lst = _core_val_ryd_list(mol)
nbf = mol.nao_nr()
cnao = numpy.empty((nbf,nbf))
if core_lst:
c = pre_nao[:,core_lst]
s1 = reduce(numpy.dot, (c.T, s, c))
cnao[:,core_lst] = c1 = numpy.dot(c, orth.lowdin(s1))
c = pre_nao[:,val_lst]
c -= reduce(numpy.dot, (c1, c1.T, s, c))
else:
c = pre_nao[:,val_lst]
s1 = reduce(numpy.dot, (c.T, s, c))
wt = pre_occ[val_lst]
cnao[:,val_lst] = numpy.dot(c, orth.weight_orth(s1, wt))
if rydbg_lst:
cvlst = core_lst + val_lst
c1 = cnao[:,cvlst]
c = pre_nao[:,rydbg_lst]
c -= reduce(numpy.dot, (c1, c1.T, s, c))
s1 = reduce(numpy.dot, (c.T, s, c))
cnao[:,rydbg_lst] = numpy.dot(c, orth.lowdin(s1))
snorm = numpy.linalg.norm(reduce(numpy.dot, (cnao.T, s, cnao)) - numpy.eye(nbf))
if snorm > 1e-9:
logger.warn(mol, 'Weak orthogonality for localized orbitals %s', snorm)
return cnao
def _iao_complementary_orbitals(mol, iao_ref):
"""Get the IAOs for complementary space (virtual orbitals).
Args:
mol (pyscf.gto.Mole): The molecule to simulate.
iao_ref (numpy.array): IAO in occupied space (float64).
Returns:
iao_comp (numpy.array): IAO in complementary space (float64).
"""
# Get the total number of AOs
norbital_total = mol.nao_nr()
# Calculate the Overlaps for total basis
s1 = mol.intor_symmetric('int1e_ovlp')
# Construct the complementary space AO
number_iaos = iao_ref.shape[1]
number_inactive = norbital_total - number_iaos
iao_com_ref = _iao_complementary_space(iao_ref, s1, number_inactive)
# Get a list of active orbitals
min_mol = iao.reference_mol(mol)
norbital_active, active_list = _iao_count_active(mol, min_mol)
# Obtain the Overlap-like matrices
s21 = s1[active_list, : ]
s2 = s21[ : , active_list]
s12 = s21.T
# Calculate P_12 = S_1^-1 * S_12 using Cholesky decomposition
s1_sqrt = scipy.linalg.cho_factor(s1)
s2_sqrt = scipy.linalg.cho_factor(s2)
p12 = scipy.linalg.cho_solve(s1_sqrt, s12)
# C~ = orth ( second_half ( S_1^-1 * S_12 * first_half ( S_2^-1 * S_21 * C ) ) )
c_tilde = scipy.linalg.cho_solve(s2_sqrt, np.dot(s21, iao_com_ref))
c_tilde = scipy.linalg.cho_solve(s1_sqrt, np.dot(s12, c_tilde))
c_tilde = np.dot(c_tilde, orth.lowdin(reduce(np.dot, (c_tilde.T, s1, c_tilde))))
# Obtain C * C^T * S1 and C~ * C~^T * S1
ccs1 = reduce(np.dot, (iao_com_ref, iao_com_ref.conj().T, s1))
ctcts1 = reduce(np.dot, (c_tilde, c_tilde.conj().T, s1))
# Calculate A = ccs1 * ctcts1 * p12 + ( 1 - ccs1 ) * ( 1 - ctcts1 ) * p12
iao_comp = (p12 + reduce(np.dot, (ccs1, ctcts1, p12)) * 2 - np.dot(ccs1, p12) - np.dot(ctcts1, p12))
iao_comp = np.dot(iao_comp, orth.lowdin(reduce(np.dot, (iao_comp.T, s1, iao_comp))))
return iao_comp
def _iao_occupied_orbitals(mol, mf):
"""Get the IAOs for occupied space.
Args:
mol (pyscf.gto.Mole): The molecule to simulate.
mf (pyscf.scf.RHF): The mean field of the molecule.
Returns:
iao_active (numpy.array): The localized orbitals for the occupied space (float64).
"""
# Get MO coefficient of occupied MOs
occupied_orbitals = mf.mo_coeff[:, mf.mo_occ > 0.5]
# Get mol data in minao basis
min_mol = iao.reference_mol(mol)
# Calculate the overlaps for total basis
s1 = mol.intor_symmetric('int1e_ovlp')
# ... for minao basis
s2 = min_mol.intor_symmetric('int1e_ovlp')
# ... between the two basis (and transpose)
s12 = gto.mole.intor_cross('int1e_ovlp', mol, min_mol)
s21 = s12.T
# Calculate P_12 = S_1^-1 * S_12 using Cholesky decomposition
s1_sqrt = scipy.linalg.cho_factor(s1)
s2_sqrt = scipy.linalg.cho_factor(s2)
p12 = scipy.linalg.cho_solve(s1_sqrt, s12)
# C~ = second_half ( S_1^-1 * S_12 * first_half ( S_2^-1 * S_21 * C ) )
c_tilde = scipy.linalg.cho_solve(s2_sqrt, np.dot(s21, occupied_orbitals))
c_tilde = scipy.linalg.cho_solve(s1_sqrt, np.dot(s12, c_tilde))
c_tilde = np.dot(c_tilde, orth.lowdin(reduce(np.dot, (c_tilde.T, s1, c_tilde))))
# Obtain C * C^T * S1 and C~ * C~^T * S1
ccs1 = reduce(np.dot, (occupied_orbitals, occupied_orbitals.conj().T, s1))
ctcts1 = reduce(np.dot, (c_tilde, c_tilde.conj().T, s1))
# Calculate A = ccs1 * ctcts1 * p12 + ( 1 - ccs1 ) * ( 1 - ctcts1 ) * p12
iao_active = (p12 + reduce(np.dot, (ccs1, ctcts1, p12)) * 2 - np.dot(ccs1, p12) - np.dot(ctcts1, p12))
# Orthogonalize A
iao_active = np.dot(iao_active, orth.lowdin(reduce(np.dot, (iao_active.T, s1, iao_active))))
return iao_active
def _nao_sub(mol, pre_occ, pre_nao, s=None):
if s is None:
if getattr(mol, 'pbc_intor', None): # whether mol object is a cell
s = mol.pbc_intor('int1e_ovlp', hermi=1)
else:
s = mol.intor_symmetric('int1e_ovlp')
core_lst, val_lst, rydbg_lst = _core_val_ryd_list(mol)
nbf = mol.nao_nr()
pre_nao = pre_nao.astype(s.dtype)
cnao = numpy.empty((nbf, nbf), dtype=s.dtype)
if core_lst:
c = pre_nao[:, core_lst].copy()
s1 = reduce(lib.dot, (c.conj().T, s, c))
cnao[:, core_lst] = c1 = lib.dot(c, orth.lowdin(s1))
c = pre_nao[:, val_lst].copy()
c -= reduce(lib.dot, (c1, c1.conj().T, s, c))
else:
c = pre_nao[:, val_lst]
if val_lst:
s1 = reduce(lib.dot, (c.conj().T, s, c))
wt = pre_occ[val_lst]
cnao[:, val_lst] = lib.dot(c, orth.weight_orth(s1, wt))
if rydbg_lst:
cvlst = core_lst + val_lst
c1 = cnao[:, cvlst].copy()
c = pre_nao[:, rydbg_lst].copy()
c -= reduce(lib.dot, (c1, c1.conj().T, s, c))
s1 = reduce(lib.dot, (c.conj().T, s, c))
cnao[:, rydbg_lst] = lib.dot(c, orth.lowdin(s1))
snorm = numpy.linalg.norm(
reduce(lib.dot, (cnao.conj().T, s, cnao)) - numpy.eye(nbf))
if snorm > 1e-9:
logger.warn(mol, 'Weak orthogonality for localized orbitals %s', snorm)
return cnao
def _nao_sub(mol, pre_occ, pre_nao, s=None):
if s is None:
if getattr(mol, 'pbc_intor', None): # whether mol object is a cell
s = mol.pbc_intor('int1e_ovlp', hermi=1)
else:
s = mol.intor_symmetric('int1e_ovlp')
core_lst, val_lst, rydbg_lst = _core_val_ryd_list(mol)
nbf = mol.nao_nr()
pre_nao = pre_nao.astype(s.dtype)
cnao = numpy.empty((nbf,nbf), dtype=s.dtype)
if core_lst:
c = pre_nao[:,core_lst].copy()
s1 = reduce(lib.dot, (c.conj().T, s, c))
cnao[:,core_lst] = c1 = lib.dot(c, orth.lowdin(s1))
c = pre_nao[:,val_lst].copy()
c -= reduce(lib.dot, (c1, c1.conj().T, s, c))
else:
c = pre_nao[:,val_lst]
if val_lst:
s1 = reduce(lib.dot, (c.conj().T, s, c))
wt = pre_occ[val_lst]
cnao[:,val_lst] = lib.dot(c, orth.weight_orth(s1, wt))
if rydbg_lst:
cvlst = core_lst + val_lst
c1 = cnao[:,cvlst].copy()
c = pre_nao[:,rydbg_lst].copy()
c -= reduce(lib.dot, (c1, c1.conj().T, s, c))
s1 = reduce(lib.dot, (c.conj().T, s, c))
cnao[:,rydbg_lst] = lib.dot(c, orth.lowdin(s1))
snorm = numpy.linalg.norm(reduce(lib.dot, (cnao.conj().T, s, cnao)) - numpy.eye(nbf))
if snorm > 1e-9:
logger.warn(mol, 'Weak orthogonality for localized orbitals %s', snorm)
return cnao
def test_orth(self):
numpy.random.seed(10)
n = 100
a = numpy.random.random((n,n))
s = numpy.dot(a.T, a)
c = orth.lowdin(s)
self.assertTrue(numpy.allclose(reduce(numpy.dot, (c.T, s, c)),
numpy.eye(n)))
x1 = numpy.dot(a, c)
x2 = orth.vec_lowdin(a)
d = numpy.dot(x1.T,x2)
d[numpy.diag_indices(n)] = 0
self.assertAlmostEqual(numpy.linalg.norm(d), 0, 9)
self.assertAlmostEqual(numpy.linalg.norm(c), 36.56738258719514, 9)
self.assertAlmostEqual(abs(c).sum(), 2655.5580057303964, 7)
def test_orth(self):
numpy.random.seed(10)
n = 100
a = numpy.random.random((n, n))
s = numpy.dot(a.T, a)
c = orth.lowdin(s)
self.assertTrue(
numpy.allclose(reduce(numpy.dot, (c.T, s, c)), numpy.eye(n)))
x1 = numpy.dot(a, c)
x2 = orth.vec_lowdin(a)
d = numpy.dot(x1.T, x2)
d[numpy.diag_indices(n)] = 0
self.assertAlmostEqual(numpy.linalg.norm(d), 0, 9)
self.assertAlmostEqual(numpy.linalg.norm(c), 36.56738258719514, 9)
self.assertAlmostEqual(abs(c).sum(), 2655.5580057303964, 7)
def _iao_atoms(mol, iao1, iao2):
"""Assign IAO to atom centers and rearrange the IAOs.
Args:
mol (pyscf.gto.Mole): The molecule to simulate.
mf (pyscf.scf.RHF): The mean field of the molecule.
iao1 (numpy.array): IAO for occupied space (float64).
iao2 (numpy.array): IAO for complementary space (float64).
Returns:
iao_combined (numpy.array): The rearranged IAO (float64).
"""
# Calclate the integrals for assignment
number_orbitals = mol.nao_nr()
r_int1e = mol.intor('cint1e_r_sph', 3)
iao_combine = np.hstack((iao1, iao2))
# Calculate atom center for each orbital
x = np.diag(reduce(np.dot,(iao_combine.T, r_int1e[0], iao_combine)))
y = np.diag(reduce(np.dot,(iao_combine.T, r_int1e[1], iao_combine)))
z = np.diag(reduce(np.dot,(iao_combine.T, r_int1e[2], iao_combine)))
# Align the coordinates
orbitals_temp = np.vstack((x, y, z))
orbitals = orbitals_temp.T
# Assign each orbital to atom center
atom_list = _dmet_atom_list(mol, orbitals)
# Prepare the orbital labels
orb_list = _dmet_orb_list(mol, atom_list)
# Rearrange the orbitals
iao_combine = iao_combine[ : , orb_list]
# Orthogonalize the orbitals
s1 = mol.intor_symmetric('int1e_ovlp')
iao_combine = np.dot(iao_combine, orth.lowdin(reduce(np.dot, (iao_combine.T, s1, iao_combine))))
return iao_combine
def simple_1iao(mol, atm_id, minao='minao'):
atm = minao_atm(mol, atm_id, minao)
c1 = pyscf.scf.addons.project_mo_nr2nr(atm, 1, mol)
s = reduce(numpy.dot, (c1.T, mol.intor_symmetric('cint1e_ovlp_sph'), c1))
return orth.lowdin(s)