#!/usr/bin/env python
# Copyright 2014-2019 The PySCF Developers. All Rights Reserved.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
from functools import reduce
import numpy
import scipy.linalg
from pyscf import lib
from pyscf import gto
from pyscf.lib import logger
from pyscf.scf import hf
from pyscf.scf import chkfile
from pyscf import __config__
WITH_META_LOWDIN = getattr(__config__, 'scf_analyze_with_meta_lowdin', True)
PRE_ORTH_METHOD = getattr(__config__, 'scf_analyze_pre_orth_method', 'ANO')
BREAKSYM = getattr(__config__, 'scf_uhf_init_guess_breaksym', True)
MO_BASE = getattr(__config__, 'MO_BASE', 1)
[docs]
def init_guess_by_minao(mol, breaksym=BREAKSYM):
'''Generate initial guess density matrix based on ANO basis, then project
the density matrix to the basis set defined by ``mol``
Returns:
Density matrices, a list of 2D ndarrays for alpha and beta spins
'''
dm = hf.init_guess_by_minao(mol)
dma = dmb = dm*.5
if breaksym:
dma, dmb = _break_dm_spin_symm(mol, (dma, dmb))
return numpy.array((dma,dmb))
[docs]
def init_guess_by_1e(mol, breaksym=BREAKSYM):
return UHF(mol).init_guess_by_1e(mol, breaksym)
[docs]
def init_guess_by_atom(mol, breaksym=BREAKSYM):
dm = hf.init_guess_by_atom(mol)
dma = dmb = dm*.5
if mol.spin == 0 and breaksym:
#Add off-diagonal part for alpha DM
dma = mol.intor_symmetric('int1e_ovlp') * 1e-2
for b0, b1, p0, p1 in mol.aoslice_by_atom():
dma[p0:p1,p0:p1] = dmb[p0:p1,p0:p1]
return numpy.array((dma,dmb))
[docs]
def init_guess_by_huckel(mol, breaksym=BREAKSYM):
return UHF(mol).init_guess_by_huckel(mol, breaksym)
[docs]
def init_guess_by_mod_huckel(mol, breaksym=BREAKSYM):
return UHF(mol).init_guess_by_mod_huckel(mol, breaksym)
[docs]
def init_guess_by_chkfile(mol, chkfile_name, project=None):
'''Read SCF chkfile and make the density matrix for UHF initial guess.
Kwargs:
project : None or bool
Whether to project chkfile's orbitals to the new basis. Note when
the geometry of the chkfile and the given molecule are very
different, this projection can produce very poor initial guess.
In PES scanning, it is recommended to switch off project.
If project is set to None, the projection is only applied when the
basis sets of the chkfile's molecule are different to the basis
sets of the given molecule (regardless whether the geometry of
the two molecules are different). Note the basis sets are
considered to be different if the two molecules are derived from
the same molecule with different ordering of atoms.
'''
from pyscf.scf import addons
chk_mol, scf_rec = chkfile.load_scf(chkfile_name)
if project is None:
project = not gto.same_basis_set(chk_mol, mol)
# Check whether the two molecules are similar
im1 = scipy.linalg.eigvalsh(mol.inertia_moment())
im2 = scipy.linalg.eigvalsh(chk_mol.inertia_moment())
# im1+1e-7 to avoid 'divide by zero' error
if abs((im1-im2)/(im1+1e-7)).max() > 0.01:
logger.warn(mol, "Large deviations found between the input "
"molecule and the molecule from chkfile\n"
"Initial guess density matrix may have large error.")
if project:
s = hf.get_ovlp(mol)
def fproj(mo):
if project:
mo = addons.project_mo_nr2nr(chk_mol, mo, mol)
norm = numpy.einsum('pi,pi->i', mo.conj(), s.dot(mo))
mo /= numpy.sqrt(norm)
return mo
mo = scf_rec['mo_coeff']
mo_occ = scf_rec['mo_occ']
if getattr(mo[0], 'ndim', None) == 1: # RHF
if numpy.iscomplexobj(mo):
raise NotImplementedError('TODO: project DHF orbital to UHF orbital')
mo_coeff = fproj(mo)
mo_occa = (mo_occ>1e-8).astype(numpy.double)
mo_occb = mo_occ - mo_occa
dm = make_rdm1([mo_coeff,mo_coeff], [mo_occa,mo_occb])
else: #UHF
if getattr(mo[0][0], 'ndim', None) == 2: # KUHF
logger.warn(mol, 'k-point UHF results are found. Density matrix '
'at Gamma point is used for the molecular SCF initial guess')
mo = mo[0]
dm = make_rdm1([fproj(mo[0]),fproj(mo[1])], mo_occ)
return dm
def _break_dm_spin_symm(mol, dm):
dma, dmb = dm
# For spin polarized system, no need to manually break spin symmetry
if mol.spin == 0 and abs(dma - dmb).max() < 1e-2:
#remove off-diagonal part of beta DM
dmb = numpy.zeros_like(dma)
for b0, b1, p0, p1 in mol.aoslice_by_atom():
dmb[...,p0:p1,p0:p1] = dma[...,p0:p1,p0:p1]
return dma, dmb
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def get_init_guess(mol, key='minao'):
return UHF(mol).get_init_guess(mol, key)
[docs]
def make_rdm1(mo_coeff, mo_occ, **kwargs):
'''One-particle density matrix in AO representation
Args:
mo_coeff : tuple of 2D ndarrays
Orbital coefficients for alpha and beta spins. Each column is one orbital.
mo_occ : tuple of 1D ndarrays
Occupancies for alpha and beta spins.
Returns:
A list of 2D ndarrays for alpha and beta spins
'''
mo_a = mo_coeff[0]
mo_b = mo_coeff[1]
dm_a = numpy.dot(mo_a*mo_occ[0], mo_a.conj().T)
dm_b = numpy.dot(mo_b*mo_occ[1], mo_b.conj().T)
# DO NOT make tag_array for DM here because the DM arrays may be modified and
# passed to functions like get_jk, get_vxc. These functions may take the tags
# (mo_coeff, mo_occ) to compute the potential if tags were found in the DM
# arrays and modifications to DM arrays may be ignored.
return lib.tag_array((dm_a, dm_b), mo_coeff=mo_coeff, mo_occ=mo_occ)
[docs]
def make_rdm2(mo_coeff, mo_occ):
'''Two-particle density matrix in AO representation
Args:
mo_coeff : tuple of 2D ndarrays
Orbital coefficients for alpha and beta spins. Each column is one orbital.
mo_occ : tuple of 1D ndarrays
Occupancies for alpha and beta spins.
Returns:
A tuple of three 4D ndarrays for alpha,alpha and alpha,beta and beta,beta spins
'''
dm1a, dm1b = make_rdm1(mo_coeff, mo_occ)
dm2aa = (numpy.einsum('ij,kl->ijkl', dm1a, dm1a)
- numpy.einsum('ij,kl->iklj', dm1a, dm1a))
dm2bb = (numpy.einsum('ij,kl->ijkl', dm1b, dm1b)
- numpy.einsum('ij,kl->iklj', dm1b, dm1b))
dm2ab = numpy.einsum('ij,kl->ijkl', dm1a, dm1b)
return dm2aa, dm2ab, dm2bb
[docs]
def get_veff(mol, dm, dm_last=0, vhf_last=0, hermi=1, vhfopt=None):
r'''Unrestricted Hartree-Fock potential matrix of alpha and beta spins,
for the given density matrix
.. math::
V_{ij}^\alpha &= \sum_{kl} (ij|kl)(\gamma_{lk}^\alpha+\gamma_{lk}^\beta)
- \sum_{kl} (il|kj)\gamma_{lk}^\alpha \\
V_{ij}^\beta &= \sum_{kl} (ij|kl)(\gamma_{lk}^\alpha+\gamma_{lk}^\beta)
- \sum_{kl} (il|kj)\gamma_{lk}^\beta
Args:
mol : an instance of :class:`Mole`
dm : a list of ndarrays
A list of density matrices, stored as (alpha,alpha,...,beta,beta,...)
Kwargs:
dm_last : ndarray or a list of ndarrays or 0
The density matrix baseline. When it is not 0, this function computes
the increment of HF potential w.r.t. the reference HF potential matrix.
vhf_last : ndarray or a list of ndarrays or 0
The reference HF potential matrix.
hermi : int
Whether J, K matrix is hermitian
| 0 : no hermitian or symmetric
| 1 : hermitian
| 2 : anti-hermitian
vhfopt :
A class which holds precomputed quantities to optimize the
computation of J, K matrices
Returns:
:math:`V_{hf} = (V^\alpha, V^\beta)`. :math:`V^\alpha` (and :math:`V^\beta`)
can be a list matrices, corresponding to the input density matrices.
Examples:
>>> import numpy
>>> from pyscf import gto, scf
>>> mol = gto.M(atom='H 0 0 0; H 0 0 1.1')
>>> dmsa = numpy.random.random((3,mol.nao_nr(),mol.nao_nr()))
>>> dmsb = numpy.random.random((3,mol.nao_nr(),mol.nao_nr()))
>>> dms = numpy.vstack((dmsa,dmsb))
>>> dms.shape
(6, 2, 2)
>>> vhfa, vhfb = scf.uhf.get_veff(mol, dms, hermi=0)
>>> vhfa.shape
(3, 2, 2)
>>> vhfb.shape
(3, 2, 2)
'''
dm = numpy.asarray(dm)
nao = dm.shape[-1]
ddm = dm - numpy.asarray(dm_last)
# dm.reshape(-1,nao,nao) to remove first dim, compress (dma,dmb)
vj, vk = hf.get_jk(mol, ddm.reshape(-1,nao,nao), hermi=hermi, vhfopt=vhfopt)
vj = vj.reshape(dm.shape)
vk = vk.reshape(dm.shape)
assert (vj.ndim >= 3 and vj.shape[0] == 2)
vhf = vj[0] + vj[1] - vk
vhf += numpy.asarray(vhf_last)
return vhf
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def get_fock(mf, h1e=None, s1e=None, vhf=None, dm=None, cycle=-1, diis=None,
diis_start_cycle=None, level_shift_factor=None, damp_factor=None):
if h1e is None: h1e = mf.get_hcore()
if vhf is None: vhf = mf.get_veff(mf.mol, dm)
f = numpy.asarray(h1e) + vhf
if f.ndim == 2:
f = (f, f)
if cycle < 0 and diis is None: # Not inside the SCF iteration
return f
if diis_start_cycle is None:
diis_start_cycle = mf.diis_start_cycle
if level_shift_factor is None:
level_shift_factor = mf.level_shift
if damp_factor is None:
damp_factor = mf.damp
if s1e is None: s1e = mf.get_ovlp()
if dm is None: dm = mf.make_rdm1()
if isinstance(level_shift_factor, (tuple, list, numpy.ndarray)):
shifta, shiftb = level_shift_factor
else:
shifta = shiftb = level_shift_factor
if isinstance(damp_factor, (tuple, list, numpy.ndarray)):
dampa, dampb = damp_factor
else:
dampa = dampb = damp_factor
if isinstance(dm, numpy.ndarray) and dm.ndim == 2:
dm = [dm*.5] * 2
if 0 <= cycle < diis_start_cycle-1 and abs(dampa)+abs(dampb) > 1e-4:
f = (hf.damping(s1e, dm[0], f[0], dampa),
hf.damping(s1e, dm[1], f[1], dampb))
if diis and cycle >= diis_start_cycle:
f = diis.update(s1e, dm, f, mf, h1e, vhf)
if abs(shifta)+abs(shiftb) > 1e-4:
f = (hf.level_shift(s1e, dm[0], f[0], shifta),
hf.level_shift(s1e, dm[1], f[1], shiftb))
return numpy.array(f)
[docs]
def get_occ(mf, mo_energy=None, mo_coeff=None):
if mo_energy is None: mo_energy = mf.mo_energy
e_idx_a = numpy.argsort(mo_energy[0])
e_idx_b = numpy.argsort(mo_energy[1])
e_sort_a = mo_energy[0][e_idx_a]
e_sort_b = mo_energy[1][e_idx_b]
nmo = mo_energy[0].size
n_a, n_b = mf.nelec
mo_occ = numpy.zeros_like(mo_energy)
mo_occ[0,e_idx_a[:n_a]] = 1
mo_occ[1,e_idx_b[:n_b]] = 1
if mf.verbose >= logger.INFO and n_a < nmo and n_b > 0 and n_b < nmo:
if e_sort_a[n_a-1]+1e-3 > e_sort_a[n_a]:
logger.warn(mf, 'alpha nocc = %d HOMO %.15g >= LUMO %.15g',
n_a, e_sort_a[n_a-1], e_sort_a[n_a])
else:
logger.info(mf, ' alpha nocc = %d HOMO = %.15g LUMO = %.15g',
n_a, e_sort_a[n_a-1], e_sort_a[n_a])
if e_sort_b[n_b-1]+1e-3 > e_sort_b[n_b]:
logger.warn(mf, 'beta nocc = %d HOMO %.15g >= LUMO %.15g',
n_b, e_sort_b[n_b-1], e_sort_b[n_b])
else:
logger.info(mf, ' beta nocc = %d HOMO = %.15g LUMO = %.15g',
n_b, e_sort_b[n_b-1], e_sort_b[n_b])
if e_sort_a[n_a-1]+1e-3 > e_sort_b[n_b]:
logger.warn(mf, 'system HOMO %.15g >= system LUMO %.15g',
e_sort_b[n_a-1], e_sort_b[n_b])
numpy.set_printoptions(threshold=nmo)
logger.debug(mf, ' alpha mo_energy =\n%s', mo_energy[0])
logger.debug(mf, ' beta mo_energy =\n%s', mo_energy[1])
numpy.set_printoptions(threshold=1000)
if mo_coeff is not None and mf.verbose >= logger.DEBUG:
ss, s = mf.spin_square((mo_coeff[0][:,mo_occ[0]>0],
mo_coeff[1][:,mo_occ[1]>0]), mf.get_ovlp())
logger.debug(mf, 'multiplicity <S^2> = %.8g 2S+1 = %.8g', ss, s)
return mo_occ
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def get_grad(mo_coeff, mo_occ, fock_ao):
'''UHF Gradients'''
occidxa = mo_occ[0] > 0
occidxb = mo_occ[1] > 0
viridxa = ~occidxa
viridxb = ~occidxb
ga = reduce(numpy.dot, (mo_coeff[0][:,viridxa].conj().T, fock_ao[0],
mo_coeff[0][:,occidxa]))
gb = reduce(numpy.dot, (mo_coeff[1][:,viridxb].conj().T, fock_ao[1],
mo_coeff[1][:,occidxb]))
return numpy.hstack((ga.ravel(), gb.ravel()))
[docs]
def energy_elec(mf, dm=None, h1e=None, vhf=None):
'''Electronic energy of Unrestricted Hartree-Fock
Note this function has side effects which cause mf.scf_summary updated.
Returns:
Hartree-Fock electronic energy and the 2-electron part contribution
'''
if dm is None: dm = mf.make_rdm1()
if h1e is None:
h1e = mf.get_hcore()
if isinstance(dm, numpy.ndarray) and dm.ndim == 2:
dm = numpy.array((dm*.5, dm*.5))
if vhf is None:
vhf = mf.get_veff(mf.mol, dm)
if h1e[0].ndim < dm[0].ndim: # get [0] because h1e and dm may not be ndarrays
h1e = (h1e, h1e)
e1 = numpy.einsum('ij,ji->', h1e[0], dm[0])
e1+= numpy.einsum('ij,ji->', h1e[1], dm[1])
e_coul =(numpy.einsum('ij,ji->', vhf[0], dm[0]) +
numpy.einsum('ij,ji->', vhf[1], dm[1])) * .5
e_elec = (e1 + e_coul).real
mf.scf_summary['e1'] = e1.real
mf.scf_summary['e2'] = e_coul.real
logger.debug(mf, 'E1 = %s Ecoul = %s', e1, e_coul.real)
return e_elec, e_coul
# mo_a and mo_b are occupied orbitals
[docs]
def spin_square(mo, s=1):
r'''Spin square and multiplicity of UHF determinant
.. math::
S^2 = \frac{1}{2}(S_+ S_- + S_- S_+) + S_z^2
where :math:`S_+ = \sum_i S_{i+}` is effective for all beta occupied
orbitals; :math:`S_- = \sum_i S_{i-}` is effective for all alpha occupied
orbitals.
1. There are two possibilities for :math:`S_+ S_-`
1) same electron :math:`S_+ S_- = \sum_i s_{i+} s_{i-}`,
.. math::
\sum_i \langle UHF|s_{i+} s_{i-}|UHF\rangle
= \sum_{pq}\langle p|s_+s_-|q\rangle \gamma_{qp} = n_\alpha
2) different electrons :math:`S_+ S_- = \sum s_{i+} s_{j-}, (i\neq j)`.
There are in total :math:`n(n-1)` terms. As a two-particle operator,
.. math::
\langle S_+ S_- \rangle = \langle ij|s_+ s_-|ij\rangle
- \langle ij|s_+ s_-|ji\rangle
= -\langle i^\alpha|j^\beta\rangle
\langle j^\beta|i^\alpha\rangle
2. Similarly, for :math:`S_- S_+`
1) same electron
.. math::
\sum_i \langle s_{i-} s_{i+}\rangle = n_\beta
2) different electrons
.. math::
\langle S_- S_+ \rangle = -\langle i^\beta|j^\alpha\rangle
\langle j^\alpha|i^\beta\rangle
3. For :math:`S_z^2`
1) same electron
.. math::
\langle s_z^2\rangle = \frac{1}{4}(n_\alpha + n_\beta)
2) different electrons
.. math::
&\frac{1}{2}\sum_{ij}(\langle ij|2s_{z1}s_{z2}|ij\rangle
-\langle ij|2s_{z1}s_{z2}|ji\rangle) \\
&=\frac{1}{4}(\langle i^\alpha|i^\alpha\rangle \langle j^\alpha|j^\alpha\rangle
- \langle i^\alpha|i^\alpha\rangle \langle j^\beta|j^\beta\rangle
- \langle i^\beta|i^\beta\rangle \langle j^\alpha|j^\alpha\rangle
+ \langle i^\beta|i^\beta\rangle \langle j^\beta|j^\beta\rangle) \\
&-\frac{1}{4}(\langle i^\alpha|j^\alpha\rangle \langle j^\alpha|i^\alpha\rangle
+ \langle i^\beta|j^\beta\rangle\langle j^\beta|i^\beta\rangle) \\
&=\frac{1}{4}(n_\alpha^2 - n_\alpha n_\beta - n_\beta n_\alpha + n_\beta^2)
-\frac{1}{4}(n_\alpha + n_\beta) \\
&=\frac{1}{4}((n_\alpha-n_\beta)^2 - (n_\alpha+n_\beta))
In total
.. math::
\langle S^2\rangle &= \frac{1}{2}
(n_\alpha-\sum_{ij}\langle i^\alpha|j^\beta\rangle \langle j^\beta|i^\alpha\rangle
+n_\beta -\sum_{ij}\langle i^\beta|j^\alpha\rangle\langle j^\alpha|i^\beta\rangle)
+ \frac{1}{4}(n_\alpha-n_\beta)^2 \\
Args:
mo : a list of 2 ndarrays
Occupied alpha and occupied beta orbitals
Kwargs:
s : ndarray
AO overlap
Returns:
A list of two floats. The first is the expectation value of S^2.
The second is the corresponding 2S+1
Examples:
>>> mol = gto.M(atom='O 0 0 0; H 0 0 1; H 0 1 0', basis='ccpvdz', charge=1, spin=1, verbose=0)
>>> mf = scf.UHF(mol)
>>> mf.kernel()
-75.623975516256706
>>> mo = (mf.mo_coeff[0][:,mf.mo_occ[0]>0], mf.mo_coeff[1][:,mf.mo_occ[1]>0])
>>> print('S^2 = %.7f, 2S+1 = %.7f' % spin_square(mo, mol.intor('int1e_ovlp_sph')))
S^2 = 0.7570150, 2S+1 = 2.0070027
'''
mo_a, mo_b = mo
nocc_a = mo_a.shape[1]
nocc_b = mo_b.shape[1]
s = reduce(numpy.dot, (mo_a.conj().T, s, mo_b))
ssxy = (nocc_a+nocc_b) * .5 - numpy.einsum('ij,ij->', s.conj(), s)
ssz = (nocc_b-nocc_a)**2 * .25
ss = (ssxy + ssz).real
s = numpy.sqrt(ss+.25) - .5
return ss, s*2+1
[docs]
def analyze(mf, verbose=logger.DEBUG, with_meta_lowdin=WITH_META_LOWDIN,
**kwargs):
'''Analyze the given SCF object: print orbital energies, occupancies;
print orbital coefficients; Mulliken population analysis; Dipole moment;
Spin density for AOs and atoms;
'''
from pyscf.lo import orth
from pyscf.tools import dump_mat
mo_energy = mf.mo_energy
mo_occ = mf.mo_occ
mo_coeff = mf.mo_coeff
nmo = len(mo_occ[0])
log = logger.new_logger(mf, verbose)
if log.verbose >= logger.NOTE:
mf.dump_scf_summary(log)
log.note('**** MO energy ****')
log.note(' alpha | beta alpha | beta')
for i in range(nmo):
log.note('MO #%-3d energy= %-18.15g | %-18.15g occ= %g | %g',
i+MO_BASE, mo_energy[0][i], mo_energy[1][i],
mo_occ[0][i], mo_occ[1][i])
ovlp_ao = mf.get_ovlp()
if log.verbose >= logger.DEBUG:
label = mf.mol.ao_labels()
if with_meta_lowdin:
log.debug(' ** MO coefficients (expansion on meta-Lowdin AOs) for alpha spin **')
orth_coeff = orth.orth_ao(mf.mol, 'meta_lowdin', s=ovlp_ao)
c_inv = numpy.dot(orth_coeff.conj().T, ovlp_ao)
dump_mat.dump_rec(mf.stdout, c_inv.dot(mo_coeff[0]), label,
start=MO_BASE, **kwargs)
log.debug(' ** MO coefficients (expansion on meta-Lowdin AOs) for beta spin **')
dump_mat.dump_rec(mf.stdout, c_inv.dot(mo_coeff[1]), label,
start=MO_BASE, **kwargs)
else:
log.debug(' ** MO coefficients (expansion on AOs) for alpha spin **')
dump_mat.dump_rec(mf.stdout, mo_coeff[0], label,
start=MO_BASE, **kwargs)
log.debug(' ** MO coefficients (expansion on AOs) for beta spin **')
dump_mat.dump_rec(mf.stdout, mo_coeff[1], label,
start=MO_BASE, **kwargs)
dm = mf.make_rdm1(mo_coeff, mo_occ)
if with_meta_lowdin:
log.note("\nTo work with the spin densities directly, `use mulliken_meta_spin()` only printing them here.\n")
mulliken_meta_spin(mf.mol, dm, s=ovlp_ao, verbose=log)
return (mf.mulliken_meta(mf.mol, dm, s=ovlp_ao, verbose=log),
mf.dip_moment(mf.mol, dm, verbose=log))
else:
log.note("\nTo work with the spin densities directly, `use mulliken_spin_pop()` only printing them here.\n")
mulliken_spin_pop(mf.mol, dm, s=ovlp_ao, verbose=log)
return (mf.mulliken_pop(mf.mol, dm, s=ovlp_ao, verbose=log),
mf.dip_moment(mf.mol, dm, verbose=log))
[docs]
def mulliken_pop(mol, dm, s=None, verbose=logger.DEBUG):
'''Mulliken population analysis
'''
if s is None: s = hf.get_ovlp(mol)
log = logger.new_logger(mol, verbose)
if isinstance(dm, numpy.ndarray) and dm.ndim == 2:
dm = numpy.array((dm*.5, dm*.5))
pop_a = numpy.einsum('ij,ji->i', dm[0], s).real
pop_b = numpy.einsum('ij,ji->i', dm[1], s).real
log.info(' ** Mulliken pop alpha | beta **')
for i, s in enumerate(mol.ao_labels()):
log.info('pop of %s %10.5f | %-10.5f',
s, pop_a[i], pop_b[i])
log.info('In total %10.5f | %-10.5f', sum(pop_a), sum(pop_b))
log.note(' ** Mulliken atomic charges ( Nelec_alpha | Nelec_beta ) **')
nelec_a = numpy.zeros(mol.natm)
nelec_b = numpy.zeros(mol.natm)
for i, s in enumerate(mol.ao_labels(fmt=None)):
nelec_a[s[0]] += pop_a[i]
nelec_b[s[0]] += pop_b[i]
chg = mol.atom_charges() - (nelec_a + nelec_b)
for ia in range(mol.natm):
symb = mol.atom_symbol(ia)
log.note('charge of %d%s = %10.5f ( %10.5f %10.5f )',
ia, symb, chg[ia], nelec_a[ia], nelec_b[ia])
return (pop_a,pop_b), chg
[docs]
def mulliken_spin_pop(mol, dm, s=None, verbose=logger.DEBUG):
r'''Mulliken spin density analysis
See Eq. 80 in https://arxiv.org/pdf/1206.2234.pdf and the surrounding
text for more details.
.. math:: M_{ij} = (D^a_{ij} - D^b_{ij}) S_{ji}
Mulliken charges
.. math:: \delta_i = \sum_j M_{ij}
Returns:
A list : spin_pop, Ms
spin_pop : nparray
Mulliken spin density on each atomic orbitals
Ms : nparray
Mulliken spin density on each atom
'''
if s is None: s = hf.get_ovlp(mol)
dma = dm[0]
dmb = dm[1]
M = dma - dmb # Spin density
log = logger.new_logger(mol, verbose)
spin_pop = numpy.einsum('ij,ji->i', M, s).real
log.info(' ** Mulliken Spin Density (per AO) **')
for i, s in enumerate(mol.ao_labels()):
log.info('spin_pop of %s %10.5f', s, spin_pop[i])
log.note(' ** Mulliken Spin Density (per atom) **')
Ms = numpy.zeros(mol.natm) # Spin density per atom
for i, s in enumerate(mol.ao_labels(fmt=None)):
Ms[s[0]] += spin_pop[i]
for ia in range(mol.natm):
symb = mol.atom_symbol(ia)
log.note('spin density of %d %s = %10.5f',
ia, symb, Ms[ia])
return spin_pop, Ms
mulliken_pop_meta_lowdin_ao = mulliken_meta
mulliken_spin_pop_meta_lowdin_ao = mulliken_meta_spin
[docs]
def canonicalize(mf, mo_coeff, mo_occ, fock=None):
'''Canonicalization diagonalizes the UHF Fock matrix within occupied,
virtual subspaces separatedly (without change occupancy).
'''
mo_occ = numpy.asarray(mo_occ)
assert (mo_occ.ndim == 2)
if fock is None:
dm = mf.make_rdm1(mo_coeff, mo_occ)
fock = mf.get_fock(dm=dm)
occidxa = mo_occ[0] == 1
occidxb = mo_occ[1] == 1
viridxa = mo_occ[0] == 0
viridxb = mo_occ[1] == 0
def eig_(fock, mo_coeff, idx, es, cs):
if numpy.count_nonzero(idx) > 0:
orb = mo_coeff[:,idx]
f1 = reduce(numpy.dot, (orb.conj().T, fock, orb))
e, c = scipy.linalg.eigh(f1)
es[idx] = e
cs[:,idx] = numpy.dot(orb, c)
mo = numpy.empty_like(mo_coeff)
mo_e = numpy.empty(mo_occ.shape)
eig_(fock[0], mo_coeff[0], occidxa, mo_e[0], mo[0])
eig_(fock[0], mo_coeff[0], viridxa, mo_e[0], mo[0])
eig_(fock[1], mo_coeff[1], occidxb, mo_e[1], mo[1])
eig_(fock[1], mo_coeff[1], viridxb, mo_e[1], mo[1])
return mo_e, mo
[docs]
def det_ovlp(mo1, mo2, occ1, occ2, ovlp):
r''' Calculate the overlap between two different determinants. It is the product
of single values of molecular orbital overlap matrix.
.. math::
S_{12} = \langle \Psi_A | \Psi_B \rangle
= (\mathrm{det}\mathbf{U}) (\mathrm{det}\mathbf{V^\dagger})
\prod\limits_{i=1}\limits^{2N} \lambda_{ii}
where :math:`\mathbf{U}, \mathbf{V}, \lambda` are unitary matrices and single
values generated by single value decomposition(SVD) of the overlap matrix
:math:`\mathbf{O}` which is the overlap matrix of two sets of molecular orbitals:
.. math::
\mathbf{U}^\dagger \mathbf{O} \mathbf{V} = \mathbf{\Lambda}
Args:
mo1, mo2 : 2D ndarrays
Molecualr orbital coefficients
occ1, occ2: 2D ndarrays
occupation numbers
Return:
A list:
the product of single values: float
(x_a, x_b): 1D ndarrays
:math:`\mathbf{U} \mathbf{\Lambda}^{-1} \mathbf{V}^\dagger`
They are used to calculate asymmetric density matrix
'''
c1_a = mo1[0][:, occ1[0]>0]
c1_b = mo1[1][:, occ1[1]>0]
c2_a = mo2[0][:, occ2[0]>0]
c2_b = mo2[1][:, occ2[1]>0]
if c1_a.shape[1] != c2_a.shape[1] or c1_b.shape[1] != c2_b.shape[1]:
raise RuntimeError('Electron numbers are not equal. Electronic coupling does not exist.')
o_a = reduce(numpy.dot, (c1_a.conj().T, ovlp, c2_a))
o_b = reduce(numpy.dot, (c1_b.conj().T, ovlp, c2_b))
u_a, s_a, vt_a = numpy.linalg.svd(o_a)
u_b, s_b, vt_b = numpy.linalg.svd(o_b)
x_a = reduce(numpy.dot, (u_a*numpy.reciprocal(s_a), vt_a))
x_b = reduce(numpy.dot, (u_b*numpy.reciprocal(s_b), vt_b))
return numpy.prod(s_a)*numpy.prod(s_b), (x_a, x_b)
[docs]
def make_asym_dm(mo1, mo2, occ1, occ2, x):
r'''One-particle asymmetric density matrix
Args:
mo1, mo2 : 2D ndarrays
Molecualr orbital coefficients
occ1, occ2: 2D ndarrays
Occupation numbers
x: 2D ndarrays
:math:`\mathbf{U} \mathbf{\Lambda}^{-1} \mathbf{V}^\dagger`.
See also :func:`det_ovlp`
Return:
A list of 2D ndarrays for alpha and beta spin
Examples:
>>> mf1 = scf.UHF(gto.M(atom='H 0 0 0; F 0 0 1.3', basis='ccpvdz')).run()
>>> mf2 = scf.UHF(gto.M(atom='H 0 0 0; F 0 0 1.4', basis='ccpvdz')).run()
>>> s = gto.intor_cross('int1e_ovlp_sph', mf1.mol, mf2.mol)
>>> det, x = det_ovlp(mf1.mo_coeff, mf1.mo_occ, mf2.mo_coeff, mf2.mo_occ, s)
>>> adm = make_asym_dm(mf1.mo_coeff, mf1.mo_occ, mf2.mo_coeff, mf2.mo_occ, x)
>>> adm.shape
(2, 19, 19)
'''
mo1_a = mo1[0][:, occ1[0]>0]
mo1_b = mo1[1][:, occ1[1]>0]
mo2_a = mo2[0][:, occ2[0]>0]
mo2_b = mo2[1][:, occ2[1]>0]
dm_a = reduce(numpy.dot, (mo1_a, x[0], mo2_a.conj().T))
dm_b = reduce(numpy.dot, (mo1_b, x[1], mo2_b.conj().T))
return numpy.array((dm_a, dm_b))
dip_moment = hf.dip_moment
[docs]
class UHF(hf.SCF):
__doc__ = hf.SCF.__doc__ + '''
Attributes for UHF:
nelec : (int, int)
If given, freeze the number of (alpha,beta) electrons to the given value.
level_shift : number or two-element list
level shift (in Eh) for alpha and beta Fock if two-element list is given.
init_guess_breaksym : logical
If given, overwrite BREAKSYM.
Examples:
>>> mol = gto.M(atom='O 0 0 0; H 0 0 1; H 0 1 0', basis='ccpvdz', charge=1, spin=1, verbose=0)
>>> mf = scf.UHF(mol)
>>> mf.kernel()
-75.623975516256706
>>> print('S^2 = %.7f, 2S+1 = %.7f' % mf.spin_square())
S^2 = 0.7570150, 2S+1 = 2.0070027
'''
_keys = set(["init_guess_breaksym"])
def __init__(self, mol):
hf.SCF.__init__(self, mol)
# self.mo_coeff => [mo_a, mo_b]
# self.mo_occ => [mo_occ_a, mo_occ_b]
# self.mo_energy => [mo_energy_a, mo_energy_b]
self.nelec = None
self.init_guess_breaksym = None
@property
def nelec(self):
if self._nelec is not None:
return self._nelec
else:
return self.mol.nelec
@nelec.setter
def nelec(self, x):
self._nelec = x
@property
def nelectron_alpha(self):
return self.nelec[0]
@nelectron_alpha.setter
def nelectron_alpha(self, x):
logger.warn(self, 'WARN: Attribute .nelectron_alpha is deprecated. '
'Set .nelec instead')
#raise RuntimeError('API updates')
self.nelec = (x, self.mol.nelectron-x)
[docs]
def dump_flags(self, verbose=None):
hf.SCF.dump_flags(self, verbose)
logger.info(self, 'number electrons alpha = %d beta = %d', *self.nelec)
[docs]
def eig(self, fock, s):
e_a, c_a = self._eigh(fock[0], s)
e_b, c_b = self._eigh(fock[1], s)
return numpy.array((e_a,e_b)), numpy.array((c_a,c_b))
get_fock = get_fock
get_occ = get_occ
[docs]
def get_grad(self, mo_coeff, mo_occ, fock=None):
if fock is None:
dm1 = self.make_rdm1(mo_coeff, mo_occ)
fock = self.get_hcore(self.mol) + self.get_veff(self.mol, dm1)
return get_grad(mo_coeff, mo_occ, fock)
[docs]
@lib.with_doc(make_rdm1.__doc__)
def make_rdm1(self, mo_coeff=None, mo_occ=None, **kwargs):
if mo_coeff is None:
mo_coeff = self.mo_coeff
if mo_occ is None:
mo_occ = self.mo_occ
return make_rdm1(mo_coeff, mo_occ, **kwargs)
[docs]
@lib.with_doc(make_rdm2.__doc__)
def make_rdm2(self, mo_coeff=None, mo_occ=None, **kwargs):
if mo_coeff is None:
mo_coeff = self.mo_coeff
if mo_occ is None:
mo_occ = self.mo_occ
return make_rdm2(mo_coeff, mo_occ, **kwargs)
energy_elec = energy_elec
[docs]
def get_init_guess(self, mol=None, key='minao'):
dm = hf.SCF.get_init_guess(self, mol, key)
if self.verbose >= logger.DEBUG1:
s = self.get_ovlp()
nelec =(numpy.einsum('ij,ji', dm[0], s).real,
numpy.einsum('ij,ji', dm[1], s).real)
logger.debug1(self, 'Nelec from initial guess = %s', nelec)
return dm
[docs]
def init_guess_by_minao(self, mol=None, breaksym=BREAKSYM):
'''Initial guess in terms of the overlap to minimal basis.'''
if mol is None: mol = self.mol
user_set_breaksym = getattr(self, "init_guess_breaksym", None)
if user_set_breaksym is not None:
breaksym = user_set_breaksym
# For spin polarized system, no need to manually break spin symmetry
if mol.spin != 0:
breaksym = False
return init_guess_by_minao(mol, breaksym)
[docs]
def init_guess_by_atom(self, mol=None, breaksym=BREAKSYM):
if mol is None: mol = self.mol
user_set_breaksym = getattr(self, "init_guess_breaksym", None)
if user_set_breaksym is not None:
breaksym = user_set_breaksym
return init_guess_by_atom(mol, breaksym)
[docs]
def init_guess_by_huckel(self, mol=None, breaksym=BREAKSYM):
if mol is None: mol = self.mol
user_set_breaksym = getattr(self, "init_guess_breaksym", None)
if user_set_breaksym is not None:
breaksym = user_set_breaksym
logger.info(self, 'Initial guess from on-the-fly Huckel, doi:10.1021/acs.jctc.8b01089.')
mo_energy, mo_coeff = hf._init_guess_huckel_orbitals(mol, updated_rule = False)
mo_energy = (mo_energy, mo_energy)
mo_coeff = (mo_coeff, mo_coeff)
mo_occ = self.get_occ(mo_energy, mo_coeff)
dma, dmb = self.make_rdm1(mo_coeff, mo_occ)
if breaksym:
dma, dmb = _break_dm_spin_symm(mol, (dma, dmb))
return numpy.array((dma,dmb))
[docs]
def init_guess_by_mod_huckel(self, mol=None, breaksym=BREAKSYM):
if mol is None: mol = self.mol
user_set_breaksym = getattr(self, "init_guess_breaksym", None)
if user_set_breaksym is not None:
breaksym = user_set_breaksym
logger.info(self, '''Initial guess from on-the-fly Huckel, doi:10.1021/acs.jctc.8b01089,
employing the updated GWH rule from doi:10.1021/ja00480a005.''')
mo_energy, mo_coeff = hf._init_guess_huckel_orbitals(mol, updated_rule = True)
mo_energy = (mo_energy, mo_energy)
mo_coeff = (mo_coeff, mo_coeff)
mo_occ = self.get_occ(mo_energy, mo_coeff)
dma, dmb = self.make_rdm1(mo_coeff, mo_occ)
if breaksym:
dma, dmb = _break_dm_spin_symm(mol, (dma, dmb))
return numpy.array((dma,dmb))
[docs]
def init_guess_by_1e(self, mol=None, breaksym=BREAKSYM):
if mol is None: mol = self.mol
user_set_breaksym = getattr(self, "init_guess_breaksym", None)
if user_set_breaksym is not None:
breaksym = user_set_breaksym
logger.info(self, 'Initial guess from hcore.')
h1e = self.get_hcore(mol)
s1e = self.get_ovlp(mol)
if isinstance(h1e, numpy.ndarray) and h1e.ndim == s1e.ndim:
h1e = (h1e, h1e)
mo_energy, mo_coeff = self.eig(h1e, s1e)
mo_occ = self.get_occ(mo_energy, mo_coeff)
dma, dmb = self.make_rdm1(mo_coeff, mo_occ)
natm = getattr(mol, 'natm', 0) # handle custom Hamiltonian
if natm > 0 and breaksym:
dma, dmb = _break_dm_spin_symm(mol, (dma, dmb))
return numpy.array((dma,dmb))
[docs]
def init_guess_by_chkfile(self, chkfile=None, project=None):
if chkfile is None: chkfile = self.chkfile
return init_guess_by_chkfile(self.mol, chkfile, project=project)
[docs]
def get_jk(self, mol=None, dm=None, hermi=1, with_j=True, with_k=True,
omega=None):
'''Coulomb (J) and exchange (K)
Args:
dm : a list of 2D arrays or a list of 3D arrays
(alpha_dm, beta_dm) or (alpha_dms, beta_dms)
'''
if mol is None: mol = self.mol
if dm is None: dm = self.make_rdm1()
if (not omega and
(self._eri is not None or mol.incore_anyway or self._is_mem_enough())):
if self._eri is None:
self._eri = mol.intor('int2e', aosym='s8')
vj, vk = hf.dot_eri_dm(self._eri, dm, hermi, with_j, with_k)
else:
vj, vk = hf.SCF.get_jk(self, mol, dm, hermi, with_j, with_k, omega)
return vj, vk
[docs]
@lib.with_doc(get_veff.__doc__)
def get_veff(self, mol=None, dm=None, dm_last=0, vhf_last=0, hermi=1):
if mol is None: mol = self.mol
if dm is None: dm = self.make_rdm1()
if isinstance(dm, numpy.ndarray) and dm.ndim == 2:
dm = numpy.asarray((dm*.5,dm*.5))
if self._eri is not None or not self.direct_scf:
vj, vk = self.get_jk(mol, dm, hermi)
vhf = vj[0] + vj[1] - vk
else:
ddm = numpy.asarray(dm) - numpy.asarray(dm_last)
vj, vk = self.get_jk(mol, ddm, hermi)
vhf = vj[0] + vj[1] - vk
vhf += numpy.asarray(vhf_last)
return vhf
[docs]
def analyze(self, verbose=None, with_meta_lowdin=WITH_META_LOWDIN,
**kwargs):
if verbose is None: verbose = self.verbose
return analyze(self, verbose, with_meta_lowdin, **kwargs)
[docs]
def mulliken_pop(self, mol=None, dm=None, s=None, verbose=logger.DEBUG):
if mol is None: mol = self.mol
if dm is None: dm = self.make_rdm1()
if s is None: s = self.get_ovlp(mol)
return mulliken_pop(mol, dm, s=s, verbose=verbose)
[docs]
def mulliken_spin_pop(self, mol=None, dm=None, s=None, verbose=logger.DEBUG):
if mol is None: mol = self.mol
if dm is None: dm = self.make_rdm1()
if s is None: s = self.get_ovlp(mol)
return mulliken_spin_pop(mol, dm, s=s, verbose=verbose)
[docs]
@lib.with_doc(spin_square.__doc__)
def spin_square(self, mo_coeff=None, s=None):
if mo_coeff is None:
mo_coeff = (self.mo_coeff[0][:,self.mo_occ[0]>0],
self.mo_coeff[1][:,self.mo_occ[1]>0])
if s is None:
s = self.get_ovlp()
return spin_square(mo_coeff, s)
canonicalize = canonicalize
[docs]
@lib.with_doc(det_ovlp.__doc__)
def det_ovlp(self, mo1, mo2, occ1, occ2, ovlp=None):
if ovlp is None: ovlp = self.get_ovlp()
return det_ovlp(mo1, mo2, occ1, occ2, ovlp)
[docs]
@lib.with_doc(make_asym_dm.__doc__)
def make_asym_dm(self, mo1, mo2, occ1, occ2, x):
return make_asym_dm(mo1, mo2, occ1, occ2, x)
def _finalize(self):
if self.mo_coeff is None or self.mo_occ is None:
# Skip spin_square (issue #1574)
return hf.SCF._finalize(self)
ss, s = self.spin_square()
if self.converged:
logger.note(self, 'converged SCF energy = %.15g '
'<S^2> = %.8g 2S+1 = %.8g', self.e_tot, ss, s)
else:
logger.note(self, 'SCF not converged.')
logger.note(self, 'SCF energy = %.15g after %d cycles '
'<S^2> = %.8g 2S+1 = %.8g',
self.e_tot, self.max_cycle, ss, s)
return self
[docs]
def convert_from_(self, mf):
'''Create UHF object based on the RHF/ROHF object'''
tgt = mf.to_uhf()
self.__dict__.update(tgt.__dict__)
return self
[docs]
def stability(self,
internal=getattr(__config__, 'scf_stability_internal', True),
external=getattr(__config__, 'scf_stability_external', False),
verbose=None,
return_status=False):
'''
Stability analysis for UHF/UKS method.
See also pyscf.scf.stability.uhf_stability function.
Args:
mf : UHF or UKS object
Kwargs:
internal : bool
Internal stability, within the UHF space.
external : bool
External stability. Including the UHF -> GHF and real -> complex
stability analysis.
return_status: bool
Whether to return `stable_i` and `stable_e`
Returns:
If return_status is False (default), the return value includes
two set of orbitals, which are more close to the stable condition.
The first corresponds to the internal stability
and the second corresponds to the external stability.
Else, another two boolean variables (indicating current status:
stable or unstable) are returned.
The first corresponds to the internal stability
and the second corresponds to the external stability.
'''
from pyscf.scf.stability import uhf_stability
return uhf_stability(self, internal, external, verbose, return_status)
[docs]
def nuc_grad_method(self):
from pyscf.grad import uhf
return uhf.Gradients(self)
[docs]
def to_ks(self, xc='HF'):
'''Convert to UKS object.
'''
from pyscf import dft
return self._transfer_attrs_(dft.UKS(self.mol, xc=xc))
def _hf1e_scf(mf, *args):
logger.info(mf, '\n')
logger.info(mf, '******** 1 electron system ********')
mf.converged = True
h1e = mf.get_hcore(mf.mol)
s1e = mf.get_ovlp(mf.mol)
if isinstance(h1e, numpy.ndarray) and h1e.ndim == s1e.ndim:
h1e = (h1e, h1e)
mf.mo_energy, mf.mo_coeff = mf.eig(h1e, s1e)
mf.mo_occ = mf.get_occ(mf.mo_energy, mf.mo_coeff)
mf.e_tot = mf.mo_energy[mf.mo_occ>0][0].real + mf.mol.energy_nuc()
mf._finalize()
return mf.e_tot
[docs]
class HF1e(UHF):
scf = _hf1e_scf
[docs]
def spin_square(self, mo_coeff=None, s=None):
return .75, 2
del (WITH_META_LOWDIN, PRE_ORTH_METHOD, BREAKSYM)