def _init_log_scheme(self):
if log.do_medium:
log.deflist([
('Scheme', 'Hirshfeld'),
('Proatomic DB', self.proatomdb),
])
biblio.cite('hirshfeld1977', 'the use of Hirshfeld partitioning')
def __init__(self,
threshold=1e-6,
maxiter=128,
nvector=6,
skip_energy=False,
prune_old_states=False):
'''
**Optional arguments:**
maxiter
The maximum number of iterations. When set to None, the SCF loop
will go one until convergence is reached.
threshold
The convergence threshold for the wavefunction
skip_energy
When set to True, the final energy is not computed. Note that some
DIIS variants need to compute the energy anyway. for these methods
this option is irrelevant.
prune_old_states
When set to True, old states are pruned from the history when their
coefficient is zero. Pruning starts at the oldest state and stops
as soon as a state is encountered with a non-zero coefficient. Even
if some newer states have a zero coefficient.
'''
biblio.cite('kudin2002', 'the EDIIS method.')
DIISSCFSolver.__init__(self, EDIIS2History, threshold, maxiter,
nvector, skip_energy, prune_old_states)
def _init_log_scheme(self):
if log.do_medium:
log.deflist([
('Scheme', 'Hirshfeld'),
('Proatomic DB', self.proatomdb),
])
biblio.cite('hirshfeld1977', 'the use of Hirshfeld partitioning')
def _init_log_scheme(self):
if log.do_medium:
log.deflist([
('Scheme', 'Minimal Basis Iterative Stockholder (MBIS)'),
('Convergence threshold', '%.1e' % self._threshold),
('Maximum iterations', self._maxiter),
])
biblio.cite('verstraelen2016', 'the use of MBIS partitioning')
def _init_log_scheme(self):
if log.do_medium:
log.deflist([
('Scheme', 'Minimal Basis Iterative Stockholder (MBIS)'),
('Convergence threshold', '%.1e' % self._threshold),
('Maximum iterations', self._maxiter),
])
biblio.cite('verstraelen2016', 'the use of MBIS partitioning')
def _init_log_scheme(self):
if log.do_medium:
log.deflist([
('Scheme', 'Iterative Stockholder'),
('Convergence threshold', '%.1e' % self._threshold),
('Maximum iterations', self._maxiter),
])
biblio.cite('lillestolen2008', 'the use of Iterative Stockholder partitioning')
def _init_log_scheme(self):
if log.do_medium:
log.deflist([
('Scheme', 'Becke'),
('Switching function', 'k=%i' % self._k),
])
biblio.cite('becke1988_multicenter', 'the use of Becke partitioning')
biblio.cite('slater1964', 'the Brag-Slater radii used in the Becke partitioning')
def _init_log_scheme(self):
if log.do_medium:
log.deflist([
('Scheme', 'Iterative Stockholder'),
('Convergence threshold', '%.1e' % self._threshold),
('Maximum iterations', self._maxiter),
])
biblio.cite('lillestolen2008',
'the use of Iterative Stockholder partitioning')
def _init_log_scheme(self):
if log.do_medium:
log.deflist([
('Scheme', 'Hirshfeld-I'),
('Convergence threshold', '%.1e' % self._threshold),
('Maximum iterations', self._maxiter),
('Proatomic DB', self._proatomdb),
])
biblio.cite('bultinck2007', 'the use of Hirshfeld-I partitioning')
def _init_log_scheme(self):
if log.do_medium:
log.deflist([
('Scheme', 'Hirshfeld-I'),
('Convergence threshold', '%.1e' % self._threshold),
('Maximum iterations', self._maxiter),
('Proatomic DB', self._proatomdb),
])
biblio.cite('bultinck2007', 'the use of Hirshfeld-I partitioning')
def solve_poisson_becke(density_decomposition):
'''Compute the electrostatic potential of a density expanded in real spherical harmonics
**Arguments:**
density_decomposition
A list of cubic splines returned by the method
AtomicGrid.get_spherical_decomposition.
The returned list of splines is a spherical decomposition of the
hartree potential (felt by a particle with the same charge unit as the
density).
'''
biblio.cite('becke1988_poisson',
'the numerical integration of the Poisson equation')
lmax = np.sqrt(len(density_decomposition)) - 1
assert lmax == int(lmax)
lmax = int(lmax)
result = []
counter = 0
for l in xrange(0, lmax + 1):
for m in xrange(-l, l + 1):
rho = density_decomposition[counter]
rtf = rho.rtransform
rgrid = RadialGrid(rtf)
radii = rtf.get_radii()
# The approach followed here is obtained after substitution of
# u = r*V in Eq. (21) in Becke's paper. After this transformation,
# the boundary conditions can be implemented such that the output
# is more accurate.
fy = -4 * np.pi * rho.y
fd = -4 * np.pi * rho.dx
f = CubicSpline(fy, fd, rtf)
b = CubicSpline(2 / radii, -2 / radii**2, rtf)
a = CubicSpline(-l * (l + 1) * radii**-2,
2 * l * (l + 1) * radii**-3, rtf)
# Derivation of boundary condition at rmax:
# Multiply differential equation with r**l and integrate. Using
# partial integration and the fact that V(r)=A/r**(l+1) for large
# r, we find -(2l+1)A=-4pi*int_0^infty r**2 r**l rho(r) and so
# V(rmax) = A/rmax**(l+1) = integrate(r**l rho(r))/(2l+1)/rmax**(l+1)
V_rmax = rgrid.integrate(
rho.y * radii**l) / radii[-1]**(l + 1) / (2 * l + 1)
# Derivation of boundary condition at rmin:
# Same as for rmax, but multiply differential equation with r**(-l-1)
# and assume that V(r)=B*r**l for small r.
V_rmin = rgrid.integrate(
rho.y * radii**(-l - 1)) * radii[0]**(l) / (2 * l + 1)
bcs = (V_rmin, None, V_rmax, None)
v = solve_ode2(b, a, f, bcs, PotentialExtrapolation(l))
result.append(v)
counter += 1
return result
def _init_log_scheme(self):
if log.do_medium:
log.deflist(
[
("Scheme", "Iterative Stockholder"),
("Convergence threshold", "%.1e" % self._threshold),
("Maximum iterations", self._maxiter),
]
)
biblio.cite("lillestolen2008", "the use of Iterative Stockholder partitioning")
def _init_log_scheme(self):
if log.do_medium:
log.deflist([
('Scheme', 'Becke'),
('Switching function', 'k=%i' % self._k),
])
biblio.cite('becke1988_multicenter',
'the use of Becke partitioning')
biblio.cite(
'slater1964',
'the Brag-Slater radii used in the Becke partitioning')
def _log_init(self):
if log.do_medium:
log('Initialized: %s' % self)
log.deflist([
('Size', self.size),
('Switching function', 'k=%i' % self._k),
])
log.blank()
# Cite reference
biblio.cite('becke1988_multicenter', 'the multicenter integration scheme used for the molecular integration grid')
biblio.cite('cordero2008', 'the covalent radii used for the Becke-Lebedev molecular integration grid')
def _log_init(self):
if log.do_high:
log('Initialized: %s' % self)
log.deflist([
('Size', self.size),
('Number of radii', self.nsphere),
('Min LL sphere', self._nlls.min()),
('Max LL sphere', self._nlls.max()),
('Radial Transform', self._rgrid.rtransform.to_string()),
])
# Cite reference
biblio.cite('lebedev1999', 'the use of Lebedev-Laikov grids (quadrature on a sphere)')
def solve_poisson_becke(density_decomposition):
'''Compute the electrostatic potential of a density expanded in real spherical harmonics
**Arguments:**
density_decomposition
A list of cubic splines returned by the method
AtomicGrid.get_spherical_decomposition.
The returned list of splines is a spherical decomposition of the
hartree potential (felt by a particle with the same charge unit as the
density).
'''
biblio.cite('becke1988_poisson', 'the numerical integration of the Poisson equation')
lmax = np.sqrt(len(density_decomposition)) - 1
assert lmax == int(lmax)
lmax = int(lmax)
result = []
counter = 0
for l in xrange(0, lmax+1):
for m in xrange(-l, l+1):
rho = density_decomposition[counter]
rtf = rho.rtransform
rgrid = RadialGrid(rtf)
radii = rtf.get_radii()
# The approach followed here is obtained after substitution of
# u = r*V in Eq. (21) in Becke's paper. After this transformation,
# the boundary conditions can be implemented such that the output
# is more accurate.
fy = -4*np.pi*rho.y
fd = -4*np.pi*rho.dx
f = CubicSpline(fy, fd, rtf)
b = CubicSpline(2/radii, -2/radii**2, rtf)
a = CubicSpline(-l*(l+1)*radii**-2, 2*l*(l+1)*radii**-3, rtf)
# Derivation of boundary condition at rmax:
# Multiply differential equation with r**l and integrate. Using
# partial integration and the fact that V(r)=A/r**(l+1) for large
# r, we find -(2l+1)A=-4pi*int_0^infty r**2 r**l rho(r) and so
# V(rmax) = A/rmax**(l+1) = integrate(r**l rho(r))/(2l+1)/rmax**(l+1)
V_rmax = rgrid.integrate(rho.y*radii**l)/radii[-1]**(l+1)/(2*l+1)
# Derivation of boundary condition at rmin:
# Same as for rmax, but multiply differential equation with r**(-l-1)
# and assume that V(r)=B*r**l for small r.
V_rmin = rgrid.integrate(rho.y*radii**(-l-1))*radii[0]**(l)/(2*l+1)
bcs = (V_rmin, None, V_rmax, None)
v = solve_ode2(b, a, f, bcs, PotentialExtrapolation(l))
result.append(v)
counter += 1
return result
def _log_init(self):
if log.do_high:
log('Initialized: %s' % self)
log.deflist([
('Size', self.size),
('Number of radii', self.nsphere),
('Min LL sphere', self._nlls.min()),
('Max LL sphere', self._nlls.max()),
('Radial Transform', self._rgrid.rtransform.to_string()),
('1D Integrator', self._rgrid.int1d),
])
# Cite reference
biblio.cite('lebedev1999', 'the use of Lebedev-Laikov grids (quadrature on a sphere)')
def __init__(self, name):
"""Initialize a LibXCEnergy instance.
Parameters
----------
name : str
The name of the functional in LibXC, without the ``lda_``, ``gga_`` or
``hyb_gga_`` prefix. (The type of functional is determined by the subclass.)
"""
name = '%s_%s' % (self.prefix, name)
self._name = name
self._libxc_wrapper = self.LibXCWrapper(name)
biblio.cite('marques2012', 'using LibXC, the library of exchange and correlation functionals')
GridObservable.__init__(self, 'libxc_%s' % name)
def __init__(self, name):
"""Initialize a LibXCEnergy instance.
Parameters
----------
name : str
The name of the functional in LibXC, without the ``lda_``, ``gga_`` or
``hyb_gga_`` prefix. (The type of functional is determined by the subclass.)
"""
name = '%s_%s' % (self.prefix, name)
self._name = name
self._libxc_wrapper = self.LibXCWrapper(name)
biblio.cite('lehtola2018', 'using LibXC, the library of exchange and correlation functionals')
GridObservable.__init__(self, 'libxc_%s' % name)
def __init__(self, *noccs, **kwargs):
r'''
**Arguments:**
nalpha, nbeta, ...
The number of electrons in each channel.
**Optional keyword arguments:**
temperature
Controls the width of the distribution (derivative)
eps
The error on the sum of the occupation number when searching for
the right Fermi level.
For each channel, the orbital occupations are assigned with the Fermi
distribution:
.. math::
n_i = \frac{1}{1 + e^{(\epsilon_i - \mu)/k_B T}}
where, for a given set of energy levels, :math:`\{\epsilon_i\}`, the
chemical potential, :math:`\mu`, is optimized as to satisfy the
following constraint:
.. math::
\sum_i n_i = n_\text{occ}
where :math:`n_\text{occ}` can be set per (spin) channel. This is
only a part of the methodology presented in [rabuck1999]_.
'''
temperature = kwargs.pop('temperature', 300)
eps = kwargs.pop('eps', 1e-8)
if len(kwargs) > 0:
raise TypeError('Unknown keyword arguments: %s' % kwargs.keys())
if temperature <= 0:
raise ValueError('The temperature must be strictly positive')
if eps <= 0:
raise ValueError(
'The root-finder threshold (eps) must be strictly positive.')
self.temperature = float(temperature)
self.eps = eps
AufbauOccModel.__init__(self, *noccs)
biblio.cite('rabuck1999',
'the Fermi broading method to assign orbital occupations')
def __init__(self, *noccs, **kwargs):
r'''
**Arguments:**
nalpha, nbeta, ...
The number of electrons in each channel.
**Optional keyword arguments:**
temperature
Controls the width of the distribution (derivative)
eps
The error on the sum of the occupation number when searching for
the right Fermi level.
For each channel, the orbital occupations are assigned with the Fermi
distribution:
.. math::
n_i = \frac{1}{1 + e^{(\epsilon_i - \mu)/k_B T}}
where, for a given set of energy levels, :math:`\{\epsilon_i\}`, the
chemical potential, :math:`\mu`, is optimized as to satisfy the
following constraint:
.. math::
\sum_i n_i = n_\text{occ}
where :math:`n_\text{occ}` can be set per (spin) channel. This is
only a part of the methodology presented in [rabuck1999]_.
'''
temperature = kwargs.pop('temperature', 300)
eps = kwargs.pop('eps', 1e-8)
if len(kwargs) > 0:
raise TypeError('Unknown keyword arguments: %s' % kwargs.keys())
if temperature <= 0:
raise ValueError('The temperature must be strictly positive')
if eps <= 0:
raise ValueError('The root-finder threshold (eps) must be strictly positive.')
self.temperature = float(temperature)
self.eps = eps
AufbauOccModel.__init__(self, *noccs)
biblio.cite('rabuck1999', 'the Fermi broading method to assign orbital occupations')
def _log_init(self):
if log.do_medium:
log('Initialized: %s' % self)
log.deflist([
('Size', self.size),
('Switching function', 'k=%i' % self._k),
])
log.blank()
# Cite reference
biblio.cite(
'becke1988_multicenter',
'the multicenter integration scheme used for the molecular integration grid'
)
biblio.cite(
'cordero2008',
'the covalent radii used for the Becke-Lebedev molecular integration grid'
)
def __init__(self, threshold=1e-6, maxiter=128, nvector=6, skip_energy=False, prune_old_states=False):
'''
**Optional arguments:**
maxiter
The maximum number of iterations. When set to None, the SCF loop
will go one until convergence is reached.
threshold
The convergence threshold for the wavefunction
skip_energy
When set to True, the final energy is not computed. Note that some
DIIS variants need to compute the energy anyway. for these methods
this option is irrelevant.
prune_old_states
When set to True, old states are pruned from the history when their
coefficient is zero. Pruning starts at the oldest state and stops
as soon as a state is encountered with a non-zero coefficient. Even
if some newer states have a zero coefficient.
'''
biblio.cite('kudin2002', 'the EDIIS method.')
DIISSCFSolver.__init__(self, EDIISHistory, threshold, maxiter, nvector, skip_energy, prune_old_states)
def do_dispersion(self):
if self.lmax < 3:
if log.do_warning:
log.warn(
'Skipping the computation of dispersion coefficients because lmax=%i<3'
% self.lmax)
return
if log.do_medium:
biblio.cite(
'tkatchenko2009',
'the method to evaluate atoms-in-molecules C6 parameters')
biblio.cite('chu2004',
'the reference C6 parameters of isolated atoms')
biblio.cite('yan1996', 'the isolated hydrogen C6 parameter')
ref_c6s = { # reference C6 values in atomic units
1: 6.499, 2: 1.42, 3: 1392.0, 4: 227.0, 5: 99.5, 6: 46.6, 7: 24.2,
8: 15.6, 9: 9.52, 10: 6.20, 11: 1518.0, 12: 626.0, 13: 528.0, 14:
305.0, 15: 185.0, 16: 134.0, 17: 94.6, 18: 64.2, 19: 3923.0, 20:
2163.0, 21: 1383.0, 22: 1044.0, 23: 832.0, 24: 602.0, 25: 552.0, 26:
482.0, 27: 408.0, 28: 373.0, 29: 253.0, 30: 284.0, 31: 498.0, 32:
354.0, 33: 246.0, 34: 210.0, 35: 162.0, 36: 130.0, 37: 4769.0, 38:
3175.0, 49: 779.0, 50: 659.0, 51: 492.0, 52: 445.0, 53: 385.0,
}
volumes, new_volumes = self._cache.load('volumes',
alloc=self.natom,
tags='o')
volume_ratios, new_volume_ratios = self._cache.load('volume_ratios',
alloc=self.natom,
tags='o')
c6s, new_c6s = self._cache.load('c6s', alloc=self.natom, tags='o')
if new_volumes or new_volume_ratios or new_c6s:
self.do_moments()
radial_moments = self._cache.load('radial_moments')
if log.do_medium:
log('Computing atomic dispersion coefficients.')
for i in xrange(self.natom):
n = self.numbers[i]
volumes[i] = radial_moments[i, 3]
ref_volume = self.proatomdb.get_record(n, 0).get_moment(3)
volume_ratios[i] = volumes[i] / ref_volume
if n in ref_c6s:
c6s[i] = (volume_ratios[i])**2 * ref_c6s[n]
else:
c6s[i] = -1 # This is just to indicate that no value is available.
def do_dispersion(self):
if self.lmax < 3:
if log.do_warning:
log.warn('Skipping the computation of dispersion coefficients because lmax=%i<3' % self.lmax)
return
if log.do_medium:
biblio.cite('tkatchenko2009', 'the method to evaluate atoms-in-molecules C6 parameters')
biblio.cite('chu2004', 'the reference C6 parameters of isolated atoms')
biblio.cite('yan1996', 'the isolated hydrogen C6 parameter')
ref_c6s = { # reference C6 values in atomic units
1: 6.499, 2: 1.42, 3: 1392.0, 4: 227.0, 5: 99.5, 6: 46.6, 7: 24.2,
8: 15.6, 9: 9.52, 10: 6.20, 11: 1518.0, 12: 626.0, 13: 528.0, 14:
305.0, 15: 185.0, 16: 134.0, 17: 94.6, 18: 64.2, 19: 3923.0, 20:
2163.0, 21: 1383.0, 22: 1044.0, 23: 832.0, 24: 602.0, 25: 552.0, 26:
482.0, 27: 408.0, 28: 373.0, 29: 253.0, 30: 284.0, 31: 498.0, 32:
354.0, 33: 246.0, 34: 210.0, 35: 162.0, 36: 130.0, 37: 4769.0, 38:
3175.0, 49: 779.0, 50: 659.0, 51: 492.0, 52: 445.0, 53: 385.0,
}
volumes, new_volumes = self._cache.load('volumes', alloc=self.natom, tags='o')
volume_ratios, new_volume_ratios = self._cache.load('volume_ratios', alloc=self.natom, tags='o')
c6s, new_c6s = self._cache.load('c6s', alloc=self.natom, tags='o')
if new_volumes or new_volume_ratios or new_c6s:
self.do_moments()
radial_moments = self._cache.load('radial_moments')
if log.do_medium:
log('Computing atomic dispersion coefficients.')
for i in xrange(self.natom):
n = self.numbers[i]
volumes[i] = radial_moments[i, 3]
ref_volume = self.proatomdb.get_record(n, 0).get_moment(3)
volume_ratios[i] = volumes[i]/ref_volume
if n in ref_c6s:
c6s[i] = (volume_ratios[i])**2*ref_c6s[n]
else:
c6s[i] = -1 # This is just to indicate that no value is available.
def setup_weights(coordinates, numbers, grid, dens=None, near=None, far=None):
'''Define a weight function for the ESPCost
**Arguments:**
coordinates
An array with shape (N, 3) containing atomic coordinates.
numbers
A vector with shape (N,) containing atomic numbers.
grid
A UniformGrid object.
**Optional arguments:**
dens
The density-based criterion. This is a three-tuple with rho, lnrho0
and sigma. rho is the atomic or the pro-atomic electron density on
the same grid as the ESP data. lnrho0 and sigma are parameters
defined in JCTC, 3, 1004 (2007), DOI:10.1021/ct600295n. The weight
function takes the form::
exp(-sigma*(ln(rho) - lnrho0)**2)
Note that the density, rho, should not contain depletions in the
atomic cores, as is often encountered with pseudo-potential
computations. In that case it is recommended to construct a
promolecular density as input for this option.
near
Exclude points near the nuclei. This is a dictionary with as items
(number, (R0, gamma)).
far
Exclude points far away. This is a two-tuple: (R0, gamma).
'''
natom, coordinates, numbers = typecheck_geo(coordinates, numbers, need_pseudo_numbers=False)
weights = np.ones(grid.shape)
# combine three possible mask functions
if dens is not None:
biblio.cite('hu2007', 'for the ESP fitting weight function')
rho, lnrho0, sigma = dens
assert (rho.shape == grid.shape).all()
multiply_dens_mask(rho, lnrho0, sigma, weights)
if near is not None:
for i in xrange(natom):
pair = near.get(numbers[i])
if pair is None:
pair = near.get(0)
if pair is None:
continue
r0, gamma = pair
if r0 > 5*angstrom:
raise ValueError('The wnear radius is excessive. Please keep it below 5 angstrom.')
multiply_near_mask(coordinates[i], grid, r0, gamma, weights)
if far is not None:
r0, gamma = far
multiply_far_mask(coordinates, grid, r0, gamma, weights)
# double that weight goes to zero at non-periodic edges
return weights
def setup_weights(coordinates, numbers, grid, dens=None, near=None, far=None):
'''Define a weight function for the ESPCost
**Arguments:**
coordinates
An array with shape (N, 3) containing atomic coordinates.
numbers
A vector with shape (N,) containing atomic numbers.
grid
A UniformGrid object.
**Optional arguments:**
dens
The density-based criterion. This is a three-tuple with rho, lnrho0
and sigma. rho is the atomic or the pro-atomic electron density on
the same grid as the ESP data. lnrho0 and sigma are parameters
defined in JCTC, 3, 1004 (2007), DOI:10.1021/ct600295n. The weight
function takes the form::
exp(-sigma*(ln(rho) - lnrho0)**2)
Note that the density, rho, should not contain depletions in the
atomic cores, as is often encountered with pseudo-potential
computations. In that case it is recommended to construct a
promolecular density as input for this option.
near
Exclude points near the nuclei. This is a dictionary with as items
(number, (R0, gamma)).
far
Exclude points far away. This is a two-tuple: (R0, gamma).
'''
natom, coordinates, numbers = typecheck_geo(coordinates,
numbers,
need_pseudo_numbers=False)
weights = np.ones(grid.shape)
# combine three possible mask functions
if dens is not None:
biblio.cite('hu2007', 'for the ESP fitting weight function')
rho, lnrho0, sigma = dens
assert (rho.shape == grid.shape).all()
multiply_dens_mask(rho, lnrho0, sigma, weights)
if near is not None:
for i in xrange(natom):
pair = near.get(numbers[i])
if pair is None:
pair = near.get(0)
if pair is None:
continue
r0, gamma = pair
if r0 > 5 * angstrom:
raise ValueError(
'The wnear radius is excessive. Please keep it below 5 angstrom.'
)
multiply_near_mask(coordinates[i], grid, r0, gamma, weights)
if far is not None:
r0, gamma = far
multiply_far_mask(coordinates, grid, r0, gamma, weights)
# double that weight goes to zero at non-periodic edges
return weights
