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 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).
'''
log.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 _load(self, filename):
fn = context.get_fn(filename)
members = []
with open(fn) as f:
state = 0
for line in f:
line = line[:line.find('#')].strip()
if len(line) > 0:
if state == 0:
# read element number
words = line.split()
number = int(words[0])
if len(words) > 1:
pseudo_number = int(words[1])
else:
pseudo_number = number
state = 1
elif state == 1:
# read rtf string
rtf = RTransform.from_string(line)
state = 2
elif state == 2:
nlls = np.array([int(w) for w in line.split()])
state = 0
members.append(
(number, pseudo_number, RadialGrid(rtf), nlls))
self._init_members_from_list(members)
def from_hdf5(cls, grp, lf):
records = []
for ds in grp.itervalues():
rtransform = RTransform.from_string(ds.attrs['rtransform'])
records.append((ds.attrs['number'], ds.attrs['pseudo_number'],
RadialGrid(rtransform), ds[:]))
return AtomicGridSpec(records)
def _init_members_from_string(self, definition):
if os.path.isfile(definition):
self._load(definition)
return
filename = context.get_fn('grids/%s.txt' % definition)
if os.path.isfile(filename):
self._load(filename)
return
name = self._simple_names.get(definition)
if name is not None:
filename = context.get_fn('grids/%s.txt' % name)
self._load(filename)
return
if definition.count(':') == 4:
words = self.name.split(':')
RTransformClass = self._simple_rtfs.get(words[0])
if RTransformClass is None:
raise ValueError('Unknown radial grid type: %s' % words[0])
rmin = float(words[1]) * angstrom
rmax = float(words[2]) * angstrom
nrad = int(words[3])
rgrid = RadialGrid(RTransformClass(rmin, rmax, nrad))
nll = int(words[4])
self._init_members_from_tuple((rgrid, nll))
else:
raise ValueError(
'Could not interpret atomic grid specification string: "%s"' %
definition)
def load_proatom_records_h5_group(f):
'''Load proatom records from the given HDF5 group'''
records = []
for grp in f.itervalues():
assert isinstance(grp, h5.Group)
if 'deriv' in grp:
deriv = grp['deriv'][:]
else:
deriv = None
records.append(ProAtomRecord(
number=grp.attrs['number'],
charge=grp.attrs['charge'],
energy=grp.attrs['energy'],
rgrid=RadialGrid(RTransform.from_string(grp.attrs['rtransform'])),
rho=grp['rho'][:],
deriv=deriv,
pseudo_number=grp.attrs.get('pseudo_number'),
ipot_energy=grp.attrs.get('ipot_energy'),
))
return records
def load_proatom_records_atdens(filename):
'''Load proatom records from the given atdens file file'''
def read_numbers(f, npoint):
numbers = []
while len(numbers) < npoint:
words = f.next().replace('D', 'E').split()
for word in words:
numbers.append(float(word))
return np.array(numbers)
from horton.grid.cext import ExpRTransform
records = []
with open(filename) as f:
# load the radii
words = f.next().split()
assert words[0] == 'RADII'
npoint = int(words[1])
radii = read_numbers(f, npoint)
# Construct a radial grid
r1 = radii[1]
r2 = radii[-1]
rtf = ExpRTransform(r1, r2, npoint-1)
rgrid = RadialGrid(rtf)
# load the proatoms
while True:
try:
words = f.next().split()
number = int(words[0])
population = int(words[1])
charge = number - population
rho = read_numbers(f, npoint)[1:]
record = ProAtomRecord(number, charge, 0.0, 0.0, rgrid, rho)
records.append(record)
except StopIteration:
break
return records
