def electrostatics_test(b, r=3, r0=None):
if r0 is None:
r0 = np.sum(b.get_positions(), axis=0) / len(a)
k1 = [(1, 1)[i] for i in b.get_pbc()]
k3 = [(1, 3)[i] for i in b.get_pbc()]
# Check multipole moments
mp = MultipoleExpansion(L_MAX, r, k3, r0)
mp.update(b, b.get_initial_charges())
# Store moments and field for later
moments = mp.get_moments()
M0_l_mp, M_L_mp = moments[1]
phi_mp, E_mp = mp.get_potential_and_field(b)
# Check if the field is the derivative of the potential
b.set_calculator(mp)
ffd, f0, err = check_forces(b)
if debug:
print("Finite differences forces:")
print(ffd)
print("Analytical forces:")
print(f0)
print("Error:")
print(err)
assert err < TOL_FOR
# ffd, f0, err = check_field(b, mp)
# if debug:
# print "Finite differences field:"
# print ffd
# print "Analytical field:"
# print f0
# print "Error:"
# print err
#
# assert err < TOL_FIELD
# Next neighbor shell by direct summation
mp = MultipoleExpansion(L_MAX, r * r * r, k1, r0)
mp.update(b, b.get_initial_charges())
phi_dir, E_dir = mp.get_potential_and_field(b)
# Multipole moments from large cell,
# transform symmetrically around the origin
#rep = [ (r-1)/2 if i else 0 for i in b.get_pbc() ]
#rep = [ (-(r-1)/2, (r-1)/2) if i else (0, 0) for i in b.get_pbc() ]
rep = [(0, (r - 1) / 2)[i] for i in b.get_pbc()]
rep = [((0, 0), (-(r - 1) / 2, (r - 1) / 2))[i] for i in b.get_pbc()]
c = b.extended_copy(tuple(rep))
M0_l, M_L = get_moments(c.get_positions(), c.get_initial_charges(), L_MAX,
r0)
#print M_L
#print M_L_mp
err_mom0 = np.max(np.abs(M0_l - M0_l_mp))
err_mom = np.max(np.abs(M_L - M_L_mp))
if debug:
print("error(mom) = ", err_mom0, err_mom)
assert err_mom0 < TOL_MOM[r]
assert err_mom < TOL_MOM[r]
# Compare fields and potentials obtained by the multipole expansion
# and from direct summation
err_phi = np.max(np.abs(phi_mp - phi_dir))
err_E = np.max(np.abs(E_mp - E_dir))
if debug:
print("error(phi) = ", err_phi)
print("error(E) = ", err_E)
assert err_phi < TOL_PHI[r]
assert err_E < TOL_E[r]
r0c = np.array( [ SX, SY, SZ ] )
# Random atoms and charges (charges between -1 and 1)
a = [ ]
for i in range(8):
a += [ ase.Atoms(
"%iH" % NAT,
positions = np.random.random([NAT,3])*SX,
charges = (2*np.random.random([NAT])-1)*CHARGE,
cell = [ SX, SY, SZ ]
) ]
# Compute moments
M = [ ]
for i in range(8):
M += [ get_moments(a[i].get_positions(), a[i].get_initial_charges(), L_MAX, r0) ]
# Construct a composite atoms object
# and compute the corresponding multipole
# expansion
b = ase.Atoms(cell=[ 2*SX, 2*SY, 2*SZ ])
Mc0_l, Mc_L = zero_moments(L_MAX)
for i, ( ca, ( M0_l, M_L ) ) in enumerate(zip(a, M)):
x = i % 2
y = (i/2) % 2
z = (i/4) % 2
dr = np.array([x*SX, y*SY, z*SZ])
ca.translate(dr)
b += ca
def _update(self, a, q):
"""
Compute multipoles, do the transformations, and compute the
electrostatic potential and field on each atom in a.
Parameters:
-----------
a: Hotbit Atoms object, or atoms object that implements the transform
and rotation interface.
q: Charges
"""
self.timer.start("multipole_to_multipole")
self.r_av = a.get_positions().copy()
self.q_a = q.copy()
nat = len(a)
r = a.get_positions()
if self.r0_v is None:
r0_v = np.sum(r, axis=0) / len(a)
else:
r0_v = self.r0_v
T0_l, T_L = get_moments(r, q, self.l_max, r0_v)
self.M = [(T0_l.copy(), T_L.copy())]
self.r0 = [r0_v]
sym_ranges = a.get_symmetry_operation_ranges()
for (s1, s2), k in zip(sym_ranges, self.k):
if s2 != np.Inf and k != 1:
print sym_ranges
print self.k
raise ValueError("For non-periodic symmetries the k-value must " "be 1.")
n1, n2, n3 = n_from_ranges(sym_ranges, self.n1, self.n2)
# Compute telescoped multipoles
level = np.ones(3, dtype=int)
for k in range(np.max(self.k) - 2):
M0_l = T0_l
M_L = T_L
T0_l = np.zeros_like(M0_l)
T_L = np.zeros_like(M_L)
if k >= self.k[0] - 2:
_n1 = [0, 1]
else:
_n1 = n1
if k >= self.k[1] - 2:
_n2 = [0, 1]
else:
_n2 = n2
if k >= self.k[2] - 2:
_n3 = [0, 1]
else:
_n3 = n3
r0_v = np.zeros(3, dtype=float)
n = 0
# Determine center of gravity
for x1 in range(*_n1):
for x2 in range(*_n2):
for x3 in range(*_n3):
x = np.array([x1, x2, x3])
r0_v += a.transform(self.r0[k], x * level)
n += 1
r0_v /= n
self.r0 += [r0_v]
# self.r0 += [ self.r0[0] ]
# Transform multipoles
for x1 in range(*_n1):
for x2 in range(*_n2):
for x3 in range(*_n3):
# Loop over all symmetry operations and compute
# telescoped multipoles
# FIXME!!! Currently only supports continuous
# symmetries, think about discrete/recurrent ones.
x = np.array([x1, x2, x3]) # + self.dx
# The origin is already okay, skip it
# if np.any(np.abs(x) > self._TOL):
r1 = a.transform(self.r0[k], x * level)
T = a.rotation(x * level)
S0_l, S_L = transform_multipole(T, self.l_max, M0_l, M_L)
multipole_to_multipole(r1 - self.r0[k], self.l_max, S0_l, S_L, T0_l, T_L)
self.M += [(T0_l.copy(), T_L.copy())]
level *= self.n2 - self.n1 + 1
self.timer.stop("multipole_to_multipole")
###
self.timer.start("multipole_to_local")
# Compute the local expansion from telescoped multipoles
L0_l, L_L = zero_moments(self.l_max)
m1, m2, m3 = n_from_ranges(sym_ranges, self.m1, self.m2)
Mi = len(self.M) - 1
for k in range(np.max(self.k) - 1):
M0_l, M_L = self.M[Mi]
if k >= self.k[0] - 1:
_m1 = [0, 1]
else:
_m1 = m1
if k >= self.k[1] - 1:
_m2 = [0, 1]
else:
_m2 = m2
if k >= self.k[2] - 1:
_m3 = [0, 1]
else:
_m3 = m3
for x1 in range(*_m1):
for x2 in range(*_m2):
for x3 in range(*_m3):
# Loop over all symmetry operations and compute the
# local expansion from the telescoped multipoles
x = np.array([x1, x2, x3]) # + self.dx
# No local expansion in the inner region
if np.any(x < self.n1) or np.any(x > self.n2):
r1 = a.transform(self.r0[Mi], x * level)
T = a.rotation(x * level)
S0_l, S_L = transform_multipole(T, self.l_max, M0_l, M_L)
multipole_to_local(-r1 + self.r0[Mi], self.l_max, S0_l, S_L, L0_l, L_L)
level /= self.n2 - self.n1 + 1
Mi -= 1
self.L = (L0_l, L_L)
self.timer.stop("multipole_to_local")
###
self.phi_a = np.zeros(nat, dtype=float)
self.E_av = np.zeros([nat, 3], dtype=float)
###
self.timer.start("local_to_local")
for i in a:
loc0_l, loc_L = local_to_local(i.position - self.r0[0], self.l_max, L0_l, L_L, 1)
self.phi_a[i.index] = loc0_l[0]
self.E_av[i.index, :] = [-loc_L[0].real, -loc_L[0].imag, loc0_l[1]]
self.timer.stop("local_to_local")
###
self.timer.start("near_field")
# Contribution of neighboring boxes
for x1 in range(*n1):
for x2 in range(*n2):
for x3 in range(*n3):
# self-interaction needs to be treated separately
if x1 != 0 or x2 != 0 or x3 != 0:
x = np.array([x1, x2, x3])
# construct a matrix with distances
r1 = a.transform(self.r0[0], x)
T = a.rotation(x)
rT = np.dot(r - self.r0[0], np.transpose(T))
dr = r.reshape(nat, 1, 3) - (r1 + rT).reshape(1, nat, 3)
abs_dr = np.sqrt(np.sum(dr * dr, axis=2))
phi = q / abs_dr
E = q.reshape(1, nat, 1) * dr / (abs_dr ** 3).reshape(nat, nat, 1)
self.phi_a += np.sum(phi, axis=1)
self.E_av += np.sum(E, axis=1)
# Self-contribution
dr = r.reshape(nat, 1, 3) - r.reshape(1, nat, 3)
abs_dr = np.sqrt(np.sum(dr * dr, axis=2))
# Avoid divide by zero
abs_dr[diag_indices_from(abs_dr)] = 1.0
phi = q / abs_dr
E = q.reshape(1, nat, 1) * dr / (abs_dr ** 3).reshape(nat, nat, 1)
phi[diag_indices_from(phi)] = 0.0
E[diag_indices_from(phi)] = 0.0
self.phi_a += np.sum(phi, axis=1)
self.E_av += np.sum(E, axis=1)
# Dipole correction for 3D sum
s1, s2, s3 = sym_ranges
if s1[1] == np.Inf and s2[1] == np.Inf and s3[1] == np.Inf:
Ml0, Mlm = self.M[0]
dip = np.array([-2 * Mlm[0].real, 2 * Mlm[0].imag, Ml0[1]])
dip *= 4 * pi / (3 * a.get_volume())
self.phi_a -= np.dot(r - self.r0[0], dip)
self.E_av += dip
self.timer.stop("near_field")
# Random atoms and charges (charges between -1 and 1)
a = []
for i in range(8):
a += [
ase.Atoms("%iH" % NAT,
positions=np.random.random([NAT, 3]) * SX,
charges=(2 * np.random.random([NAT]) - 1) * CHARGE,
cell=[SX, SY, SZ])
]
# Compute moments
M = []
for i in range(8):
M += [
get_moments(a[i].get_positions(), a[i].get_initial_charges(),
L_MAX, r0)
]
# Construct a composite atoms object
# and compute the corresponding multipole
# expansion
b = ase.Atoms(cell=[2 * SX, 2 * SY, 2 * SZ])
Mc0_l, Mc_L = zero_moments(L_MAX)
for i, (ca, (M0_l, M_L)) in enumerate(zip(a, M)):
x = i % 2
y = (i / 2) % 2
z = (i / 4) % 2
dr = np.array([x * SX, y * SY, z * SZ])
ca.translate(dr)
def _update(self, a, q):
"""
Compute multipoles, do the transformations, and compute the
electrostatic potential and field on each atom in a.
Parameters:
-----------
a: Hotbit Atoms object, or atoms object that implements the transform
and rotation interface.
q: Charges
"""
self.timer.start('multipole_to_multipole')
self.r_av = a.get_positions().copy()
self.q_a = q.copy()
nat = len(a)
r = a.get_positions()
if self.r0_v is None:
r0_v = np.sum(r, axis=0) / len(a)
else:
r0_v = self.r0_v
T0_l, T_L = get_moments(r, q, self.l_max, r0_v)
self.M = [(T0_l.copy(), T_L.copy())]
self.r0 = [r0_v]
sym_ranges = a.get_symmetry_operation_ranges()
for (s1, s2), k in zip(sym_ranges, self.k):
if s2 != np.Inf and k != 1:
print(sym_ranges)
print(self.k)
raise ValueError(
'For non-periodic symmetries the k-value must '
'be 1.')
n1, n2, n3 = n_from_ranges(sym_ranges, self.n1, self.n2)
# Compute telescoped multipoles
level = np.ones(3, dtype=int)
for k in range(np.max(self.k) - 2):
M0_l = T0_l
M_L = T_L
T0_l = np.zeros_like(M0_l)
T_L = np.zeros_like(M_L)
if k >= self.k[0] - 2:
_n1 = [0, 1]
else:
_n1 = n1
if k >= self.k[1] - 2:
_n2 = [0, 1]
else:
_n2 = n2
if k >= self.k[2] - 2:
_n3 = [0, 1]
else:
_n3 = n3
r0_v = np.zeros(3, dtype=float)
n = 0
# Determine center of gravity
for x1 in range(*_n1):
for x2 in range(*_n2):
for x3 in range(*_n3):
x = np.array([x1, x2, x3])
r0_v += a.transform(self.r0[k], x * level)
n += 1
r0_v /= n
self.r0 += [r0_v]
#self.r0 += [ self.r0[0] ]
# Transform multipoles
for x1 in range(*_n1):
for x2 in range(*_n2):
for x3 in range(*_n3):
# Loop over all symmetry operations and compute
# telescoped multipoles
# FIXME!!! Currently only supports continuous
# symmetries, think about discrete/recurrent ones.
x = np.array([x1, x2, x3]) #+ self.dx
# The origin is already okay, skip it
#if np.any(np.abs(x) > self._TOL):
r1 = a.transform(self.r0[k], x * level)
T = a.rotation(x * level)
S0_l, S_L = transform_multipole(
T, self.l_max, M0_l, M_L)
multipole_to_multipole(r1 - self.r0[k], self.l_max,
S0_l, S_L, T0_l, T_L)
self.M += [(T0_l.copy(), T_L.copy())]
level *= self.n2 - self.n1 + 1
self.timer.stop('multipole_to_multipole')
###
self.timer.start('multipole_to_local')
# Compute the local expansion from telescoped multipoles
L0_l, L_L = zero_moments(self.l_max)
m1, m2, m3 = n_from_ranges(sym_ranges, self.m1, self.m2)
Mi = len(self.M) - 1
for k in range(np.max(self.k) - 1):
M0_l, M_L = self.M[Mi]
if k >= self.k[0] - 1:
_m1 = [0, 1]
else:
_m1 = m1
if k >= self.k[1] - 1:
_m2 = [0, 1]
else:
_m2 = m2
if k >= self.k[2] - 1:
_m3 = [0, 1]
else:
_m3 = m3
for x1 in range(*_m1):
for x2 in range(*_m2):
for x3 in range(*_m3):
# Loop over all symmetry operations and compute the
# local expansion from the telescoped multipoles
x = np.array([x1, x2, x3]) #+ self.dx
# No local expansion in the inner region
if np.any(x < self.n1) or np.any(x > self.n2):
r1 = a.transform(self.r0[Mi], x * level)
T = a.rotation(x * level)
S0_l, S_L = transform_multipole(
T, self.l_max, M0_l, M_L)
multipole_to_local(-r1 + self.r0[Mi], self.l_max,
S0_l, S_L, L0_l, L_L)
level //= self.n2 - self.n1 + 1
Mi -= 1
self.L = (L0_l, L_L)
self.timer.stop('multipole_to_local')
###
self.phi_a = np.zeros(nat, dtype=float)
self.E_av = np.zeros([nat, 3], dtype=float)
###
self.timer.start('local_to_local')
for i in a:
loc0_l, loc_L = local_to_local(i.position - self.r0[0], self.l_max,
L0_l, L_L, 1)
self.phi_a[i.index] = loc0_l[0]
self.E_av[i.index, :] = [-loc_L[0].real, -loc_L[0].imag, loc0_l[1]]
self.timer.stop('local_to_local')
###
self.timer.start('near_field')
# Contribution of neighboring boxes
for x1 in range(*n1):
for x2 in range(*n2):
for x3 in range(*n3):
# self-interaction needs to be treated separately
if x1 != 0 or x2 != 0 or x3 != 0:
x = np.array([x1, x2, x3])
# construct a matrix with distances
r1 = a.transform(self.r0[0], x)
T = a.rotation(x)
rT = np.dot(r - self.r0[0], np.transpose(T))
dr = r.reshape(nat, 1, 3) - \
(r1+rT).reshape(1, nat, 3)
abs_dr = np.sqrt(np.sum(dr * dr, axis=2))
phi = q / abs_dr
E = q.reshape(1, nat, 1)*dr/ \
(abs_dr**3).reshape(nat, nat, 1)
self.phi_a += np.sum(phi, axis=1)
self.E_av += np.sum(E, axis=1)
# Self-contribution
dr = r.reshape(nat, 1, 3) - r.reshape(1, nat, 3)
abs_dr = np.sqrt(np.sum(dr * dr, axis=2))
# Avoid divide by zero
abs_dr[diag_indices_from(abs_dr)] = 1.0
phi = q / abs_dr
E = q.reshape(1, nat, 1) * dr / (abs_dr**3).reshape(nat, nat, 1)
phi[diag_indices_from(phi)] = 0.0
E[diag_indices_from(phi)] = 0.0
self.phi_a += np.sum(phi, axis=1)
self.E_av += np.sum(E, axis=1)
# Dipole correction for 3D sum
s1, s2, s3 = sym_ranges
if s1[1] == np.Inf and s2[1] == np.Inf and s3[1] == np.Inf:
Ml0, Mlm = self.M[0]
dip = np.array([-2 * Mlm[0].real, 2 * Mlm[0].imag, Ml0[1]])
dip *= 4 * pi / (3 * a.get_volume())
self.phi_a -= np.dot(r - self.r0[0], dip)
self.E_av += dip
self.timer.stop('near_field')
def electrostatics_test(b, r=3, r0=None):
if r0 is None:
r0 = np.sum(b.get_positions(), axis=0)/len(a)
k1 = [ (1,1)[i] for i in b.get_pbc() ]
k3 = [ (1,3)[i] for i in b.get_pbc() ]
# Check multipole moments
mp = MultipoleExpansion(L_MAX, r, k3, r0)
mp.update(b, b.get_initial_charges())
# Store moments and field for later
moments = mp.get_moments()
M0_l_mp, M_L_mp = moments[1]
phi_mp, E_mp = mp.get_potential_and_field(b)
# Check if the field is the derivative of the potential
b.set_calculator(mp)
ffd, f0, err = check_forces(b)
if debug:
print "Finite differences forces:"
print ffd
print "Analytical forces:"
print f0
print "Error:"
print err
assert err < TOL_FOR
# ffd, f0, err = check_field(b, mp)
# if debug:
# print "Finite differences field:"
# print ffd
# print "Analytical field:"
# print f0
# print "Error:"
# print err
#
# assert err < TOL_FIELD
# Next neighbor shell by direct summation
mp = MultipoleExpansion(L_MAX, r*r*r, k1, r0)
mp.update(b, b.get_initial_charges())
phi_dir, E_dir = mp.get_potential_and_field(b)
# Multipole moments from large cell,
# transform symmetrically around the origin
#rep = [ (r-1)/2 if i else 0 for i in b.get_pbc() ]
#rep = [ (-(r-1)/2, (r-1)/2) if i else (0, 0) for i in b.get_pbc() ]
rep = [ (0,(r-1)/2)[i] for i in b.get_pbc() ]
rep = [ ((0,0),(-(r-1)/2, (r-1)/2))[i] for i in b.get_pbc() ]
c = b.extended_copy(tuple(rep))
M0_l, M_L = get_moments(c.get_positions(), c.get_initial_charges(), L_MAX, r0)
#print M_L
#print M_L_mp
err_mom0 = np.max(np.abs(M0_l-M0_l_mp))
err_mom = np.max(np.abs(M_L-M_L_mp))
if debug:
print "error(mom) = ", err_mom0, err_mom
assert err_mom0 < TOL_MOM[r]
assert err_mom < TOL_MOM[r]
# Compare fields and potentials obtained by the multipole expansion
# and from direct summation
err_phi = np.max(np.abs(phi_mp-phi_dir))
err_E = np.max(np.abs(E_mp-E_dir))
if debug:
print "error(phi) = ", err_phi
print "error(E) = ", err_E
assert err_phi < TOL_PHI[r]
assert err_E < TOL_E[r]
