def get_phi(self, k2):
"""Get the solution of the Helmholtz equation for a Gaussian."""
# This should lead to very big errors
r = np.sqrt(self.r2)
k = sqrt(k2)
sigma = 1. / sqrt(2 * self.a)
p = sigma * k / sqrt(2)
i = 1j
rhop = r / sqrt(2) / sigma + i * p
rhom = r / sqrt(2) / sigma - i * p
# h = np.sin(k * r)
# the gpaw-Gaussian is sqrt(4 * pi) times a 3D normalized Gaussian
return sqrt(4 * pi) * exp(-p**2) / r / 2 * (
np.cos(k * r) * (cerf(rhop) + cerf(rhom)) + i * np.sin(k * r) *
(2 + cerf(rhop) - cerf(rhom)))
def get_phi(self, k2):
"""Get the solution of the Helmholtz equation for a Gaussian."""
# This should lead to very big errors
r = np.sqrt(self.r2)
k = sqrt(k2)
sigma = 1. / sqrt(2 * self.a)
p = sigma * k / sqrt(2)
i = 1j
rhop = r / sqrt(2) / sigma + i * p
rhom = r / sqrt(2) / sigma - i * p
h = np.sin(k * r)
# the gpaw-Gaussian is sqrt(4 * pi) times a 3D normalized Gaussian
return sqrt(4 * pi) * exp(-p**2) / r / 2 * (
np.cos(k * r) * (cerf(rhop) + cerf(rhom)) +
i * np.sin(k * r) * (2 + cerf(rhop) - cerf(rhom)))
# 1 0.842701
# 3 0.999978
# I 0. + 1.65043 I
# 1 + I 1.31615 + 0.190453 I
# 0.3 + 3*I 1467.69 - 166.561 I
# 3 - 0.3*I 1.00001 - 0.0000228553 I
values = [
[ 1, 0.84270079295+0j],
[ 3, 0.999977909503+0j],
[ 1j, 1.6504257588j],
[ 1+1j, 1.3161512816979477+0.19045346923783471j ],
[ 0.3 + 3j, 1467.69028322-166.560924526j ],
[ 3 + 0.3j, 0.99997602085736015+2.1863701577230078e-06j]]
maxerror = 1.e-10
for test in values:
z, res = test
try:
r = z.real
except:
r = z
error = abs(res / cerf(z) - 1.)
if error < maxerror:
print 'z=', z, ' ok (error=', error, ')'
else:
print z, res, cerf(z), erf(r), error
string = ('error for erf(' + str(z) +') = ' + str(error) +
' > ' + str(maxerror))
assert(error < maxerror), string
# Mathematica says:
# z erf(z)
# 1 0.842701
# 3 0.999978
# I 0. + 1.65043 I
# 1 + I 1.31615 + 0.190453 I
# 0.3 + 3*I 1467.69 - 166.561 I
# 3 - 0.3*I 1.00001 - 0.0000228553 I
values = [[1, 0.84270079295 + 0j], [3, 0.999977909503 + 0j],
[1j, 1.6504257588j],
[1 + 1j, 1.3161512816979477 + 0.19045346923783471j],
[0.3 + 3j, 1467.69028322 - 166.560924526j],
[3 + 0.3j, 0.99997602085736015 + 2.1863701577230078e-06j]]
maxerror = 1.e-10
for test in values:
z, res = test
try:
r = z.real
except:
r = z
error = abs(res / cerf(z) - 1.)
if error < maxerror:
print('z=', z, ' ok (error=', error, ')')
else:
print(z, res, cerf(z), erf(r), error)
string = ('error for erf(' + str(z) + ') = ' + str(error) + ' > ' +
str(maxerror))
assert (error < maxerror), string