def test_spherical_average_of_spherical_harmonic(self):
r"""Test spherical average of spherical harmonic is zero."""
# construct helper function
def func(sph_points):
# Spherical harmonic of order 6 and magnetic 0
r, phi, theta = sph_points.T
return (np.sqrt(13) / (np.sqrt(np.pi) * 32) *
(231 * np.cos(theta)**6.0 - 315 * np.cos(theta)**4.0 +
105 * np.cos(theta)**2.0 - 5.0))
# Construct Radial Grid and atomic grid with spherical harmonics of degree 10
# for all points.
oned_grid = np.arange(0.0, 5.0, 0.001)
rad = OneDGrid(oned_grid, np.ones(len(oned_grid)), (0, np.inf))
atgrid = AtomGrid(rad, degrees=[10])
spherical_pts = atgrid.convert_cartesian_to_spherical(atgrid.points)
func_values = func(spherical_pts)
spherical_avg = atgrid.spherical_average(func_values)
# Test that the spherical average of a spherical harmonic is zero.
numb_pts = 1000
random_rad_pts = np.random.uniform(0.02, np.pi, size=(numb_pts, 3))
spherical_avg2 = spherical_avg(random_rad_pts[:, 0])
assert_allclose(spherical_avg2, 0.0, atol=1e-4)
def test_spherical_average_of_gaussian(self):
r"""Test spherical average of a Gaussian (radial) function is itself and its integral."""
# construct helper function
def func(sph_points):
return np.exp(-sph_points[:, 0]**2.0)
# Construct Radial Grid and atomic grid with spherical harmonics of degree 10
# for all points.
oned_grid = np.arange(0.0, 5.0, 0.001)
rad = OneDGrid(oned_grid, np.ones(len(oned_grid)), (0, np.inf))
atgrid = AtomGrid(rad, degrees=[5])
spherical_pts = atgrid.convert_cartesian_to_spherical(atgrid.points)
func_values = func(spherical_pts)
spherical_avg = atgrid.spherical_average(func_values)
# Test that the spherical average of a Gaussian is itself
numb_pts = 1000
random_rad_pts = np.random.uniform(0.0, 1.5, size=(numb_pts, 3))
spherical_avg2 = spherical_avg(random_rad_pts[:, 0])
func_vals = func(random_rad_pts)
assert_allclose(spherical_avg2, func_vals, atol=1e-4)
# Test the integral of spherical average is the integral of Gaussian e^(-x^2)e^(-y^2)...
# from -infinity to infinity which is equal to pi^(3/2)
integral = (
4.0 * np.pi *
np.trapz(y=spherical_avg(oned_grid) * oned_grid**2.0, x=oned_grid))
actual_integral = np.sqrt(np.pi)**3.0
assert_allclose(actual_integral, integral)
