def intYdYdphi_ey(l1, m1, l2, m2):
"""Calculates::
pi 2pi
/ / * -1 d Y(u,v)
A = | | Y (u,v) sin(u)*cos(v) --- l'm' sin(u) dv du
LL' / / lm dv
0 0
where u = theta and v = phi in the usual notation. Note that the result
is only non-zero if `|l1-l2|` is odd and `|m1-m2|` = 1 (stricter rule applies).
"""
if abs(l1-l2) % 2 != 1 or abs(m1-m2) != 1:
return 0.0
scale = C(l1,m1)*C(l2,m2)*np.pi*1j*m2
if abs(m1) == abs(m2)+1 and l1 > l2: # and (l1-l2)%2 == 1 which is always true
return scale*2*lmfact(l2,m2)
elif abs(m1) == abs(m2)-1 and l1 < l2: # and (l2-l1)%2 == 1 which is always true
return scale*2*lmfact(l1,m1)
else:
return 0.0
def intYdYdphi_ey(l1, m1, l2, m2):
"""Calculates::
pi 2pi
/ / * -1 d Y(u,v)
A = | | Y (u,v) sin(u)*cos(v) --- l'm' sin(u) dv du
LL' / / lm dv
0 0
where u = theta and v = phi in the usual notation. Note that the result
is only non-zero if `|l1-l2|` is odd and `|m1-m2|` = 1 (stricter rule applies).
"""
if abs(l1-l2) % 2 != 1 or abs(m1-m2) != 1:
return 0.0
scale = C(l1,m1)*C(l2,m2)*np.pi*1j*m2
if abs(m1) == abs(m2)+1 and l1 > l2: # and (l1-l2)%2 == 1 which is always true
return scale*2*lmfact(l2,m2)
elif abs(m1) == abs(m2)-1 and l1 < l2: # and (l2-l1)%2 == 1 which is always true
return scale*2*lmfact(l1,m1)
else:
return 0.0
import numpy as np
from gpaw.sphere import lmfact
# Highest l for which associated Legendre polynomials are implemented
lmax = 9
# Integral norm of the associated Legendre polynomials
ilegendre = lambda l, m: 2. / (2. * l + 1.) * lmfact(l, m)
def legendre(l, m, w):
if m < 0 or l < m:
return np.zeros_like(w)
if (l, m) == (0, 0):
return np.ones_like(w)
elif (l, m) == (1, 0):
return w.copy()
elif (l, m) == (1, 1):
return (1 - w**2)**0.5
elif (l, m) == (2, 0):
return 1.5 * w**2 - 0.5
elif (l, m) == (2, 1):
return 3 * (1 - w**2)**0.5 * w
elif (l, m) == (2, 2):
return 3 * (1 - w**2)
elif (l, m) == (3, 0):
return 2.5 * w**3 - 1.5 * w
elif (l, m) == (3, 1):
return (1 - w**2)**0.5 * (7.5 * w**2 - 1.5)
elif (l, m) == (3, 2):
import numpy as np
from gpaw.utilities import fact
from gpaw.sphere import lmfact
from gpaw.sphere.legendre import ilegendre, legendre, dlegendre
# Define the Heaviside function
heaviside = lambda x: (1.0 + np.sign(x)) / 2.0
# Define spherical harmoncics and normalization coefficient
C = lambda l, m: ((2. * l + 1.) / (4 * np.pi * lmfact(l, m)))**0.5
def Y(l, m, theta, phi):
if m == 0:
return C(l, m) * legendre(l, abs(m), np.cos(theta))
elif m > 0:
return C(l, m) * legendre(l, abs(m), np.cos(theta)) * np.cos(
m * phi) * 2**0.5
else:
return C(l, m) * legendre(l, abs(m), np.cos(theta)) * np.sin(
abs(m) * phi) * 2**0.5
# Define theta-derivative of spherical harmoncics
def dYdtheta(l, m, theta, phi):
if m == 0:
return C(l, m) * dlegendre(l, abs(m), np.cos(theta))
elif m > 0:
return C(l, m) * dlegendre(l, abs(m), np.cos(theta)) * np.cos(
m * phi) * 2**0.5
import numpy as np
from gpaw.utilities import fact
from gpaw.sphere import lmfact
from gpaw.sphere.legendre import ilegendre, legendre, dlegendre
# Define the Heaviside function
heaviside = lambda x: (1.0+np.sign(x))/2.0
# Define spherical harmoncics and normalization coefficient
C = lambda l,m: ((2.*l+1.)/(4*np.pi*lmfact(l,m)))**0.5
def Y(l,m,theta,phi):
if m == 0:
#return _Y(l,m,theta,phi)
return C(l,m)*legendre(l,abs(m),np.cos(theta))
elif m > 0:
#return (-1)**m*np.real(_Y(l,abs(m),theta,phi))*2**0.5
#return (-1)**m*(_Y(l,m,theta,phi)+_Y(l,m,theta,phi).conj())/2**0.5
#return (-1)**m*(_Y(l,m,theta,phi)+(-1)**m*_Y(l,-m,theta,phi))/2**0.5
#return (-1)**m*_C(l,m)*legendre(l,abs(m),np.cos(theta))*np.cos(m*phi)*2**0.5
return C(l,m)*legendre(l,abs(m),np.cos(theta))*np.cos(m*phi)*2**0.5
else:
#return (-1)**m*np.imag(_Y(l,abs(m),theta,phi))*2**0.5
#return (-1)**m*(_Y(l,abs(m),theta,phi)-_Y(l,abs(m),theta,phi).conj())/(2**0.5*1j)
#return (-1)**m*(_Y(l,abs(m),theta,phi)-(-1)**abs(m)*_Y(l,-abs(m),theta,phi))/(2**0.5*1j)
#return -_C(l,m)*legendre(l,abs(m),np.cos(theta))*np.sin(m*phi)*2**0.5
return -C(l,m)*legendre(l,abs(m),np.cos(theta))*np.sin(m*phi)*2**0.5
# Define theta-derivative of spherical harmoncics
import numpy as np
from gpaw.sphere import lmfact
# Highest l for which associated Legendre polynomials are implemented
lmax = 9
# Integral norm of the associated Legendre polynomials
ilegendre = lambda l,m: 2./(2.*l+1.)*lmfact(l,m)
def legendre(l, m, w):
if m < 0 or l < m:
return np.zeros_like(w)
if (l,m) == (0,0):
return np.ones_like(w)
elif (l,m) == (1,0):
return w.copy()
elif (l,m) == (1,1):
return (1-w**2)**0.5
elif (l,m) == (2,0):
return 1.5*w**2-0.5
elif (l,m) == (2,1):
return 3*(1-w**2)**0.5*w
elif (l,m) == (2,2):
return 3*(1-w**2)
elif (l,m) == (3,0):
return 2.5*w**3-1.5*w
elif (l,m) == (3,1):
return (1-w**2)**0.5*(7.5*w**2-1.5)
elif (l,m) == (3,2):
import numpy as np
from gpaw.utilities import fact
from gpaw.sphere import lmfact
from gpaw.sphere.legendre import ilegendre, legendre, dlegendre
# Define the Heaviside function
heaviside = lambda x: (1.0+np.sign(x))/2.0
# Define spherical harmoncics and normalization coefficient
C = lambda l,m: (-1)**((m+abs(m))//2)*((2.*l+1.)/(4*np.pi*lmfact(l,m)))**0.5
Y = lambda l,m,theta,phi: C(l,m)*legendre(l,abs(m),np.cos(theta))*np.exp(1j*m*phi)
# Define theta-derivative of spherical harmoncics
dYdtheta = lambda l,m,theta,phi: C(l,m)*dlegendre(l,abs(m),np.cos(theta))*np.exp(1j*m*phi)
# Define phi-derivative of spherical harmoncics
dYdphi = lambda l,m,theta,phi: 1j*m*C(l,m)*legendre(l,abs(m),np.cos(theta))*np.exp(1j*m*phi)
# -------------------------------------------------------------------
def intYY(l1, m1, l2, m2):
"""Calculates::
pi 2pi
/ / *
A = | | Y (u,v) Y (u,v) sin(u) dv du
LL' / / lm l'm'
0 0
from math import factorial as fact
import numpy as np
from gpaw.sphere import lmfact
from gpaw.sphere.legendre import ilegendre, legendre, dlegendre
# Define the Heaviside function
heaviside = lambda x: (1.0+np.sign(x))/2.0
# Define spherical harmoncics and normalization coefficient
C = lambda l,m: (-1)**((m+abs(m))//2)*((2.*l+1.)/(4*np.pi*lmfact(l,m)))**0.5
Y = lambda l,m,theta,phi: C(l,m)*legendre(l,abs(m),np.cos(theta))*np.exp(1j*m*phi)
# Define theta-derivative of spherical harmoncics
dYdtheta = lambda l,m,theta,phi: C(l,m)*dlegendre(l,abs(m),np.cos(theta))*np.exp(1j*m*phi)
# Define phi-derivative of spherical harmoncics
dYdphi = lambda l,m,theta,phi: 1j*m*C(l,m)*legendre(l,abs(m),np.cos(theta))*np.exp(1j*m*phi)
# -------------------------------------------------------------------
def intYY(l1, m1, l2, m2):
"""Calculates::
pi 2pi
/ / *
A = | | Y (u,v) Y (u,v) sin(u) dv du
LL' / / lm l'm'
0 0
where u = theta and v = phi in the usual notation. Note that the result
import numpy as np
from gpaw.utilities import fact
from gpaw.sphere import lmfact
from gpaw.sphere.legendre import ilegendre, legendre, dlegendre
# Define the Heaviside function
heaviside = lambda x: (1.0 + np.sign(x)) / 2.0
# Define spherical harmoncics and normalization coefficient
C = lambda l, m: (-1)**((m + abs(m)) // 2) * ((2. * l + 1.) /
(4 * np.pi * lmfact(l, m)))**0.5
Y = lambda l, m, theta, phi: C(l, m) * legendre(l, abs(m), np.cos(theta)
) * np.exp(1j * m * phi)
# Define theta-derivative of spherical harmoncics
dYdtheta = lambda l, m, theta, phi: C(l, m) * dlegendre(
l, abs(m), np.cos(theta)) * np.exp(1j * m * phi)
# Define phi-derivative of spherical harmoncics
dYdphi = lambda l, m, theta, phi: 1j * m * C(l, m) * legendre(
l, abs(m), np.cos(theta)) * np.exp(1j * m * phi)
# -------------------------------------------------------------------
def intYY(l1, m1, l2, m2):
"""Calculates::
pi 2pi
/ / *