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
else:
return C(l,m)*dlegendre(l,abs(m),np.cos(theta))*np.sin(abs(m)*phi)*2**0.5
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
else:
return C(l, m) * dlegendre(l, abs(m), np.cos(theta)) * np.sin(
abs(m) * phi) * 2**0.5
def dYdtheta(l,m,theta,phi):
if m == 0:
#return _dYdtheta(l,m,theta,phi)
return C(l,m)*dlegendre(l,abs(m),np.cos(theta))
elif m > 0:
#return (-1)**m*np.real(_dYdtheta(l,abs(m),theta,phi))*2**0.5
#return (-1)**m*(_dYdtheta(l,m,theta,phi)+_dYdtheta(l,m,theta,phi).conj())/2**0.5
#return (-1)**m*(_dYdtheta(l,m,theta,phi)+(-1)**m*_dYdtheta(l,-m,theta,phi))/2**0.5
#return (-1)**m*_C(l,m)*dlegendre(l,abs(m),np.cos(theta))*np.cos(m*phi)*2**0.5
return C(l,m)*dlegendre(l,abs(m),np.cos(theta))*np.cos(m*phi)*2**0.5
else:
#return (-1)**m*np.imag(_dYdtheta(l,abs(m),theta,phi))*2**0.5
#return (-1)**m*(_dYdtheta(l,abs(m),theta,phi)-_dYdtheta(l,abs(m),theta,phi).conj())/(2**0.5*1j)
#return (-1)**m*(_dYdtheta(l,abs(m),theta,phi)-(-1)**abs(m)*_dYdtheta(l,-abs(m),theta,phi))/(2**0.5*1j)
#return -_C(l,m)*dlegendre(l,abs(m),np.cos(theta))*np.sin(m*phi)*2**0.5
return -C(l,m)*dlegendre(l,abs(m),np.cos(theta))*np.sin(m*phi)*2**0.5
def dYdtheta(l, m, theta, phi):
if m == 0:
#return _dYdtheta(l,m,theta,phi)
return C(l, m) * dlegendre(l, abs(m), np.cos(theta))
elif m > 0:
#return (-1)**m*np.real(_dYdtheta(l,abs(m),theta,phi))*2**0.5
#return (-1)**m*(_dYdtheta(l,m,theta,phi)+_dYdtheta(l,m,theta,phi).conj())/2**0.5
#return (-1)**m*(_dYdtheta(l,m,theta,phi)+(-1)**m*_dYdtheta(l,-m,theta,phi))/2**0.5
#return (-1)**m*_C(l,m)*dlegendre(l,abs(m),np.cos(theta))*np.cos(m*phi)*2**0.5
return C(l, m) * dlegendre(l, abs(m), np.cos(theta)) * np.cos(
m * phi) * 2**0.5
else:
#return (-1)**m*np.imag(_dYdtheta(l,abs(m),theta,phi))*2**0.5
#return (-1)**m*(_dYdtheta(l,abs(m),theta,phi)-_dYdtheta(l,abs(m),theta,phi).conj())/(2**0.5*1j)
#return (-1)**m*(_dYdtheta(l,abs(m),theta,phi)-(-1)**abs(m)*_dYdtheta(l,-abs(m),theta,phi))/(2**0.5*1j)
#return -_C(l,m)*dlegendre(l,abs(m),np.cos(theta))*np.sin(m*phi)*2**0.5
return -C(l, m) * dlegendre(l, abs(m), np.cos(theta)) * np.sin(
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: (-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
