def radial_sto(n, zeta, l, r):
"""Build radial part of slater type orbital"""
# Stolen from gpaw all_electron.py (intialize_wave_functions)
#
# STOs are defined as
#
# N Y_l^m r^{n-1} exp(-zeta * r)
#
# N = (2 zeta)^n sqrt(2 zeta / (2n)!)
assert n > 0
radial = r**(n + l + 0.5)
radial *= exp(-zeta * r)
N = (2 * zeta)**(n + l) * sqrt(2 * zeta / fact(2 * (n + l)))
radial *= N # perhaps also do Y_l^m normalization
return radial
def radial_sto(n, zeta, l, r):
"""Build radial part of slater type orbital"""
# Stolen from gpaw all_electron.py (intialize_wave_functions)
#
# STOs are defined as
#
# N Y_l^m r^{n-1} exp(-zeta * r)
#
# N = (2 zeta)^n sqrt(2 zeta / (2n)!)
assert n > 0
radial = r**(n + l + 0.5)
radial *= exp(-zeta * r)
N = (2 * zeta)**(n + l) * sqrt(2 * zeta / fact(2 * (n + l)))
radial *= N # perhaps also do Y_l^m normalization
return radial
def __init__(self, gd, v, scale=1.0, n=1, dtype=float, allocate=True):
h = (gd.h_cv**2).sum(1)**0.5
d = gd.xxxiucell_cv[:,v]
A=np.zeros((2*n+1,2*n+1))
for i,io in enumerate(range(-n,n+1)):
for j in range(2*n+1):
A[i,j]=io**j/float(fact(j))
A[n,0]=1.
coefs=np.linalg.inv(A)[1]
coefs=np.delete(coefs,len(coefs)//2)
offs=np.delete(np.arange(-n,n+1),n)
coef_p = []
offset_pc = []
for i in range(3):
if abs(d[i])>1e-11:
coef_p.extend(list(coefs * d[i] / h[i] * scale))
offset = np.zeros((2*n, 3), int)
offset[:,i]=offs
offset_pc.extend(offset)
FDOperator.__init__(self, coef_p, offset_pc, gd, dtype, allocate)
def __init__(self, gd, v, scale=1.0, n=1, dtype=float, allocate=True):
h = (gd.h_cv**2).sum(1)**0.5
d = gd.xxxiucell_cv[:, v]
A = np.zeros((2 * n + 1, 2 * n + 1))
for i, io in enumerate(range(-n, n + 1)):
for j in range(2 * n + 1):
A[i, j] = io**j / float(fact(j))
A[n, 0] = 1.
coefs = np.linalg.inv(A)[1]
coefs = np.delete(coefs, len(coefs) // 2)
offs = np.delete(np.arange(-n, n + 1), n)
coef_p = []
offset_pc = []
for i in range(3):
if abs(d[i]) > 1e-11:
coef_p.extend(list(coefs * d[i] / h[i] * scale))
offset = np.zeros((2 * n, 3), int)
offset[:, i] = offs
offset_pc.extend(offset)
FDOperator.__init__(self, coef_p, offset_pc, gd, dtype, allocate)
def intYdYdtheta_ey(l1, m1, l2, m2):
"""Calculates::
pi 2pi
/ / * d Y(u,v)
A = | | Y (u,v) cos(u)*sin(v) --- l'm' sin(u) dv du
LL' / / lm du
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.sign(m1-m2)*np.pi/1j
if abs(m1) == abs(m2)+1:
if l1+1 == l2:
return scale*2/(2.0*l2+1)*(l2+1)/(2.0*l2-1.0)*fact(l2+abs(m2))/fact(l2-abs(m2)-1)*(l2-abs(m2)-1)
elif l1-1 == l2:
return scale*2/(2.0*l2+1)*((l2+1)*fact(l2+abs(m2))/fact(l2-abs(m2))*(l2-abs(m2))-l2/(2.0*l2+3.0)*fact(l2+abs(m2)+1)/fact(l2-abs(m2))*(l2-abs(m2)+1))
elif l1-l2 > 2: # and (l1-l2)%2 == 1 which is always true
return -scale*2*abs(m2)*fact(l2+abs(m2))/fact(l2-abs(m2))
else:
return 0.0
else:
assert abs(m1) == abs(m2)-1
if l1 == l2+1:
return -scale*2/(2.0*l1+1.0)*(l1-1)/(2.0*l1-1.0)*fact(l1+abs(m1))/fact(l1-abs(m1)-1)*(l1-abs(m1)-1)
elif l1 == l2-1:
return -scale*2/(2.0*l1+1.0)*((l1+1)/(2.0*l1+3.0)*fact(l1+abs(m1)+1)/fact(l1-abs(m1))*(l1-abs(m1)+1)-(abs(m1)+1)*fact(l1+abs(m1))/fact(l1-abs(m1))*(l1-abs(m1)))
elif l2-l1 > 2: # and (l2-l1)%2 == 1 which is always true
return scale*2*(abs(m1)+1)*fact(l1+abs(m1))/fact(l1-abs(m1))
else:
return 0.0
def intYdYdtheta_ey(l1, m1, l2, m2):
"""Calculates::
pi 2pi
/ / * d Y(u,v)
A = | | Y (u,v) cos(u)*sin(v) --- l'm' sin(u) dv du
LL' / / lm du
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.sign(m1 - m2) * np.pi / 1j
if abs(m1) == abs(m2) + 1:
if l1 + 1 == l2:
return scale * 2 / (2.0 * l2 + 1) * (l2 + 1) / (
2.0 * l2 - 1.0) * fact(l2 + abs(m2)) / fact(l2 - abs(m2) -
1) * (l2 -
abs(m2) - 1)
elif l1 - 1 == l2:
return scale * 2 / (2.0 * l2 + 1) * (
(l2 + 1) * fact(l2 + abs(m2)) / fact(l2 - abs(m2)) *
(l2 - abs(m2)) - l2 / (2.0 * l2 + 3.0) *
fact(l2 + abs(m2) + 1) / fact(l2 - abs(m2)) *
(l2 - abs(m2) + 1))
elif l1 - l2 > 2: # and (l1-l2)%2 == 1 which is always true
return -scale * 2 * abs(m2) * fact(l2 + abs(m2)) / fact(l2 -
abs(m2))
else:
return 0.0
else:
assert abs(m1) == abs(m2) - 1
if l1 == l2 + 1:
return -scale * 2 / (2.0 * l1 + 1.0) * (l1 - 1) / (
2.0 * l1 - 1.0) * fact(l1 + abs(m1)) / fact(l1 - abs(m1) -
1) * (l1 -
abs(m1) - 1)
elif l1 == l2 - 1:
return -scale * 2 / (2.0 * l1 + 1.0) * (
(l1 + 1) / (2.0 * l1 + 3.0) * fact(l1 + abs(m1) + 1) /
fact(l1 - abs(m1)) * (l1 - abs(m1) + 1) -
(abs(m1) + 1) * fact(l1 + abs(m1)) / fact(l1 - abs(m1)) *
(l1 - abs(m1)))
elif l2 - l1 > 2: # and (l2-l1)%2 == 1 which is always true
return scale * 2 * (abs(m1) +
1) * fact(l1 + abs(m1)) / fact(l1 - abs(m1))
else:
return 0.0