def spherical_bessel_J(n, var, algorithm="maxima"):
r"""
Returns the spherical Bessel function of the first kind for
integers n >= 1.
Reference: AS 10.1.8 page 437 and AS 10.1.15 page 439.
EXAMPLES::
sage: spherical_bessel_J(2,x)
((3/x^2 - 1)*sin(x) - 3*cos(x)/x)/x
sage: spherical_bessel_J(1, 5.2, algorithm='scipy')
-0.12277149950007...
sage: spherical_bessel_J(1, 3, algorithm='scipy')
0.345677499762355...
"""
if algorithm == "scipy":
from scipy.special.specfun import sphj
return sphj(int(n), float(var))[1][-1]
elif algorithm == "maxima":
_init()
return meval("spherical_bessel_j(%s,%s)" % (ZZ(n), var))
else:
raise ValueError("unknown algorithm '%s'" % algorithm)
def spherical_bessel_J(n, var, algorithm="maxima"):
r"""
Returns the spherical Bessel function of the first kind for
integers n >= 1.
Reference: AS 10.1.8 page 437 and AS 10.1.15 page 439.
EXAMPLES::
sage: spherical_bessel_J(2,x)
((3/x^2 - 1)*sin(x) - 3*cos(x)/x)/x
sage: spherical_bessel_J(1, 5.2, algorithm='scipy')
-0.12277149950007...
sage: spherical_bessel_J(1, 3, algorithm='scipy')
0.345677499762355...
"""
if algorithm == "scipy":
from scipy.special.specfun import sphj
return CDF(sphj(int(n), float(var))[1][-1])
elif algorithm == 'maxima':
_init()
return meval("spherical_bessel_j(%s,%s)"%(ZZ(n),var))
else:
raise ValueError("unknown algorithm '%s'"%algorithm)
def two_phi_planewave_integrals(k_Gv, setup=None, Gstart=0, Gend=None,
rgd=None, phi_jg=None,
phit_jg=None,l_j=None):
"""Calculate PAW-correction matrix elements with planewaves.
::
/ _ _ ik.r _ ~ _ ik.r ~ _
| dr [phi (r) e phi (r) - phi (r) e phi (r)]
/ 1 2 1 2
ll - / 2 ~ ~
= 4 * pi \sum_lm i Y (k) | dr r [ phi (r) phi (r) - phi (r) phi (r) j (kr)
lm / 1 2 1 2 ll
/
* | d\Omega Y Y Y
/ l1m1 l2m2 lm
"""
if Gend is None:
Gend = len(k_Gv)
if setup is not None:
rgd = setup.rgd
l_j = setup.l_j
# Obtain the phi_j and phit_j
phi_jg = []
phit_jg = []
rcut2 = 2 * max(setup.rcut_j)
gcut2 = rgd.ceil(rcut2)
for phi_g, phit_g in zip(setup.data.phi_jg, setup.data.phit_jg):
phi_g = phi_g.copy()
phit_g = phit_g.copy()
phi_g[gcut2:] = phit_g[gcut2:] = 0.
phi_jg.append(phi_g)
phit_jg.append(phit_g)
else:
assert rgd is not None
assert phi_jg is not None
assert l_j is not None
ng = rgd.N
r_g = rgd.r_g
dr_g = rgd.dr_g
# Construct L (l**2 + m) and j (nl) index
L_i = []
j_i = []
for j, l in enumerate(l_j):
for m in range(2 * l + 1):
L_i.append(l**2 + m)
j_i.append(j)
ni = len(L_i)
nj = len(l_j)
lmax = max(l_j) * 2 + 1
if setup is not None:
assert ni == setup.ni and nj == setup.nj
# Initialize
npw = k_Gv.shape[0]
R_jj = np.zeros((nj, nj))
R_ii = np.zeros((ni, ni))
phi_Gii = np.zeros((npw, ni, ni), dtype=complex)
j_lg = np.zeros((lmax, ng))
# Store (phi_j1 * phi_j2 - phit_j1 * phit_j2 ) for further use
tmp_jjg = np.zeros((nj, nj, ng))
for j1 in range(nj):
for j2 in range(nj):
tmp_jjg[j1, j2] = (phi_jg[j1] * phi_jg[j2] -
phit_jg[j1] * phit_jg[j2])
# Loop over G vectors
for iG in range(Gstart, Gend):
kk = k_Gv[iG]
k = np.sqrt(np.dot(kk, kk)) # calculate length of q+G
# Calculating spherical bessel function
for g, r in enumerate(r_g):
j_lg[:, g] = sphj(lmax - 1, k * r)[1]
for li in range(lmax):
# Radial part
for j1 in range(nj):
for j2 in range(nj):
R_jj[j1, j2] = np.dot(r_g**2*dr_g,
tmp_jjg[j1, j2] * j_lg[li])
for mi in range(2 * li + 1):
# Angular part
for i1 in range(ni):
L1 = L_i[i1]
j1 = j_i[i1]
for i2 in range(ni):
L2 = L_i[i2]
j2 = j_i[i2]
R_ii[i1, i2] = G_LLL[L1, L2, li**2+mi] * R_jj[j1, j2]
phi_Gii[iG] += R_ii * Y(li**2 + mi,
kk[0]/k, kk[1]/k, kk[2]/k) * (-1j)**li
phi_Gii *= 4 * pi
return phi_Gii.reshape(npw, ni*ni)
def two_phi_planewave_integrals(k_Gv,
setup=None,
Gstart=0,
Gend=None,
rgd=None,
phi_jg=None,
phit_jg=None,
l_j=None):
"""Calculate PAW-correction matrix elements with planewaves.
::
/ _ _ ik.r _ ~ _ ik.r ~ _
| dr [phi (r) e phi (r) - phi (r) e phi (r)]
/ 1 2 1 2
ll - / 2 ~ ~
= 4 * pi \sum_lm i Y (k) | dr r [ phi (r) phi (r) - phi (r) phi (r) j (kr)
lm / 1 2 1 2 ll
/
* | d\Omega Y Y Y
/ l1m1 l2m2 lm
"""
if Gend is None:
Gend = len(k_Gv)
if setup is not None:
rgd = setup.rgd
l_j = setup.l_j
# Obtain the phi_j and phit_j
phi_jg = []
phit_jg = []
rcut2 = 2 * max(setup.rcut_j)
gcut2 = rgd.ceil(rcut2)
for phi_g, phit_g in zip(setup.data.phi_jg, setup.data.phit_jg):
phi_g = phi_g.copy()
phit_g = phit_g.copy()
phi_g[gcut2:] = phit_g[gcut2:] = 0.
phi_jg.append(phi_g)
phit_jg.append(phit_g)
else:
assert rgd is not None
assert phi_jg is not None
assert l_j is not None
ng = rgd.N
r_g = rgd.r_g
dr_g = rgd.dr_g
# Construct L (l**2 + m) and j (nl) index
L_i = []
j_i = []
for j, l in enumerate(l_j):
for m in range(2 * l + 1):
L_i.append(l**2 + m)
j_i.append(j)
ni = len(L_i)
nj = len(l_j)
lmax = max(l_j) * 2 + 1
if setup is not None:
assert ni == setup.ni and nj == setup.nj
# Initialize
npw = k_Gv.shape[0]
R_jj = np.zeros((nj, nj))
R_ii = np.zeros((ni, ni))
phi_Gii = np.zeros((npw, ni, ni), dtype=complex)
j_lg = np.zeros((lmax, ng))
# Store (phi_j1 * phi_j2 - phit_j1 * phit_j2 ) for further use
tmp_jjg = np.zeros((nj, nj, ng))
for j1 in range(nj):
for j2 in range(nj):
tmp_jjg[j1,
j2] = (phi_jg[j1] * phi_jg[j2] - phit_jg[j1] * phit_jg[j2])
# Loop over G vectors
for iG in range(Gstart, Gend):
kk = k_Gv[iG]
k = np.sqrt(np.dot(kk, kk)) # calculate length of q+G
# Calculating spherical bessel function
for g, r in enumerate(r_g):
j_lg[:, g] = sphj(lmax - 1, k * r)[1]
for li in range(lmax):
# Radial part
for j1 in range(nj):
for j2 in range(nj):
R_jj[j1, j2] = np.dot(r_g**2 * dr_g,
tmp_jjg[j1, j2] * j_lg[li])
for mi in range(2 * li + 1):
# Angular part
for i1 in range(ni):
L1 = L_i[i1]
j1 = j_i[i1]
for i2 in range(ni):
L2 = L_i[i2]
j2 = j_i[i2]
R_ii[i1, i2] = G_LLL[L1, L2, li**2 + mi] * R_jj[j1, j2]
phi_Gii[iG] += R_ii * Y(li**2 + mi, kk[0] / k, kk[1] / k,
kk[2] / k) * (-1j)**li
phi_Gii *= 4 * pi
return phi_Gii.reshape(npw, ni * ni)
def two_phi_nabla_planewave_integrals(k_Gv, setup=None, Gstart=0, Gend=None,
rgd=None, phi_jg=None,
phit_jg=None, l_j=None):
"""Calculate PAW-correction matrix elements with planewaves and gradient.
::
/ _ _ ik.r d _ ~ _ ik.r d ~ _
| dr [phi (r) e -- phi (r) - phi (r) e -- phi (r)]
/ 1 dx 2 1 dx 2
and similar for y and z."""
if Gend is None:
Gend = len(k_Gv)
if setup is not None:
rgd = setup.rgd
l_j = setup.l_j
# Obtain the phi_j and phit_j
phi_jg = []
phit_jg = []
rcut2 = 2 * max(setup.rcut_j)
gcut2 = rgd.ceil(rcut2)
for phi_g, phit_g in zip(setup.data.phi_jg, setup.data.phit_jg):
phi_g = phi_g.copy()
phit_g = phit_g.copy()
phi_g[gcut2:] = phit_g[gcut2:] = 0.
phi_jg.append(phi_g)
phit_jg.append(phit_g)
else:
assert rgd is not None
assert phi_jg is not None
assert l_j is not None
ng = rgd.N
r_g = rgd.r_g
dr_g = rgd.dr_g
# Construct L (l**2 + m) and j (nl) index
L_i = []
j_i = []
for j, l in enumerate(l_j):
for m in range(2 * l + 1):
L_i.append(l**2 + m)
j_i.append(j)
ni = len(L_i)
nj = len(l_j)
lmax = max(l_j) * 2 + 1
ljdef = 3
l2max = max(l_j) * (max(l_j) > ljdef) + ljdef * (max(l_j) <= ljdef)
G_LLL = gaunt(l2max)
Y_LLv = nabla(2 * l2max)
if setup is not None:
assert ni == setup.ni and nj == setup.nj
# Initialize
npw = k_Gv.shape[0]
R1_jj = np.zeros((nj, nj))
R2_jj = np.zeros((nj, nj))
R_vii = np.zeros((3, ni, ni))
phi_vGii = np.zeros((3, npw, ni, ni), dtype=complex)
j_lg = np.zeros((lmax, ng))
# Store (phi_j1 * dphidr_j2 - phit_j1 * dphitdr_j2) for further use
tmp_jjg = np.zeros((nj, nj, ng))
tmpder_jjg = np.zeros((nj, nj, ng))
for j1 in range(nj):
for j2 in range(nj):
dphidr_g = np.empty_like(phi_jg[j2])
rgd.derivative(phi_jg[j2], dphidr_g)
dphitdr_g = np.empty_like(phit_jg[j2])
rgd.derivative(phit_jg[j2], dphitdr_g)
tmpder_jjg[j1, j2] = (phi_jg[j1] * dphidr_g -
phit_jg[j1] * dphitdr_g)
tmp_jjg[j1, j2] = (phi_jg[j1] * phi_jg[j2] -
phit_jg[j1] * phit_jg[j2])
# Loop over G vectors
for iG in range(Gstart, Gend):
kk = k_Gv[iG]
k = np.sqrt(np.dot(kk, kk)) # calculate length of q+G
# Calculating spherical bessel function
for g, r in enumerate(r_g):
j_lg[:, g] = sphj(lmax - 1, k * r)[1]
for li in range(lmax):
# Radial part
for j1 in range(nj):
for j2 in range(nj):
R1_jj[j1, j2] = np.dot(r_g**2 * dr_g,
tmpder_jjg[j1, j2] * j_lg[li])
R2_jj[j1, j2] = np.dot(r_g * dr_g,
tmp_jjg[j1, j2] * j_lg[li])
for mi in range(2 * li + 1):
if k == 0:
Ytmp = Y(li**2 + mi, 1.0, 0, 0)
else:
# Note the spherical bessel gives
# zero when k == 0 for li != 0
Ytmp = Y(li**2 + mi, kk[0] / k, kk[1] / k, kk[2] / k)
for v in range(3):
Lv = 1 + (v + 2) % 3
# Angular part
for i1 in range(ni):
L1 = L_i[i1]
j1 = j_i[i1]
for i2 in range(ni):
L2 = L_i[i2]
j2 = j_i[i2]
l2 = l_j[j2]
R_vii[v, i1, i2] = ((4 * pi / 3)**0.5 *
np.dot(G_LLL[L1, L2],
G_LLL[Lv, li**2 + mi])
* (R1_jj[j1, j2] - l2 *
R2_jj[j1, j2]))
R_vii[v, i1, i2] += (R2_jj[j1, j2] *
np.dot(G_LLL[L1, li**2 + mi],
Y_LLv[:, L2, v]))
phi_vGii[:, iG] += (R_vii * Ytmp * (-1j)**li)
phi_vGii *= 4 * pi
return phi_vGii.reshape(3, npw, ni * ni)