def logarithmic_derivative(self, l, energies, rcut):
rgd = self.rgd
ch = Channel(l)
gcut = rgd.round(rcut)
N = 0
if l < len(self.waves_l):
# Nonlocal PAW stuff:
waves = self.waves_l[l]
if len(waves) > 0:
pt_ng = waves.pt_ng
dH_nn = waves.dH_nn
dS_nn = waves.dS_nn
N = len(pt_ng)
u_g = rgd.zeros()
u_ng = rgd.zeros(N)
dudr_n = np.empty(N)
logderivs = []
d0 = 42.0
offset = 0
for e in energies:
dudr = ch.integrate_outwards(u_g, rgd, self.vtr_g, gcut, e)[0]
u = u_g[gcut]
if N:
for n in range(N):
dudr_n[n] = ch.integrate_outwards(u_ng[n],
rgd,
self.vtr_g,
gcut,
e,
pt_g=pt_ng[n])[0]
A_nn = (dH_nn - e * dS_nn) / (4 * pi)
B_nn = rgd.integrate(pt_ng[:, None] * u_ng, -1)
c_n = rgd.integrate(pt_ng * u_g, -1)
d_n = np.linalg.solve(
np.dot(A_nn, B_nn) + np.eye(N), np.dot(A_nn, c_n))
u -= np.dot(u_ng[:, gcut], d_n)
dudr -= np.dot(dudr_n, d_n)
d1 = np.arctan(dudr / u) / pi + offset
if d1 > d0:
offset -= 1
d1 -= 1
logderivs.append(d1)
d0 = d1
return np.array(logderivs)
def __init__(self, aea, waves, l0, rcmax=0):
self.aea = aea
self.l0 = l0
self.gd = aea.gd
if len(waves) == 0:
lmax = -1
else:
lmax = max([l for n, l, rc in waves])
self.waves_l = [PAWWaves(l, aea.channels[l].basis)
for l in range(lmax + 1)]
self.rcmax = rcmax
for n, l, rc in waves:
ch = aea.channels[l]
e = ch.e_n[n - l - 1]
f = ch.f_n[n - l - 1]
print n,l,rc,e,f
phi_g = ch.basis.expand(ch.C_nb[0])
self.waves_l[l].add(phi_g, e, f, rc)
self.rcmax = max(rc, self.rcmax)
self.gcmax = self.gd.get_index(self.rcmax)
self.alpha = log(1.0e4) / self.rcmax**2 # exp(-alpha*rcmax^2)=1.0e-4
self.alpha = round(self.alpha, 2)
self.ghat_g = (np.exp(-self.alpha * self.gd.r_g**2) *
(self.alpha / pi)**1.5)
f0 = 0.0
if l0 == 0:
f0 = 1.0
self.zeropot = Channel(l0, 0, [f0], aea.channels[l0].basis)
self.vtr_g = None
def add_waves(self, rc):
if isinstance(rc, float):
radii = [rc]
else:
radii = rc
if self.lmax >= 0:
radii += [radii[-1]] * (self.lmax + 1 - len(radii))
del radii[self.lmax + 1:] # remove unused radii
self.rcmax = max(radii)
self.waves_l = []
for l in range(self.lmax + 1):
rcut = radii[l]
waves = PAWWaves(self.rgd, l, rcut)
e = -1.0
for n in self.states[l]:
if isinstance(n, int):
# Bound state:
ch = self.aea.channels[l]
e = ch.e_n[n - l - 1]
f = ch.f_n[n - l - 1]
phi_g = ch.phi_ng[n - l - 1]
else:
if n is None:
e += 1.0
else:
e = n
n = -1
f = 0.0
phi_g = self.rgd.zeros()
gc = self.rgd.round(2.5 * self.rcmax)
ch = Channel(l)
a = ch.integrate_outwards(phi_g, self.rgd,
self.aea.vr_sg[0], gc, e,
self.aea.scalar_relativistic,
self.aea.Z)[1]
phi_g[1:gc + 1] /= self.rgd.r_g[1:gc + 1]
phi_g[0] = a
phi_g /= (self.rgd.integrate(phi_g**2) / (4 * pi))**0.5
waves.add(phi_g, n, e, f)
self.waves_l.append(waves)
def add_waves(self, rc):
if isinstance(rc, float):
radii = [rc]
else:
radii = rc
self.rcmax = max(radii)
if self.lmax >= 0:
radii += [radii[-1]] * (self.lmax + 1 - len(radii))
self.waves_l = []
for l in range(self.lmax + 1):
rcut = radii[l]
waves = PAWWaves(self.rgd, l, rcut)
e = -1.0
for n in self.states[l]:
if isinstance(n, int):
# Bound state:
ch = self.aea.channels[l]
e = ch.e_n[n - l - 1]
f = ch.f_n[n - l - 1]
phi_g = ch.phi_ng[n - l - 1]
else:
if n is None:
e += 1.0
else:
e = n
n = -1
f = 0.0
phi_g = self.rgd.zeros()
gc = self.rgd.round(1.5 * rcut)
ch = Channel(l)
a = ch.integrate_outwards(phi_g, self.rgd,
self.aea.vr_sg[0], gc, e,
self.aea.scalar_relativistic,
self.aea.Z)[1]
phi_g[1:gc + 1] /= self.rgd.r_g[1:gc + 1]
phi_g[0] = a
phi_g /= (self.rgd.integrate(phi_g**2) / (4 * pi))**0.5
waves.add(phi_g, n, e, f)
self.waves_l.append(waves)
def match_local_potential(self, l0, r0, P, e0):
self.log('Local potential matching %s-scattering at e=%.3f eV' %
('spdfg'[l0], e0 * Hartree) +
' and r=%.2f Bohr' % r0)
g0 = self.rgd.ceil(r0)
gc = g0 + 20
ch = Channel(l0)
phi_g = self.rgd.zeros()
ch.integrate_outwards(phi_g, self.rgd, self.aea.vr_sg[0], gc, e0,
self.aea.scalar_relativistic)
phi_g[1:gc] /= self.rgd.r_g[1:gc]
if l0 == 0:
phi_g[0] = phi_g[1]
#phit_g, c = self.rgd.pseudize_normalized(phi_g, g0, l=l0, points=P)
phit_g, c = self.rgd.pseudize(phi_g, g0, l=l0, points=P)
r_g = self.rgd.r_g[1:g0]
dgdr_g = 1 / self.rgd.dr_g
d2gdr2_g = self.rgd.d2gdr2()
a_g = phit_g.copy()
a_g[1:] /= self.rgd.r_g[1:]**l0
a_g[0] = c
dadg_g = self.rgd.zeros()
d2adg2_g = self.rgd.zeros()
dadg_g[1:-1] = 0.5 * (a_g[2:] - a_g[:-2])
d2adg2_g[1:-1] = a_g[2:] - 2 * a_g[1:-1] + a_g[:-2]
q_g = (((l0 + 1) * dgdr_g + 0.5 * self.rgd.r_g * d2gdr2_g) * dadg_g +
0.5 * self.rgd.r_g * d2adg2_g * dgdr_g**2)
q_g[:g0] /= a_g[:g0]
q_g += e0 * self.rgd.r_g
q_g[0] = 0.0
self.vtr_g = self.aea.vr_sg[0].copy()
self.vtr_g[0] = 0.0
self.vtr_g[1:g0] = q_g[1:g0]#e0 * r_g - t_g * r_g**(l0 + 1) / phit_g[1:g0]
self.l0 = l0
self.e0 = e0
self.r0 = r0
self.nderiv0 = P
def logarithmic_derivative(self, l, energies, rcut):
rgd = self.rgd
ch = Channel(l)
gcut = rgd.round(rcut)
N = 0
if l < len(self.waves_l):
# Nonlocal PAW stuff:
waves = self.waves_l[l]
if len(waves) > 0:
pt_ng = waves.pt_ng
dH_nn = waves.dH_nn
dS_nn = waves.dS_nn
N = len(pt_ng)
u_g = rgd.zeros()
u_ng = rgd.zeros(N)
dudr_n = np.empty(N)
logderivs = []
for e in energies:
dudr = ch.integrate_outwards(u_g, rgd, self.vtr_g, gcut, e)
u = u_g[gcut]
if N:
for n in range(N):
dudr_n[n] = ch.integrate_outwards(u_ng[n], rgd,
self.vtr_g, gcut, e,
pt_g=pt_ng[n])
A_nn = (dH_nn - e * dS_nn) / (4 * pi)
B_nn = rgd.integrate(pt_ng[:, None] * u_ng, -1)
c_n = rgd.integrate(pt_ng * u_g, -1)
d_n = np.linalg.solve(np.dot(A_nn, B_nn) + np.eye(N),
np.dot(A_nn, c_n))
u -= np.dot(u_ng[:, gcut], d_n)
dudr -= np.dot(dudr_n, d_n)
logderivs.append(dudr / u)
return logderivs
def match_local_potential(self, r0, P):
l0 = self.l0
self.log('Local potential matching %s-scattering at e=0.0 eV' %
'spdfg'[l0] + ' and r=%.2f Bohr' % r0)
g0 = self.rgd.ceil(r0)
gc = g0 + 20
e0 = 0.0
ch = Channel(l0)
phi_g = self.rgd.zeros()
a = ch.integrate_outwards(phi_g, self.rgd, self.aea.vr_sg[0], gc, e0,
self.aea.scalar_relativistic, self.aea.Z)[1]
phi_g[1:gc] /= self.rgd.r_g[1:gc]
phi_g[0] = a
phit_g, c = self.rgd.pseudize(phi_g, g0, l=l0, points=P)
dgdr_g = 1 / self.rgd.dr_g
d2gdr2_g = self.rgd.d2gdr2()
a_g = phit_g.copy()
a_g[1:] /= self.rgd.r_g[1:]**l0
a_g[0] = c
dadg_g = self.rgd.zeros()
d2adg2_g = self.rgd.zeros()
dadg_g[1:-1] = 0.5 * (a_g[2:] - a_g[:-2])
d2adg2_g[1:-1] = a_g[2:] - 2 * a_g[1:-1] + a_g[:-2]
q_g = (((l0 + 1) * dgdr_g + 0.5 * self.rgd.r_g * d2gdr2_g) * dadg_g +
0.5 * self.rgd.r_g * d2adg2_g * dgdr_g**2)
q_g[:g0] /= a_g[:g0]
q_g += e0 * self.rgd.r_g
q_g[0] = 0.0
self.vtr_g = self.aea.vr_sg[0].copy()
self.vtr_g[0] = 0.0
self.vtr_g[1:g0] = q_g[1:g0]
def match_local_potential(self, r0, P):
l0 = self.l0
self.log('Local potential matching %s-scattering at e=0.0 eV' %
'spdfg'[l0] + ' and r=%.2f Bohr' % r0)
g0 = self.rgd.ceil(r0)
gc = g0 + 20
e0 = 0.0
ch = Channel(l0)
phi_g = self.rgd.zeros()
a = ch.integrate_outwards(phi_g, self.rgd, self.aea.vr_sg[0], gc, e0,
self.aea.scalar_relativistic, self.aea.Z)[1]
phi_g[1:gc] /= self.rgd.r_g[1:gc]
phi_g[0] = a
phit_g, c = self.rgd.pseudize(phi_g, g0, l=l0, points=P)
dgdr_g = 1 / self.rgd.dr_g
d2gdr2_g = self.rgd.d2gdr2()
a_g = phit_g.copy()
a_g[1:] /= self.rgd.r_g[1:]**l0
a_g[0] = c
dadg_g = self.rgd.zeros()
d2adg2_g = self.rgd.zeros()
dadg_g[1:-1] = 0.5 * (a_g[2:] - a_g[:-2])
d2adg2_g[1:-1] = a_g[2:] - 2 * a_g[1:-1] + a_g[:-2]
q_g = (((l0 + 1) * dgdr_g + 0.5 * self.rgd.r_g * d2gdr2_g) * dadg_g +
0.5 * self.rgd.r_g * d2adg2_g * dgdr_g**2)
q_g[:g0] /= a_g[:g0]
q_g += e0 * self.rgd.r_g
q_g[0] = 0.0
self.vtr_g = self.aea.vr_sg[0].copy()
self.vtr_g[0] = 0.0
self.vtr_g[1:g0] = q_g[1:g0]
def create_basis_function(self, l, n, tailnorm, scale):
rgd = self.rgd
waves = self.waves_l[l]
# Find cutoff radii:
n_g = np.add.accumulate(waves.phit_ng[n]**2 * rgd.r_g**2 * rgd.dr_g)
norm = n_g[-1]
g2 = (norm - n_g > tailnorm * norm).sum()
r2 = rgd.r_g[g2]
r1 = max(0.6 * r2, waves.rcut)
g1 = rgd.ceil(r1)
# Set up confining potential:
r = rgd.r_g[g1:g2]
vtr_g = self.vtr_g.copy()
vtr_g[g1:g2] += scale * np.exp((r2 - r1) / (r1 - r)) / (r - r2)**2
vtr_g[g2:] = np.inf
# Nonlocal PAW stuff:
pt_ng = waves.pt_ng
dH_nn = waves.dH_nn
dS_nn = waves.dS_nn
N = len(pt_ng)
u_g = rgd.zeros()
u_ng = rgd.zeros(N)
duodr_n = np.empty(N)
a_n = np.empty(N)
e = waves.e_n[n]
e0 = e
ch = Channel(l)
while True:
duodr, a = ch.integrate_outwards(u_g, rgd, vtr_g, g1, e)
for n in range(N):
duodr_n[n], a_n[n] = ch.integrate_outwards(u_ng[n],
rgd,
vtr_g,
g1,
e,
pt_g=pt_ng[n])
A_nn = (dH_nn - e * dS_nn) / (4 * pi)
B_nn = rgd.integrate(pt_ng[:, None] * u_ng, -1)
c_n = rgd.integrate(pt_ng * u_g, -1)
d_n = np.linalg.solve(
np.dot(A_nn, B_nn) + np.eye(N), np.dot(A_nn, c_n))
u_g[:g1 + 1] -= np.dot(d_n, u_ng[:, :g1 + 1])
a -= np.dot(d_n, a_n)
duodr -= np.dot(duodr_n, d_n)
uo = u_g[g1]
duidr = ch.integrate_inwards(u_g, rgd, vtr_g, g1, e, gmax=g2)
ui = u_g[g1]
A = duodr / uo - duidr / ui
u_g[g1:] *= uo / ui
x = (norm / rgd.integrate(u_g**2, -2) * (4 * pi))**0.5
u_g *= x
a *= x
if abs(A) < 1e-5:
break
e += 0.5 * A * u_g[g1]**2
u_g[1:] /= rgd.r_g[1:]
u_g[0] = a * 0.0**l
return u_g, r1, r2, e - e0
def create_basis_function(self, l, n, tailnorm, scale):
rgd = self.rgd
waves = self.waves_l[l]
# Find cutoff radii:
n_g = np.add.accumulate(waves.phit_ng[n]**2 * rgd.r_g**2 * rgd.dr_g)
norm = n_g[-1]
g2 = (norm - n_g > tailnorm * norm).sum()
r2 = rgd.r_g[g2]
r1 = max(0.6 * r2, waves.rcut)
g1 = rgd.ceil(r1)
# Set up confining potential:
r = rgd.r_g[g1:g2]
vtr_g = self.vtr_g.copy()
vtr_g[g1:g2] += scale * np.exp((r2 - r1) / (r1 - r)) / (r - r2)**2
vtr_g[g2:] = np.inf
# Nonlocal PAW stuff:
pt_ng = waves.pt_ng
dH_nn = waves.dH_nn
dS_nn = waves.dS_nn
N = len(pt_ng)
u_g = rgd.zeros()
u_ng = rgd.zeros(N)
duodr_n = np.empty(N)
a_n = np.empty(N)
e = waves.e_n[n]
e0 = e
ch = Channel(l)
while True:
duodr, a = ch.integrate_outwards(u_g, rgd, vtr_g, g1, e)
for n in range(N):
duodr_n[n], a_n[n] = ch.integrate_outwards(u_ng[n], rgd,
vtr_g, g1, e,
pt_g=pt_ng[n])
A_nn = (dH_nn - e * dS_nn) / (4 * pi)
B_nn = rgd.integrate(pt_ng[:, None] * u_ng, -1)
c_n = rgd.integrate(pt_ng * u_g, -1)
d_n = np.linalg.solve(np.dot(A_nn, B_nn) + np.eye(N),
np.dot(A_nn, c_n))
u_g[:g1 + 1] -= np.dot(d_n, u_ng[:, :g1 + 1])
a -= np.dot(d_n, a_n)
duodr -= np.dot(duodr_n, d_n)
uo = u_g[g1]
duidr = ch.integrate_inwards(u_g, rgd, vtr_g, g1, e, gmax=g2)
ui = u_g[g1]
A = duodr / uo - duidr / ui
u_g[g1:] *= uo / ui
x = (norm / rgd.integrate(u_g**2, -2) * (4 * pi))**0.5
u_g *= x
a *= x
if abs(A) < 1e-5:
break
e += 0.5 * A * u_g[g1]**2
u_g[1:] /= rgd.r_g[1:]
u_g[0] = a * 0.0**l
return u_g, r1, r2, e - e0