def poisson(self, n_g, l=0): # Old C version
vr_g = self.zeros()
nrdr_g = n_g * self.r_g * self.dr_g
beta = self.a / self.b
ng = int(round(1.0 / self.b))
assert abs(ng - 1 / self.b) < 1e-5
hartree(l, nrdr_g, beta, ng, vr_g)
#vrp_g = self.purepythonpoisson(n_g,l)
#assert abs(vr_g-vrp_g).max() < 1e-12
return vr_g
def atomic_exact_exchange(atom, type='all'):
"""Returns the exact exchange energy of the atom defined by the
instantiated AllElectron object 'atom'
"""
G_LLL = gaunt(lmax=max(atom.l_j)) # Make gaunt coeff. list
Nj = len(atom.n_j) # The total number of orbitals
# determine relevant states for chosen type of exchange contribution
if type == 'all':
nstates = mstates = range(Nj)
else:
Njcore = core_states(atom.symbol) # The number of core orbitals
if type == 'val-val':
nstates = mstates = range(Njcore, Nj)
elif type == 'core-core':
nstates = mstates = range(Njcore)
elif type == 'val-core':
nstates = range(Njcore, Nj)
mstates = range(Njcore)
else:
raise RuntimeError('Unknown type of exchange: ', type)
# Arrays for storing the potential (times radius)
vr = np.zeros(atom.N)
vrl = np.zeros(atom.N)
# do actual calculation of exchange contribution
Exx = 0.0
for j1 in nstates:
# angular momentum of first state
l1 = atom.l_j[j1]
for j2 in mstates:
# angular momentum of second state
l2 = atom.l_j[j2]
# joint occupation number
f12 = 0.5 * (atom.f_j[j1] / (2. * l1 + 1) * atom.f_j[j2] /
(2. * l2 + 1))
# electron density times radius times length element
nrdr = atom.u_j[j1] * atom.u_j[j2] * atom.dr
nrdr[1:] /= atom.r[1:]
# potential times radius
vr[:] = 0.0
# L summation
for l in range(l1 + l2 + 1):
# get potential for current l-value
hartree(l, nrdr, atom.r, vrl)
# take all m1 m2 and m values of Gaunt matrix of the form
# G(L1,L2,L) where L = {l,m}
G2 = G_LLL[l1**2:(l1 + 1)**2, l2**2:(l2 + 1)**2,
l**2:(l + 1)**2]**2
# add to total potential
vr += vrl * np.sum(G2)
# add to total exchange the contribution from current two states
Exx += -.5 * f12 * np.dot(vr, nrdr)
# double energy if mixed contribution
if type == 'val-core':
Exx *= 2.
# return exchange energy
return Exx
def constructX(gen):
"""Construct the X_p^a matrix for the given atom.
The X_p^a matrix describes the valence-core interactions of the
partial waves.
"""
# initialize attributes
uv_j = gen.vu_j # soft valence states * r:
lv_j = gen.vl_j # their repective l quantum numbers
Nvi = 0
for l in lv_j:
Nvi += 2 * l + 1 # total number of valence states (including m)
# number of core and valence orbitals (j only, i.e. not m-number)
Njcore = gen.njcore
Njval = len(lv_j)
# core states * r:
uc_j = gen.u_j[:Njcore]
r, dr, N, beta = gen.r, gen.dr, gen.N, gen.beta
# potential times radius
vr = np.zeros(N)
# initialize X_ii matrix
X_ii = np.zeros((Nvi, Nvi))
# make gaunt coeff. list
lmax = max(gen.l_j[:Njcore] + gen.vl_j)
gaunt = make_gaunt(lmax=lmax)
# sum over core states
for jc in range(Njcore):
lc = gen.l_j[jc]
# sum over first valence state index
i1 = 0
for jv1 in range(Njval):
lv1 = lv_j[jv1]
# electron density 1 times radius times length element
n1c = uv_j[jv1] * uc_j[jc] * dr
n1c[1:] /= r[1:]
# sum over second valence state index
i2 = 0
for jv2 in range(Njval):
lv2 = lv_j[jv2]
# electron density 2
n2c = uv_j[jv2] * uc_j[jc] * dr
n2c[1:] /= r[1:]
# sum expansion in angular momenta
for l in range(min(lv1, lv2) + lc + 1):
# Int density * potential * r^2 * dr:
hartree(l, n2c, beta, N, vr)
nv = np.dot(n1c, vr)
# expansion coefficients
A_mm = X_ii[i1:i1 + 2 * lv1 + 1, i2:i2 + 2 * lv2 + 1]
for mc in range(2 * lc + 1):
for m in range(2 * l + 1):
G1c = gaunt[lv1**2:(lv1 + 1)**2,
lc**2 + mc, l**2 + m]
G2c = gaunt[lv2**2:(lv2 + 1)**2,
lc**2 + mc, l**2 + m]
A_mm += nv * np.outer(G1c, G2c)
i2 += 2 * lv2 + 1
i1 += 2 * lv1 + 1
# pack X_ii matrix
X_p = pack2(X_ii)
return X_p
def atomic_exact_exchange(atom, type = 'all'):
"""Returns the exact exchange energy of the atom defined by the
instantiated AllElectron object 'atom'
"""
gaunt = make_gaunt(lmax=max(atom.l_j)) # Make gaunt coeff. list
Nj = len(atom.n_j) # The total number of orbitals
# determine relevant states for chosen type of exchange contribution
if type == 'all':
nstates = mstates = range(Nj)
else:
Njcore = core_states(atom.symbol) # The number of core orbitals
if type == 'val-val':
nstates = mstates = range(Njcore, Nj)
elif type == 'core-core':
nstates = mstates = range(Njcore)
elif type == 'val-core':
nstates = range(Njcore,Nj)
mstates = range(Njcore)
else:
raise RuntimeError('Unknown type of exchange: ', type)
# Arrays for storing the potential (times radius)
vr = np.zeros(atom.N)
vrl = np.zeros(atom.N)
# do actual calculation of exchange contribution
Exx = 0.0
for j1 in nstates:
# angular momentum of first state
l1 = atom.l_j[j1]
for j2 in mstates:
# angular momentum of second state
l2 = atom.l_j[j2]
# joint occupation number
f12 = .5 * atom.f_j[j1] / (2. * l1 + 1) * \
atom.f_j[j2] / (2. * l2 + 1)
# electron density times radius times length element
nrdr = atom.u_j[j1] * atom.u_j[j2] * atom.dr
nrdr[1:] /= atom.r[1:]
# potential times radius
vr[:] = 0.0
# L summation
for l in range(l1 + l2 + 1):
# get potential for current l-value
hartree(l, nrdr, atom.beta, atom.N, vrl)
# take all m1 m2 and m values of Gaunt matrix of the form
# G(L1,L2,L) where L = {l,m}
G2 = gaunt[l1**2:(l1+1)**2, l2**2:(l2+1)**2, l**2:(l+1)**2]**2
# add to total potential
vr += vrl * np.sum(G2)
# add to total exchange the contribution from current two states
Exx += -.5 * f12 * np.dot(vr, nrdr)
# double energy if mixed contribution
if type == 'val-core': Exx *= 2.
# return exchange energy
return Exx
def poisson(self, n_g, l=0):
vr_g = self.zeros()
nrdr_g = n_g * self.r_g * self.dr_g
hartree(l, nrdr_g, self.r_g, vr_g)
return vr_g
def run(self, core='', rcut=1.0, extra=None,
logderiv=False, vbar=None, exx=False, name=None,
normconserving='', filter=(0.4, 1.75), rcutcomp=None,
write_xml=True, use_restart_file=True,
empty_states=''):
self.name = name
self.core = core
if type(rcut) is float:
rcut_l = [rcut]
else:
rcut_l = rcut
rcutmax = max(rcut_l)
rcutmin = min(rcut_l)
self.rcut_l = rcut_l
if rcutcomp is None:
rcutcomp = rcutmin
self.rcutcomp = rcutcomp
hfilter, xfilter = filter
Z = self.Z
n_j = self.n_j
l_j = self.l_j
f_j = self.f_j
e_j = self.e_j
if vbar is None:
vbar = ('poly', rcutmin * 0.9)
vbar_type, rcutvbar = vbar
normconserving_l = [x in normconserving for x in 'spdf']
# Parse core string:
j = 0
if core.startswith('['):
a, core = core.split(']')
core_symbol = a[1:]
j = len(configurations[core_symbol][1])
while core != '':
assert n_j[j] == int(core[0])
assert l_j[j] == 'spdf'.find(core[1])
if j != self.jcorehole:
assert f_j[j] == 2 * (2 * l_j[j] + 1)
j += 1
core = core[2:]
njcore = j
self.njcore = njcore
lmaxocc = max(l_j[njcore:])
while empty_states != '':
n = int(empty_states[0])
l = 'spdf'.find(empty_states[1])
assert n == 1 + l + l_j.count(l)
n_j.append(n)
l_j.append(l)
f_j.append(0.0)
e_j.append(-0.01)
empty_states = empty_states[2:]
if 2 in l_j[njcore:]:
# We have a bound valence d-state. Add bound s- and
# p-states if not already there:
for l in [0, 1]:
if l not in l_j[njcore:]:
n_j.append(1 + l + l_j.count(l))
l_j.append(l)
f_j.append(0.0)
e_j.append(-0.01)
if l_j[njcore:] == [0] and Z > 2:
# We have only a bound valence s-state and we are not
# hydrogen and not helium. Add bound p-state:
n_j.append(n_j[njcore])
l_j.append(1)
f_j.append(0.0)
e_j.append(-0.01)
nj = len(n_j)
self.Nv = sum(f_j[njcore:])
self.Nc = sum(f_j[:njcore])
# Do all-electron calculation:
AllElectron.run(self, use_restart_file)
# Highest occupied atomic orbital:
self.emax = max(e_j)
N = self.N
r = self.r
dr = self.dr
d2gdr2 = self.d2gdr2
beta = self.beta
dv = r**2 * dr
t = self.text
t()
t('Generating PAW setup')
if core != '':
t('Frozen core:', core)
# So far - no ghost-states:
self.ghost = False
# Calculate the kinetic energy of the core states:
Ekincore = 0.0
j = 0
for f, e, u in zip(f_j[:njcore], e_j[:njcore], self.u_j[:njcore]):
u = np.where(abs(u) < 1e-160, 0, u) # XXX Numeric!
k = e - np.sum((u**2 * self.vr * dr)[1:] / r[1:])
Ekincore += f * k
if j == self.jcorehole:
self.Ekincorehole = k
j += 1
# Calculate core density:
if njcore == 0:
nc = np.zeros(N)
else:
uc_j = self.u_j[:njcore]
uc_j = np.where(abs(uc_j) < 1e-160, 0, uc_j) # XXX Numeric!
nc = np.dot(f_j[:njcore], uc_j**2) / (4 * pi)
nc[1:] /= r[1:]**2
nc[0] = nc[1]
self.nc = nc
# Calculate core kinetic energy density
if njcore == 0:
tauc = np.zeros(N)
else:
tauc = self.radial_kinetic_energy_density(f_j[:njcore],
l_j[:njcore],
self.u_j[:njcore])
t('Kinetic energy of the core from tauc =',
np.dot(tauc * r * r, dr) * 4 * pi)
lmax = max(l_j[njcore:])
# Order valence states with respect to angular momentum
# quantum number:
self.n_ln = n_ln = []
self.f_ln = f_ln = []
self.e_ln = e_ln = []
for l in range(lmax + 1):
n_n = []
f_n = []
e_n = []
for j in range(njcore, nj):
if l_j[j] == l:
n_n.append(n_j[j])
f_n.append(f_j[j])
e_n.append(e_j[j])
n_ln.append(n_n)
f_ln.append(f_n)
e_ln.append(e_n)
# Add extra projectors:
if extra is not None:
if len(extra) == 0:
lmaxextra = 0
else:
lmaxextra = max(extra.keys())
if lmaxextra > lmax:
for l in range(lmax, lmaxextra):
n_ln.append([])
f_ln.append([])
e_ln.append([])
lmax = lmaxextra
for l in extra:
nn = -1
for e in extra[l]:
n_ln[l].append(nn)
f_ln[l].append(0.0)
e_ln[l].append(e)
nn -= 1
else:
# Automatic:
# Make sure we have two projectors for each occupied channel:
for l in range(lmaxocc + 1):
if len(n_ln[l]) < 2 and not normconserving_l[l]:
# Only one - add one more:
assert len(n_ln[l]) == 1
n_ln[l].append(-1)
f_ln[l].append(0.0)
e_ln[l].append(1.0 + e_ln[l][0])
if lmaxocc < 2 and lmaxocc == lmax:
# Add extra projector for l = lmax + 1:
n_ln.append([-1])
f_ln.append([0.0])
e_ln.append([0.0])
lmax += 1
self.lmax = lmax
rcut_l.extend([rcutmin] * (lmax + 1 - len(rcut_l)))
t('Cutoffs:')
for rc, s in zip(rcut_l, 'spdf'):
t('rc(%s)=%.3f' % (s, rc))
t('rc(vbar)=%.3f' % rcutvbar)
t('rc(comp)=%.3f' % rcutcomp)
t('rc(nct)=%.3f' % rcutmax)
t()
t('Kinetic energy of the core states: %.6f' % Ekincore)
# Allocate arrays:
self.u_ln = u_ln = [] # phi * r
self.s_ln = s_ln = [] # phi-tilde * r
self.q_ln = q_ln = [] # p-tilde * r
for l in range(lmax + 1):
nn = len(n_ln[l])
u_ln.append(np.zeros((nn, N)))
s_ln.append(np.zeros((nn, N)))
q_ln.append(np.zeros((nn, N)))
# Fill in all-electron wave functions:
for l in range(lmax + 1):
# Collect all-electron wave functions:
u_n = [self.u_j[j] for j in range(njcore, nj) if l_j[j] == l]
for n, u in enumerate(u_n):
u_ln[l][n] = u
# Grid-index corresponding to rcut:
gcut_l = [1 + int(rc * N / (rc + beta)) for rc in rcut_l]
rcutfilter = xfilter * rcutmax
self.rcutfilter = rcutfilter
gcutfilter = 1 + int(rcutfilter * N / (rcutfilter + beta))
gcutmax = 1 + int(rcutmax * N / (rcutmax + beta))
# Outward integration of unbound states stops at 3 * rcut:
gmax = int(3 * rcutmax * N / (3 * rcutmax + beta))
assert gmax > gcutfilter
# Calculate unbound extra states:
c2 = -(r / dr)**2
c10 = -d2gdr2 * r**2
for l, (n_n, e_n, u_n) in enumerate(zip(n_ln, e_ln, u_ln)):
for n, e, u in zip(n_n, e_n, u_n):
if n < 0:
u[:] = 0.0
shoot(u, l, self.vr, e, self.r2dvdr, r, dr, c10, c2,
self.scalarrel, gmax=gmax)
u *= 1.0 / u[gcut_l[l]]
charge = Z - self.Nv - self.Nc
t('Charge: %.1f' % charge)
t('Core electrons: %.1f' % self.Nc)
t('Valence electrons: %.1f' % self.Nv)
# Construct smooth wave functions:
coefs = []
for l, (u_n, s_n) in enumerate(zip(u_ln, s_ln)):
nodeless = True
gc = gcut_l[l]
for u, s in zip(u_n, s_n):
s[:] = u
if normconserving_l[l]:
A = np.zeros((5, 5))
A[:4, 0] = 1.0
A[:4, 1] = r[gc - 2:gc + 2]**2
A[:4, 2] = A[:4, 1]**2
A[:4, 3] = A[:4, 1] * A[:4, 2]
A[:4, 4] = A[:4, 2]**2
A[4, 4] = 1.0
a = u[gc - 2:gc + 3] / r[gc - 2:gc + 3]**(l + 1)
a = np.log(a)
def f(x):
a[4] = x
b = solve(A, a)
r1 = r[:gc]
r2 = r1**2
rl1 = r1**(l + 1)
y = b[0] + r2 * (b[1] + r2 * (b[2] + r2 * (b[3] + r2
* b[4])))
y = np.exp(y)
s[:gc] = rl1 * y
return np.dot(s**2, dr) - 1
x1 = 0.0
x2 = 0.001
f1 = f(x1)
f2 = f(x2)
while abs(f1) > 1e-6:
x0 = (x1 / f1 - x2 / f2) / (1 / f1 - 1 / f2)
f0 = f(x0)
if abs(f1) < abs(f2):
x2, f2 = x1, f1
x1, f1 = x0, f0
else:
A = np.ones((4, 4))
A[:, 0] = 1.0
A[:, 1] = r[gc - 2:gc + 2]**2
A[:, 2] = A[:, 1]**2
A[:, 3] = A[:, 1] * A[:, 2]
a = u[gc - 2:gc + 2] / r[gc - 2:gc + 2]**(l + 1)
if 0:#l < 2 and nodeless:
a = np.log(a)
a = solve(A, a)
r1 = r[:gc]
r2 = r1**2
rl1 = r1**(l + 1)
y = a[0] + r2 * (a[1] + r2 * (a[2] + r2 * (a[3])))
if 0:#l < 2 and nodeless:
y = np.exp(y)
s[:gc] = rl1 * y
coefs.append(a)
if nodeless:
if not np.alltrue(s[1:gc] > 0.0):
raise RuntimeError(
'Error: The %d%s pseudo wave has a node!' %
(n_ln[l][0], 'spdf'[l]))
# Only the first state for each l must be nodeless:
nodeless = False
# Calculate pseudo core density:
gcutnc = 1 + int(rcutmax * N / (rcutmax + beta))
self.nct = nct = nc.copy()
A = np.ones((4, 4))
A[0] = 1.0
A[1] = r[gcutnc - 2:gcutnc + 2]**2
A[2] = A[1]**2
A[3] = A[1] * A[2]
a = nc[gcutnc - 2:gcutnc + 2]
a = solve(np.transpose(A), a)
r2 = r[:gcutnc]**2
nct[:gcutnc] = a[0] + r2 * (a[1] + r2 * (a[2] + r2 * a[3]))
t('Pseudo-core charge: %.6f' % (4 * pi * np.dot(nct, dv)))
# ... and the pseudo core kinetic energy density:
tauct = tauc.copy()
a = tauc[gcutnc - 2:gcutnc + 2]
a = solve(np.transpose(A), a)
tauct[:gcutnc] = a[0] + r2 * (a[1] + r2 * (a[2] + r2 * a[3]))
# ... and the soft valence density:
nt = np.zeros(N)
for f_n, s_n in zip(f_ln, s_ln):
nt += np.dot(f_n, s_n**2) / (4 * pi)
nt[1:] /= r[1:]**2
nt[0] = nt[1]
nt += nct
self.nt = nt
# Calculate the shape function:
x = r / rcutcomp
gaussian = np.zeros(N)
self.gamma = gamma = 10.0
gaussian[:gmax] = np.exp(-gamma * x[:gmax]**2)
gt = 4 * (gamma / rcutcomp**2)**1.5 / sqrt(pi) * gaussian
t('Shape function alpha=%.3f' % (gamma / rcutcomp**2))
norm = np.dot(gt, dv)
#print norm, norm-1
assert abs(norm - 1) < 1e-2
gt /= norm
# Calculate smooth charge density:
Nt = np.dot(nt, dv)
rhot = nt - (Nt + charge / (4 * pi)) * gt
t('Pseudo-electron charge', 4 * pi * Nt)
vHt = np.zeros(N)
hartree(0, rhot * r * dr, self.beta, self.N, vHt)
vHt[1:] /= r[1:]
vHt[0] = vHt[1]
vXCt = np.zeros(N)
extra_xc_data = {}
if self.xc.type != 'GLLB':
Exct = self.xc.calculate_spherical(self.rgd,
nt.reshape((1, -1)),
vXCt.reshape((1, -1)))
else:
Exct = self.xc.get_smooth_xc_potential_and_energy_1d(vXCt)
# Calculate extra-stuff for non-local functionals
self.xc.get_extra_setup_data(extra_xc_data)
vt = vHt + vXCt
# Construct zero potential:
gc = 1 + int(rcutvbar * N / (rcutvbar + beta))
if vbar_type == 'f':
assert lmax == 2
uf = np.zeros(N)
l = 3
# Solve for all-electron f-state:
eps = 0.0
shoot(uf, l, self.vr, eps, self.r2dvdr, r, dr, c10, c2,
self.scalarrel, gmax=gmax)
uf *= 1.0 / uf[gc]
# Fit smooth pseudo f-state polynomium:
A = np.ones((4, 4))
A[:, 0] = 1.0
A[:, 1] = r[gc - 2:gc + 2]**2
A[:, 2] = A[:, 1]**2
A[:, 3] = A[:, 1] * A[:, 2]
a = uf[gc - 2:gc + 2] / r[gc - 2:gc + 2]**(l + 1)
a0, a1, a2, a3 = solve(A, a)
r1 = r[:gc]
r2 = r1**2
rl1 = r1**(l + 1)
y = a0 + r2 * (a1 + r2 * (a2 + r2 * a3))
sf = uf.copy()
sf[:gc] = rl1 * y
# From 0 to gc, use analytic formula for kinetic energy operator:
r4 = r2**2
r6 = r4 * r2
enumerator = (a0 * l * (l + 1) +
a1 * (l + 2) * (l + 3) * r2 +
a2 * (l + 4) * (l + 5) * r4 +
a3 * (l + 6) * (l + 7) * r6)
denominator = a0 + a1 * r2 + a2 * r4 + a3 * r6
ekin_over_phit = - 0.5 * (enumerator / denominator - l * (l + 1))
ekin_over_phit[1:] /= r2[1:]
vbar = eps - vt
vbar[:gc] -= ekin_over_phit
vbar[0] = vbar[1] # Actually we can collect the terms into
# a single fraction without poles, so as to avoid doing this,
# but this is good enough
# From gc to gmax, use finite-difference formula for kinetic
# energy operator:
vbar[gc:gmax] -= self.kin(l, sf)[gc:gmax] / sf[gc:gmax]
vbar[gmax:] = 0.0
else:
assert vbar_type == 'poly'
A = np.ones((2, 2))
A[0] = 1.0
A[1] = r[gc - 1:gc + 1]**2
a = vt[gc - 1:gc + 1]
a = solve(np.transpose(A), a)
r2 = r**2
vbar = a[0] + r2 * a[1] - vt
vbar[gc:] = 0.0
vt += vbar
# Construct projector functions:
for l, (e_n, s_n, q_n) in enumerate(zip(e_ln, s_ln, q_ln)):
for e, s, q in zip(e_n, s_n, q_n):
q[:] = self.kin(l, s) + (vt - e) * s
q[gcutmax:] = 0.0
filter = Filter(r, dr, gcutfilter, hfilter).filter
vbar = filter(vbar * r)
# Calculate matrix elements:
self.dK_lnn = dK_lnn = []
self.dH_lnn = dH_lnn = []
self.dO_lnn = dO_lnn = []
for l, (e_n, u_n, s_n, q_n) in enumerate(zip(e_ln, u_ln,
s_ln, q_ln)):
A_nn = np.inner(s_n, q_n * dr)
# Do a LU decomposition of A:
nn = len(e_n)
L_nn = np.identity(nn, float)
U_nn = A_nn.copy()
# Keep all bound states normalized
if sum([n > 0 for n in n_ln[l]]) <= 1:
for i in range(nn):
for j in range(i + 1, nn):
L_nn[j, i] = 1.0 * U_nn[j, i] / U_nn[i, i]
U_nn[j, :] -= U_nn[i, :] * L_nn[j, i]
dO_nn = (np.inner(u_n, u_n * dr) -
np.inner(s_n, s_n * dr))
e_nn = np.zeros((nn, nn))
e_nn.ravel()[::nn + 1] = e_n
dH_nn = np.dot(dO_nn, e_nn) - A_nn
q_n[:] = np.dot(inv(np.transpose(U_nn)), q_n)
s_n[:] = np.dot(inv(L_nn), s_n)
u_n[:] = np.dot(inv(L_nn), u_n)
dO_nn = np.dot(np.dot(inv(L_nn), dO_nn),
inv(np.transpose(L_nn)))
dH_nn = np.dot(np.dot(inv(L_nn), dH_nn),
inv(np.transpose(L_nn)))
ku_n = [self.kin(l, u, e) for u, e in zip(u_n, e_n)]
ks_n = [self.kin(l, s) for s in s_n]
dK_nn = 0.5 * (np.inner(u_n, ku_n * dr) -
np.inner(s_n, ks_n * dr))
dK_nn += np.transpose(dK_nn).copy()
dK_lnn.append(dK_nn)
dO_lnn.append(dO_nn)
dH_lnn.append(dH_nn)
for n, q in enumerate(q_n):
q[:] = filter(q, l) * r**(l + 1)
A_nn = np.inner(s_n, q_n * dr)
q_n[:] = np.dot(inv(np.transpose(A_nn)), q_n)
self.vt = vt
self.vbar = vbar
t('state eigenvalue norm')
t('--------------------------------')
for l, (n_n, f_n, e_n) in enumerate(zip(n_ln, f_ln, e_ln)):
for n in range(len(e_n)):
if n_n[n] > 0:
f = '(%d)' % f_n[n]
t('%d%s%-4s: %12.6f %12.6f' % (
n_n[n], 'spdf'[l], f, e_n[n],
np.dot(s_ln[l][n]**2, dr)))
else:
t('*%s : %12.6f' % ('spdf'[l], e_n[n]))
t('--------------------------------')
self.logd = {}
if logderiv:
ni = 300
self.elog = np.linspace(-5.0, 1.0, ni)
# Calculate logarithmic derivatives:
gld = gcutmax + 10
self.rlog = r[gld]
assert gld < gmax
t('Calculating logarithmic derivatives at r=%.3f' % r[gld])
t('(skip with [Ctrl-C])')
try:
u = np.zeros(N)
for l in range(4):
self.logd[l] = (np.empty(ni), np.empty(ni))
if l <= lmax:
dO_nn = dO_lnn[l]
dH_nn = dH_lnn[l]
q_n = q_ln[l]
fae = open(self.symbol + '.ae.ld.' + 'spdf'[l], 'w')
fps = open(self.symbol + '.ps.ld.' + 'spdf'[l], 'w')
for i, e in enumerate(self.elog):
# All-electron logarithmic derivative:
u[:] = 0.0
shoot(u, l, self.vr, e, self.r2dvdr, r, dr, c10, c2,
self.scalarrel, gmax=gld)
dudr = 0.5 * (u[gld + 1] - u[gld - 1]) / dr[gld]
ld = dudr / u[gld] - 1.0 / r[gld]
print >> fae, e, ld
self.logd[l][0][i] = ld
# PAW logarithmic derivative:
s = self.integrate(l, vt, e, gld)
if l <= lmax:
A_nn = dH_nn - e * dO_nn
s_n = [self.integrate(l, vt, e, gld, q)
for q in q_n]
B_nn = np.inner(q_n, s_n * dr)
a_n = np.dot(q_n, s * dr)
B_nn = np.dot(A_nn, B_nn)
B_nn.ravel()[::len(a_n) + 1] += 1.0
c_n = solve(B_nn, np.dot(A_nn, a_n))
s -= np.dot(c_n, s_n)
dsdr = 0.5 * (s[gld + 1] - s[gld - 1]) / dr[gld]
ld = dsdr / s[gld] - 1.0 / r[gld]
print >> fps, e, ld
self.logd[l][1][i] = ld
except KeyboardInterrupt:
pass
self.write(nc,'nc')
self.write(nt, 'nt')
self.write(nct, 'nct')
self.write(vbar, 'vbar')
self.write(vt, 'vt')
self.write(tauc, 'tauc')
self.write(tauct, 'tauct')
for l, (n_n, f_n, u_n, s_n, q_n) in enumerate(zip(n_ln, f_ln,
u_ln, s_ln, q_ln)):
for n, f, u, s, q in zip(n_n, f_n, u_n, s_n, q_n):
if n < 0:
self.write(u, 'ae', n=n, l=l)
self.write(s, 'ps', n=n, l=l)
self.write(q, 'proj', n=n, l=l)
# Test for ghost states:
for h in [0.05]:
self.diagonalize(h)
self.vn_j = vn_j = []
self.vl_j = vl_j = []
self.vf_j = vf_j = []
self.ve_j = ve_j = []
self.vu_j = vu_j = []
self.vs_j = vs_j = []
self.vq_j = vq_j = []
j_ln = [[0 for f in f_n] for f_n in f_ln]
j = 0
for l, n_n in enumerate(n_ln):
for n, nn in enumerate(n_n):
if nn > 0:
vf_j.append(f_ln[l][n])
vn_j.append(nn)
vl_j.append(l)
ve_j.append(e_ln[l][n])
vu_j.append(u_ln[l][n])
vs_j.append(s_ln[l][n])
vq_j.append(q_ln[l][n])
j_ln[l][n] = j
j += 1
for l, n_n in enumerate(n_ln):
for n, nn in enumerate(n_n):
if nn < 0:
vf_j.append(0)
vn_j.append(nn)
vl_j.append(l)
ve_j.append(e_ln[l][n])
vu_j.append(u_ln[l][n])
vs_j.append(s_ln[l][n])
vq_j.append(q_ln[l][n])
j_ln[l][n] = j
j += 1
nj = j
self.dK_jj = np.zeros((nj, nj))
for l, j_n in enumerate(j_ln):
for n1, j1 in enumerate(j_n):
for n2, j2 in enumerate(j_n):
self.dK_jj[j1, j2] = self.dK_lnn[l][n1, n2]
if exx:
X_p = constructX(self)
ExxC = atomic_exact_exchange(self, 'core-core')
else:
X_p = None
ExxC = None
sqrt4pi = sqrt(4 * pi)
setup = SetupData(self.symbol, self.xc.name, self.name,
readxml=False)
def divide_by_r(x_g, l):
r = self.r
#for x_g, l in zip(x_jg, l_j):
p = x_g.copy()
p[1:] /= self.r[1:]
# XXXXX go to higher order!!!!!
if l == 0:#l_j[self.jcorehole] == 0:
p[0] = (p[2] +
(p[1] - p[2]) * (r[0] - r[2]) / (r[1] - r[2]))
return p
def divide_all_by_r(x_jg):
return [divide_by_r(x_g, l) for x_g, l in zip(x_jg, vl_j)]
setup.l_j = vl_j
setup.n_j = vn_j
setup.f_j = vf_j
setup.eps_j = ve_j
setup.rcut_j = [rcut_l[l] for l in vl_j]
setup.nc_g = nc * sqrt4pi
setup.nct_g = nct * sqrt4pi
setup.nvt_g = (nt - nct) * sqrt4pi
setup.e_kinetic_core = Ekincore
setup.vbar_g = vbar * sqrt4pi
setup.tauc_g = tauc * sqrt4pi
setup.tauct_g = tauct * sqrt4pi
setup.extra_xc_data = extra_xc_data
setup.Z = Z
setup.Nc = self.Nc
setup.Nv = self.Nv
setup.e_kinetic = self.Ekin
setup.e_xc = self.Exc
setup.e_electrostatic = self.Epot
setup.e_total = self.Epot + self.Exc + self.Ekin
setup.rgd = self.rgd
setup.rcgauss = self.rcutcomp / sqrt(self.gamma)
setup.e_kin_jj = self.dK_jj
setup.ExxC = ExxC
setup.phi_jg = divide_all_by_r(vu_j)
setup.phit_jg = divide_all_by_r(vs_j)
setup.pt_jg = divide_all_by_r(vq_j)
setup.X_p = X_p
if self.jcorehole is not None:
setup.has_corehole = True
setup.lcorehole = l_j[self.jcorehole] # l_j or vl_j ????? XXX
setup.ncorehole = n_j[self.jcorehole]
setup.phicorehole_g = divide_by_r(self.u_j[self.jcorehole],
setup.lcorehole)
setup.core_hole_e = self.e_j[self.jcorehole]
setup.core_hole_e_kin = self.Ekincorehole
setup.fcorehole = self.fcorehole
if self.ghost:
raise RuntimeError('Ghost!')
if self.scalarrel:
reltype = 'scalar-relativistic'
else:
reltype = 'non-relativistic'
attrs = [('type', reltype), ('name', 'gpaw-%s' % version)]
data = 'Frozen core: '+ (self.core or 'none')
setup.generatorattrs = attrs
setup.generatordata = data
self.id_j = []
for l, n in zip(vl_j, vn_j):
if n > 0:
id = '%s-%d%s' % (self.symbol, n, 'spdf'[l])
else:
id = '%s-%s%d' % (self.symbol, 'spdf'[l], -n)
self.id_j.append(id)
setup.id_j = self.id_j
if write_xml:
setup.write_xml()
return setup
def run(self,
core='',
rcut=1.0,
extra=None,
logderiv=False,
vbar=None,
exx=False,
name=None,
normconserving='',
filter=(0.4, 1.75),
rcutcomp=None,
write_xml=True,
use_restart_file=True,
empty_states=''):
self.name = name
self.core = core
if type(rcut) is float:
rcut_l = [rcut]
else:
rcut_l = rcut
rcutmax = max(rcut_l)
rcutmin = min(rcut_l)
self.rcut_l = rcut_l
if rcutcomp is None:
rcutcomp = rcutmin
self.rcutcomp = rcutcomp
hfilter, xfilter = filter
Z = self.Z
n_j = self.n_j
l_j = self.l_j
f_j = self.f_j
e_j = self.e_j
if vbar is None:
vbar = ('poly', rcutmin * 0.9)
vbar_type, rcutvbar = vbar
normconserving_l = [x in normconserving for x in 'spdf']
# Parse core string:
j = 0
if core.startswith('['):
a, core = core.split(']')
core_symbol = a[1:]
j = len(configurations[core_symbol][1])
while core != '':
assert n_j[j] == int(core[0])
assert l_j[j] == 'spdf'.find(core[1])
if j != self.jcorehole:
assert f_j[j] == 2 * (2 * l_j[j] + 1)
j += 1
core = core[2:]
njcore = j
self.njcore = njcore
lmaxocc = max(l_j[njcore:])
while empty_states != '':
n = int(empty_states[0])
l = 'spdf'.find(empty_states[1])
print l_j
assert n == 1 + l + l_j.count(l)
n_j.append(n)
l_j.append(l)
f_j.append(0.0)
e_j.append(-0.01)
empty_states = empty_states[2:]
if 2 in l_j[njcore:]:
# We have a bound valence d-state. Add bound s- and
# p-states if not already there:
for l in [0, 1]:
if l not in l_j[njcore:]:
n_j.append(1 + l + l_j.count(l))
l_j.append(l)
f_j.append(0.0)
e_j.append(-0.01)
if l_j[njcore:] == [0] and Z > 2:
# We have only a bound valence s-state and we are not
# hydrogen and not helium. Add bound p-state:
n_j.append(n_j[njcore])
l_j.append(1)
f_j.append(0.0)
e_j.append(-0.01)
nj = len(n_j)
self.Nv = sum(f_j[njcore:])
self.Nc = sum(f_j[:njcore])
# Do all-electron calculation:
AllElectron.run(self, use_restart_file)
# Highest occupied atomic orbital:
self.emax = max(e_j)
N = self.N
r = self.r
dr = self.dr
d2gdr2 = self.d2gdr2
beta = self.beta
dv = r**2 * dr
t = self.text
t()
t('Generating PAW setup')
if core != '':
t('Frozen core:', core)
# So far - no ghost-states:
self.ghost = False
# Calculate the kinetic energy of the core states:
Ekincore = 0.0
j = 0
for f, e, u in zip(f_j[:njcore], e_j[:njcore], self.u_j[:njcore]):
u = np.where(abs(u) < 1e-160, 0, u) # XXX Numeric!
k = e - np.sum((u**2 * self.vr * dr)[1:] / r[1:])
Ekincore += f * k
if j == self.jcorehole:
self.Ekincorehole = k
j += 1
# Calculate core density:
if njcore == 0:
nc = np.zeros(N)
else:
uc_j = self.u_j[:njcore]
uc_j = np.where(abs(uc_j) < 1e-160, 0, uc_j) # XXX Numeric!
nc = np.dot(f_j[:njcore], uc_j**2) / (4 * pi)
nc[1:] /= r[1:]**2
nc[0] = nc[1]
self.nc = nc
# Calculate core kinetic energy density
if njcore == 0:
tauc = np.zeros(N)
else:
tauc = self.radial_kinetic_energy_density(f_j[:njcore],
l_j[:njcore],
self.u_j[:njcore])
t('Kinetic energy of the core from tauc =',
np.dot(tauc * r * r, dr) * 4 * pi)
lmax = max(l_j[njcore:])
# Order valence states with respect to angular momentum
# quantum number:
self.n_ln = n_ln = []
self.f_ln = f_ln = []
self.e_ln = e_ln = []
for l in range(lmax + 1):
n_n = []
f_n = []
e_n = []
for j in range(njcore, nj):
if l_j[j] == l:
n_n.append(n_j[j])
f_n.append(f_j[j])
e_n.append(e_j[j])
n_ln.append(n_n)
f_ln.append(f_n)
e_ln.append(e_n)
# Add extra projectors:
if extra is not None:
if len(extra) == 0:
lmaxextra = 0
else:
lmaxextra = max(extra.keys())
if lmaxextra > lmax:
for l in range(lmax, lmaxextra):
n_ln.append([])
f_ln.append([])
e_ln.append([])
lmax = lmaxextra
for l in extra:
nn = -1
for e in extra[l]:
n_ln[l].append(nn)
f_ln[l].append(0.0)
e_ln[l].append(e)
nn -= 1
else:
# Automatic:
# Make sure we have two projectors for each occupied channel:
for l in range(lmaxocc + 1):
if len(n_ln[l]) < 2 and not normconserving_l[l]:
# Only one - add one more:
assert len(n_ln[l]) == 1
n_ln[l].append(-1)
f_ln[l].append(0.0)
e_ln[l].append(1.0 + e_ln[l][0])
if lmaxocc < 2 and lmaxocc == lmax:
# Add extra projector for l = lmax + 1:
n_ln.append([-1])
f_ln.append([0.0])
e_ln.append([0.0])
lmax += 1
self.lmax = lmax
rcut_l.extend([rcutmin] * (lmax + 1 - len(rcut_l)))
t('Cutoffs:')
for rc, s in zip(rcut_l, 'spdf'):
t('rc(%s)=%.3f' % (s, rc))
t('rc(vbar)=%.3f' % rcutvbar)
t('rc(comp)=%.3f' % rcutcomp)
t()
t('Kinetic energy of the core states: %.6f' % Ekincore)
# Allocate arrays:
self.u_ln = u_ln = [] # phi * r
self.s_ln = s_ln = [] # phi-tilde * r
self.q_ln = q_ln = [] # p-tilde * r
for l in range(lmax + 1):
nn = len(n_ln[l])
u_ln.append(np.zeros((nn, N)))
s_ln.append(np.zeros((nn, N)))
q_ln.append(np.zeros((nn, N)))
# Fill in all-electron wave functions:
for l in range(lmax + 1):
# Collect all-electron wave functions:
u_n = [self.u_j[j] for j in range(njcore, nj) if l_j[j] == l]
for n, u in enumerate(u_n):
u_ln[l][n] = u
# Grid-index corresponding to rcut:
gcut_l = [1 + int(rc * N / (rc + beta)) for rc in rcut_l]
rcutfilter = xfilter * rcutmax
self.rcutfilter = rcutfilter
gcutfilter = 1 + int(rcutfilter * N / (rcutfilter + beta))
gcutmax = 1 + int(rcutmax * N / (rcutmax + beta))
# Outward integration of unbound states stops at 3 * rcut:
gmax = int(3 * rcutmax * N / (3 * rcutmax + beta))
assert gmax > gcutfilter
# Calculate unbound extra states:
c2 = -(r / dr)**2
c10 = -d2gdr2 * r**2
for l, (n_n, e_n, u_n) in enumerate(zip(n_ln, e_ln, u_ln)):
for n, e, u in zip(n_n, e_n, u_n):
if n < 0:
u[:] = 0.0
shoot(u,
l,
self.vr,
e,
self.r2dvdr,
r,
dr,
c10,
c2,
self.scalarrel,
gmax=gmax)
u *= 1.0 / u[gcut_l[l]]
charge = Z - self.Nv - self.Nc
t('Charge: %.1f' % charge)
t('Core electrons: %.1f' % self.Nc)
t('Valence electrons: %.1f' % self.Nv)
# Construct smooth wave functions:
coefs = []
for l, (u_n, s_n) in enumerate(zip(u_ln, s_ln)):
nodeless = True
gc = gcut_l[l]
for u, s in zip(u_n, s_n):
s[:] = u
if normconserving_l[l]:
A = np.zeros((5, 5))
A[:4, 0] = 1.0
A[:4, 1] = r[gc - 2:gc + 2]**2
A[:4, 2] = A[:4, 1]**2
A[:4, 3] = A[:4, 1] * A[:4, 2]
A[:4, 4] = A[:4, 2]**2
A[4, 4] = 1.0
a = u[gc - 2:gc + 3] / r[gc - 2:gc + 3]**(l + 1)
a = np.log(a)
def f(x):
a[4] = x
b = solve(A, a)
r1 = r[:gc]
r2 = r1**2
rl1 = r1**(l + 1)
y = b[0] + r2 * (b[1] + r2 * (b[2] + r2 *
(b[3] + r2 * b[4])))
y = np.exp(y)
s[:gc] = rl1 * y
return np.dot(s**2, dr) - 1
x1 = 0.0
x2 = 0.001
f1 = f(x1)
f2 = f(x2)
while abs(f1) > 1e-6:
x0 = (x1 / f1 - x2 / f2) / (1 / f1 - 1 / f2)
f0 = f(x0)
if abs(f1) < abs(f2):
x2, f2 = x1, f1
x1, f1 = x0, f0
else:
A = np.ones((4, 4))
A[:, 0] = 1.0
A[:, 1] = r[gc - 2:gc + 2]**2
A[:, 2] = A[:, 1]**2
A[:, 3] = A[:, 1] * A[:, 2]
a = u[gc - 2:gc + 2] / r[gc - 2:gc + 2]**(l + 1)
if 0: #l < 2 and nodeless:
a = np.log(a)
a = solve(A, a)
r1 = r[:gc]
r2 = r1**2
rl1 = r1**(l + 1)
y = a[0] + r2 * (a[1] + r2 * (a[2] + r2 * (a[3])))
if 0: #l < 2 and nodeless:
y = np.exp(y)
s[:gc] = rl1 * y
coefs.append(a)
if nodeless:
if not np.alltrue(s[1:gc] > 0.0):
raise RuntimeError(
'Error: The %d%s pseudo wave has a node!' %
(n_ln[l][0], 'spdf'[l]))
# Only the first state for each l must be nodeless:
nodeless = False
# Calculate pseudo core density:
gcutnc = 1 + int(rcutmax * N / (rcutmax + beta))
self.nct = nct = nc.copy()
A = np.ones((4, 4))
A[0] = 1.0
A[1] = r[gcutnc - 2:gcutnc + 2]**2
A[2] = A[1]**2
A[3] = A[1] * A[2]
a = nc[gcutnc - 2:gcutnc + 2]
a = solve(np.transpose(A), a)
r2 = r[:gcutnc]**2
nct[:gcutnc] = a[0] + r2 * (a[1] + r2 * (a[2] + r2 * a[3]))
t('Pseudo-core charge: %.6f' % (4 * pi * np.dot(nct, dv)))
# ... and the pseudo core kinetic energy density:
tauct = tauc.copy()
a = tauc[gcutnc - 2:gcutnc + 2]
a = solve(np.transpose(A), a)
tauct[:gcutnc] = a[0] + r2 * (a[1] + r2 * (a[2] + r2 * a[3]))
# ... and the soft valence density:
nt = np.zeros(N)
for f_n, s_n in zip(f_ln, s_ln):
nt += np.dot(f_n, s_n**2) / (4 * pi)
nt[1:] /= r[1:]**2
nt[0] = nt[1]
nt += nct
self.nt = nt
# Calculate the shape function:
x = r / rcutcomp
gaussian = np.zeros(N)
self.gamma = gamma = 10.0
gaussian[:gmax] = np.exp(-gamma * x[:gmax]**2)
gt = 4 * (gamma / rcutcomp**2)**1.5 / sqrt(pi) * gaussian
norm = np.dot(gt, dv)
#print norm, norm-1
assert abs(norm - 1) < 1e-2
gt /= norm
# Calculate smooth charge density:
Nt = np.dot(nt, dv)
rhot = nt - (Nt + charge / (4 * pi)) * gt
t('Pseudo-electron charge', 4 * pi * Nt)
vHt = np.zeros(N)
hartree(0, rhot * r * dr, self.beta, self.N, vHt)
vHt[1:] /= r[1:]
vHt[0] = vHt[1]
vXCt = np.zeros(N)
extra_xc_data = {}
if self.xc.type != 'GLLB':
Exct = self.xc.calculate_spherical(self.rgd, nt.reshape((1, -1)),
vXCt.reshape((1, -1)))
else:
Exct = self.xc.get_smooth_xc_potential_and_energy_1d(vXCt)
# Calculate extra-stuff for non-local functionals
self.xc.get_extra_setup_data(extra_xc_data)
vt = vHt + vXCt
# Construct zero potential:
gc = 1 + int(rcutvbar * N / (rcutvbar + beta))
if vbar_type == 'f':
assert lmax == 2
uf = np.zeros(N)
l = 3
# Solve for all-electron f-state:
eps = 0.0
shoot(uf,
l,
self.vr,
eps,
self.r2dvdr,
r,
dr,
c10,
c2,
self.scalarrel,
gmax=gmax)
uf *= 1.0 / uf[gc]
# Fit smooth pseudo f-state polynomium:
A = np.ones((4, 4))
A[:, 0] = 1.0
A[:, 1] = r[gc - 2:gc + 2]**2
A[:, 2] = A[:, 1]**2
A[:, 3] = A[:, 1] * A[:, 2]
a = uf[gc - 2:gc + 2] / r[gc - 2:gc + 2]**(l + 1)
a0, a1, a2, a3 = solve(A, a)
r1 = r[:gc]
r2 = r1**2
rl1 = r1**(l + 1)
y = a0 + r2 * (a1 + r2 * (a2 + r2 * a3))
sf = uf.copy()
sf[:gc] = rl1 * y
# From 0 to gc, use analytic formula for kinetic energy operator:
r4 = r2**2
r6 = r4 * r2
enumerator = (a0 * l * (l + 1) + a1 * (l + 2) * (l + 3) * r2 + a2 *
(l + 4) * (l + 5) * r4 + a3 * (l + 6) * (l + 7) * r6)
denominator = a0 + a1 * r2 + a2 * r4 + a3 * r6
ekin_over_phit = -0.5 * (enumerator / denominator - l * (l + 1))
ekin_over_phit[1:] /= r2[1:]
vbar = eps - vt
vbar[:gc] -= ekin_over_phit
vbar[0] = vbar[1] # Actually we can collect the terms into
# a single fraction without poles, so as to avoid doing this,
# but this is good enough
# From gc to gmax, use finite-difference formula for kinetic
# energy operator:
vbar[gc:gmax] -= self.kin(l, sf)[gc:gmax] / sf[gc:gmax]
vbar[gmax:] = 0.0
else:
assert vbar_type == 'poly'
A = np.ones((2, 2))
A[0] = 1.0
A[1] = r[gc - 1:gc + 1]**2
a = vt[gc - 1:gc + 1]
a = solve(np.transpose(A), a)
r2 = r**2
vbar = a[0] + r2 * a[1] - vt
vbar[gc:] = 0.0
vt += vbar
# Construct projector functions:
for l, (e_n, s_n, q_n) in enumerate(zip(e_ln, s_ln, q_ln)):
for e, s, q in zip(e_n, s_n, q_n):
q[:] = self.kin(l, s) + (vt - e) * s
q[gcutmax:] = 0.0
filter = Filter(r, dr, gcutfilter, hfilter).filter
vbar = filter(vbar * r)
# Calculate matrix elements:
self.dK_lnn = dK_lnn = []
self.dH_lnn = dH_lnn = []
self.dO_lnn = dO_lnn = []
for l, (e_n, u_n, s_n, q_n) in enumerate(zip(e_ln, u_ln, s_ln, q_ln)):
A_nn = np.inner(s_n, q_n * dr)
# Do a LU decomposition of A:
nn = len(e_n)
L_nn = np.identity(nn, float)
U_nn = A_nn.copy()
# Keep all bound states normalized
if sum([n > 0 for n in n_ln[l]]) <= 1:
for i in range(nn):
for j in range(i + 1, nn):
L_nn[j, i] = 1.0 * U_nn[j, i] / U_nn[i, i]
U_nn[j, :] -= U_nn[i, :] * L_nn[j, i]
dO_nn = (np.inner(u_n, u_n * dr) - np.inner(s_n, s_n * dr))
e_nn = np.zeros((nn, nn))
e_nn.ravel()[::nn + 1] = e_n
dH_nn = np.dot(dO_nn, e_nn) - A_nn
q_n[:] = np.dot(inv(np.transpose(U_nn)), q_n)
s_n[:] = np.dot(inv(L_nn), s_n)
u_n[:] = np.dot(inv(L_nn), u_n)
dO_nn = np.dot(np.dot(inv(L_nn), dO_nn), inv(np.transpose(L_nn)))
dH_nn = np.dot(np.dot(inv(L_nn), dH_nn), inv(np.transpose(L_nn)))
ku_n = [self.kin(l, u, e) for u, e in zip(u_n, e_n)]
ks_n = [self.kin(l, s) for s in s_n]
dK_nn = 0.5 * (np.inner(u_n, ku_n * dr) - np.inner(s_n, ks_n * dr))
dK_nn += np.transpose(dK_nn).copy()
dK_lnn.append(dK_nn)
dO_lnn.append(dO_nn)
dH_lnn.append(dH_nn)
for n, q in enumerate(q_n):
q[:] = filter(q, l) * r**(l + 1)
A_nn = np.inner(s_n, q_n * dr)
q_n[:] = np.dot(inv(np.transpose(A_nn)), q_n)
self.vt = vt
self.vbar = vbar
t('state eigenvalue norm')
t('--------------------------------')
for l, (n_n, f_n, e_n) in enumerate(zip(n_ln, f_ln, e_ln)):
for n in range(len(e_n)):
if n_n[n] > 0:
f = '(%d)' % f_n[n]
t('%d%s%-4s: %12.6f %12.6f' %
(n_n[n], 'spdf'[l], f, e_n[n], np.dot(s_ln[l][n]**2,
dr)))
else:
t('*%s : %12.6f' % ('spdf'[l], e_n[n]))
t('--------------------------------')
self.logd = {}
if logderiv:
ni = 300
self.elog = np.linspace(-5.0, 1.0, ni)
# Calculate logarithmic derivatives:
gld = gcutmax + 10
self.rlog = r[gld]
assert gld < gmax
t('Calculating logarithmic derivatives at r=%.3f' % r[gld])
t('(skip with [Ctrl-C])')
try:
u = np.zeros(N)
for l in range(4):
self.logd[l] = (np.empty(ni), np.empty(ni))
if l <= lmax:
dO_nn = dO_lnn[l]
dH_nn = dH_lnn[l]
q_n = q_ln[l]
fae = open(self.symbol + '.ae.ld.' + 'spdf'[l], 'w')
fps = open(self.symbol + '.ps.ld.' + 'spdf'[l], 'w')
for i, e in enumerate(self.elog):
# All-electron logarithmic derivative:
u[:] = 0.0
shoot(u,
l,
self.vr,
e,
self.r2dvdr,
r,
dr,
c10,
c2,
self.scalarrel,
gmax=gld)
dudr = 0.5 * (u[gld + 1] - u[gld - 1]) / dr[gld]
ld = dudr / u[gld] - 1.0 / r[gld]
print >> fae, e, ld
self.logd[l][0][i] = ld
# PAW logarithmic derivative:
s = self.integrate(l, vt, e, gld)
if l <= lmax:
A_nn = dH_nn - e * dO_nn
s_n = [
self.integrate(l, vt, e, gld, q) for q in q_n
]
B_nn = np.inner(q_n, s_n * dr)
a_n = np.dot(q_n, s * dr)
B_nn = np.dot(A_nn, B_nn)
B_nn.ravel()[::len(a_n) + 1] += 1.0
c_n = solve(B_nn, np.dot(A_nn, a_n))
s -= np.dot(c_n, s_n)
dsdr = 0.5 * (s[gld + 1] - s[gld - 1]) / dr[gld]
ld = dsdr / s[gld] - 1.0 / r[gld]
print >> fps, e, ld
self.logd[l][1][i] = ld
except KeyboardInterrupt:
pass
self.write(nc, 'nc')
self.write(nt, 'nt')
self.write(nct, 'nct')
self.write(vbar, 'vbar')
self.write(vt, 'vt')
self.write(tauc, 'tauc')
self.write(tauct, 'tauct')
for l, (n_n, f_n, u_n, s_n,
q_n) in enumerate(zip(n_ln, f_ln, u_ln, s_ln, q_ln)):
for n, f, u, s, q in zip(n_n, f_n, u_n, s_n, q_n):
if n < 0:
self.write(u, 'ae', n=n, l=l)
self.write(s, 'ps', n=n, l=l)
self.write(q, 'proj', n=n, l=l)
# Test for ghost states:
for h in [0.05]:
self.diagonalize(h)
self.vn_j = vn_j = []
self.vl_j = vl_j = []
self.vf_j = vf_j = []
self.ve_j = ve_j = []
self.vu_j = vu_j = []
self.vs_j = vs_j = []
self.vq_j = vq_j = []
j_ln = [[0 for f in f_n] for f_n in f_ln]
j = 0
for l, n_n in enumerate(n_ln):
for n, nn in enumerate(n_n):
if nn > 0:
vf_j.append(f_ln[l][n])
vn_j.append(nn)
vl_j.append(l)
ve_j.append(e_ln[l][n])
vu_j.append(u_ln[l][n])
vs_j.append(s_ln[l][n])
vq_j.append(q_ln[l][n])
j_ln[l][n] = j
j += 1
for l, n_n in enumerate(n_ln):
for n, nn in enumerate(n_n):
if nn < 0:
vf_j.append(0)
vn_j.append(nn)
vl_j.append(l)
ve_j.append(e_ln[l][n])
vu_j.append(u_ln[l][n])
vs_j.append(s_ln[l][n])
vq_j.append(q_ln[l][n])
j_ln[l][n] = j
j += 1
nj = j
self.dK_jj = np.zeros((nj, nj))
for l, j_n in enumerate(j_ln):
for n1, j1 in enumerate(j_n):
for n2, j2 in enumerate(j_n):
self.dK_jj[j1, j2] = self.dK_lnn[l][n1, n2]
if exx:
X_p = constructX(self)
ExxC = atomic_exact_exchange(self, 'core-core')
else:
X_p = None
ExxC = None
sqrt4pi = sqrt(4 * pi)
setup = SetupData(self.symbol, self.xc.name, self.name, readxml=False)
def divide_by_r(x_g, l):
r = self.r
#for x_g, l in zip(x_jg, l_j):
p = x_g.copy()
p[1:] /= self.r[1:]
# XXXXX go to higher order!!!!!
if l == 0: #l_j[self.jcorehole] == 0:
p[0] = (p[2] + (p[1] - p[2]) * (r[0] - r[2]) / (r[1] - r[2]))
return p
def divide_all_by_r(x_jg):
return [divide_by_r(x_g, l) for x_g, l in zip(x_jg, vl_j)]
setup.l_j = vl_j
setup.n_j = vn_j
setup.f_j = vf_j
setup.eps_j = ve_j
setup.rcut_j = [rcut_l[l] for l in vl_j]
setup.nc_g = nc * sqrt4pi
setup.nct_g = nct * sqrt4pi
setup.nvt_g = (nt - nct) * sqrt4pi
setup.e_kinetic_core = Ekincore
setup.vbar_g = vbar * sqrt4pi
setup.tauc_g = tauc * sqrt4pi
setup.tauct_g = tauct * sqrt4pi
setup.extra_xc_data = extra_xc_data
setup.Z = Z
setup.Nc = self.Nc
setup.Nv = self.Nv
setup.e_kinetic = self.Ekin
setup.e_xc = self.Exc
setup.e_electrostatic = self.Epot
setup.e_total = self.Epot + self.Exc + self.Ekin
setup.beta = self.beta
setup.ng = self.N
setup.rcgauss = self.rcutcomp / sqrt(self.gamma)
setup.e_kin_jj = self.dK_jj
setup.ExxC = ExxC
setup.phi_jg = divide_all_by_r(vu_j)
setup.phit_jg = divide_all_by_r(vs_j)
setup.pt_jg = divide_all_by_r(vq_j)
setup.X_p = X_p
if self.jcorehole is not None:
setup.has_corehole = True
setup.lcorehole = l_j[self.jcorehole] # l_j or vl_j ????? XXX
setup.ncorehole = n_j[self.jcorehole]
setup.phicorehole_g = divide_by_r(self.u_j[self.jcorehole],
setup.lcorehole)
setup.core_hole_e = self.e_j[self.jcorehole]
setup.core_hole_e_kin = self.Ekincorehole
setup.fcorehole = self.fcorehole
if self.ghost:
raise RuntimeError('Ghost!')
if self.scalarrel:
reltype = 'scalar-relativistic'
else:
reltype = 'non-relativistic'
attrs = [('type', reltype), ('name', 'gpaw-%s' % version)]
data = 'Frozen core: ' + (self.core or 'none')
setup.generatorattrs = attrs
setup.generatordata = data
self.id_j = []
for l, n in zip(vl_j, vn_j):
if n > 0:
id = '%s-%d%s' % (self.symbol, n, 'spdf'[l])
else:
id = '%s-%s%d' % (self.symbol, 'spdf'[l], -n)
self.id_j.append(id)
setup.id_j = self.id_j
if write_xml:
setup.write_xml()
return setup
def calculate(self):
ESIC = 0
xc = self.paw.hamiltonian.xc
assert xc.type == 'LDA'
# Calculate the contribution from the core orbitals
for a in self.paw.density.D_asp:
setup = self.paw.density.setups[a]
# TODO: Use XC which has been used to calculate the actual
# calculation.
# TODO: Loop over setups, not atoms.
print('Atom core SIC for ', setup.symbol)
print('%10s%10s%10s' % ('E_xc[n_i]', 'E_Ha[n_i]', 'E_SIC'))
g = Generator(setup.symbol, xcname='LDA', nofiles=True, txt=None)
g.run(**parameters[setup.symbol])
njcore = g.njcore
for f, l, e, u in zip(g.f_j[:njcore], g.l_j[:njcore],
g.e_j[:njcore], g.u_j[:njcore]):
# Calculate orbital density
# NOTE: It's spherically symmetrized!
#n = np.dot(self.f_j,
assert l == 0, ('Not tested for l>0 core states')
na = np.where(abs(u) < 1e-160, 0, u)**2 / (4 * pi)
na[1:] /= g.r[1:]**2
na[0] = na[1]
nb = np.zeros(g.N)
v_sg = np.zeros((2, g.N))
e_g = np.zeros(g.N)
vHr = np.zeros(g.N)
Exc = xc.calculate_spherical(g.rgd, np.array([na, nb]), v_sg)
hartree(0, na * g.r * g.dr, g.r, vHr)
EHa = 2 * pi * np.dot(vHr * na * g.r, g.dr)
print(
('%10.2f%10.2f%10.2f' % (Exc * Hartree, EHa * Hartree, -f *
(EHa + Exc) * Hartree)))
ESIC += -f * (EHa + Exc)
sic = SIC(finegrid=True, coulomb_factor=1, xc_factor=1)
sic.initialize(self.paw.density, self.paw.hamiltonian, self.paw.wfs)
sic.set_positions(self.paw.atoms.get_scaled_positions())
print('Valence electron sic ')
print('%10s%10s%10s%10s%10s%10s' %
('spin', 'k-point', 'band', 'E_xc[n_i]', 'E_Ha[n_i]', 'E_SIC'))
assert len(
self.paw.wfs.kpt_u) == 1, ('Not tested for bulk calculations')
for s, spin in sic.spin_s.items():
spin.initialize_orbitals()
spin.update_optimal_states()
spin.update_potentials()
n = 0
for xc, c in zip(spin.exc_m, spin.ecoulomb_m):
print(('%10i%10i%10i%10.2f%10.2f%10.2f' %
(s, 0, n, -xc * Hartree, -c * Hartree, 2 *
(xc + c) * Hartree)))
n += 1
ESIC += spin.esic
print('Total correction for self-interaction energy:')
print('%10.2f eV' % (ESIC * Hartree))
print('New total energy:')
total = (ESIC * Hartree + self.paw.get_potential_energy() +
self.paw.get_reference_energy())
print('%10.2f eV' % total)
return total
def constructX(gen):
"""Construct the X_p^a matrix for the given atom.
The X_p^a matrix describes the valence-core interactions of the
partial waves.
"""
# initialize attributes
uv_j = gen.vu_j # soft valence states * r:
lv_j = gen.vl_j # their repective l quantum numbers
Nvi = 0
for l in lv_j:
Nvi += 2 * l + 1 # total number of valence states (including m)
# number of core and valence orbitals (j only, i.e. not m-number)
Njcore = gen.njcore
Njval = len(lv_j)
# core states * r:
uc_j = gen.u_j[:Njcore]
r, dr, N, beta = gen.r, gen.dr, gen.N, gen.beta
# potential times radius
vr = np.zeros(N)
# initialize X_ii matrix
X_ii = np.zeros((Nvi, Nvi))
# make gaunt coeff. list
lmax = max(gen.l_j[:Njcore] + gen.vl_j)
gaunt = make_gaunt(lmax=lmax)
# sum over core states
for jc in range(Njcore):
lc = gen.l_j[jc]
# sum over first valence state index
i1 = 0
for jv1 in range(Njval):
lv1 = lv_j[jv1]
# electron density 1 times radius times length element
n1c = uv_j[jv1] * uc_j[jc] * dr
n1c[1:] /= r[1:]
# sum over second valence state index
i2 = 0
for jv2 in range(Njval):
lv2 = lv_j[jv2]
# electron density 2
n2c = uv_j[jv2] * uc_j[jc] * dr
n2c[1:] /= r[1:]
# sum expansion in angular momenta
for l in range(min(lv1, lv2) + lc + 1):
# Int density * potential * r^2 * dr:
hartree(l, n2c, r, vr)
nv = np.dot(n1c, vr)
# expansion coefficients
A_mm = X_ii[i1:i1 + 2 * lv1 + 1, i2:i2 + 2 * lv2 + 1]
for mc in range(2 * lc + 1):
for m in range(2 * l + 1):
G1c = gaunt[lv1**2:(lv1 + 1)**2, lc**2 + mc,
l**2 + m]
G2c = gaunt[lv2**2:(lv2 + 1)**2, lc**2 + mc,
l**2 + m]
A_mm += nv * np.outer(G1c, G2c)
i2 += 2 * lv2 + 1
i1 += 2 * lv1 + 1
# pack X_ii matrix
X_p = pack2(X_ii)
return X_p
def run(self, use_restart_file=True):
# beta g
# r = ------, g = 0, 1, ..., N - 1
# N - g
#
# rN
# g = --------
# beta + r
t = self.text
N = self.N
beta = self.beta
t(N, 'radial gridpoints.')
self.rgd = AERadialGridDescriptor(beta / N, 1.0 / N, N)
g = np.arange(N, dtype=float)
self.r = self.rgd.r_g
self.dr = self.rgd.dr_g
self.d2gdr2 = self.rgd.d2gdr2()
# Number of orbitals:
nj = len(self.n_j)
# Radial wave functions multiplied by radius:
self.u_j = np.zeros((nj, self.N))
# Effective potential multiplied by radius:
self.vr = np.zeros(N)
# Electron density:
self.n = np.zeros(N)
# Always spinpaired nspins=1
self.xc = XC(self.xcname)
# Initialize for non-local functionals
if self.xc.type == 'GLLB':
self.xc.pass_stuff_1d(self)
self.xc.initialize_1d()
n_j = self.n_j
l_j = self.l_j
f_j = self.f_j
e_j = self.e_j
Z = self.Z # nuclear charge
r = self.r # radial coordinate
dr = self.dr # dr/dg
n = self.n # electron density
vr = self.vr # effective potential multiplied by r
vHr = np.zeros(self.N)
self.vXC = np.zeros(self.N)
restartfile = '%s/%s.restart' % (tempdir, self.symbol)
if self.xc.type == 'GLLB' or not use_restart_file:
# Do not start from initial guess when doing
# non local XC!
# This is because we need wavefunctions as well
# on the first iteration.
fd = None
else:
try:
fd = open(restartfile, 'rb')
except IOError:
fd = None
else:
try:
n[:] = pickle.load(fd)
except (ValueError, IndexError):
fd = None
else:
norm = np.dot(n * r**2, dr) * 4 * pi
if abs(norm - sum(f_j)) > 0.01:
fd = None
else:
t('Using old density for initial guess.')
n *= sum(f_j) / norm
if fd is None:
self.initialize_wave_functions()
n[:] = self.calculate_density()
bar = '|------------------------------------------------|'
t(bar)
niter = 0
nitermax = 117
qOK = log(1e-10)
mix = 0.4
# orbital_free needs more iterations and coefficient
if self.orbital_free:
mix = 0.01
nitermax = 2000
e_j[0] /= self.tw_coeff
if Z > 10: #help convergence for third row elements
mix = 0.002
nitermax = 10000
vrold = None
while True:
# calculate hartree potential
hartree(0, n * r * dr, r, vHr)
# add potential from nuclear point charge (v = -Z / r)
vHr -= Z
# calculated exchange correlation potential and energy
self.vXC[:] = 0.0
if self.xc.type == 'GLLB':
# Update the potential to self.vXC an the energy to self.Exc
Exc = self.xc.get_xc_potential_and_energy_1d(self.vXC)
else:
Exc = self.xc.calculate_spherical(self.rgd, n.reshape((1, -1)),
self.vXC.reshape((1, -1)))
# calculate new total Kohn-Sham effective potential and
# admix with old version
vr[:] = (vHr + self.vXC * r)
if self.orbital_free:
vr /= self.tw_coeff
if niter > 0:
vr[:] = mix * vr + (1 - mix) * vrold
vrold = vr.copy()
# solve Kohn-Sham equation and determine the density change
self.solve()
dn = self.calculate_density() - n
n += dn
# estimate error from the square of the density change integrated
q = log(np.sum((r * dn)**2))
# print progress bar
if niter == 0:
q0 = q
b0 = 0
else:
b = int((q0 - q) / (q0 - qOK) * 50)
if b > b0:
self.txt.write(bar[b0:min(b, 50)])
self.txt.flush()
b0 = b
# check if converged and break loop if so
if q < qOK:
self.txt.write(bar[b0:])
self.txt.flush()
break
niter += 1
if niter > nitermax:
raise RuntimeError('Did not converge!')
tau = self.calculate_kinetic_energy_density()
t()
t('Converged in %d iteration%s.' % (niter, 's'[:niter != 1]))
try:
fd = open(restartfile, 'wb')
except IOError:
pass
else:
pickle.dump(n, fd)
try:
os.chmod(restartfile, 0o666)
except OSError:
pass
Ekin = 0
for f, e in zip(f_j, e_j):
Ekin += f * e
e_coulomb = 2 * pi * np.dot(n * r * (vHr - Z), dr)
Ekin += -4 * pi * np.dot(n * vr * r, dr)
if self.orbital_free:
# e and vr are not scaled back
# instead Ekin is scaled for total energy (printed and inside setup)
Ekin *= self.tw_coeff
t()
t('Lambda:{0}'.format(self.tw_coeff))
t('Correct eigenvalue:{0}'.format(e_j[0] * self.tw_coeff))
t()
t()
t('Energy contributions:')
t('-------------------------')
t('Kinetic: %+13.6f' % Ekin)
t('XC: %+13.6f' % Exc)
t('Potential: %+13.6f' % e_coulomb)
t('-------------------------')
t('Total: %+13.6f' % (Ekin + Exc + e_coulomb))
self.ETotal = Ekin + Exc + e_coulomb
t()
t('state eigenvalue ekin rmax')
t('-----------------------------------------------')
for m, l, f, e, u in zip(n_j, l_j, f_j, e_j, self.u_j):
# Find kinetic energy:
k = e - np.sum((
np.where(abs(u) < 1e-160, 0, u)**2 * # XXXNumeric!
vr * dr)[1:] / r[1:])
# Find outermost maximum:
g = self.N - 4
while u[g - 1] >= u[g]:
g -= 1
x = r[g - 1:g + 2]
y = u[g - 1:g + 2]
A = np.transpose(np.array([x**i for i in range(3)]))
c, b, a = np.linalg.solve(A, y)
assert a < 0.0
rmax = -0.5 * b / a
s = 'spdf'[l]
t('%d%s^%-4.1f: %12.6f %12.6f %12.3f' % (m, s, f, e, k, rmax))
t('-----------------------------------------------')
t('(units: Bohr and Hartree)')
for m, l, u in zip(n_j, l_j, self.u_j):
self.write(u, 'ae', n=m, l=l)
self.write(n, 'n')
self.write(vr, 'vr')
self.write(vHr, 'vHr')
self.write(self.vXC, 'vXC')
self.write(tau, 'tau')
self.Ekin = Ekin
self.e_coulomb = e_coulomb
self.Exc = Exc
def calculate(self):
ESIC = 0
xc = self.paw.hamiltonian.xc
assert xc.type == 'LDA'
# Calculate the contribution from the core orbitals
for a in self.paw.density.D_asp:
setup = self.paw.density.setups[a]
# TODO: Use XC which has been used to calculate the actual
# calculation.
# TODO: Loop over setups, not atoms.
print('Atom core SIC for ', setup.symbol)
print('%10s%10s%10s' % ('E_xc[n_i]', 'E_Ha[n_i]', 'E_SIC'))
g = Generator(setup.symbol, xcname='LDA', nofiles=True, txt=None)
g.run(**parameters[setup.symbol])
njcore = g.njcore
for f, l, e, u in zip(g.f_j[:njcore], g.l_j[:njcore],
g.e_j[:njcore], g.u_j[:njcore]):
# Calculate orbital density
# NOTE: It's spherically symmetrized!
#n = np.dot(self.f_j,
assert l == 0, ('Not tested for l>0 core states')
na = np.where(abs(u) < 1e-160, 0,u)**2 / (4 * pi)
na[1:] /= g.r[1:]**2
na[0] = na[1]
nb = np.zeros(g.N)
v_sg = np.zeros((2, g.N))
e_g = np.zeros(g.N)
vHr = np.zeros(g.N)
Exc = xc.calculate_spherical(g.rgd, np.array([na, nb]), v_sg)
hartree(0, na * g.r * g.dr, g.r, vHr)
EHa = 2*pi*np.dot(vHr*na*g.r , g.dr)
print(('%10.2f%10.2f%10.2f' % (Exc * Hartree, EHa * Hartree,
-f*(EHa+Exc) * Hartree)))
ESIC += -f*(EHa+Exc)
sic = SIC(finegrid=True, coulomb_factor=1, xc_factor=1)
sic.initialize(self.paw.density, self.paw.hamiltonian, self.paw.wfs)
sic.set_positions(self.paw.atoms.get_scaled_positions())
print('Valence electron sic ')
print('%10s%10s%10s%10s%10s%10s' % ('spin', 'k-point', 'band',
'E_xc[n_i]', 'E_Ha[n_i]', 'E_SIC'))
assert len(self.paw.wfs.kpt_u)==1, ('Not tested for bulk calculations')
for s, spin in sic.spin_s.items():
spin.initialize_orbitals()
spin.update_optimal_states()
spin.update_potentials()
n = 0
for xc, c in zip(spin.exc_m, spin.ecoulomb_m):
print(('%10i%10i%10i%10.2f%10.2f%10.2f' %
(s, 0, n, -xc * Hartree, -c * Hartree,
2 * (xc + c) * Hartree)))
n += 1
ESIC += spin.esic
print('Total correction for self-interaction energy:')
print('%10.2f eV' % (ESIC * Hartree))
print('New total energy:')
total = (ESIC * Hartree + self.paw.get_potential_energy() +
self.paw.get_reference_energy())
print('%10.2f eV' % total)
return total
def run(self, use_restart_file=True):
# beta g
# r = ------, g = 0, 1, ..., N - 1
# N - g
#
# rN
# g = --------
# beta + r
t = self.text
N = self.N
beta = self.beta
t(N, 'radial gridpoints.')
self.rgd = AERadialGridDescriptor(beta / N, 1.0 / N, N)
g = np.arange(N, dtype=float)
self.r = self.rgd.r_g
self.dr = self.rgd.dr_g
self.d2gdr2 = self.rgd.d2gdr2()
# Number of orbitals:
nj = len(self.n_j)
# Radial wave functions multiplied by radius:
self.u_j = np.zeros((nj, self.N))
# Effective potential multiplied by radius:
self.vr = np.zeros(N)
# Electron density:
self.n = np.zeros(N)
# Always spinpaired nspins=1
self.xc = XC(self.xcname)
# Initialize for non-local functionals
if self.xc.type == 'GLLB':
self.xc.pass_stuff_1d(self)
self.xc.initialize_1d()
n_j = self.n_j
l_j = self.l_j
f_j = self.f_j
e_j = self.e_j
Z = self.Z # nuclear charge
r = self.r # radial coordinate
dr = self.dr # dr/dg
n = self.n # electron density
vr = self.vr # effective potential multiplied by r
vHr = np.zeros(self.N)
self.vXC = np.zeros(self.N)
restartfile = '%s/%s.restart' % (tempdir, self.symbol)
if self.xc.type == 'GLLB' or not use_restart_file:
# Do not start from initial guess when doing
# non local XC!
# This is because we need wavefunctions as well
# on the first iteration.
fd = None
else:
try:
fd = open(restartfile, 'r')
except IOError:
fd = None
else:
try:
n[:] = pickle.load(fd)
except (ValueError, IndexError):
fd = None
else:
norm = np.dot(n * r**2, dr) * 4 * pi
if abs(norm - sum(f_j)) > 0.01:
fd = None
else:
t('Using old density for initial guess.')
n *= sum(f_j) / norm
if fd is None:
self.initialize_wave_functions()
n[:] = self.calculate_density()
bar = '|------------------------------------------------|'
t(bar)
niter = 0
nitermax = 117
qOK = log(1e-10)
mix = 0.4
# orbital_free needs more iterations and coefficient
if self.orbital_free:
#qOK = log(1e-14)
e_j[0] /= self.tf_coeff
mix = 0.01
nitermax = 1000
vrold = None
while True:
# calculate hartree potential
hartree(0, n * r * dr, r, vHr)
# add potential from nuclear point charge (v = -Z / r)
vHr -= Z
# calculated exchange correlation potential and energy
self.vXC[:] = 0.0
if self.xc.type == 'GLLB':
# Update the potential to self.vXC an the energy to self.Exc
Exc = self.xc.get_xc_potential_and_energy_1d(self.vXC)
else:
Exc = self.xc.calculate_spherical(self.rgd,
n.reshape((1, -1)),
self.vXC.reshape((1, -1)))
# calculate new total Kohn-Sham effective potential and
# admix with old version
vr[:] = (vHr + self.vXC * r) / self.tf_coeff
if niter > 0:
vr[:] = mix * vr + (1 - mix) * vrold
vrold = vr.copy()
# solve Kohn-Sham equation and determine the density change
self.solve()
dn = self.calculate_density() - n
n += dn
# estimate error from the square of the density change integrated
q = log(np.sum((r * dn)**2))
# print progress bar
if niter == 0:
q0 = q
b0 = 0
else:
b = int((q0 - q) / (q0 - qOK) * 50)
if b > b0:
self.txt.write(bar[b0:min(b, 50)])
self.txt.flush()
b0 = b
# check if converged and break loop if so
if q < qOK:
self.txt.write(bar[b0:])
self.txt.flush()
break
niter += 1
if niter > nitermax:
raise RuntimeError('Did not converge!')
tau = self.calculate_kinetic_energy_density()
t()
t('Converged in %d iteration%s.' % (niter, 's'[:niter != 1]))
try:
fd = open(restartfile, 'w')
except IOError:
pass
else:
pickle.dump(n, fd)
try:
os.chmod(restartfile, 0666)
except OSError:
pass
Ekin = 0
if self.orbital_free:
e_j[0] *= self.tf_coeff
vr *= self.tf_coeff
for f, e in zip(f_j, e_j):
Ekin += f * e
Epot = 2 * pi * np.dot(n * r * (vHr - Z), dr)
Ekin += -4 * pi * np.dot(n * vr * r, dr)
t()
t('Energy contributions:')
t('-------------------------')
t('Kinetic: %+13.6f' % Ekin)
t('XC: %+13.6f' % Exc)
t('Potential: %+13.6f' % Epot)
t('-------------------------')
t('Total: %+13.6f' % (Ekin + Exc + Epot))
self.ETotal = Ekin + Exc + Epot
t()
t('state eigenvalue ekin rmax')
t('-----------------------------------------------')
for m, l, f, e, u in zip(n_j, l_j, f_j, e_j, self.u_j):
# Find kinetic energy:
k = e - np.sum((np.where(abs(u) < 1e-160, 0, u)**2 * # XXXNumeric!
vr * dr)[1:] / r[1:])
# Find outermost maximum:
g = self.N - 4
while u[g - 1] >= u[g]:
g -= 1
x = r[g - 1:g + 2]
y = u[g - 1:g + 2]
A = np.transpose(np.array([x**i for i in range(3)]))
c, b, a = np.linalg.solve(A, y)
assert a < 0.0
rmax = -0.5 * b / a
s = 'spdf'[l]
t('%d%s^%-4.1f: %12.6f %12.6f %12.3f' % (m, s, f, e, k, rmax))
t('-----------------------------------------------')
t('(units: Bohr and Hartree)')
for m, l, u in zip(n_j, l_j, self.u_j):
self.write(u, 'ae', n=m, l=l)
self.write(n, 'n')
self.write(vr, 'vr')
self.write(vHr, 'vHr')
self.write(self.vXC, 'vXC')
self.write(tau, 'tau')
self.Ekin = Ekin
self.Epot = Epot
self.Exc = Exc
def get_spherical_ks_potential(self,a):
#self.paw.restore_state()
print "XC:", self.paw.hamiltonian.xc.name
assert self.paw.hamiltonian.xc.type == 'LDA'
# If the calculation is just loaded, density needs to be interpolated
if self.paw.density.nt_sg is None:
print "Interpolating density"
self.paw.density.interpolate()
# Get xccorr for atom a
setup = self.paw.density.setups[a]
xccorr = setup.xc_correction
# Get D_sp for atom a
D_sp = self.paw.density.D_asp[a]
# density a function of L and partial wave radial pair density coefficient
D_sLq = np.inner(D_sp, xccorr.B_pqL.T)
# The 'spherical' spherical harmonic
Y0 = 1.0/sqrt(4*pi)
# Generate cartesian fine grid xc-potential
print "Generate cartesian fine grid xc-potential"
gd = self.paw.density.finegd
vxct_sg = gd.zeros(1)
xc = self.paw.hamiltonian.xc
xc.calculate(gd, self.paw.density.nt_sg, vxct_sg)
# ---------------------------------------------
# The Electrostatic potential
# ---------------------------------------------
# V_ES(r) = Vt_ES(r) - Vt^a(r) + V^a(r), where
# Vt^a = P[ nt^a(r) + \sum Q_L g_L(r) ]
# V^a = P[ -Z\delta(r) + n^a(r) ]
# P = Poisson solution
# ---------------------------------------------
print "Evaluating ES Potential..."
# Make sure that the calculation has ES potential
# TODO
if self.paw.hamiltonian.vHt_g == None:
raise "Converge tha Hartree potential first."
# Interpolate the smooth electrostatic potential from fine grid to radial grid
radHt_g = self.grid_to_radial(a, gd, self.paw.hamiltonian.vHt_g)
radHt_g *= xccorr.rgd.r_g
print "D_sp", D_sp
# Calculate the difference in density and pseudo density
dn_g = (Y0 * np.dot(D_sLq[0, 0], (xccorr.n_qg - xccorr.nt_qg)) +
xccorr.nc_g - xccorr.nct_g)
# Add the compensation charge contribution
dn_g -= Y0 * self.paw.density.Q_aL[a][0] * setup.g_lg[0]
# Calculate the Hartree potential for this
vHr = np.zeros((xccorr.ng,))
hartree(0, dn_g * xccorr.rgd.r_g * xccorr.rgd.dr_g, setup.beta, setup.ng, vHr)
# Add the core potential contribution
vHr -= setup.Z
radHt_g += vHr
# --------------------------------------------
# The XC potential
# --------------------------------------------
# V_xc = Vt_xc(r) - Vt_xc^a(r) + V_xc^a(r)
# --------------------------------------------
print "Evaluating xc potential"
# Interpolate the smooth xc potential from fine grid to radial grid
radvxct_g = self.grid_to_radial(a, gd, vxct_sg[0])
# Arrays for evaluating radial xc potential slice
e_g = np.zeros((xccorr.ng,))
vxc_sg = np.zeros((len(D_sp), xccorr.ng))
for n, Y_L in enumerate(xccorr.Y_nL):
n_sLg = np.dot(D_sLq, xccorr.n_qg)
n_sLg[:, 0] += sqrt(4 * pi) * xccorr.nc_g
vxc_sg[:] = 0.0
xc.calculate_radial(xccorr.rgd, n_sLg, Y_L, vxc_sg)
radvxct_g += weight_n[n] * vxc_sg[0]
nt_sLg = np.dot(D_sLq, xccorr.nt_qg)
nt_sLg[:, 0] += sqrt(4 * pi) *xccorr.nct_g
vxc_sg[:] = 0.0
xc.calculate_radial(xccorr.rgd, nt_sLg, Y_L, vxc_sg)
radvxct_g -= weight_n[n] * vxc_sg[0]
radvks_g = radvxct_g*xccorr.rgd.r_g + radHt_g
return (xccorr.rgd.r_g, radvks_g)
def H(n_g, l):
yrrdr_g = np.zeros(gcut2)
nrdr_g = n_g * r_g * dr_g
hartree(l, nrdr_g, beta, ng, yrrdr_g)
yrrdr_g *= r_g * dr_g
return yrrdr_g
def poisson(self, n_g, l=0):
vr_g = self.zeros()
nrdr_g = n_g * self.r_g * self.dr_g
hartree(l, nrdr_g, self.r_g, vr_g)
return vr_g
def get_spherical_ks_potential(self, a):
#self.paw.restore_state()
print "XC:", self.paw.hamiltonian.xc.name
assert self.paw.hamiltonian.xc.type == 'LDA'
# If the calculation is just loaded, density needs to be interpolated
if self.paw.density.nt_sg is None:
print "Interpolating density"
self.paw.density.interpolate()
# Get xccorr for atom a
setup = self.paw.density.setups[a]
xccorr = setup.xc_correction
# Get D_sp for atom a
D_sp = self.paw.density.D_asp[a]
# density a function of L and partial wave radial pair density coefficient
D_sLq = np.inner(D_sp, xccorr.B_pqL.T)
# The 'spherical' spherical harmonic
Y0 = 1.0 / sqrt(4 * pi)
# Generate cartesian fine grid xc-potential
print "Generate cartesian fine grid xc-potential"
gd = self.paw.density.finegd
vxct_sg = gd.zeros(1)
xc = self.paw.hamiltonian.xc
xc.calculate(gd, self.paw.density.nt_sg, vxct_sg)
# ---------------------------------------------
# The Electrostatic potential
# ---------------------------------------------
# V_ES(r) = Vt_ES(r) - Vt^a(r) + V^a(r), where
# Vt^a = P[ nt^a(r) + \sum Q_L g_L(r) ]
# V^a = P[ -Z\delta(r) + n^a(r) ]
# P = Poisson solution
# ---------------------------------------------
print "Evaluating ES Potential..."
# Make sure that the calculation has ES potential
# TODO
if self.paw.hamiltonian.vHt_g == None:
raise "Converge tha Hartree potential first."
# Interpolate the smooth electrostatic potential from fine grid to radial grid
radHt_g = self.grid_to_radial(a, gd, self.paw.hamiltonian.vHt_g)
radHt_g *= xccorr.rgd.r_g
print "D_sp", D_sp
# Calculate the difference in density and pseudo density
dn_g = (Y0 * np.dot(D_sLq[0, 0], (xccorr.n_qg - xccorr.nt_qg)) +
xccorr.nc_g - xccorr.nct_g)
# Add the compensation charge contribution
dn_g -= Y0 * self.paw.density.Q_aL[a][0] * setup.g_lg[0]
# Calculate the Hartree potential for this
vHr = np.zeros((xccorr.ng, ))
hartree(0, dn_g * xccorr.rgd.r_g * xccorr.rgd.dr_g, setup.beta,
setup.ng, vHr)
# Add the core potential contribution
vHr -= setup.Z
radHt_g += vHr
# --------------------------------------------
# The XC potential
# --------------------------------------------
# V_xc = Vt_xc(r) - Vt_xc^a(r) + V_xc^a(r)
# --------------------------------------------
print "Evaluating xc potential"
# Interpolate the smooth xc potential from fine grid to radial grid
radvxct_g = self.grid_to_radial(a, gd, vxct_sg[0])
# Arrays for evaluating radial xc potential slice
e_g = np.zeros((xccorr.ng, ))
vxc_sg = np.zeros((len(D_sp), xccorr.ng))
for n, Y_L in enumerate(xccorr.Y_nL):
n_sLg = np.dot(D_sLq, xccorr.n_qg)
n_sLg[:, 0] += sqrt(4 * pi) * xccorr.nc_g
vxc_sg[:] = 0.0
xc.calculate_radial(xccorr.rgd, n_sLg, Y_L, vxc_sg)
radvxct_g += weight_n[n] * vxc_sg[0]
nt_sLg = np.dot(D_sLq, xccorr.nt_qg)
nt_sLg[:, 0] += sqrt(4 * pi) * xccorr.nct_g
vxc_sg[:] = 0.0
xc.calculate_radial(xccorr.rgd, nt_sLg, Y_L, vxc_sg)
radvxct_g -= weight_n[n] * vxc_sg[0]
radvks_g = radvxct_g * xccorr.rgd.r_g + radHt_g
return (xccorr.rgd.r_g, radvks_g)