def get_radial_potential(calc, a, ai):
"""Calculates dV/dr / r for the effective potential.
Below, f_g denotes dV/dr = minus the radial force"""
rgd = a.xc_correction.rgd
r_g = rgd.r_g
r_g[0] = 1.0e-12
dr_g = rgd.dr_g
Ns = calc.wfs.nspins
D_sp = calc.density.D_asp[ai]
B_pq = a.xc_correction.B_pqL[:, :, 0]
n_qg = a.xc_correction.n_qg
D_sq = np.dot(D_sp, B_pq)
n_sg = np.dot(D_sq, n_qg) / (4 * np.pi)**0.5
n_sg[:] += a.xc_correction.nc_g / Ns
# Coulomb force from nucleus
fc_g = a.Z / r_g**2
# Hartree force
rho_g = 4 * np.pi * r_g**2 * dr_g * np.sum(n_sg, axis=0)
fh_g = -np.array([np.sum(rho_g[:ig]) for ig in range(len(r_g))]) / r_g**2
# xc force
xc = XC(calc.get_xc_functional())
v_sg = np.zeros_like(n_sg)
xc.calculate_spherical(a.xc_correction.rgd, n_sg, v_sg)
fxc_sg = np.array([a.xc_correction.rgd.derivative(v_sg[s])
for s in range(Ns)])
fxc_g = np.sum(fxc_sg, axis=0) / Ns
# f_sg = np.tile(fc_g, (Ns, 1)) + np.tile(fh_g, (Ns, 1)) + fxc_sg
f_sg = np.tile(fc_g + fh_g + fxc_g, (Ns, 1))
return f_sg[:] / r_g
class AllElectronAtom:
def __init__(self, symbol, xc='LDA', spinpol=False, dirac=False,
configuration=None,
log=None):
"""All-electron calculation for spherically symmetric atom.
symbol: str (or int)
Chemical symbol (or atomic number).
xc: str
Name of XC-functional.
spinpol: bool
If true, do spin-polarized calculation. Default is spin-paired.
dirac: bool
Solve Dirac equation instead of Schrödinger equation.
configuration: list
Electronic configuration for symbol, format as in
gpaw.atom.configurations
log: stream
Text output."""
if isinstance(symbol, int):
symbol = chemical_symbols[symbol]
self.symbol = symbol
self.Z = atomic_numbers[symbol]
self.nspins = 1 + int(bool(spinpol))
self.dirac = bool(dirac)
if configuration is not None:
self.configuration = copy.deepcopy(configuration)
else:
self.configuration = None
self.scalar_relativistic = False
if isinstance(xc, str):
self.xc = XC(xc)
else:
self.xc = xc
self.fd = log or sys.stdout
self.vr_sg = None # potential * r
self.n_sg = 0.0 # density
self.rgd = None # radial grid descriptor
# Energies:
self.ekin = None
self.eeig = None
self.eH = None
self.eZ = None
self.channels = None
self.initialize_configuration(self.configuration)
self.log('Z: ', self.Z)
self.log('Name: ', atomic_names[self.Z])
self.log('Symbol: ', symbol)
self.log('XC-functional: ', self.xc.name)
self.log('Equation: ', ['Schrödinger', 'Dirac'][self.dirac])
self.method = 'Gaussian basis-set'
def log(self, *args, **kwargs):
print(file=self.fd, *args, **kwargs)
def initialize_configuration(self, configuration=None):
self.f_lsn = {}
if configuration is None:
configuration = configurations[self.symbol][1]
for n, l, f, e in configuration:
if l not in self.f_lsn:
self.f_lsn[l] = [[] for s in range(self.nspins)]
if self.nspins == 1:
self.f_lsn[l][0].append(f)
else:
# Use Hund's rule:
f0 = min(f, 2 * l + 1)
self.f_lsn[l][0].append(f0)
self.f_lsn[l][1].append(f - f0)
if 0:
n = 2 + len(self.f_lsn[2][0])
if self.f_lsn[0][0][n] == 2:
self.f_lsn[0][0][n] = 1
self.f_lsn[2][0][n - 3] += 1
def add(self, n, l, df=+1, s=None):
"""Add (remove) electrons."""
if s is None:
if self.nspins == 1:
s = 0
else:
self.add(n, l, 0.5 * df, 0)
self.add(n, l, 0.5 * df, 1)
return
if l not in self.f_lsn:
self.f_lsn[l] = [[] for x in range(self.nspins)]
f_n = self.f_lsn[l][s]
if len(f_n) < n - l:
f_n.extend([0] * (n - l - len(f_n)))
f_n[n - l - 1] += df
def initialize(self, ngpts=2000, rcut=50.0,
alpha1=0.01, alpha2=None, ngauss=50,
eps=1.0e-7):
"""Initialize basis sets and radial grid.
ngpts: int
Number of grid points for radial grid.
rcut: float
Cutoff for radial grid.
alpha1: float
Smallest exponent for gaussian.
alpha2: float
Largest exponent for gaussian.
ngauss: int
Number of gaussians.
eps: float
Cutoff for eigenvalues of overlap matrix."""
if alpha2 is None:
alpha2 = 50.0 * self.Z**2
# Use grid with r(0)=0, r(1)=a and r(ngpts)=rcut:
a = 1 / alpha2**0.5 / 20
b = (rcut - a * ngpts) / (rcut * ngpts)
b = 1 / round(1 / b)
self.rgd = AERadialGridDescriptor(a, b, ngpts)
self.log('Grid points: %d (%.5f, %.5f, %.5f, ..., %.3f, %.3f)' %
((self.rgd.N,) + tuple(self.rgd.r_g[[0, 1, 2, -2, -1]])))
# Distribute exponents between alpha1 and alpha2:
alpha_B = alpha1 * (alpha2 / alpha1)**np.linspace(0, 1, ngauss)
self.log('Exponents: %d (%.3f, %.3f, ..., %.3f, %.3f)' %
((ngauss,) + tuple(alpha_B[[0, 1, -2, -1]])))
# Maximum l value:
lmax = max(self.f_lsn.keys())
self.channels = []
nb_l = []
if not self.dirac:
for l in range(lmax + 1):
basis = GaussianBasis(l, alpha_B, self.rgd, eps)
nb_l.append(len(basis))
for s in range(self.nspins):
self.channels.append(Channel(l, s, self.f_lsn[l][s],
basis))
else:
for K in range(1, lmax + 2):
leff = (K**2 - (self.Z / c)**2)**0.5 - 1
basis = GaussianBasis(leff, alpha_B, self.rgd, eps)
nb_l.append(len(basis))
for k, l in [(-K, K - 1), (K, K)]:
if l > lmax:
continue
f_n = self.f_lsn[l][0]
j = abs(k) - 0.5
f_n = (2 * j + 1) / (4 * l + 2) * np.array(f_n)
self.channels.append(DiracChannel(k, f_n, basis))
self.log('Basis functions: %s (%s)' %
(', '.join([str(nb) for nb in nb_l]),
', '.join('spdf'[:lmax + 1])))
self.vr_sg = self.rgd.zeros(self.nspins)
self.vr_sg[:] = -self.Z
def solve(self):
"""Diagonalize Schrödinger equation."""
self.eeig = 0.0
for channel in self.channels:
if self.method == 'Gaussian basis-set':
channel.solve(self.vr_sg[channel.s])
else:
channel.solve2(self.vr_sg[channel.s], self.scalar_relativistic,
self.Z)
self.eeig += channel.get_eigenvalue_sum()
def calculate_density(self):
"""Calculate elctron density and kinetic energy."""
self.n_sg = self.rgd.zeros(self.nspins)
for channel in self.channels:
self.n_sg[channel.s] += channel.calculate_density()
def calculate_electrostatic_potential(self):
"""Calculate electrostatic potential and energy."""
n_g = self.n_sg.sum(0)
self.vHr_g = self.rgd.poisson(n_g)
self.eH = 0.5 * self.rgd.integrate(n_g * self.vHr_g, -1)
self.eZ = -self.Z * self.rgd.integrate(n_g, -1)
def calculate_xc_potential(self):
self.vxc_sg = self.rgd.zeros(self.nspins)
self.exc = self.xc.calculate_spherical(self.rgd, self.n_sg,
self.vxc_sg)
def step(self):
self.solve()
self.calculate_density()
self.calculate_electrostatic_potential()
self.calculate_xc_potential()
self.vr_sg = self.vxc_sg * self.rgd.r_g
self.vr_sg += self.vHr_g
self.vr_sg -= self.Z
self.ekin = (self.eeig -
self.rgd.integrate((self.vr_sg * self.n_sg).sum(0), -1))
def run(self, mix=0.4, maxiter=117, dnmax=1e-9):
if self.channels is None:
self.initialize()
if self.dirac:
equation = 'Dirac'
elif self.scalar_relativistic:
equation = 'scalar-relativistic Schrödinger'
else:
equation = 'non-relativistic Schrödinger'
self.log('\nSolving %s equation using %s:' % (equation, self.method))
dn = self.Z
vr_old_sg = None
n_old_sg = None
for iter in range(maxiter):
self.log('.', end='')
self.fd.flush()
if iter > 0:
self.vr_sg *= mix
self.vr_sg += (1 - mix) * vr_old_sg
dn = self.rgd.integrate(abs(self.n_sg - n_old_sg).sum(0))
if dn <= dnmax:
self.log('\nConverged in', iter, 'steps')
break
vr_old_sg = self.vr_sg
n_old_sg = self.n_sg
self.step()
self.summary()
if self.method != 'Gaussian basis-set':
for channel in self.channels:
assert channel.solve2ok
if dn > dnmax:
raise RuntimeError('Did not converge!')
def refine(self):
self.method = 'finite difference'
self.run(dnmax=1e-6, mix=0.14, maxiter=200)
def summary(self):
self.write_states()
self.write_energies()
def write_states(self):
self.log('\n state occupation eigenvalue <r>')
if self.dirac:
self.log(' nl(j) [Hartree] [eV] [Bohr]')
else:
self.log(' nl [Hartree] [eV] [Bohr]')
self.log('-----------------------------------------------------')
states = []
for ch in self.channels:
for n, f in enumerate(ch.f_n):
states.append((ch.e_n[n], ch, n))
states.sort()
for e, ch, n in states:
name = str(n + ch.l + 1) + ch.name
if self.nspins == 2:
name += '(%s)' % '+-'[ch.s]
n_g = ch.calculate_density(n)
rave = self.rgd.integrate(n_g, 1)
self.log(' %-7s %6.3f %13.6f %13.5f %6.3f' %
(name, ch.f_n[n], e, e * units.Hartree, rave))
def write_energies(self):
self.log('\nEnergies: [Hartree] [eV]')
self.log('--------------------------------------------')
for text, e in [('kinetic ', self.ekin),
('coulomb (e-e)', self.eH),
('coulomb (e-n)', self.eZ),
('xc ', self.exc),
('total ',
self.ekin + self.eH + self.eZ + self.exc)]:
self.log(' %s %+13.6f %+13.5f' % (text, e, units.Hartree * e))
self.calculate_exx()
self.log('\nExact exchange energy: %.6f Hartree, %.5f eV' %
(self.exx, self.exx * units.Hartree))
def get_channel(self, l=None, s=0, k=None):
if self.dirac:
for channel in self.channels:
if channel.k == k:
return channel
else:
for channel in self.channels:
if channel.l == l and channel.s == s:
return channel
raise ValueError
def get_orbital(self, n, l=None, s=0, k=None):
channel = self.get_channel(l, s, k)
return channel.basis.expand(channel.C_nb[n])
def plot_wave_functions(self, rc=4.0):
import matplotlib.pyplot as plt
for ch in self.channels:
for n in range(len(ch.f_n)):
fr_g = ch.basis.expand(ch.C_nb[n]) * self.rgd.r_g
# fr_g = ch.phi_ng[n]
name = str(n + ch.l + 1) + ch.name
lw = 2
if self.nspins == 2:
name += '(%s)' % '+-'[ch.s]
if ch.s == 1:
lw = 1
if self.dirac and ch.k > 0:
lw = 1
ls = ['-', '--', '-.', ':'][ch.l]
n_g = ch.calculate_density(n)
rave = self.rgd.integrate(n_g, 1)
gave = self.rgd.round(rave)
fr_g *= np.sign(fr_g[gave], 0)
plt.plot(self.rgd.r_g, fr_g,
ls=ls, lw=lw, color=colors[n + ch.l], label=name)
plt.legend(loc='best')
plt.xlabel('r [Bohr]')
plt.ylabel('$r\\phi(r)$')
plt.axis(xmax=rc)
plt.show()
def logarithmic_derivative(self, l, energies, rcut):
ch = Channel(l)
gcut = self.rgd.round(rcut)
u_g = self.rgd.empty()
logderivs = []
d0 = 42.0
offset = 0
for e in energies:
dudr = ch.integrate_outwards(u_g, self.rgd, self.vr_sg[0],
gcut, e, self.scalar_relativistic,
self.Z)[0]
d1 = np.arctan(dudr / u_g[gcut]) / pi + offset
if d1 > d0:
offset -= 1
d1 -= 1
logderivs.append(d1)
d0 = d1
return np.array(logderivs)
def calculate_exx(self, s=None):
if s is None:
self.exx = sum(self.calculate_exx(s)
for s in range(self.nspins)) / self.nspins
return self.exx
states = []
lmax = 0
for ch in self.channels:
l = ch.l
for n, phi_g in enumerate(ch.phi_ng):
f = ch.f_n[n]
if f > 0 and ch.s == s:
states.append((l, f * self.nspins / 2.0 / (2 * l + 1),
phi_g))
if l > lmax:
lmax = l
G_LLL = gaunt(lmax)
exx = 0.0
j1 = 0
for l1, f1, phi1_g in states:
f = 1.0
for l2, f2, phi2_g in states[j1:]:
n_g = phi1_g * phi2_g
for l in range((l1 + l2) % 2, l1 + l2 + 1, 2):
G = (G_LLL[l1**2:(l1 + 1)**2,
l2**2:(l2 + 1)**2,
l**2:(l + 1)**2]**2).sum()
vr_g = self.rgd.poisson(n_g, l)
e = f * self.rgd.integrate(vr_g * n_g, -1) / 4 / pi
exx -= e * G * f1 * f2
f = 2.0
j1 += 1
return exx
from gpaw.atom.radialgd import EquidistantRadialGridDescriptor
from gpaw.grid_descriptor import GridDescriptor
from gpaw.xc import XC
import numpy as np
from gpaw.test import equal
rgd = EquidistantRadialGridDescriptor(0.01, 100)
for name in ['LDA', 'PBE']:
xc = XC(name)
for nspins in [1, 2]:
n = rgd.zeros(nspins)
v = rgd.zeros(nspins)
n[:] = np.exp(-rgd.r_g**2)
n[-1] *= 2
E = xc.calculate_spherical(rgd, n, v)
i = 23
x = v[-1, i] * rgd.dv_g[i]
n[-1, i] += 0.000001
Ep = xc.calculate_spherical(rgd, n, v)
n[-1, i] -= 0.000002
Em = xc.calculate_spherical(rgd, n, v)
x2 = (Ep - Em) / 0.000002
print(name, nspins, E, x, x2, x - x2)
equal(x, x2, 1e-9)
n[-1, i] += 0.000001
if nspins == 1:
ns = rgd.empty(2)
ns[:] = n / 2
Es = xc.calculate_spherical(rgd, ns, 0 * ns)
equal(E, Es, 1e-13)
class C_GLLBScr(Contribution):
def __init__(self, nlfunc, weight, functional='GGA_X_B88', metallic=False):
Contribution.__init__(self, nlfunc, weight)
self.functional = functional
self.old_coeffs = None
self.iter = 0
self.metallic = metallic
def get_name(self):
return 'SCREENING'
def get_desc(self):
return '(' + self.functional + ')'
# Initialize GLLBScr functional
def initialize_1d(self):
self.ae = self.nlfunc.ae
self.xc = XC(self.functional)
self.v_g = np.zeros(self.ae.N)
self.e_g = np.zeros(self.ae.N)
# Calcualte the GLLB potential and energy 1d
def add_xc_potential_and_energy_1d(self, v_g):
self.v_g[:] = 0.0
self.e_g[:] = 0.0
self.xc.calculate_spherical(self.ae.rgd, self.ae.n.reshape((1, -1)),
self.v_g.reshape((1, -1)), self.e_g)
v_g += 2 * self.weight * self.e_g / (self.ae.n + 1e-10)
Exc = self.weight * np.sum(self.e_g * self.ae.rgd.dv_g)
return Exc
def initialize(self):
self.occupations = self.nlfunc.occupations
self.xc = XC(self.functional)
# Always 1 spin, no matter what calculation nspins is
self.vt_sg = self.nlfunc.finegd.empty(1)
self.e_g = self.nlfunc.finegd.empty()#.ravel()
def get_coefficient_calculator(self):
return self
def f(self, f):
return sqrt(f)
def get_coefficients(self, e_j, f_j):
homo_e = max( [ np.where(f>1e-3, e, -1000) for f,e in zip(f_j, e_j)] )
return [ f * K_G * self.f( max(0, homo_e - e)) for e,f in zip(e_j, f_j) ]
def get_coefficients_1d(self, smooth=False, lumo_perturbation = False):
homo_e = max( [ np.where(f>1e-3, e, -1000) for f,e in zip(self.ae.f_j, self.ae.e_j)])
if not smooth:
if lumo_perturbation:
lumo_e = min( [ np.where(f<1e-3, e, 1000) for f,e in zip(self.ae.f_j, self.ae.e_j)])
return np.array([ f * K_G * (self.f( max(0, lumo_e - e)) - self.f(max(0, homo_e -e)))
for e,f in zip(self.ae.e_j, self.ae.f_j) ])
else:
return np.array([ f * K_G * (self.f( max(0, homo_e - e)))
for e,f in zip(self.ae.e_j, self.ae.f_j) ])
else:
return [ [ f * K_G * self.f( max(0, homo_e - e))
for e,f in zip(e_n, f_n) ]
for e_n, f_n in zip(self.ae.e_ln, self.ae.f_ln) ]
def get_coefficients_by_kpt(self, kpt_u, lumo_perturbation=False, homolumo=None, nspins=1):
if not hasattr(kpt_u[0],'orbitals_ready'):
kpt_u[0].orbitals_ready = True
return None
#if kpt_u[0].psit_nG is None or isinstance(kpt_u[0].psit_nG,
# TarFileReference):
# if kpt_u[0].C_nM==None:
# return None
if homolumo == None:
if self.metallic:
# For metallic systems, the calculated fermi level represents
# the most accurate estimate for reference-energy
eref_lumo_s = eref_s = nspins * [ self.occupations.get_fermi_level() ]
else:
# Find h**o and lumo levels for each spin
eref_s = []
eref_lumo_s = []
for s in range(nspins):
h**o, lumo = self.occupations.get_homo_lumo_by_spin(self.nlfunc.wfs, s)
eref_s.append(h**o)
eref_lumo_s.append(lumo)
else:
eref_s, eref_lumo_s = homolumo
if not isinstance(eref_s, (list, tuple)):
eref_s = [ eref_s ]
eref_lumo_s = [ eref_lumo_s ]
# The parameter ee might sometimes be set to small thereshold value to
# achieve convergence on small systems with degenerate H**O.
if len(kpt_u) > nspins:
ee = 0.0
else:
ee = 0.05 / 27.21
if lumo_perturbation:
return [np.array([
f * K_G * (self.f( np.where(eref_lumo_s[kpt.s] - e>ee, eref_lumo_s[kpt.s]-e,0))
-self.f( np.where(eref_s[kpt.s] - e>ee, eref_s[kpt.s]-e,0)))
for e, f in zip(kpt.eps_n, kpt.f_n) ])
for kpt in kpt_u ]
else:
coeff = [ np.array([ f * K_G * self.f( np.where(eref_s[kpt.s] - e>ee, eref_s[kpt.s]-e,0))
for e, f in zip(kpt.eps_n, kpt.f_n) ])
for kpt in kpt_u ]
return coeff
def calculate_spinpaired(self, e_g, n_g, v_g):
self.e_g[:] = 0.0
self.vt_sg[:] = 0.0
self.xc.calculate(self.nlfunc.finegd, n_g[None, ...], self.vt_sg,
self.e_g)
self.e_g[:] = np.where(n_g<1e-10, 0, self.e_g)
v_g += self.weight * 2 * self.e_g / (n_g + 1e-10)
e_g += self.weight * self.e_g
def calculate_spinpolarized(self, e_g, n_sg, v_sg):
# Calculate spinpolarized exchange screening as two spin-paired calculations n=2*n_s
for n, v in [ (n_sg[0], v_sg[0]), (n_sg[1], v_sg[1]) ]:
self.e_g[:] = 0.0
self.vt_sg[:] = 0.0
self.xc.calculate(self.nlfunc.finegd, 2*n[None, ...], self.vt_sg, self.e_g)
self.e_g[:] = np.where(n<1e-10, 0, self.e_g)
v += self.weight * 2 * self.e_g / (2 * n + 1e-9)
e_g += self.weight * self.e_g / 2
def calculate_energy_and_derivatives(self, setup, D_sp, H_sp, a, addcoredensity=True):
# Get the XC-correction instance
c = setup.xc_correction
nspins = self.nlfunc.nspins
E = 0
for D_p, dEdD_p in zip(D_sp, H_sp):
D_Lq = np.dot(c.B_pqL.T, nspins*D_p)
n_Lg = np.dot(D_Lq, c.n_qg)
if addcoredensity:
n_Lg[0] += c.nc_g * sqrt(4 * pi)
nt_Lg = np.dot(D_Lq, c.nt_qg)
if addcoredensity:
nt_Lg[0] += c.nct_g * sqrt(4 * pi)
dndr_Lg = np.zeros((c.Lmax, c.ng))
dntdr_Lg = np.zeros((c.Lmax, c.ng))
for L in range(c.Lmax):
c.rgd.derivative(n_Lg[L], dndr_Lg[L])
c.rgd.derivative(nt_Lg[L], dntdr_Lg[L])
vt_g = np.zeros(c.ng)
v_g = np.zeros(c.ng)
e_g = np.zeros(c.ng)
deda2_g = np.zeros(c.ng)
for y, (w, Y_L) in enumerate(zip(weight_n, c.Y_nL)):
# Cut gradient releated coefficient to match the setup's Lmax
A_Li = rnablaY_nLv[y, :c.Lmax]
# Expand pseudo density
nt_g = np.dot(Y_L, nt_Lg)
# Expand pseudo density gradient
a1x_g = np.dot(A_Li[:, 0], nt_Lg)
a1y_g = np.dot(A_Li[:, 1], nt_Lg)
a1z_g = np.dot(A_Li[:, 2], nt_Lg)
a2_g = a1x_g**2 + a1y_g**2 + a1z_g**2
a2_g[1:] /= c.rgd.r_g[1:]**2
a2_g[0] = a2_g[1]
a1_g = np.dot(Y_L, dntdr_Lg)
a2_g += a1_g**2
vt_g[:] = 0.0
e_g[:] = 0.0
# Calculate pseudo GGA energy density (potential is discarded)
self.xc.kernel.calculate(e_g, nt_g.reshape((1, -1)),
vt_g.reshape((1, -1)),
a2_g.reshape((1, -1)),
deda2_g.reshape((1, -1)))
# Calculate pseudo GLLB-potential from GGA-energy density
vt_g[:] = 2 * e_g / (nt_g + 1e-10)
dEdD_p -= self.weight * w * np.dot(np.dot(c.B_pqL, Y_L),
np.dot(c.nt_qg, vt_g * c.rgd.dv_g))
E -= w * np.dot(e_g, c.rgd.dv_g) / nspins
# Expand density
n_g = np.dot(Y_L, n_Lg)
# Expand density gradient
a1x_g = np.dot(A_Li[:, 0], n_Lg)
a1y_g = np.dot(A_Li[:, 1], n_Lg)
a1z_g = np.dot(A_Li[:, 2], n_Lg)
a2_g = a1x_g**2 + a1y_g**2 + a1z_g**2
a2_g[1:] /= c.rgd.r_g[1:]**2
a2_g[0] = a2_g[1]
a1_g = np.dot(Y_L, dndr_Lg)
a2_g += a1_g**2
v_g[:] = 0.0
e_g[:] = 0.0
# Calculate GGA energy density (potential is discarded)
self.xc.kernel.calculate(e_g, n_g.reshape((1, -1)),
v_g.reshape((1, -1)),
a2_g.reshape((1, -1)),
deda2_g.reshape((1, -1)))
# Calculate GLLB-potential from GGA-energy density
v_g[:] = 2 * e_g / (n_g + 1e-10)
dEdD_p += self.weight * w * np.dot(np.dot(c.B_pqL, Y_L),
np.dot(c.n_qg, v_g * c.rgd.dv_g))
E += w * np.dot(e_g, c.rgd.dv_g) / nspins
return E * self.weight
def add_smooth_xc_potential_and_energy_1d(self, vt_g):
self.v_g[:] = 0.0
self.e_g[:] = 0.0
self.xc.calculate_spherical(self.ae.rgd, self.ae.nt.reshape((1, -1)),
self.v_g.reshape((1, -1)), self.e_g)
vt_g += 2 * self.weight * self.e_g / (self.ae.nt + 1e-10)
return self.weight * np.sum(self.e_g * self.ae.rgd.dv_g)
def initialize_from_atomic_orbitals(self, basis_functions):
# GLLBScr needs only density which is already initialized
pass
def add_extra_setup_data(self, dict):
# GLLBScr has not any special data
pass
def read(self, reader):
# GLLBScr has no special data to be read
pass
def write(self, writer, natoms):
# GLLBScr has no special data to be written
pass
class C_XC(Contribution):
def __init__(self, nlfunc, weight, functional = 'LDA'):
Contribution.__init__(self, nlfunc, weight)
self.functional = functional
def get_name(self):
return 'XC'
def get_desc(self):
return "("+self.functional+")"
def initialize(self):
self.xc = XC(self.functional)
self.vt_sg = self.nlfunc.finegd.empty(self.nlfunc.nspins)
self.e_g = self.nlfunc.finegd.empty()
def initialize_1d(self):
self.ae = self.nlfunc.ae
self.xc = XC(self.functional)
self.v_g = np.zeros(self.ae.N)
def calculate_spinpaired(self, e_g, n_g, v_g):
self.e_g[:] = 0.0
self.vt_sg[:] = 0.0
self.xc.calculate(self.nlfunc.finegd, n_g[None, ...], self.vt_sg,
self.e_g)
v_g += self.weight * self.vt_sg[0]
e_g += self.weight * self.e_g
def calculate_spinpolarized(self, e_g, n_sg, v_sg):
self.e_g[:] = 0.0
self.vt_sg[:] = 0.0
self.xc.calculate(self.nlfunc.finegd, n_sg, self.vt_sg, self.e_g)
#self.xc.get_energy_and_potential(na_g, self.vt_sg[0], nb_g, self.vt_sg[1], e_g=self.e_g)
v_sg[0] += self.weight * self.vt_sg[0]
v_sg[1] += self.weight * self.vt_sg[1]
e_g += self.weight * self.e_g
def calculate_energy_and_derivatives(self, setup, D_sp, H_sp, a, addcoredensity=True):
E = self.xc.calculate_paw_correction(setup, D_sp, H_sp, True, a)
E += setup.xc_correction.Exc0
print("E", E)
return E
def add_xc_potential_and_energy_1d(self, v_g):
self.v_g[:] = 0.0
Exc = self.xc.calculate_spherical(self.ae.rgd,
self.ae.n.reshape((1, -1)),
self.v_g.reshape((1, -1)))
v_g += self.weight * self.v_g
return self.weight * Exc
def add_smooth_xc_potential_and_energy_1d(self, vt_g):
self.v_g[:] = 0.0
Exc = self.xc.calculate_spherical(self.ae.rgd,
self.ae.nt.reshape((1, -1)),
self.v_g.reshape((1, -1)))
vt_g += self.weight * self.v_g
return self.weight * Exc
def initialize_from_atomic_orbitals(self, basis_functions):
# LDA needs only density, which is already initialized
pass
def add_extra_setup_data(self, dict):
# LDA has not any special data
pass
def write(self, writer, natoms):
# LDA has not any special data to be written
pass
def read(self, reader):
# LDA has not any special data to be read
pass
class C_XC(Contribution):
def __init__(self, nlfunc, weight, functional='LDA'):
Contribution.__init__(self, nlfunc, weight)
self.functional = functional
def get_name(self):
return 'XC'
def get_desc(self):
return "(" + self.functional + ")"
def initialize(self):
self.xc = XC(self.functional)
self.vt_sg = self.nlfunc.finegd.empty(self.nlfunc.nspins)
self.e_g = self.nlfunc.finegd.empty()
def initialize_1d(self):
self.ae = self.nlfunc.ae
self.xc = XC(self.functional)
self.v_g = np.zeros(self.ae.N)
def calculate_spinpaired(self, e_g, n_g, v_g):
self.e_g[:] = 0.0
self.vt_sg[:] = 0.0
self.xc.calculate(self.nlfunc.finegd, n_g[None, ...], self.vt_sg,
self.e_g)
v_g += self.weight * self.vt_sg[0]
e_g += self.weight * self.e_g
def calculate_spinpolarized(self, e_g, n_sg, v_sg):
self.e_g[:] = 0.0
self.vt_sg[:] = 0.0
self.xc.calculate(self.nlfunc.finegd, n_sg, self.vt_sg, self.e_g)
#self.xc.get_energy_and_potential(na_g, self.vt_sg[0], nb_g, self.vt_sg[1], e_g=self.e_g)
v_sg[0] += self.weight * self.vt_sg[0]
v_sg[1] += self.weight * self.vt_sg[1]
e_g += self.weight * self.e_g
def calculate_energy_and_derivatives(self,
setup,
D_sp,
H_sp,
a,
addcoredensity=True):
E = self.xc.calculate_paw_correction(setup, D_sp, H_sp, True, a)
E += setup.xc_correction.Exc0
print("E", E)
return E
def add_xc_potential_and_energy_1d(self, v_g):
self.v_g[:] = 0.0
Exc = self.xc.calculate_spherical(self.ae.rgd,
self.ae.n.reshape((1, -1)),
self.v_g.reshape((1, -1)))
v_g += self.weight * self.v_g
return self.weight * Exc
def add_smooth_xc_potential_and_energy_1d(self, vt_g):
self.v_g[:] = 0.0
Exc = self.xc.calculate_spherical(self.ae.rgd,
self.ae.nt.reshape((1, -1)),
self.v_g.reshape((1, -1)))
vt_g += self.weight * self.v_g
return self.weight * Exc
def initialize_from_atomic_orbitals(self, basis_functions):
# LDA needs only density, which is already initialized
pass
def add_extra_setup_data(self, dict):
# LDA has not any special data
pass
def write(self, writer, natoms):
# LDA has not any special data to be written
pass
def read(self, reader):
# LDA has not any special data to be read
pass
class C_GLLBScr(Contribution):
def __init__(self,
nlfunc,
weight,
functional='GGA_X_B88',
width=None,
eps=0.05,
damp=1e-10):
Contribution.__init__(self, nlfunc, weight)
self.functional = functional
self.old_coeffs = None
self.iter = 0
self.damp = damp
if width is not None:
width = width / 27.21
self.eps = eps / 27.21
self.width = width
def get_name(self):
return 'SCREENING'
def set_damp(self, damp):
self.damp = damp
def get_desc(self):
return '(' + self.functional + ')'
# Initialize GLLBScr functional
def initialize_1d(self):
self.ae = self.nlfunc.ae
self.xc = XC(self.functional)
self.v_g = np.zeros(self.ae.N)
self.e_g = np.zeros(self.ae.N)
# Calcualte the GLLB potential and energy 1d
def add_xc_potential_and_energy_1d(self, v_g):
self.v_g[:] = 0.0
self.e_g[:] = 0.0
self.xc.calculate_spherical(self.ae.rgd, self.ae.n.reshape((1, -1)),
self.v_g.reshape((1, -1)), self.e_g)
v_g += 2 * self.weight * self.e_g / (self.ae.n + self.damp)
Exc = self.weight * np.sum(self.e_g * self.ae.rgd.dv_g)
return Exc
def initialize(self):
self.occupations = self.nlfunc.occupations
self.xc = XC(self.functional)
# Always 1 spin, no matter what calculation nspins is
self.vt_sg = self.nlfunc.finegd.empty(1)
self.e_g = self.nlfunc.finegd.empty() #.ravel()
def get_coefficient_calculator(self):
return self
def f(self, f):
if self.width is None:
if f > self.eps:
return sqrt(f)
else:
return 0.0
else:
dEH = -f
w = self.width
if dEH / w < -100:
return sqrt(f)
Knew = -0.5 * erf(sqrt((max(0.0,dEH)-dEH)/w)) * \
sqrt(w*pi) * exp(-dEH/w)
Knew += 0.5 * sqrt(w * pi) * exp(-dEH / w)
Knew += sqrt(max(0.0, dEH) - dEH) * exp(max(0.0, dEH) / w)
#print dEH, w, dEH/w, Knew, f**0.5
return Knew
def get_coefficients(self, e_j, f_j):
homo_e = max([np.where(f > 1e-3, e, -1000) for f, e in zip(f_j, e_j)])
return [f * K_G * self.f(homo_e - e) for e, f in zip(e_j, f_j)]
def get_coefficients_1d(self, smooth=False, lumo_perturbation=False):
homo_e = max([
np.where(f > 1e-3, e, -1000)
for f, e in zip(self.ae.f_j, self.ae.e_j)
])
if not smooth:
if lumo_perturbation:
lumo_e = min([
np.where(f < 1e-3, e, 1000)
for f, e in zip(self.ae.f_j, self.ae.e_j)
])
return np.array([
f * K_G * (self.f(lumo_e - e) - self.f(homo_e - e))
for e, f in zip(self.ae.e_j, self.ae.f_j)
])
else:
return np.array([
f * K_G * (self.f(homo_e - e))
for e, f in zip(self.ae.e_j, self.ae.f_j)
])
else:
return [[f * K_G * self.f(homo_e - e) for e, f in zip(e_n, f_n)]
for e_n, f_n in zip(self.ae.e_ln, self.ae.f_ln)]
def get_coefficients_by_kpt(self,
kpt_u,
lumo_perturbation=False,
homolumo=None,
nspins=1):
#if not hasattr(kpt_u[0],'orbitals_ready'):
# kpt_u[0].orbitals_ready = True
# return None
#if not hasattr(self.occupations, 'nvalence'):
# print "occupations not ready"
# return None
#if self.occupations.nvalence is None:
# return None
#if kpt_u[0].psit_nG is None or isinstance(kpt_u[0].psit_nG,
# TarFileReference):
# if kpt_u[0].C_nM is None:
# return None
if homolumo is None:
# Find h**o and lumo levels for each spin
eref_s = []
eref_lumo_s = []
for s in range(nspins):
h**o, lumo = self.nlfunc.wfs.get_homo_lumo(s)
eref_s.append(h**o)
eref_lumo_s.append(lumo)
else:
eref_s, eref_lumo_s = homolumo
if not isinstance(eref_s, (list, tuple)):
eref_s = [eref_s]
eref_lumo_s = [eref_lumo_s]
# The parameter ee might sometimes be set to small thereshold value to
# achieve convergence on small systems with degenerate H**O.
if len(kpt_u) > nspins:
ee = 0.0
else:
ee = 0.05 / 27.21
if lumo_perturbation:
return [
np.array([
f * K_G * (self.f(eref_lumo_s[kpt.s] - e) -
self.f(eref_s[kpt.s] - e))
for e, f in zip(kpt.eps_n, kpt.f_n)
]) for kpt in kpt_u
]
else:
coeff = [
np.array([
f * K_G * self.f(eref_s[kpt.s] - e)
for e, f in zip(kpt.eps_n, kpt.f_n)
]) for kpt in kpt_u
]
#print coeff
return coeff
def calculate_spinpaired(self, e_g, n_g, v_g):
self.e_g[:] = 0.0
self.vt_sg[:] = 0.0
self.xc.calculate(self.nlfunc.finegd, n_g[None, ...], self.vt_sg,
self.e_g)
self.e_g[:] = np.where(n_g < self.damp, 0, self.e_g)
v_g += self.weight * 2 * self.e_g / (n_g + self.damp)
e_g += self.weight * self.e_g
def calculate_spinpolarized(self, e_g, n_sg, v_sg):
# Calculate spinpolarized exchange screening as two spin-paired calculations n=2*n_s
for n, v in [(n_sg[0], v_sg[0]), (n_sg[1], v_sg[1])]:
self.e_g[:] = 0.0
self.vt_sg[:] = 0.0
self.xc.calculate(self.nlfunc.finegd, 2 * n[None, ...], self.vt_sg,
self.e_g)
self.e_g[:] = np.where(n < self.damp, 0, self.e_g)
v += self.weight * 2 * self.e_g / (2 * n + self.damp)
e_g += self.weight * self.e_g / 2
def calculate_energy_and_derivatives(self,
setup,
D_sp,
H_sp,
a,
addcoredensity=True):
# Get the XC-correction instance
c = setup.xc_correction
nspins = self.nlfunc.nspins
E = 0
for D_p, dEdD_p in zip(D_sp, H_sp):
D_Lq = np.dot(c.B_pqL.T, nspins * D_p)
n_Lg = np.dot(D_Lq, c.n_qg)
if addcoredensity:
n_Lg[0] += c.nc_g * sqrt(4 * pi)
nt_Lg = np.dot(D_Lq, c.nt_qg)
if addcoredensity:
nt_Lg[0] += c.nct_g * sqrt(4 * pi)
dndr_Lg = np.zeros((c.Lmax, c.ng))
dntdr_Lg = np.zeros((c.Lmax, c.ng))
for L in range(c.Lmax):
c.rgd.derivative(n_Lg[L], dndr_Lg[L])
c.rgd.derivative(nt_Lg[L], dntdr_Lg[L])
vt_g = np.zeros(c.ng)
v_g = np.zeros(c.ng)
e_g = np.zeros(c.ng)
deda2_g = np.zeros(c.ng)
for y, (w, Y_L) in enumerate(zip(weight_n, c.Y_nL)):
# Cut gradient releated coefficient to match the setup's Lmax
A_Li = rnablaY_nLv[y, :c.Lmax]
# Expand pseudo density
nt_g = np.dot(Y_L, nt_Lg)
# Expand pseudo density gradient
a1x_g = np.dot(A_Li[:, 0], nt_Lg)
a1y_g = np.dot(A_Li[:, 1], nt_Lg)
a1z_g = np.dot(A_Li[:, 2], nt_Lg)
a2_g = a1x_g**2 + a1y_g**2 + a1z_g**2
a2_g[1:] /= c.rgd.r_g[1:]**2
a2_g[0] = a2_g[1]
a1_g = np.dot(Y_L, dntdr_Lg)
a2_g += a1_g**2
vt_g[:] = 0.0
e_g[:] = 0.0
# Calculate pseudo GGA energy density (potential is discarded)
self.xc.kernel.calculate(e_g, nt_g.reshape((1, -1)),
vt_g.reshape((1, -1)),
a2_g.reshape((1, -1)),
deda2_g.reshape((1, -1)))
# Calculate pseudo GLLB-potential from GGA-energy density
vt_g[:] = 2 * e_g / (nt_g + self.damp)
dEdD_p -= self.weight * w * np.dot(
np.dot(c.B_pqL, Y_L), np.dot(c.nt_qg, vt_g * c.rgd.dv_g))
E -= w * np.dot(e_g, c.rgd.dv_g) / nspins
# Expand density
n_g = np.dot(Y_L, n_Lg)
# Expand density gradient
a1x_g = np.dot(A_Li[:, 0], n_Lg)
a1y_g = np.dot(A_Li[:, 1], n_Lg)
a1z_g = np.dot(A_Li[:, 2], n_Lg)
a2_g = a1x_g**2 + a1y_g**2 + a1z_g**2
a2_g[1:] /= c.rgd.r_g[1:]**2
a2_g[0] = a2_g[1]
a1_g = np.dot(Y_L, dndr_Lg)
a2_g += a1_g**2
v_g[:] = 0.0
e_g[:] = 0.0
# Calculate GGA energy density (potential is discarded)
self.xc.kernel.calculate(e_g, n_g.reshape((1, -1)),
v_g.reshape((1, -1)),
a2_g.reshape((1, -1)),
deda2_g.reshape((1, -1)))
# Calculate GLLB-potential from GGA-energy density
v_g[:] = 2 * e_g / (n_g + self.damp)
dEdD_p += self.weight * w * np.dot(
np.dot(c.B_pqL, Y_L), np.dot(c.n_qg, v_g * c.rgd.dv_g))
E += w * np.dot(e_g, c.rgd.dv_g) / nspins
return E * self.weight
def add_smooth_xc_potential_and_energy_1d(self, vt_g):
self.v_g[:] = 0.0
self.e_g[:] = 0.0
self.xc.calculate_spherical(self.ae.rgd, self.ae.nt.reshape((1, -1)),
self.v_g.reshape((1, -1)), self.e_g)
vt_g += 2 * self.weight * self.e_g / (self.ae.nt + self.damp)
return self.weight * np.sum(self.e_g * self.ae.rgd.dv_g)
def initialize_from_atomic_orbitals(self, basis_functions):
# GLLBScr needs only density which is already initialized
pass
def add_extra_setup_data(self, dict):
# GLLBScr has not any special data
pass
def read(self, reader):
# GLLBScr has no special data to be read
pass
def write(self, writer):
# GLLBScr has no special data to be written
pass
class AllElectronAtom:
def __init__(self, symbol, xc='LDA', spinpol=False, dirac=False,
log=sys.stdout):
"""All-electron calculation for spherically symmetric atom.
symbol: str (or int)
Chemical symbol (or atomic number).
xc: str
Name of XC-functional.
spinpol: bool
If true, do spin-polarized calculation. Default is spin-paired.
dirac: bool
Solve Dirac equation instead of Schrödinger equation.
log: stream
Text output."""
if isinstance(symbol, int):
symbol = chemical_symbols[symbol]
self.symbol = symbol
self.Z = atomic_numbers[symbol]
self.nspins = 1 + int(bool(spinpol))
self.dirac = bool(dirac)
self.scalar_relativistic = False
if isinstance(xc, str):
self.xc = XC(xc)
else:
self.xc = xc
if log is None:
log = devnull
self.fd = log
self.vr_sg = None # potential * r
self.n_sg = 0.0 # density
self.rgd = None # radial grid descriptor
# Energies:
self.ekin = None
self.eeig = None
self.eH = None
self.eZ = None
self.channels = None
self.initialize_configuration()
self.log('Z: ', self.Z)
self.log('Name: ', atomic_names[self.Z])
self.log('Symbol: ', symbol)
self.log('XC-functional: ', self.xc.name)
self.log('Equation: ', ['Schrödinger', 'Dirac'][self.dirac])
self.method = 'Gaussian basis-set'
def log(self, *args, **kwargs):
prnt(file=self.fd, *args, **kwargs)
def initialize_configuration(self):
self.f_lsn = {}
for n, l, f, e in configurations[self.symbol][1]:
if l not in self.f_lsn:
self.f_lsn[l] = [[] for s in range(self.nspins)]
if self.nspins == 1:
self.f_lsn[l][0].append(f)
else:
# Use Hund's rule:
f0 = min(f, 2 * l + 1)
self.f_lsn[l][0].append(f0)
self.f_lsn[l][1].append(f - f0)
def add(self, n, l, df=+1, s=None):
"""Add (remove) electrons."""
if s is None:
if self.nspins == 1:
s = 0
else:
self.add(n, l, 0.5 * df, 0)
self.add(n, l, 0.5 * df, 1)
return
if l not in self.f_lsn:
self.f_lsn[l] = [[] for x in range(self.nspins)]
f_n = self.f_lsn[l][s]
if len(f_n) < n - l:
f_n.extend([0] * (n - l - len(f_n)))
f_n[n - l - 1] += df
def initialize(self, ngpts=2000, rcut=50.0,
alpha1=0.01, alpha2=None, ngauss=50,
eps=1.0e-7):
"""Initialize basis sets and radial grid.
ngpts: int
Number of grid points for radial grid.
rcut: float
Cutoff for radial grid.
alpha1: float
Smallest exponent for gaussian.
alpha2: float
Largest exponent for gaussian.
ngauss: int
Number of gaussians.
eps: float
Cutoff for eigenvalues of overlap matrix."""
if alpha2 is None:
alpha2 = 50.0 * self.Z**2
# Use grid with r(0)=0, r(1)=a and r(ngpts)=rcut:
a = 1 / alpha2**0.5 / 20
b = (rcut - a * ngpts) / (rcut * ngpts)
b = 1 / round(1 / b)
self.rgd = AERadialGridDescriptor(a, b, ngpts)
self.log('Grid points: %d (%.5f, %.5f, %.5f, ..., %.3f, %.3f)' %
((self.rgd.N,) + tuple(self.rgd.r_g[[0, 1, 2, -2, -1]])))
# Distribute exponents between alpha1 and alpha2:
alpha_B = alpha1 * (alpha2 / alpha1)**np.linspace(0, 1, ngauss)
self.log('Exponents: %d (%.3f, %.3f, ..., %.3f, %.3f)' %
((ngauss,) + tuple(alpha_B[[0, 1, -2, -1]])))
# Maximum l value:
lmax = max(self.f_lsn.keys())
self.channels = []
nb_l = []
if not self.dirac:
for l in range(lmax + 1):
basis = GaussianBasis(l, alpha_B, self.rgd, eps)
nb_l.append(len(basis))
for s in range(self.nspins):
self.channels.append(Channel(l, s, self.f_lsn[l][s],
basis))
else:
for K in range(1, lmax + 2):
leff = (K**2 - (self.Z / c)**2)**0.5 - 1
basis = GaussianBasis(leff, alpha_B, self.rgd, eps)
nb_l.append(len(basis))
for k, l in [(-K, K - 1), (K, K)]:
if l > lmax:
continue
f_n = self.f_lsn[l][0]
j = abs(k) - 0.5
f_n = (2 * j + 1) / (4 * l + 2) * np.array(f_n)
self.channels.append(DiracChannel(k, f_n, basis))
self.log('Basis functions: %s (%s)' %
(', '.join([str(nb) for nb in nb_l]),
', '.join('spdf'[:lmax + 1])))
self.vr_sg = self.rgd.zeros(self.nspins)
self.vr_sg[:] = -self.Z
def solve(self):
"""Diagonalize Schrödinger equation."""
self.eeig = 0.0
for channel in self.channels:
if self.method == 'Gaussian basis-set':
channel.solve(self.vr_sg[channel.s])
else:
channel.solve2(self.vr_sg[channel.s], self.scalar_relativistic)
self.eeig += channel.get_eigenvalue_sum()
def calculate_density(self):
"""Calculate elctron density and kinetic energy."""
self.n_sg = self.rgd.zeros(self.nspins)
for channel in self.channels:
self.n_sg[channel.s] += channel.calculate_density()
def calculate_electrostatic_potential(self):
"""Calculate electrostatic potential and energy."""
n_g = self.n_sg.sum(0)
self.vHr_g = self.rgd.poisson(n_g)
self.eH = 0.5 * self.rgd.integrate(n_g * self.vHr_g, -1)
self.eZ = -self.Z * self.rgd.integrate(n_g, -1)
def calculate_xc_potential(self):
self.vxc_sg = self.rgd.zeros(self.nspins)
self.exc = self.xc.calculate_spherical(self.rgd, self.n_sg, self.vxc_sg)
def step(self):
self.solve()
self.calculate_density()
self.calculate_electrostatic_potential()
self.calculate_xc_potential()
self.vr_sg = self.vxc_sg * self.rgd.r_g
self.vr_sg += self.vHr_g
self.vr_sg -= self.Z
self.ekin = (self.eeig -
self.rgd.integrate((self.vr_sg * self.n_sg).sum(0), -1))
def run(self, mix=0.4, maxiter=117, dnmax=1e-9):
if self.channels is None:
self.initialize()
if self.dirac:
equation = 'Dirac'
elif self.scalar_relativistic:
equation = 'scalar-relativistic Schrödinger'
else:
equation = 'non-relativistic Schrödinger'
self.log('\nSolving %s equation using %s:' % (equation, self.method))
dn = self.Z
for iter in range(maxiter):
self.log('.', end='')
self.fd.flush()
if iter > 0:
self.vr_sg *= mix
self.vr_sg += (1 - mix) * vr_old_sg
dn = self.rgd.integrate(abs(self.n_sg - n_old_sg).sum(0))
if dn <= dnmax:
self.log('\nConverged in', iter, 'steps')
break
vr_old_sg = self.vr_sg
n_old_sg = self.n_sg
self.step()
self.summary()
if dn > dnmax:
raise RuntimeError('Did not converge!')
def refine(self):
self.method = 'finite difference'
self.run(dnmax=1e-6, mix=0.14, maxiter=200)
def summary(self):
self.write_states()
self.write_energies()
def write_states(self):
self.log('\n state occupation eigenvalue <r>')
if self.dirac:
self.log(' nl(j) [Hartree] [eV] [Bohr]')
else:
self.log(' nl [Hartree] [eV] [Bohr]')
self.log('-----------------------------------------------------')
states = []
for ch in self.channels:
for n, f in enumerate(ch.f_n):
states.append((ch.e_n[n], ch, n))
states.sort()
for e, ch, n in states:
name = str(n + ch.l + 1) + ch.name
if self.nspins == 2:
name += '(%s)' % '+-'[ch.s]
n_g = ch.calculate_density(n)
rave = self.rgd.integrate(n_g, 1)
self.log(' %-7s %6.3f %13.6f %13.5f %6.3f' %
(name, ch.f_n[n], e, e * units.Hartree, rave))
def write_energies(self):
self.log('\nEnergies: [Hartree] [eV]')
self.log('--------------------------------------------')
for text, e in [('kinetic ', self.ekin),
('coulomb (e-e)', self.eH),
('coulomb (e-n)', self.eZ),
('xc ', self.exc),
('total ',
self.ekin + self.eH + self.eZ + self.exc)]:
self.log(' %s %+13.6f %+13.5f' % (text, e, units.Hartree * e))
self.calculate_exx()
self.log('\nExact exchange energy: %.6f Hartree, %.5f eV' %
(self.exx, self.exx * units.Hartree))
def get_channel(self, l=None, s=0, k=None):
if self.dirac:
for channel in self.channels:
if channel.k == k:
return channel
else:
for channel in self.channels:
if channel.l == l and channel.s == s:
return channel
raise ValueError
def get_orbital(self, n, l=None, s=0, k=None):
channel = self.get_channel(l, s, k)
return channel.basis.expand(channel.C_nb[n])
def plot_wave_functions(self, rc=4.0):
import matplotlib.pyplot as plt
for ch in self.channels:
for n in range(len(ch.f_n)):
fr_g = ch.basis.expand(ch.C_nb[n]) * self.rgd.r_g
name = str(n + ch.l + 1) + ch.name
lw = 2
if self.nspins == 2:
name += '(%s)' % '+-'[ch.s]
if ch.s == 1:
lw = 1
if self.dirac and ch.k > 0:
lw = 1
ls = ['-', '--', '-.', ':'][ch.l]
n_g = ch.calculate_density(n)
rave = self.rgd.integrate(n_g, 1)
gave = self.rgd.round(rave)
fr_g *= cmp(fr_g[gave], 0)
plt.plot(self.rgd.r_g, fr_g,
ls=ls, lw=lw, color=colors[n + ch.l], label=name)
plt.legend(loc='best')
plt.xlabel('r [Bohr]')
plt.ylabel('$r\\phi(r)$')
plt.axis(xmax=rc)
plt.show()
def logarithmic_derivative(self, l, energies, rcut):
ch = Channel(l)
gcut = self.rgd.round(rcut)
u_g = self.rgd.empty()
logderivs = []
for e in energies:
dudr = ch.integrate_outwards(u_g, self.rgd, self.vr_sg[0],
gcut, e, self.scalar_relativistic)
logderivs.append(dudr / u_g[gcut])
return logderivs
def calculate_exx(self, s=None):
if s is None:
self.exx = sum(self.calculate_exx(s)
for s in range(self.nspins)) / self.nspins
return self.exx
states = []
lmax = 0
for ch in self.channels:
l = ch.l
for n, phi_g in enumerate(ch.phi_ng):
f = ch.f_n[n]
if f > 0 and ch.s == s:
states.append((l, f * self.nspins / 2.0 / (2 * l + 1),
phi_g))
if l > lmax:
lmax = l
G_LLL = make_gaunt(lmax)
exx = 0.0
j1 = 0
for l1, f1, phi1_g in states:
f = 1.0
for l2, f2, phi2_g in states[j1:]:
n_g = phi1_g * phi2_g
for l in range((l1 + l2) % 2, l1 + l2 + 1, 2):
G = (G_LLL[l1**2:(l1 + 1)**2,
l2**2:(l2 + 1)**2,
l**2:(l + 1)**2]**2).sum()
vr_g = self.rgd.poisson(n_g, l)
e = f * self.rgd.integrate(vr_g * n_g, -1) / 4 / pi
exx -= e * G * f1 * f2
f = 2.0
j1 += 1
return exx
from gpaw.xc import XC
import numpy as np
from gpaw.test import equal
r = 0.01 * np.arange(100)
dr = 0.01 * np.ones(100)
rgd = RadialGridDescriptor(r, dr)
for name in ['LDA', 'PBE']:
xc = XC(name)
for nspins in [1, 2]:
n = rgd.zeros(nspins)
v = rgd.zeros(nspins)
n[:] = np.exp(-r**2)
n[-1] *= 2
E = xc.calculate_spherical(rgd, n, v)
i = 23
x = v[-1, i] * rgd.dv_g[i]
n[-1, i] += 0.000001
Ep = xc.calculate_spherical(rgd, n, v)
n[-1, i] -= 0.000002
Em = xc.calculate_spherical(rgd, n, v)
x2 = (Ep - Em) / 0.000002
print name, nspins, E, x, x2, x - x2
equal(x, x2, 1e-9)
n[-1, i] += 0.000001
if nspins == 1:
ns = rgd.empty(2)
ns[:] = n / 2
Es = xc.calculate_spherical(rgd, ns, 0 * ns)
equal(E, Es, 1e-13)
class AllElectron:
"""Object for doing an atomic DFT calculation."""
def __init__(self,
symbol,
xcname='LDA',
scalarrel=True,
corehole=None,
configuration=None,
nofiles=True,
txt='-',
gpernode=150,
orbital_free=False,
tw_coeff=1.):
"""Do an atomic DFT calculation.
Example::
a = AllElectron('Fe')
a.run()
"""
if txt is None:
txt = devnull
elif txt == '-':
txt = sys.stdout
elif isinstance(txt, str):
txt = open(txt, 'w')
self.txt = txt
self.symbol = symbol
self.xcname = xcname
self.scalarrel = scalarrel
self.nofiles = nofiles
# Get reference state:
self.Z, nlfe_j = configurations[symbol]
# Collect principal quantum numbers, angular momentum quantum
# numbers, occupation numbers and eigenvalues (j is a combined
# index for n and l):
self.n_j = [n for n, l, f, e in nlfe_j]
self.l_j = [l for n, l, f, e in nlfe_j]
self.f_j = [f for n, l, f, e in nlfe_j]
self.e_j = [e for n, l, f, e in nlfe_j]
if configuration is not None:
j = 0
for conf in configuration.split(','):
if conf[0].isdigit():
n = int(conf[0])
l = 'spdf'.find(conf[1])
if len(conf) == 2:
f = 1.0
else:
f = float(conf[2:])
#try:
assert n == self.n_j[j]
assert l == self.l_j[j]
self.f_j[j] = f
#except IndexError:
# self.n_j.append(n)
# self.l_j.append(l)
# self.f_j.append(f)
# self.e_j.append(self.e_j[-1])
j += 1
else:
j += {'He': 1, 'Ne': 3, 'Ar': 5, 'Kr': 8, 'Xe': 11}[conf]
maxnodes = max([n - l - 1 for n, l in zip(self.n_j, self.l_j)])
self.N = (maxnodes + 1) * gpernode
self.beta = 0.4
self.orbital_free = orbital_free
self.tw_coeff = tw_coeff
if self.orbital_free:
self.n_j = [1]
self.l_j = [0]
self.f_j = [self.Z]
self.e_j = [self.e_j[-1]]
t = self.text
t()
if scalarrel:
t('Scalar-relativistic atomic ', end='')
else:
t('Atomic ', end='')
t('%s calculation for %s (%s, Z=%d)' %
(xcname, symbol, atomic_names[self.Z], self.Z))
if corehole is not None:
self.ncorehole, self.lcorehole, self.fcorehole = corehole
# Find j for core hole and adjust occupation:
for j in range(len(self.f_j)):
if (self.n_j[j] == self.ncorehole
and self.l_j[j] == self.lcorehole):
assert self.f_j[j] == 2 * (2 * self.lcorehole + 1)
self.f_j[j] -= self.fcorehole
self.jcorehole = j
break
coreholestate = '%d%s' % (self.ncorehole, 'spdf'[self.lcorehole])
t('Core hole in %s state (%s occupation: %.1f)' %
(coreholestate, coreholestate, self.f_j[self.jcorehole]))
else:
self.jcorehole = None
self.fcorehole = 0
def text(self, *args, **kwargs):
self.txt.write(
kwargs.get('sep', ' ').join([str(arg) for arg in args]) +
kwargs.get('end', '\n'))
def initialize_wave_functions(self):
r = self.r
dr = self.dr
# Initialize with Slater function:
for l, e, u in zip(self.l_j, self.e_j, self.u_j):
if self.symbol in ['Hf', 'Ta', 'W', 'Re', 'Os', 'Ir', 'Pt', 'Au']:
a = sqrt(-4.0 * e)
else:
a = sqrt(-2.0 * e)
u[:] = r**(1 + l) * np.exp(-a * r)
norm = np.dot(u**2, dr)
u *= 1.0 / sqrt(norm)
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 write(self, array, name=None, n=None, l=None):
if self.nofiles:
return
if name:
name = self.symbol + '.' + name
else:
name = self.symbol
if l is not None:
assert n is not None
if n > 0:
# Bound state:
name += '.%d%s' % (n, 'spdf'[l])
else:
name += '.x%d%s' % (-n, 'spdf'[l])
f = open(name, 'w')
for r, a in zip(self.r, array):
print(r, a, file=f)
def calculate_density(self):
"""Return the electron charge density divided by 4 pi"""
n = np.dot(self.f_j,
np.where(abs(self.u_j) < 1e-160, 0, self.u_j)**2) / (4 * pi)
n[1:] /= self.r[1:]**2
n[0] = n[1]
return n
def calculate_kinetic_energy_density(self):
"""Return the kinetic energy density"""
return self.radial_kinetic_energy_density(self.f_j, self.l_j, self.u_j)
def radial_kinetic_energy_density(self, f_j, l_j, u_j):
"""Kinetic energy density from a restricted set of wf's
"""
shape = np.shape(u_j[0])
dudr = np.zeros(shape)
tau = np.zeros(shape)
for f, l, u in zip(f_j, l_j, u_j):
self.rgd.derivative(u, dudr)
# contribution from angular derivatives
if l > 0:
tau += f * l * (l + 1) * np.where(abs(u) < 1e-160, 0, u)**2
# contribution from radial derivatives
dudr = u - self.r * dudr
tau += f * np.where(abs(dudr) < 1e-160, 0, dudr)**2
tau[1:] /= self.r[1:]**4
tau[0] = tau[1]
return 0.5 * tau / (4 * pi)
def calculate_kinetic_energy_density2(self):
"""Return the kinetic energy density
calculation over R(r)=u(r)/r
slower convergence with # of radial grid points for
Ekin of H than radial_kinetic_energy_density
"""
shape = self.u_j.shape[1]
R = np.zeros(shape)
dRdr = np.zeros(shape)
tau = np.zeros(shape)
for f, l, u in zip(self.f_j, self.l_j, self.u_j):
R[1:] = u[1:] / self.r[1:]
if l == 0:
# estimate value at origin by Taylor series to first order
d1 = self.r[1]
d2 = self.r[2]
R[0] = .5 * (R[1] + R[2] + (R[1] - R[2]) * (d1 + d2) /
(d2 - d1))
else:
R[0] = 0
self.rgd.derivative(R, dRdr)
# contribution from radial derivatives
tau += f * np.where(abs(dRdr) < 1e-160, 0, dRdr)**2
# contribution from angular derivatives
if l > 0:
R[1:] = R[1:] / self.r[1:]
if l == 1:
R[0] = R[1]
else:
R[0] = 0
tau += f * l * (l + 1) * np.where(abs(R) < 1e-160, 0, R)**2
return 0.5 * tau / (4 * pi)
def solve(self):
"""Solve the Schrodinger equation
::
2
d u 1 dv du u l(l + 1)
- --- - ---- -- (-- - -) + [-------- + 2M(v - e)] u = 0,
2 2 dr dr r 2
dr 2Mc r
where the relativistic mass::
1
M = 1 - --- (v - e)
2
2c
and the fine-structure constant alpha = 1/c = 1/137.036
is set to zero for non-scalar-relativistic calculations.
On the logaritmic radial grids defined by::
beta g
r = ------, g = 0, 1, ..., N - 1
N - g
rN
g = --------, r = [0; oo[
beta + r
the Schrodinger equation becomes::
2
d u du
--- c + -- c + u c = 0
2 2 dg 1 0
dg
with the vectors c0, c1, and c2 defined by::
2 dg 2
c = -r (--)
2 dr
2 2
d g 2 r dg dv
c = - --- r - ---- -- --
1 2 2 dr dr
dr 2Mc
2 r dv
c = l(l + 1) + 2M(v - e)r + ---- --
0 2 dr
2Mc
"""
r = self.r
dr = self.dr
vr = self.vr
c2 = -(r / dr)**2
c10 = -self.d2gdr2 * r**2 # first part of c1 vector
if self.scalarrel:
self.r2dvdr = np.zeros(self.N)
self.rgd.derivative(vr, self.r2dvdr)
self.r2dvdr *= r
self.r2dvdr -= vr
else:
self.r2dvdr = None
# solve for each quantum state separately
for j, (n, l, e,
u) in enumerate(zip(self.n_j, self.l_j, self.e_j, self.u_j)):
nodes = n - l - 1 # analytically expected number of nodes
delta = -0.2 * e
nn, A = shoot(u, l, vr, e, self.r2dvdr, r, dr, c10, c2,
self.scalarrel)
# adjust eigenenergy until u has the correct number of nodes
while nn != nodes:
diff = np.sign(nn - nodes)
while diff == np.sign(nn - nodes):
e -= diff * delta
nn, A = shoot(u, l, vr, e, self.r2dvdr, r, dr, c10, c2,
self.scalarrel)
delta /= 2
# adjust eigenenergy until u is smooth at the turning point
de = 1.0
while abs(de) > 1e-9:
norm = np.dot(np.where(abs(u) < 1e-160, 0, u)**2, dr)
u *= 1.0 / sqrt(norm)
de = 0.5 * A / norm
x = abs(de / e)
if x > 0.1:
de *= 0.1 / x
e -= de
assert e < 0.0
nn, A = shoot(u, l, vr, e, self.r2dvdr, r, dr, c10, c2,
self.scalarrel)
self.e_j[j] = e
u *= 1.0 / sqrt(np.dot(np.where(abs(u) < 1e-160, 0, u)**2, dr))
def solve_confined(self, j, rc, vconf=None):
"""Solve the Schroedinger equation in a confinement potential.
Solves the Schroedinger equation like the solve method, but with a
number of differences. Before invoking this method, run solve() to
get initial guesses.
Parameters:
j: solves only for the state given by j
rc: solution cutoff. Solution will be zero outside this.
vconf: added to the potential (use this as confinement potential)
Returns: a tuple containing the solution u and its energy e.
Unlike the solve method, this method will not alter any attributes of
this object.
"""
r = self.r
dr = self.dr
vr = self.vr.copy()
if vconf is not None:
vr += vconf * r
c2 = -(r / dr)**2
c10 = -self.d2gdr2 * r**2 # first part of c1 vector
if j is None:
n, l, e, u = 3, 2, -0.15, self.u_j[-1].copy()
else:
n = self.n_j[j]
l = self.l_j[j]
e = self.e_j[j]
u = self.u_j[j].copy()
nn, A = shoot_confined(u,
l,
vr,
e,
self.r2dvdr,
r,
dr,
c10,
c2,
self.scalarrel,
rc=rc,
beta=self.beta)
assert nn == n - l - 1 # run() should have been called already
# adjust eigenenergy until u is smooth at the turning point
de = 1.0
while abs(de) > 1e-9:
norm = np.dot(np.where(abs(u) < 1e-160, 0, u)**2, dr)
u *= 1.0 / sqrt(norm)
de = 0.5 * A / norm
x = abs(de / e)
if x > 0.1:
de *= 0.1 / x
e -= de
assert e < 0.0
nn, A = shoot_confined(u,
l,
vr,
e,
self.r2dvdr,
r,
dr,
c10,
c2,
self.scalarrel,
rc=rc,
beta=self.beta)
u *= 1.0 / sqrt(np.dot(np.where(abs(u) < 1e-160, 0, u)**2, dr))
return u, e
def kin(self, l, u, e=None): # XXX move to Generator
r = self.r[1:]
dr = self.dr[1:]
c0 = 0.5 * l * (l + 1) / r**2
c1 = -0.5 * self.d2gdr2[1:]
c2 = -0.5 * dr**-2
if e is not None and self.scalarrel:
x = 0.5 * alpha**2
Mr = r * (1.0 + x * e) - x * self.vr[1:]
c0 += (
(Mr - r) *
(self.vr[1:] - e * r) + 0.5 * x * self.r2dvdr[1:] / Mr) / r**2
c1 -= 0.5 * x * self.r2dvdr[1:] / (Mr * dr * r)
fp = c2 + 0.5 * c1
fm = c2 - 0.5 * c1
f0 = c0 - 2 * c2
kr = np.zeros(self.N)
kr[1:] = f0 * u[1:] + fm * u[:-1]
kr[1:-1] += fp[:-1] * u[2:]
kr[0] = 0.0
return kr
def r2g(self, r):
"""Convert radius to index of the radial grid."""
return int(r * self.N / (self.beta + r))
def get_confinement_potential(self, alpha, ri, rc):
r"""Create a smooth confinement potential.
Returns a (potential) function which is zero inside the radius ri
and goes to infinity smoothly at rc, after which point it is nan.
The potential is given by::
alpha / rc - ri \
V(r) = -------- exp ( - --------- ) for ri < r < rc
rc - r \ r - ri /
"""
i_ri = self.r2g(ri)
i_rc = self.r2g(rc)
if self.r[i_rc] == rc:
# Avoid division by zero in the odd case that rc coincides
# exactly with a grid point (which actually happens sometimes)
i_rc -= 1
potential = np.zeros(np.shape(self.r))
r = self.r[i_ri + 1:i_rc + 1]
exponent = -(rc - ri) / (r - ri)
denom = rc - r
value = np.exp(exponent) / denom
potential[i_ri + 1:i_rc + 1] = value
potential[i_rc + 1:] = np.inf
return alpha * potential
class AllElectron:
"""Object for doing an atomic DFT calculation."""
def __init__(self, symbol, xcname='LDA', scalarrel=False,
corehole=None, configuration=None, nofiles=True,
txt='-', gpernode=150, orbital_free=False, tf_coeff=1.):
"""Do an atomic DFT calculation.
Example::
a = AllElectron('Fe')
a.run()
"""
if txt is None:
txt = devnull
elif txt == '-':
txt = sys.stdout
elif isinstance(txt, str):
txt = open(txt, 'w')
self.txt = txt
self.symbol = symbol
self.xcname = xcname
self.scalarrel = scalarrel
self.nofiles = nofiles
# Get reference state:
self.Z, nlfe_j = configurations[symbol]
# Collect principal quantum numbers, angular momentum quantum
# numbers, occupation numbers and eigenvalues (j is a combined
# index for n and l):
self.n_j = [n for n, l, f, e in nlfe_j]
self.l_j = [l for n, l, f, e in nlfe_j]
self.f_j = [f for n, l, f, e in nlfe_j]
self.e_j = [e for n, l, f, e in nlfe_j]
if configuration is not None:
j = 0
for conf in configuration.split(','):
if conf[0].isdigit():
n = int(conf[0])
l = 'spdf'.find(conf[1])
if len(conf) == 2:
f = 1.0
else:
f = float(conf[2:])
#try:
assert n == self.n_j[j]
assert l == self.l_j[j]
self.f_j[j] = f
#except IndexError:
# self.n_j.append(n)
# self.l_j.append(l)
# self.f_j.append(f)
# self.e_j.append(self.e_j[-1])
j += 1
else:
j += {'He': 1,
'Ne': 3,
'Ar': 5,
'Kr': 8,
'Xe': 11}[conf]
maxnodes = max([n - l - 1 for n, l in zip(self.n_j, self.l_j)])
self.N = (maxnodes + 1) * gpernode
self.beta = 0.4
self.orbital_free = orbital_free
self.tf_coeff = tf_coeff
if self.orbital_free:
self.n_j = [1]
self.l_j = [0]
self.f_j = [self.Z]
self.e_j = [self.e_j[-1]]
t = self.text
t()
if scalarrel:
t('Scalar-relativistic atomic ', end='')
else:
t('Atomic ', end='')
t('%s calculation for %s (%s, Z=%d)' % (xcname, symbol,
atomic_names[self.Z], self.Z))
if corehole is not None:
self.ncorehole, self.lcorehole, self.fcorehole = corehole
# Find j for core hole and adjust occupation:
for j in range(len(self.f_j)):
if (self.n_j[j] == self.ncorehole and
self.l_j[j] == self.lcorehole):
assert self.f_j[j] == 2 * (2 * self.lcorehole + 1)
self.f_j[j] -= self.fcorehole
self.jcorehole = j
break
coreholestate = '%d%s' % (self.ncorehole, 'spdf'[self.lcorehole])
t('Core hole in %s state (%s occupation: %.1f)' % (
coreholestate, coreholestate, self.f_j[self.jcorehole]))
else:
self.jcorehole = None
self.fcorehole = 0
def text(self, *args, **kwargs):
self.txt.write(kwargs.get('sep', ' ').join([str(arg)
for arg in args]) +
kwargs.get('end', '\n'))
def initialize_wave_functions(self):
r = self.r
dr = self.dr
# Initialize with Slater function:
for l, e, u in zip(self.l_j, self.e_j, self.u_j):
if self.symbol in ['Hf', 'Ta', 'W', 'Re', 'Os',
'Ir', 'Pt', 'Au']:
a = sqrt(-4.0 * e)
else:
a = sqrt(-2.0 * e)
u[:] = r**(1 + l) * np.exp(-a * r)
norm = np.dot(u**2, dr)
u *= 1.0 / sqrt(norm)
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 write(self, array, name=None, n=None, l=None):
if self.nofiles:
return
if name:
name = self.symbol + '.' + name
else:
name = self.symbol
if l is not None:
assert n is not None
if n > 0:
# Bound state:
name += '.%d%s' % (n, 'spdf'[l])
else:
name += '.x%d%s' % (-n, 'spdf'[l])
f = open(name, 'w')
for r, a in zip(self.r, array):
print(r, a, file=f)
def calculate_density(self):
"""Return the electron charge density divided by 4 pi"""
n = np.dot(self.f_j,
np.where(abs(self.u_j) < 1e-160, 0,
self.u_j)**2) / (4 * pi)
n[1:] /= self.r[1:]**2
n[0] = n[1]
return n
def calculate_kinetic_energy_density(self):
"""Return the kinetic energy density"""
return self.radial_kinetic_energy_density(self.f_j, self.l_j, self.u_j)
def radial_kinetic_energy_density(self, f_j, l_j, u_j):
"""Kinetic energy density from a restricted set of wf's
"""
shape = np.shape(u_j[0])
dudr = np.zeros(shape)
tau = np.zeros(shape)
for f, l, u in zip(f_j, l_j, u_j):
self.rgd.derivative(u, dudr)
# contribution from angular derivatives
if l > 0:
tau += f * l * (l + 1) * np.where(abs(u) < 1e-160, 0, u)**2
# contribution from radial derivatives
dudr = u - self.r * dudr
tau += f * np.where(abs(dudr) < 1e-160, 0, dudr)**2
tau[1:] /= self.r[1:]**4
tau[0] = tau[1]
return 0.5 * tau / (4 * pi)
def calculate_kinetic_energy_density2(self):
"""Return the kinetic energy density
calculation over R(r)=u(r)/r
slower convergence with # of radial grid points for
Ekin of H than radial_kinetic_energy_density
"""
shape = self.u_j.shape[1]
R = np.zeros(shape)
dRdr = np.zeros(shape)
tau = np.zeros(shape)
for f, l, u in zip(self.f_j, self.l_j, self.u_j):
R[1:] = u[1:] / self.r[1:]
if l == 0:
# estimate value at origin by Taylor series to first order
d1 = self.r[1]
d2 = self.r[2]
R[0] = .5 * (R[1] + R[2] + (R[1] - R[2]) *
(d1 + d2) / (d2 - d1))
else:
R[0] = 0
self.rgd.derivative(R, dRdr)
# contribution from radial derivatives
tau += f * np.where(abs(dRdr) < 1e-160, 0, dRdr)**2
# contribution from angular derivatives
if l > 0:
R[1:] = R[1:] / self.r[1:]
if l == 1:
R[0] = R[1]
else:
R[0] = 0
tau += f * l * (l + 1) * np.where(abs(R) < 1e-160, 0, R)**2
return 0.5 * tau / (4 * pi)
def solve(self):
"""Solve the Schrodinger equation
::
2
d u 1 dv du u l(l + 1)
- --- - ---- -- (-- - -) + [-------- + 2M(v - e)] u = 0,
2 2 dr dr r 2
dr 2Mc r
where the relativistic mass::
1
M = 1 - --- (v - e)
2
2c
and the fine-structure constant alpha = 1/c = 1/137.036
is set to zero for non-scalar-relativistic calculations.
On the logaritmic radial grids defined by::
beta g
r = ------, g = 0, 1, ..., N - 1
N - g
rN
g = --------, r = [0; oo[
beta + r
the Schrodinger equation becomes::
2
d u du
--- c + -- c + u c = 0
2 2 dg 1 0
dg
with the vectors c0, c1, and c2 defined by::
2 dg 2
c = -r (--)
2 dr
2 2
d g 2 r dg dv
c = - --- r - ---- -- --
1 2 2 dr dr
dr 2Mc
2 r dv
c = l(l + 1) + 2M(v - e)r + ---- --
0 2 dr
2Mc
"""
r = self.r
dr = self.dr
vr = self.vr
c2 = -(r / dr)**2
c10 = -self.d2gdr2 * r**2 # first part of c1 vector
if self.scalarrel:
self.r2dvdr = np.zeros(self.N)
self.rgd.derivative(vr, self.r2dvdr)
self.r2dvdr *= r
self.r2dvdr -= vr
else:
self.r2dvdr = None
# solve for each quantum state separately
for j, (n, l, e, u) in enumerate(zip(self.n_j, self.l_j,
self.e_j, self.u_j)):
nodes = n - l - 1 # analytically expected number of nodes
delta = -0.2 * e
nn, A = shoot(u, l, vr, e, self.r2dvdr, r, dr, c10, c2,
self.scalarrel)
# adjust eigenenergy until u has the correct number of nodes
while nn != nodes:
diff = cmp(nn, nodes)
while diff == cmp(nn, nodes):
e -= diff * delta
nn, A = shoot(u, l, vr, e, self.r2dvdr, r, dr, c10, c2,
self.scalarrel)
delta /= 2
# adjust eigenenergy until u is smooth at the turning point
de = 1.0
while abs(de) > 1e-9:
norm = np.dot(np.where(abs(u) < 1e-160, 0, u)**2, dr)
u *= 1.0 / sqrt(norm)
de = 0.5 * A / norm
x = abs(de / e)
if x > 0.1:
de *= 0.1 / x
e -= de
assert e < 0.0
nn, A = shoot(u, l, vr, e, self.r2dvdr, r, dr, c10, c2,
self.scalarrel)
self.e_j[j] = e
u *= 1.0 / sqrt(np.dot(np.where(abs(u) < 1e-160, 0, u)**2, dr))
def solve_confined(self, j, rc, vconf=None):
"""Solve the Schroedinger equation in a confinement potential.
Solves the Schroedinger equation like the solve method, but with a
number of differences. Before invoking this method, run solve() to
get initial guesses.
Parameters:
j: solves only for the state given by j
rc: solution cutoff. Solution will be zero outside this.
vconf: added to the potential (use this as confinement potential)
Returns: a tuple containing the solution u and its energy e.
Unlike the solve method, this method will not alter any attributes of
this object.
"""
r = self.r
dr = self.dr
vr = self.vr.copy()
if vconf is not None:
vr += vconf * r
c2 = -(r / dr)**2
c10 = -self.d2gdr2 * r**2 # first part of c1 vector
if j is None:
n, l, e, u = 3, 2, -0.15, self.u_j[-1].copy()
else:
n = self.n_j[j]
l = self.l_j[j]
e = self.e_j[j]
u = self.u_j[j].copy()
nn, A = shoot_confined(u, l, vr, e, self.r2dvdr, r, dr, c10, c2,
self.scalarrel, rc=rc, beta=self.beta)
assert nn == n - l - 1 # run() should have been called already
# adjust eigenenergy until u is smooth at the turning point
de = 1.0
while abs(de) > 1e-9:
norm = np.dot(np.where(abs(u) < 1e-160, 0, u)**2, dr)
u *= 1.0 / sqrt(norm)
de = 0.5 * A / norm
x = abs(de / e)
if x > 0.1:
de *= 0.1 / x
e -= de
assert e < 0.0
nn, A = shoot_confined(u, l, vr, e, self.r2dvdr, r, dr, c10, c2,
self.scalarrel, rc=rc, beta=self.beta)
u *= 1.0 / sqrt(np.dot(np.where(abs(u) < 1e-160, 0, u)**2, dr))
return u, e
def kin(self, l, u, e=None): # XXX move to Generator
r = self.r[1:]
dr = self.dr[1:]
c0 = 0.5 * l * (l + 1) / r**2
c1 = -0.5 * self.d2gdr2[1:]
c2 = -0.5 * dr**-2
if e is not None and self.scalarrel:
x = 0.5 * alpha**2
Mr = r * (1.0 + x * e) - x * self.vr[1:]
c0 += ((Mr - r) * (self.vr[1:] - e * r) +
0.5 * x * self.r2dvdr[1:] / Mr) / r**2
c1 -= 0.5 * x * self.r2dvdr[1:] / (Mr * dr * r)
fp = c2 + 0.5 * c1
fm = c2 - 0.5 * c1
f0 = c0 - 2 * c2
kr = np.zeros(self.N)
kr[1:] = f0 * u[1:] + fm * u[:-1]
kr[1:-1] += fp[:-1] * u[2:]
kr[0] = 0.0
return kr
def r2g(self, r):
"""Convert radius to index of the radial grid."""
return int(r * self.N / (self.beta + r))
def get_confinement_potential(self, alpha, ri, rc):
"""Create a smooth confinement potential.
Returns a (potential) function which is zero inside the radius ri
and goes to infinity smoothly at rc, after which point it is nan.
The potential is given by::
alpha / rc - ri \
V(r) = -------- exp ( - --------- ) for ri < r < rc
rc - r \ r - ri /
"""
i_ri = self.r2g(ri)
i_rc = self.r2g(rc)
if self.r[i_rc] == rc:
# Avoid division by zero in the odd case that rc coincides
# exactly with a grid point (which actually happens sometimes)
i_rc -= 1
potential = np.zeros(np.shape(self.r))
r = self.r[i_ri + 1:i_rc + 1]
exponent = - (rc - ri) / (r - ri)
denom = rc - r
value = np.exp(exponent) / denom
potential[i_ri + 1:i_rc + 1] = value
potential[i_rc + 1:] = np.inf
return alpha * potential
class AllElectronAtom:
def __init__(self,
symbol,
xc='LDA',
spinpol=False,
dirac=False,
log=sys.stdout):
"""All-electron calculation for spherically symmetric atom.
symbol: str (or int)
Chemical symbol (or atomic number).
xc: str
Name of XC-functional.
spinpol: bool
If true, do spin-polarized calculation. Default is spin-paired.
dirac: bool
Solve Dirac equation instead of Schrödinger equation.
log: stream
Text output."""
if isinstance(symbol, int):
symbol = chemical_symbols[symbol]
self.symbol = symbol
self.Z = atomic_numbers[symbol]
self.nspins = 1 + int(bool(spinpol))
self.dirac = bool(dirac)
if isinstance(xc, str):
self.xc = XC(xc)
else:
self.xc = xc
if log is None:
log = devnull
self.fd = log
self.vr_sg = None # potential * r
self.n_sg = 0.0 # density
self.gd = None # radial grid descriptor
# Energies:
self.ekin = None
self.eeig = None
self.eH = None
self.eZ = None
self.channels = None
self.initialize_configuration()
self.log('Z: ', self.Z)
self.log('Name: ', atomic_names[self.Z])
self.log('Symbol: ', symbol)
self.log('XC-functional: ', self.xc.name)
self.log('Equation: ', ['Schrödinger', 'Dirac'][self.dirac])
def log(self, *args, **kwargs):
self.fd.write(
kwargs.get('sep', ' ').join([str(arg) for arg in args]) +
kwargs.get('end', '\n'))
def initialize_configuration(self):
self.f_lsn = {}
for n, l, f, e in configurations[self.symbol][1]:
if l not in self.f_lsn:
self.f_lsn[l] = [[] for s in range(self.nspins)]
if self.nspins == 1:
self.f_lsn[l][0].append(f)
else:
# Use Hund's rule:
f0 = min(f, 2 * l + 1)
self.f_lsn[l][0].append(f0)
self.f_lsn[l][1].append(f - f0)
def add(self, n, l, df=+1, s=None):
"""Add (remove) electrons."""
if s is None:
if self.nspins == 1:
s = 0
else:
self.add(n, l, 0.5 * df, 0)
self.add(n, l, 0.5 * df, 1)
return
if l not in self.f_lsn:
self.f_lsn[l] = [[] for x in range(self.nspins)]
f_n = self.f_lsn[l][s]
if len(f_n) < n - l:
f_n.extend([0] * (n - l - len(f_n)))
f_n[n - l - 1] += df
def initialize(self,
ngpts=1000,
rcut=50.0,
alpha1=0.01,
alpha2=None,
ngauss=50,
eps=1.0e-7):
"""Initialize basis sets and radial grid.
ngpts: int
Number of grid points for radial grid.
rcut: float
Cutoff for radial grid.
alpha1: float
Smallest exponent for gaussian.
alpha2: float
Largest exponent for gaussian.
ngauss: int
Number of gaussians.
eps: float
Cutoff for eigenvalues of overlap matrix."""
if alpha2 is None:
alpha2 = 50.0 * self.Z**2
self.gd = GridDescriptor(r1=1 / alpha2**0.5 / 50, rN=rcut, N=ngpts)
self.log('Grid points: %d (%.5f, %.5f, %.5f, ..., %.3f, %.3f)' %
((self.gd.N, ) + tuple(self.gd.r_g[[0, 1, 2, -2, -1]])))
# Distribute exponents between alpha1 and alpha2:
alpha_B = alpha1 * (alpha2 / alpha1)**np.linspace(0, 1, ngauss)
self.log('Exponents: %d (%.3f, %.3f, ..., %.3f, %.3f)' %
((ngauss, ) + tuple(alpha_B[[0, 1, -2, -1]])))
# Maximum l value:
lmax = max(self.f_lsn.keys())
self.channels = []
nb_l = []
if not self.dirac:
for l in range(lmax + 1):
basis = GaussianBasis(l, alpha_B, self.gd, eps)
nb_l.append(len(basis))
for s in range(self.nspins):
self.channels.append(Channel(l, s, self.f_lsn[l][s],
basis))
else:
for K in range(1, lmax + 2):
leff = (K**2 - (self.Z / c)**2)**0.5 - 1
basis = GaussianBasis(leff, alpha_B, self.gd, eps)
nb_l.append(len(basis))
for k, l in [(-K, K - 1), (K, K)]:
if l > lmax:
continue
f_n = self.f_lsn[l][0]
j = abs(k) - 0.5
f_n = (2 * j + 1) / (4 * l + 2) * np.array(f_n)
self.channels.append(DiracChannel(k, f_n, basis))
self.log('Basis functions: %s (%s)' %
(', '.join([str(nb)
for nb in nb_l]), ', '.join('spdf'[:lmax + 1])))
self.vr_sg = self.gd.zeros(self.nspins)
self.vr_sg[:] = -self.Z
def solve(self):
"""Diagonalize Schrödinger equation."""
self.eeig = 0.0
for channel in self.channels:
channel.solve(self.vr_sg[channel.s])
self.eeig += channel.get_eigenvalue_sum()
def calculate_density(self):
"""Calculate elctron density and kinetic energy."""
self.n_sg = self.gd.zeros(self.nspins)
for channel in self.channels:
self.n_sg[channel.s] += channel.calculate_density()
def calculate_electrostatic_potential(self):
"""Calculate electrostatic potential and energy."""
n_g = self.n_sg.sum(0)
self.vHr_g = self.gd.poisson(n_g)
self.eH = 0.5 * self.gd.integrate(n_g * self.vHr_g, -1)
self.eZ = -self.Z * self.gd.integrate(n_g, -1)
def calculate_xc_potential(self):
self.vxc_sg = self.gd.zeros(self.nspins)
self.exc = self.xc.calculate_spherical(self.gd, self.n_sg, self.vxc_sg)
def step(self):
self.solve()
self.calculate_density()
self.calculate_electrostatic_potential()
self.calculate_xc_potential()
self.vr_sg = self.vxc_sg * self.gd.r_g
self.vr_sg += self.vHr_g
self.vr_sg -= self.Z
self.ekin = (self.eeig - self.gd.integrate(
(self.vr_sg * self.n_sg).sum(0), -1))
def run(self, mix=0.4, maxiter=117, dnmax=1e-9):
if self.channels is None:
self.initialize()
dn = self.Z
pb = ProgressBar(log(dnmax / dn), 0, 53, self.fd)
self.log()
for iter in range(maxiter):
if iter > 1:
self.vr_sg *= mix
self.vr_sg += (1 - mix) * vr_old_sg
dn = self.gd.integrate(abs(self.n_sg - n_old_sg).sum(0))
pb(log(dnmax / dn))
if dn <= dnmax:
break
vr_old_sg = self.vr_sg
n_old_sg = self.n_sg
self.step()
self.summary()
if dn > dnmax:
raise RuntimeError('Did not converge!')
def summary(self):
self.write_states()
self.write_energies()
def write_states(self):
self.log('\n state occupation eigenvalue <r>')
if self.dirac:
self.log(' nl(j) [Hartree] [eV] [Bohr]')
else:
self.log(' nl [Hartree] [eV] [Bohr]')
self.log('=====================================================')
states = []
for ch in self.channels:
for n, f in enumerate(ch.f_n):
states.append((ch.e_n[n], ch, n))
states.sort()
for e, ch, n in states:
name = str(n + ch.l + 1) + ch.name
if self.nspins == 2:
name += '(%s)' % '+-'[ch.s]
n_g = ch.calculate_density(n)
rave = self.gd.integrate(n_g, 1)
self.log(' %-7s %6.3f %13.6f %13.5f %6.3f' %
(name, ch.f_n[n], e, e * units.Hartree, rave))
self.log('=====================================================')
def write_energies(self):
self.log('\nEnergies: [Hartree] [eV]')
self.log('============================================')
for text, e in [
('kinetic ', self.ekin), ('coulomb (e-e)', self.eH),
('coulomb (e-n)', self.eZ), ('xc ', self.exc),
('total ', self.ekin + self.eH + self.eZ + self.exc)
]:
self.log(' %s %+13.6f %+13.5f' % (text, e, units.Hartree * e))
self.log('============================================')
def get_channel(self, l=None, s=0, k=None):
if self.dirac:
for channel in self.channels:
if channel.k == k:
return channel
else:
for channel in self.channels:
if channel.l == l and channel.s == s:
return channel
raise ValueError
def get_orbital(self, n, l=None, s=0, k=None):
channel = self.get_channel(l, s, k)
return channel.basis.expand(channel.C_nb[n])
def plot_wave_functions(self, rc=4.0):
import matplotlib.pyplot as plt
colors = 'krgbycm'
for ch in self.channels:
for n in range(len(ch.f_n)):
fr_g = ch.basis.expand(ch.C_nb[n]) * self.gd.r_g
name = str(n + ch.l + 1) + ch.name
lw = 2
if self.nspins == 2:
name += '(%s)' % '+-'[ch.s]
if ch.s == 1:
lw = 1
if self.dirac and ch.k > 0:
lw = 1
ls = ['-', '--', '-.', ':'][ch.l]
n_g = ch.calculate_density(n)
rave = self.gd.integrate(n_g, 1)
gave = self.gd.get_index(rave)
fr_g *= cmp(fr_g[gave], 0)
plt.plot(self.gd.r_g,
fr_g,
ls=ls,
lw=lw,
color=colors[n + ch.l],
label=name)
plt.legend(loc='best')
plt.axis(xmax=rc)
plt.show()
def logarithmic_derivative(self, l, energies, rcut):
vr = splrep(self.gd.r_g, self.vr_sg[0])
def v(r):
return splev(r, vr) / r
def f(y, r, e):
if r == 0:
return [y[1], -2.0]
return [y[1], 2 * (v(r) - e) * y[0]]
logderivs = []
for e in energies:
u, dudr = odeint(f, [0, 1], [0, rcut], (e, ))[1, :]
logderivs.append(dudr / u)
return logderivs
class AllElectronAtom:
def __init__(self, symbol, xc='LDA', spinpol=False, dirac=False,
log=sys.stdout):
"""All-electron calculation for spherically symmetric atom.
symbol: str (or int)
Chemical symbol (or atomic number).
xc: str
Name of XC-functional.
spinpol: bool
If true, do spin-polarized calculation. Default is spin-paired.
dirac: bool
Solve Dirac equation instead of Schrödinger equation.
log: stream
Text output."""
if isinstance(symbol, int):
symbol = chemical_symbols[symbol]
self.symbol = symbol
self.Z = atomic_numbers[symbol]
self.nspins = 1 + int(bool(spinpol))
self.dirac = bool(dirac)
if isinstance(xc, str):
self.xc = XC(xc)
else:
self.xc = xc
if log is None:
log = devnull
self.fd = log
self.vr_sg = None # potential * r
self.n_sg = 0.0 # density
self.gd = None # radial grid descriptor
# Energies:
self.ekin = None
self.eeig = None
self.eH = None
self.eZ = None
self.channels = None
self.initialize_configuration()
self.log('Z: ', self.Z)
self.log('Name: ', atomic_names[self.Z])
self.log('Symbol: ', symbol)
self.log('XC-functional: ', self.xc.name)
self.log('Equation: ', ['Schrödinger', 'Dirac'][self.dirac])
def log(self, *args, **kwargs):
self.fd.write(kwargs.get('sep', ' ').join([str(arg) for arg in args]) +
kwargs.get('end', '\n'))
def initialize_configuration(self):
self.f_lsn = {}
for n, l, f, e in configurations[self.symbol][1]:
if l not in self.f_lsn:
self.f_lsn[l] = [[] for s in range(self.nspins)]
if self.nspins == 1:
self.f_lsn[l][0].append(f)
else:
# Use Hund's rule:
f0 = min(f, 2 * l + 1)
self.f_lsn[l][0].append(f0)
self.f_lsn[l][1].append(f - f0)
def add(self, n, l, df=+1, s=None):
"""Add (remove) electrons."""
if s is None:
if self.nspins == 1:
s = 0
else:
self.add(n, l, 0.5 * df, 0)
self.add(n, l, 0.5 * df, 1)
return
if l not in self.f_lsn:
self.f_lsn[l] = [[] for x in range(self.nspins)]
f_n = self.f_lsn[l][s]
if len(f_n) < n - l:
f_n.extend([0] * (n - l - len(f_n)))
f_n[n - l - 1] += df
def initialize(self, ngpts=1000, rcut=50.0,
alpha1=0.01, alpha2=None, ngauss=50,
eps=1.0e-7):
"""Initialize basis sets and radial grid.
ngpts: int
Number of grid points for radial grid.
rcut: float
Cutoff for radial grid.
alpha1: float
Smallest exponent for gaussian.
alpha2: float
Largest exponent for gaussian.
ngauss: int
Number of gaussians.
eps: float
Cutoff for eigenvalues of overlap matrix."""
if alpha2 is None:
alpha2 = 50.0 * self.Z**2
self.gd = GridDescriptor(r1=1 / alpha2**0.5 / 50, rN=rcut, N=ngpts)
self.log('Grid points: %d (%.5f, %.5f, %.5f, ..., %.3f, %.3f)' %
((self.gd.N,) + tuple(self.gd.r_g[[0, 1, 2, -2, -1]])))
# Distribute exponents between alpha1 and alpha2:
alpha_B = alpha1 * (alpha2 / alpha1)**np.linspace(0, 1, ngauss)
self.log('Exponents: %d (%.3f, %.3f, ..., %.3f, %.3f)' %
((ngauss,) + tuple(alpha_B[[0, 1, -2, -1]])))
# Maximum l value:
lmax = max(self.f_lsn.keys())
self.channels = []
nb_l = []
if not self.dirac:
for l in range(lmax + 1):
basis = GaussianBasis(l, alpha_B, self.gd, eps)
nb_l.append(len(basis))
for s in range(self.nspins):
self.channels.append(Channel(l, s, self.f_lsn[l][s],
basis))
else:
for K in range(1, lmax + 2):
leff = (K**2 - (self.Z / c)**2)**0.5 - 1
basis = GaussianBasis(leff, alpha_B, self.gd, eps)
nb_l.append(len(basis))
for k, l in [(-K, K - 1), (K, K)]:
if l > lmax:
continue
f_n = self.f_lsn[l][0]
j = abs(k) - 0.5
f_n = (2 * j + 1) / (4 * l + 2) * np.array(f_n)
self.channels.append(DiracChannel(k, f_n, basis))
self.log('Basis functions: %s (%s)' %
(', '.join([str(nb) for nb in nb_l]),
', '.join('spdf'[:lmax + 1])))
self.vr_sg = self.gd.zeros(self.nspins)
self.vr_sg[:] = -self.Z
def solve(self):
"""Diagonalize Schrödinger equation."""
self.eeig = 0.0
for channel in self.channels:
channel.solve(self.vr_sg[channel.s])
self.eeig += channel.get_eigenvalue_sum()
def calculate_density(self):
"""Calculate elctron density and kinetic energy."""
self.n_sg = self.gd.zeros(self.nspins)
for channel in self.channels:
self.n_sg[channel.s] += channel.calculate_density()
def calculate_electrostatic_potential(self):
"""Calculate electrostatic potential and energy."""
n_g = self.n_sg.sum(0)
self.vHr_g = self.gd.poisson(n_g)
self.eH = 0.5 * self.gd.integrate(n_g * self.vHr_g, -1)
self.eZ = -self.Z * self.gd.integrate(n_g, -1)
def calculate_xc_potential(self):
self.vxc_sg = self.gd.zeros(self.nspins)
self.exc = self.xc.calculate_spherical(self.gd, self.n_sg, self.vxc_sg)
def step(self):
self.solve()
self.calculate_density()
self.calculate_electrostatic_potential()
self.calculate_xc_potential()
self.vr_sg = self.vxc_sg * self.gd.r_g
self.vr_sg += self.vHr_g
self.vr_sg -= self.Z
self.ekin = (self.eeig -
self.gd.integrate((self.vr_sg * self.n_sg).sum(0), -1))
def run(self, mix=0.4, maxiter=117, dnmax=1e-9):
if self.channels is None:
self.initialize()
dn = self.Z
pb = ProgressBar(log(dnmax / dn), 0, 53, self.fd)
self.log()
for iter in range(maxiter):
if iter > 1:
self.vr_sg *= mix
self.vr_sg += (1 - mix) * vr_old_sg
dn = self.gd.integrate(abs(self.n_sg - n_old_sg).sum(0))
pb(log(dnmax / dn))
if dn <= dnmax:
break
vr_old_sg = self.vr_sg
n_old_sg = self.n_sg
self.step()
self.summary()
if dn > dnmax:
raise RuntimeError('Did not converge!')
def summary(self):
self.write_states()
self.write_energies()
def write_states(self):
self.log('\n state occupation eigenvalue <r>')
if self.dirac:
self.log(' nl(j) [Hartree] [eV] [Bohr]')
else:
self.log(' nl [Hartree] [eV] [Bohr]')
self.log('=====================================================')
states = []
for ch in self.channels:
for n, f in enumerate(ch.f_n):
states.append((ch.e_n[n], ch, n))
states.sort()
for e, ch, n in states:
name = str(n + ch.l + 1) + ch.name
if self.nspins == 2:
name += '(%s)' % '+-'[ch.s]
n_g = ch.calculate_density(n)
rave = self.gd.integrate(n_g, 1)
self.log(' %-7s %6.3f %13.6f %13.5f %6.3f' %
(name, ch.f_n[n], e, e * units.Hartree, rave))
self.log('=====================================================')
def write_energies(self):
self.log('\nEnergies: [Hartree] [eV]')
self.log('============================================')
for text, e in [('kinetic ', self.ekin),
('coulomb (e-e)', self.eH),
('coulomb (e-n)', self.eZ),
('xc ', self.exc),
('total ',
self.ekin + self.eH + self.eZ + self.exc)]:
self.log(' %s %+13.6f %+13.5f' % (text, e, units.Hartree * e))
self.log('============================================')
def get_channel(self, l=None, s=0, k=None):
if self.dirac:
for channel in self.channels:
if channel.k == k:
return channel
else:
for channel in self.channels:
if channel.l == l and channel.s == s:
return channel
raise ValueError
def get_orbital(self, n, l=None, s=0, k=None):
channel = self.get_channel(l, s, k)
return channel.basis.expand(channel.C_nb[n])
def plot_wave_functions(self, rc=4.0):
import matplotlib.pyplot as plt
colors = 'krgbycm'
for ch in self.channels:
for n in range(len(ch.f_n)):
fr_g = ch.basis.expand(ch.C_nb[n]) * self.gd.r_g
name = str(n + ch.l + 1) + ch.name
lw = 2
if self.nspins == 2:
name += '(%s)' % '+-'[ch.s]
if ch.s == 1:
lw = 1
if self.dirac and ch.k > 0:
lw = 1
ls = ['-', '--', '-.', ':'][ch.l]
n_g = ch.calculate_density(n)
rave = self.gd.integrate(n_g, 1)
gave = self.gd.get_index(rave)
fr_g *= cmp(fr_g[gave], 0)
plt.plot(self.gd.r_g, fr_g,
ls=ls, lw=lw, color=colors[n + ch.l], label=name)
plt.legend(loc='best')
plt.axis(xmax=rc)
plt.show()
def logarithmic_derivative(self, l, energies, rcut):
vr = splrep(self.gd.r_g, self.vr_sg[0])
def v(r):
return splev(r, vr) / r
def f(y, r, e):
if r == 0:
return [y[1], -2.0]
return [y[1], 2 * (v(r) - e) * y[0]]
logderivs = []
for e in energies:
u, dudr = odeint(f, [0, 1], [0, rcut], (e,))[1, :]
logderivs.append(dudr / u)
return logderivs
class C_XC(Contribution):
def __init__(self, nlfunc, weight, functional = 'LDA'):
Contribution.__init__(self, nlfunc, weight)
self.functional = functional
def get_name(self):
return 'XC'
def get_desc(self):
return "("+self.functional+")"
def initialize(self):
self.xc = XC(self.functional)
self.vt_sg = self.nlfunc.finegd.empty(self.nlfunc.nspins)
self.e_g = self.nlfunc.finegd.empty()
def initialize_1d(self):
self.ae = self.nlfunc.ae
self.xc = XC(self.functional)
self.v_g = np.zeros(self.ae.N)
def calculate_spinpaired(self, e_g, n_g, v_g):
self.e_g[:] = 0.0
self.vt_sg[:] = 0.0
self.xc.calculate(self.nlfunc.finegd, n_g[None, ...], self.vt_sg,
self.e_g)
v_g += self.weight * self.vt_sg[0]
e_g += self.weight * self.e_g
def calculate_spinpolarized(self, e_g, na_g, va_g, nb_g, vb_g):
self.e_g[:] = 0.0
self.vt_sg[:] = 0.0
self.xc.get_energy_and_potential(na_g, self.vt_sg[0], nb_g, self.vt_sg[1], e_g=self.e_g)
va_g += self.weight * self.vt_sg[0]
vb_g += self.weight * self.vt_sg[1]
e_g += (self.weight * self.e_g).ravel()
def calculate_energy_and_derivatives(self, D_sp, H_sp, a):
# Get the XC-correction instance
c = self.nlfunc.setups[a].xc_correction
assert self.nlfunc.nspins == 1
D_p = D_sp[0]
dEdD_p = H_sp[0][:]
D_Lq = dot3(c.B_pqL.T, D_p)
n_Lg = np.dot(D_Lq, c.n_qg)
n_Lg[0] += c.nc_g * sqrt(4 * pi)
nt_Lg = np.dot(D_Lq, c.nt_qg)
nt_Lg[0] += c.nct_g * sqrt(4 * pi)
dndr_Lg = np.zeros((c.Lmax, c.ng))
dntdr_Lg = np.zeros((c.Lmax, c.ng))
for L in range(c.Lmax):
c.rgd.derivative(n_Lg[L], dndr_Lg[L])
c.rgd.derivative(nt_Lg[L], dntdr_Lg[L])
E = 0
vt_g = np.zeros(c.ng)
v_g = np.zeros(c.ng)
e_g = np.zeros(c.ng)
y = 0
for w, Y_L in zip(weight_n, c.Y_nL):
A_Li = rnablaY_nLv[y, :c.Lmax]
a1x_g = np.dot(A_Li[:, 0], n_Lg)
a1y_g = np.dot(A_Li[:, 1], n_Lg)
a1z_g = np.dot(A_Li[:, 2], n_Lg)
a2_g = a1x_g**2 + a1y_g**2 + a1z_g**2
a2_g[1:] /= c.rgd.r_g[1:]**2
a2_g[0] = a2_g[1]
a1_g = np.dot(Y_L, dndr_Lg)
a2_g += a1_g**2
deda2_g = np.zeros(c.ng)
v_g[:] = 0.0
e_g[:] = 0.0
n_g = np.dot(Y_L, n_Lg)
self.xc.kernel.calculate(e_g, n_g.reshape((1, -1)),
v_g.reshape((1, -1)),
a2_g.reshape((1, -1)),
deda2_g.reshape((1, -1)))
E += w * np.dot(e_g, c.rgd.dv_g)
x_g = -2.0 * deda2_g * c.rgd.dv_g * a1_g
c.rgd.derivative2(x_g, x_g)
x_g += v_g * c.rgd.dv_g
dEdD_p += self.weight * w * np.dot(dot3(c.B_pqL, Y_L),
np.dot(c.n_qg, x_g))
x_g = 8.0 * pi * deda2_g * c.rgd.dr_g
dEdD_p += w * np.dot(dot3(c.B_pqL,
A_Li[:, 0]),
np.dot(c.n_qg, x_g * a1x_g))
dEdD_p += w * np.dot(dot3(c.B_pqL,
A_Li[:, 1]),
np.dot(c.n_qg, x_g * a1y_g))
dEdD_p += w * np.dot(dot3(c.B_pqL,
A_Li[:, 2]),
np.dot(c.n_qg, x_g * a1z_g))
n_g = np.dot(Y_L, nt_Lg)
a1x_g = np.dot(A_Li[:, 0], nt_Lg)
a1y_g = np.dot(A_Li[:, 1], nt_Lg)
a1z_g = np.dot(A_Li[:, 2], nt_Lg)
a2_g = a1x_g**2 + a1y_g**2 + a1z_g**2
a2_g[1:] /= c.rgd.r_g[1:]**2
a2_g[0] = a2_g[1]
a1_g = np.dot(Y_L, dntdr_Lg)
a2_g += a1_g**2
v_g = np.zeros(c.ng)
e_g = np.zeros(c.ng)
deda2_g = np.zeros(c.ng)
v_g[:] = 0.0
e_g[:] = 0.0
self.xc.kernel.calculate(e_g, n_g.reshape((1, -1)),
v_g.reshape((1, -1)),
a2_g.reshape((1, -1)),
deda2_g.reshape((1, -1)))
E -= w * np.dot(e_g, c.dv_g)
x_g = -2.0 * deda2_g * c.dv_g * a1_g
c.rgd.derivative2(x_g, x_g)
x_g += v_g * c.dv_g
B_Lqp = c.B_pqL.T
dEdD_p -= w * np.dot(dot3(c.B_pqL, Y_L),
np.dot(c.nt_qg, x_g))
x_g = 8.0 * pi * deda2_g * c.rgd.dr_g
dEdD_p -= w * np.dot(dot3(c.B_pqL,
A_Li[:, 0]),
np.dot(c.nt_qg, x_g * a1x_g))
dEdD_p -= w * np.dot(dot3(c.B_pqL,
A_Li[:, 1]),
np.dot(c.nt_qg, x_g * a1y_g))
dEdD_p -= w * np.dot(dot3(c.B_pqL,
A_Li[:, 2]),
np.dot(c.nt_qg, x_g * a1z_g))
y += 1
return (E) * self.weight
def add_xc_potential_and_energy_1d(self, v_g):
self.v_g[:] = 0.0
Exc = self.xc.calculate_spherical(self.ae.rgd,
self.ae.n.reshape((1, -1)),
self.v_g.reshape((1, -1)))
v_g += self.weight * self.v_g
return self.weight * Exc
def add_smooth_xc_potential_and_energy_1d(self, vt_g):
self.v_g[:] = 0.0
Exc = self.xc.calculate_spherical(self.ae.rgd,
self.ae.nt.reshape((1, -1)),
self.v_g.reshape((1, -1)))
vt_g += self.weight * self.v_g
return self.weight * Exc
def initialize_from_atomic_orbitals(self, basis_functions):
# LDA needs only density, which is already initialized
pass
def add_extra_setup_data(self, dict):
# LDA has not any special data
pass
def write(self, writer, natoms):
# LDA has not any special data to be written
pass
def read(self, reader):
# LDA has not any special data to be read
pass
class C_GLLBScr(Contribution):
def __init__(self, nlfunc, weight, functional='GGA_X_B88'):
Contribution.__init__(self, nlfunc, weight)
self.functional = functional
self.old_coeffs = None
self.iter = 0
def get_name(self):
return 'SCREENING'
def get_desc(self):
return '(' + self.functional + ')'
# Initialize GLLBScr functional
def initialize_1d(self):
self.ae = self.nlfunc.ae
self.xc = XC(self.functional)
self.v_g = np.zeros(self.ae.N)
self.e_g = np.zeros(self.ae.N)
# Calcualte the GLLB potential and energy 1d
def add_xc_potential_and_energy_1d(self, v_g):
self.v_g[:] = 0.0
self.e_g[:] = 0.0
self.xc.calculate_spherical(self.ae.rgd, self.ae.n.reshape((1, -1)),
self.v_g.reshape((1, -1)), self.e_g)
v_g += 2 * self.weight * self.e_g / (self.ae.n + 1e-10)
Exc = self.weight * np.sum(self.e_g * self.ae.rgd.dv_g)
return Exc
def initialize(self):
self.occupations = self.nlfunc.occupations
self.xc = XC(self.functional)
self.vt_sg = self.nlfunc.finegd.empty(self.nlfunc.nspins)
self.e_g = self.nlfunc.finegd.empty()#.ravel()
def get_coefficient_calculator(self):
return self
def f(self, f):
return sqrt(f)
def get_coefficients_1d(self, smooth=False, lumo_perturbation = False):
homo_e = max( [ np.where(f>1e-3, e, -1000) for f,e in zip(self.ae.f_j, self.ae.e_j)])
if not smooth:
if lumo_perturbation:
lumo_e = min( [ np.where(f<1e-3, e, 1000) for f,e in zip(self.ae.f_j, self.ae.e_j)])
return np.array([ f * K_G * (self.f( max(0, lumo_e - e)) - self.f(max(0, homo_e -e)))
for e,f in zip(self.ae.e_j, self.ae.f_j) ])
else:
return np.array([ f * K_G * (self.f( max(0, homo_e - e)))
for e,f in zip(self.ae.e_j, self.ae.f_j) ])
else:
return [ [ f * K_G * self.f( max(0, homo_e - e))
for e,f in zip(e_n, f_n) ]
for e_n, f_n in zip(self.ae.e_ln, self.ae.f_ln) ]
def get_coefficients_by_kpt(self, kpt_u, lumo_perturbation=False, homolumo=None):
if kpt_u[0].psit_nG is None or isinstance(kpt_u[0].psit_nG,
TarFileReference):
return None
if homolumo == None:
e_ref, e_ref_lumo = self.occupations.get_homo_lumo(self.nlfunc.wfs)
else:
e_ref, e_ref_lumo = homolumo
# The parameter ee might sometimes be set to small thereshold value to
# achieve convergence on systems with degenerate H**O.
if len(kpt_u) > 1:
ee = 0.0
else:
ee = 0.1 / 27.21
if lumo_perturbation:
return [np.array([
f * K_G * (self.f( np.where(e_ref_lumo - e>ee, e_ref_lumo-e,0))
-self.f( np.where(e_ref - e>ee, e_ref-e,0)))
for e, f in zip(kpt.eps_n, kpt.f_n) ])
for kpt in kpt_u ]
else:
coeff = [ np.array([ f * K_G * self.f( np.where(e_ref - e>ee, e_ref-e,0))
for e, f in zip(kpt.eps_n, kpt.f_n) ])
for kpt in kpt_u ]
if self.old_coeffs is None:
self.old_coeffs = coeff
else:
# Mix the coefficients with 25%
mix = 0.25
self.old_coeffs = [ (1-mix) * old + mix * new for new, old in zip(coeff, self.old_coeffs) ]
return self.old_coeffs
def calculate_spinpaired(self, e_g, n_g, v_g):
self.e_g[:] = 0.0
self.vt_sg[:] = 0.0
self.xc.calculate(self.nlfunc.finegd, n_g[None, ...], self.vt_sg,
self.e_g)
v_g += self.weight * 2 * self.e_g / (n_g + 1e-10)
e_g += self.weight * self.e_g
def calculate_spinpolarized(self, e_g, na_g, va_g, nb_g, vb_g,
a2_g=None, aa2_g=None, ab2_g=None, deda2_g=None,
dedaa2_g=None, dedab2_g=None):
raise NotImplementedError
def calculate_energy_and_derivatives(self, D_sp, H_sp, a):
# Get the XC-correction instance
c = self.nlfunc.setups[a].xc_correction
assert self.nlfunc.nspins == 1
D_p = D_sp[0]
dEdD_p = H_sp[0][:]
D_Lq = np.dot(c.B_pqL.T, D_p)
n_Lg = np.dot(D_Lq, c.n_qg)
n_Lg[0] += c.nc_g * sqrt(4 * pi)
nt_Lg = np.dot(D_Lq, c.nt_qg)
nt_Lg[0] += c.nct_g * sqrt(4 * pi)
dndr_Lg = np.zeros((c.Lmax, c.ng))
dntdr_Lg = np.zeros((c.Lmax, c.ng))
for L in range(c.Lmax):
c.rgd.derivative(n_Lg[L], dndr_Lg[L])
c.rgd.derivative(nt_Lg[L], dntdr_Lg[L])
E = 0
vt_g = np.zeros(c.ng)
v_g = np.zeros(c.ng)
e_g = np.zeros(c.ng)
deda2_g = np.zeros(c.ng)
for y, (w, Y_L) in enumerate(zip(weight_n, c.Y_nL)):
# Cut gradient releated coefficient to match the setup's Lmax
A_Li = rnablaY_nLv[y, :c.Lmax]
# Expand pseudo density
nt_g = np.dot(Y_L, nt_Lg)
# Expand pseudo density gradient
a1x_g = np.dot(A_Li[:, 0], nt_Lg)
a1y_g = np.dot(A_Li[:, 1], nt_Lg)
a1z_g = np.dot(A_Li[:, 2], nt_Lg)
a2_g = a1x_g**2 + a1y_g**2 + a1z_g**2
a2_g[1:] /= c.rgd.r_g[1:]**2
a2_g[0] = a2_g[1]
a1_g = np.dot(Y_L, dntdr_Lg)
a2_g += a1_g**2
vt_g[:] = 0.0
e_g[:] = 0.0
# Calculate pseudo GGA energy density (potential is discarded)
self.xc.kernel.calculate(e_g, nt_g.reshape((1, -1)),
vt_g.reshape((1, -1)),
a2_g.reshape((1, -1)),
deda2_g.reshape((1, -1)))
# Calculate pseudo GLLB-potential from GGA-energy density
vt_g[:] = 2 * e_g / (nt_g + 1e-10)
dEdD_p -= self.weight * w * np.dot(np.dot(c.B_pqL, Y_L),
np.dot(c.nt_qg, vt_g * c.rgd.dv_g))
E -= w * np.dot(e_g, c.rgd.dv_g)
# Expand density
n_g = np.dot(Y_L, n_Lg)
# Expand density gradient
a1x_g = np.dot(A_Li[:, 0], n_Lg)
a1y_g = np.dot(A_Li[:, 1], n_Lg)
a1z_g = np.dot(A_Li[:, 2], n_Lg)
a2_g = a1x_g**2 + a1y_g**2 + a1z_g**2
a2_g[1:] /= c.rgd.r_g[1:]**2
a2_g[0] = a2_g[1]
a1_g = np.dot(Y_L, dndr_Lg)
a2_g += a1_g**2
v_g[:] = 0.0
e_g[:] = 0.0
# Calculate GGA energy density (potential is discarded)
self.xc.kernel.calculate(e_g, n_g.reshape((1, -1)),
v_g.reshape((1, -1)),
a2_g.reshape((1, -1)),
deda2_g.reshape((1, -1)))
# Calculate GLLB-potential from GGA-energy density
v_g[:] = 2 * e_g / (n_g + 1e-10)
dEdD_p += self.weight * w * np.dot(np.dot(c.B_pqL, Y_L),
np.dot(c.n_qg, v_g * c.rgd.dv_g))
E += w * np.dot(e_g, c.rgd.dv_g)
return (E) * self.weight
def add_smooth_xc_potential_and_energy_1d(self, vt_g):
self.v_g[:] = 0.0
self.e_g[:] = 0.0
self.xc.calculate_spherical(self.ae.rgd, self.ae.nt.reshape((1, -1)),
self.v_g.reshape((1, -1)), self.e_g)
vt_g += 2 * self.weight * self.e_g / (self.ae.nt + 1e-10)
return self.weight * np.sum(self.e_g * self.ae.rgd.dv_g)
def initialize_from_atomic_orbitals(self, basis_functions):
# GLLBScr needs only density which is already initialized
pass
def add_extra_setup_data(self, dict):
# GLLBScr has not any special data
pass
def read(self, reader):
# GLLBScr has no special data to be read
pass
def write(self, writer, natoms):
# GLLBScr has no special data to be written
pass