def calculate(self, e_g, n_sg, dedn_sg,
sigma_xg=None, dedsigma_xg=None,
tau_sg=None, dedtau_sg=None):
n_g = n_sg[0]
m_vg = n_sg[1:4]
m_g = (m_vg**2).sum(0)**0.5
nnew_sg = np.empty((2,) + n_g.shape)
nnew_sg[:] = n_g
nnew_sg[0] += m_g
nnew_sg[1] -= m_g
nnew_sg *= 0.5
vnew_sg = np.zeros_like(nnew_sg)
LibXC.calculate(self, e_g, nnew_sg, vnew_sg)
dedn_sg[0] += 0.5 * vnew_sg.sum(0)
vnew_sg /= np.where(m_g < 1e-15, 1, m_g)
dedn_sg[1:4] += 0.5 * vnew_sg[0] * m_vg
dedn_sg[1:4] -= 0.5 * vnew_sg[1] * m_vg
def calculate(self,
e_g,
n_sg,
dedn_sg,
sigma_xg=None,
dedsigma_xg=None,
tau_sg=None,
dedtau_sg=None):
n_g = n_sg[0]
m_vg = n_sg[1:4]
m_g = (m_vg**2).sum(0)**0.5
nnew_sg = np.empty((2, ) + n_g.shape)
nnew_sg[:] = n_g
nnew_sg[0] += m_g
nnew_sg[1] -= m_g
nnew_sg *= 0.5
vnew_sg = np.zeros_like(nnew_sg)
LibXC.calculate(self, e_g, nnew_sg, vnew_sg)
dedn_sg[0] += 0.5 * vnew_sg.sum(0)
vnew_sg /= np.where(m_g < 1e-15, 1, m_g)
dedn_sg[1:4] += 0.5 * vnew_sg[0] * m_vg
dedn_sg[1:4] -= 0.5 * vnew_sg[1] * m_vg
class VDWFunctional(GGA):
"""Base class for vdW-DF."""
def __init__(
self,
name,
world=None,
q0cut=5.0,
phi0=0.5,
ds=1.0,
Dmax=20.0,
nD=201,
ndelta=21,
soft_correction=False,
kernel=None,
Zab=None,
vdwcoef=1.0,
verbose=False,
energy_only=False,
):
"""vdW-DF.
parameters:
name: str
Name of functional.
world: MPI communicator
Communicator to parallelize over. Defaults to gpaw.mpi.world.
q0cut: float
Maximum value for q0.
phi0: float
Smooth value for phi(0,0).
ds: float
Cutoff for smooth kernel.
Dmax: float
Maximum value for D.
nD: int
Number of values for D in kernel-table.
ndelta: int
Number of values for delta in kernel-table.
soft_correction: bool
Correct for soft kernel.
kernel:
Which kernel to use.
Zab:
parameter in nonlocal kernel.
vdwcoef: float
Scaling of vdW energy.
verbose: bool
Print useful information.
"""
if world is None:
self.world = mpi.world
else:
self.world = world
self.q0cut = q0cut
self.phi0 = phi0
self.ds = ds
self.delta_i = np.linspace(0, 1.0, ndelta)
self.D_j = np.linspace(0, Dmax, nD)
self.verbose = verbose
self.read_table()
self.soft_correction = soft_correction
if soft_correction:
dD = self.D_j[1]
self.C_soft = np.dot(self.D_j ** 2, self.phi_ij[0]) * 4 * pi * dD
self.gd = None
self.energy_only = energy_only
self.timer = nulltimer
if name == "vdW-DF":
assert kernel is None and Zab is None
kernel = LibXC("GGA_X_PBE_R+LDA_C_PW")
Zab = -0.8491
elif name == "vdW-DF2":
assert kernel is None and Zab is None
kernel = LibXC("GGA_X_RPW86+LDA_C_PW")
Zab = -1.887
elif name == "optPBE-vdW":
assert kernel is None and Zab is None
kernel = LibXC("GGA_X_OPTPBE_VDW+LDA_C_PW")
Zab = -0.8491
elif name == "optB88-vdW":
assert kernel is None and Zab is None
kernel = LibXC("GGA_X_OPTB88_VDW+LDA_C_PW")
Zab = -0.8491
elif name == "C09-vdW":
assert kernel is None and Zab is None
kernel = LibXC("GGA_X_C09X+LDA_C_PW")
Zab = -0.8491
else:
assert kernel is not None and Zab is not None
self.Zab = Zab
GGA.__init__(self, kernel)
self.vdwcoef = vdwcoef
self.name = name
self.LDAc = LibXC("LDA_C_PW")
def get_setup_name(self):
return "revPBE"
def initialize(self, density, hamiltonian, wfs, occupations):
self.timer = wfs.timer
def get_Ecnl(self):
return self.Ecnl
def calculate_gga(self, e_g, n_sg, dedn_sg, sigma_xg, dedsigma_xg):
GGA.calculate_gga(self, e_g, n_sg, dedn_sg, sigma_xg, dedsigma_xg)
eLDAc_g = self.gd.empty()
vLDAc_sg = self.gd.zeros(1)
if self.vdwcoef == 0.0:
return
if len(n_sg) == 1:
self.LDAc.calculate(eLDAc_g, n_sg, vLDAc_sg)
e = self.get_non_local_energy(n_sg[0], sigma_xg[0], eLDAc_g, vLDAc_sg[0], dedn_sg[0], dedsigma_xg[0])
else:
n_sg = n_sg.sum(0)
n_sg.shape = (1,) + n_sg.shape
self.LDAc.calculate(eLDAc_g, n_sg, vLDAc_sg)
v_g = np.zeros_like(e_g)
deda2nl_g = np.zeros_like(e_g)
a2_g = sigma_xg[0] + 2 * sigma_xg[1] + sigma_xg[2]
e = self.get_non_local_energy(n_sg[0], a2_g, eLDAc_g, vLDAc_sg[0], v_g, deda2nl_g)
dedsigma_xg[0] += self.vdwcoef * deda2nl_g
dedsigma_xg[1] += self.vdwcoef * 2 * deda2nl_g
dedsigma_xg[2] += self.vdwcoef * deda2nl_g
dedn_sg += self.vdwcoef * v_g
if self.gd.comm.rank == 0:
e_g[0, 0, 0] += self.vdwcoef * e / self.gd.dv
def get_non_local_energy(self, n_g=None, a2_g=None, e_LDAc_g=None, v_LDAc_g=None, v_g=None, deda2_g=None):
"""Calculate non-local correlation energy.
parameters:
n_g: ndarray
Density.
a2_g: ndarray
Absolute value of the gradient of the density - squared.
e_LDAc_g: ndarray
LDA correlation energy density.
"""
gd = self.gd
n_g = n_g.clip(1e-7, np.inf)
# Calculate q0 and cut it off smoothly at q0cut:
kF_g = (3 * pi ** 2 * n_g) ** (1.0 / 3.0)
q0_g, dhdx_g = hRPS(kF_g - 4 * pi / 3 * e_LDAc_g / n_g - self.Zab / 36 / kF_g * a2_g / n_g ** 2, self.q0cut)
if self.verbose:
print ("VDW: q0 (min, mean, max): (%f, %f, %f)" % (q0_g.min(), q0_g.mean(), q0_g.max()))
if self.soft_correction:
dEcnl = gd.integrate(n_g ** 2 / q0_g ** 3) * 0.5 * self.C_soft
else:
dEcnl = 0.0
# Distribute density and q0 to all processors:
n_g = gd.collect(n_g, broadcast=True)
q0_g = gd.collect(q0_g, broadcast=True)
if not self.energy_only:
self.dhdx_g = gd.collect(dhdx_g, broadcast=True)
Ecnl = self.calculate_6d_integral(n_g, q0_g, a2_g, e_LDAc_g, v_LDAc_g, v_g, deda2_g)
Ecnl += dEcnl
self.Ecnl = Ecnl
return Ecnl
def read_table(self):
name = "phi-%.3f-%.3f-%.3f-%d-%d.pckl" % (self.phi0, self.ds, self.D_j[-1], len(self.delta_i), len(self.D_j))
if "GPAW_VDW" in os.environ:
print "Use of GPAW_VDW is deprecated."
print "Put", name, "in your GPAW_SETUP_PATH directory."
dirs = [os.environ["GPAW_VDW"]]
else:
dirs = setup_paths + ["."]
for dir in dirs:
filename = os.path.join(dir, name)
if os.path.isfile(filename):
self.phi_ij = pickle.load(open(filename))
if self.verbose:
print "VDW: using", filename
return
print "VDW: Could not find table file:", name
self.make_table(name)
def make_table(self, name):
print "VDW: Generating vdW-DF kernel ..."
print "VDW:",
ndelta = len(self.delta_i)
nD = len(self.D_j)
self.phi_ij = np.zeros((ndelta, nD))
for i in range(self.world.rank, ndelta, self.world.size):
print ndelta - i,
sys.stdout.flush()
delta = self.delta_i[i]
for j in range(nD - 1, -1, -1):
D = self.D_j[j]
d = D * (1.0 + delta)
dp = D * (1.0 - delta)
if d ** 2 + dp ** 2 > self.ds ** 2:
self.phi_ij[i, j] = phi(d, dp)
else:
P = np.polyfit(
[0, self.D_j[j + 1] ** 2, self.D_j[j + 2] ** 2],
[self.phi0, self.phi_ij[i, j + 1], self.phi_ij[i, j + 2]],
2,
)
self.phi_ij[i, : j + 3] = np.polyval(P, self.D_j[: j + 3] ** 2)
break
self.world.sum(self.phi_ij)
print
print "VDW: Done!"
if self.world.rank == 0:
pickle.dump(self.phi_ij, open(name, "w"), pickle.HIGHEST_PROTOCOL)
def make_prl_plot(self, multiply_by_4_pi_D_squared=True):
import pylab as plt
x = np.linspace(0, 8.0, 100)
for delta in [0, 0.5, 0.9]:
y = [self.phi(D * (1.0 + delta), D * (1.0 - delta)) for D in x]
if multiply_by_4_pi_D_squared:
y *= 4 * pi * x ** 2
plt.plot(x, y, label=r"$\delta=%.1f$" % delta)
plt.legend(loc="best")
plt.plot(x, np.zeros(len(x)), "k-")
plt.xlabel("D")
plt.ylabel(r"$4\pi D^2 \phi(\rm{Hartree})$")
plt.show()
def phi(self, d, dp):
"""Kernel function.
Uses bi-linear interpolation and returns zero for D > Dmax.
"""
P = self.phi_ij
D = (d + dp) / 2.0
if D < 1e-14:
return P[0, 0]
if D >= self.D_j[-1]:
return 0.0
delta = abs((d - dp) / (2 * D))
ddelta = self.delta_i[1]
x = delta / ddelta
i = int(x)
if i == len(self.delta_i) - 1:
i -= 1
x = 1.0
else:
x -= i
dD = self.D_j[1]
y = D / dD
j = int(y)
y -= j
return x * (y * P[i + 1, j + 1] + (1 - y) * P[i + 1, j]) + (1 - x) * (y * P[i, j + 1] + (1 - y) * P[i, j])
class BEEVDWKernel(XCKernel):
"""Kernel for BEEVDW functionals."""
def __init__(self, bee, xcoefs, ldac, ggac):
"""BEEVDW kernel.
parameters:
bee : str
choose BEE1 or BEE2 exchange basis expansion.
xcoefs : array
coefficients for exchange.
ldac : float
coefficient for LDA correlation.
pbec : float
coefficient for PBE correlation.
"""
if bee == 'BEE2':
self.BEE = BEE2(xcoefs)
self.GGAc = LibXC('GGA_C_PBE')
self.xtype = 'GGA'
self.type = 'GGA'
elif bee == 'BEE3':
self.BEE = LibXC('MGGA_X_MBEEFVDW')
self.GGAc = LibXC('GGA_C_PBE_SOL')
self.xtype = 'MGGA'
self.type = 'MGGA'
else:
raise ValueError('Unknown BEE exchange: %s', bee)
self.LDAc = LibXC('LDA_C_PW')
self.ldac = ldac
self.ggac = ggac
if bee in ['BEE1', 'BEE2']:
self.ggac -= 1.0
self.name = 'BEEVDW'
def calculate(self,
e_g,
n_sg,
dedn_sg,
sigma_xg=None,
dedsigma_xg=None,
tau_sg=None,
dedtau_sg=None):
if debug:
self.check_arguments(e_g, n_sg, dedn_sg, sigma_xg, dedsigma_xg,
tau_sg, dedtau_sg)
if self.xtype == 'GGA':
self.BEE.calculate(e_g, n_sg, dedn_sg, sigma_xg, dedsigma_xg)
elif self.xtype == 'MGGA':
self.BEE.calculate(e_g, n_sg, dedn_sg, sigma_xg, dedsigma_xg,
tau_sg, dedtau_sg)
else:
raise ValueError('Unexpected value of xtype:', self.xtype)
e0_g = np.empty_like(e_g)
dedn0_sg = np.empty_like(dedn_sg)
dedsigma0_xg = np.empty_like(dedsigma_xg)
for coef, kernel in [(self.ldac, self.LDAc), (self.ggac, self.GGAc)]:
dedn0_sg[:] = 0.0
kernel.calculate(e0_g, n_sg, dedn0_sg, sigma_xg, dedsigma0_xg)
e_g += coef * e0_g
dedn_sg += coef * dedn0_sg
if kernel.type == 'GGA':
dedsigma_xg += coef * dedsigma0_xg
class CXGGAKernel:
def __init__(self, just_kidding=False):
self.just_kidding = just_kidding
self.type = 'GGA'
self.lda_c = LibXC('LDA_C_PW')
if self.just_kidding:
self.name = 'purepython rPW86_with_%s' % self.lda_c.name
else:
self.name = 'purepython cx'
def calculate(self, e_g, n_sg, v_sg, sigma_xg, dedsigma_xg):
e_g[:] = 0.0
dedsigma_xg[:] = 0.0
self.lda_c.calculate(e_g, n_sg, v_sg, sigma_xg, dedsigma_xg)
for arr in [n_sg, v_sg, sigma_xg, dedsigma_xg]:
assert len(arr) == 1
self._exchange(n_sg[0], sigma_xg[0], e_g, v_sg[0], dedsigma_xg[0])
def _exchange(self, rho, grho, sx, v1x, v2x):
"""Calculate cx local exchange.
Note that this *adds* to the energy density sx so that it can
be called after LDA correlation part without ruining anything.
Also it adds to v1x and v2x as is normal in GPAW."""
tol = 1e-20
rho[rho < tol] = tol
grho[grho < tol] = tol
alp = 0.021789
beta = 1.15
a = 1.851
b = 17.33
c = 0.163
mu_LM = 0.09434
s_prefactor = 6.18733545256027
Ax = -0.738558766382022 # = -3./4. * (3./pi)**(1./3)
four_thirds = 4. / 3.
grad_rho = np.sqrt(grho)
# eventually we need s to power 12. Clip to avoid overflow
# (We have individual tolerances on both rho and grho, but
# they are not sufficient to guarantee this)
s_1 = (grad_rho / (s_prefactor * rho**four_thirds)).clip(0.0, 1e20)
s_2 = s_1 * s_1
s_3 = s_2 * s_1
s_4 = s_3 * s_1
s_5 = s_4 * s_1
s_6 = s_5 * s_1
fs_rPW86 = (1.0 + a * s_2 + b * s_4 + c * s_6)**(1. / 15.)
if self.just_kidding:
fs = fs_rPW86
else:
fs = (1.0 + mu_LM * s_2) / (1.0 + alp * s_6) \
+ alp * s_6 / (beta + alp * s_6) * fs_rPW86
# the energy density for the exchange.
sx[:] += Ax * rho**four_thirds * fs
df_rPW86_ds = (1. / (15. * fs_rPW86**14.0)) * \
(2 * a * s_1 + 4 * b * s_3 + 6 * c * s_5)
if self.just_kidding:
df_ds = df_rPW86_ds # XXXXXXXXXXXXXXXXXXXX
else:
df_ds = 1. / (1. + alp * s_6)**2 \
* (2.0 * mu_LM * s_1 * (1. + alp * s_6) -
6.0 * alp * s_5 * (1. + mu_LM * s_2)) \
+ alp * s_6 / (beta + alp * s_6) * df_rPW86_ds \
+ 6.0 * alp * s_5 * fs_rPW86 / (beta + alp * s_6) \
* (1. - alp * s_6 / (beta + alp * s_6))
# de/dn. This is the partial derivative of sx wrt. n, for s constant
v1x[:] += Ax * four_thirds * (rho**(1. / 3.) * fs - grad_rho /
(s_prefactor * rho) * df_ds)
# de/d(nabla n). The other partial derivative
v2x[:] += 0.5 * Ax * df_ds / (s_prefactor * grad_rho)
class VDWFunctionalBase:
"""Base class for vdW-DF."""
def __init__(self,
world=None,
Zab=-0.8491,
vdwcoef=1.0,
q0cut=5.0,
phi0=0.5,
ds=1.0,
Dmax=20.0,
nD=201,
ndelta=21,
soft_correction=False,
setup_name='revPBE',
verbose=False,
energy_only=False):
"""vdW-DF.
parameters:
name: str
Name of functional.
world: MPI communicator
Communicator to parallelize over. Defaults to gpaw.mpi.world.
q0cut: float
Maximum value for q0.
phi0: float
Smooth value for phi(0,0).
ds: float
Cutoff for smooth kernel.
Dmax: float
Maximum value for D.
nD: int
Number of values for D in kernel-table.
ndelta: int
Number of values for delta in kernel-table.
soft_correction: bool
Correct for soft kernel.
kernel:
Which kernel to use.
Zab:
parameter in nonlocal kernel.
vdwcoef: float
Scaling of vdW energy.
verbose: bool
Print useful information.
"""
if world is None:
self.world = mpi.world
else:
self.world = world
self.Zab = Zab
self.vdwcoef = vdwcoef
self.q0cut = q0cut
self.phi0 = phi0
self.ds = ds
self.delta_i = np.linspace(0, 1.0, ndelta)
self.D_j = np.linspace(0, Dmax, nD)
self.verbose = verbose
self.read_table()
self.soft_correction = soft_correction
if soft_correction:
dD = self.D_j[1]
self.C_soft = np.dot(self.D_j**2, self.phi_ij[0]) * 4 * pi * dD
self.gd = None
self.energy_only = energy_only
self.timer = nulltimer
self.LDAc = LibXC('LDA_C_PW')
self.setup_name = setup_name
def get_setup_name(self):
return self.setup_name
def get_Ecnl(self):
return self.Ecnl
def calculate_impl(self, gd, n_sg, v_sg, e_g):
sigma_xg, dedsigma_xg, gradn_svg = gga_vars(gd, self.grad_v, n_sg)
self.calculate_exchange(e_g, n_sg, v_sg, sigma_xg, dedsigma_xg)
self.calculate_correlation(e_g, n_sg, v_sg, sigma_xg, dedsigma_xg)
add_gradient_correction(self.grad_v, gradn_svg, sigma_xg, dedsigma_xg,
v_sg)
def calculate_exchange(self, e_g, n_sg, dedn_sg, sigma_xg, dedsigma_xg):
raise NotImplementedError
def calculate_correlation(self, e_g, n_sg, dedn_sg, sigma_xg, dedsigma_xg):
eLDAc_g = self.gd.empty()
vLDAc_sg = self.gd.zeros(1)
if self.vdwcoef == 0.0:
return
if len(n_sg) == 1:
self.LDAc.calculate(eLDAc_g, n_sg, vLDAc_sg)
e = self.get_non_local_energy(n_sg[0], sigma_xg[0], eLDAc_g,
vLDAc_sg[0], dedn_sg[0],
dedsigma_xg[0])
else:
n_sg = n_sg.sum(0)
n_sg.shape = (1, ) + n_sg.shape
self.LDAc.calculate(eLDAc_g, n_sg, vLDAc_sg)
v_g = np.zeros_like(e_g)
deda2nl_g = np.zeros_like(e_g)
a2_g = sigma_xg[0] + 2 * sigma_xg[1] + sigma_xg[2]
e = self.get_non_local_energy(n_sg[0], a2_g, eLDAc_g, vLDAc_sg[0],
v_g, deda2nl_g)
dedsigma_xg[0] += self.vdwcoef * deda2nl_g
dedsigma_xg[1] += self.vdwcoef * 2 * deda2nl_g
dedsigma_xg[2] += self.vdwcoef * deda2nl_g
dedn_sg += self.vdwcoef * v_g
if self.gd.comm.rank == 0:
e_g[0, 0, 0] += self.vdwcoef * e / self.gd.dv
def get_non_local_energy(self,
n_g=None,
a2_g=None,
e_LDAc_g=None,
v_LDAc_g=None,
v_g=None,
deda2_g=None):
"""Calculate non-local correlation energy.
parameters:
n_g: ndarray
Density.
a2_g: ndarray
Absolute value of the gradient of the density - squared.
e_LDAc_g: ndarray
LDA correlation energy density.
"""
gd = self.gd
n_g = n_g.clip(1e-7, np.inf)
# Calculate q0 and cut it off smoothly at q0cut:
kF_g = (3 * pi**2 * n_g)**(1.0 / 3.0)
q0_g, dhdx_g = hRPS(
kF_g - 4 * pi / 3 * e_LDAc_g / n_g -
self.Zab / 36 / kF_g * a2_g / n_g**2, self.q0cut)
if self.verbose:
print(('VDW: q0 (min, mean, max): (%f, %f, %f)' %
(q0_g.min(), q0_g.mean(), q0_g.max())))
if self.soft_correction:
dEcnl = -gd.integrate(n_g**2 / q0_g**3) * 0.5 * self.C_soft
else:
dEcnl = 0.0
# Distribute density and q0 to all processors:
n_g = gd.collect(n_g, broadcast=True)
q0_g = gd.collect(q0_g, broadcast=True)
if not self.energy_only:
self.dhdx_g = gd.collect(dhdx_g, broadcast=True)
Ecnl = self.calculate_6d_integral(n_g, q0_g, a2_g, e_LDAc_g, v_LDAc_g,
v_g, deda2_g)
Ecnl += dEcnl
self.Ecnl = Ecnl
return Ecnl
def read_table(self):
name = ('phi-%.3f-%.3f-%.3f-%d-%d.txt' %
(self.phi0, self.ds, self.D_j[-1], len(
self.delta_i), len(self.D_j)))
dirs = setup_paths + ['.']
for dir in dirs:
filename = os.path.join(dir, name)
if os.path.isfile(filename):
self.phi_ij = np.loadtxt(filename)
if self.verbose:
print('VDW: using', filename)
return
if sys.version_info[0] == 2:
oldname = name[:-3] + 'pckl'
for dir in dirs:
filename = os.path.join(dir, oldname)
if os.path.isfile(filename):
self.phi_ij = pickle.load(open(filename, 'rb'))
if self.verbose:
print('VDW: using', filename)
return
print('VDW: Could not find table file:', name)
self.make_table(name)
def make_table(self, name):
print('VDW: Generating vdW-DF kernel ...')
print('VDW:', end=' ')
ndelta = len(self.delta_i)
nD = len(self.D_j)
self.phi_ij = np.zeros((ndelta, nD))
for i in range(self.world.rank, ndelta, self.world.size):
print(ndelta - i, end=' ')
sys.stdout.flush()
delta = self.delta_i[i]
for j in range(nD - 1, -1, -1):
D = self.D_j[j]
d = D * (1.0 + delta)
dp = D * (1.0 - delta)
if d**2 + dp**2 > self.ds**2:
with seterr(divide='ignore'):
self.phi_ij[i, j] = phi(d, dp)
else:
P = np.polyfit([0, self.D_j[j + 1]**2, self.D_j[j + 2]**2],
[
self.phi0, self.phi_ij[i, j + 1],
self.phi_ij[i, j + 2]
], 2)
self.phi_ij[i, :j + 3] = np.polyval(P, self.D_j[:j + 3]**2)
break
self.world.sum(self.phi_ij)
print()
print('VDW: Done!')
header = ('phi0={0:.3f}, ds={1:.3f}, Dmax={2:.3f}, nD={3}, ndelta={4}'.
format(self.phi0, self.ds, self.D_j[-1], len(self.delta_i),
len(self.D_j)))
if self.world.rank == 0:
np.savetxt(name, self.phi_ij, header=header)
def make_prl_plot(self, multiply_by_4_pi_D_squared=True):
import pylab as plt
x = np.linspace(0, 8.0, 100)
for delta in [0, 0.5, 0.9]:
y = [self.phi(D * (1.0 + delta), D * (1.0 - delta)) for D in x]
if multiply_by_4_pi_D_squared:
y *= 4 * pi * x**2
plt.plot(x, y, label=r'$\delta=%.1f$' % delta)
plt.legend(loc='best')
plt.plot(x, np.zeros(len(x)), 'k-')
plt.xlabel('D')
plt.ylabel(r'$4\pi D^2 \phi(\rm{Hartree})$')
plt.show()
def phi(self, d, dp):
"""Kernel function.
Uses bi-linear interpolation and returns zero for D > Dmax.
"""
P = self.phi_ij
D = (d + dp) / 2.0
if D < 1e-14:
return P[0, 0]
if D >= self.D_j[-1]:
return 0.0
delta = abs((d - dp) / (2 * D))
ddelta = self.delta_i[1]
x = delta / ddelta
i = int(x)
if i == len(self.delta_i) - 1:
i -= 1
x = 1.0
else:
x -= i
dD = self.D_j[1]
y = D / dD
j = int(y)
y -= j
return (x * (y * P[i + 1, j + 1] + (1 - y) * P[i + 1, j]) + (1 - x) *
(y * P[i, j + 1] + (1 - y) * P[i, j]))
class TB09Kernel:
name = 'TB09'
type = 'MGGA'
alpha = -0.012
beta = 1.023
def __init__(self, c=None):
self.tb09 = LibXC('MGGA_X_TB09').xc.tb09
self.ldac = LibXC('LDA_C_PW')
self.fixedc = c is not None # calculate c or use fixed value
self.c = c # amount of "exact exchange"
self.n = 0 # Lebedev quadrature point number (0-49)
self.sign = 1.0 # sign of PAW correction: +1 for AE and -1 for PS
self.I = None # integral from Eq. (3)
def calculate(self, e_g, n_sg, dedn_sg, sigma_xg, dedsigma_xg, tau_sg,
dedtau_sg):
ns = len(n_sg)
n_sg[n_sg < 1e-6] = 1e-6
if n_sg.ndim == 4:
if not self.fixedc:
if self.c is None:
# We don't have the integral yet - just use 1.0:
self.c = 1.0
else:
self.I = self.world.sum(self.I)
self.c = (self.alpha + self.beta *
(self.I / self.gd.volume)**0.5)
# Start calculation of c for use in the next SCF step:
if ns == 1:
gradn_g = sigma_xg[0]**0.5
else:
gradn_g = (sigma_xg[0] + 2 * sigma_xg[1] +
sigma_xg[2])**0.5
self.I = self.gd.integrate(gradn_g / n_sg.sum(0))
# The domain is not distributed like the PAW corrections:
self.I /= self.world.size
lapl_sg = self.gd.empty(ns)
for n_g, lapl_g in zip(n_sg, lapl_sg):
self.lapl.apply(n_g, lapl_g)
else:
rgd = self.rgd
lapl_sg = []
for n_Lg in self.n_sLg:
lapl_g = rgd.laplace(np.dot(self.Y_L, n_Lg))
l = 0
L1 = 0
while L1 < len(self.Y_L):
L2 = L1 + 2 * l + 1
n_g = np.dot(self.Y_L[L1:L2], n_Lg[L1:L2])
with seterr(divide='ignore', invalid='ignore'):
lapl_g -= l * (l + 1) * n_g / rgd.r_g**2
lapl_g[0] = 0.0
L1 = L2
l += 1
lapl_sg.append(lapl_g)
if not self.fixedc:
# PAW corrections to integral:
w = self.sign * weight_n[self.n]
if ns == 1:
gradn_g = sigma_xg[0]**0.5
else:
gradn_g = (sigma_xg[0] + 2 * sigma_xg[1] +
sigma_xg[2])**0.5
self.I += w * rgd.integrate(gradn_g / n_sg.sum(0))
self.n += 1
if self.n == len(weight_n):
self.n = 0
self.sign = -self.sign
# dedn_sg[:] = 0.0
sigma_xg[sigma_xg < 1e-10] = 1e-10
tau_sg[tau_sg < 1e-10] = 1e-10
for n_g, sigma_g, lapl_g, tau_g, v_g in zip(n_sg, sigma_xg[::2],
lapl_sg, tau_sg, dedn_sg):
self.tb09(self.c, n_g.ravel(), sigma_g, lapl_g, tau_g, v_g,
dedsigma_xg)
self.ldac.calculate(e_g, n_sg, dedn_sg)
e_g[:] = 0.0
dedsigma_xg[:] = 0.0
dedtau_sg[:] = 0.0
class BEEVDWKernel(XCKernel):
"""Kernel for BEEVDW functionals."""
def __init__(self, bee, xcoefs, ldac, ggac):
"""BEEVDW kernel.
parameters:
bee : str
choose BEE1 or BEE2 exchange basis expansion.
xcoefs : array
coefficients for exchange.
ldac : float
coefficient for LDA correlation.
pbec : float
coefficient for PBE correlation.
"""
if bee == 'BEE2':
self.BEE = BEE2(xcoefs)
self.GGAc = LibXC('GGA_C_PBE')
self.xtype = 'GGA'
self.type = 'GGA'
elif bee == 'BEE3':
self.BEE = LibXC('MGGA_X_MBEEFVDW')
self.GGAc = LibXC('GGA_C_PBE_SOL')
self.xtype = 'MGGA'
self.type = 'MGGA'
else:
raise ValueError('Unknown BEE exchange: %s', bee)
self.LDAc = LibXC('LDA_C_PW')
self.ldac = ldac
self.ggac = ggac
if bee in ['BEE1', 'BEE2']:
self.ggac -= 1.0
self.name = 'BEEVDW'
def calculate(self, e_g, n_sg, dedn_sg,
sigma_xg=None, dedsigma_xg=None,
tau_sg=None, dedtau_sg=None):
if debug:
self.check_arguments(e_g, n_sg, dedn_sg, sigma_xg, dedsigma_xg,
tau_sg, dedtau_sg)
if self.xtype == 'GGA':
self.BEE.calculate(e_g, n_sg, dedn_sg, sigma_xg, dedsigma_xg)
elif self.xtype == 'MGGA':
self.BEE.calculate(e_g, n_sg, dedn_sg, sigma_xg, dedsigma_xg,
tau_sg, dedtau_sg)
else:
raise ValueError('Unexpected value of xtype:', self.xtype)
e0_g = np.empty_like(e_g)
dedn0_sg = np.empty_like(dedn_sg)
dedsigma0_xg = np.empty_like(dedsigma_xg)
for coef, kernel in [(self.ldac, self.LDAc),
(self.ggac, self.GGAc)]:
dedn0_sg[:] = 0.0
kernel.calculate(e0_g, n_sg, dedn0_sg, sigma_xg, dedsigma0_xg)
e_g += coef * e0_g
dedn_sg += coef * dedn0_sg
if kernel.type == 'GGA':
dedsigma_xg += coef * dedsigma0_xg
class VDWFunctional(GGA):
"""Base class for vdW-DF."""
def __init__(self,
name,
world=None,
q0cut=5.0,
phi0=0.5,
ds=1.0,
Dmax=20.0,
nD=201,
ndelta=21,
soft_correction=False,
kernel=None,
Zab=None,
vdwcoef=1.0,
verbose=False,
energy_only=False):
"""vdW-DF.
parameters:
name: str
Name of functional.
world: MPI communicator
Communicator to parallelize over. Defaults to gpaw.mpi.world.
q0cut: float
Maximum value for q0.
phi0: float
Smooth value for phi(0,0).
ds: float
Cutoff for smooth kernel.
Dmax: float
Maximum value for D.
nD: int
Number of values for D in kernel-table.
ndelta: int
Number of values for delta in kernel-table.
soft_correction: bool
Correct for soft kernel.
kernel:
Which kernel to use.
Zab:
parameter in nonlocal kernel.
vdwcoef: float
Scaling of vdW energy.
verbose: bool
Print useful information.
"""
if world is None:
self.world = mpi.world
else:
self.world = world
self.q0cut = q0cut
self.phi0 = phi0
self.ds = ds
self.delta_i = np.linspace(0, 1.0, ndelta)
self.D_j = np.linspace(0, Dmax, nD)
self.verbose = verbose
self.read_table()
self.soft_correction = soft_correction
if soft_correction:
dD = self.D_j[1]
self.C_soft = np.dot(self.D_j**2, self.phi_ij[0]) * 4 * pi * dD
self.gd = None
self.energy_only = energy_only
self.timer = nulltimer
if name == 'vdW-DF':
assert kernel is None and Zab is None
kernel = LibXC('GGA_X_PBE_R+LDA_C_PW')
Zab = -0.8491
elif name == 'vdW-DF2':
assert kernel is None and Zab is None
kernel = LibXC('GGA_X_PW86+LDA_C_PW')
Zab = -1.887
self.Zab = Zab
GGA.__init__(self, kernel)
self.vdwcoef = vdwcoef
self.name = name
self.LDAc = LibXC('LDA_C_PW')
def get_setup_name(self):
return 'revPBE'
def initialize(self, density, hamiltonian, wfs, occupations):
self.timer = wfs.timer
def calculate_gga(self, e_g, n_sg, dedn_sg, sigma_xg, dedsigma_xg):
GGA.calculate_gga(self, e_g, n_sg, dedn_sg, sigma_xg, dedsigma_xg)
eLDAc_g = self.gd.empty()
vLDAc_sg = self.gd.zeros(1)
if self.vdwcoef == 0.0:
return
if len(n_sg) == 1:
self.LDAc.calculate(eLDAc_g, n_sg, vLDAc_sg)
e = self.get_non_local_energy(n_sg[0], sigma_xg[0], eLDAc_g,
vLDAc_sg[0], dedn_sg[0],
dedsigma_xg[0])
else:
n_sg = n_sg.sum(0)
n_sg.shape = (1, ) + n_sg.shape
self.LDAc.calculate(eLDAc_g, n_sg, vLDAc_sg)
v_g = np.zeros_like(e_g)
deda2nl_g = np.zeros_like(e_g)
a2_g = sigma_xg[0] + 2 * sigma_xg[1] + sigma_xg[2]
e = self.get_non_local_energy(n_sg[0], a2_g, eLDAc_g, vLDAc_sg[0],
v_g, deda2nl_g)
dedsigma_xg[0] += self.vdwcoef * deda2nl_g
dedsigma_xg[1] += self.vdwcoef * 2 * deda2nl_g
dedsigma_xg[2] += self.vdwcoef * deda2nl_g
dedn_sg += self.vdwcoef * v_g
if self.gd.comm.rank == 0:
e_g[0, 0, 0] += self.vdwcoef * e / self.gd.dv
def get_non_local_energy(self,
n_g=None,
a2_g=None,
e_LDAc_g=None,
v_LDAc_g=None,
v_g=None,
deda2_g=None):
"""Calculate non-local correlation energy.
parameters:
n_g: ndarray
Density.
a2_g: ndarray
Absolute value of the gradient of the density - squared.
e_LDAc_g: ndarray
LDA correlation energy density.
"""
gd = self.gd
n_g = n_g.clip(1e-7, np.inf)
# Calculate q0 and cut it off smoothly at q0cut:
kF_g = (3 * pi**2 * n_g)**(1.0 / 3.0)
q0_g, dhdx_g = hRPS(
kF_g - 4 * pi / 3 * e_LDAc_g / n_g -
self.Zab / 36 / kF_g * a2_g / n_g**2, self.q0cut)
if self.verbose:
print('VDW: q0 (min, mean, max): (%f, %f, %f)' %
(q0_g.min(), q0_g.mean(), q0_g.max()))
if self.soft_correction:
dEcnl = gd.integrate(n_g**2 / q0_g**3) * 0.5 * self.C_soft
else:
dEcnl = 0.0
# Distribute density and q0 to all processors:
n_g = gd.collect(n_g, broadcast=True)
q0_g = gd.collect(q0_g, broadcast=True)
if not self.energy_only:
self.dhdx_g = gd.collect(dhdx_g, broadcast=True)
Ecnl = self.calculate_6d_integral(n_g, q0_g, a2_g, e_LDAc_g, v_LDAc_g,
v_g, deda2_g)
return Ecnl + dEcnl
def read_table(self):
name = ('phi-%.3f-%.3f-%.3f-%d-%d.pckl' %
(self.phi0, self.ds, self.D_j[-1], len(
self.delta_i), len(self.D_j)))
if 'GPAW_VDW' in os.environ:
print 'Use of GPAW_VDW is deprecated.'
print 'Put', name, 'in your GPAW_SETUP_PATH directory.'
dirs = [os.environ['GPAW_VDW']]
else:
dirs = setup_paths + ['.']
for dir in dirs:
filename = os.path.join(dir, name)
if os.path.isfile(filename):
self.phi_ij = pickle.load(open(filename))
if self.verbose:
print 'VDW: using', filename
return
print 'VDW: Could not find table file:', name
self.make_table(name)
def make_table(self, name):
print 'VDW: Generating vdW-DF kernel ...'
print 'VDW:',
ndelta = len(self.delta_i)
nD = len(self.D_j)
self.phi_ij = np.zeros((ndelta, nD))
for i in range(self.world.rank, ndelta, self.world.size):
print ndelta - i,
sys.stdout.flush()
delta = self.delta_i[i]
for j in range(nD - 1, -1, -1):
D = self.D_j[j]
d = D * (1.0 + delta)
dp = D * (1.0 - delta)
if d**2 + dp**2 > self.ds**2:
self.phi_ij[i, j] = phi(d, dp)
else:
P = np.polyfit([0, self.D_j[j + 1]**2, self.D_j[j + 2]**2],
[
self.phi0, self.phi_ij[i, j + 1],
self.phi_ij[i, j + 2]
], 2)
self.phi_ij[i, :j + 3] = np.polyval(P, self.D_j[:j + 3]**2)
break
self.world.sum(self.phi_ij)
print
print 'VDW: Done!'
if self.world.rank == 0:
pickle.dump(self.phi_ij, open(name, 'w'), pickle.HIGHEST_PROTOCOL)
def make_prl_plot(self, multiply_by_4_pi_D_squared=True):
import pylab as plt
x = np.linspace(0, 8.0, 100)
for delta in [0, 0.5, 0.9]:
y = [self.phi(D * (1.0 + delta), D * (1.0 - delta)) for D in x]
if multiply_by_4_pi_D_squared:
y *= 4 * pi * x**2
plt.plot(x, y, label=r'$\delta=%.1f$' % delta)
plt.legend(loc='best')
plt.plot(x, np.zeros(len(x)), 'k-')
plt.xlabel('D')
plt.ylabel(r'$4\pi D^2 \phi(\rm{Hartree})$')
plt.show()
def phi(self, d, dp):
"""Kernel function.
Uses bi-linear interpolation and returns zero for D > Dmax.
"""
P = self.phi_ij
D = (d + dp) / 2.0
if D < 1e-14:
return P[0, 0]
if D >= self.D_j[-1]:
return 0.0
delta = abs((d - dp) / (2 * D))
ddelta = self.delta_i[1]
x = delta / ddelta
i = int(x)
if i == len(self.delta_i) - 1:
i -= 1
x = 1.0
else:
x -= i
dD = self.D_j[1]
y = D / dD
j = int(y)
y -= j
return (x * (y * P[i + 1, j + 1] + (1 - y) * P[i + 1, j]) + (1 - x) *
(y * P[i, j + 1] + (1 - y) * P[i, j]))
