frommathimport pi
importnumpyasnp
fromgpaw.xc.ldaimport (calculate_paw_correction, stress_integral,
stress_lda_term)
fromgpaw.utilities.blasimport axpy
fromgpaw.fd_operatorsimport Gradient
fromgpaw.sphere.lebedevimport Y_nL, weight_n
fromgpaw.xc.pawcorrectionimport rnablaY_nLv
fromgpaw.xc.functionalimport XCFunctional
defstress_gga_term(gd, sigma_xg, gradn_svg, dedsigma_xg):
P = 0
nspins = gradn_svg.shape[0]
for sigma_g, dedsigma_g in zip(sigma_xg, dedsigma_xg):
P -= 2 * stress_integral(gd, sigma_g, dedsigma_g)
stress_vv = P * np.eye(3)
for v1 in range(3):
for v2 in range(3):
stress_vv[v1, v2] -= 2 * stress_integral(
gd, gradn_svg[0, v1] * gradn_svg[0, v2], dedsigma_xg[0]
)
if nspins == 2:
stress_vv[v1, v2] -= 2 * stress_integral(
gd, gradn_svg[0, v1] * gradn_svg[1, v2], dedsigma_xg[1]
)
stress_vv[v1, v2] -= 2 * stress_integral(
gd, gradn_svg[1, v1] * gradn_svg[1, v2], dedsigma_xg[2]
)
return stress_vv
classGGARadialExpansion:
def__init__(self, rcalc, *args):
self.rcalc = rcalc
self.args = args
def__call__(self, rgd, D_sLq, n_qg, nc0_sg):
n_sLg = np.dot(D_sLq, n_qg)
n_sLg[:, 0] += nc0_sg
dndr_sLg = np.empty_like(n_sLg)
for n_Lg, dndr_Lg in zip(n_sLg, dndr_sLg):
for n_g, dndr_g in zip(n_Lg, dndr_Lg):
rgd.derivative(n_g, dndr_g)
nspins, Lmax, nq = D_sLq.shape
dEdD_sqL = np.zeros((nspins, nq, Lmax))
E = 0.0
for n, Y_L in enumerate(Y_nL[:, :Lmax]):
w = weight_n[n]
rnablaY_Lv = rnablaY_nLv[n, :Lmax]
e_g, dedn_sg, b_vsg, dedsigma_xg = \
self.rcalc(rgd, n_sLg, Y_L, dndr_sLg, rnablaY_Lv, n,
*self.args)
dEdD_sqL += np.dot(rgd.dv_g * dedn_sg,
n_qg.T)[:, :, np.newaxis] * (w * Y_L)
dedsigma_xg *= rgd.dr_g
B_vsg = dedsigma_xg[::2] * b_vsg
if nspins == 2:
B_vsg += 0.5 * dedsigma_xg[1] * b_vsg[:, ::-1]
B_vsq = np.dot(B_vsg, n_qg.T)
dEdD_sqL += 8 * pi * w * np.inner(rnablaY_Lv, B_vsq.T).T
E += w * rgd.integrate(e_g)
return E, dEdD_sqL
# First part of gga_calculate_radial - initializes some quantities.
defradial_gga_vars(rgd, n_sLg, Y_L, dndr_sLg, rnablaY_Lv):
nspins = len(n_sLg)
n_sg = np.dot(Y_L, n_sLg)
a_sg = np.dot(Y_L, dndr_sLg)
b_vsg = np.dot(rnablaY_Lv.T, n_sLg)
sigma_xg = rgd.empty(2 * nspins - 1)
sigma_xg[::2] = (b_vsg ** 2).sum(0)
if nspins == 2:
sigma_xg[1] = (b_vsg[:, 0] * b_vsg[:, 1]).sum(0)
sigma_xg[:, 1:] /= rgd.r_g[1:] ** 2
sigma_xg[:, 0] = sigma_xg[:, 1]
sigma_xg[::2] += a_sg ** 2
if nspins == 2:
sigma_xg[1] += a_sg[0] * a_sg[1]
e_g = rgd.empty()
dedn_sg = rgd.zeros(nspins)
dedsigma_xg = rgd.zeros(2 * nspins - 1)
return e_g, n_sg, dedn_sg, sigma_xg, dedsigma_xg, a_sg, b_vsg
defadd_radial_gradient_correction(rgd, sigma_xg, dedsigma_xg, a_sg):
nspins = len(a_sg)
vv_sg = sigma_xg[:nspins] # reuse array
for s in range(nspins):
rgd.derivative2(-2 * rgd.dv_g * dedsigma_xg[2 * s] * a_sg[s],
vv_sg[s])
if nspins == 2:
v_g = sigma_xg[2]
rgd.derivative2(rgd.dv_g * dedsigma_xg[1] * a_sg[1], v_g)
vv_sg[0] -= v_g
rgd.derivative2(rgd.dv_g * dedsigma_xg[1] * a_sg[0], v_g)
vv_sg[1] -= v_g
vv_sg[:, 1:] /= rgd.dv_g[1:]
vv_sg[:, 0] = vv_sg[:, 1]
return vv_sg
classGGARadialCalculator:
def__init__(self, kernel):
self.kernel = kernel
def__call__(self, rgd, n_sLg, Y_L, dndr_sLg, rnablaY_Lv, n):
(e_g, n_sg, dedn_sg, sigma_xg, dedsigma_xg, a_sg,
b_vsg) = radial_gga_vars(rgd, n_sLg, Y_L, dndr_sLg, rnablaY_Lv)
self.kernel.calculate(e_g, n_sg, dedn_sg, sigma_xg, dedsigma_xg)
vv_sg = add_radial_gradient_correction(rgd, sigma_xg,
dedsigma_xg, a_sg)
return e_g, dedn_sg + vv_sg, b_vsg, dedsigma_xg
defcalculate_sigma(gd, grad_v, n_sg):
r"""Calculate sigma(r) and grad n(r).
_ __ _ 2 __ _
Returns sigma(r) = |\/ n(r)| and \/ n (r).
With multiple spins, sigma has the three elements
_ __ _ 2
sigma (r) = |\/ n (r)| ,
0 up
_ __ _ __ _
sigma (r) = \/ n (r) . \/ n (r) ,
1 up dn
_ __ _ 2
sigma (r) = |\/ n (r)| .
2 dn
"""
nspins = len(n_sg)
gradn_svg = gd.empty((nspins, 3))
sigma_xg = gd.zeros(nspins * 2 - 1)
for v in range(3):
for s in range(nspins):
grad_v[v](n_sg[s], gradn_svg[s, v])
axpy(1.0, gradn_svg[s, v]**2, sigma_xg[2 * s])
if nspins == 2:
axpy(1.0, gradn_svg[0, v] * gradn_svg[1, v], sigma_xg[1])
return sigma_xg, gradn_svg
defadd_gradient_correction(grad_v, gradn_svg, sigma_xg, dedsigma_xg, v_sg):
r"""Add gradient correction to potential.
::
__ / de(r) __ \
v (r) += -2 \/ . | --------- \/ n(r) |
xc \ dsigma(r) /
Adds arbitrary data to sigma_xg. Be sure to pass a copy if you need
sigma_xg after calling this function.
"""
nspins = len(v_sg)
# vv_g is a calculation buffer. Its contents will be corrupted.
vv_g = sigma_xg[0]
for v in range(3):
for s in range(nspins):
grad_v[v](dedsigma_xg[2 * s] * gradn_svg[s, v], vv_g)
vv_g *= 2.0
v_sg[s] -= vv_g
# axpy(-2.0, vv_g, v_sg[s])
if nspins == 2:
grad_v[v](dedsigma_xg[1] * gradn_svg[s, v], vv_g)
v_sg[1 - s] -= vv_g
# axpy(-1.0, vv_g, v_sg[1 - s])
# TODO: can the number of gradient evaluations be reduced?
defgga_vars(gd, grad_v, n_sg):
nspins = len(n_sg)
sigma_xg, gradn_svg = calculate_sigma(gd, grad_v, n_sg)
dedsigma_xg = gd.empty(nspins * 2 - 1)
return sigma_xg, dedsigma_xg, gradn_svg
defget_gradient_ops(gd, nn, xp):
return [Gradient(gd, v, n=nn, xp=xp).apply for v in range(3)]
[docs]
classGGA(XCFunctional):
def__init__(self, kernel, stencil=2):
XCFunctional.__init__(self, kernel.name, kernel.type)
self.kernel = kernel
self.stencil_range = stencil
defset_grid_descriptor(self, gd):
XCFunctional.set_grid_descriptor(self, gd)
self.grad_v = get_gradient_ops(gd, self.stencil_range, self.xp)
[docs]
deftodict(self):
d = super().todict()
d['stencil'] = self.stencil_range
return d
[docs]
defget_description(self):
return ('{} with {} nearest neighbor stencil'
.format(self.name, self.stencil_range))
defcalculate_impl(self, gd, n_sg, v_sg, e_g):
sigma_xg, dedsigma_xg, gradn_svg = gga_vars(gd, self.grad_v, n_sg)
self.kernel.calculate(e_g, n_sg, v_sg, sigma_xg, dedsigma_xg)
add_gradient_correction(self.grad_v, gradn_svg, sigma_xg,
dedsigma_xg, v_sg)
defcalculate_paw_correction(self, setup, D_sp, dEdD_sp=None,
addcoredensity=True, a=None):
rcalc = GGARadialCalculator(self.kernel)
expansion = GGARadialExpansion(rcalc)
return calculate_paw_correction(expansion,
setup, D_sp, dEdD_sp,
addcoredensity, a)
defstress_tensor_contribution(self, n_sg, skip_sum=False):
sigma_xg, gradn_svg = calculate_sigma(self.gd, self.grad_v, n_sg)
nspins = len(n_sg)
dedsigma_xg = self.gd.empty(nspins * 2 - 1)
v_sg = self.gd.zeros(nspins)
e_g = self.gd.empty()
self.kernel.calculate(e_g, n_sg, v_sg, sigma_xg, dedsigma_xg)
stress_vv = stress_lda_term(self.gd, e_g, n_sg, v_sg)
stress_vv[:] += stress_gga_term(self.gd, sigma_xg, gradn_svg,
dedsigma_xg)
if not skip_sum:
self.gd.comm.sum(stress_vv)
return stress_vv
defcalculate_spherical(self, rgd, n_sg, v_sg, e_g=None):
dndr_sg = np.empty_like(n_sg)
for n_g, dndr_g in zip(n_sg, dndr_sg):
rgd.derivative(n_g, dndr_g)
if e_g is None:
e_g = rgd.empty()
rcalc = GGARadialCalculator(self.kernel)
e_g[:], dedn_sg = rcalc(rgd, n_sg[:, np.newaxis],
[1.0],
dndr_sg[:, np.newaxis],
np.zeros((1, 3)), n=None)[:2]
v_sg[:] = dedn_sg
return rgd.integrate(e_g)
classPurePythonGGAKernel:
def__init__(self, name):
self.type = 'GGA'
self.name, self.kappa, self.mu, self.beta = pbe_constants(name)
defcalculate(self, e_g, n_sg, dedn_sg,
sigma_xg, dedsigma_xg,
tau_sg=None, dedtau_sg=None):
e_g[:] = 0.
dedsigma_xg[:] = 0.
# spin-paired:
if len(n_sg) == 1:
n = n_sg[0]
n[n < 1e-20] = 1e-40
# exchange
res = gga_x(self.name, 0, n, sigma_xg[0], self.kappa, self.mu)
ex, rs, dexdrs, dexda2 = res
# correlation
res = gga_c(self.name, 0, n, sigma_xg[0], 0, self.beta)
ec, rs_, decdrs, decda2, decdzeta = res
e_g[:] += n * (ex + ec)
dedn_sg[:] += ex + ec - rs * (dexdrs + decdrs) / 3.
dedsigma_xg[:] += n * (dexda2 + decda2)
# spin-polarized:
else:
na = 2. * n_sg[0]
na[na < 1e-20] = 1e-40
nb = 2. * n_sg[1]
nb[nb < 1e-20] = 1e-40
n = 0.5 * (na + nb)
zeta = 0.5 * (na - nb) / n
# exchange
exa, rsa, dexadrs, dexada2 = gga_x(
self.name, 1, na, 4.0 * sigma_xg[0], self.kappa, self.mu)
exb, rsb, dexbdrs, dexbda2 = gga_x(
self.name, 1, nb, 4.0 * sigma_xg[2], self.kappa, self.mu)
a2 = sigma_xg[0] + 2.0 * sigma_xg[1] + sigma_xg[2]
# correlation
ec, rs, decdrs, decda2, decdzeta = gga_c(
self.name, 1, n, a2, zeta, self.beta)
e_g[:] += 0.5 * (na * exa + nb * exb) + n * ec
dedn_sg[0][:] += (exa + ec - (rsa * dexadrs + rs * decdrs) / 3.0
- (zeta - 1.0) * decdzeta)
dedn_sg[1][:] += (exb + ec - (rsb * dexbdrs + rs * decdrs) / 3.0
- (zeta + 1.0) * decdzeta)
dedsigma_xg[0][:] += 2.0 * na * dexada2 + n * decda2
dedsigma_xg[1][:] += 2.0 * n * decda2
dedsigma_xg[2][:] += 2.0 * nb * dexbda2 + n * decda2
defpbe_constants(name):
if name == 'pyPBE':
name = 'PBE'
kappa = 0.804
mu = 0.2195149727645171
beta = 0.06672455060314922
elif name in ['pyPBEsol', 'pyzvPBEsol']:
name = name[2:]
kappa = 0.804
mu = 10. / 81.
beta = 0.046
elif name == 'pyRPBE':
name = 'RPBE'
kappa = 0.804
mu = 0.2195149727645171
beta = 0.06672455060314922
else:
raise NotImplementedError(name)
return name, kappa, mu, beta
# a2 = |grad n|^2
defgga_x(name, spin, n, a2, kappa, mu, dedmu_g=None):
# if dedmu is given, calculate also d(e_x)/d(mu)
assert spin in [0, 1]
C0I, C1, C2, C3, CC1, CC2, IF2, GAMMA = gga_constants()
rs = (C0I / n)**(1 / 3.)
# lda part
ex = C1 / rs
dexdrs = -ex / rs
# gga part
c = (C2 * rs / n)**2.
s2 = a2 * c
if name in ['PBE', 'PBEsol', 'zvPBEsol', 'QNA']:
x = 1.0 + mu * s2 / kappa
Fx = 1.0 + kappa - kappa / x
dFxds2 = mu / (x**2.)
elif name == 'RPBE':
arg = np.maximum(-mu * s2 / kappa, -5.e2)
x = np.exp(arg)
Fx = 1.0 + kappa * (1.0 - x)
dFxds2 = mu * x
else:
raise NotImplementedError
ds2drs = 8.0 * c * a2 / rs
dexdrs = dexdrs * Fx + ex * dFxds2 * ds2drs
dexda2 = ex * dFxds2 * c
if dedmu_g is not None:
dedmu_g[:] = ex * s2 / (1 + mu * s2 / kappa)**2
ex *= Fx
return ex, rs, dexdrs, dexda2
defgga_c(name, spin, n, a2, zeta, BETA, decdbeta_g=None):
# If decdbeta is specified, calculate also d(e_c)/d(beta)
assert spin in [0, 1]
fromgpaw.xc.ldaimport G
C0I, C1, C2, C3, CC1, CC2, IF2, GAMMA = gga_constants()
rs = (C0I / n)**(1 / 3.)
if name == 'zvPBEsol':
zv_a = 1.8
zv_o = 9. / 2.
zv_x = 1. / 6.
# lda part
ec, decdrs_0 = G(rs**0.5, 0.031091, 0.21370, 7.5957,
3.5876, 1.6382, 0.49294)
if spin == 0:
decdrs = decdrs_0
decdzeta = 0. # dummy
else:
e1, decdrs_1 = G(rs**0.5, 0.015545, 0.20548, 14.1189,
6.1977, 3.3662, 0.62517)
alpha, dalphadrs = G(rs**0.5, 0.016887, 0.11125, 10.357,
3.6231, 0.88026, 0.49671)
alpha *= -1.
dalphadrs *= -1.
zp = 1.0 + zeta
zm = 1.0 - zeta
xp = zp**(1 / 3.)
xm = zm**(1 / 3.)
f = CC1 * (zp * xp + zm * xm - 2.0)
f1 = CC2 * (xp - xm)
zeta3 = zeta * zeta * zeta
zeta4 = zeta * zeta * zeta * zeta
x = 1.0 - zeta4
decdrs = (decdrs_0 * (1.0 - f * zeta4) +
decdrs_1 * f * zeta4 +
dalphadrs * f * x * IF2)
decdzeta = (4.0 * zeta3 * f * (e1 - ec - alpha * IF2) +
f1 * (zeta4 * e1 - zeta4 * ec + x * alpha * IF2))
ec += alpha * IF2 * f * x + (e1 - ec) * f * zeta4
# gga part
n2 = n * n
if spin == 1:
phi = 0.5 * (xp * xp + xm * xm)
phi2 = phi * phi
phi3 = phi * phi2
t2 = C3 * a2 * rs / (n2 * phi2)
y = -ec / (GAMMA * phi3)
if name == 'zvPBEsol':
u3 = t2**(3. / 2.) * phi3 * (rs / 3.)**(-3. * zv_x)
zvarg = -zv_a * u3 * abs(zeta)**zv_o
zvf = np.exp(zvarg)
else:
t2 = C3 * a2 * rs / n2
y = -ec / GAMMA
x = np.exp(y)
A = np.zeros_like(x)
indices = np.nonzero(y)
if isinstance(BETA, np.ndarray):
A[indices] = (BETA[indices] / (GAMMA * (x[indices] - 1.0)))
else:
A[indices] = (BETA / (GAMMA * (x[indices] - 1.0)))
At2 = A * t2
nom = 1.0 + At2
denom = nom + At2 * At2
X = 1.0 + BETA * t2 * nom / (denom * GAMMA)
H = GAMMA * np.log(X)
tmp = (GAMMA * BETA / (denom * (BETA * t2 * nom + GAMMA * denom)))
tmp2 = A * A * x / BETA
dAdrs = tmp2 * decdrs
if spin == 1:
H *= phi3
tmp *= phi3
dAdrs /= phi3
if name == 'zvPBEsol':
H_ = H.copy()
H *= zvf
dHdt2 = (1.0 + 2.0 * At2) * tmp
dHdA = -At2 * t2 * t2 * (2.0 + At2) * tmp
decdrs += dHdt2 * 7.0 * t2 / rs + dHdA * dAdrs
decda2 = dHdt2 * C3 * rs / n2
if spin == 1:
dphidzeta = np.zeros_like(x)
ind1 = np.nonzero(xp)
ind2 = np.nonzero(xm)
dphidzeta[ind1] += 1.0 / (3.0 * xp[ind1])
dphidzeta[ind2] -= 1.0 / (3.0 * xm[ind2])
dAdzeta = tmp2 * (decdzeta - 3.0 * ec * dphidzeta / phi) / phi3
decdzeta += ((3.0 * H / phi - dHdt2 * 2.0 * t2 / phi) * dphidzeta
+ dHdA * dAdzeta)
decda2 /= phi2
if name == 'zvPBEsol':
u3_ = (t2**(3. / 2.) *
phi3 * rs**(-3. * zv_x - 1.) / 3.**(-3. * zv_x))
zvarg_ = -zv_a * u3_ * abs(zeta)**zv_o
dzvfdrs = -3. * zv_x * zvf * zvarg_
decdrs *= zvf
decdrs += H_ * dzvfdrs
dt2da2 = C3 * rs / n2
assert np.shape(dt2da2) == np.shape(t2)
dzvfda2 = dt2da2 * zvf * zvarg * 3. / (2. * t2)
decda2 *= zvf
decda2 += H_ * dzvfda2
dadz = zv_o * abs(zeta)**(zv_o - 1.)
dbdphi = 3. * u3 / phi
dPdzeta = u3 * dadz + abs(zeta)**(zv_o) * dbdphi * dphidzeta
dzvfdzeta = -zv_a * zvf * dPdzeta
decdzeta *= zvf
decdzeta += H_ * dzvfdzeta
ec += H
if decdbeta_g is not None:
if spin == 0:
phi3 = 1.0
Y = GAMMA * phi3
decdbeta_g[:] = Y / X * t2 / GAMMA
decdbeta_g *= ((1 + 2 * At2) / (1 + At2 + At2**2) -
(1 + At2) * (At2 + 2 * At2**2) / (1 + At2 + At2**2)**2)
return ec, rs, decdrs, decda2, decdzeta
defgga_constants():
fromgpaw.xc.ldaimport lda_constants
C0I, C1, CC1, CC2, IF2 = lda_constants()
C2 = 0.26053088059892404
C3 = 0.10231023756535741
GAMMA = 0.0310906908697
return C0I, C1, C2, C3, CC1, CC2, IF2, GAMMA