class HybridXC(HybridXCBase):
orbital_dependent = True
def __init__(self, name, hybrid=None, xc=None,
alpha=None,
gamma_point=1,
method='standard',
bandstructure=False,
logfilename='-', bands=None,
fcut=1e-10,
molecule=False,
qstride=1,
world=None):
"""Mix standard functionals with exact exchange.
name: str
Name of functional: EXX, PBE0, HSE03, HSE06
hybrid: float
Fraction of exact exchange.
xc: str or XCFunctional object
Standard DFT functional with scaled down exchange.
method: str
Use 'standard' standard formula and 'acdf for
adiabatic-connection dissipation fluctuation formula.
alpha: float
XXX describe
gamma_point: bool
0: Skip k2-k1=0 interactions.
1: Use the alpha method.
2: Integrate the gamma point.
bandstructure: bool
Calculate bandstructure instead of just the total energy.
bands: list of int
List of bands to calculate bandstructure for. Default is
all bands.
molecule: bool
Decouple electrostatic interactions between periodically
repeated images.
fcut: float
Threshold for empty band.
"""
self.alpha = alpha
self.fcut = fcut
self.gamma_point = gamma_point
self.method = method
self.bandstructure = bandstructure
self.bands = bands
self.fd = logfilename
self.write_timing_information = True
HybridXCBase.__init__(self, name, hybrid, xc)
# EXX energies:
self.exx = None # total
self.evv = None # valence-valence (pseudo part)
self.evvacdf = None # valence-valence (pseudo part)
self.devv = None # valence-valence (PAW correction)
self.evc = None # valence-core
self.ecc = None # core-core
self.exx_skn = None # bandstructure
self.qlatest = None
if world is None:
world = mpi.world
self.world = world
self.molecule = molecule
if isinstance(qstride, int):
qstride = [qstride] * 3
self.qstride_c = np.asarray(qstride)
self.timer = Timer()
def log(self, *args, **kwargs):
prnt(file=self.fd, *args, **kwargs)
self.fd.flush()
def calculate_radial(self, rgd, n_sLg, Y_L, v_sg,
dndr_sLg=None, rnablaY_Lv=None,
tau_sg=None, dedtau_sg=None):
return self.xc.calculate_radial(rgd, n_sLg, Y_L, v_sg,
dndr_sLg, rnablaY_Lv)
def calculate_paw_correction(self, setup, D_sp, dEdD_sp=None,
addcoredensity=True, a=None):
return self.xc.calculate_paw_correction(setup, D_sp, dEdD_sp,
addcoredensity, a)
def initialize(self, dens, ham, wfs, occupations):
assert wfs.bd.comm.size == 1
self.xc.initialize(dens, ham, wfs, occupations)
self.dens = dens
self.wfs = wfs
# Make a k-point descriptor that is not distributed
# (self.kd.comm is serial_comm):
self.kd = wfs.kd.copy()
self.fd = logfile(self.fd, self.world.rank)
wfs.initialize_wave_functions_from_restart_file()
def set_positions(self, spos_ac):
self.spos_ac = spos_ac
def calculate(self, gd, n_sg, v_sg=None, e_g=None):
# Normal XC contribution:
exc = self.xc.calculate(gd, n_sg, v_sg, e_g)
# Add EXX contribution:
return exc + self.exx * self.hybrid
def calculate_exx(self):
"""Non-selfconsistent calculation."""
self.timer.start('EXX')
self.timer.start('Initialization')
kd = self.kd
wfs = self.wfs
if fftw.FFTPlan is fftw.NumpyFFTPlan:
self.log('NOT USING FFTW !!')
self.log('Spins:', self.wfs.nspins)
W = max(1, self.wfs.kd.comm.size // self.wfs.nspins)
# Are the k-points distributed?
kparallel = (W > 1)
# Find number of occupied bands:
self.nocc_sk = np.zeros((self.wfs.nspins, kd.nibzkpts), int)
for kpt in self.wfs.kpt_u:
for n, f in enumerate(kpt.f_n):
if abs(f) < self.fcut:
self.nocc_sk[kpt.s, kpt.k] = n
break
else:
self.nocc_sk[kpt.s, kpt.k] = self.wfs.bd.nbands
self.wfs.kd.comm.sum(self.nocc_sk)
noccmin = self.nocc_sk.min()
noccmax = self.nocc_sk.max()
self.log('Number of occupied bands (min, max): %d, %d' %
(noccmin, noccmax))
self.log('Number of valence electrons:', self.wfs.setups.nvalence)
if self.bandstructure:
self.log('Calculating eigenvalue shifts.')
# allocate array for eigenvalue shifts:
self.exx_skn = np.zeros((self.wfs.nspins,
kd.nibzkpts,
self.wfs.bd.nbands))
if self.bands is None:
noccmax = self.wfs.bd.nbands
else:
noccmax = max(max(self.bands) + 1, noccmax)
N_c = self.kd.N_c
vol = wfs.gd.dv * wfs.gd.N_c.prod()
if self.alpha is None:
alpha = 6 * vol**(2 / 3.0) / pi**2
else:
alpha = self.alpha
if self.gamma_point == 1:
if alpha == 0.0:
qvol = (2*np.pi)**3 / vol / N_c.prod()
self.gamma = 4*np.pi * (3*qvol / (4*np.pi))**(1/3.) / qvol
else:
self.gamma = self.calculate_gamma(vol, alpha)
else:
kcell_cv = wfs.gd.cell_cv.copy()
kcell_cv[0] *= N_c[0]
kcell_cv[1] *= N_c[1]
kcell_cv[2] *= N_c[2]
self.gamma = madelung(kcell_cv) * vol * N_c.prod() / (4 * np.pi)
self.log('Value of alpha parameter: %.3f Bohr^2' % alpha)
self.log('Value of gamma parameter: %.3f Bohr^2' % self.gamma)
# Construct all possible q=k2-k1 vectors:
Nq_c = (N_c - 1) // self.qstride_c
i_qc = np.indices(Nq_c * 2 + 1, float).transpose(
(1, 2, 3, 0)).reshape((-1, 3))
self.bzq_qc = (i_qc - Nq_c) / N_c * self.qstride_c
self.q0 = ((Nq_c * 2 + 1).prod() - 1) // 2 # index of q=(0,0,0)
assert not self.bzq_qc[self.q0].any()
# Count number of pairs for each q-vector:
self.npairs_q = np.zeros(len(self.bzq_qc), int)
for s in range(kd.nspins):
for k1 in range(kd.nibzkpts):
for k2 in range(kd.nibzkpts):
for K2, q, n1_n, n2 in self.indices(s, k1, k2):
self.npairs_q[q] += len(n1_n)
self.npairs0 = self.npairs_q.sum() # total number of pairs
self.log('Number of pairs:', self.npairs0)
# Distribute q-vectors to Q processors:
Q = self.world.size // self.wfs.kd.comm.size
myrank = self.world.rank // self.wfs.kd.comm.size
rank = 0
N = 0
myq = []
nq = 0
for q, n in enumerate(self.npairs_q):
if n > 0:
nq += 1
if rank == myrank:
myq.append(q)
N += n
if N >= (rank + 1.0) * self.npairs0 / Q:
rank += 1
assert len(myq) > 0, 'Too few q-vectors for too many processes!'
self.bzq_qc = self.bzq_qc[myq]
try:
self.q0 = myq.index(self.q0)
except ValueError:
self.q0 = None
self.log('%d x %d x %d k-points' % tuple(self.kd.N_c))
self.log('Distributing %d IBZ k-points over %d process(es).' %
(kd.nibzkpts, self.wfs.kd.comm.size))
self.log('Distributing %d q-vectors over %d process(es).' % (nq, Q))
# q-point descriptor for my q-vectors:
qd = KPointDescriptor(self.bzq_qc)
# Plane-wave descriptor for all wave-functions:
self.pd = PWDescriptor(wfs.pd.ecut, wfs.gd,
dtype=wfs.pd.dtype, kd=kd)
# Plane-wave descriptor pair-densities:
self.pd2 = PWDescriptor(self.dens.pd2.ecut, self.dens.gd,
dtype=wfs.dtype, kd=qd)
self.log('Cutoff energies:')
self.log(' Wave functions: %10.3f eV' %
(self.pd.ecut * Hartree))
self.log(' Density: %10.3f eV' %
(self.pd2.ecut * Hartree))
# Calculate 1/|G+q|^2 with special treatment of |G+q|=0:
G2_qG = self.pd2.G2_qG
if self.q0 is None:
if self.omega is None:
self.iG2_qG = [1.0 / G2_G for G2_G in G2_qG]
else:
self.iG2_qG = [(1.0 / G2_G *
(1 - np.exp(-G2_G / (4 * self.omega**2))))
for G2_G in G2_qG]
else:
G2_qG[self.q0][0] = 117.0 # avoid division by zero
if self.omega is None:
self.iG2_qG = [1.0 / G2_G for G2_G in G2_qG]
self.iG2_qG[self.q0][0] = self.gamma
else:
self.iG2_qG = [(1.0 / G2_G *
(1 - np.exp(-G2_G / (4 * self.omega**2))))
for G2_G in G2_qG]
self.iG2_qG[self.q0][0] = 1 / (4 * self.omega**2)
G2_qG[self.q0][0] = 0.0 # restore correct value
# Compensation charges:
self.ghat = PWLFC([setup.ghat_l for setup in wfs.setups], self.pd2)
self.ghat.set_positions(self.spos_ac)
if self.molecule:
self.initialize_gaussian()
self.log('Value of beta parameter: %.3f 1/Bohr^2' % self.beta)
self.timer.stop('Initialization')
# Ready ... set ... go:
self.t0 = time()
self.npairs = 0
self.evv = 0.0
self.evvacdf = 0.0
for s in range(self.wfs.nspins):
kpt1_q = [KPoint(self.wfs, noccmax).initialize(kpt)
for kpt in self.wfs.kpt_u if kpt.s == s]
kpt2_q = kpt1_q[:]
if len(kpt1_q) == 0:
# No s-spins on this CPU:
continue
# Send and receive ranks:
srank = self.wfs.kd.get_rank_and_index(
s, (kpt1_q[0].k - 1) % kd.nibzkpts)[0]
rrank = self.wfs.kd.get_rank_and_index(
s, (kpt1_q[-1].k + 1) % kd.nibzkpts)[0]
# Shift k-points kd.nibzkpts - 1 times:
for i in range(kd.nibzkpts):
if i < kd.nibzkpts - 1:
if kparallel:
kpt = kpt2_q[-1].next(self.wfs)
kpt.start_receiving(rrank)
kpt2_q[0].start_sending(srank)
else:
kpt = kpt2_q[0]
self.timer.start('Calculate')
for kpt1, kpt2 in zip(kpt1_q, kpt2_q):
# Loop over all k-points that k2 can be mapped to:
for K2, q, n1_n, n2 in self.indices(s, kpt1.k, kpt2.k):
self.apply(K2, q, kpt1, kpt2, n1_n, n2)
self.timer.stop('Calculate')
if i < kd.nibzkpts - 1:
self.timer.start('Wait')
if kparallel:
kpt.wait()
kpt2_q[0].wait()
self.timer.stop('Wait')
kpt2_q.pop(0)
kpt2_q.append(kpt)
self.evv = self.world.sum(self.evv)
self.evvacdf = self.world.sum(self.evvacdf)
self.calculate_exx_paw_correction()
if self.method == 'standard':
self.exx = self.evv + self.devv + self.evc + self.ecc
elif self.method == 'acdf':
self.exx = self.evvacdf + self.devv + self.evc + self.ecc
else:
1 / 0
self.log('Exact exchange energy:')
for txt, e in [
('core-core', self.ecc),
('valence-core', self.evc),
('valence-valence (pseudo, acdf)', self.evvacdf),
('valence-valence (pseudo, standard)', self.evv),
('valence-valence (correction)', self.devv),
('total (%s)' % self.method, self.exx)]:
self.log(' %-36s %14.6f eV' % (txt + ':', e * Hartree))
self.log('Total time: %10.3f seconds' % (time() - self.t0))
self.npairs = self.world.sum(self.npairs)
assert self.npairs == self.npairs0
self.timer.stop('EXX')
self.timer.write(self.fd)
def calculate_gamma(self, vol, alpha):
if self.molecule:
return 0.0
N_c = self.kd.N_c
offset_c = (N_c + 1) % 2 * 0.5 / N_c
bzq_qc = monkhorst_pack(N_c) + offset_c
qd = KPointDescriptor(bzq_qc)
pd = PWDescriptor(self.wfs.pd.ecut, self.wfs.gd, kd=qd)
gamma = (vol / (2 * pi)**2 * sqrt(pi / alpha) *
self.kd.nbzkpts)
for G2_G in pd.G2_qG:
if G2_G[0] < 1e-7:
G2_G = G2_G[1:]
gamma -= np.dot(np.exp(-alpha * G2_G), G2_G**-1)
return gamma / self.qstride_c.prod()
def indices(self, s, k1, k2):
"""Generator for (K2, q, n1, n2) indices for (k1, k2) pair.
s: int
Spin index.
k1: int
Index of k-point in the IBZ.
k2: int
Index of k-point in the IBZ.
Returns (K, q, n1_n, n2), where K then index of the k-point in
the BZ that k2 is mapped to, q is the index of the q-vector
between K and k1, and n1_n is a list of bands that should be
combined with band n2."""
for K, k in enumerate(self.kd.bz2ibz_k):
if k == k2:
for K, q, n1_n, n2 in self._indices(s, k1, k2, K):
yield K, q, n1_n, n2
def _indices(self, s, k1, k2, K2):
k1_c = self.kd.ibzk_kc[k1]
k2_c = self.kd.bzk_kc[K2]
q_c = k2_c - k1_c
q = abs(self.bzq_qc - q_c).sum(1).argmin()
if abs(self.bzq_qc[q] - q_c).sum() > 1e-7:
return
if self.gamma_point == 0 and q == self.q0:
return
nocc1 = self.nocc_sk[s, k1]
nocc2 = self.nocc_sk[s, k2]
# Is k2 in the IBZ?
is_ibz2 = (self.kd.ibz2bz_k[k2] == K2)
for n2 in range(self.wfs.bd.nbands):
# Find range of n1's (from n1a to n1b-1):
if is_ibz2:
# We get this combination twice, so let's only do half:
if k1 >= k2:
n1a = n2
else:
n1a = n2 + 1
else:
n1a = 0
n1b = self.wfs.bd.nbands
if self.bandstructure:
if n2 >= nocc2:
n1b = min(n1b, nocc1)
else:
if n2 >= nocc2:
break
n1b = min(n1b, nocc1)
if self.bands is not None:
assert self.bandstructure
n1_n = []
for n1 in range(n1a, n1b):
if (n1 in self.bands and n2 < nocc2 or
is_ibz2 and n2 in self.bands and n1 < nocc1):
n1_n.append(n1)
n1_n = np.array(n1_n)
else:
n1_n = np.arange(n1a, n1b)
if len(n1_n) == 0:
continue
yield K2, q, n1_n, n2
def apply(self, K2, q, kpt1, kpt2, n1_n, n2):
k20_c = self.kd.ibzk_kc[kpt2.k]
k2_c = self.kd.bzk_kc[K2]
if k2_c.any():
self.timer.start('Initialize plane waves')
eik2r_R = self.wfs.gd.plane_wave(k2_c)
eik20r_R = self.wfs.gd.plane_wave(k20_c)
self.timer.stop('Initialize plane waves')
else:
eik2r_R = 1.0
eik20r_R = 1.0
w1 = self.kd.weight_k[kpt1.k]
w2 = self.kd.weight_k[kpt2.k]
# Is k2 in the 1. BZ?
is_ibz2 = (self.kd.ibz2bz_k[kpt2.k] == K2)
e_n = self.calculate_interaction(n1_n, n2, kpt1, kpt2, q, K2,
eik20r_R, eik2r_R,
is_ibz2)
e_n *= 1.0 / self.kd.nbzkpts / self.wfs.nspins * self.qstride_c.prod()
if q == self.q0:
e_n[n1_n == n2] *= 0.5
f1_n = kpt1.f_n[n1_n]
eps1_n = kpt1.eps_n[n1_n]
f2 = kpt2.f_n[n2]
eps2 = kpt2.eps_n[n2]
s_n = np.sign(eps2 - eps1_n)
evv = (f1_n * f2 * e_n).sum()
evvacdf = 0.5 * (f1_n * (1 - s_n) * e_n +
f2 * (1 + s_n) * e_n).sum()
self.evv += evv * w1
self.evvacdf += evvacdf * w1
if is_ibz2:
self.evv += evv * w2
self.evvacdf += evvacdf * w2
if self.bandstructure:
x = self.wfs.nspins
self.exx_skn[kpt1.s, kpt1.k, n1_n] += x * f2 * e_n
if is_ibz2:
self.exx_skn[kpt2.s, kpt2.k, n2] += x * np.dot(f1_n, e_n)
def calculate_interaction(self, n1_n, n2, kpt1, kpt2, q, k,
eik20r_R, eik2r_R, is_ibz2):
"""Calculate Coulomb interactions.
For all n1 in the n1_n list, calculate interaction with n2."""
# number of plane waves:
ng1 = self.wfs.ng_k[kpt1.k]
ng2 = self.wfs.ng_k[kpt2.k]
# Transform to real space and apply symmetry operation:
self.timer.start('IFFT1')
if is_ibz2:
u2_R = self.pd.ifft(kpt2.psit_nG[n2, :ng2], kpt2.k)
else:
psit2_R = self.pd.ifft(kpt2.psit_nG[n2, :ng2], kpt2.k) * eik20r_R
self.timer.start('Symmetry transform')
u2_R = self.kd.transform_wave_function(psit2_R, k) / eik2r_R
self.timer.stop()
self.timer.stop()
# Calculate pair densities:
nt_nG = self.pd2.zeros(len(n1_n), q=q)
for n1, nt_G in zip(n1_n, nt_nG):
self.timer.start('IFFT2')
u1_R = self.pd.ifft(kpt1.psit_nG[n1, :ng1], kpt1.k)
self.timer.stop()
nt_R = u1_R.conj() * u2_R
self.timer.start('FFT')
nt_G[:] = self.pd2.fft(nt_R, q)
self.timer.stop()
s = self.kd.sym_k[k]
time_reversal = self.kd.time_reversal_k[k]
k2_c = self.kd.ibzk_kc[kpt2.k]
self.timer.start('Compensation charges')
Q_anL = {} # coefficients for shape functions
for a, P1_ni in kpt1.P_ani.items():
P1_ni = P1_ni[n1_n]
if is_ibz2:
P2_i = kpt2.P_ani[a][n2]
else:
b = self.kd.symmetry.a_sa[s, a]
S_c = (np.dot(self.spos_ac[a], self.kd.symmetry.op_scc[s]) -
self.spos_ac[b])
assert abs(S_c.round() - S_c).max() < 1e-5
if self.ghat.dtype == complex:
x = np.exp(2j * pi * np.dot(k2_c, S_c))
else:
x = 1.0
P2_i = np.dot(self.wfs.setups[a].R_sii[s],
kpt2.P_ani[b][n2]) * x
if time_reversal:
P2_i = P2_i.conj()
D_np = []
for P1_i in P1_ni:
D_ii = np.outer(P1_i.conj(), P2_i)
D_np.append(pack(D_ii))
Q_anL[a] = np.dot(D_np, self.wfs.setups[a].Delta_pL)
self.timer.start('Expand')
if q != self.qlatest:
self.f_IG = self.ghat.expand(q)
self.qlatest = q
self.timer.stop('Expand')
# Add compensation charges:
self.ghat.add(nt_nG, Q_anL, q, self.f_IG)
self.timer.stop('Compensation charges')
if self.molecule and n2 in n1_n:
nn = (n1_n == n2).nonzero()[0][0]
nt_nG[nn] -= self.ngauss_G
else:
nn = None
iG2_G = self.iG2_qG[q]
# Calculate energies:
e_n = np.empty(len(n1_n))
for n, nt_G in enumerate(nt_nG):
e_n[n] = -4 * pi * np.real(self.pd2.integrate(nt_G, nt_G * iG2_G))
self.npairs += 1
if nn is not None:
e_n[nn] -= 2 * (self.pd2.integrate(nt_nG[nn], self.vgauss_G) +
(self.beta / 2 / pi)**0.5)
if self.write_timing_information:
t = (time() - self.t0) / len(n1_n)
self.log('Time for first pair-density: %10.3f seconds' % t)
self.log('Estimated total time: %10.3f seconds' %
(t * self.npairs0 / self.world.size))
self.write_timing_information = False
return e_n
def calculate_exx_paw_correction(self):
self.timer.start('PAW correction')
self.devv = 0.0
self.evc = 0.0
self.ecc = 0.0
deg = 2 // self.wfs.nspins # spin degeneracy
for a, D_sp in self.dens.D_asp.items():
setup = self.wfs.setups[a]
for D_p in D_sp:
D_ii = unpack2(D_p)
ni = len(D_ii)
for i1 in range(ni):
for i2 in range(ni):
A = 0.0
for i3 in range(ni):
p13 = packed_index(i1, i3, ni)
for i4 in range(ni):
p24 = packed_index(i2, i4, ni)
A += setup.M_pp[p13, p24] * D_ii[i3, i4]
self.devv -= D_ii[i1, i2] * A / deg
self.evc -= np.dot(D_p, setup.X_p)
self.ecc += setup.ExxC
if not self.bandstructure:
self.timer.stop('PAW correction')
return
Q = self.world.size // self.wfs.kd.comm.size
self.exx_skn *= Q
for kpt in self.wfs.kpt_u:
for a, D_sp in self.dens.D_asp.items():
setup = self.wfs.setups[a]
for D_p in D_sp:
D_ii = unpack2(D_p)
ni = len(D_ii)
P_ni = kpt.P_ani[a]
for i1 in range(ni):
for i2 in range(ni):
A = 0.0
for i3 in range(ni):
p13 = packed_index(i1, i3, ni)
for i4 in range(ni):
p24 = packed_index(i2, i4, ni)
A += setup.M_pp[p13, p24] * D_ii[i3, i4]
self.exx_skn[kpt.s, kpt.k] -= \
(A * P_ni[:, i1].conj() * P_ni[:, i2]).real
p12 = packed_index(i1, i2, ni)
self.exx_skn[kpt.s, kpt.k] -= \
(P_ni[:, i1].conj() * setup.X_p[p12] *
P_ni[:, i2]).real / self.wfs.nspins
self.world.sum(self.exx_skn)
self.exx_skn *= self.hybrid / Q
self.timer.stop('PAW correction')
def initialize_gaussian(self):
"""Calculate gaussian compensation charge and its potential.
Used to decouple electrostatic interactions between
periodically repeated images for molecular calculations.
Charge containing one electron::
(beta/pi)^(3/2)*exp(-beta*r^2),
its Fourier transform::
exp(-G^2/(4*beta)),
and its potential::
erf(beta^0.5*r)/r.
"""
gd = self.wfs.gd
# Set exponent of exp-function to -19 on the boundary:
self.beta = 4 * 19 * (gd.icell_cv**2).sum(1).max()
# Calculate gaussian:
G_Gv = self.pd2.get_reciprocal_vectors()
G2_G = self.pd2.G2_qG[0]
C_v = gd.cell_cv.sum(0) / 2 # center of cell
self.ngauss_G = np.exp(-1.0 / (4 * self.beta) * G2_G +
1j * np.dot(G_Gv, C_v)) / gd.dv
# Calculate potential from gaussian:
R_Rv = gd.get_grid_point_coordinates().transpose((1, 2, 3, 0))
r_R = ((R_Rv - C_v)**2).sum(3)**0.5
if (gd.N_c % 2 == 0).all():
r_R[tuple(gd.N_c // 2)] = 1.0 # avoid dividing by zero
v_R = erf(self.beta**0.5 * r_R) / r_R
if (gd.N_c % 2 == 0).all():
v_R[tuple(gd.N_c // 2)] = (4 * self.beta / pi)**0.5
self.vgauss_G = self.pd2.fft(v_R)
# Compare self-interaction to analytic result:
assert abs(0.5 * self.pd2.integrate(self.ngauss_G, self.vgauss_G) -
(self.beta / 2 / pi)**0.5) < 1e-6
class TDDFT(object):
"""
Time-dependent DFT+Hartree-Fock in Kohn-Sham orbitals basis:
calc: GPAW calculator (setups='sg15')
nbands (int): number of bands in calculation
"""
def __init__(self, calc, nbands=None, Fock=False):
self.calc = calc
self.Fock = Fock
self.K = calc.get_ibz_k_points() # reduced Brillioun zone
self.NK = self.K.shape[0]
self.wk = calc.get_k_point_weights(
) # weight of reduced Brillioun zone
if nbands is None:
self.nbands = calc.get_number_of_bands()
else:
self.nbands = nbands
self.nvalence = int(calc.get_number_of_electrons() / 2)
self.EK = [
calc.get_eigenvalues(k)[:self.nbands] for k in range(self.NK)
] # bands energy
self.EK = np.array(self.EK) / Hartree
self.shape = tuple(
calc.get_number_of_grid_points()) # shape of real space grid
self.density = calc.get_pseudo_density(
) * Bohr**3 # density at zero time
# array of u_nk (periodic part of Kohn-Sham orbitals,only reduced Brillion zone)
self.ukn = np.zeros((
self.NK,
self.nbands,
) + self.shape,
dtype=np.complex)
for k in range(self.NK):
kpt = calc.wfs.kpt_u[k]
for n in range(self.nbands):
psit_G = kpt.psit_nG[n]
psit_R = calc.wfs.pd.ifft(psit_G, kpt.q)
self.ukn[k, n] = psit_R
self.icell = 2.0 * np.pi * calc.wfs.gd.icell_cv # inverse cell
self.cell = calc.wfs.gd.cell_cv # cell
self.r = calc.wfs.gd.get_grid_point_coordinates()
for i in range(3):
self.r[i] -= self.cell[i, i] / 2.
self.volume = np.abs(np.linalg.det(
calc.wfs.gd.cell_cv)) # volume of cell
self.norm = calc.wfs.gd.dv #
self.Fermi = calc.get_fermi_level() / Hartree #Fermi level
#desriptors at q=gamma for Hartree
self.kdH = KPointDescriptor([[0, 0, 0]])
self.pdH = PWDescriptor(ecut=calc.wfs.pd.ecut,
gd=calc.wfs.gd,
kd=self.kdH,
dtype=complex)
#desriptors at q=gamma for Fock
self.kdF = KPointDescriptor([[0, 0, 0]])
self.pdF = PWDescriptor(ecut=calc.wfs.pd.ecut / 4.,
gd=calc.wfs.gd,
kd=self.kdF,
dtype=complex)
#Fermi-Dirac temperature
self.temperature = calc.occupations.width
#calculate pair-density matrices
if Fock:
self.M = np.zeros((self.nbands, self.nbands, self.NK, self.NK,
self.pdF.get_reciprocal_vectors().shape[0]),
dtype=np.complex)
indexes = [(n, k)
for n, k in product(range(self.nbands), range(self.NK))]
for i1 in range(len(indexes)):
n1, k1 = indexes[i1]
for i2 in range(i1, len(indexes)):
n2, k2 = indexes[i1]
self.M[n1, n2, k1, k2] = self.pdF.fft(
self.ukn[k1, n1].conj() * self.ukn[k2, n2])
self.M[n2, n1, k2, k1] = self.M[n1, n2, k1, k2].conj()
self.M *= calc.wfs.gd.dv
#Fermi-Dirac distribution
self.f = 1 / (1 + np.exp((self.EK - self.Fermi) / self.temperature))
self.Hartree_elements = np.zeros(
(self.NK, self.nbands, self.NK, self.nbands, self.nbands),
dtype=np.complex)
self.LDAx_elements = np.zeros(
(self.NK, self.nbands, self.NK, self.nbands, self.nbands),
dtype=np.complex)
self.LDAc_elements = np.zeros(
(self.NK, self.nbands, self.NK, self.nbands, self.nbands),
dtype=np.complex)
G = self.pdH.get_reciprocal_vectors()
G2 = np.linalg.norm(G, axis=1)**2
G2[G2 == 0] = np.inf
matrix = np.zeros((self.NK, self.nbands, self.nbands),
dtype=np.complex)
for k in tqdm(range(self.NK)):
for n in range(self.nbands):
density = 2 * np.abs(self.ukn[k, n])**2
operator = xc.VLDAx(density)
self.LDAx_elements[k, n] = operator_matrix_periodic(
matrix, operator, self.ukn.conj(), self.ukn) * self.norm
operator = xc.VLDAc(density)
self.LDAc_elements[k, n] = operator_matrix_periodic(
matrix, operator, self.ukn.conj(), self.ukn) * self.norm
density = self.pdH.fft(density)
operator = 4 * np.pi * self.pdH.ifft(density / G2)
self.Hartree_elements[k, n] = operator_matrix_periodic(
matrix, operator, self.ukn.conj(), self.ukn) * self.norm
self.wavefunction = np.zeros((self.NK, self.nbands, self.nbands),
dtype=np.complex)
self.Kinetic = np.zeros((self.NK, self.nbands, self.nbands),
dtype=np.complex)
self.dipole = self.get_dipole_matrix()
for k in range(self.NK):
self.wavefunction[k] = np.eye(self.nbands)
self.Kinetic[k] = np.diag(self.EK[k])
self.VH0 = self.get_Hartree_matrix(self.wavefunction)
self.VLDAc0 = self.get_LDA_correlation_matrix(self.wavefunction)
self.VLDAx0 = self.get_LDA_exchange_matrix(self.wavefunction)
self.Full_BZ = calc.get_bz_k_points()
self.IBZ_map = calc.get_bz_to_ibz_map()
def get_dipole_matrix(self, direction=[1, 0, 0]):
"""
return two-dimensional numpy complex array of dipole matrix elements(
"""
direction /= np.linalg.norm(direction)
r = np.sum(direction[:, None, None, None] * self.r, axis=0)
dipole = np.zeros((self.NK, self.nbands, self.nbands),
dtype=np.complex)
dipole = operator_matrix_periodic(
dipole, r, self.ukn.conj(),
self.ukn) * self.norm #!!!!!! no direction
return dipole
def get_density(self, wavefunction):
"""
return numpy array of electron density in real space at each k-point of full Brillioun zone
wavefunction: numpy array [N_kpoint X N_band X N_band] of wavefunction in basis of Kohn-Sham orbital
"""
if wavefunction is None:
return self.density
density = np.zeros(self.shape, dtype=np.float)
for k in range(self.NK):
for n in range(self.nbands):
for m in range(self.nbands):
density += 2 * self.wk[k] * self.f[k, n] * np.abs(
wavefunction[k, m, n] * self.ukn[k, m])**2
return density
def get_Hartree_potential(self, wavefunction):
"""
return numpy array of Hartree potential in real space at each k-point of full Brillioun zone
wavefunction: numpy array [N_kpoint X N_band X N_band] of wavefunction in basis of Kohn-Sham orbital
"""
density = self.get_density(wavefunction)
VH = np.zeros(self.shape)
G = self.pdH.get_reciprocal_vectors()
G2 = np.linalg.norm(G, axis=1)**2
G2[G2 == 0] = np.inf
nG = self.pdH.fft(density)
return -4 * np.pi * self.pdH.ifft(nG / G2)
def get_coloumb_potential(self, q):
"""
return coloumb potential in plane wave space V= 4 pi /(|q+G|**2)
q: [qx,qy,qz] vector in units of reciprocal space
"""
G = self.pdF.get_reciprocal_vectors() + np.dot(q, self.icell)
G2 = np.linalg.norm(G, axis=1)**2
G2[G2 == 0] = np.inf
return 4 * np.pi / G2
def get_Hartree_matrix(self, wavefunction=None):
"""
return numpy array [N_kpoint X N_band X N_band] of Hartree potential matrix elements
wavefunction: numpy array [N_kpoint X N_band X N_band] of wavefunction in basis of Kohn-Sham orbital
"""
VH = self.get_Hartree_potential(wavefunction)
VH_matrix = np.zeros((self.NK, self.nbands, self.nbands),
dtype=np.complex)
VH_matrix = operator_matrix_periodic(VH_matrix, VH, self.ukn.conj(),
self.ukn) * self.norm
return VH_matrix
def get_Fock_matrix(self, wavefunction=None):
"""
return numpy array [N_kpoint X N_band X N_band] of Fock potential matrix elements
wavefunction: numpy array [N_kpoint X N_band X N_band] of wavefunction in basis of Kohn-Sham orbital
"""
VF_matrix = np.zeros((self.NK, self.nbands, self.nbands),
dtype=np.complex)
if self.Fock:
if wavefunction is None:
wavefunction = np.zeros((self.NK, self.nbands, self.nbands))
for k in range(self.NK):
wavefunction[k] = np.eye(self.nbands)
K = self.Full_BZ
NK = K.shape[0]
NG = self.pdF.get_reciprocal_vectors().shape[0]
V = np.zeros((self.NK, NK, NG))
for k in range(self.NK):
for q in range(NK):
kq = K[q] - self.K[k]
V[k, q] = self.get_coloumb_potential(kq)
VF_matrix = Fock_matrix(VF_matrix, V, self.M.conj(), self.M,
self.IBZ_map, self.nvalence)
return VF_matrix / self.volume
def get_LDA_exchange_matrix(self, wavefunction=None):
"""
return numpy array [N_kpoint X N_band X N_band] of LDA exchange potential matrix elements
wavefunction: numpy array [N_kpoint X N_band X N_band] of wavefunction in basis of Kohn-Sham orbital
"""
density = self.get_density(wavefunction)
exchange = xc.VLDAx(density)
LDAx_matrix = np.zeros((self.NK, self.nbands, self.nbands),
dtype=np.complex)
LDAx_matrix = operator_matrix_periodic(
LDAx_matrix, exchange, self.ukn.conj(), self.ukn) * self.norm
return LDAx_matrix
def get_LDA_correlation_matrix(self, wavefunction=None):
"""
return numpy array [N_kpoint X N_band X N_band] of LDA correlation potential matrix elements
wavefunction: numpy array [N_kpoint X N_band X N_band] of wavefunction in basis of Kohn-Sham orbital
"""
density = self.get_density(wavefunction)
correlation = xc.VLDAc(density)
LDAc_matrix = np.zeros((self.NK, self.nbands, self.nbands),
dtype=np.complex)
LDAc_matrix = operator_matrix_periodic(
LDAc_matrix, correlation, self.ukn.conj(), self.ukn) * self.norm
return LDAc_matrix
def occupation(self, wavefunction):
return 2 * np.sum(self.wk[:, None, None] * self.f[:, None, :] *
np.abs(wavefunction)**2,
axis=2)
def fast_Hartree_matrix(self, wavefunction):
return np.einsum('kn,knqij->qij', self.occupation(wavefunction),
self.Hartree_elements) - self.VH0
def fast_LDA_correlation_matrix(self, wavefunction):
return np.einsum('kn,knqij->qij', self.occupation(wavefunction),
self.LDAc_elements) - self.VLDAc0
def fast_LDA_exchange_matrix(self, wavefunction):
return np.einsum('kn,knqij->qij', self.occupation(wavefunction),
self.LDAx_elements) - self.VLDAx0
def propagate(self, dt, steps, E, operator, corrections=10):
self.wavefunction = np.zeros((self.NK, self.nbands, self.nbands),
dtype=np.complex)
for k in range(self.NK):
self.wavefunction[k] = np.eye(self.nbands)
H = np.copy(self.Kinetic) + E[0] * self.dipole
operator_macro = np.array(
[operator[k].diagonal() for k in range(self.NK)])
self.macro_dipole = np.zeros(steps, dtype=np.complex)
for t in tqdm(range(steps)):
wavefunction_next = np.copy(self.wavefunction)
error = np.inf
while error > 1e-8:
wavefunction_check = np.copy(wavefunction_next)
H_next = self.Kinetic + E[t] * self.dipole
H_next += self.fast_Hartree_matrix(wavefunction_next)
H_next += self.fast_LDA_correlation_matrix(wavefunction_next)
H_next += self.fast_LDA_exchange_matrix(wavefunction_next)
H_mid = 0.5 * (H + H_next)
for k in range(self.NK):
H_left = np.eye(self.nbands) + 0.5j * dt * H_mid[k]
H_right = np.eye(self.nbands) - 0.5j * dt * H_mid[k]
wavefunction_next[k] = linalg.solve(
H_left, H_right @ self.wavefunction[k])
error = np.abs(np.sum(wavefunction_next - wavefunction_check))
self.wavefunction = np.copy(wavefunction_next)
H = np.copy(H_next)
self.macro_dipole[t] = np.sum(
self.occupation(self.wavefunction) * operator_macro)
lfcr = PWLFC([[s]], pdr)
c_axi = {0: np.zeros((1, 2 * l + 1), complex)}
c_axi[0][0, 0] = 1.9
cr_axi = {0: np.zeros((1, 2 * l + 1))}
cr_axi[0][0, 0] = 1.9
b = pd.zeros(1, dtype=complex)
br = pdr.zeros(1)
lfc.set_positions(spos_ac)
lfc.add(b, c_axi)
lfcr.set_positions(spos_ac)
lfcr.add(br, cr_axi)
a = pd.ifft(b)
ar = pdr.ifft(br)
equal(abs(a-ar).max(), 0, 1e-14)
if l == 0:
a = a[:, ::-1].copy()
b0 = pd.fft(a)
br0 = pdr.fft(a.real)
lfc.integrate(b0, c_axi)
lfcr.integrate(br0, cr_axi)
assert abs(c_axi[0][0]-cr_axi[0][0]).max() < 1e-14
c_axiv = {0: np.zeros((1, 2 * l + 1, 3), complex)}
cr_axiv = {0: np.zeros((1, 2 * l + 1, 3))}
lfc.derivative(b0, c_axiv)
class TDDFT(object):
"""
Time-dependent DFT+Hartree-Fock in Kohn-Sham orbitals basis:
calc: GPAW calculator (setups='sg15')
nbands (int): number of bands in calculation
"""
def __init__(self,calc,nbands=None):
self.calc=calc
self.K=calc.get_ibz_k_points() # reduced Brillioun zone
self.NK=self.K.shape[0]
self.wk=calc.get_k_point_weights() # weight of reduced Brillioun zone
if nbands is None:
self.nbands=calc.get_number_of_bands()
else:
self.nbands=nbands
self.nvalence=int(calc.get_number_of_electrons()/2)
self.EK=[calc.get_eigenvalues(k)[:self.nbands] for k in range(self.NK)] # bands energy
self.EK=np.array(self.EK)/Hartree
self.shape=tuple(calc.get_number_of_grid_points()) # shape of real space grid
self.density=calc.get_pseudo_density()*Bohr**3 # density at zero time
# array of u_nk (periodic part of Kohn-Sham orbitals,only reduced Brillion zone)
self.ukn=np.zeros((self.NK,self.nbands,)+self.shape,dtype=np.complex)
for k in range(self.NK):
kpt = calc.wfs.kpt_u[k]
for n in range(self.nbands):
psit_G = kpt.psit_nG[n]
psit_R = calc.wfs.pd.ifft(psit_G, kpt.q)
self.ukn[k,n]=psit_R
self.icell=2.0 * np.pi * calc.wfs.gd.icell_cv # inverse cell
self.cell = calc.wfs.gd.cell_cv # cell
self.r=calc.wfs.gd.get_grid_point_coordinates()
for i in range(3):
self.r[i]-=self.cell[i,i]/2.
self.volume = np.abs(np.linalg.det(calc.wfs.gd.cell_cv)) # volume of cell
self.norm=calc.wfs.gd.dv #
self.Fermi=calc.get_fermi_level()/Hartree #Fermi level
#desriptors at q=gamma for Hartree
self.kd=KPointDescriptor([[0,0,0]])
self.pd=PWDescriptor(ecut=calc.wfs.pd.ecut,gd=calc.wfs.gd,kd=self.kd,dtype=complex)
#Fermi-Dirac temperature
self.temperature=calc.occupations.width
#Fermi-Dirac distribution
self.f=1/(1+np.exp((self.EK-self.Fermi)/self.temperature))
self.Hartree_elements=np.zeros((self.NK,self.nbands,self.NK,self.nbands,self.nbands),dtype=np.complex)
self.LDAx_elements=np.zeros((self.NK,self.nbands,self.NK,self.nbands,self.nbands),dtype=np.complex)
self.LDAc_elements=np.zeros((self.NK,self.nbands,self.NK,self.nbands,self.nbands),dtype=np.complex)
G=self.pd.get_reciprocal_vectors()
G2=np.linalg.norm(G,axis=1)**2;G2[G2==0]=np.inf
matrix=np.zeros((self.NK,self.nbands,self.nbands),dtype=np.complex)
for k in tqdm(range(self.NK)):
for n in range(self.nbands):
density=self.norm*np.abs(self.ukn[k,n])**2
operator=xc.VLDAx(density)
self.LDAx_elements[k,n]=operator_matrix_periodic(matrix,operator,self.ukn.conj(),self.ukn)*self.norm
operator=xc.VLDAc(density)
self.LDAc_elements[k,n]=operator_matrix_periodic(matrix,operator,self.ukn.conj(),self.ukn)*self.norm
density=self.pd.fft(density)
operator=4*np.pi*self.pd.ifft(density/G2)
self.Hartree_elements[k,n]=operator_matrix_periodic(matrix,operator,self.ukn.conj(),self.ukn)*self.norm
self.wavefunction=np.zeros((self.NK,self.nbands,self.nbands),dtype=np.complex)
self.Kinetic=np.zeros((self.NK,self.nbands,self.nbands),dtype=np.complex)
for k in range(self.NK):
self.wavefunction[k]=np.eye(self.nbands)
self.Kinetic[k]=np.diag(self.EK[k])
self.VH0=np.einsum('kn,knqij->qij',self.occupation(self.wavefunction),self.Hartree_elements)
self.VLDAc0=np.einsum('kn,knqij->qij',self.occupation(self.wavefunction),self.LDAc_elements)
self.VLDAx0=np.einsum('kn,knqij->qij',self.occupation(self.wavefunction),self.LDAx_elements)
self.Full_BZ=calc.get_bz_k_points()
self.IBZ_map=calc.get_bz_to_ibz_map()
def get_transition_matrix(self,direction):
direction/=np.linalg.norm(direction)
self.dipole=np.zeros((self.NK,self.nbands,self.nbands),dtype=np.complex)
for k in range(self.NK):
kpt = self.calc.wfs.kpt_u[k]
G=self.calc.wfs.pd.get_reciprocal_vectors(q=k,add_q=True)
G=np.sum(G*direction[None,:],axis=1)
for n in range(self.nvalence):
for m in range(self.nvalence,self.nbands):
wfn=kpt.psit_nG[n];wfm=kpt.psit_nG[m]
self.dipole[k,n,m]=self.calc.wfs.pd.integrate(wfm,G*wfn)/(self.EK[k,n]-self.EK[k,m])
self.dipole[k,m,n]=self.dipole[k,n,m].conj()
return self.dipole
def occupation(self,wavefunction):
return 2*np.sum(self.wk[:,None,None]*self.f[:,None,:]*np.abs(wavefunction)**2,axis=2)
def fast_Hartree_matrix(self,wavefunction):
return np.einsum('kn,knqij->qij',self.occupation(wavefunction),self.Hartree_elements)-self.VH0
def fast_LDA_correlation_matrix(self,wavefunction):
return np.einsum('kn,knqij->qij',self.occupation(wavefunction),self.LDAc_elements)-self.VLDAc0
def fast_LDA_exchange_matrix(self,wavefunction):
return np.einsum('kn,knqij->qij',self.occupation(wavefunction),self.LDAx_elements)-self.VLDAx0
def propagate(self,dt,steps,E,direction,corrections=10):
dipole=self.get_transition_matrix(direction)
self.time_occupation=np.zeros((steps,self.nbands),dtype=np.complex)
self.polarization=np.zeros(steps,dtype=np.complex)
self.time_occupation[0]=np.sum(self.occupation(self.wavefunction),axis=0)
for k in range(self.NK):
operator=np.linalg.multi_dot([self.wavefunction[k].T.conj(),dipole[k],self.wavefunction[k]])
self.polarization[0]+=self.wk[k]*np.sum(operator.diagonal())
for t in tqdm(range(1,steps)):
H = self.Kinetic+E[t]*self.dipole
H+= self.fast_Hartree_matrix(self.wavefunction)
H+= self.fast_LDA_correlation_matrix(self.wavefunction)
H+= self.fast_LDA_exchange_matrix(self.wavefunction)
for k in range(self.NK):
H_left = np.eye(self.nbands)+0.5j*dt*H[k]
H_right= np.eye(self.nbands)-0.5j*dt*H[k]
self.wavefunction[k]=linalg.solve(H_left, [email protected][k])
operator=np.linalg.multi_dot([self.wavefunction[k].T.conj(),dipole[k],self.wavefunction[k]])
self.polarization[t]+=self.wk[k]*np.sum(operator.diagonal())
self.time_occupation[t]=np.sum(self.occupation(self.wavefunction),axis=0)
class HybridXC(HybridXCBase):
orbital_dependent = True
def __init__(self,
name,
hybrid=None,
xc=None,
alpha=None,
gamma_point=1,
method='standard',
bandstructure=False,
logfilename='-',
bands=None,
fcut=1e-10,
molecule=False,
qstride=1,
world=None):
"""Mix standard functionals with exact exchange.
name: str
Name of functional: EXX, PBE0, HSE03, HSE06
hybrid: float
Fraction of exact exchange.
xc: str or XCFunctional object
Standard DFT functional with scaled down exchange.
method: str
Use 'standard' standard formula and 'acdf for
adiabatic-connection dissipation fluctuation formula.
alpha: float
XXX describe
gamma_point: bool
0: Skip k2-k1=0 interactions.
1: Use the alpha method.
2: Integrate the gamma point.
bandstructure: bool
Calculate bandstructure instead of just the total energy.
bands: list of int
List of bands to calculate bandstructure for. Default is
all bands.
molecule: bool
Decouple electrostatic interactions between periodically
repeated images.
fcut: float
Threshold for empty band.
"""
self.alpha = alpha
self.fcut = fcut
self.gamma_point = gamma_point
self.method = method
self.bandstructure = bandstructure
self.bands = bands
self.fd = logfilename
self.write_timing_information = True
HybridXCBase.__init__(self, name, hybrid, xc)
# EXX energies:
self.exx = None # total
self.evv = None # valence-valence (pseudo part)
self.evvacdf = None # valence-valence (pseudo part)
self.devv = None # valence-valence (PAW correction)
self.evc = None # valence-core
self.ecc = None # core-core
self.exx_skn = None # bandstructure
self.qlatest = None
if world is None:
world = mpi.world
self.world = world
self.molecule = molecule
if isinstance(qstride, int):
qstride = [qstride] * 3
self.qstride_c = np.asarray(qstride)
self.timer = Timer()
def log(self, *args, **kwargs):
prnt(file=self.fd, *args, **kwargs)
self.fd.flush()
def calculate_radial(self,
rgd,
n_sLg,
Y_L,
v_sg,
dndr_sLg=None,
rnablaY_Lv=None,
tau_sg=None,
dedtau_sg=None):
return self.xc.calculate_radial(rgd, n_sLg, Y_L, v_sg, dndr_sLg,
rnablaY_Lv)
def calculate_paw_correction(self,
setup,
D_sp,
dEdD_sp=None,
addcoredensity=True,
a=None):
return self.xc.calculate_paw_correction(setup, D_sp, dEdD_sp,
addcoredensity, a)
def initialize(self, dens, ham, wfs, occupations):
assert wfs.bd.comm.size == 1
self.xc.initialize(dens, ham, wfs, occupations)
self.dens = dens
self.wfs = wfs
# Make a k-point descriptor that is not distributed
# (self.kd.comm is serial_comm):
self.kd = wfs.kd.copy()
self.fd = logfile(self.fd, self.world.rank)
wfs.initialize_wave_functions_from_restart_file()
def set_positions(self, spos_ac):
self.spos_ac = spos_ac
def calculate(self, gd, n_sg, v_sg=None, e_g=None):
# Normal XC contribution:
exc = self.xc.calculate(gd, n_sg, v_sg, e_g)
# Add EXX contribution:
return exc + self.exx * self.hybrid
def calculate_exx(self):
"""Non-selfconsistent calculation."""
self.timer.start('EXX')
self.timer.start('Initialization')
kd = self.kd
wfs = self.wfs
if fftw.FFTPlan is fftw.NumpyFFTPlan:
self.log('NOT USING FFTW !!')
self.log('Spins:', self.wfs.nspins)
W = max(1, self.wfs.kd.comm.size // self.wfs.nspins)
# Are the k-points distributed?
kparallel = (W > 1)
# Find number of occupied bands:
self.nocc_sk = np.zeros((self.wfs.nspins, kd.nibzkpts), int)
for kpt in self.wfs.kpt_u:
for n, f in enumerate(kpt.f_n):
if abs(f) < self.fcut:
self.nocc_sk[kpt.s, kpt.k] = n
break
else:
self.nocc_sk[kpt.s, kpt.k] = self.wfs.bd.nbands
self.wfs.kd.comm.sum(self.nocc_sk)
noccmin = self.nocc_sk.min()
noccmax = self.nocc_sk.max()
self.log('Number of occupied bands (min, max): %d, %d' %
(noccmin, noccmax))
self.log('Number of valence electrons:', self.wfs.setups.nvalence)
if self.bandstructure:
self.log('Calculating eigenvalue shifts.')
# allocate array for eigenvalue shifts:
self.exx_skn = np.zeros(
(self.wfs.nspins, kd.nibzkpts, self.wfs.bd.nbands))
if self.bands is None:
noccmax = self.wfs.bd.nbands
else:
noccmax = max(max(self.bands) + 1, noccmax)
N_c = self.kd.N_c
vol = wfs.gd.dv * wfs.gd.N_c.prod()
if self.alpha is None:
alpha = 6 * vol**(2 / 3.0) / pi**2
else:
alpha = self.alpha
if self.gamma_point == 1:
if alpha == 0.0:
qvol = (2 * np.pi)**3 / vol / N_c.prod()
self.gamma = 4 * np.pi * (3 * qvol /
(4 * np.pi))**(1 / 3.) / qvol
else:
self.gamma = self.calculate_gamma(vol, alpha)
else:
kcell_cv = wfs.gd.cell_cv.copy()
kcell_cv[0] *= N_c[0]
kcell_cv[1] *= N_c[1]
kcell_cv[2] *= N_c[2]
self.gamma = madelung(kcell_cv) * vol * N_c.prod() / (4 * np.pi)
self.log('Value of alpha parameter: %.3f Bohr^2' % alpha)
self.log('Value of gamma parameter: %.3f Bohr^2' % self.gamma)
# Construct all possible q=k2-k1 vectors:
Nq_c = (N_c - 1) // self.qstride_c
i_qc = np.indices(Nq_c * 2 + 1, float).transpose((1, 2, 3, 0)).reshape(
(-1, 3))
self.bzq_qc = (i_qc - Nq_c) / N_c * self.qstride_c
self.q0 = ((Nq_c * 2 + 1).prod() - 1) // 2 # index of q=(0,0,0)
assert not self.bzq_qc[self.q0].any()
# Count number of pairs for each q-vector:
self.npairs_q = np.zeros(len(self.bzq_qc), int)
for s in range(kd.nspins):
for k1 in range(kd.nibzkpts):
for k2 in range(kd.nibzkpts):
for K2, q, n1_n, n2 in self.indices(s, k1, k2):
self.npairs_q[q] += len(n1_n)
self.npairs0 = self.npairs_q.sum() # total number of pairs
self.log('Number of pairs:', self.npairs0)
# Distribute q-vectors to Q processors:
Q = self.world.size // self.wfs.kd.comm.size
myrank = self.world.rank // self.wfs.kd.comm.size
rank = 0
N = 0
myq = []
nq = 0
for q, n in enumerate(self.npairs_q):
if n > 0:
nq += 1
if rank == myrank:
myq.append(q)
N += n
if N >= (rank + 1.0) * self.npairs0 / Q:
rank += 1
assert len(myq) > 0, 'Too few q-vectors for too many processes!'
self.bzq_qc = self.bzq_qc[myq]
try:
self.q0 = myq.index(self.q0)
except ValueError:
self.q0 = None
self.log('%d x %d x %d k-points' % tuple(self.kd.N_c))
self.log('Distributing %d IBZ k-points over %d process(es).' %
(kd.nibzkpts, self.wfs.kd.comm.size))
self.log('Distributing %d q-vectors over %d process(es).' % (nq, Q))
# q-point descriptor for my q-vectors:
qd = KPointDescriptor(self.bzq_qc)
# Plane-wave descriptor for all wave-functions:
self.pd = PWDescriptor(wfs.pd.ecut, wfs.gd, dtype=wfs.pd.dtype, kd=kd)
# Plane-wave descriptor pair-densities:
self.pd2 = PWDescriptor(self.dens.pd2.ecut,
self.dens.gd,
dtype=wfs.dtype,
kd=qd)
self.log('Cutoff energies:')
self.log(' Wave functions: %10.3f eV' %
(self.pd.ecut * Hartree))
self.log(' Density: %10.3f eV' %
(self.pd2.ecut * Hartree))
# Calculate 1/|G+q|^2 with special treatment of |G+q|=0:
G2_qG = self.pd2.G2_qG
if self.q0 is None:
if self.omega is None:
self.iG2_qG = [1.0 / G2_G for G2_G in G2_qG]
else:
self.iG2_qG = [
(1.0 / G2_G * (1 - np.exp(-G2_G / (4 * self.omega**2))))
for G2_G in G2_qG
]
else:
G2_qG[self.q0][0] = 117.0 # avoid division by zero
if self.omega is None:
self.iG2_qG = [1.0 / G2_G for G2_G in G2_qG]
self.iG2_qG[self.q0][0] = self.gamma
else:
self.iG2_qG = [
(1.0 / G2_G * (1 - np.exp(-G2_G / (4 * self.omega**2))))
for G2_G in G2_qG
]
self.iG2_qG[self.q0][0] = 1 / (4 * self.omega**2)
G2_qG[self.q0][0] = 0.0 # restore correct value
# Compensation charges:
self.ghat = PWLFC([setup.ghat_l for setup in wfs.setups], self.pd2)
self.ghat.set_positions(self.spos_ac)
if self.molecule:
self.initialize_gaussian()
self.log('Value of beta parameter: %.3f 1/Bohr^2' % self.beta)
self.timer.stop('Initialization')
# Ready ... set ... go:
self.t0 = time()
self.npairs = 0
self.evv = 0.0
self.evvacdf = 0.0
for s in range(self.wfs.nspins):
kpt1_q = [
KPoint(self.wfs, noccmax).initialize(kpt)
for kpt in self.wfs.kpt_u if kpt.s == s
]
kpt2_q = kpt1_q[:]
if len(kpt1_q) == 0:
# No s-spins on this CPU:
continue
# Send and receive ranks:
srank = self.wfs.kd.get_rank_and_index(s, (kpt1_q[0].k - 1) %
kd.nibzkpts)[0]
rrank = self.wfs.kd.get_rank_and_index(s, (kpt1_q[-1].k + 1) %
kd.nibzkpts)[0]
# Shift k-points kd.nibzkpts - 1 times:
for i in range(kd.nibzkpts):
if i < kd.nibzkpts - 1:
if kparallel:
kpt = kpt2_q[-1].next(self.wfs)
kpt.start_receiving(rrank)
kpt2_q[0].start_sending(srank)
else:
kpt = kpt2_q[0]
self.timer.start('Calculate')
for kpt1, kpt2 in zip(kpt1_q, kpt2_q):
# Loop over all k-points that k2 can be mapped to:
for K2, q, n1_n, n2 in self.indices(s, kpt1.k, kpt2.k):
self.apply(K2, q, kpt1, kpt2, n1_n, n2)
self.timer.stop('Calculate')
if i < kd.nibzkpts - 1:
self.timer.start('Wait')
if kparallel:
kpt.wait()
kpt2_q[0].wait()
self.timer.stop('Wait')
kpt2_q.pop(0)
kpt2_q.append(kpt)
self.evv = self.world.sum(self.evv)
self.evvacdf = self.world.sum(self.evvacdf)
self.calculate_exx_paw_correction()
if self.method == 'standard':
self.exx = self.evv + self.devv + self.evc + self.ecc
elif self.method == 'acdf':
self.exx = self.evvacdf + self.devv + self.evc + self.ecc
else:
1 / 0
self.log('Exact exchange energy:')
for txt, e in [('core-core', self.ecc), ('valence-core', self.evc),
('valence-valence (pseudo, acdf)', self.evvacdf),
('valence-valence (pseudo, standard)', self.evv),
('valence-valence (correction)', self.devv),
('total (%s)' % self.method, self.exx)]:
self.log(' %-36s %14.6f eV' % (txt + ':', e * Hartree))
self.log('Total time: %10.3f seconds' % (time() - self.t0))
self.npairs = self.world.sum(self.npairs)
assert self.npairs == self.npairs0
self.timer.stop('EXX')
self.timer.write(self.fd)
def calculate_gamma(self, vol, alpha):
if self.molecule:
return 0.0
N_c = self.kd.N_c
offset_c = (N_c + 1) % 2 * 0.5 / N_c
bzq_qc = monkhorst_pack(N_c) + offset_c
qd = KPointDescriptor(bzq_qc)
pd = PWDescriptor(self.wfs.pd.ecut, self.wfs.gd, kd=qd)
gamma = (vol / (2 * pi)**2 * sqrt(pi / alpha) * self.kd.nbzkpts)
for G2_G in pd.G2_qG:
if G2_G[0] < 1e-7:
G2_G = G2_G[1:]
gamma -= np.dot(np.exp(-alpha * G2_G), G2_G**-1)
return gamma / self.qstride_c.prod()
def indices(self, s, k1, k2):
"""Generator for (K2, q, n1, n2) indices for (k1, k2) pair.
s: int
Spin index.
k1: int
Index of k-point in the IBZ.
k2: int
Index of k-point in the IBZ.
Returns (K, q, n1_n, n2), where K then index of the k-point in
the BZ that k2 is mapped to, q is the index of the q-vector
between K and k1, and n1_n is a list of bands that should be
combined with band n2."""
for K, k in enumerate(self.kd.bz2ibz_k):
if k == k2:
for K, q, n1_n, n2 in self._indices(s, k1, k2, K):
yield K, q, n1_n, n2
def _indices(self, s, k1, k2, K2):
k1_c = self.kd.ibzk_kc[k1]
k2_c = self.kd.bzk_kc[K2]
q_c = k2_c - k1_c
q = abs(self.bzq_qc - q_c).sum(1).argmin()
if abs(self.bzq_qc[q] - q_c).sum() > 1e-7:
return
if self.gamma_point == 0 and q == self.q0:
return
nocc1 = self.nocc_sk[s, k1]
nocc2 = self.nocc_sk[s, k2]
# Is k2 in the IBZ?
is_ibz2 = (self.kd.ibz2bz_k[k2] == K2)
for n2 in range(self.wfs.bd.nbands):
# Find range of n1's (from n1a to n1b-1):
if is_ibz2:
# We get this combination twice, so let's only do half:
if k1 >= k2:
n1a = n2
else:
n1a = n2 + 1
else:
n1a = 0
n1b = self.wfs.bd.nbands
if self.bandstructure:
if n2 >= nocc2:
n1b = min(n1b, nocc1)
else:
if n2 >= nocc2:
break
n1b = min(n1b, nocc1)
if self.bands is not None:
assert self.bandstructure
n1_n = []
for n1 in range(n1a, n1b):
if (n1 in self.bands and n2 < nocc2
or is_ibz2 and n2 in self.bands and n1 < nocc1):
n1_n.append(n1)
n1_n = np.array(n1_n)
else:
n1_n = np.arange(n1a, n1b)
if len(n1_n) == 0:
continue
yield K2, q, n1_n, n2
def apply(self, K2, q, kpt1, kpt2, n1_n, n2):
k20_c = self.kd.ibzk_kc[kpt2.k]
k2_c = self.kd.bzk_kc[K2]
if k2_c.any():
self.timer.start('Initialize plane waves')
eik2r_R = self.wfs.gd.plane_wave(k2_c)
eik20r_R = self.wfs.gd.plane_wave(k20_c)
self.timer.stop('Initialize plane waves')
else:
eik2r_R = 1.0
eik20r_R = 1.0
w1 = self.kd.weight_k[kpt1.k]
w2 = self.kd.weight_k[kpt2.k]
# Is k2 in the 1. BZ?
is_ibz2 = (self.kd.ibz2bz_k[kpt2.k] == K2)
e_n = self.calculate_interaction(n1_n, n2, kpt1, kpt2, q, K2, eik20r_R,
eik2r_R, is_ibz2)
e_n *= 1.0 / self.kd.nbzkpts / self.wfs.nspins * self.qstride_c.prod()
if q == self.q0:
e_n[n1_n == n2] *= 0.5
f1_n = kpt1.f_n[n1_n]
eps1_n = kpt1.eps_n[n1_n]
f2 = kpt2.f_n[n2]
eps2 = kpt2.eps_n[n2]
s_n = np.sign(eps2 - eps1_n)
evv = (f1_n * f2 * e_n).sum()
evvacdf = 0.5 * (f1_n * (1 - s_n) * e_n + f2 * (1 + s_n) * e_n).sum()
self.evv += evv * w1
self.evvacdf += evvacdf * w1
if is_ibz2:
self.evv += evv * w2
self.evvacdf += evvacdf * w2
if self.bandstructure:
x = self.wfs.nspins
self.exx_skn[kpt1.s, kpt1.k, n1_n] += x * f2 * e_n
if is_ibz2:
self.exx_skn[kpt2.s, kpt2.k, n2] += x * np.dot(f1_n, e_n)
def calculate_interaction(self, n1_n, n2, kpt1, kpt2, q, k, eik20r_R,
eik2r_R, is_ibz2):
"""Calculate Coulomb interactions.
For all n1 in the n1_n list, calculate interaction with n2."""
# number of plane waves:
ng1 = self.wfs.ng_k[kpt1.k]
ng2 = self.wfs.ng_k[kpt2.k]
# Transform to real space and apply symmetry operation:
self.timer.start('IFFT1')
if is_ibz2:
u2_R = self.pd.ifft(kpt2.psit_nG[n2, :ng2], kpt2.k)
else:
psit2_R = self.pd.ifft(kpt2.psit_nG[n2, :ng2], kpt2.k) * eik20r_R
self.timer.start('Symmetry transform')
u2_R = self.kd.transform_wave_function(psit2_R, k) / eik2r_R
self.timer.stop()
self.timer.stop()
# Calculate pair densities:
nt_nG = self.pd2.zeros(len(n1_n), q=q)
for n1, nt_G in zip(n1_n, nt_nG):
self.timer.start('IFFT2')
u1_R = self.pd.ifft(kpt1.psit_nG[n1, :ng1], kpt1.k)
self.timer.stop()
nt_R = u1_R.conj() * u2_R
self.timer.start('FFT')
nt_G[:] = self.pd2.fft(nt_R, q)
self.timer.stop()
s = self.kd.sym_k[k]
time_reversal = self.kd.time_reversal_k[k]
k2_c = self.kd.ibzk_kc[kpt2.k]
self.timer.start('Compensation charges')
Q_anL = {} # coefficients for shape functions
for a, P1_ni in kpt1.P_ani.items():
P1_ni = P1_ni[n1_n]
if is_ibz2:
P2_i = kpt2.P_ani[a][n2]
else:
b = self.kd.symmetry.a_sa[s, a]
S_c = (np.dot(self.spos_ac[a], self.kd.symmetry.op_scc[s]) -
self.spos_ac[b])
assert abs(S_c.round() - S_c).max() < 1e-5
if self.ghat.dtype == complex:
x = np.exp(2j * pi * np.dot(k2_c, S_c))
else:
x = 1.0
P2_i = np.dot(self.wfs.setups[a].R_sii[s],
kpt2.P_ani[b][n2]) * x
if time_reversal:
P2_i = P2_i.conj()
D_np = []
for P1_i in P1_ni:
D_ii = np.outer(P1_i.conj(), P2_i)
D_np.append(pack(D_ii))
Q_anL[a] = np.dot(D_np, self.wfs.setups[a].Delta_pL)
self.timer.start('Expand')
if q != self.qlatest:
self.f_IG = self.ghat.expand(q)
self.qlatest = q
self.timer.stop('Expand')
# Add compensation charges:
self.ghat.add(nt_nG, Q_anL, q, self.f_IG)
self.timer.stop('Compensation charges')
if self.molecule and n2 in n1_n:
nn = (n1_n == n2).nonzero()[0][0]
nt_nG[nn] -= self.ngauss_G
else:
nn = None
iG2_G = self.iG2_qG[q]
# Calculate energies:
e_n = np.empty(len(n1_n))
for n, nt_G in enumerate(nt_nG):
e_n[n] = -4 * pi * np.real(self.pd2.integrate(nt_G, nt_G * iG2_G))
self.npairs += 1
if nn is not None:
e_n[nn] -= 2 * (self.pd2.integrate(nt_nG[nn], self.vgauss_G) +
(self.beta / 2 / pi)**0.5)
if self.write_timing_information:
t = (time() - self.t0) / len(n1_n)
self.log('Time for first pair-density: %10.3f seconds' % t)
self.log('Estimated total time: %10.3f seconds' %
(t * self.npairs0 / self.world.size))
self.write_timing_information = False
return e_n
def calculate_exx_paw_correction(self):
self.timer.start('PAW correction')
self.devv = 0.0
self.evc = 0.0
self.ecc = 0.0
deg = 2 // self.wfs.nspins # spin degeneracy
for a, D_sp in self.dens.D_asp.items():
setup = self.wfs.setups[a]
for D_p in D_sp:
D_ii = unpack2(D_p)
ni = len(D_ii)
for i1 in range(ni):
for i2 in range(ni):
A = 0.0
for i3 in range(ni):
p13 = packed_index(i1, i3, ni)
for i4 in range(ni):
p24 = packed_index(i2, i4, ni)
A += setup.M_pp[p13, p24] * D_ii[i3, i4]
self.devv -= D_ii[i1, i2] * A / deg
self.evc -= np.dot(D_p, setup.X_p)
self.ecc += setup.ExxC
if not self.bandstructure:
self.timer.stop('PAW correction')
return
Q = self.world.size // self.wfs.kd.comm.size
self.exx_skn *= Q
for kpt in self.wfs.kpt_u:
for a, D_sp in self.dens.D_asp.items():
setup = self.wfs.setups[a]
for D_p in D_sp:
D_ii = unpack2(D_p)
ni = len(D_ii)
P_ni = kpt.P_ani[a]
for i1 in range(ni):
for i2 in range(ni):
A = 0.0
for i3 in range(ni):
p13 = packed_index(i1, i3, ni)
for i4 in range(ni):
p24 = packed_index(i2, i4, ni)
A += setup.M_pp[p13, p24] * D_ii[i3, i4]
self.exx_skn[kpt.s, kpt.k] -= \
(A * P_ni[:, i1].conj() * P_ni[:, i2]).real
p12 = packed_index(i1, i2, ni)
self.exx_skn[kpt.s, kpt.k] -= \
(P_ni[:, i1].conj() * setup.X_p[p12] *
P_ni[:, i2]).real / self.wfs.nspins
self.world.sum(self.exx_skn)
self.exx_skn *= self.hybrid / Q
self.timer.stop('PAW correction')
def initialize_gaussian(self):
"""Calculate gaussian compensation charge and its potential.
Used to decouple electrostatic interactions between
periodically repeated images for molecular calculations.
Charge containing one electron::
(beta/pi)^(3/2)*exp(-beta*r^2),
its Fourier transform::
exp(-G^2/(4*beta)),
and its potential::
erf(beta^0.5*r)/r.
"""
gd = self.wfs.gd
# Set exponent of exp-function to -19 on the boundary:
self.beta = 4 * 19 * (gd.icell_cv**2).sum(1).max()
# Calculate gaussian:
G_Gv = self.pd2.get_reciprocal_vectors()
G2_G = self.pd2.G2_qG[0]
C_v = gd.cell_cv.sum(0) / 2 # center of cell
self.ngauss_G = np.exp(-1.0 / (4 * self.beta) * G2_G +
1j * np.dot(G_Gv, C_v)) / gd.dv
# Calculate potential from gaussian:
R_Rv = gd.get_grid_point_coordinates().transpose((1, 2, 3, 0))
r_R = ((R_Rv - C_v)**2).sum(3)**0.5
if (gd.N_c % 2 == 0).all():
r_R[tuple(gd.N_c // 2)] = 1.0 # avoid dividing by zero
v_R = erf(self.beta**0.5 * r_R) / r_R
if (gd.N_c % 2 == 0).all():
v_R[tuple(gd.N_c // 2)] = (4 * self.beta / pi)**0.5
self.vgauss_G = self.pd2.fft(v_R)
# Compare self-interaction to analytic result:
assert abs(0.5 * self.pd2.integrate(self.ngauss_G, self.vgauss_G) -
(self.beta / 2 / pi)**0.5) < 1e-6
lfc1 = LFC(gd, [[s]], kd, dtype=complex)
lfc2 = PWLFC([[s]], pd)
c_axi = {0: np.zeros((1, 2 * l + 1), complex)}
c_axi[0][0, 0] = 1.9 - 4.5j
c_axiv = {0: np.zeros((1, 2 * l + 1, 3), complex)}
b1 = gd.zeros(1, dtype=complex)
b2 = pd.zeros(1, dtype=complex)
for lfc, b in [(lfc1, b1), (lfc2, b2)]:
lfc.set_positions(spos_ac)
lfc.add(b, c_axi, 0)
b2 = pd.ifft(b2[0]) * eikr
equal(abs(b2-b1[0]).max(), 0, 0.001)
b1 = eikr[None]
b2 = pd.fft(b1[0] * 0 + 1).reshape((1, -1))
results = []
results2 = []
for lfc, b in [(lfc1, b1), (lfc2, b2)]:
lfc.integrate(b, c_axi, 0)
results.append(c_axi[0][0].copy())
lfc.derivative(b, c_axiv, 0)
results2.append(c_axiv[0][0].copy())
equal(abs(np.ptp(results2, 0)).max(), 0, 1e-7)
equal(abs(np.ptp(results, 0)).max(), 0, 3e-8)
lfcr = PWLFC([[s]], pdr)
c_axi = {0: np.zeros((1, 2 * l + 1), complex)}
c_axi[0][0, 0] = 1.9
cr_axi = {0: np.zeros((1, 2 * l + 1))}
cr_axi[0][0, 0] = 1.9
b = pd.zeros(1, dtype=complex)
br = pdr.zeros(1)
lfc.set_positions(spos_ac)
lfc.add(b, c_axi)
lfcr.set_positions(spos_ac)
lfcr.add(br, cr_axi)
a = pd.ifft(b)
ar = pdr.ifft(br)
equal(abs(a-ar).max(), 0, 1e-14)
if l == 0:
a = a[:, ::-1].copy()
b0 = pd.fft(a)
br0 = pdr.fft(a.real)
lfc.integrate(b0, c_axi)
lfcr.integrate(br0, cr_axi)
assert abs(c_axi[0][0]-cr_axi[0][0]).max() < 1e-14
c_axiv = {0: np.zeros((1, 2 * l + 1, 3), complex)}
cr_axiv = {0: np.zeros((1, 2 * l + 1, 3))}
lfc.derivative(b0, c_axiv)
class HybridXC(XCFunctional):
orbital_dependent = True
def __init__(self, name, hybrid=None, xc=None, finegrid=False, alpha=None):
"""Mix standard functionals with exact exchange.
name: str
Name of hybrid functional.
hybrid: float
Fraction of exact exchange.
xc: str or XCFunctional object
Standard DFT functional with scaled down exchange.
finegrid: boolean
Use fine grid for energy functional evaluations?
"""
if name == 'EXX':
assert hybrid is None and xc is None
hybrid = 1.0
xc = XC(XCNull())
elif name == 'PBE0':
assert hybrid is None and xc is None
hybrid = 0.25
xc = XC('HYB_GGA_XC_PBEH')
elif name == 'B3LYP':
assert hybrid is None and xc is None
hybrid = 0.2
xc = XC('HYB_GGA_XC_B3LYP')
if isinstance(xc, str):
xc = XC(xc)
self.hybrid = hybrid
self.xc = xc
self.type = xc.type
self.alpha = alpha
self.exx = 0.0
XCFunctional.__init__(self, name)
def get_setup_name(self):
return 'PBE'
def calculate_radial(self,
rgd,
n_sLg,
Y_L,
v_sg,
dndr_sLg=None,
rnablaY_Lv=None,
tau_sg=None,
dedtau_sg=None):
return self.xc.calculate_radial(rgd, n_sLg, Y_L, v_sg, dndr_sLg,
rnablaY_Lv)
def initialize(self, density, hamiltonian, wfs, occupations):
self.xc.initialize(density, hamiltonian, wfs, occupations)
self.nspins = wfs.nspins
self.setups = wfs.setups
self.density = density
self.kpt_u = wfs.kpt_u
self.gd = density.gd
self.kd = wfs.kd
self.bd = wfs.bd
N_c = self.gd.N_c
N = self.gd.N_c.prod()
vol = self.gd.dv * N
if self.alpha is None:
self.alpha = 6 * vol**(2 / 3.0) / pi**2
self.gamma = (vol / (2 * pi)**2 * sqrt(pi / self.alpha) *
self.kd.nbzkpts)
ecut = 0.5 * pi**2 / (self.gd.h_cv**2).sum(1).max()
if self.kd.N_c is None:
self.bzk_kc = np.zeros((1, 3))
dfghdfgh
else:
n = self.kd.N_c * 2 - 1
bzk_kc = np.indices(n).transpose((1, 2, 3, 0))
bzk_kc.shape = (-1, 3)
bzk_kc -= self.kd.N_c - 1
self.bzk_kc = bzk_kc.astype(float) / self.kd.N_c
self.pwd = PWDescriptor(ecut, self.gd, self.bzk_kc)
n = 0
for k_c, Gpk2_G in zip(self.bzk_kc[:], self.pwd.G2_qG):
if (k_c > -0.5).all() and (k_c <= 0.5).all(): #XXX???
if k_c.any():
self.gamma -= np.dot(np.exp(-self.alpha * Gpk2_G),
Gpk2_G**-1)
else:
self.gamma -= np.dot(np.exp(-self.alpha * Gpk2_G[1:]),
Gpk2_G[1:]**-1)
n += 1
assert n == self.kd.N_c.prod()
self.ghat = LFC(self.gd, [setup.ghat_l for setup in density.setups],
dtype=complex)
self.ghat.set_k_points(self.bzk_kc)
self.fullkd = KPointDescriptor(self.kd.bzk_kc, nspins=1)
class S:
id_a = []
def set_symmetry(self, s):
pass
self.fullkd.set_symmetry(Atoms(pbc=True), S(), False)
self.fullkd.set_communicator(world)
self.pt = LFC(self.gd, [setup.pt_j for setup in density.setups],
dtype=complex)
self.pt.set_k_points(self.fullkd.ibzk_kc)
self.interpolator = density.interpolator
def set_positions(self, spos_ac):
self.ghat.set_positions(spos_ac)
self.pt.set_positions(spos_ac)
def calculate(self, gd, n_sg, v_sg=None, e_g=None):
# Normal XC contribution:
exc = self.xc.calculate(gd, n_sg, v_sg, e_g)
# Add EXX contribution:
return exc + self.exx
def calculate_exx(self):
"""Non-selfconsistent calculation."""
kd = self.kd
K = self.fullkd.nibzkpts
assert self.nspins == 1
Q = K // world.size
assert Q * world.size == K
parallel = (world.size > self.nspins)
self.exx = 0.0
self.exx_skn = np.zeros((self.nspins, K, self.bd.nbands))
kpt_u = []
for k in range(world.rank * Q, (world.rank + 1) * Q):
k_c = self.fullkd.ibzk_kc[k]
for k1, k1_c in enumerate(kd.bzk_kc):
if abs(k1_c - k_c).max() < 1e-10:
break
# Index of symmetry related point in the irreducible BZ
ik = kd.kibz_k[k1]
kpt = self.kpt_u[ik]
# KPoint from ground-state calculation
phase_cd = np.exp(2j * pi * self.gd.sdisp_cd * k_c[:, np.newaxis])
kpt2 = KPoint0(kpt.weight, kpt.s, k, None, phase_cd)
kpt2.psit_nG = np.empty_like(kpt.psit_nG)
kpt2.f_n = kpt.f_n / kpt.weight / K * 2
for n, psit_G in enumerate(kpt2.psit_nG):
psit_G[:] = kd.transform_wave_function(kpt.psit_nG[n], k1)
kpt2.P_ani = self.pt.dict(len(kpt.psit_nG))
self.pt.integrate(kpt2.psit_nG, kpt2.P_ani, k)
kpt_u.append(kpt2)
for s in range(self.nspins):
kpt1_q = [KPoint(self.fullkd, kpt) for kpt in kpt_u if kpt.s == s]
kpt2_q = kpt1_q[:]
if len(kpt1_q) == 0:
# No s-spins on this CPU:
continue
# Send rank:
srank = self.fullkd.get_rank_and_index(s, (kpt1_q[0].k - 1) % K)[0]
# Receive rank:
rrank = self.fullkd.get_rank_and_index(s,
(kpt1_q[-1].k + 1) % K)[0]
# Shift k-points K // 2 times:
for i in range(K // 2 + 1):
if i < K // 2:
if parallel:
kpt = kpt2_q[-1].next()
kpt.start_receiving(rrank)
kpt2_q[0].start_sending(srank)
else:
kpt = kpt2_q[0]
for kpt1, kpt2 in zip(kpt1_q, kpt2_q):
if 2 * i == K:
self.apply(kpt1, kpt2, invert=(kpt1.k > kpt2.k))
else:
self.apply(kpt1, kpt2)
self.apply(kpt1, kpt2, invert=True)
if i < K // 2:
if parallel:
kpt.wait()
kpt2_q[0].wait()
kpt2_q.pop(0)
kpt2_q.append(kpt)
self.exx = world.sum(self.exx)
world.sum(self.exx_skn)
self.exx += self.calculate_paw_correction()
def apply(self, kpt1, kpt2, invert=False):
#print world.rank,kpt1.k,kpt2.k,invert
k1_c = self.fullkd.ibzk_kc[kpt1.k]
k2_c = self.fullkd.ibzk_kc[kpt2.k]
if invert:
k2_c = -k2_c
k12_c = k1_c - k2_c
N_c = self.gd.N_c
eikr_R = np.exp(2j * pi * np.dot(np.indices(N_c).T, k12_c / N_c).T)
for q, k_c in enumerate(self.bzk_kc):
if abs(k_c + k12_c).max() < 1e-9:
q0 = q
break
for q, k_c in enumerate(self.bzk_kc):
if abs(k_c - k12_c).max() < 1e-9:
q00 = q
break
Gpk2_G = self.pwd.G2_qG[q0]
if Gpk2_G[0] == 0:
Gpk2_G = Gpk2_G.copy()
Gpk2_G[0] = 1.0 / self.gamma
N = N_c.prod()
vol = self.gd.dv * N
nspins = self.nspins
same = (kpt1.k == kpt2.k)
for n1, psit1_R in enumerate(kpt1.psit_nG):
f1 = kpt1.f_n[n1]
for n2, psit2_R in enumerate(kpt2.psit_nG):
if same and n2 > n1:
continue
f2 = kpt2.f_n[n2]
nt_R = self.calculate_pair_density(n1, n2, kpt1, kpt2, q0,
invert)
nt_G = self.pwd.fft(nt_R * eikr_R) / N
vt_G = nt_G.copy()
vt_G *= -pi * vol / Gpk2_G
e = np.vdot(nt_G, vt_G).real * nspins * self.hybrid
if same and n1 == n2:
e /= 2
self.exx += e * f1 * f2
self.ekin -= 2 * e * f1 * f2
self.exx_skn[kpt1.s, kpt1.k, n1] += f2 * e
self.exx_skn[kpt2.s, kpt2.k, n2] += f1 * e
calculate_potential = not True
if calculate_potential:
vt_R = self.pwd.ifft(vt_G).conj() * eikr_R * N / vol
if kpt1 is kpt2 and not invert and n1 == n2:
kpt1.vt_nG[n1] = 0.5 * f1 * vt_R
if invert:
kpt1.Htpsit_nG[n1] += \
f2 * nspins * psit2_R.conj() * vt_R
else:
kpt1.Htpsit_nG[n1] += f2 * nspins * psit2_R * vt_R
if kpt1 is not kpt2:
if invert:
kpt2.Htpsit_nG[n2] += (f1 * nspins *
psit1_R.conj() * vt_R)
else:
kpt2.Htpsit_nG[n2] += (f1 * nspins * psit1_R *
vt_R.conj())
def calculate_paw_correction(self):
exx = 0
deg = 2 // self.nspins # spin degeneracy
for a, D_sp in self.density.D_asp.items():
setup = self.setups[a]
for D_p in D_sp:
D_ii = unpack2(D_p)
ni = len(D_ii)
for i1 in range(ni):
for i2 in range(ni):
A = 0.0
for i3 in range(ni):
p13 = packed_index(i1, i3, ni)
for i4 in range(ni):
p24 = packed_index(i2, i4, ni)
A += setup.M_pp[p13, p24] * D_ii[i3, i4]
p12 = packed_index(i1, i2, ni)
exx -= self.hybrid / deg * D_ii[i1, i2] * A
if setup.X_p is not None:
exx -= self.hybrid * np.dot(D_p, setup.X_p)
exx += self.hybrid * setup.ExxC
return exx
def calculate_pair_density(self, n1, n2, kpt1, kpt2, q, invert):
if invert:
nt_G = kpt1.psit_nG[n1].conj() * kpt2.psit_nG[n2].conj()
else:
nt_G = kpt1.psit_nG[n1].conj() * kpt2.psit_nG[n2]
Q_aL = {}
for a, P1_ni in kpt1.P_ani.items():
P1_i = P1_ni[n1]
P2_i = kpt2.P_ani[a][n2]
if invert:
D_ii = np.outer(P1_i.conj(), P2_i.conj())
else:
D_ii = np.outer(P1_i.conj(), P2_i)
D_p = pack(D_ii)
Q_aL[a] = np.dot(D_p, self.setups[a].Delta_pL)
self.ghat.add(nt_G, Q_aL, q)
return nt_G
class HybridXC(XCFunctional):
orbital_dependent = True
def __init__(self, name, hybrid=None, xc=None, finegrid=False,
alpha=None):
"""Mix standard functionals with exact exchange.
name: str
Name of hybrid functional.
hybrid: float
Fraction of exact exchange.
xc: str or XCFunctional object
Standard DFT functional with scaled down exchange.
finegrid: boolean
Use fine grid for energy functional evaluations?
"""
if name == 'EXX':
assert hybrid is None and xc is None
hybrid = 1.0
xc = XC(XCNull())
elif name == 'PBE0':
assert hybrid is None and xc is None
hybrid = 0.25
xc = XC('HYB_GGA_XC_PBEH')
elif name == 'B3LYP':
assert hybrid is None and xc is None
hybrid = 0.2
xc = XC('HYB_GGA_XC_B3LYP')
if isinstance(xc, str):
xc = XC(xc)
self.hybrid = hybrid
self.xc = xc
self.type = xc.type
self.alpha = alpha
self.exx = 0.0
XCFunctional.__init__(self, name)
def get_setup_name(self):
return 'PBE'
def calculate_radial(self, rgd, n_sLg, Y_L, v_sg,
dndr_sLg=None, rnablaY_Lv=None,
tau_sg=None, dedtau_sg=None):
return self.xc.calculate_radial(rgd, n_sLg, Y_L, v_sg,
dndr_sLg, rnablaY_Lv)
def initialize(self, density, hamiltonian, wfs, occupations):
self.xc.initialize(density, hamiltonian, wfs, occupations)
self.nspins = wfs.nspins
self.setups = wfs.setups
self.density = density
self.kpt_u = wfs.kpt_u
self.gd = density.gd
self.kd = wfs.kd
self.bd = wfs.bd
N_c = self.gd.N_c
N = self.gd.N_c.prod()
vol = self.gd.dv * N
if self.alpha is None:
self.alpha = 6 * vol**(2 / 3.0) / pi**2
self.gamma = (vol / (2 * pi)**2 * sqrt(pi / self.alpha) *
self.kd.nbzkpts)
ecut = 0.5 * pi**2 / (self.gd.h_cv**2).sum(1).max()
if self.kd.N_c is None:
self.bzk_kc = np.zeros((1, 3))
dfghdfgh
else:
n = self.kd.N_c * 2 - 1
bzk_kc = np.indices(n).transpose((1, 2, 3, 0))
bzk_kc.shape = (-1, 3)
bzk_kc -= self.kd.N_c - 1
self.bzk_kc = bzk_kc.astype(float) / self.kd.N_c
self.pwd = PWDescriptor(ecut, self.gd, self.bzk_kc)
n = 0
for k_c, Gpk2_G in zip(self.bzk_kc[:], self.pwd.G2_qG):
if (k_c > -0.5).all() and (k_c <= 0.5).all(): #XXX???
if k_c.any():
self.gamma -= np.dot(np.exp(-self.alpha * Gpk2_G),
Gpk2_G**-1)
else:
self.gamma -= np.dot(np.exp(-self.alpha * Gpk2_G[1:]),
Gpk2_G[1:]**-1)
n += 1
assert n == self.kd.N_c.prod()
self.ghat = LFC(self.gd,
[setup.ghat_l for setup in density.setups],
dtype=complex
)
self.ghat.set_k_points(self.bzk_kc)
self.fullkd = KPointDescriptor(self.kd.bzk_kc, nspins=1)
class S:
id_a = []
def set_symmetry(self, s): pass
self.fullkd.set_symmetry(Atoms(pbc=True), S(), False)
self.fullkd.set_communicator(world)
self.pt = LFC(self.gd, [setup.pt_j for setup in density.setups],
dtype=complex)
self.pt.set_k_points(self.fullkd.ibzk_kc)
self.interpolator = density.interpolator
def set_positions(self, spos_ac):
self.ghat.set_positions(spos_ac)
self.pt.set_positions(spos_ac)
def calculate(self, gd, n_sg, v_sg=None, e_g=None):
# Normal XC contribution:
exc = self.xc.calculate(gd, n_sg, v_sg, e_g)
# Add EXX contribution:
return exc + self.exx
def calculate_exx(self):
"""Non-selfconsistent calculation."""
kd = self.kd
K = self.fullkd.nibzkpts
assert self.nspins == 1
Q = K // world.size
assert Q * world.size == K
parallel = (world.size > self.nspins)
self.exx = 0.0
self.exx_skn = np.zeros((self.nspins, K, self.bd.nbands))
kpt_u = []
for k in range(world.rank * Q, (world.rank + 1) * Q):
k_c = self.fullkd.ibzk_kc[k]
for k1, k1_c in enumerate(kd.bzk_kc):
if abs(k1_c - k_c).max() < 1e-10:
break
# Index of symmetry related point in the irreducible BZ
ik = kd.kibz_k[k1]
kpt = self.kpt_u[ik]
# KPoint from ground-state calculation
phase_cd = np.exp(2j * pi * self.gd.sdisp_cd * k_c[:, np.newaxis])
kpt2 = KPoint0(kpt.weight, kpt.s, k, None, phase_cd)
kpt2.psit_nG = np.empty_like(kpt.psit_nG)
kpt2.f_n = kpt.f_n / kpt.weight / K * 2
for n, psit_G in enumerate(kpt2.psit_nG):
psit_G[:] = kd.transform_wave_function(kpt.psit_nG[n], k1)
kpt2.P_ani = self.pt.dict(len(kpt.psit_nG))
self.pt.integrate(kpt2.psit_nG, kpt2.P_ani, k)
kpt_u.append(kpt2)
for s in range(self.nspins):
kpt1_q = [KPoint(self.fullkd, kpt) for kpt in kpt_u if kpt.s == s]
kpt2_q = kpt1_q[:]
if len(kpt1_q) == 0:
# No s-spins on this CPU:
continue
# Send rank:
srank = self.fullkd.get_rank_and_index(s, (kpt1_q[0].k - 1) % K)[0]
# Receive rank:
rrank = self.fullkd.get_rank_and_index(s, (kpt1_q[-1].k + 1) % K)[0]
# Shift k-points K // 2 times:
for i in range(K // 2 + 1):
if i < K // 2:
if parallel:
kpt = kpt2_q[-1].next()
kpt.start_receiving(rrank)
kpt2_q[0].start_sending(srank)
else:
kpt = kpt2_q[0]
for kpt1, kpt2 in zip(kpt1_q, kpt2_q):
if 2 * i == K:
self.apply(kpt1, kpt2, invert=(kpt1.k > kpt2.k))
else:
self.apply(kpt1, kpt2)
self.apply(kpt1, kpt2, invert=True)
if i < K // 2:
if parallel:
kpt.wait()
kpt2_q[0].wait()
kpt2_q.pop(0)
kpt2_q.append(kpt)
self.exx = world.sum(self.exx)
world.sum(self.exx_skn)
self.exx += self.calculate_paw_correction()
def apply(self, kpt1, kpt2, invert=False):
#print world.rank,kpt1.k,kpt2.k,invert
k1_c = self.fullkd.ibzk_kc[kpt1.k]
k2_c = self.fullkd.ibzk_kc[kpt2.k]
if invert:
k2_c = -k2_c
k12_c = k1_c - k2_c
N_c = self.gd.N_c
eikr_R = np.exp(2j * pi * np.dot(np.indices(N_c).T, k12_c / N_c).T)
for q, k_c in enumerate(self.bzk_kc):
if abs(k_c + k12_c).max() < 1e-9:
q0 = q
break
for q, k_c in enumerate(self.bzk_kc):
if abs(k_c - k12_c).max() < 1e-9:
q00 = q
break
Gpk2_G = self.pwd.G2_qG[q0]
if Gpk2_G[0] == 0:
Gpk2_G = Gpk2_G.copy()
Gpk2_G[0] = 1.0 / self.gamma
N = N_c.prod()
vol = self.gd.dv * N
nspins = self.nspins
same = (kpt1.k == kpt2.k)
for n1, psit1_R in enumerate(kpt1.psit_nG):
f1 = kpt1.f_n[n1]
for n2, psit2_R in enumerate(kpt2.psit_nG):
if same and n2 > n1:
continue
f2 = kpt2.f_n[n2]
nt_R = self.calculate_pair_density(n1, n2, kpt1, kpt2, q0,
invert)
nt_G = self.pwd.fft(nt_R * eikr_R) / N
vt_G = nt_G.copy()
vt_G *= -pi * vol / Gpk2_G
e = np.vdot(nt_G, vt_G).real * nspins * self.hybrid
if same and n1 == n2:
e /= 2
self.exx += e * f1 * f2
self.ekin -= 2 * e * f1 * f2
self.exx_skn[kpt1.s, kpt1.k, n1] += f2 * e
self.exx_skn[kpt2.s, kpt2.k, n2] += f1 * e
calculate_potential = not True
if calculate_potential:
vt_R = self.pwd.ifft(vt_G).conj() * eikr_R * N / vol
if kpt1 is kpt2 and not invert and n1 == n2:
kpt1.vt_nG[n1] = 0.5 * f1 * vt_R
if invert:
kpt1.Htpsit_nG[n1] += \
f2 * nspins * psit2_R.conj() * vt_R
else:
kpt1.Htpsit_nG[n1] += f2 * nspins * psit2_R * vt_R
if kpt1 is not kpt2:
if invert:
kpt2.Htpsit_nG[n2] += (f1 * nspins *
psit1_R.conj() * vt_R)
else:
kpt2.Htpsit_nG[n2] += (f1 * nspins *
psit1_R * vt_R.conj())
def calculate_paw_correction(self):
exx = 0
deg = 2 // self.nspins # spin degeneracy
for a, D_sp in self.density.D_asp.items():
setup = self.setups[a]
for D_p in D_sp:
D_ii = unpack2(D_p)
ni = len(D_ii)
for i1 in range(ni):
for i2 in range(ni):
A = 0.0
for i3 in range(ni):
p13 = packed_index(i1, i3, ni)
for i4 in range(ni):
p24 = packed_index(i2, i4, ni)
A += setup.M_pp[p13, p24] * D_ii[i3, i4]
p12 = packed_index(i1, i2, ni)
exx -= self.hybrid / deg * D_ii[i1, i2] * A
if setup.X_p is not None:
exx -= self.hybrid * np.dot(D_p, setup.X_p)
exx += self.hybrid * setup.ExxC
return exx
def calculate_pair_density(self, n1, n2, kpt1, kpt2, q, invert):
if invert:
nt_G = kpt1.psit_nG[n1].conj() * kpt2.psit_nG[n2].conj()
else:
nt_G = kpt1.psit_nG[n1].conj() * kpt2.psit_nG[n2]
Q_aL = {}
for a, P1_ni in kpt1.P_ani.items():
P1_i = P1_ni[n1]
P2_i = kpt2.P_ani[a][n2]
if invert:
D_ii = np.outer(P1_i.conj(), P2_i.conj())
else:
D_ii = np.outer(P1_i.conj(), P2_i)
D_p = pack(D_ii)
Q_aL[a] = np.dot(D_p, self.setups[a].Delta_pL)
self.ghat.add(nt_G, Q_aL, q)
return nt_G
lfc1 = LFC(gd, [[s]], kd, dtype=complex)
lfc2 = PWLFC([[s]], pd)
c_axi = {0: np.zeros((1, 2 * l + 1), complex)}
c_axi[0][0, 0] = 1.9 - 4.5j
c_axiv = {0: np.zeros((1, 2 * l + 1, 3), complex)}
b1 = gd.zeros(1, dtype=complex)
b2 = pd.zeros(1, dtype=complex)
for lfc, b in [(lfc1, b1), (lfc2, b2)]:
lfc.set_positions(spos_ac)
lfc.add(b, c_axi, 0)
b2 = pd.ifft(b2[0]) * eikr
equal(abs(b2 - b1[0]).max(), 0, 0.001)
b1 = eikr[None]
b2 = pd.fft(b1[0] * 0 + 1).reshape((1, -1))
results = []
results2 = []
for lfc, b in [(lfc1, b1), (lfc2, b2)]:
lfc.integrate(b, c_axi, 0)
results.append(c_axi[0][0].copy())
lfc.derivative(b, c_axiv, 0)
results2.append(c_axiv[0][0].copy())
equal(abs(np.ptp(results2, 0)).max(), 0, 1e-7)
equal(abs(np.ptp(results, 0)).max(), 0, 3e-8)
