def get_jk(mf, cell, dm, hermi=1, vhfopt=None, kpt=np.zeros(3), kpt_band=None):
dm = np.asarray(dm)
nao = dm.shape[-1]
coords = gen_grid.gen_uniform_grids(cell)
if kpt_band is None:
kpt1 = kpt2 = kpt
aoR_k1 = aoR_k2 = numint.eval_ao(cell, coords, kpt)
else:
kpt1 = kpt_band
kpt2 = kpt
aoR_k1 = numint.eval_ao(cell, coords, kpt1)
aoR_k2 = numint.eval_ao(cell, coords, kpt2)
vkR_k1k2 = get_vkR(mf, cell, aoR_k1, aoR_k2, kpt1, kpt2)
ngs, nao = aoR_k1.shape
def contract(dm):
vjR_k2 = get_vjR(cell, dm, aoR_k2)
vj = (cell.vol/ngs) * np.dot(aoR_k1.T.conj(), vjR_k2.reshape(-1,1)*aoR_k1)
#:vk = (cell.vol/ngs) * np.einsum('rs,Rp,Rqs,Rr->pq', dm, aoR_k1.conj(),
#: vkR_k1k2, aoR_k2)
aoR_dm_k2 = np.dot(aoR_k2, dm)
tmp_Rq = np.einsum('Rqs,Rs->Rq', vkR_k1k2, aoR_dm_k2)
vk = (cell.vol/ngs) * np.dot(aoR_k1.T.conj(), tmp_Rq)
return vj, vk
if dm.ndim == 2:
vj, vk = contract(dm)
else:
jk = [contract(x) for x in dm.reshape(-1,nao,nao)]
vj = lib.asarray([x[0] for x in jk])
vk = lib.asarray([x[1] for x in jk])
return vj.reshape(dm.shape), vk.reshape(dm.shape)
def get_j_kpts(mf, cell, dm_kpts, kpts, kpt_band=None):
coords = gen_grid.gen_uniform_grids(cell)
nkpts = len(kpts)
ngs = len(coords)
dm_kpts = np.asarray(dm_kpts)
nao = dm_kpts.shape[-1]
ni = numint._KNumInt(kpts)
aoR_kpts = ni.eval_ao(cell, coords, kpts)
if kpt_band is not None:
aoR_kband = numint.eval_ao(cell, coords, kpt_band)
dms = dm_kpts.reshape(-1,nkpts,nao,nao)
nset = dms.shape[0]
vjR = [get_vjR(cell, dms[i], aoR_kpts) for i in range(nset)]
if kpt_band is not None:
vj_kpts = [cell.vol/ngs * lib.dot(aoR_kband.T.conj()*vjR[i], aoR_kband)
for i in range(nset)]
if dm_kpts.ndim == 3: # One set of dm_kpts for KRHF
vj_kpts = vj_kpts[0]
return lib.asarray(vj_kpts)
else:
vj_kpts = []
for i in range(nset):
vj = [cell.vol/ngs * lib.dot(aoR_k.T.conj()*vjR[i], aoR_k)
for aoR_k in aoR_kpts]
vj_kpts.append(lib.asarray(vj))
return lib.asarray(vj_kpts).reshape(dm_kpts.shape)
def get_j(cell, dm, hermi=1, vhfopt=None, kpt=np.zeros(3), kpt_band=None):
dm = np.asarray(dm)
nao = dm.shape[-1]
coords = gen_grid.gen_uniform_grids(cell)
if kpt_band is None:
kpt1 = kpt2 = kpt
aoR_k1 = aoR_k2 = numint.eval_ao(cell, coords, kpt)
else:
kpt1 = kpt_band
kpt2 = kpt
aoR_k1 = numint.eval_ao(cell, coords, kpt1)
aoR_k2 = numint.eval_ao(cell, coords, kpt2)
ngs, nao = aoR_k1.shape
def contract(dm):
vjR_k2 = get_vjR(cell, dm, aoR_k2)
vj = (cell.vol/ngs) * np.dot(aoR_k1.T.conj(), vjR_k2.reshape(-1,1)*aoR_k1)
return vj
if dm.ndim == 2:
vj = contract(dm)
else:
vj = lib.asarray([contract(x) for x in dm.reshape(-1,nao,nao)])
return vj.reshape(dm.shape)
def get_jk_kpts(mf, cell, dm_kpts, kpts, kpt_band=None):
coords = gen_grid.gen_uniform_grids(cell)
nkpts = len(kpts)
ngs = len(coords)
dm_kpts = np.asarray(dm_kpts)
nao = dm_kpts.shape[-1]
dms = dm_kpts.reshape(-1,nkpts,nao,nao)
nset = dms.shape[0]
ni = numint._KNumInt(kpts)
aoR_kpts = ni.eval_ao(cell, coords, kpts)
if kpt_band is not None:
aoR_kband = numint.eval_ao(cell, coords, kpt_band)
# J
vjR = [get_vjR_kpts(cell, dms[i], aoR_kpts) for i in range(nset)]
if kpt_band is not None:
vj_kpts = [cell.vol/ngs * lib.dot(aoR_kband.T.conj()*vjR[i], aoR_kband)
for i in range(nset)]
else:
vj_kpts = []
for i in range(nset):
vj = [cell.vol/ngs * lib.dot(aoR_k.T.conj()*vjR[i], aoR_k)
for aoR_k in aoR_kpts]
vj_kpts.append(lib.asarray(vj))
vj_kpts = lib.asarray(vj_kpts)
vjR = None
# K
weight = 1./nkpts * (cell.vol/ngs)
vk_kpts = np.zeros_like(vj_kpts)
if kpt_band is not None:
for k2, kpt2 in enumerate(kpts):
aoR_dms = [lib.dot(aoR_kpts[k2], dms[i,k2]) for i in range(nset)]
vkR_k1k2 = get_vkR(mf, cell, aoR_kband, aoR_kpts[k2],
kpt_band, kpt2)
#:vk_kpts = 1./nkpts * (cell.vol/ngs) * np.einsum('rs,Rp,Rqs,Rr->pq',
#: dm_kpts[k2], aoR_kband.conj(),
#: vkR_k1k2, aoR_kpts[k2])
for i in range(nset):
tmp_Rq = np.einsum('Rqs,Rs->Rq', vkR_k1k2, aoR_dms[i])
vk_kpts[i] += weight * lib.dot(aoR_kband.T.conj(), tmp_Rq)
vkR_k1k2 = None
if dm_kpts.ndim == 3:
vj_kpts = vj_kpts[0]
vk_kpts = vk_kpts[0]
return lib.asarray(vj_kpts), lib.asarray(vk_kpts)
else:
for k2, kpt2 in enumerate(kpts):
aoR_dms = [lib.dot(aoR_kpts[k2], dms[i,k2]) for i in range(nset)]
for k1, kpt1 in enumerate(kpts):
vkR_k1k2 = get_vkR(mf, cell, aoR_kpts[k1], aoR_kpts[k2],
kpt1, kpt2)
for i in range(nset):
tmp_Rq = np.einsum('Rqs,Rs->Rq', vkR_k1k2, aoR_dms[i])
vk_kpts[i,k1] += weight * lib.dot(aoR_kpts[k1].T.conj(), tmp_Rq)
vkR_k1k2 = None
return vj_kpts.reshape(dm_kpts.shape), vk_kpts.reshape(dm_kpts.shape)
def aoR_loop(self, cell, gs=None, kpts=None, kpt_band=None):
if kpts is None: kpts = self.kpts
kpts = numpy.asarray(kpts)
if gs is None:
gs = self.gs
else:
self.gs = gs
ngrids = numpy.prod(numpy.asarray(gs)*2+1)
if (self._numint.cell is None or id(cell) != id(self._numint.cell) or
self._numint._deriv != 0 or
self._numint._kpts.shape != kpts.shape or
abs(self._numint._kpts - kpts).sum() > 1e-9 or
self._numint._coords.shape[0] != ngrids or
(gs is not None and numpy.any(gs != self.gs))):
nkpts = len(kpts)
coords = gen_grid.gen_uniform_grids(cell, gs)
nao = cell.nao_nr()
blksize = int(self.max_memory*1e6/(nkpts*nao*16*numint.BLKSIZE))*numint.BLKSIZE
blksize = min(max(blksize, numint.BLKSIZE), ngrids)
try:
self._numint.cache_ao(cell, kpts, 0, coords, nao, blksize)
except IOError as e:
sys.stderr.write('HDF5 file cannot be opened twice in different mode.\n')
raise e
with h5py.File(self._numint._ao.name, 'r') as f:
if kpt_band is None:
for k in range(len(kpts)):
aoR = f['ao/%d'%k].value
yield k, aoR
else:
kpt_band = numpy.reshape(kpt_band, 3)
where = numpy.argmin(pyscf.lib.norm(kpts-kpt_band,axis=1))
if abs(kpts[where]-kpt_band).sum() > 1e-9:
where = None
coords = gen_grid.gen_uniform_grids(cell, gs)
yield 0, numint.eval_ao(cell, coords, kpt_band, deriv=deriv)
else:
yield where, f['ao/%d'%where].value
def get_mo_pairs_G_old(cell, mo_coeffs, kpts=None, q=None):
'''Calculate forward (G|ij) and "inverse" (ij|G) FFT of all MO pairs.
TODO: - Implement simplifications for real orbitals.
Args:
mo_coeff: length-2 list of (nao,nmo) ndarrays
The two sets of MO coefficients to use in calculating the
product |ij).
Returns:
mo_pairs_G, mo_pairs_invG : (ngs, nmoi*nmoj) ndarray
The FFTs of the real-space MO pairs.
'''
coords = gen_uniform_grids(cell)
if kpts is None:
q = np.zeros(3)
aoR = eval_ao(cell, coords)
ngs = aoR.shape[0]
if np.array_equal(mo_coeffs[0], mo_coeffs[1]):
nmoi = nmoj = mo_coeffs[0].shape[1]
moiR = mojR = einsum('ri,ia->ra', aoR, mo_coeffs[0])
else:
nmoi = mo_coeffs[0].shape[1]
nmoj = mo_coeffs[1].shape[1]
moiR = einsum('ri,ia->ra', aoR, mo_coeffs[0])
mojR = einsum('ri,ia->ra', aoR, mo_coeffs[1])
else:
if q is None:
q = kpts[1]-kpts[0]
aoR_ki = eval_ao(cell, coords, kpt=kpts[0])
aoR_kj = eval_ao(cell, coords, kpt=kpts[1])
ngs = aoR_ki.shape[0]
nmoi = mo_coeffs[0].shape[1]
nmoj = mo_coeffs[1].shape[1]
moiR = einsum('ri,ia->ra', aoR_ki, mo_coeffs[0])
mojR = einsum('ri,ia->ra', aoR_kj, mo_coeffs[1])
mo_pairs_R = np.einsum('ri,rj->rij', np.conj(moiR), mojR)
mo_pairs_G = np.zeros([ngs,nmoi*nmoj], np.complex128)
mo_pairs_invG = np.zeros([ngs,nmoi*nmoj], np.complex128)
fac = np.exp(-1j*np.dot(coords, q))
for i in xrange(nmoi):
for j in xrange(nmoj):
mo_pairs_G[:,i*nmoj+j] = tools.fftk(mo_pairs_R[:,i,j], cell.gs, fac)
mo_pairs_invG[:,i*nmoj+j] = np.conj(tools.fftk(np.conj(mo_pairs_R[:,i,j]), cell.gs,
fac.conj()))
return mo_pairs_G, mo_pairs_invG
def get_mo_pairs_G(cell, mo_coeffs, kpts=None, q=None):
'''Calculate forward (G|ij) FFT of all MO pairs.
TODO: - Implement simplifications for real orbitals.
Args:
mo_coeff: length-2 list of (nao,nmo) ndarrays
The two sets of MO coefficients to use in calculating the
product |ij).
Returns:
mo_pairs_G : (ngs, nmoi*nmoj) ndarray
The FFT of the real-space MO pairs.
'''
coords = gen_grid.gen_uniform_grids(cell)
if kpts is None:
q = np.zeros(3)
aoR = numint.eval_ao(cell, coords)
ngs = aoR.shape[0]
if np.array_equal(mo_coeffs[0], mo_coeffs[1]):
nmoi = nmoj = mo_coeffs[0].shape[1]
moiR = mojR = einsum('ri,ia->ra', aoR, mo_coeffs[0])
else:
nmoi = mo_coeffs[0].shape[1]
nmoj = mo_coeffs[1].shape[1]
moiR = einsum('ri,ia->ra', aoR, mo_coeffs[0])
mojR = einsum('ri,ia->ra', aoR, mo_coeffs[1])
else:
if q is None:
q = kpts[1] - kpts[0]
aoR_ki = numint.eval_ao(cell, coords, kpt=kpts[0])
aoR_kj = numint.eval_ao(cell, coords, kpt=kpts[1])
ngs = aoR_ki.shape[0]
nmoi = mo_coeffs[0].shape[1]
nmoj = mo_coeffs[1].shape[1]
moiR = einsum('ri,ia->ra', aoR_ki, mo_coeffs[0])
mojR = einsum('ri,ia->ra', aoR_kj, mo_coeffs[1])
#mo_pairs_R = einsum('ri,rj->rij', np.conj(moiR), mojR)
mo_pairs_G = np.zeros([ngs, nmoi * nmoj], np.complex128)
fac = np.exp(-1j * np.dot(coords, q))
for i in xrange(nmoi):
for j in xrange(nmoj):
mo_pairs_R_ij = np.conj(moiR[:, i]) * mojR[:, j]
mo_pairs_G[:, i * nmoj + j] = tools.fftk(mo_pairs_R_ij, cell.gs,
fac)
return mo_pairs_G
def get_pp_nl(cell, kpt=np.zeros(3)):
coords = gen_grid.gen_uniform_grids(cell)
aoR = numint.eval_ao(cell, coords, kpt)
nao = cell.nao_nr()
SI = cell.get_SI()
aokG = tools.fftk(np.asarray(aoR.T, order='C'),
cell.gs, np.exp(-1j*np.dot(coords, kpt))).T
ngs = len(aokG)
fakemol = pyscf.gto.Mole()
fakemol._atm = np.zeros((1,pyscf.gto.ATM_SLOTS), dtype=np.int32)
fakemol._bas = np.zeros((1,pyscf.gto.BAS_SLOTS), dtype=np.int32)
ptr = pyscf.gto.PTR_ENV_START
fakemol._env = np.zeros(ptr+10)
fakemol._bas[0,pyscf.gto.NPRIM_OF ] = 1
fakemol._bas[0,pyscf.gto.NCTR_OF ] = 1
fakemol._bas[0,pyscf.gto.PTR_EXP ] = ptr+3
fakemol._bas[0,pyscf.gto.PTR_COEFF] = ptr+4
Gv = np.asarray(cell.Gv+kpt)
G_rad = lib.norm(Gv, axis=1)
vppnl = np.zeros((nao,nao), dtype=np.complex128)
for ia in range(cell.natm):
symb = cell.atom_symbol(ia)
if symb not in cell._pseudo:
continue
pp = cell._pseudo[symb]
for l, proj in enumerate(pp[5:]):
rl, nl, hl = proj
if nl > 0:
hl = np.asarray(hl)
fakemol._bas[0,pyscf.gto.ANG_OF] = l
fakemol._env[ptr+3] = .5*rl**2
fakemol._env[ptr+4] = rl**(l+1.5)*np.pi**1.25
pYlm_part = pyscf.dft.numint.eval_ao(fakemol, Gv, deriv=0)
pYlm = np.empty((nl,l*2+1,ngs))
for k in range(nl):
qkl = pseudo.pp._qli(G_rad*rl, l, k)
pYlm[k] = pYlm_part.T * qkl
# pYlm is real
SPG_lmi = np.einsum('g,nmg->nmg', SI[ia].conj(), pYlm)
SPG_lm_aoG = np.einsum('nmg,gp->nmp', SPG_lmi, aokG)
tmp = np.einsum('ij,jmp->imp', hl, SPG_lm_aoG)
vppnl += np.einsum('imp,imq->pq', SPG_lm_aoG.conj(), tmp)
vppnl *= (1./ngs**2)
if aoR.dtype == np.double:
return vppnl.real
else:
return vppnl
def get_pp_nl(cell, kpt=np.zeros(3)):
coords = gen_grid.gen_uniform_grids(cell)
aoR = numint.eval_ao(cell, coords, kpt)
nao = cell.nao_nr()
SI = cell.get_SI()
aokG = tools.fftk(np.asarray(aoR.T, order='C'), cell.gs,
np.exp(-1j * np.dot(coords, kpt))).T
ngs = len(aokG)
fakemol = pyscf.gto.Mole()
fakemol._atm = np.zeros((1, pyscf.gto.ATM_SLOTS), dtype=np.int32)
fakemol._bas = np.zeros((1, pyscf.gto.BAS_SLOTS), dtype=np.int32)
ptr = pyscf.gto.PTR_ENV_START
fakemol._env = np.zeros(ptr + 10)
fakemol._bas[0, pyscf.gto.NPRIM_OF] = 1
fakemol._bas[0, pyscf.gto.NCTR_OF] = 1
fakemol._bas[0, pyscf.gto.PTR_EXP] = ptr + 3
fakemol._bas[0, pyscf.gto.PTR_COEFF] = ptr + 4
Gv = np.asarray(cell.Gv + kpt)
G_rad = lib.norm(Gv, axis=1)
vppnl = np.zeros((nao, nao), dtype=np.complex128)
for ia in range(cell.natm):
symb = cell.atom_symbol(ia)
if symb not in cell._pseudo:
continue
pp = cell._pseudo[symb]
for l, proj in enumerate(pp[5:]):
rl, nl, hl = proj
if nl > 0:
hl = np.asarray(hl)
fakemol._bas[0, pyscf.gto.ANG_OF] = l
fakemol._env[ptr + 3] = .5 * rl**2
fakemol._env[ptr + 4] = rl**(l + 1.5) * np.pi**1.25
pYlm_part = pyscf.dft.numint.eval_ao(fakemol, Gv, deriv=0)
pYlm = np.empty((nl, l * 2 + 1, ngs))
for k in range(nl):
qkl = pseudo.pp._qli(G_rad * rl, l, k)
pYlm[k] = pYlm_part.T * qkl
# pYlm is real
SPG_lmi = np.einsum('g,nmg->nmg', SI[ia].conj(), pYlm)
SPG_lm_aoG = np.einsum('nmg,gp->nmp', SPG_lmi, aokG)
tmp = np.einsum('ij,jmp->imp', hl, SPG_lm_aoG)
vppnl += np.einsum('imp,imq->pq', SPG_lm_aoG.conj(), tmp)
vppnl *= (1. / ngs**2)
if aoR.dtype == np.double:
return vppnl.real
else:
return vppnl
def precompute_exx(cell, kpts):
from pyscf.pbc import gto as pbcgto
from pyscf.pbc.dft import gen_grid
log = lib.logger.Logger(cell.stdout, cell.verbose)
log.debug("# Precomputing Wigner-Seitz EXX kernel")
Nk = get_monkhorst_pack_size(cell, kpts)
log.debug("# Nk = %s", Nk)
kcell = pbcgto.Cell()
kcell.atom = 'H 0. 0. 0.'
kcell.spin = 1
kcell.unit = 'B'
kcell.verbose = 0
kcell.a = cell.lattice_vectors() * Nk
Lc = 1.0/lib.norm(np.linalg.inv(kcell.a), axis=0)
log.debug("# Lc = %s", Lc)
Rin = Lc.min() / 2.0
log.debug("# Rin = %s", Rin)
# ASE:
alpha = 5./Rin # sqrt(-ln eps) / Rc, eps ~ 10^{-11}
log.info("WS alpha = %s", alpha)
kcell.gs = np.array([2*int(L*alpha*3.0) for L in Lc]) # ~ [60,60,60]
# QE:
#alpha = 3./Rin * np.sqrt(0.5)
#kcell.gs = (4*alpha*np.linalg.norm(kcell.a,axis=1)).astype(int)
log.debug("# kcell.gs FFT = %s", kcell.gs)
kcell.build(False,False)
rs = gen_grid.gen_uniform_grids(kcell)
kngs = len(rs)
log.debug("# kcell kngs = %d", kngs)
corners = np.dot(np.indices((2,2,2)).reshape((3,8)).T, kcell.a)
#vR = np.empty(kngs)
#for i, rv in enumerate(rs):
# # Minimum image convention to corners of kcell parallelepiped
# r = lib.norm(rv-corners, axis=1).min()
# if np.isclose(r, 0.):
# vR[i] = 2*alpha / np.sqrt(np.pi)
# else:
# vR[i] = scipy.special.erf(alpha*r) / r
r = np.min([lib.norm(rs-c, axis=1) for c in corners], axis=0)
vR = scipy.special.erf(alpha*r) / (r+1e-200)
vR[r<1e-9] = 2*alpha / np.sqrt(np.pi)
vG = (kcell.vol/kngs) * fft(vR, kcell.gs)
ws_exx = {'alpha': alpha,
'kcell': kcell,
'q' : kcell.Gv,
'vq' : vG}
log.debug("# Finished precomputing")
return ws_exx
def precompute_exx(cell, kpts):
from pyscf.pbc import gto as pbcgto
from pyscf.pbc.dft import gen_grid
log = lib.logger.Logger(cell.stdout, cell.verbose)
log.debug("# Precomputing Wigner-Seitz EXX kernel")
Nk = get_monkhorst_pack_size(cell, kpts)
log.debug("# Nk = %s", Nk)
kcell = pbcgto.Cell()
kcell.atom = 'H 0. 0. 0.'
kcell.spin = 1
kcell.unit = 'B'
kcell.verbose = 0
kcell.a = cell.lattice_vectors() * Nk
Lc = 1.0/lib.norm(np.linalg.inv(kcell.a), axis=0)
log.debug("# Lc = %s", Lc)
Rin = Lc.min() / 2.0
log.debug("# Rin = %s", Rin)
# ASE:
alpha = 5./Rin # sqrt(-ln eps) / Rc, eps ~ 10^{-11}
log.info("WS alpha = %s", alpha)
kcell.mesh = np.array([4*int(L*alpha*3.0) for L in Lc]) # ~ [60,60,60]
# QE:
#alpha = 3./Rin * np.sqrt(0.5)
#kcell.mesh = (4*alpha*np.linalg.norm(kcell.a,axis=1)).astype(int)
log.debug("# kcell.mesh FFT = %s", kcell.mesh)
kcell.build(False,False)
rs = gen_grid.gen_uniform_grids(kcell)
kngs = len(rs)
log.debug("# kcell kngs = %d", kngs)
corners = np.dot(np.indices((2,2,2)).reshape((3,8)).T, kcell.a)
#vR = np.empty(kngs)
#for i, rv in enumerate(rs):
# # Minimum image convention to corners of kcell parallelepiped
# r = lib.norm(rv-corners, axis=1).min()
# if np.isclose(r, 0.):
# vR[i] = 2*alpha / np.sqrt(np.pi)
# else:
# vR[i] = scipy.special.erf(alpha*r) / r
r = np.min([lib.norm(rs-c, axis=1) for c in corners], axis=0)
vR = scipy.special.erf(alpha*r) / (r+1e-200)
vR[r<1e-9] = 2*alpha / np.sqrt(np.pi)
vG = (kcell.vol/kngs) * fft(vR, kcell.mesh)
ws_exx = {'alpha': alpha,
'kcell': kcell,
'q' : kcell.Gv,
'vq' : vG}
log.debug("# Finished precomputing")
return ws_exx
def get_nuc(cell, kpt=np.zeros(3)):
'''Get the bare periodic nuc-el AO matrix, with G=0 removed.
See Martin (12.16)-(12.21).
'''
coords = gen_grid.gen_uniform_grids(cell)
aoR = numint.eval_ao(cell, coords, kpt)
chargs = cell.atom_charges()
SI = cell.get_SI()
coulG = tools.get_coulG(cell)
vneG = -np.dot(chargs, SI) * coulG
vneR = tools.ifft(vneG, cell.gs).real
vne = np.dot(aoR.T.conj(), vneR.reshape(-1, 1) * aoR)
return vne
def get_nuc(cell, kpt=np.zeros(3)):
'''Get the bare periodic nuc-el AO matrix, with G=0 removed.
See Martin (12.16)-(12.21).
'''
coords = gen_grid.gen_uniform_grids(cell)
aoR = numint.eval_ao(cell, coords, kpt)
chargs = cell.atom_charges()
SI = cell.get_SI()
coulG = tools.get_coulG(cell)
vneG = -np.dot(chargs,SI) * coulG
vneR = tools.ifft(vneG, cell.gs).real
vne = np.dot(aoR.T.conj(), vneR.reshape(-1,1)*aoR)
return vne
def get_pp(cell, kpt=np.zeros(3)):
'''Get the periodic pseudotential nuc-el AO matrix, with G=0 removed.
'''
coords = gen_grid.gen_uniform_grids(cell)
aoR = numint.eval_ao(cell, coords, kpt)
nao = cell.nao_nr()
SI = cell.get_SI()
vlocG = pseudo.get_vlocG(cell)
vlocG[:, 0] = 0
vpplocG = -np.sum(SI * vlocG, axis=0)
# vpploc evaluated in real-space
vpplocR = tools.ifft(vpplocG, cell.mesh)
vpploc = np.dot(aoR.T.conj(), vpplocR.reshape(-1, 1) * aoR)
# vppnonloc evaluated in reciprocal space
aokG = np.empty(aoR.shape, np.complex128)
expmikr = np.exp(-1j * np.dot(coords, kpt))
for i in range(nao):
aokG[:, i] = tools.fftk(aoR[:, i], cell.mesh, expmikr)
ngrids = len(aokG)
vppnl = np.zeros((nao, nao), dtype=np.complex128)
hs, projGs = pseudo.get_projG(cell, kpt)
for ia, [h_ia, projG_ia] in enumerate(zip(hs, projGs)):
for l, h in enumerate(h_ia):
nl = h.shape[0]
for m in range(-l, l + 1):
SPG_lm_aoG = np.zeros((nl, nao), dtype=np.complex128)
for i in range(nl):
SPG_lmi = SI[ia, :] * projG_ia[l][m][i]
SPG_lm_aoG[i, :] = np.einsum('g,gp->p', SPG_lmi.conj(),
aokG)
for i in range(nl):
for j in range(nl):
# Note: There is no (-1)^l here.
vppnl += h[i, j] * np.einsum('p,q->pq',
SPG_lm_aoG[i, :].conj(),
SPG_lm_aoG[j, :])
vppnl *= (1. / ngrids**2)
ovlp = cell.pbc_intor('int1e_ovlp_sph', hermi=1, kpts=kpt)
vpploc += 1. / cell.vol * np.sum(pseudo.get_alphas(cell)) * ovlp
return vpploc + vppnl
def get_pp(cell, kpt=np.zeros(3)):
'''Get the periodic pseudotential nuc-el AO matrix, with G=0 removed.
'''
coords = gen_grid.gen_uniform_grids(cell)
aoR = numint.eval_ao(cell, coords, kpt)
nao = cell.nao_nr()
SI = cell.get_SI()
vlocG = pseudo.get_vlocG(cell)
vlocG[:,0] = 0
vpplocG = -np.sum(SI * vlocG, axis=0)
# vpploc evaluated in real-space
vpplocR = tools.ifft(vpplocG, cell.mesh)
vpploc = np.dot(aoR.T.conj(), vpplocR.reshape(-1,1)*aoR)
# vppnonloc evaluated in reciprocal space
aokG = np.empty(aoR.shape, np.complex128)
expmikr = np.exp(-1j*np.dot(coords,kpt))
for i in range(nao):
aokG[:,i] = tools.fftk(aoR[:,i], cell.mesh, expmikr)
ngrids = len(aokG)
vppnl = np.zeros((nao,nao), dtype=np.complex128)
hs, projGs = pseudo.get_projG(cell, kpt)
for ia, [h_ia,projG_ia] in enumerate(zip(hs,projGs)):
for l, h in enumerate(h_ia):
nl = h.shape[0]
for m in range(-l,l+1):
SPG_lm_aoG = np.zeros((nl,nao), dtype=np.complex128)
for i in range(nl):
SPG_lmi = SI[ia,:] * projG_ia[l][m][i]
SPG_lm_aoG[i,:] = np.einsum('g,gp->p', SPG_lmi.conj(), aokG)
for i in range(nl):
for j in range(nl):
# Note: There is no (-1)^l here.
vppnl += h[i,j]*np.einsum('p,q->pq',
SPG_lm_aoG[i,:].conj(),
SPG_lm_aoG[j,:])
vppnl *= (1./ngrids**2)
ovlp = cell.pbc_intor('int1e_ovlp_sph', hermi=1, kpts=kpt)
vpploc += 1./cell.vol * np.sum(pseudo.get_alphas(cell)) * ovlp
return vpploc + vppnl
def get_jk(mf,
cell,
dm,
hermi=1,
vhfopt=None,
kpt=np.zeros(3),
kpts_band=None):
dm = np.asarray(dm)
nao = dm.shape[-1]
coords = gen_grid.gen_uniform_grids(cell)
if kpts_band is None:
kpt1 = kpt2 = kpt
aoR_k1 = aoR_k2 = numint.eval_ao(cell, coords, kpt)
else:
kpt1 = kpts_band
kpt2 = kpt
aoR_k1 = numint.eval_ao(cell, coords, kpt1)
aoR_k2 = numint.eval_ao(cell, coords, kpt2)
vkR_k1k2 = get_vkR(mf, cell, aoR_k1, aoR_k2, kpt1, kpt2)
ngs, nao = aoR_k1.shape
def contract(dm):
vjR_k2 = get_vjR(cell, dm, aoR_k2)
vj = (cell.vol / ngs) * np.dot(aoR_k1.T.conj(),
vjR_k2.reshape(-1, 1) * aoR_k1)
#:vk = (cell.vol/ngs) * np.einsum('rs,Rp,Rqs,Rr->pq', dm, aoR_k1.conj(),
#: vkR_k1k2, aoR_k2)
aoR_dm_k2 = np.dot(aoR_k2, dm)
tmp_Rq = np.einsum('Rqs,Rs->Rq', vkR_k1k2, aoR_dm_k2)
vk = (cell.vol / ngs) * np.dot(aoR_k1.T.conj(), tmp_Rq)
return vj, vk
if dm.ndim == 2:
vj, vk = contract(dm)
else:
jk = [contract(x) for x in dm.reshape(-1, nao, nao)]
vj = lib.asarray([x[0] for x in jk])
vk = lib.asarray([x[1] for x in jk])
return vj.reshape(dm.shape), vk.reshape(dm.shape)
def get_vkR(mf, cell, aoR_k1, aoR_k2, kpt1, kpt2):
coords = gen_grid.gen_uniform_grids(cell)
ngs, nao = aoR_k1.shape
expmikr = np.exp(-1j*np.dot(kpt1-kpt2,coords.T))
coulG = tools.get_coulG(cell, kpt1-kpt2, exx=True, mf=mf)
def prod(ij):
i, j = divmod(ij, nao)
rhoR = aoR_k1[:,i] * aoR_k2[:,j].conj()
rhoG = tools.fftk(rhoR, cell.gs, expmikr)
vG = coulG*rhoG
vR = tools.ifftk(vG, cell.gs, expmikr.conj())
return vR
if aoR_k1.dtype == np.double and aoR_k2.dtype == np.double:
vR = numpy.asarray([prod(ij).real for ij in range(nao**2)])
else:
vR = numpy.asarray([prod(ij) for ij in range(nao**2)])
return vR.reshape(nao,nao,-1).transpose(2,0,1)
def get_vkR(mf, cell, aoR_k1, aoR_k2, kpt1, kpt2):
coords = gen_grid.gen_uniform_grids(cell)
ngs, nao = aoR_k1.shape
expmikr = np.exp(-1j * np.dot(kpt1 - kpt2, coords.T))
coulG = tools.get_coulG(cell, kpt1 - kpt2, exx=True, mf=mf)
def prod(ij):
i, j = divmod(ij, nao)
rhoR = aoR_k1[:, i] * aoR_k2[:, j].conj()
rhoG = tools.fftk(rhoR, cell.gs, expmikr)
vG = coulG * rhoG
vR = tools.ifftk(vG, cell.gs, expmikr.conj())
return vR
if aoR_k1.dtype == np.double and aoR_k2.dtype == np.double:
vR = numpy.asarray([prod(ij).real for ij in range(nao**2)])
else:
vR = numpy.asarray([prod(ij) for ij in range(nao**2)])
return vR.reshape(nao, nao, -1).transpose(2, 0, 1)
def get_vkR(mf, cell, aoR_k1, aoR_k2, kpt1, kpt2):
'''Get the real-space 2-index "exchange" potential V_{i,k1; j,k2}(r)
where {i,k1} = exp^{i k1 r) |i> , {j,k2} = exp^{-i k2 r) <j|
'''
coords = gen_grid.gen_uniform_grids(cell)
ngrids, nao = aoR_k1.shape
expmikr = np.exp(-1j*np.dot(kpt1-kpt2,coords.T))
coulG = tools.get_coulG(cell, kpt1-kpt2, exx=True, mf=mf)
def prod(ij):
i, j = divmod(ij, nao)
rhoR = aoR_k1[:,i] * aoR_k2[:,j].conj()
rhoG = tools.fftk(rhoR, cell.mesh, expmikr)
vG = coulG*rhoG
vR = tools.ifftk(vG, cell.mesh, expmikr.conj())
return vR
if aoR_k1.dtype == np.double and aoR_k2.dtype == np.double:
vR = numpy.asarray([prod(ij).real for ij in range(nao**2)])
else:
vR = numpy.asarray([prod(ij) for ij in range(nao**2)])
return vR.reshape(nao,nao,-1).transpose(2,0,1)
def get_pp_loc_part2(cell, kpt=np.zeros(3)):
coords = gen_grid.gen_uniform_grids(cell)
aoR = numint.eval_ao(cell, coords, kpt)
nao = cell.nao_nr()
SI = cell.get_SI()
G = lib.norm(cell.Gv, axis=1)
vlocG = np.zeros((cell.natm,len(G)))
for ia in range(cell.natm):
Zia = cell.atom_charge(ia)
symb = cell.atom_symbol(ia)
if symb not in cell._pseudo:
vlocG[ia] = 0
continue
pp = cell._pseudo[symb]
rloc, nexp, cexp = pp[1:3+1]
G_red = G*rloc
cfacs = np.array(
[1*G_red**0,
3 - G_red**2,
15 - 10*G_red**2 + G_red**4,
105 - 105*G_red**2 + 21*G_red**4 - G_red**6])
with np.errstate(divide='ignore'):
# Note the signs -- potential here is positive
vlocG[ia,:] = (# 4*np.pi * Zia * np.exp(-0.5*G_red**2)/G**2
- (2*np.pi)**(3/2.)*rloc**3*np.exp(-0.5*G_red**2)*(
np.dot(cexp, cfacs[:nexp])) )
vpplocG = -np.sum(SI * vlocG, axis=0)
vpplocR = tools.ifft(vpplocG, cell.gs).real
vpploc = np.dot(aoR.T.conj(), vpplocR.reshape(-1,1)*aoR)
if aoR.dtype == np.double:
return vpploc.real
else:
return vpploc
def get_ao_pairs_G(cell, kpt=np.zeros(3)):
'''Calculate forward (G|ij) and "inverse" (ij|G) FFT of all AO pairs.
Args:
cell : instance of :class:`Cell`
Returns:
ao_pairs_G, ao_pairs_invG : (ngrids, nao*(nao+1)/2) ndarray
The FFTs of the real-space AO pairs.
'''
coords = gen_uniform_grids(cell)
aoR = eval_ao(cell, coords, kpt) # shape = (coords, nao)
ngrids, nao = aoR.shape
gamma_point = abs(kpt).sum() < 1e-9
if gamma_point:
npair = nao * (nao + 1) // 2
ao_pairs_G = np.empty([ngrids, npair], np.complex128)
ij = 0
for i in range(nao):
for j in range(i + 1):
ao_ij_R = np.conj(aoR[:, i]) * aoR[:, j]
ao_pairs_G[:, ij] = tools.fft(ao_ij_R, cell.mesh)
#ao_pairs_invG[:,ij] = ngrids*tools.ifft(ao_ij_R, cell.mesh)
ij += 1
ao_pairs_invG = ao_pairs_G.conj()
else:
ao_pairs_G = np.zeros([ngrids, nao, nao], np.complex128)
for i in range(nao):
for j in range(nao):
ao_ij_R = np.conj(aoR[:, i]) * aoR[:, j]
ao_pairs_G[:, i, j] = tools.fft(ao_ij_R, cell.mesh)
ao_pairs_invG = ao_pairs_G.transpose(0, 2,
1).conj().reshape(-1, nao**2)
ao_pairs_G = ao_pairs_G.reshape(-1, nao**2)
return ao_pairs_G, ao_pairs_invG
def get_pp_loc_part2(cell, kpt=np.zeros(3)):
coords = gen_grid.gen_uniform_grids(cell)
aoR = numint.eval_ao(cell, coords, kpt)
nao = cell.nao_nr()
SI = cell.get_SI()
G = lib.norm(cell.Gv, axis=1)
vlocG = np.zeros((cell.natm,len(G)))
for ia in range(cell.natm):
Zia = cell.atom_charge(ia)
symb = cell.atom_symbol(ia)
if symb not in cell._pseudo:
vlocG[ia] = 0
continue
pp = cell._pseudo[symb]
rloc, nexp, cexp = pp[1:3+1]
G_red = G*rloc
cfacs = np.array(
[1*G_red**0,
3 - G_red**2,
15 - 10*G_red**2 + G_red**4,
105 - 105*G_red**2 + 21*G_red**4 - G_red**6])
with np.errstate(divide='ignore'):
# Note the signs -- potential here is positive
vlocG[ia,:] = (# 4*np.pi * Zia * np.exp(-0.5*G_red**2)/G**2
- (2*np.pi)**(3/2.)*rloc**3*np.exp(-0.5*G_red**2)*(
np.dot(cexp, cfacs[:nexp])) )
vpplocG = -np.sum(SI * vlocG, axis=0)
vpplocR = tools.ifft(vpplocG, cell.gs).real
vpploc = np.dot(aoR.T.conj(), vpplocR.reshape(-1,1)*aoR)
if aoR.dtype == np.double:
return vpploc.real
else:
return vpploc
def get_ao_pairs_G(cell, kpt=np.zeros(3)):
'''Calculate forward (G|ij) and "inverse" (ij|G) FFT of all AO pairs.
Args:
cell : instance of :class:`Cell`
Returns:
ao_pairs_G, ao_pairs_invG : (ngs, nao*(nao+1)/2) ndarray
The FFTs of the real-space AO pairs.
'''
coords = gen_uniform_grids(cell)
aoR = eval_ao(cell, coords, kpt) # shape = (coords, nao)
ngs, nao = aoR.shape
gamma_point = abs(kpt).sum() < 1e-9
if gamma_point:
npair = nao*(nao+1)//2
ao_pairs_G = np.empty([ngs, npair], np.complex128)
ij = 0
for i in range(nao):
for j in range(i+1):
ao_ij_R = np.conj(aoR[:,i]) * aoR[:,j]
ao_pairs_G[:,ij] = tools.fft(ao_ij_R, cell.gs)
#ao_pairs_invG[:,ij] = ngs*tools.ifft(ao_ij_R, cell.gs)
ij += 1
ao_pairs_invG = ao_pairs_G.conj()
else:
ao_pairs_G = np.zeros([ngs, nao,nao], np.complex128)
for i in range(nao):
for j in range(nao):
ao_ij_R = np.conj(aoR[:,i]) * aoR[:,j]
ao_pairs_G[:,i,j] = tools.fft(ao_ij_R, cell.gs)
ao_pairs_invG = ao_pairs_G.transpose(0,2,1).conj().reshape(-1,nao**2)
ao_pairs_G = ao_pairs_G.reshape(-1,nao**2)
return ao_pairs_G, ao_pairs_invG
mygs = 4 # grid density
alat = 4.0
orbs_to_show = set((('2s', ''), ('2p', 'x'), ('3d', 'xy'), ('3d', 'z^2')))
assert len(orbs_to_show) == 4 # !!!! hard-code 4 plots
gs = np.array([mygs] * 3)
basis = bfd_basis()
cell = gto.M(a=alat * np.eye(3),
atom='C 2.0 2.0 2.0',
verbose=3,
gs=gs,
pseudo={'C': 'bfd'},
basis=basis)
coords = gen_grid.gen_uniform_grids(cell)
aoR = numint.eval_ao(cell, coords)
grid_shape = 2 * gs + 1
fig = plt.figure()
iplot = 0
bas_labels = cell.ao_labels(fmt=False)
for ibas in range(len(bas_labels)):
iatom, alabel, b_l, b_m = bas_labels[ibas]
if (b_l, b_m) not in orbs_to_show:
continue
iplot += 1
ax = fig.add_subplot(2, 2, iplot, projection='3d')
def test_components(pseudo=None):
# The molecular calculation
mol = gto.Mole()
mol.unit = 'B'
L = 60
mol.atom.extend([
['He', (L / 2., L / 2., L / 2.)],
])
mol.basis = 'sto-3g'
mol.build()
m = rks.RKS(mol)
m.xc = 'LDA,VWN_RPA'
print(m.scf()) # -2.90705411168
dm = m.make_rdm1()
# The periodic calculation
cell = pbcgto.Cell()
cell.unit = 'B'
cell.a = np.diag([L, L, L])
cell.gs = np.array([80, 80, 80])
cell.atom = mol.atom
cell.basis = mol.basis
cell.pseudo = pseudo
cell.build()
# These should match reasonably well (roughly with accuracy of normalization)
print "Kinetic energy"
tao = pbchf.get_t(cell)
tao2 = mol.intor_symmetric('cint1e_kin_sph')
print np.dot(np.ravel(tao), np.ravel(dm)) # 2.82793077196
print np.dot(np.ravel(tao2), np.ravel(dm)) # 2.82352636524
print "Overlap"
sao = pbchf.get_ovlp(cell)
print np.dot(np.ravel(sao), np.ravel(dm)) # 1.99981725342
print np.dot(np.ravel(m.get_ovlp()), np.ravel(dm)) # 2.0
# The next two entries should *not* match, since G=0 component is removed
print "Coulomb (G!=0)"
jao = pbchf.get_j(cell, dm)
print np.dot(np.ravel(dm), np.ravel(jao)) # 4.03425518427
print np.dot(np.ravel(dm), np.ravel(m.get_j(dm))) # 4.22285177049
# The next two entries should *not* match, since G=0 component is removed
print "Nuc-el (G!=0)"
if cell.pseudo:
vppao = pbchf.get_pp(cell)
print np.dot(np.ravel(dm), np.ravel(vppao))
else:
neao = pbchf.get_nuc(cell)
print np.dot(np.ravel(dm), np.ravel(neao)) # -6.50203360062
vne = mol.intor_symmetric('cint1e_nuc_sph')
print np.dot(np.ravel(dm), np.ravel(vne)) # -6.68702326551
print "Normalization"
coords = gen_uniform_grids(cell)
aoR = eval_ao(cell, coords)
rhoR = eval_rho(cell, aoR, dm)
print cell.vol / len(rhoR) * np.sum(rhoR) # 1.99981725342 (should be 2.0)
print "(Hartree + vne) * DM"
print np.dot(np.ravel(dm), np.ravel(m.get_j(dm))) + np.dot(
np.ravel(dm), np.ravel(vne))
if cell.pseudo:
print np.einsum("ij,ij", dm, vppao + jao + pbchf.get_jvloc_G0(cell))
else:
print np.einsum("ij,ij", dm, neao + jao)
ew_cut = (40, 40, 40)
ew_eta = 0.05
for ew_eta in [0.1, 0.5, 1.]:
ew = pbchf.ewald(cell, ew_eta, ew_cut)
print "Ewald (eta, energy)", ew_eta, ew # should be same for all eta
print "Ewald divergent terms summation", ew
# These two should now match if the box is reasonably big to
# remove images, and ngs is big.
print "Total coulomb (analytic) ",
print(.5 * np.dot(np.ravel(dm), np.ravel(m.get_j(dm))) +
np.dot(np.ravel(dm), np.ravel(vne))) # -4.57559738004
if not cell.pseudo:
print "Total coulomb (fft coul + ewald)",
print np.einsum("ij,ij", dm, neao + .5 * jao) + ew # -4.57948259115
# Exc
cell.ew_eta, cell.ew_cut = ew_eta, ew_cut
mf = pbcdft.RKS(cell)
mf.xc = 'LDA,VWN_RPA'
rks.get_veff(mf, cell, dm)
print "Exc", mf._exc # -1.05967570089
def precompute_exx(cell, kpts):
from pyscf.pbc import gto as pbcgto
from pyscf.pbc.dft import gen_grid
log = lib.logger.Logger(cell.stdout, cell.verbose)
log.debug("# Precomputing Wigner-Seitz EXX kernel")
Nk = get_monkhorst_pack_size(cell, kpts)
log.debug("# Nk = %s", Nk)
kcell = pbcgto.Cell()
kcell.atom = 'H 0. 0. 0.'
kcell.spin = 1
kcell.unit = 'B'
kcell.verbose = 0
kcell.a = cell.lattice_vectors() * Nk
Lc = 1.0 / lib.norm(np.linalg.inv(kcell.a), axis=0)
log.debug("# Lc = %s", Lc)
Rin = Lc.min() / 2.0
log.debug("# Rin = %s", Rin)
# ASE:
alpha = 5. / Rin # sqrt(-ln eps) / Rc, eps ~ 10^{-11}
log.info("WS alpha = %s", alpha)
kcell.mesh = np.array([4 * int(L * alpha * 3.0)
for L in Lc]) # ~ [120,120,120]
# QE:
#alpha = 3./Rin * np.sqrt(0.5)
#kcell.mesh = (4*alpha*np.linalg.norm(kcell.a,axis=1)).astype(int)
log.debug("# kcell.mesh FFT = %s", kcell.mesh)
rs = gen_grid.gen_uniform_grids(kcell)
kngs = len(rs)
log.debug("# kcell kngs = %d", kngs)
corners_coord = lib.cartesian_prod(([0, 1], [0, 1], [0, 1]))
corners = np.dot(corners_coord, kcell.a)
#vR = np.empty(kngs)
#for i, rv in enumerate(rs):
# # Minimum image convention to corners of kcell parallelepiped
# r = lib.norm(rv-corners, axis=1).min()
# if np.isclose(r, 0.):
# vR[i] = 2*alpha / np.sqrt(np.pi)
# else:
# vR[i] = scipy.special.erf(alpha*r) / r
r = np.min([lib.norm(rs - c, axis=1) for c in corners], axis=0)
vR = scipy.special.erf(alpha * r) / (r + 1e-200)
vR[r < 1e-9] = 2 * alpha / np.sqrt(np.pi)
vG = (kcell.vol / kngs) * fft(vR, kcell.mesh)
if abs(vG.imag).max() > 1e-6:
# vG should be real in regular lattice. If imaginary part is observed,
# this probably means a ws cell was built from a unconventional
# lattice. The SR potential erfc(alpha*r) for the charge in the center
# of ws cell decays to the region out of ws cell. The Ewald-sum based
# on the minimum image convention cannot be used to build the kernel
# Eq (12) of PRB 87, 165122
raise RuntimeError('Unconventional lattice was found')
ws_exx = {
'alpha': alpha,
'kcell': kcell,
'q': kcell.Gv,
'vq': vG.real.copy()
}
log.debug("# Finished precomputing")
return ws_exx
def get_pp(cell, kpt=np.zeros(3)):
'''Get the periodic pseudotential nuc-el AO matrix
'''
import pyscf.dft
from pyscf.pbc import tools
from pyscf.pbc.dft import gen_grid
from pyscf.pbc.dft import numint
coords = gen_grid.gen_uniform_grids(cell)
aoR = numint.eval_ao(cell, coords, kpt)
nao = cell.nao_nr()
SI = cell.get_SI()
vlocG = get_vlocG(cell)
vpplocG = -np.sum(SI * vlocG, axis=0)
vpplocG[0] = np.sum(get_alphas(cell)) # from get_jvloc_G0 function
# vpploc evaluated in real-space
vpplocR = tools.ifft(vpplocG, cell.gs).real
vpploc = np.dot(aoR.T.conj(), vpplocR.reshape(-1, 1) * aoR)
# vppnonloc evaluated in reciprocal space
aokG = tools.fftk(np.asarray(aoR.T, order='C'), cell.gs,
np.exp(-1j * np.dot(coords, kpt))).T
ngs = len(aokG)
fakemol = pyscf.gto.Mole()
fakemol._atm = np.zeros((1, pyscf.gto.ATM_SLOTS), dtype=np.int32)
fakemol._bas = np.zeros((1, pyscf.gto.BAS_SLOTS), dtype=np.int32)
ptr = pyscf.gto.PTR_ENV_START
fakemol._env = np.zeros(ptr + 10)
fakemol._bas[0, pyscf.gto.NPRIM_OF] = 1
fakemol._bas[0, pyscf.gto.NCTR_OF] = 1
fakemol._bas[0, pyscf.gto.PTR_EXP] = ptr + 3
fakemol._bas[0, pyscf.gto.PTR_COEFF] = ptr + 4
Gv = np.asarray(cell.Gv + kpt)
G_rad = lib.norm(Gv, axis=1)
vppnl = np.zeros((nao, nao), dtype=np.complex128)
for ia in range(cell.natm):
symb = cell.atom_symbol(ia)
if symb not in cell._pseudo:
continue
pp = cell._pseudo[symb]
for l, proj in enumerate(pp[5:]):
rl, nl, hl = proj
if nl > 0:
hl = np.asarray(hl)
fakemol._bas[0, pyscf.gto.ANG_OF] = l
fakemol._env[ptr + 3] = .5 * rl**2
fakemol._env[ptr + 4] = rl**(l + 1.5) * np.pi**1.25
pYlm_part = pyscf.dft.numint.eval_ao(fakemol, Gv, deriv=0)
pYlm = np.empty((nl, l * 2 + 1, ngs))
for k in range(nl):
qkl = _qli(G_rad * rl, l, k)
pYlm[k] = pYlm_part.T * qkl
# pYlm is real
SPG_lmi = np.einsum('g,nmg->nmg', SI[ia].conj(), pYlm)
SPG_lm_aoG = np.einsum('nmg,gp->nmp', SPG_lmi, aokG)
tmp = np.einsum('ij,jmp->imp', hl, SPG_lm_aoG)
vppnl += np.einsum('imp,imq->pq', SPG_lm_aoG.conj(), tmp)
vppnl *= (1. / ngs**2)
if aoR.dtype == np.double:
return vpploc.real + vppnl.real
else:
return vpploc + vppnl
def get_pp(cell, kpt=np.zeros(3)):
'''Get the periodic pseudotential nuc-el AO matrix
'''
import pyscf.dft
from pyscf.pbc import tools
from pyscf.pbc.dft import gen_grid
from pyscf.pbc.dft import numint
coords = gen_grid.gen_uniform_grids(cell)
aoR = numint.eval_ao(cell, coords, kpt)
nao = cell.nao_nr()
SI = cell.get_SI()
vlocG = get_vlocG(cell)
vpplocG = -np.sum(SI * vlocG, axis=0)
vpplocG[0] = np.sum(get_alphas(cell)) # from get_jvloc_G0 function
# vpploc evaluated in real-space
vpplocR = tools.ifft(vpplocG, cell.gs).real
vpploc = np.dot(aoR.T.conj(), vpplocR.reshape(-1,1)*aoR)
# vppnonloc evaluated in reciprocal space
aokG = tools.fftk(np.asarray(aoR.T, order='C'),
cell.gs, np.exp(-1j*np.dot(coords, kpt))).T
ngs = len(aokG)
fakemol = pyscf.gto.Mole()
fakemol._atm = np.zeros((1,pyscf.gto.ATM_SLOTS), dtype=np.int32)
fakemol._bas = np.zeros((1,pyscf.gto.BAS_SLOTS), dtype=np.int32)
ptr = pyscf.gto.PTR_ENV_START
fakemol._env = np.zeros(ptr+10)
fakemol._bas[0,pyscf.gto.NPRIM_OF ] = 1
fakemol._bas[0,pyscf.gto.NCTR_OF ] = 1
fakemol._bas[0,pyscf.gto.PTR_EXP ] = ptr+3
fakemol._bas[0,pyscf.gto.PTR_COEFF] = ptr+4
Gv = np.asarray(cell.Gv+kpt)
G_rad = lib.norm(Gv, axis=1)
vppnl = np.zeros((nao,nao), dtype=np.complex128)
for ia in range(cell.natm):
symb = cell.atom_symbol(ia)
if symb not in cell._pseudo:
continue
pp = cell._pseudo[symb]
for l, proj in enumerate(pp[5:]):
rl, nl, hl = proj
if nl > 0:
hl = np.asarray(hl)
fakemol._bas[0,pyscf.gto.ANG_OF] = l
fakemol._env[ptr+3] = .5*rl**2
fakemol._env[ptr+4] = rl**(l+1.5)*np.pi**1.25
pYlm_part = pyscf.dft.numint.eval_ao(fakemol, Gv, deriv=0)
pYlm = np.empty((nl,l*2+1,ngs))
for k in range(nl):
qkl = _qli(G_rad*rl, l, k)
pYlm[k] = pYlm_part.T * qkl
# pYlm is real
SPG_lmi = np.einsum('g,nmg->nmg', SI[ia].conj(), pYlm)
SPG_lm_aoG = np.einsum('nmg,gp->nmp', SPG_lmi, aokG)
tmp = np.einsum('ij,jmp->imp', hl, SPG_lm_aoG)
vppnl += np.einsum('imp,imq->pq', SPG_lm_aoG.conj(), tmp)
vppnl *= (1./ngs**2)
if aoR.dtype == np.double:
return vpploc.real + vppnl.real
else:
return vpploc + vppnl
def test_components(pseudo=None):
# The molecular calculation
mol = gto.Mole()
mol.unit = 'B'
L = 60
mol.atom.extend([['He', (L/2.,L/2.,L/2.)], ])
mol.basis = 'sto-3g'
mol.build()
m = rks.RKS(mol)
m.xc = 'LDA,VWN_RPA'
print(m.scf()) # -2.90705411168
dm = m.make_rdm1()
# The periodic calculation
cell = pbcgto.Cell()
cell.unit = 'B'
cell.a = np.diag([L,L,L])
cell.gs = np.array([80,80,80])
cell.atom = mol.atom
cell.basis = mol.basis
cell.pseudo = pseudo
cell.build()
# These should match reasonably well (roughly with accuracy of normalization)
print "Kinetic energy"
tao = pbchf.get_t(cell)
tao2 = mol.intor_symmetric('cint1e_kin_sph')
print np.dot(np.ravel(tao), np.ravel(dm)) # 2.82793077196
print np.dot(np.ravel(tao2), np.ravel(dm)) # 2.82352636524
print "Overlap"
sao = pbchf.get_ovlp(cell)
print np.dot(np.ravel(sao), np.ravel(dm)) # 1.99981725342
print np.dot(np.ravel(m.get_ovlp()), np.ravel(dm)) # 2.0
# The next two entries should *not* match, since G=0 component is removed
print "Coulomb (G!=0)"
jao = pbchf.get_j(cell, dm)
print np.dot(np.ravel(dm),np.ravel(jao)) # 4.03425518427
print np.dot(np.ravel(dm),np.ravel(m.get_j(dm))) # 4.22285177049
# The next two entries should *not* match, since G=0 component is removed
print "Nuc-el (G!=0)"
if cell.pseudo:
vppao = pbchf.get_pp(cell)
print np.dot(np.ravel(dm), np.ravel(vppao))
else:
neao = pbchf.get_nuc(cell)
print np.dot(np.ravel(dm), np.ravel(neao)) # -6.50203360062
vne = mol.intor_symmetric('cint1e_nuc_sph')
print np.dot(np.ravel(dm), np.ravel(vne)) # -6.68702326551
print "Normalization"
coords = gen_uniform_grids(cell)
aoR = eval_ao(cell, coords)
rhoR = eval_rho(cell, aoR, dm)
print cell.vol/len(rhoR)*np.sum(rhoR) # 1.99981725342 (should be 2.0)
print "(Hartree + vne) * DM"
print np.dot(np.ravel(dm),np.ravel(m.get_j(dm)))+np.dot(np.ravel(dm), np.ravel(vne))
if cell.pseudo:
print np.einsum("ij,ij", dm, vppao + jao + pbchf.get_jvloc_G0(cell))
else:
print np.einsum("ij,ij", dm, neao + jao)
ew_cut = (40,40,40)
ew_eta = 0.05
for ew_eta in [0.1, 0.5, 1.]:
ew = pbchf.ewald(cell, ew_eta, ew_cut)
print "Ewald (eta, energy)", ew_eta, ew # should be same for all eta
print "Ewald divergent terms summation", ew
# These two should now match if the box is reasonably big to
# remove images, and ngs is big.
print "Total coulomb (analytic) ",
print (.5*np.dot(np.ravel(dm), np.ravel(m.get_j(dm)))
+ np.dot(np.ravel(dm), np.ravel(vne)) ) # -4.57559738004
if not cell.pseudo:
print "Total coulomb (fft coul + ewald)",
print np.einsum("ij,ij", dm, neao + .5*jao) + ew # -4.57948259115
# Exc
cell.ew_eta, cell.ew_cut = ew_eta, ew_cut
mf = pbcdft.RKS(cell)
mf.xc = 'LDA,VWN_RPA'
rks.get_veff(mf, cell, dm)
print "Exc", mf._exc # -1.05967570089
def ao_on_grid(cell):
from pyscf.pbc.dft import gen_grid, numint
coords = gen_grid.gen_uniform_grids(cell)
aoR = numint.eval_ao(cell, coords)
return aoR
def get_jk_kpts(mf, cell, dm_kpts, kpts, kpts_band=None):
coords = gen_grid.gen_uniform_grids(cell)
nkpts = len(kpts)
ngs = len(coords)
dm_kpts = np.asarray(dm_kpts)
nao = dm_kpts.shape[-1]
dms = dm_kpts.reshape(-1, nkpts, nao, nao)
nset = dms.shape[0]
ni = numint._KNumInt(kpts)
aoR_kpts = ni.eval_ao(cell, coords, kpts)
if kpts_band is not None:
aoR_kband = numint.eval_ao(cell, coords, kpts_band)
# J
vjR = [get_vjR_kpts(cell, dms[i], aoR_kpts) for i in range(nset)]
if kpts_band is not None:
vj_kpts = [
cell.vol / ngs * lib.dot(aoR_kband.T.conj() * vjR[i], aoR_kband)
for i in range(nset)
]
else:
vj_kpts = []
for i in range(nset):
vj = [
cell.vol / ngs * lib.dot(aoR_k.T.conj() * vjR[i], aoR_k)
for aoR_k in aoR_kpts
]
vj_kpts.append(lib.asarray(vj))
vj_kpts = lib.asarray(vj_kpts)
vjR = None
# K
weight = 1. / nkpts * (cell.vol / ngs)
vk_kpts = np.zeros_like(vj_kpts)
if kpts_band is not None:
for k2, kpt2 in enumerate(kpts):
aoR_dms = [lib.dot(aoR_kpts[k2], dms[i, k2]) for i in range(nset)]
vkR_k1k2 = get_vkR(mf, cell, aoR_kband, aoR_kpts[k2], kpts_band,
kpt2)
#:vk_kpts = 1./nkpts * (cell.vol/ngs) * np.einsum('rs,Rp,Rqs,Rr->pq',
#: dm_kpts[k2], aoR_kband.conj(),
#: vkR_k1k2, aoR_kpts[k2])
for i in range(nset):
tmp_Rq = np.einsum('Rqs,Rs->Rq', vkR_k1k2, aoR_dms[i])
vk_kpts[i] += weight * lib.dot(aoR_kband.T.conj(), tmp_Rq)
vkR_k1k2 = None
if dm_kpts.ndim == 3:
vj_kpts = vj_kpts[0]
vk_kpts = vk_kpts[0]
return lib.asarray(vj_kpts), lib.asarray(vk_kpts)
else:
for k2, kpt2 in enumerate(kpts):
aoR_dms = [lib.dot(aoR_kpts[k2], dms[i, k2]) for i in range(nset)]
for k1, kpt1 in enumerate(kpts):
vkR_k1k2 = get_vkR(mf, cell, aoR_kpts[k1], aoR_kpts[k2], kpt1,
kpt2)
for i in range(nset):
tmp_Rq = np.einsum('Rqs,Rs->Rq', vkR_k1k2, aoR_dms[i])
vk_kpts[i, k1] += weight * lib.dot(aoR_kpts[k1].T.conj(),
tmp_Rq)
vkR_k1k2 = None
return vj_kpts.reshape(dm_kpts.shape), vk_kpts.reshape(dm_kpts.shape)