def test_ft_ao_with_kpts(self):
numpy.random.seed(1)
kpt = numpy.random.random(3)
coords = pdft.gen_grid.gen_uniform_grids(cell)
aoR = pdft.numint.eval_ao(cell, coords, kpt=kpt)
ngs, nao = aoR.shape
expmikr = numpy.exp(-1j * numpy.dot(coords, kpt))
ref = numpy.asarray(
[tools.fftk(aoR[:, i], cell.gs, expmikr) for i in range(nao)])
ref = ref.T * (cell.vol / ngs)
dat = ft_ao.ft_ao(cell, cell.Gv, kpt=kpt)
self.assertAlmostEqual(numpy.linalg.norm(ref[:, 0] - dat[:, 0]),
1.3359899490499813e-10, 9)
self.assertAlmostEqual(numpy.linalg.norm(ref[:, 1] - dat[:, 1]),
0.0042404556036939756, 4)
self.assertAlmostEqual(numpy.linalg.norm(ref[:, 2:] - dat[:, 2:]),
4.8856357999633564e-14, 9)
coords = pdft.gen_grid.gen_uniform_grids(cell1)
aoR = pdft.numint.eval_ao(cell1, coords, kpt=kpt)
ngs, nao = aoR.shape
expmikr = numpy.exp(-1j * numpy.dot(coords, kpt))
ref = numpy.asarray(
[tools.fftk(aoR[:, i], cell1.gs, expmikr) for i in range(nao)])
ref = ref.T * (cell1.vol / ngs)
dat = ft_ao.ft_ao(cell1, cell1.Gv, kpt=kpt)
self.assertAlmostEqual(numpy.linalg.norm(ref[:, 0] - dat[:, 0]), 0, 5)
self.assertAlmostEqual(numpy.linalg.norm(ref[:, 1] - dat[:, 1]), 0, 3)
self.assertAlmostEqual(numpy.linalg.norm(ref[:, 2:] - dat[:, 2:]), 0,
3)
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 : (ngrids, 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)
ngrids = 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])
ngrids = 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([ngrids, nmoi * nmoj], np.complex128)
mo_pairs_invG = np.zeros([ngrids, nmoi * nmoj], np.complex128)
fac = np.exp(-1j * np.dot(coords, q))
for i in range(nmoi):
for j in range(nmoj):
mo_pairs_G[:, i * nmoj + j] = tools.fftk(mo_pairs_R[:, i, j],
cell.mesh, fac)
mo_pairs_invG[:, i * nmoj + j] = np.conj(
tools.fftk(np.conj(mo_pairs_R[:, i, j]), cell.mesh,
fac.conj()))
return mo_pairs_G, mo_pairs_invG
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 test_ft_aoao_with_kpts(self):
numpy.random.seed(1)
kpti, kptj = numpy.random.random((2, 3))
dat = ft_ao.ft_aopair(cell, cell.Gv, kpti_kptj=(kpti, kptj))
self.assertAlmostEqual(finger(dat),
-0.80184732435570638 + 2.4078835207597176j, 9)
coords = pdft.gen_grid.gen_uniform_grids(cell)
aoi = pdft.numint.eval_ao(cell, coords, kpt=kpti)
aoj = pdft.numint.eval_ao(cell, coords, kpt=kptj)
ngs, nao = aoj.shape
q = kptj - kpti
expmikr = numpy.exp(-1j * numpy.dot(coords, q))
ref = numpy.asarray([
tools.fftk(aoi[:, i].conj() * aoj[:, j], cell.gs, expmikr)
for i in range(nao) for j in range(nao)
])
ref = ref.reshape(nao, nao, -1).transpose(2, 0, 1) * (cell.vol / ngs)
self.assertAlmostEqual(numpy.linalg.norm(ref[:, 0, 0] - dat[:, 0, 0]),
0, 5)
self.assertAlmostEqual(numpy.linalg.norm(ref[:, 1, 1] - dat[:, 1, 1]),
0.023225471785938184, 4)
self.assertAlmostEqual(
numpy.linalg.norm(ref[:, 2:, 2:] - dat[:, 2:, 2:]), 0, 9)
self.assertAlmostEqual(
numpy.linalg.norm(ref[:, 0, 2:] - dat[:, 0, 2:]), 0, 9)
self.assertAlmostEqual(
numpy.linalg.norm(ref[:, 2:, 0] - dat[:, 2:, 0]), 0, 9)
coords = pdft.gen_grid.gen_uniform_grids(cell1)
aoi = pdft.numint.eval_ao(cell1, coords, kpt=kpti)
aoj = pdft.numint.eval_ao(cell1, coords, kpt=kptj)
ngs, nao = aoj.shape
q = kptj - kpti
dat = ft_ao.ft_aopair(cell1, cell1.Gv, kpti_kptj=(kpti, kptj), q=q)
self.assertAlmostEqual(finger(dat),
0.72664436503332241 + 3.2542145296611373j, 9)
expmikr = numpy.exp(-1j * numpy.dot(coords, q))
ref = numpy.asarray([
tools.fftk(aoi[:, i].conj() * aoj[:, j], cell1.gs, expmikr)
for i in range(nao) for j in range(nao)
])
ref = ref.reshape(nao, nao, -1).transpose(2, 0, 1) * (cell1.vol / ngs)
self.assertAlmostEqual(numpy.linalg.norm(ref[:, 0, 0] - dat[:, 0, 0]),
0, 7)
self.assertAlmostEqual(numpy.linalg.norm(ref[:, 1, 1] - dat[:, 1, 1]),
0, 7)
self.assertAlmostEqual(
numpy.linalg.norm(ref[:, 2:, 2:] - dat[:, 2:, 2:]), 0, 7)
self.assertAlmostEqual(
numpy.linalg.norm(ref[:, 0, 2:] - dat[:, 0, 2:]), 0, 7)
self.assertAlmostEqual(
numpy.linalg.norm(ref[:, 2:, 0] - dat[:, 2:, 0]), 0, 7)
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
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).
Kwargs:
kpts : (nkpts, 3) ndarray
The sampled k-points; may be required for G=0 correction.
Returns:
vR : (ngs, nao, nao) ndarray
The real-space "exchange" potential at every grid point, for all
AO pairs.
Note:
This is essentially a density-fitting or resolution-of-the-identity.
The returned object is of size ngs*nao**2 and could be precomputed and
saved in vhfopt.
'''
coords = pyscf.pbc.dft.gen_grid.gen_uniform_grids(cell)
ngs, nao = aoR_k1.shape
coulG = tools.get_coulG(cell, kpt1-kpt2, exx=True, mf=mf)
vR = np.zeros((ngs,nao,nao), dtype=np.complex128)
for i in range(nao):
for j in range(nao):
rhoR = aoR_k1[:,i] * aoR_k2[:,j].conj()
rhoG = tools.fftk(rhoR, cell.gs, coords, kpt1-kpt2)
vG = coulG*rhoG
vR[:,i,j] = tools.ifftk(vG, cell.gs, coords, kpt1-kpt2)
return vR
def test_ft_aoao_pair_vs_fft(self):
numpy.random.seed(1)
kpti, kptj = numpy.random.random((2, 3))
coords = pdft.gen_grid.gen_uniform_grids(cell1)
aoi = pdft.numint.eval_ao(cell1, coords, kpt=kpti)
aoj = pdft.numint.eval_ao(cell1, coords, kpt=kptj)
ngrids, nao = aoj.shape
q = kptj - kpti
dat = ft_ao.ft_aopair(cell1, cell1.Gv, kpti_kptj=(kpti, kptj), q=q)
self.assertAlmostEqual(finger(dat),
0.72664436503332241 + 3.2542145296611373j, 9)
expmikr = numpy.exp(-1j * numpy.dot(coords, q))
ref = numpy.asarray([
tools.fftk(aoi[:, i].conj() * aoj[:, j], cell1.mesh, expmikr)
for i in range(nao) for j in range(nao)
])
ref = ref.reshape(nao, nao, -1).transpose(2, 0,
1) * (cell1.vol / ngrids)
self.assertAlmostEqual(numpy.linalg.norm(ref[:, 0, 0] - dat[:, 0, 0]),
0, 7)
self.assertAlmostEqual(numpy.linalg.norm(ref[:, 1, 1] - dat[:, 1, 1]),
0, 7)
self.assertAlmostEqual(
numpy.linalg.norm(ref[:, 2:, 2:] - dat[:, 2:, 2:]), 0, 7)
self.assertAlmostEqual(
numpy.linalg.norm(ref[:, 0, 2:] - dat[:, 0, 2:]), 0, 7)
self.assertAlmostEqual(
numpy.linalg.norm(ref[:, 2:, 0] - dat[:, 2:, 0]), 0, 7)
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
def get_hcore(cell, kpts):
'''Part of the nuclear gradients of core Hamiltonian'''
h1 = np.asarray(cell.pbc_intor('int1e_ipkin', kpts=kpts))
dtype = h1.dtype
if cell._pseudo:
SI = cell.get_SI()
nao = cell.nao_nr()
Gv = cell.Gv
natom = cell.natm
coords = cell.get_uniform_grids()
ngrids, nkpts = len(coords), len(kpts)
vlocG = get_vlocG(cell)
vpplocG = -np.einsum('ij,ij->j', SI, vlocG)
vpplocG[0] = np.sum(get_alphas(cell))
vpplocR = tools.ifft(vpplocG, cell.mesh).real
fakemol = _make_fakemol()
ptr = mole.PTR_ENV_START
for kn, kpt in enumerate(kpts):
aos = eval_ao_kpts(cell, coords, kpt, deriv=1)[0]
vloc = np.einsum('agi,g,gj->aij', aos[1:].conj(), vpplocR, aos[0])
expir = np.exp(-1j * np.dot(coords, kpt))
aokG = np.asarray([
tools.fftk(np.asarray(ao.T, order='C'), cell.mesh, expir).T
for ao in aos
])
Gk = Gv + kpt
G_rad = lib.norm(Gk, axis=1)
vnl = np.zeros(vloc.shape, dtype=np.complex128)
for ia in range(natom):
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, mole.ANG_OF] = l
fakemol._env[ptr + 3] = .5 * rl**2
fakemol._env[ptr + 4] = rl**(l + 1.5) * np.pi**1.25
pYlm_part = fakemol.eval_gto('GTOval', Gk)
pYlm = np.empty((nl, l * 2 + 1, ngrids))
for k in range(nl):
qkl = _qli(G_rad * rl, l, k)
pYlm[k] = pYlm_part.T * qkl
SPG_lmi = np.einsum('g,nmg->nmg', SI[ia].conj(), pYlm)
SPG_lm_aoG = np.einsum('nmg,agp->anmp', SPG_lmi, aokG)
tmp = np.einsum('ij,ajmp->aimp', hl, SPG_lm_aoG[1:])
vnl += np.einsum('aimp,imq->apq', tmp.conj(),
SPG_lm_aoG[0])
vnl *= (1. / ngrids**2)
if dtype == np.float64:
h1[kn, :] += vloc.real + vnl.real
else:
h1[kn, :] += vloc + vnl
else:
raise NotImplementedError
return h1
def get_pp_o1(cell, kpt=np.zeros(3)):
'''Get the periodic pseudotential nuc-el AO matrix, with G=0 removed.
'''
coords = pyscf.pbc.dft.gen_grid.gen_uniform_grids(cell)
aoR = pyscf.pbc.dft.numint.eval_ao(cell, coords, kpt)
nao = cell.nao_nr()
SI = cell.get_SI()
vlocG = pseudo.get_vlocG(cell)
vpplocG = -np.sum(SI * vlocG, axis=0)
# vpploc evaluated in real-space
vpplocR = tools.ifft(vpplocG, cell.gs)
vpploc = np.dot(aoR.T.conj(), vpplocR.reshape(-1,1)*aoR)
# vppnonloc evaluated in reciprocal space
aokG = np.empty(aoR.shape, np.complex128)
for i in range(nao):
aokG[:,i] = tools.fftk(aoR[:,i], cell.gs, coords, kpt)
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)
fakemol._env = np.zeros(5)
fakemol._bas[0,pyscf.gto.NPRIM_OF ] = 1
fakemol._bas[0,pyscf.gto.NCTR_OF ] = 1
fakemol._bas[0,pyscf.gto.PTR_EXP ] = 3
fakemol._bas[0,pyscf.gto.PTR_COEFF] = 4
Gv = np.asarray(cell.Gv+kpt)
G_rad = pyscf.lib.norm(Gv, axis=1)
vppnl = np.zeros((nao,nao), dtype=np.complex128)
for ia in range(cell.natm):
pp = cell._pseudo[cell.atom_symbol(ia)]
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[3] = .5*rl**2
fakemol._env[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)
return vpploc + vppnl
def hcore_deriv(atm_id):
shl0, shl1, p0, p1 = aoslices[atm_id]
symb = cell.atom_symbol(atm_id)
fakemol = _make_fakemol()
vloc_g = 1j * np.einsum('ga,g->ag', Gv, SI[atm_id] * vlocG[atm_id])
nkpts, nao = h1.shape[0], h1.shape[2]
hcore = np.zeros([3, nkpts, nao, nao], dtype=h1.dtype)
for kn, kpt in enumerate(kpts):
ao = eval_ao_kpts(cell, coords, kpt)[0]
rho = np.einsum('gi,gj->gij', ao.conj(), ao)
for ax in range(3):
vloc_R = tools.ifft(vloc_g[ax], mesh).real
vloc = np.einsum('gij,g->ij', rho, vloc_R)
hcore[ax, kn] += vloc
rho = None
aokG = tools.fftk(np.asarray(ao.T, order='C'), mesh,
np.exp(-1j * np.dot(coords, kpt))).T
ao = None
Gk = Gv + kpt
G_rad = lib.norm(Gk, axis=1)
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, mole.ANG_OF] = l
fakemol._env[ptr + 3] = .5 * rl**2
fakemol._env[ptr + 4] = rl**(l + 1.5) * np.pi**1.25
pYlm_part = fakemol.eval_gto('GTOval', Gk)
pYlm = np.empty((nl, l * 2 + 1, ngrids))
for k in range(nl):
qkl = _qli(G_rad * rl, l, k)
pYlm[k] = pYlm_part.T * qkl
SPG_lmi = np.einsum('g,nmg->nmg', SI[atm_id].conj(), pYlm)
SPG_lm_aoG = np.einsum('nmg,gp->nmp', SPG_lmi, aokG)
SPG_lmi_G = 1j * np.einsum('nmg, ga->anmg', SPG_lmi, Gv)
SPG_lm_G_aoG = np.einsum('anmg, gp->anmp', SPG_lmi_G, aokG)
tmp_1 = np.einsum('ij,ajmp->aimp', hl, SPG_lm_G_aoG)
tmp_2 = np.einsum('ij,jmp->imp', hl, SPG_lm_aoG)
vppnl = np.einsum(
'imp,aimq->apq', SPG_lm_aoG.conj(), tmp_1) + np.einsum(
'aimp,imq->apq', SPG_lm_G_aoG.conj(), tmp_2)
vppnl *= (1. / ngrids**2)
if dtype == np.float64:
hcore[:, kn] += vppnl.real
else:
hcore[:, kn] += vppnl
hcore[:, kn, p0:p1] -= h1[kn, :, p0:p1]
hcore[:, kn, :, p0:p1] -= h1[kn, :, p0:p1].transpose(0, 2,
1).conj()
return hcore
def test_ft_ao_with_kpts(self):
numpy.random.seed(1)
kpt = numpy.random.random(3)
coords = pdft.gen_grid.gen_uniform_grids(cell)
aoR = pdft.numint.eval_ao(cell, coords, kpt=kpt)
ngs, nao = aoR.shape
expmikr = numpy.exp(-1j*numpy.dot(coords,kpt))
ref = numpy.asarray([tools.fftk(aoR[:,i], cell.gs, expmikr) for i in range(nao)])
ref = ref.T * (cell.vol/ngs)
dat = ft_ao.ft_ao(cell, cell.Gv, kpt=kpt)
self.assertAlmostEqual(numpy.linalg.norm(ref[:,0]-dat[:,0]) , 1.3359899490499813e-10, 9)
self.assertAlmostEqual(numpy.linalg.norm(ref[:,1]-dat[:,1]) , 0.0042404556036939756 , 4)
self.assertAlmostEqual(numpy.linalg.norm(ref[:,2:]-dat[:,2:]), 4.8856357999633564e-14, 9)
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 test_ft_aoao_with_kpts(self):
numpy.random.seed(1)
kpti, kptj = numpy.random.random((2,3))
coords = pdft.gen_grid.gen_uniform_grids(cell)
aoi = pdft.numint.eval_ao(cell, coords, kpt=kpti)
aoj = pdft.numint.eval_ao(cell, coords, kpt=kptj)
ngs, nao = aoj.shape
q = kptj - kpti
ref = numpy.asarray([tools.fftk(aoi[:,i].conj()*aoj[:,j], cell.gs, coords, q)
for i in range(nao) for j in range(nao)])
ref = ref.reshape(nao,nao,-1).transpose(2,0,1) * (cell.vol/ngs)
dat = ft_ao.ft_aopair(cell, cell.Gv, kpti_kptj=(kpti,kptj))
self.assertAlmostEqual(numpy.linalg.norm(ref[:,0,0]-dat[:,0,0]) , 1.8912693795904546e-06, 7)
self.assertAlmostEqual(numpy.linalg.norm(ref[:,1,1]-dat[:,1,1]) , 0.023225471785938184 , 4)
self.assertAlmostEqual(numpy.linalg.norm(ref[:,2:,2:]-dat[:,2:,2:]), 3.9231124086361633e-14, 9)
self.assertAlmostEqual(numpy.linalg.norm(ref[:,0,2:]-dat[:,0,2:]) , 3.6949758392853562e-11, 9)
self.assertAlmostEqual(numpy.linalg.norm(ref[:,2:,0]-dat[:,2:,0]) , 4.1245047267152665e-11, 9)
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_vkR(mydf, cell, aoR_k1, aoR_k2, kpt1, kpt2, coords, gs, exxdiv):
"""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|
Kwargs:
kpt1, kpt2 : (3,) ndarray
The sampled k-points; may be required for G=0 correction.
Returns:
vR : (ngs, nao, nao) ndarray
The real-space "exchange" potential at every grid point, for all
AO pairs.
Note:
This is essentially a density-fitting or resolution-of-the-identity.
The returned object is of size ngs*nao**2
"""
ngs, nao = aoR_k1.shape
expmikr = np.exp(-1j * np.dot(kpt1 - kpt2, coords.T))
mydf.exxdiv = exxdiv
coulG = tools.get_coulG(cell, kpt1 - kpt2, True, mydf, gs)
aoR_k1 = np.asarray(aoR_k1.T, order="C")
aoR_k2 = np.asarray(aoR_k2.T, order="C")
vR = np.empty((nao, nao, ngs), dtype=np.complex128)
for i in range(nao):
rhoR = aoR_k1 * aoR_k2[i].conj()
rhoG = tools.fftk(rhoR, gs, expmikr)
vG = rhoG * coulG
vR[:, i] = tools.ifftk(vG, gs, expmikr.conj())
vR = vR.transpose(2, 0, 1)
if aoR_k1.dtype == np.double and aoR_k2.dtype == np.double:
return vR.real
else:
return vR
def get_vkR(mydf, cell, aoR_k1, aoR_k2, kpt1, kpt2, coords, gs, exxdiv):
'''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|
Kwargs:
kpt1, kpt2 : (3,) ndarray
The sampled k-points; may be required for G=0 correction.
Returns:
vR : (ngs, nao, nao) ndarray
The real-space "exchange" potential at every grid point, for all
AO pairs.
Note:
This is essentially a density-fitting or resolution-of-the-identity.
The returned object is of size ngs*nao**2
'''
ngs, nao = aoR_k1.shape
expmikr = np.exp(-1j * np.dot(kpt1 - kpt2, coords.T))
mydf.exxdiv = exxdiv
coulG = tools.get_coulG(cell, kpt1 - kpt2, True, mydf, gs)
aoR_k1 = np.asarray(aoR_k1.T, order='C')
aoR_k2 = np.asarray(aoR_k2.T, order='C')
vR = np.empty((nao, nao, ngs), dtype=np.complex128)
for i in range(nao):
rhoR = aoR_k1 * aoR_k2[i].conj()
rhoG = tools.fftk(rhoR, gs, expmikr)
vG = rhoG * coulG
vR[:, i] = tools.ifftk(vG, gs, expmikr.conj())
vR = vR.transpose(2, 0, 1)
if aoR_k1.dtype == np.double and aoR_k2.dtype == np.double:
return vR.real
else:
return vR
def get_t_pw(cell, kpt=np.zeros(3)):
'''Get the kinetic energy AO matrix using the PW resolution.
Note: Incurs error due to finite resolution of the gradient operator.
'''
coords = pyscf.pbc.dft.gen_grid.gen_uniform_grids(cell)
aoR = pyscf.pbc.dft.numint.eval_ao(cell, coords, kpt, deriv=0)
nao = cell.nao_nr()
kG = kpt + cell.Gv
abskG2 = np.einsum('gi,gi->g', kG, kG)
aokG = np.empty(aoR.shape, np.complex128)
TaokG = np.empty(aoR.shape, np.complex128)
nao = cell.nao_nr()
for i in range(nao):
aokG[:,i] = tools.fftk(aoR[:,i], cell.gs, coords, kpt)
TaokG[:,i] = 0.5*abskG2*aokG[:,i]
ngs = len(aokG)
t = np.dot(aokG.T.conj(), TaokG)
t *= (cell.vol/ngs**2)
return t
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
'''
from pyscf.pbc import tools
coords = cell.get_uniform_grids()
aoR = cell.pbc_eval_gto('GTOval', coords, kpt=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.mesh).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.mesh, np.exp(-1j*np.dot(coords, kpt))).T
ngrids = len(aokG)
fakemol = mole.Mole()
fakemol._atm = np.zeros((1,mole.ATM_SLOTS), dtype=np.int32)
fakemol._bas = np.zeros((1,mole.BAS_SLOTS), dtype=np.int32)
ptr = mole.PTR_ENV_START
fakemol._env = np.zeros(ptr+10)
fakemol._bas[0,mole.NPRIM_OF ] = 1
fakemol._bas[0,mole.NCTR_OF ] = 1
fakemol._bas[0,mole.PTR_EXP ] = ptr+3
fakemol._bas[0,mole.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,mole.ANG_OF] = l
fakemol._env[ptr+3] = .5*rl**2
fakemol._env[ptr+4] = rl**(l+1.5)*np.pi**1.25
pYlm_part = fakemol.eval_gto('GTOval', Gv)
pYlm = np.empty((nl,l*2+1,ngrids))
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./ngrids**2)
if aoR.dtype == np.double:
return vpploc.real + vppnl.real
else:
return vpploc + vppnl
