def test_psi_vmat(self):
pcm = ddcosmo.DDCOSMO(mol)
pcm.lmax = 2
r_vdw = ddcosmo.get_atomic_radii(pcm)
fi = ddcosmo.make_fi(pcm, r_vdw)
ui = 1 - fi
ui[ui<0] = 0
grids = dft.gen_grid.Grids(mol).build()
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(pcm.lebedev_order)
ylm_1sph = numpy.vstack(sph.real_sph_vec(coords_1sph, pcm.lmax, True))
cached_pol = ddcosmo.cache_fake_multipoles(grids, r_vdw, pcm.lmax)
numpy.random.seed(1)
nao = mol.nao_nr()
dm = numpy.random.random((nao,nao))
dm = dm + dm.T
natm = mol.natm
nlm = (pcm.lmax+1)**2
LX = numpy.random.random((natm,nlm))
L = ddcosmo.make_L(pcm, r_vdw, ylm_1sph, fi)
psi, vmat = ddcosmo.make_psi_vmat(pcm, dm, r_vdw, ui, grids,
ylm_1sph, cached_pol, LX, L)[:2]
psi_ref = make_psi(pcm.mol, dm, r_vdw, pcm.lmax)
self.assertAlmostEqual(abs(psi_ref - psi).max(), 0, 12)
LS = numpy.linalg.solve(L.T.reshape(natm*nlm,-1),
psi_ref.ravel()).reshape(natm,nlm)
vmat_ref = make_vmat(pcm, r_vdw, pcm.lebedev_order, pcm.lmax, LX, LS)
self.assertAlmostEqual(abs(vmat_ref - vmat).max(), 0, 12)
def get_phi1(pcmojb):
pcmobj.grids.build()
mol = pcmobj.mol
natm = mol.natm
lmax = pcmobj.lmax
r_vdw = pcmobj.get_atomic_radii()
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(pcmobj.lebedev_order)
ylm_1sph = numpy.vstack(sph.real_sph_vec(coords_1sph, lmax, True))
fi = ddcosmo.make_fi(pcmobj, r_vdw)
ui = 1 - fi
ui[ui<0] = 0
nexposed = numpy.count_nonzero(ui==1)
nbury = numpy.count_nonzero(ui==0)
on_shell = numpy.count_nonzero(ui>0) - nexposed
nlm = (lmax+1)**2
Lmat = ddcosmo.make_L(pcmobj, r_vdw, ylm_1sph, fi)
Lmat = Lmat.reshape(natm*nlm,-1)
cached_pol = ddcosmo.cache_fake_multipoles(pcmobj.grids, r_vdw, lmax)
phi = ddcosmo.make_phi(pcmobj, dm, r_vdw, ui)
L_X = numpy.linalg.solve(Lmat, phi.ravel()).reshape(natm,-1)
psi, vmat, L_S = \
ddcosmo.make_psi_vmat(pcmobj, dm, r_vdw, ui, pcmobj.grids, ylm_1sph,
cached_pol, L_X, Lmat)
phi1 = ddcosmo_grad.make_phi1(pcmobj, dm, r_vdw, ui)
phi1 = numpy.einsum('izjx,jx->iz', phi1, L_S)
return L_S, phi, phi1
def test_psi_vmat(self):
pcm = ddcosmo.DDCOSMO(mol)
pcm.lmax = 2
pcm.eps = 0
r_vdw = ddcosmo.get_atomic_radii(pcm)
fi = ddcosmo.make_fi(pcm, r_vdw)
ui = 1 - fi
ui[ui < 0] = 0
grids = dft.gen_grid.Grids(mol).build()
pcm.grids = grids
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(
pcm.lebedev_order)
ylm_1sph = numpy.vstack(sph.real_sph_vec(coords_1sph, pcm.lmax, True))
cached_pol = ddcosmo.cache_fake_multipoles(grids, r_vdw, pcm.lmax)
numpy.random.seed(1)
nao = mol.nao_nr()
dm = numpy.random.random((nao, nao))
dm = dm + dm.T
natm = mol.natm
nlm = (pcm.lmax + 1)**2
LX = numpy.random.random((natm, nlm))
L = ddcosmo.make_L(pcm, r_vdw, ylm_1sph, fi)
psi, vmat = ddcosmo.make_psi_vmat(pcm, dm, r_vdw, ui, ylm_1sph,
cached_pol, LX, L)[:2]
psi_ref = make_psi(pcm.mol, dm, r_vdw, pcm.lmax)
self.assertAlmostEqual(abs(psi_ref - psi).max(), 0, 12)
LS = numpy.linalg.solve(L.reshape(natm * nlm, -1).T,
psi_ref.ravel()).reshape(natm, nlm)
vmat_ref = make_vmat(pcm, r_vdw, pcm.lebedev_order, pcm.lmax, LX, LS)
self.assertAlmostEqual(abs(vmat_ref - vmat).max(), 0, 12)
def get_phi1(pcmojb):
pcmobj.grids.build()
mol = pcmobj.mol
natm = mol.natm
lmax = pcmobj.lmax
r_vdw = pcmobj.get_atomic_radii()
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(
pcmobj.lebedev_order)
ylm_1sph = numpy.vstack(sph.real_sph_vec(coords_1sph, lmax, True))
fi = ddcosmo.make_fi(pcmobj, r_vdw)
ui = 1 - fi
ui[ui < 0] = 0
nexposed = numpy.count_nonzero(ui == 1)
nbury = numpy.count_nonzero(ui == 0)
on_shell = numpy.count_nonzero(ui > 0) - nexposed
nlm = (lmax + 1)**2
Lmat = ddcosmo.make_L(pcmobj, r_vdw, ylm_1sph, fi)
Lmat = Lmat.reshape(natm * nlm, -1)
cached_pol = ddcosmo.cache_fake_multipoles(pcmobj.grids, r_vdw,
lmax)
phi = ddcosmo.make_phi(pcmobj, dm, r_vdw, ui, ylm_1sph)
L_X = numpy.linalg.solve(Lmat, phi.ravel()).reshape(natm, -1)
psi, vmat, L_S = \
ddcosmo.make_psi_vmat(pcmobj, dm, r_vdw, ui, ylm_1sph,
cached_pol, L_X, Lmat)
phi1 = ddcosmo_grad.make_phi1(pcmobj, dm, r_vdw, ui, ylm_1sph)
phi1 = numpy.einsum('izjx,jx->iz', phi1, L_S)
return L_S, phi, phi1
def test_L1(self):
pcmobj = ddcosmo.DDCOSMO(mol0)
r_vdw = pcmobj.get_atomic_radii()
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(pcmobj.lebedev_order)
ylm_1sph = numpy.vstack(sph.real_sph_vec(coords_1sph, pcmobj.lmax, True))
fi = ddcosmo.make_fi(pcmobj, r_vdw)
L1 = ddcosmo_grad.make_L1(pcmobj, r_vdw, ylm_1sph, fi)
pcmobj = ddcosmo.DDCOSMO(mol1)
fi = ddcosmo.make_fi(pcmobj, r_vdw)
L_1 = ddcosmo.make_L(pcmobj, r_vdw, ylm_1sph, fi)
pcmobj = ddcosmo.DDCOSMO(mol2)
fi = ddcosmo.make_fi(pcmobj, r_vdw)
L_2 = ddcosmo.make_L(pcmobj, r_vdw, ylm_1sph, fi)
self.assertAlmostEqual(abs((L_2-L_1)/dx - L1[0,2]).max(), 0, 7)
def test_L1(self):
pcmobj = ddcosmo.DDCOSMO(mol0)
r_vdw = pcmobj.get_atomic_radii()
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(pcmobj.lebedev_order)
ylm_1sph = numpy.vstack(sph.real_sph_vec(coords_1sph, pcmobj.lmax, True))
fi = ddcosmo.make_fi(pcmobj, r_vdw)
L1 = ddcosmo_grad.make_L1(pcmobj, r_vdw, ylm_1sph, fi)
pcmobj = ddcosmo.DDCOSMO(mol1)
fi = ddcosmo.make_fi(pcmobj, r_vdw)
L_1 = ddcosmo.make_L(pcmobj, r_vdw, ylm_1sph, fi)
pcmobj = ddcosmo.DDCOSMO(mol2)
fi = ddcosmo.make_fi(pcmobj, r_vdw)
L_2 = ddcosmo.make_L(pcmobj, r_vdw, ylm_1sph, fi)
self.assertAlmostEqual(abs((L_2-L_1)/dx - L1[0,2]).max(), 0, 7)
def test_solvent_nuc(self):
def get_nuc(mol):
pcm = ddcosmo.DDCOSMO(mol)
pcm.lmax = 2
pcm.eps = 0
natm = mol.natm
nao = mol.nao
nlm = (pcm.lmax + 1)**2
r_vdw = ddcosmo.get_atomic_radii(pcm)
fi = ddcosmo.make_fi(pcm, r_vdw)
ui = 1 - fi
ui[ui < 0] = 0
pcm.grids = grids = dft.gen_grid.Grids(mol).run(level=0)
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(
pcm.lebedev_order)
ylm_1sph = numpy.vstack(
sph.real_sph_vec(coords_1sph, pcm.lmax, True))
cached_pol = ddcosmo.cache_fake_multipoles(grids, r_vdw, pcm.lmax)
L = ddcosmo.make_L(pcm, r_vdw, ylm_1sph, fi)
return nuc_part(pcm, r_vdw, ui, ylm_1sph, cached_pol, L)
pcm = ddcosmo.DDCOSMO(mol0)
pcm.lmax = 2
pcm.eps = 0
natm = mol0.natm
nao = mol0.nao
nlm = (pcm.lmax + 1)**2
r_vdw = ddcosmo.get_atomic_radii(pcm)
fi = ddcosmo.make_fi(pcm, r_vdw)
ui = 1 - fi
ui[ui < 0] = 0
pcm.grids = grids = dft.gen_grid.Grids(mol0).run(level=0)
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(
pcm.lebedev_order)
ylm_1sph = numpy.vstack(sph.real_sph_vec(coords_1sph, pcm.lmax, True))
cached_pol = ddcosmo.cache_fake_multipoles(grids, r_vdw, pcm.lmax)
L = ddcosmo.make_L(pcm, r_vdw, ylm_1sph, fi)
dvmat = nuc_part1(pcm, r_vdw, ui, ylm_1sph, cached_pol, L)
vmat1 = get_nuc(mol1)
vmat2 = get_nuc(mol2)
self.assertAlmostEqual(
abs((vmat2 - vmat1) / dx - dvmat[0, 2]).max(), 0, 8)
nao = mol0.nao
numpy.random.seed(19)
dm = numpy.random.random((nao, nao))
vref = pcm._get_vind(dm)[1]
vmat = 0.5 * get_nuc(mol0)
vmat += pcm._B_dot_x(dm)
self.assertAlmostEqual(abs(vmat - vref).max(), 0, 14)
dm1 = numpy.random.random((2, nao, nao))
de = _ddcosmo_tdscf_grad._grad_ne(pcm, dm1, r_vdw, ui, ylm_1sph,
cached_pol, L)
ref = numpy.einsum('azij,nij->naz', dvmat, dm1)
self.assertAlmostEqual(abs(de - ref).max(), 0, 12)
def test_fi(self):
pcmobj = ddcosmo.DDCOSMO(mol0)
fi1 = ddcosmo_grad.make_fi1(pcmobj, pcmobj.get_atomic_radii())
ui1 = -fi1
fi = ddcosmo.make_fi(pcmobj, pcmobj.get_atomic_radii())
ui = 1 - fi
ui1[:,:,ui<0] = 0
pcmobj = ddcosmo.DDCOSMO(mol1)
fi_1 = ddcosmo.make_fi(pcmobj, pcmobj.get_atomic_radii())
ui_1 = 1 - fi_1
ui_1[ui_1<0] = 0
pcmobj = ddcosmo.DDCOSMO(mol2)
fi_2 = ddcosmo.make_fi(pcmobj, pcmobj.get_atomic_radii())
ui_2 = 1 - fi_2
ui_2[ui_2<0] = 0
self.assertAlmostEqual(abs((fi_2-fi_1)/dx - fi1[0,2]).max(), 0, 6)
self.assertAlmostEqual(abs((ui_2-ui_1)/dx - ui1[0,2]).max(), 0, 6)
def test_fi(self):
pcmobj = ddcosmo.DDCOSMO(mol0)
fi1 = ddcosmo_grad.make_fi1(pcmobj, pcmobj.get_atomic_radii())
ui1 = -fi1
fi = ddcosmo.make_fi(pcmobj, pcmobj.get_atomic_radii())
ui = 1 - fi
ui1[:, :, ui < 0] = 0
pcmobj = ddcosmo.DDCOSMO(mol1)
fi_1 = ddcosmo.make_fi(pcmobj, pcmobj.get_atomic_radii())
ui_1 = 1 - fi_1
ui_1[ui_1 < 0] = 0
pcmobj = ddcosmo.DDCOSMO(mol2)
fi_2 = ddcosmo.make_fi(pcmobj, pcmobj.get_atomic_radii())
ui_2 = 1 - fi_2
ui_2[ui_2 < 0] = 0
self.assertAlmostEqual(abs((fi_2 - fi_1) / dx - fi1[0, 2]).max(), 0, 6)
self.assertAlmostEqual(abs((ui_2 - ui_1) / dx - ui1[0, 2]).max(), 0, 6)
def test_L_x(self):
pcm = ddcosmo.DDCOSMO(mol)
r_vdw = ddcosmo.get_atomic_radii(pcm)
n = mol.natm * (pcm.lmax+1)**2
Lref = make_L(pcm, r_vdw, pcm.lebedev_order, pcm.lmax, pcm.eta).reshape(n,n)
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(pcm.lebedev_order)
ylm_1sph = numpy.vstack(sph.real_sph_vec(coords_1sph, pcm.lmax, True))
fi = ddcosmo.make_fi(pcm, r_vdw)
L = ddcosmo.make_L(pcm, r_vdw, ylm_1sph, fi).reshape(n,n)
numpy.random.seed(1)
x = numpy.random.random(n)
self.assertTrue(abs(Lref.dot(n)-L.dot(n)).max() < 1e-12)
def gen_ddpcm_solver(pcmobj, verbose=None):
mol = pcmobj.mol
if pcmobj.grids.coords is None:
pcmobj.grids.build(with_non0tab=True)
natm = mol.natm
lmax = pcmobj.lmax
r_vdw = ddcosmo.get_atomic_radii(pcmobj)
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(
pcmobj.lebedev_order)
ylm_1sph = numpy.vstack(sph.real_sph_vec(coords_1sph, lmax, True))
fi = ddcosmo.make_fi(pcmobj, r_vdw)
ui = 1 - fi
ui[ui < 0] = 0
nexposed = numpy.count_nonzero(ui == 1)
nbury = numpy.count_nonzero(ui == 0)
on_shell = numpy.count_nonzero(ui > 0) - nexposed
logger.debug(pcmobj, 'Num points exposed %d', nexposed)
logger.debug(pcmobj, 'Num points buried %d', nbury)
logger.debug(pcmobj, 'Num points on shell %d', on_shell)
nlm = (lmax + 1)**2
Lmat = ddcosmo.make_L(pcmobj, r_vdw, ylm_1sph, fi)
Lmat = Lmat.reshape(natm * nlm, -1)
Amat = make_A(pcmobj, r_vdw, ylm_1sph, ui).reshape(natm * nlm, -1)
fac = 2 * numpy.pi * (pcmobj.eps + 1) / (pcmobj.eps - 1)
A_diele = Amat + fac * numpy.eye(natm * nlm)
A_inf = Amat + 2 * numpy.pi * numpy.eye(natm * nlm)
cached_pol = ddcosmo.cache_fake_multipoles(pcmobj.grids, r_vdw, lmax)
def gen_vind(dm):
phi = ddcosmo.make_phi(pcmobj, dm, r_vdw, ui)
phi = numpy.linalg.solve(A_diele, A_inf.dot(phi.ravel()))
Xvec = numpy.linalg.solve(Lmat, phi.ravel()).reshape(natm, -1)
psi, vmat = ddcosmo.make_psi_vmat(pcmobj, dm, r_vdw, ui, pcmobj.grids,
ylm_1sph, cached_pol, Xvec, Lmat)[:2]
dielectric = pcmobj.eps
f_epsilon = (dielectric - 1.) / dielectric
epcm = .5 * f_epsilon * numpy.einsum('jx,jx', psi, Xvec)
vpcm = .5 * f_epsilon * vmat
return epcm, vpcm
return gen_vind
def test_B_dot_x(self):
pcm = ddcosmo.DDCOSMO(mol)
pcm.lmax = 2
pcm.eps = 0
natm = mol.natm
nao = mol.nao
nlm = (pcm.lmax + 1)**2
r_vdw = ddcosmo.get_atomic_radii(pcm)
fi = ddcosmo.make_fi(pcm, r_vdw)
ui = 1 - fi
ui[ui < 0] = 0
grids = dft.gen_grid.Grids(mol).run(level=0)
pcm.grids = grids
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(
pcm.lebedev_order)
ylm_1sph = numpy.vstack(sph.real_sph_vec(coords_1sph, pcm.lmax, True))
cached_pol = ddcosmo.cache_fake_multipoles(grids, r_vdw, pcm.lmax)
L = ddcosmo.make_L(pcm, r_vdw, ylm_1sph, fi)
B = make_B(pcm, r_vdw, ui, ylm_1sph, cached_pol, L)
numpy.random.seed(19)
dm = numpy.random.random((2, nao, nao))
Bx = numpy.einsum('ijkl,xkl->xij', B, dm)
phi = ddcosmo.make_phi(pcm, dm, r_vdw, ui, ylm_1sph, with_nuc=False)
Xvec = numpy.linalg.solve(L.reshape(natm * nlm, -1),
phi.reshape(-1, natm * nlm).T)
Xvec = Xvec.reshape(natm, nlm, -1).transpose(2, 0, 1)
psi, vref, LS = ddcosmo.make_psi_vmat(pcm,
dm,
r_vdw,
ui,
ylm_1sph,
cached_pol,
Xvec,
L,
with_nuc=False)
self.assertAlmostEqual(abs(Bx - vref).max(), 0, 12)
e1 = numpy.einsum('nij,nij->n', psi, Xvec)
e2 = numpy.einsum('nij,nij->n', phi, LS)
e3 = numpy.einsum('nij,nij->n', dm, vref) * .5
self.assertAlmostEqual(abs(e1 - e2).max(), 0, 12)
self.assertAlmostEqual(abs(e1 - e3).max(), 0, 12)
vmat = pcm._B_dot_x(dm)
self.assertEqual(vmat.shape, (2, nao, nao))
self.assertAlmostEqual(abs(vmat - vref * .5).max(), 0, 12)
self.assertAlmostEqual(lib.fp(vmat), -17.383712106418606, 12)
def test_L_x(self):
pcm = ddcosmo.DDCOSMO(mol)
r_vdw = ddcosmo.get_atomic_radii(pcm)
n = mol.natm * (pcm.lmax + 1)**2
Lref = make_L(pcm, r_vdw, pcm.lebedev_order, pcm.lmax,
pcm.eta).reshape(n, n)
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(
pcm.lebedev_order)
ylm_1sph = numpy.vstack(sph.real_sph_vec(coords_1sph, pcm.lmax, True))
fi = ddcosmo.make_fi(pcm, r_vdw)
L = ddcosmo.make_L(pcm, r_vdw, ylm_1sph, fi).reshape(n, n)
numpy.random.seed(1)
x = numpy.random.random(n)
self.assertTrue(abs(Lref.dot(n) - L.dot(n)).max() < 1e-12)
def gen_ddpcm_solver(pcmobj, verbose=None):
mol = pcmobj.mol
if pcmobj.grids.coords is None:
pcmobj.grids.build(with_non0tab=True)
natm = mol.natm
lmax = pcmobj.lmax
r_vdw = ddcosmo.get_atomic_radii(pcmobj)
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(pcmobj.lebedev_order)
ylm_1sph = numpy.vstack(sph.real_sph_vec(coords_1sph, lmax, True))
fi = ddcosmo.make_fi(pcmobj, r_vdw)
ui = 1 - fi
ui[ui<0] = 0
nexposed = numpy.count_nonzero(ui==1)
nbury = numpy.count_nonzero(ui==0)
on_shell = numpy.count_nonzero(ui>0) - nexposed
logger.debug(pcmobj, 'Num points exposed %d', nexposed)
logger.debug(pcmobj, 'Num points buried %d', nbury)
logger.debug(pcmobj, 'Num points on shell %d', on_shell)
nlm = (lmax+1)**2
Lmat = ddcosmo.make_L(pcmobj, r_vdw, ylm_1sph, fi)
Lmat = Lmat.reshape(natm*nlm,-1)
Amat = make_A(pcmobj, r_vdw, ylm_1sph, ui).reshape(natm*nlm,-1)
fac = 2*numpy.pi * (pcmobj.eps+1) / (pcmobj.eps-1)
A_diele = Amat + fac * numpy.eye(natm*nlm)
A_inf = Amat + 2*numpy.pi * numpy.eye(natm*nlm)
cached_pol = ddcosmo.cache_fake_multipoles(pcmobj.grids, r_vdw, lmax)
def gen_vind(dm):
phi = ddcosmo.make_phi(pcmobj, dm, r_vdw, ui)
phi = numpy.linalg.solve(A_diele, A_inf.dot(phi.ravel()))
L_X = numpy.linalg.solve(Lmat, phi.ravel()).reshape(natm,-1)
psi, vmat = ddcosmo.make_psi_vmat(pcmobj, dm, r_vdw, ui, pcmobj.grids,
ylm_1sph, cached_pol, L_X, Lmat)[:2]
dielectric = pcmobj.eps
f_epsilon = (dielectric-1.)/dielectric
epcm = .5 * f_epsilon * numpy.einsum('jx,jx', psi, L_X)
return epcm, .5 * f_epsilon * vmat
return gen_vind
def test_phi(self):
pcm = ddcosmo.DDCOSMO(mol)
r_vdw = ddcosmo.get_atomic_radii(pcm)
fi = ddcosmo.make_fi(pcm, r_vdw)
ui = 1 - fi
ui[ui<0] = 0
numpy.random.seed(1)
nao = mol.nao_nr()
dm = numpy.random.random((nao,nao))
dm = dm + dm.T
v_phi = make_v_phi(mol, dm, r_vdw, pcm.lebedev_order)
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(pcm.lebedev_order)
ylm_1sph = numpy.vstack(sph.real_sph_vec(coords_1sph, pcm.lmax, True))
phi = -numpy.einsum('n,xn,jn,jn->jx', weights_1sph, ylm_1sph, ui, v_phi)
phi1 = ddcosmo.make_phi(pcm, dm, r_vdw, ui)
self.assertTrue(abs(phi - phi1).max() < 1e-12)
def kernel(pcmobj, dm, verbose=None):
mol = pcmobj.mol
natm = mol.natm
lmax = pcmobj.lmax
if pcmobj.grids.coords is None:
pcmobj.grids.build(with_non0tab=True)
if not (isinstance(dm, numpy.ndarray) and dm.ndim == 2):
# UHF density matrix
dm = dm[0] + dm[1]
r_vdw = ddcosmo.get_atomic_radii(pcmobj)
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(
pcmobj.lebedev_order)
ylm_1sph = numpy.vstack(sph.real_sph_vec(coords_1sph, lmax, True))
fi = ddcosmo.make_fi(pcmobj, r_vdw)
ui = 1 - fi
ui[ui < 0] = 0
cached_pol = ddcosmo.cache_fake_multipoles(pcmobj.grids, r_vdw, lmax)
nlm = (lmax + 1)**2
L0 = ddcosmo.make_L(pcmobj, r_vdw, ylm_1sph, fi)
L0 = L0.reshape(natm * nlm, -1)
L1 = make_L1(pcmobj, r_vdw, ylm_1sph, fi)
phi0 = ddcosmo.make_phi(pcmobj, dm, r_vdw, ui)
phi1 = make_phi1(pcmobj, dm, r_vdw, ui)
L0_X = numpy.linalg.solve(L0, phi0.ravel()).reshape(natm, -1)
psi0, vmat, L0_S = \
ddcosmo.make_psi_vmat(pcmobj, dm, r_vdw, ui, pcmobj.grids, ylm_1sph,
cached_pol, L0_X, L0)
e_psi1 = make_e_psi1(pcmobj, dm, r_vdw, ui, pcmobj.grids, ylm_1sph,
cached_pol, L0_X, L0)
dielectric = pcmobj.eps
if dielectric > 0:
f_epsilon = (dielectric - 1.) / dielectric
else:
f_epsilon = 1
de = .5 * f_epsilon * e_psi1
de += .5 * f_epsilon * numpy.einsum('jx,azjx->az', L0_S, phi1)
de -= .5 * f_epsilon * numpy.einsum('aziljm,il,jm->az', L1, L0_S, L0_X)
return de
def build(self):
if self.grids.coords is None:
self.grids.build(with_non0tab=True)
mol = self.mol
natm = mol.natm
lmax = self.lmax
r_vdw = ddcosmo.get_atomic_radii(self)
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(
self.lebedev_order)
ylm_1sph = numpy.vstack(sph.real_sph_vec(coords_1sph, lmax, True))
fi = ddcosmo.make_fi(self, r_vdw)
ui = 1 - fi
ui[ui < 0] = 0
nexposed = numpy.count_nonzero(ui == 1)
nbury = numpy.count_nonzero(ui == 0)
on_shell = numpy.count_nonzero(ui > 0) - nexposed
logger.debug(self, 'Num points exposed %d', nexposed)
logger.debug(self, 'Num points buried %d', nbury)
logger.debug(self, 'Num points on shell %d', on_shell)
nlm = (lmax + 1)**2
Lmat = ddcosmo.make_L(self, r_vdw, ylm_1sph, fi)
Lmat = Lmat.reshape(natm * nlm, -1)
Amat = make_A(self, r_vdw, ylm_1sph, ui).reshape(natm * nlm, -1)
fac = 2 * numpy.pi * (self.eps + 1) / (self.eps - 1)
A_diele = Amat + fac * numpy.eye(natm * nlm)
A_inf = Amat + 2 * numpy.pi * numpy.eye(natm * nlm)
cached_pol = ddcosmo.cache_fake_multipoles(self.grids, r_vdw, lmax)
self._intermediates = {
'r_vdw': r_vdw,
'ylm_1sph': ylm_1sph,
'ui': ui,
'Lmat': Lmat,
'A_diele': A_diele,
'A_inf': A_inf,
'cached_pol': cached_pol,
}
def kernel(pcmobj, dm, verbose=None):
mol = pcmobj.mol
natm = mol.natm
lmax = pcmobj.lmax
if pcmobj.grids.coords is None:
pcmobj.grids.build(with_non0tab=True)
if not (isinstance(dm, numpy.ndarray) and dm.ndim == 2):
# UHF density matrix
dm = dm[0] + dm[1]
r_vdw = ddcosmo.get_atomic_radii(pcmobj)
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(pcmobj.lebedev_order)
ylm_1sph = numpy.vstack(sph.real_sph_vec(coords_1sph, lmax, True))
fi = ddcosmo.make_fi(pcmobj, r_vdw)
ui = 1 - fi
ui[ui<0] = 0
cached_pol = ddcosmo.cache_fake_multipoles(pcmobj.grids, r_vdw, lmax)
nlm = (lmax+1)**2
L0 = ddcosmo.make_L(pcmobj, r_vdw, ylm_1sph, fi)
L0 = L0.reshape(natm*nlm,-1)
L1 = make_L1(pcmobj, r_vdw, ylm_1sph, fi)
phi0 = ddcosmo.make_phi(pcmobj, dm, r_vdw, ui)
phi1 = make_phi1(pcmobj, dm, r_vdw, ui)
L0_X = numpy.linalg.solve(L0, phi0.ravel()).reshape(natm,-1)
psi0, vmat, L0_S = \
ddcosmo.make_psi_vmat(pcmobj, dm, r_vdw, ui, pcmobj.grids, ylm_1sph,
cached_pol, L0_X, L0)
e_psi1 = make_e_psi1(pcmobj, dm, r_vdw, ui, pcmobj.grids, ylm_1sph,
cached_pol, L0_X, L0)
dielectric = pcmobj.eps
if dielectric > 0:
f_epsilon = (dielectric-1.)/dielectric
else:
f_epsilon = 1
de = .5 * f_epsilon * e_psi1
de+= .5 * f_epsilon * numpy.einsum('jx,azjx->az', L0_S, phi1)
de-= .5 * f_epsilon * numpy.einsum('aziljm,il,jm->az', L1, L0_S, L0_X)
return de
def getB(mol):
pcm = ddcosmo.DDCOSMO(mol)
pcm.lmax = 2
pcm.eps = 0
natm = mol.natm
nao = mol.nao
nlm = (pcm.lmax + 1)**2
r_vdw = ddcosmo.get_atomic_radii(pcm)
fi = ddcosmo.make_fi(pcm, r_vdw)
ui = 1 - fi
ui[ui < 0] = 0
pcm.grids = grids = dft.gen_grid.Grids(mol).run(level=0)
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(
pcm.lebedev_order)
ylm_1sph = numpy.vstack(
sph.real_sph_vec(coords_1sph, pcm.lmax, True))
cached_pol = ddcosmo.cache_fake_multipoles(grids, r_vdw, pcm.lmax)
L = ddcosmo.make_L(pcm, r_vdw, ylm_1sph, fi)
return make_B(pcm, r_vdw, ui, ylm_1sph, cached_pol, L)
def test_phi(self):
pcm = ddcosmo.DDCOSMO(mol)
r_vdw = ddcosmo.get_atomic_radii(pcm)
fi = ddcosmo.make_fi(pcm, r_vdw)
ui = 1 - fi
ui[ui < 0] = 0
numpy.random.seed(1)
nao = mol.nao_nr()
dm = numpy.random.random((nao, nao))
dm = dm + dm.T
v_phi = make_v_phi(mol, dm, r_vdw, pcm.lebedev_order)
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(
pcm.lebedev_order)
ylm_1sph = numpy.vstack(sph.real_sph_vec(coords_1sph, pcm.lmax, True))
phi = -numpy.einsum('n,xn,jn,jn->jx', weights_1sph, ylm_1sph, ui,
v_phi)
phi1 = ddcosmo.make_phi(pcm, dm, r_vdw, ui, ylm_1sph)
self.assertTrue(abs(phi - phi1).max() < 1e-12)
def make_vmat(pcm, r_vdw, lebedev_order, lmax, LX, LS):
mol = pcm.mol
grids = dft.gen_grid.Grids(mol)
atom_grids_tab = grids.gen_atomic_grids(mol)
grids.build()
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(lebedev_order)
ao = dft.numint.eval_ao(mol, grids.coords)
nao = ao.shape[1]
vmat = numpy.zeros((nao, nao))
i1 = 0
for ia in range(mol.natm):
xnj, w = atom_grids_tab[mol.atom_symbol(ia)]
i0, i1 = i1, i1 + w.size
r = lib.norm(xnj, axis=1)
Ys = sph.real_sph_vec(xnj / r.reshape(-1, 1), lmax, True)
p1 = 0
for l in range(lmax + 1):
fac = 4 * numpy.pi / (l * 2 + 1)
p0, p1 = p1, p1 + (l * 2 + 1)
rr = numpy.zeros_like(r)
rr[r <= r_vdw[ia]] = r[r <= r_vdw[ia]]**l / r_vdw[ia]**(l + 1)
rr[r > r_vdw[ia]] = r_vdw[ia]**l / r[r > r_vdw[ia]]**(l + 1)
eta_nj = fac * numpy.einsum('n,mn,m->n', rr, Ys[l], LX[ia, p0:p1])
vmat -= numpy.einsum('n,np,nq->pq', grids.weights[i0:i1] * eta_nj,
ao[i0:i1], ao[i0:i1])
atom_coords = mol.atom_coords()
Ylm_sphere = numpy.vstack(sph.real_sph_vec(coords_1sph, lmax, True))
fi = ddcosmo.make_fi(pcm, r_vdw)
ui = 1 - fi
ui[ui < 0] = 0
xi_nj = numpy.einsum('n,jn,xn,jx->jn', weights_1sph, ui, Ylm_sphere, LS)
pmol = mol.copy()
for ia in range(mol.natm):
for i, c in enumerate(coords_1sph):
r = atom_coords[ia] + r_vdw[ia] * c
pmol.set_rinv_orig(r)
vmat += pmol.intor('int1e_rinv') * xi_nj[ia, i]
return vmat
def make_vmat(pcm, r_vdw, lebedev_order, lmax, LX, LS):
mol = pcm.mol
grids = dft.gen_grid.Grids(mol)
atom_grids_tab = grids.gen_atomic_grids(mol)
grids.build()
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(lebedev_order)
ao = dft.numint.eval_ao(mol, grids.coords)
nao = ao.shape[1]
vmat = numpy.zeros((nao,nao))
i1 = 0
for ia in range(mol.natm):
xnj, w = atom_grids_tab[mol.atom_symbol(ia)]
i0, i1 = i1, i1 + w.size
r = lib.norm(xnj, axis=1)
Ys = sph.real_sph_vec(xnj/r.reshape(-1,1), lmax, True)
p1 = 0
for l in range(lmax+1):
fac = 4*numpy.pi/(l*2+1)
p0, p1 = p1, p1 + (l*2+1)
rr = numpy.zeros_like(r)
rr[r<=r_vdw[ia]] = r[r<=r_vdw[ia]]**l / r_vdw[ia]**(l+1)
rr[r> r_vdw[ia]] = r_vdw[ia]**l / r[r>r_vdw[ia]]**(l+1)
eta_nj = fac * numpy.einsum('n,mn,m->n', rr, Ys[l], LX[ia,p0:p1])
vmat -= numpy.einsum('n,np,nq->pq', grids.weights[i0:i1] * eta_nj,
ao[i0:i1], ao[i0:i1])
atom_coords = mol.atom_coords()
Ylm_sphere = numpy.vstack(sph.real_sph_vec(coords_1sph, lmax, True))
fi = ddcosmo.make_fi(pcm, r_vdw)
ui = 1 - fi
ui[ui<0] = 0
xi_nj = numpy.einsum('n,jn,xn,jx->jn', weights_1sph, ui, Ylm_sphere, LS)
pmol = mol.copy()
for ia in range(mol.natm):
for i,c in enumerate(coords_1sph):
r = atom_coords[ia] + r_vdw[ia] * c
pmol.set_rinv_orig(r)
vmat += pmol.intor('int1e_rinv') * xi_nj[ia,i]
return vmat
def make_L(pcmobj, r_vdw, lebedev_order, lmax, eta=0.1):
mol = pcmobj.mol
natm = mol.natm
nlm = (lmax + 1)**2
leb_coords, leb_weights = ddcosmo.make_grids_one_sphere(lebedev_order)
nleb_grid = leb_weights.size
atom_coords = mol.atom_coords()
Ylm_sphere = numpy.vstack(sph.real_sph_vec(leb_coords, lmax, True))
fi = ddcosmo.make_fi(pcmobj, r_vdw)
L_diag = numpy.zeros((natm, nlm))
p1 = 0
for l in range(lmax + 1):
p0, p1 = p1, p1 + (l * 2 + 1)
L_diag[:, p0:p1] = 4 * numpy.pi / (l * 2 + 1)
L_diag /= r_vdw.reshape(-1, 1)
L = numpy.diag(L_diag.ravel()).reshape(natm, nlm, natm, nlm)
for ja in range(natm):
for ka in range(natm):
if ja == ka:
continue
vjk = r_vdw[ja] * leb_coords + atom_coords[ja] - atom_coords[ka]
v = lib.norm(vjk, axis=1)
tjk = v / r_vdw[ka]
sjk = vjk / v.reshape(-1, 1)
Ys = sph.real_sph_vec(sjk, lmax, True)
# scale the weight, see JCTC 9, 3637, Eq (16)
wjk = pcmobj.regularize_xt(tjk, eta, r_vdw[ka])
wjk[fi[ja] > 1] /= fi[ja, fi[ja] > 1]
tt = numpy.ones_like(wjk)
p1 = 0
for l in range(lmax + 1):
fac = 4 * numpy.pi / (l * 2 + 1) / r_vdw[ka]
p0, p1 = p1, p1 + (l * 2 + 1)
val = numpy.einsum('n,xn,n,mn->xm', leb_weights, Ylm_sphere,
wjk * tt, Ys[l])
L[ja, :, ka, p0:p1] += -fac * val
tt *= tjk
return L.reshape(natm * nlm, natm * nlm)
def make_fi1(pcmobj, r_vdw):
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(pcmobj.lebedev_order)
mol = pcmobj.mol
eta = pcmobj.eta
natm = mol.natm
atom_coords = mol.atom_coords()
ngrid_1sph = coords_1sph.shape[0]
fi1 = numpy.zeros((natm,3,natm,ngrid_1sph))
for ia in range(natm):
for ja in ddcosmo.atoms_with_vdw_overlap(ia, atom_coords, r_vdw):
v = r_vdw[ia]*coords_1sph + atom_coords[ia] - atom_coords[ja]
rv = lib.norm(v, axis=1)
t = rv / r_vdw[ja]
xt1 = regularize_xt1(t, eta*r_vdw[ja])
s_ij = v.T / rv
xt1 = 1./r_vdw[ja] * xt1 * s_ij
fi1[ia,:,ia] += xt1
fi1[ja,:,ia] -= xt1
fi = ddcosmo.make_fi(pcmobj, r_vdw)
fi1[:,:,fi<1e-20] = 0
return fi1
def make_L(pcmobj, r_vdw, lebedev_order, lmax, eta=0.1):
mol = pcmobj.mol
natm = mol.natm
nlm = (lmax+1)**2
leb_coords, leb_weights = ddcosmo.make_grids_one_sphere(lebedev_order)
nleb_grid = leb_weights.size
atom_coords = mol.atom_coords()
Ylm_sphere = numpy.vstack(sph.real_sph_vec(leb_coords, lmax, True))
fi = ddcosmo.make_fi(pcmobj, r_vdw)
L_diag = numpy.zeros((natm,nlm))
p1 = 0
for l in range(lmax+1):
p0, p1 = p1, p1 + (l*2+1)
L_diag[:,p0:p1] = 4*numpy.pi/(l*2+1)
L_diag /= r_vdw.reshape(-1,1)
L = numpy.diag(L_diag.ravel()).reshape(natm,nlm,natm,nlm)
for ja in range(natm):
for ka in range(natm):
if ja == ka:
continue
vjk = r_vdw[ja] * leb_coords + atom_coords[ja] - atom_coords[ka]
v = lib.norm(vjk, axis=1)
tjk = v / r_vdw[ka]
sjk = vjk / v.reshape(-1,1)
Ys = sph.real_sph_vec(sjk, lmax, True)
# scale the weight, see JCTC 9, 3637, Eq (16)
wjk = pcmobj.regularize_xt(tjk, eta, r_vdw[ka])
wjk[fi[ja]>1] /= fi[ja,fi[ja]>1]
tt = numpy.ones_like(wjk)
p1 = 0
for l in range(lmax+1):
fac = 4*numpy.pi/(l*2+1) / r_vdw[ka]
p0, p1 = p1, p1 + (l*2+1)
val = numpy.einsum('n,xn,n,mn->xm', leb_weights, Ylm_sphere, wjk*tt, Ys[l])
L[ja,:,ka,p0:p1] += -fac * val
tt *= tjk
return L.reshape(natm*nlm,natm*nlm)
def make_fi1(pcmobj, r_vdw):
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(pcmobj.lebedev_order)
mol = pcmobj.mol
eta = pcmobj.eta
natm = mol.natm
atom_coords = mol.atom_coords()
ngrid_1sph = coords_1sph.shape[0]
fi1 = numpy.zeros((natm,3,natm,ngrid_1sph))
for ia in range(natm):
for ja in ddcosmo.atoms_with_vdw_overlap(ia, atom_coords, r_vdw):
v = r_vdw[ia]*coords_1sph + atom_coords[ia] - atom_coords[ja]
rv = lib.norm(v, axis=1)
t = rv / r_vdw[ja]
xt1 = regularize_xt1(t, eta*r_vdw[ja])
s_ij = v.T / rv
xt1 = 1./r_vdw[ja] * xt1 * s_ij
fi1[ia,:,ia] += xt1
fi1[ja,:,ia] -= xt1
fi = ddcosmo.make_fi(pcmobj, r_vdw)
fi1[:,:,fi<1e-20] = 0
return fi1
def B1_dot_x(pcmobj, dm, r_vdw, ui, ylm_1sph, cached_pol, L):
mol = pcmobj.mol
mol = pcmobj.mol
natm = mol.natm
nao = mol.nao
lmax = pcmobj.lmax
nlm = (lmax + 1)**2
dms = numpy.asarray(dm)
is_single_dm = dms.ndim == 2
dms = dms.reshape(-1, nao, nao)
n_dm = dms.shape[0]
atom_coords = mol.atom_coords()
atom_charges = mol.atom_charges()
aoslices = mol.aoslice_by_atom()
grids = pcmobj.grids
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(
pcmobj.lebedev_order)
ngrid_1sph = coords_1sph.shape[0]
extern_point_idx = ui > 0
fi0 = ddcosmo.make_fi(pcmobj, r_vdw)
fi1 = ddcosmo_grad.make_fi1(pcmobj, pcmobj.get_atomic_radii())
fi1[:, :, ui == 0] = 0
ui1 = -fi1
Bx = numpy.zeros((natm, 3, nao, nao))
ao = mol.eval_gto('GTOval', grids.coords)
aow = numpy.einsum('gi,g->gi', ao, grids.weights)
aopair = numpy.einsum('gi,gj->gij', ao, aow)
den = numpy.einsum('gij,ij->g', aopair, dm)
psi0 = numpy.zeros((natm, nlm))
i1 = 0
for ia in range(natm):
fak_pol, leak_idx = cached_pol[mol.atom_symbol(ia)]
fac_pol = ddcosmo._vstack_factor_fak_pol(fak_pol, lmax)
i0, i1 = i1, i1 + fac_pol.shape[1]
psi0[ia] = -numpy.einsum('mn,n->m', fac_pol, den[i0:i1])
LS0 = numpy.linalg.solve(L.reshape(natm * nlm, -1).T, psi0.ravel())
LS0 = LS0.reshape(natm, nlm)
phi0 = numpy.zeros((natm, nlm))
phi1 = numpy.zeros((natm, 3, natm, nlm))
int3c2e = mol._add_suffix('int3c2e')
int3c2e_ip1 = mol._add_suffix('int3c2e_ip1')
cintopt = gto.moleintor.make_cintopt(mol._atm, mol._bas, mol._env, int3c2e)
cintopt_ip1 = gto.moleintor.make_cintopt(mol._atm, mol._bas, mol._env,
int3c2e_ip1)
for ia in range(natm):
cav_coords = atom_coords[ia] + r_vdw[ia] * coords_1sph
#fakemol = gto.fakemol_for_charges(cav_coords[ui[ia]>0])
fakemol = gto.fakemol_for_charges(cav_coords)
v_nj = df.incore.aux_e2(mol,
fakemol,
intor=int3c2e,
aosym='s1',
cintopt=cintopt)
v_phi = numpy.einsum('pqg,pq->g', v_nj, dm)
phi0[ia] = numpy.einsum('n,ln,n,n->l', weights_1sph, ylm_1sph, ui[ia],
v_phi)
phi1[:, :, ia] += lib.einsum('n,ln,azn,n->azl', weights_1sph, ylm_1sph,
ui1[:, :, ia], v_phi)
Bx += lib.einsum('l,n,ln,azn,ijn->azij', LS0[ia], weights_1sph,
ylm_1sph, ui1[:, :, ia], v_nj)
wtmp = lib.einsum('n,ln,n->ln', weights_1sph, ylm_1sph, ui[ia])
v_e1_nj = df.incore.aux_e2(mol,
fakemol,
intor=int3c2e_ip1,
comp=3,
aosym='s1',
cintopt=cintopt_ip1)
vtmp = lib.einsum('l,ln,xijn->xij', LS0[ia], wtmp, v_e1_nj)
Bx[ia] += vtmp
Bx[ia] += vtmp.transpose(0, 2, 1)
for ja in range(natm):
shl0, shl1, p0, p1 = aoslices[ja]
Bx[ja, :, p0:p1, :] -= vtmp[:, p0:p1]
Bx[ja, :, :, p0:p1] -= vtmp[:, p0:p1].transpose(0, 2, 1)
tmp = numpy.einsum('xijn,ij->xn', v_e1_nj[:, p0:p1], dm[p0:p1])
tmp += numpy.einsum('xijn,ji->xn', v_e1_nj[:, p0:p1], dm[:, p0:p1])
phitmp = numpy.einsum('ln,xn->xl', wtmp, tmp)
phi1[ja, :, ia] -= phitmp
phi1[ia, :, ia] += phitmp
Xvec0 = numpy.linalg.solve(L.reshape(natm * nlm, -1), phi0.ravel())
Xvec0 = Xvec0.reshape(natm, nlm)
L1 = ddcosmo_grad.make_L1(pcmobj, r_vdw, ylm_1sph, fi0)
phi1 -= lib.einsum('aziljm,jm->azil', L1, Xvec0)
Xvec1 = numpy.linalg.solve(L.reshape(natm * nlm, -1),
phi1.reshape(-1, natm * nlm).T)
Xvec1 = Xvec1.T.reshape(natm, 3, natm, nlm)
psi1 = numpy.zeros((natm, 3, natm, nlm))
i1 = 0
for ia, (coords, weight,
weight1) in enumerate(rks_grad.grids_response_cc(grids)):
i0, i1 = i1, i1 + weight.size
ao = mol.eval_gto('GTOval_sph_deriv1', coords)
aow = numpy.einsum('gi,g->gi', ao[0], weight)
aopair1 = lib.einsum('xgi,gj->xgij', ao[1:], aow)
aow = numpy.einsum('gi,zxg->zxgi', ao[0], weight1)
aopair0 = lib.einsum('zxgi,gj->zxgij', aow, ao[0])
den0 = numpy.einsum('zxgij,ij->zxg', aopair0, dm)
den1 = numpy.empty((natm, 3, weight.size))
for ja in range(natm):
shl0, shl1, p0, p1 = aoslices[ja]
den1[ja] = numpy.einsum('xgij,ij->xg', aopair1[:, :, p0:p1],
dm[p0:p1, :])
den1[ja] += numpy.einsum('xgij,ji->xg', aopair1[:, :, p0:p1],
dm[:, p0:p1])
fak_pol, leak_idx = cached_pol[mol.atom_symbol(ia)]
fac_pol = ddcosmo._vstack_factor_fak_pol(fak_pol, lmax)
scaled_weights = lib.einsum('azm,mn->azn', Xvec1[:, :, ia], fac_pol)
scaled_weights *= weight
aow = numpy.einsum('gi,azg->azgi', ao[0], scaled_weights)
Bx -= numpy.einsum('gi,azgj->azij', ao[0], aow)
tmp = numpy.einsum('mn,zxn->zxm', fac_pol, den1)
psi1[:, :, ia] -= numpy.einsum('mn,zxn->zxm', fac_pol, den0)
psi1[ia, :, ia] -= tmp.sum(axis=0)
for ja in range(natm):
psi1[ja, :, ia] += tmp[ja]
eta_nj = lib.einsum('mn,m->n', fac_pol, Xvec0[ia])
Bx -= lib.einsum('n,zxnpq->zxpq', eta_nj, aopair0)
vtmp = lib.einsum('n,xnpq->xpq', eta_nj, aopair1)
Bx[ia] -= vtmp
Bx[ia] -= vtmp.transpose(0, 2, 1)
for ja in range(natm):
shl0, shl1, q0, q1 = aoslices[ja]
Bx[ja, :, q0:q1, :] += vtmp[:, q0:q1]
Bx[ja, :, :, q0:q1] += vtmp[:, q0:q1].transpose(0, 2, 1)
psi1 -= numpy.einsum('il,aziljm->azjm', LS0, L1)
LS1 = numpy.linalg.solve(
L.reshape(natm * nlm, -1).T,
psi1.reshape(-1, natm * nlm).T)
LS1 = LS1.T.reshape(natm, 3, natm, nlm)
cav_coords = (atom_coords.reshape(natm, 1, 3) +
numpy.einsum('r,gx->rgx', r_vdw, coords_1sph))
cav_coords = cav_coords[extern_point_idx]
fakemol = gto.fakemol_for_charges(cav_coords)
v_nj = df.incore.aux_e2(mol,
fakemol,
intor=int3c2e,
aosym='s1',
cintopt=cintopt)
v_phi = numpy.zeros((natm, ngrid_1sph, nao, nao))
v_phi[extern_point_idx] += v_nj.transpose(2, 0, 1)
Bx += lib.einsum('azjx,n,xn,jn,jnpq->azpq', LS1, weights_1sph, ylm_1sph,
ui, v_phi)
return Bx
def make_B1(pcmobj, r_vdw, ui, ylm_1sph, cached_pol, L):
'''1st order'''
mol = pcmobj.mol
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(
pcmobj.lebedev_order)
ngrid_1sph = coords_1sph.shape[0]
mol = pcmobj.mol
natm = mol.natm
nao = mol.nao
lmax = pcmobj.lmax
nlm = (lmax + 1)**2
atom_coords = mol.atom_coords()
atom_charges = mol.atom_charges()
grids = pcmobj.grids
extern_point_idx = ui > 0
cav_coords = (atom_coords.reshape(natm, 1, 3) +
numpy.einsum('r,gx->rgx', r_vdw, coords_1sph))
cav_coords = cav_coords[extern_point_idx]
int3c2e = mol._add_suffix('int3c2e')
cintopt = gto.moleintor.make_cintopt(mol._atm, mol._bas, mol._env, int3c2e)
fakemol = gto.fakemol_for_charges(cav_coords)
v_nj = df.incore.aux_e2(mol,
fakemol,
intor=int3c2e,
aosym='s1',
cintopt=cintopt)
nao_pair = v_nj.shape[0]
v_phi = numpy.zeros((natm, ngrid_1sph, nao, nao))
v_phi[extern_point_idx] += v_nj.transpose(2, 0, 1)
phi0 = numpy.einsum('n,xn,jn,jnpq->jxpq', weights_1sph, ylm_1sph, ui,
v_phi)
Xvec0 = numpy.linalg.solve(L.reshape(natm * nlm, -1),
phi0.reshape(natm * nlm, -1))
Xvec0 = Xvec0.reshape(natm, nlm, nao, nao)
ao = mol.eval_gto('GTOval', grids.coords)
aow = numpy.einsum('gi,g->gi', ao, grids.weights)
aopair = numpy.einsum('gi,gj->gij', ao, aow)
psi0 = numpy.zeros((natm, nlm, nao, nao))
i1 = 0
for ia in range(natm):
fak_pol, leak_idx = cached_pol[mol.atom_symbol(ia)]
i0, i1 = i1, i1 + fak_pol[0].shape[1]
p1 = 0
for l in range(lmax + 1):
fac = 4 * numpy.pi / (l * 2 + 1)
p0, p1 = p1, p1 + (l * 2 + 1)
psi0[ia, p0:p1] = -fac * numpy.einsum('mn,nij->mij', fak_pol[l],
aopair[i0:i1])
fi0 = ddcosmo.make_fi(pcmobj, r_vdw)
fi1 = ddcosmo_grad.make_fi1(pcmobj, pcmobj.get_atomic_radii())
fi1[:, :, ui == 0] = 0
ui1 = -fi1
phi1 = numpy.zeros(ui1.shape[:3] + (nlm, nao, nao))
int3c2e = mol._add_suffix('int3c2e')
int3c2e_ip1 = mol._add_suffix('int3c2e_ip1')
aoslices = mol.aoslice_by_atom()
for ia in range(natm):
cav_coords = atom_coords[ia] + r_vdw[ia] * coords_1sph
#fakemol = gto.fakemol_for_charges(cav_coords[ui[ia]>0])
fakemol = gto.fakemol_for_charges(cav_coords)
v_nj = df.incore.aux_e2(mol, fakemol, intor=int3c2e, aosym='s1')
phi1[:, :, ia] += lib.einsum('n,ln,azn,ijn->azlij', weights_1sph,
ylm_1sph, ui1[:, :, ia], v_nj)
v_e1_nj = df.incore.aux_e2(mol,
fakemol,
intor=int3c2e_ip1,
comp=3,
aosym='s1')
v_e2_nj = v_e1_nj + v_e1_nj.transpose(0, 2, 1, 3)
phi1[ia, :, ia] += lib.einsum('n,ln,n,xijn->xlij', weights_1sph,
ylm_1sph, ui[ia], v_e2_nj)
for ja in range(natm):
shl0, shl1, p0, p1 = aoslices[ja]
v = numpy.einsum('n,ln,n,xijn->xlij', weights_1sph, ylm_1sph,
ui[ia], v_e1_nj[:, p0:p1])
phi1[ja, :, ia, :, p0:p1, :] -= v
phi1[ja, :, ia, :, :, p0:p1] -= v.transpose(0, 1, 3, 2)
psi1 = numpy.zeros((natm, 3, natm, nlm, nao, nao))
i1 = 0
for ia, (coords, weight,
weight1) in enumerate(rks_grad.grids_response_cc(grids)):
i0, i1 = i1, i1 + weight.size
ao = mol.eval_gto('GTOval_sph_deriv1', coords)
aow = numpy.einsum('gi,g->gi', ao[0], weight)
aopair1 = lib.einsum('xgi,gj->xgij', ao[1:], aow)
aow = numpy.einsum('gi,zxg->zxgi', ao[0], weight1)
aopair0 = lib.einsum('zxgi,gj->zxgij', aow, ao[0])
fak_pol, leak_idx = cached_pol[mol.atom_symbol(ia)]
p1 = 0
for l in range(lmax + 1):
fac = 4 * numpy.pi / (l * 2 + 1)
p0, p1 = p1, p1 + (l * 2 + 1)
psi1[:, :, ia, p0:p1] -= fac * numpy.einsum(
'mn,zxnij->zxmij', fak_pol[l], aopair0)
vtmp = fac * numpy.einsum('mn,xnij->xmij', fak_pol[l], aopair1)
psi1[ia, :, ia, p0:p1] -= vtmp
psi1[ia, :, ia, p0:p1] -= vtmp.transpose(0, 1, 3, 2)
for ja in range(natm):
shl0, shl1, q0, q1 = aoslices[ja]
psi1[ja, :, ia, p0:p1, q0:q1, :] += vtmp[:, :, q0:q1]
psi1[ja, :, ia, p0:p1, :,
q0:q1] += vtmp[:, :, q0:q1].transpose(0, 1, 3, 2)
L1 = ddcosmo_grad.make_L1(pcmobj, r_vdw, ylm_1sph, fi0)
Xvec1 = numpy.linalg.solve(
L.reshape(natm * nlm, -1),
phi1.transpose(2, 3, 0, 1, 4, 5).reshape(natm * nlm, -1))
Xvec1 = Xvec1.reshape(natm, nlm, natm, 3, nao,
nao).transpose(2, 3, 0, 1, 4, 5)
LS0 = numpy.linalg.solve(
L.reshape(natm * nlm, -1).T, psi0.reshape(natm * nlm, -1))
LS0 = LS0.reshape(natm, nlm, nao, nao)
B = lib.einsum('ixnlpq,nlrs->ixpqrs', psi1, Xvec0)
B += lib.einsum('nlpq,ixnlrs->ixpqrs', psi0, Xvec1)
B -= lib.einsum('ilpq,aziljm,jmrs->azpqrs', LS0, L1, Xvec0)
B = B + B.transpose(0, 1, 4, 5, 2, 3)
return B
def nuc_part1(pcmobj, r_vdw, ui, ylm_1sph, cached_pol, L):
'''1st order'''
mol = pcmobj.mol
natm = mol.natm
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(
pcmobj.lebedev_order)
ngrid_1sph = coords_1sph.shape[0]
lmax = pcmobj.lmax
nlm = (lmax + 1)**2
nao = mol.nao
atom_coords = mol.atom_coords()
atom_charges = mol.atom_charges()
grids = pcmobj.grids
aoslices = mol.aoslice_by_atom()
extern_point_idx = ui > 0
fi0 = ddcosmo.make_fi(pcmobj, r_vdw)
fi1 = ddcosmo_grad.make_fi1(pcmobj, pcmobj.get_atomic_radii())
fi1[:, :, ui == 0] = 0
ui1 = -fi1
vmat1 = numpy.zeros((natm, 3, nao, nao))
cav_coords = (atom_coords.reshape(natm, 1, 3) +
numpy.einsum('r,gx->rgx', r_vdw, coords_1sph))
v_phi = numpy.zeros((natm, ngrid_1sph))
for ia in range(natm):
# Note (-) sign is not applied to atom_charges, because (-) is explicitly
# included in rhs and L matrix
d_rs = atom_coords.reshape(-1, 1, 3) - cav_coords[ia]
v_phi[ia] = numpy.einsum('z,zp->p', atom_charges,
1. / lib.norm(d_rs, axis=2))
phi0 = -numpy.einsum('n,xn,jn,jn->jx', weights_1sph, ylm_1sph, ui, v_phi)
Xvec0 = numpy.linalg.solve(L.reshape(natm * nlm, -1), phi0.ravel())
Xvec0 = Xvec0.reshape(natm, nlm)
ngrid_1sph = weights_1sph.size
v_phi0 = numpy.empty((natm, ngrid_1sph))
for ia in range(natm):
cav_coords = atom_coords[ia] + r_vdw[ia] * coords_1sph
d_rs = atom_coords.reshape(-1, 1, 3) - cav_coords
v_phi0[ia] = numpy.einsum('z,zp->p', atom_charges,
1. / lib.norm(d_rs, axis=2))
phi1 = -numpy.einsum('n,ln,azjn,jn->azjl', weights_1sph, ylm_1sph, ui1,
v_phi0)
for ia in range(natm):
cav_coords = atom_coords[ia] + r_vdw[ia] * coords_1sph
for ja in range(natm):
rs = atom_coords[ja] - cav_coords
d_rs = lib.norm(rs, axis=1)
v_phi = atom_charges[ja] * numpy.einsum('px,p->px', rs,
1. / d_rs**3)
tmp = numpy.einsum('n,ln,n,nx->xl', weights_1sph, ylm_1sph, ui[ia],
v_phi)
phi1[ja, :, ia] += tmp # response of the other atoms
phi1[ia, :, ia] -= tmp # response of cavity grids
L1 = ddcosmo_grad.make_L1(pcmobj, r_vdw, ylm_1sph, fi0)
phi1 -= lib.einsum('aziljm,jm->azil', L1, Xvec0)
Xvec1 = numpy.linalg.solve(L.reshape(natm * nlm, -1),
phi1.reshape(-1, natm * nlm).T)
Xvec1 = Xvec1.T.reshape(natm, 3, natm, nlm)
i1 = 0
for ia, (coords, weight,
weight1) in enumerate(rks_grad.grids_response_cc(grids)):
i0, i1 = i1, i1 + weight.size
ao = mol.eval_gto('GTOval_sph_deriv1', coords)
aow = numpy.einsum('gi,g->gi', ao[0], weight)
aopair1 = lib.einsum('xgi,gj->xgij', ao[1:], aow)
aow = numpy.einsum('gi,zxg->zxgi', ao[0], weight1)
aopair0 = lib.einsum('zxgi,gj->zxgij', aow, ao[0])
fak_pol, leak_idx = cached_pol[mol.atom_symbol(ia)]
fac_pol = ddcosmo._vstack_factor_fak_pol(fak_pol, lmax)
vmat1 -= numpy.einsum('m,mn,zxnij->zxij', Xvec0[ia], fac_pol, aopair0)
vtmp = numpy.einsum('m,mn,xnij->xij', Xvec0[ia], fac_pol, aopair1)
vmat1[ia, :] -= vtmp
vmat1[ia, :] -= vtmp.transpose(0, 2, 1)
for ja in range(natm):
shl0, shl1, p0, p1 = aoslices[ja]
vmat1[ja, :, p0:p1, :] += vtmp[:, p0:p1]
vmat1[ja, :, :, p0:p1] += vtmp[:, p0:p1].transpose(0, 2, 1)
scaled_weights = lib.einsum('azm,mn->azn', Xvec1[:, :, ia], fac_pol)
scaled_weights *= weight
aow = numpy.einsum('gi,azg->azgi', ao[0], scaled_weights)
vmat1 -= numpy.einsum('gi,azgj->azij', ao[0], aow)
psi0 = numpy.zeros((natm, nlm))
for ia in range(natm):
psi0[ia,
0] += numpy.sqrt(4 * numpy.pi) / r_vdw[ia] * mol.atom_charge(ia)
LS0 = numpy.linalg.solve(L.reshape(natm * nlm, -1).T, psi0.ravel())
LS0 = LS0.reshape(natm, nlm)
LS1 = numpy.einsum('il,aziljm->azjm', LS0, L1)
LS1 = numpy.linalg.solve(
L.reshape(natm * nlm, -1).T,
LS1.reshape(-1, natm * nlm).T)
LS1 = LS1.T.reshape(natm, 3, natm, nlm)
int3c2e = mol._add_suffix('int3c2e')
int3c2e_ip1 = mol._add_suffix('int3c2e_ip1')
cintopt = gto.moleintor.make_cintopt(mol._atm, mol._bas, mol._env, int3c2e)
for ia in range(natm):
cav_coords = atom_coords[ia] + r_vdw[ia] * coords_1sph
#fakemol = gto.fakemol_for_charges(cav_coords[ui[ia]>0])
fakemol = gto.fakemol_for_charges(cav_coords)
wtmp = lib.einsum('l,n,ln->ln', LS0[ia], weights_1sph, ylm_1sph)
v_nj = df.incore.aux_e2(mol,
fakemol,
intor=int3c2e,
aosym='s1',
cintopt=cintopt)
vmat1 -= numpy.einsum('azl,n,ln,n,pqn->azpq', LS1[:, :, ia],
weights_1sph, ylm_1sph, ui[ia], v_nj)
vmat1 += lib.einsum('ln,azn,ijn->azij', wtmp, ui1[:, :, ia], v_nj)
v_e1_nj = df.incore.aux_e2(mol,
fakemol,
intor=int3c2e_ip1,
comp=3,
aosym='s1')
vtmp = lib.einsum('ln,n,xijn->xij', wtmp, ui[ia], v_e1_nj)
vmat1[ia] += vtmp
vmat1[ia] += vtmp.transpose(0, 2, 1)
for ja in range(natm):
shl0, shl1, p0, p1 = aoslices[ja]
vmat1[ja, :, p0:p1, :] -= vtmp[:, p0:p1]
vmat1[ja, :, :, p0:p1] -= vtmp[:, p0:p1].transpose(0, 2, 1)
return vmat1
def tda_grad(td, z):
'''ddcosmo TDA gradients'''
mol = td.mol
mf = td._scf
mo_coeff = mf.mo_coeff
mo_energy = mf.mo_energy
mo_occ = mf.mo_occ
nao, nmo = mo_coeff.shape
nocc = (mo_occ > 0).sum()
nvir = nmo - nocc
z = z[0].reshape(nocc, nvir).T * numpy.sqrt(2)
orbv = mo_coeff[:, nocc:]
orbo = mo_coeff[:, :nocc]
r_vdw = ddcosmo.get_atomic_radii(td.with_solvent)
fi = ddcosmo.make_fi(td.with_solvent, r_vdw)
ui = 1 - fi
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(
td.with_solvent.lebedev_order)
ylm_1sph = numpy.vstack(
sph.real_sph_vec(coords_1sph, td.with_solvent.lmax, True))
grids = td.with_solvent.grids
cached_pol = ddcosmo.cache_fake_multipoles(grids, r_vdw,
td.with_solvent.lmax)
L = ddcosmo.make_L(td.with_solvent, r_vdw, ylm_1sph, fi)
def fvind(x):
v_mo = numpy.einsum('iabj,xai->xbj', g[:nocc, nocc:, nocc:, :nocc], x)
v_mo += numpy.einsum('aibj,xai->xbj', g[nocc:, :nocc, nocc:, :nocc], x)
return v_mo
h1 = rhf_grad.get_hcore(mol)
s1 = rhf_grad.get_ovlp(mol)
eri1 = -mol.intor('int2e_ip1', aosym='s1', comp=3)
eri1 = eri1.reshape(3, nao, nao, nao, nao)
eri0 = ao2mo.kernel(mol, mo_coeff)
eri0 = ao2mo.restore(1, eri0, nmo).reshape(nmo, nmo, nmo, nmo)
g = eri0 * 2 - eri0.transpose(0, 3, 2, 1)
zeta = lib.direct_sum('i+j->ij', mo_energy, mo_energy) * .5
zeta[nocc:, :nocc] = mo_energy[:nocc]
zeta[:nocc, nocc:] = mo_energy[nocc:]
dielectric = td.with_solvent.eps
if dielectric > 0:
f_epsilon = (dielectric - 1.) / dielectric
else:
f_epsilon = 1
pcm_nuc = .5 * f_epsilon * nuc_part1(td.with_solvent, r_vdw, ui, ylm_1sph,
cached_pol, L)
B0 = .5 * f_epsilon * make_B(td.with_solvent, r_vdw, ui, ylm_1sph,
cached_pol, L)
B0 = lib.einsum('pqrs,pi,qj,rk,sl->ijkl', B0, mo_coeff, mo_coeff, mo_coeff,
mo_coeff)
g += B0 * 2
B1 = .5 * f_epsilon * make_B1(td.with_solvent, r_vdw, ui, ylm_1sph,
cached_pol, L)
offsetdic = mol.offset_nr_by_atom()
de = numpy.zeros((mol.natm, 3))
for ia in range(mol.natm):
shl0, shl1, p0, p1 = offsetdic[ia]
mol.set_rinv_origin(mol.atom_coord(ia))
h1ao = -mol.atom_charge(ia) * mol.intor('int1e_iprinv', comp=3)
h1ao[:, p0:p1] += h1[:, p0:p1]
h1ao = h1ao + h1ao.transpose(0, 2, 1)
h1ao += pcm_nuc[ia]
h1mo = numpy.einsum('pi,xpq,qj->xij', mo_coeff, h1ao, mo_coeff)
s1mo = numpy.einsum('pi,xpq,qj->xij', mo_coeff[p0:p1], s1[:, p0:p1],
mo_coeff)
s1mo = s1mo + s1mo.transpose(0, 2, 1)
f1 = h1mo - numpy.einsum('xpq,pq->xpq', s1mo, zeta)
f1 -= numpy.einsum('klpq,xlk->xpq', g[:nocc, :nocc],
s1mo[:, :nocc, :nocc])
eri1a = eri1.copy()
eri1a[:, :p0] = 0
eri1a[:, p1:] = 0
eri1a = eri1a + eri1a.transpose(0, 2, 1, 3, 4)
eri1a = eri1a + eri1a.transpose(0, 3, 4, 1, 2)
g1 = lib.einsum('xpqrs,pi,qj,rk,sl->xijkl', eri1a, mo_coeff, mo_coeff,
mo_coeff, mo_coeff)
tmp1 = lib.einsum('xpqrs,pi,qj,rk,sl->xijkl', B1[ia], mo_coeff,
mo_coeff, mo_coeff, mo_coeff)
g1 = g1 * 2 - g1.transpose(0, 1, 4, 3, 2)
g1 += tmp1 * 2
f1 += numpy.einsum('xkkpq->xpq', g1[:, :nocc, :nocc])
f1ai = f1[:, nocc:, :nocc].copy()
c1 = s1mo * -.5
c1vo = cphf.solve(fvind, mo_energy, mo_occ, f1ai, max_cycle=50)[0]
c1[:, nocc:, :nocc] = c1vo
c1[:, :nocc,
nocc:] = -(s1mo[:, nocc:, :nocc] + c1vo).transpose(0, 2, 1)
f1 += numpy.einsum('kapq,xak->xpq', g[:nocc, nocc:], c1vo)
f1 += numpy.einsum('akpq,xak->xpq', g[nocc:, :nocc], c1vo)
e1 = numpy.einsum('xaijb,ai,bj->x', g1[:, nocc:, :nocc, :nocc, nocc:],
z, z)
e1 += numpy.einsum('xab,ai,bi->x', f1[:, nocc:, nocc:], z, z)
e1 -= numpy.einsum('xij,ai,aj->x', f1[:, :nocc, :nocc], z, z)
g1 = numpy.einsum('pjkl,xpi->xijkl', g, c1)
g1 += numpy.einsum('ipkl,xpj->xijkl', g, c1)
g1 += numpy.einsum('ijpl,xpk->xijkl', g, c1)
g1 += numpy.einsum('ijkp,xpl->xijkl', g, c1)
e1 += numpy.einsum('xaijb,ai,bj->x', g1[:, nocc:, :nocc, :nocc, nocc:],
z, z)
de[ia] = e1
return de
def test_B1(self):
def getB(mol):
pcm = ddcosmo.DDCOSMO(mol)
pcm.lmax = 2
pcm.eps = 0
natm = mol.natm
nao = mol.nao
nlm = (pcm.lmax + 1)**2
r_vdw = ddcosmo.get_atomic_radii(pcm)
fi = ddcosmo.make_fi(pcm, r_vdw)
ui = 1 - fi
ui[ui < 0] = 0
pcm.grids = grids = dft.gen_grid.Grids(mol).run(level=0)
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(
pcm.lebedev_order)
ylm_1sph = numpy.vstack(
sph.real_sph_vec(coords_1sph, pcm.lmax, True))
cached_pol = ddcosmo.cache_fake_multipoles(grids, r_vdw, pcm.lmax)
L = ddcosmo.make_L(pcm, r_vdw, ylm_1sph, fi)
return make_B(pcm, r_vdw, ui, ylm_1sph, cached_pol, L)
pcm = ddcosmo.DDCOSMO(mol0)
pcm.lmax = 2
pcm.eps = 0
natm = mol0.natm
nao = mol0.nao
nlm = (pcm.lmax + 1)**2
r_vdw = ddcosmo.get_atomic_radii(pcm)
fi = ddcosmo.make_fi(pcm, r_vdw)
ui = 1 - fi
ui[ui < 0] = 0
pcm.grids = grids = dft.gen_grid.Grids(mol0).run(level=0)
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(
pcm.lebedev_order)
ylm_1sph = numpy.vstack(sph.real_sph_vec(coords_1sph, pcm.lmax, True))
cached_pol = ddcosmo.cache_fake_multipoles(grids, r_vdw, pcm.lmax)
L = ddcosmo.make_L(pcm, r_vdw, ylm_1sph, fi)
dB = make_B1(pcm, r_vdw, ui, ylm_1sph, cached_pol, L)
B1 = getB(mol1)
B2 = getB(mol2)
self.assertAlmostEqual(abs((B2 - B1) / dx - dB[0, 2]).max(), 0, 8)
nao = mol0.nao
numpy.random.seed(1)
dm1 = numpy.random.random((2, nao, nao))
dm2 = numpy.random.random((2, nao, nao))
dm = dm1[0]
ref = numpy.einsum('azpqrs,npq->nazrs', dB, dm1)
v = B1_dot_x(pcm, dm, r_vdw, ui, ylm_1sph, cached_pol, L)
self.assertAlmostEqual(abs(v - ref[0]).max(), 0, 12)
de = _ddcosmo_tdscf_grad._grad_ee(pcm, dm1, dm2, r_vdw, ui, ylm_1sph,
cached_pol, L)
ref = numpy.einsum('nazij,nij->naz', ref, dm2)
self.assertAlmostEqual(abs(de - ref).max(), 0, 12)
numpy.random.seed(1)
dm = numpy.random.random((nao, nao))
dm = dm + dm.T
ref = ddcosmo_grad.kernel(pcm, dm)
dielectric = pcm.eps
if dielectric > 0:
f_epsilon = (dielectric - 1.) / dielectric
else:
f_epsilon = 1
de = _ddcosmo_tdscf_grad._grad_nn(pcm, r_vdw, ui, ylm_1sph, cached_pol,
L)
de += _ddcosmo_tdscf_grad._grad_ne(pcm, dm, r_vdw, ui, ylm_1sph,
cached_pol, L)
de += .5 * _ddcosmo_tdscf_grad._grad_ee(pcm, dm, dm, r_vdw, ui,
ylm_1sph, cached_pol, L)
de *= .5 * f_epsilon
self.assertAlmostEqual(abs(de - ref).max(), 0, 12)
