def dia(magobj, gauge_orig=None):
'''Part of rotational g-tensors. It is the direct second derivatives of
the Lagrangian (corresponding to the zeroth order wavefunction). Unit
hbar/mu_N is not included. This part may be different to the conventional
dia-magnetic contributions of rotational g-tensors.
'''
mol = magobj.mol
im, mass_center = rhf_g.inertia_tensor(mol)
if gauge_orig is None:
# Eq. (35) of JCP, 105, 2804
e2 = uhf_mag.dia(magobj, gauge_orig)
e2 -= uhf_mag.dia(magobj, mass_center)
e2 = rhf_g._safe_solve(im, e2)
return -4 * nist.PROTON_MASS_AU * e2
else:
dm0a, dm0b = magobj._scf.make_rdm1()
dm0 = dm0a + dm0b
with mol.with_common_origin(gauge_orig):
int_r = mol.intor('int1e_r', comp=3)
cm = mass_center - gauge_orig
e2 = numpy.einsum('xpq,qp,y->xy', int_r, dm0, cm)
e2 = numpy.eye(3) * e2.trace() - e2
e2 *= .5
e2 = rhf_g._safe_solve(im, e2)
return -2 * nist.PROTON_MASS_AU * e2
def dia(magobj, gauge_orig=None):
'''Part of rotational g-tensors. It is the direct second derivatives of
the Lagrangian (corresponding to the zeroth order wavefunction). Unit
hbar/mu_N is not included. This part may be different to the conventional
dia-magnetic contributions of rotational g-tensors.
'''
mol = magobj.mol
im, mass_center = rhf_g.inertia_tensor(mol)
if gauge_orig is None:
# Eq. (35) of JCP, 105, 2804
e2 = uhf_mag.dia(magobj, gauge_orig)
e2 -= uhf_mag.dia(magobj, mass_center)
e2 = rhf_g._safe_solve(im, e2)
return -4 * nist.PROTON_MASS_AU * e2
else:
dm0a, dm0b = magobj._scf.make_rdm1()
dm0 = dm0a + dm0b
with mol.with_common_origin(gauge_orig):
int_r = mol.intor('int1e_r', comp=3)
cm = mass_center - gauge_orig
e2 = numpy.einsum('xpq,qp,y->xy', int_r, dm0, cm)
e2 = numpy.eye(3) * e2.trace() - e2
e2 *= .5
e2 = rhf_g._safe_solve(im, e2)
return -2 * nist.PROTON_MASS_AU * e2
def para(nsrobj, mo10=None, mo_coeff=None, mo_occ=None, shielding_nuc=None):
'''Paramagnetic part of NSR shielding tensors.
'''
if shielding_nuc is None: shielding_nuc = nsrobj.shielding_nuc
# The first order Hamiltonian for rotation part is the same to the
# first order Hamiltonian for magnetic field except a factor of 2.
nsr_para = rhf_nmr.para(nsrobj, mo10, mo_coeff, mo_occ,
shielding_nuc)[0] * 2
mol = nsrobj.mol
im, mass_center = inertia_tensor(mol)
nsr_para = _safe_solve(im, nsr_para)
unit = _atom_gyro_list(mol)[shielding_nuc] * nist.ALPHA**2
return numpy.einsum('ixy,i->ixy', nsr_para, unit)
def dia(magobj, gauge_orig=None):
'''Part of rotational g-tensors. It is the direct second derivatives of
the Lagrangian (corresponding to the zeroth order wavefunction). Unit
hbar/mu_N is not included. This part may be different to the conventional
dia-magnetic contributions of rotational g-tensors.
'''
if gauge_orig is None:
mol = magobj.mol
im, mass_center = rhf_g.inertia_tensor(mol)
# Eq. (35) of JCP, 105, 2804
e2 = rks_mag.dia(magobj, gauge_orig)
e2 -= rks_mag.dia(magobj, mass_center)
e2 = rhf_g._safe_solve(im, e2)
return -4 * nist.PROTON_MASS_AU * e2
else:
return rhf_g.dia(magobj, gauge_orig)
def dia(magobj, gauge_orig=None):
'''Part of rotational g-tensors. It is the direct second derivatives of
the Lagrangian (corresponding to the zeroth order wavefunction). Unit
hbar/mu_N is not included. This part may be different to the conventional
dia-magnetic contributions of rotational g-tensors.
'''
if gauge_orig is None:
mol = magobj.mol
im, mass_center = rhf_g.inertia_tensor(mol)
# Eq. (35) of JCP, 105, 2804
e2 = uks_mag.dia(magobj, gauge_orig)
e2 -= uks_mag.dia(magobj, mass_center)
e2 = rhf_g._safe_solve(im, e2)
return -4 * nist.PROTON_MASS_AU * e2
else:
return uhf_g.dia(magobj, gauge_orig)
def para(magobj, gauge_orig=None, h1=None, s1=None, with_cphf=None):
'''Part of rotational g-tensors from the first order wavefunctions. Unit
hbar/mu_N is not included. This part may be different to the conventional
para-magnetic contributions of rotational g-tensors.
'''
mol = magobj.mol
im, mass_center = rhf_g.inertia_tensor(mol)
if gauge_orig is None:
# The first order Hamiltonian for rotation part is the same to the
# first order Hamiltonian for magnetic field except a factor of 2. It can
# be computed using the magnetizability code.
mag_para = uhf_mag.para(magobj, gauge_orig, h1, s1, with_cphf) * 2
else:
mf = magobj._scf
mo_energy = mf.mo_energy
mo_coeff = mf.mo_coeff
mo_occ = mf.mo_occ
orboa = mo_coeff[0][:, mo_occ[0] > 0]
orbob = mo_coeff[1][:, mo_occ[1] > 0]
# for magnetic field
with mol.with_common_origin(mass_center):
h10 = .5 * mol.intor('int1e_cg_irxp', 3)
h10a = lib.einsum('xpq,pi,qj->xij', h10, mo_coeff[0].conj(), orboa)
h10b = lib.einsum('xpq,pi,qj->xij', h10, mo_coeff[1].conj(), orbob)
# for rotation part
with mol.with_common_origin(gauge_orig):
h01 = -mol.intor('int1e_cg_irxp', 3)
h01a = lib.einsum('xpq,pi,qj->xij', h01, mo_coeff[0].conj(), orboa)
h01b = lib.einsum('xpq,pi,qj->xij', h01, mo_coeff[1].conj(), orbob)
s10a = numpy.zeros_like(h10a)
s10b = numpy.zeros_like(h10b)
mo10 = uhf_nmr._solve_mo1_uncoupled(mo_energy, mo_occ, (h10a, h10b),
(s10a, s10b))[0]
mag_para = numpy.einsum('xji,yji->xy', mo10[0].conj(), h01a)
mag_para += numpy.einsum('xji,yji->xy', mo10[1].conj(), h01b)
mag_para = mag_para + mag_para.conj()
mag_para = rhf_g._safe_solve(im, mag_para)
# unit = hbar/mu_N, mu_N is nuclear magneton
unit = -2 * nist.PROTON_MASS_AU
return mag_para * unit
def para(magobj, gauge_orig=None, h1=None, s1=None, with_cphf=None):
'''Part of rotational g-tensors from the first order wavefunctions. Unit
hbar/mu_N is not included. This part may be different to the conventional
para-magnetic contributions of rotational g-tensors.
'''
mol = magobj.mol
im, mass_center = rhf_g.inertia_tensor(mol)
if gauge_orig is None:
# The first order Hamiltonian for rotation part is the same to the
# first order Hamiltonian for magnetic field except a factor of 2. It can
# be computed using the magnetizability code.
mag_para = uhf_mag.para(magobj, gauge_orig, h1, s1, with_cphf) * 2
else:
mf = magobj._scf
mo_energy = mf.mo_energy
mo_coeff = mf.mo_coeff
mo_occ = mf.mo_occ
orboa = mo_coeff[0][:,mo_occ[0]>0]
orbob = mo_coeff[1][:,mo_occ[1]>0]
# for magnetic field
with mol.with_common_origin(mass_center):
h10 = .5 * mol.intor('int1e_cg_irxp', 3)
h10a = lib.einsum('xpq,pi,qj->xij', h10, mo_coeff[0].conj(), orboa)
h10b = lib.einsum('xpq,pi,qj->xij', h10, mo_coeff[1].conj(), orbob)
# for rotation part
with mol.with_common_origin(gauge_orig):
h01 = -mol.intor('int1e_cg_irxp', 3)
h01a = lib.einsum('xpq,pi,qj->xij', h01, mo_coeff[0].conj(), orboa)
h01b = lib.einsum('xpq,pi,qj->xij', h01, mo_coeff[1].conj(), orbob)
s10a = numpy.zeros_like(h10a)
s10b = numpy.zeros_like(h10b)
mo10 = uhf_nmr._solve_mo1_uncoupled(mo_energy, mo_occ,
(h10a,h10b), (s10a,s10b))[0]
mag_para = numpy.einsum('xji,yji->xy', mo10[0].conj(), h01a)
mag_para+= numpy.einsum('xji,yji->xy', mo10[1].conj(), h01b)
mag_para = mag_para + mag_para.conj()
mag_para = rhf_g._safe_solve(im, mag_para)
# unit = hbar/mu_N, mu_N is nuclear magneton
unit = -2 * nist.PROTON_MASS_AU
return mag_para * unit
def nuc(mol, shielding_nuc):
'''Nuclear contributions'''
im, mass_center = inertia_tensor(mol)
charges = mol.atom_charges()
coords = mol.atom_coords()
nsr_nuc = []
for n, atm_id in enumerate(shielding_nuc):
rkl = coords - coords[atm_id]
d = numpy.linalg.norm(rkl, axis=1)
d[atm_id] = 1e100
e11 = numpy.einsum('z,zx,zy->xy', charges / d**3, rkl, rkl)
e11 = numpy.eye(3) * e11.trace() - e11
nsr_nuc.append(e11)
nsr_nuc = _safe_solve(im, numpy.asarray(nsr_nuc))
unit = _atom_gyro_list(mol)[shielding_nuc] * nist.ALPHA**2
return numpy.einsum('ixy,i->ixy', nsr_nuc, unit)
def dia(nsrobj, gauge_orig=None, shielding_nuc=None, dm0=None):
'''Diamagnetic part of NSR tensors.
'''
if shielding_nuc is None: shielding_nuc = nsrobj.shielding_nuc
if dm0 is None: dm0 = nsrobj._scf.make_rdm1()
mol = nsrobj.mol
im, mass_center = inertia_tensor(mol)
if gauge_orig is None:
ao_coords = rhf_mag._get_ao_coords(mol)
# Eq. (34) of JCP, 105, 2804
nsr_dia = rhf_nmr.dia(nsrobj, gauge_orig, shielding_nuc, dm0)
for n, atm_id in enumerate(shielding_nuc):
coord = mol.atom_coord(atm_id)
with mol.with_common_origin(coord):
with mol.with_rinv_origin(coord):
# a11part = (B dot) -1/2 frac{\vec{r}_N}{r_N^3} r_N (dot mu)
h11 = mol.intor('int1e_cg_a11part', comp=9)
e11 = numpy.einsum('xpq,qp->x', h11, dm0).reshape(3, 3)
nsr_dia[n] -= e11 - numpy.eye(3) * e11.trace()
nsr_dia[n] *= 2
else:
nsr_dia = []
for n, atm_id in enumerate(shielding_nuc):
coord = mol.atom_coord(atm_id)
with mol.with_rinv_origin(coord):
with mol.with_common_origin(gauge_orig):
# a11part = (B dot) -1/2 frac{\vec{r}_N}{r_N^3} (r-R_c) (dot mu)
h11 = mol.intor('int1e_cg_a11part', comp=9)
e11 = numpy.einsum('xpq,qp->x', h11, dm0).reshape(3, 3)
with mol.with_common_origin(coord):
# a11part = (B dot) -1/2 frac{\vec{r}_N}{r_N^3} r_N (dot mu)
h11 = mol.intor('int1e_cg_a11part', comp=9)
# e11 ~ (B dot) -1/2 frac{\vec{r}_N}{r_N^3} (R_N-R_c) (dot mu)
e11 -= numpy.einsum('xpq,qp->x', h11, dm0).reshape(3, 3)
e11 = e11 - numpy.eye(3) * e11.trace()
nsr_dia.append(e11)
nsr_dia = _safe_solve(im, numpy.asarray(nsr_dia))
unit = _atom_gyro_list(mol)[shielding_nuc] * nist.ALPHA**2
return numpy.einsum('ixy,i->ixy', nsr_dia, unit)
