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 kernel(self, mo1=None):
cput0 = (time.clock(), time.time())
self.check_sanity()
#self.dump_flags()
dm0 = self._scf.make_rdm1()
#print('dm0 from kernel', dm0.shape)
#dm0 = dm0[0] - dm0[1]
tt_dia = self.dia(dm0)
tt_para = self.para(mo1, self._scf.mo_coeff, self._scf.mo_occ)
im, mass_center = rhf_rotg.inertia_tensor(self.mol)
im_inv = numpy.linalg.inv(im)
#print('im', im)
#print('im_inv', im_inv)
To_MHz = 1804742.77343363
esr_para = -1.0 * numpy.einsum('ij,jk->ik', im_inv,
tt_para) * To_MHz * 2
esr_dia = -1.0 * numpy.einsum('ij,jk->ik', im_inv, tt_dia) * To_MHz * 2
print('esr_dia', esr_dia)
esr_tensor = esr_para + esr_dia
#print('esr_para', esr_para) for debug
#print('esr_dia', esr_dia)
#print('esr_tot', esr_tot)
esr_tot, v = self.align(esr_tensor)
esr_dia = reduce(numpy.dot, (v.T, esr_dia, v))
esr_para = reduce(numpy.dot, (v.T, esr_para, v))
_write(self, esr_dia, 'esr-tensor diamagnetic terms (in MHz)')
_write(self, esr_para, 'esr-tensor paramagnetic terms (in MHz)')
_write(self, esr_tot, 'esr-tensor total (in MHz)')
return esr_tot
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)
def dia(gobj, dm0, gauge_orig=None):
'''Note the side effects of set_common_origin'''
if isinstance(dm0, numpy.ndarray) and dm0.ndim == 2: # RHF DM
return numpy.zeros((3, 3))
mol = gobj.mol
effspin = mol.spin * .5
muB = .5 # Bohr magneton
dma, dmb = dm0
totdm = dma + dmb
spindm = dma - dmb
alpha2 = nist.ALPHA**2
#Many choices of qed_fac, see JPC, 101, 3388
#qed_fac = (nist.G_ELECTRON - 1)
#qed_fac = nist.G_ELECTRON / 2
qed_fac = 1
assert (not mol.has_ecp())
if gauge_orig is not None:
mol.set_common_origin(gauge_orig)
e11 = numpy.zeros((3, 3))
im, mass_center = rhf_rotg.inertia_tensor(mol)
for ia in range(mol.natm):
Z = koseki_charge(mol.atom_charge(ia))
R = mol.atom_coord(ia) - mass_center
with mol.with_rinv_origin(R):
h11 = mol.intor('int1e_drinv', comp=3) * Z # * mol.atom_charge(ia)
t1 = numpy.einsum('xij,ij->x', h11, spindm)
e11 += numpy.dot(R, t1) * numpy.eye(3) - numpy.kron(R, t1).reshape(
3, 3)
#GIAO part of dia-magnetic constribution
#print('kron', numpy.kron(R, t1).reshape(3,3))
#h22 = mol.intor('int1e_a01gp', comp=9)
#e11 -= Z * numpy.einsum('xij,ij->x', h22, spindm).reshape(3,3)
gdia = e11 * alpha2 / effspin / 4.
return gdia
