In addition to NMR chemical shieldings and spin-spin couplings, other magnetic properties available in Q-Chem are
hyperfine interaction tensors,
the nuclear quadrupole interaction from electric field gradient tensors, and
the electronic g-tensor,
The hyperfine interaction tensor describes the interaction the interaction of unpaired electron spin with an atom’s nuclear spin levels:
which is broken down into Fermi contact (FC), spin-dipole (SD), and orbital Zeeman/spin-orbit coupling (OZ/SOC) terms:
where the Fermi contact (FC) contribution is
and the spin-dipole (SD) contribution is
for a nucleus . The orbital Zeeman/spin-orbit coupling cross-term (OZ/SOC) is currently not available.
Hyperfine interaction tensors are available for all SCF-based methods with an unrestricted (not restricted open-shell) reference. Post-HF methods are unavailable.
Another sensitive probe of the individual nuclear environments in a molecule is the nuclear quadrupole interaction (NQI), which is a measure of how a nucleus’ quadrupole moment interacts with the local electric field gradient:
for a nucleus . Diagonalizing the tensor gives three principal values, ordered , which are components of the asymmetry parameter eta:
The electronic g-tensor is a measure of the electron describes the coupling of unpaired electron spins with an external magnetic field, represented by the phenomenological Hamiltonian
where is the Bohr magneton, is the intrinsic molecular spin vector, and is the incident magnetic field vector.
The g-tensor is comprised of the Spin-Zeeman term and the g-tensor shift that includes the relativistic mass correction , diamagnetic spin-orbit coupling and paramagnetic spin-orbit coupling terms
For the Spin-Zeeman term the contribution is isotropic and equals the free electron g-factor. The relativistic interaction terms are added as perturbations following the Breit-Pauli ansatz resulting the the following expressions. The relativistic mass correction shift term is
with as the fine-structure constant, as spin density and as kinetic energy integrals. The diamagnetic spin-orbit term is currently not implemented in Q-Chem and therefore excluded but typically also only of minor importance for lighter elements or first to second row transition metal systems.
The paramagnetic spin-orbit coupling term is a second-order term in the perturbation series but constitutes the main contribution to the g-tensor shift
where is the spin-orbit coupling interaction where a spin-orbit mean-field approach302 is used by default and the orbital Zeeman interaction
with as angular momentum.
In this implementation the paramagnetic spin-orbit coupling term is evaluated using a response theory approach, as first demonstrated by Gauss et al.351, but with a computational approach following that used in the Q-Chem polarization code811. At the moment the g-tensor is only implemented at the CCSD level.
Only one keyword is necessary in the $rem section to activate the magnetic property module.
All other options are controlled through the $magnet input section, which has the same key-value format as the $rem section (see section 3.4). Current options are:
For both hyperfine and EFG tensors, the results for all nuclei are automatically calculated.
Calculation of g-tensor is activated by specifying the G_TENSOR keyword in the $rem section. Example 10.11.4.4 illustrates g-tensor calculation for water cation.
$molecule 1 2 N 0.0000000000 0.0000000000 0.0000000000 C 1.4467530000 0.0000000000 0.0000000000 C 1.9682482963 0.0000000000 1.4334965024 O 1.2385450522 0.0000000000 2.4218667010 H 1.7988742211 -0.8959881458 -0.5223754133 H 1.7997303368 0.8930070757 -0.5235632630 H -0.4722340827 -0.0025218132 0.8996536532 H -0.5080000000 0.0766867527 -0.8765335943 O 3.3107284257 -0.0000000000 1.5849828121 H 3.9426948542 -0.0000000000 0.7289954096 $end $rem METHOD = hf BASIS = def2-sv(p) SCF_CONVERGENCE = 11 THRESH = 14 SYMMETRY = false SYM_IGNORE = true MAGNET = true $end $magnet hyperfine = true electric = true $end
$molecule 1 2 O 0.00000000 0.00000000 0.13475163 H 0.00000000 -1.70748899 -1.06930309 H 0.00000000 1.70748899 -1.06930309 $end $rem INPUT_BOHR = true METHOD = ccsd BASIS = 3-21g CC_REF_PROP = true G_TENSOR = true N_FROZEN_CORE = 0 SYM_IGNORE = true NO_REORIENT = true SCF_CONVERGENCE = 12 CC_CONVERGENCE = 12 $end $gauge_origin 0.000000 0.000000 0.0172393 $end