It is now possible to calculate certain open-shell magnetic field-related properties in Q-Chem: the hyperfine interaction tensor (HFI), the electric field gradient tensor (EFG), and the g-tensor.
The hyperfine interaction tensor describes the interaction the interaction of unpaired electron spin with an atom’s nuclear spin levels:
where the Fermi contact (FC) contribution is
and the spin-dipole (SD) contribution is
for a nucleus .
Another sensitive probe of the individual nuclear environments in a molecule is the nuclear quadrupolar interaction (NQI), arising from the interaction of a nuclei’s quadrupole moment with an applied electric field gradient (EFG), calculated as
for a nucleus . Diagonalizing the tensor gives three principal values, ordered , which are components of the asymmetry parameter eta:
Both the hyperfine and EFG tensors are automatically calculated for all possible nuclei. All SCF-based methods (HF and DFT) are available with restricted and unrestricted references. Restricted open-shell references and post-HF methods are unavailable.
The g-tensor describes the coupling of unpaired electron spins with an external magnetic field
where is the Bohr magneton, is spin and the 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 approach242 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.283, but with a computational approach following that used in the Q-Chem polarization code683. 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:
Calculation of g-tensor is activated by specifying the G_TENSOR keyword in the $rem section. Example 10.12.3.1 illustrates g-tensor calculation for water cation.
$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 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
$comment Test for ccsd g-tensor $end $rem input_bohr = true jobtype = sp 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 $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 $gauge_origin 0.000000 0.000000 0.0172393 $end