# 10.12.3 Additional Magnetic Field-Related Properties

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:

 $A_{ab}^{\text{tot}}(N)=A_{ab}^{\text{FC}}(N)\delta_{ab}+A_{ab}^{\text{SD}}(N),$ (10.51)

where the Fermi contact (FC) contribution is

 $A^{\text{FC}}(N)=\frac{\alpha}{2}\frac{1}{S}\frac{8\pi}{3}g_{e}g_{N}\mu_{N}% \sum_{\mu\nu}P_{\mu\nu}^{\alpha-\beta}\left\langle\chi_{\mu}|\delta(\mathbf{r}% _{N})|\chi_{\nu}\right\rangle$ (10.52)

and the spin-dipole (SD) contribution is

 $A_{ab}^{\text{SD}}(N)=\frac{\alpha}{2}\frac{1}{S}g_{e}g_{N}\mu_{N}\sum_{\mu\nu% }P_{\mu\nu}^{\alpha-\beta}\left\langle\chi_{\mu}\left|\frac{3r_{N,a}r_{N,b}-% \delta_{ab}r_{N}^{2}}{r_{N}^{5}}\right|\chi_{\nu}\right\rangle$ (10.53)

for a nucleus $N$.

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

 $\displaystyle Q_{ab}(N)$ $\displaystyle=\frac{\partial^{2}V_{eN}}{\partial X_{N,a}\partial X_{N,b}}+% \frac{\partial^{2}V_{NN}}{\partial X_{N,a}\partial X_{N,b}}$ (10.54) $\displaystyle\begin{split}\displaystyle=-\sum_{\mu\nu}P_{\mu\nu}^{\alpha+\beta% }\left\langle\chi_{\mu}\left|\frac{3r_{N,a}r_{N,b}-\delta_{ab}r_{N}^{2}}{r_{N}% ^{5}}\right|\chi_{\nu}\right\rangle\\ \displaystyle+\sum_{A\neq N}Z_{A}\frac{3R_{AN,a}R_{AN,b}-\delta_{ab}R_{AN}^{2}% }{R_{AN}^{5}}\end{split}$

for a nucleus $N$. Diagonalizing the tensor gives three principal values, ordered $|Q_{1}|\leq|Q_{2}|\leq|Q_{3}|$, which are components of the asymmetry parameter eta:

 $\eta=\frac{Q_{1}-Q_{2}}{Q_{3}}$ (10.55)

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

 $H^{g-tensor}=\mu_{B}\mathbf{S}\cdot\mathbf{g}\cdot\mathbf{B}$ (10.56)

where $\mu_{B}$ is the Bohr magneton, $\mathbf{S}$ is spin and $\mathbf{B}$ 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 $\mathbf{g}^{rmc}$, diamagnetic spin-orbit coupling $\mathbf{g}^{dso}$ and paramagnetic spin-orbit coupling $\mathbf{g}^{pso}$ terms

 $\mathbf{g}=g_{e}\mathbf{I}+\mathbf{g}^{rmc}+\mathbf{g}^{dso}+\mathbf{g}^{pso}.$ (10.57)

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 $g^{rmc}$ is

 $g^{rmc}_{pq}=-\frac{\alpha^{2}g_{e}}{2S}\delta_{pq}\sum_{\mu\nu}P^{\alpha-% \beta}_{\mu\nu}T_{\mu\nu}$ (10.58)

with $\alpha$ as the fine-structure constant, $P^{\alpha-\beta}$ as spin density and $T$ as kinetic energy integrals. The diamagnetic spin-orbit term $g^{dso}$ 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 $g^{pso}$ is a second-order term in the perturbation series but constitutes the main contribution to the g-tensor shift

 $g^{pso}=\frac{1}{\alpha S}\sum_{N}\frac{\left\langle\Psi_{0}\left|h^{SO}\right% |\Psi_{N}\right\rangle\left\langle\Psi_{N}\left|h^{OZ}\right|\Psi_{0}\right% \rangle}{E_{N}-E_{0}}$ (10.59)

where $h^{SO}$ is the spin-orbit coupling interaction where a spin-orbit mean-field approach242 is used by default and $h^{OZ}$ the orbital Zeeman interaction

 $h^{OZ}=\mu_{B}\mathbf{L}\cdot\mathbf{B}$ (10.60)

with $\mathbf{L}$ 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.

## 10.12.3.1 Job Control and Examples

Only one keyword is necessary in the $rem section to activate the magnetic property module. MAGNET Activate the magnetic property module. TYPE: LOGICAL DEFAULT: FALSE OPTIONS: FALSE (or 0) Don’t activate the magnetic property module. TRUE (or 1) Activate the magnetic property module. RECOMMENDATION: None. All other options are controlled through the$magnet input section, which has the same key-value format as the $rem section (see section 3.4). Current options are: HYPERFINE Activate the calculation of hyperfine interaction tensors. INPUT SECTION:$magnet
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE (or 0) Don’t calculate hyperfine interaction tensors. TRUE (or 1) Calculate hyperfine interaction tensors.
RECOMMENDATION:
None. Due to the nature of the property, which requires the spin density $\rho^{\alpha-\beta}(\mathbf{r})\equiv\rho^{\alpha}(\mathbf{r})-\rho^{\beta}(% \mathbf{r})$, this is not meaningful for restricted (RHF) references. Only UHF (not ROHF) is available.

ELECTRIC
Activate the calculation of electric field gradient tensors.
INPUT SECTION: $magnet TYPE: LOGICAL DEFAULT: FALSE OPTIONS: FALSE (or 0) Don’t calculate EFG tensors and nuclear quadrupole parameters. TRUE (or 1) Calculate EFG tensors and nuclear quadrupole parameters. RECOMMENDATION: None. Calculation of g-tensor is activated by specifying the G_TENSOR keyword in the$rem section. Example 10.12.3.1 illustrates g-tensor calculation for water cation.

G_TENSOR
Activates g-tensor calculation.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE (or 0) Don’t calculate g-tensor TRUE (or 1) Calculate g-tensor.
RECOMMENDATION:
None.

Example 10.32  Calculating hyperfine and EFG tensors for the glycine 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


Example 10.33  Calculating g-tensor for the water cation.

$comment Test for ccsd g-tensor$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