# 10.12.3 Additional Magnetic Field-Related Properties

(May 16, 2021)

In addition to NMR chemical shieldings and spin-spin couplings, other magnetic properties available in Q-Chem are

• hyperfine interaction tensors,

• the electronic g-tensor,

## 10.12.3.1 Hyperfine Interaction

The hyperfine interaction tensor describes the interaction the interaction of unpaired electron spin with an atom’s nuclear spin levels:

 $\hat{H}_{\text{HFI}}/h=\mathbf{\hat{S}}\cdot\mathbf{A}\cdot\mathbf{\hat{I}},$ (10.63)

which is broken down into Fermi contact (FC), spin-dipole (SD), and orbital Zeeman/spin-orbit coupling (OZ/SOC) terms:

 $A_{ab}^{\text{tot}}(N)=A_{ab}^{\text{FC}}(N)\delta_{ab}+A_{ab}^{\text{SD}}(N)+% A_{ab}^{\text{OZ/SOC}},$ (10.64)

where the Fermi contact (FC) contribution is

 $A^{\text{FC}}(N)=\frac{\alpha}{2}\frac{1}{S}\frac{8\pi}{3}g_{e}g_{N}\mu_{N}% \sum_{\mu\nu}P_{\mu\nu}^{\alpha-\beta}\left\langle\chi_{\mu}|\delta(\mathbf{r}% _{N})|\chi_{\nu}\right\rangle$ (10.65)

and the spin-dipole (SD) contribution is

 $A_{ab}^{\text{SD}}(N)=\frac{\alpha}{2}\frac{1}{S}g_{e}g_{N}\mu_{N}\sum_{\mu\nu% }P_{\mu\nu}^{\alpha-\beta}\left\langle\chi_{\mu}\left|\frac{3r_{N,a}r_{N,b}-% \delta_{ab}r_{N}^{2}}{r_{N}^{5}}\right|\chi_{\nu}\right\rangle$ (10.66)

for a nucleus $N$. 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:

 $\hat{H}_{\text{NQI}}/h=\mathbf{\hat{I}}\cdot\mathbf{Q}\cdot\mathbf{\hat{I}},$ (10.67)
 $\displaystyle Q_{ab}(N)$ $\displaystyle=\frac{\partial^{2}V_{eN}}{\partial X_{N,a}\partial X_{N,b}}+% \frac{\partial^{2}V_{NN}}{\partial X_{N,a}\partial X_{N,b}}$ (10.68) $\displaystyle\begin{split}\displaystyle=-\sum_{\mu\nu}P_{\mu\nu}^{\alpha+\beta% }\left\langle\chi_{\mu}\left|\frac{3r_{N,a}r_{N,b}-\delta_{ab}r_{N}^{2}}{r_{N}% ^{5}}\right|\chi_{\nu}\right\rangle\\ \displaystyle+\sum_{A\neq N}Z_{A}\frac{3R_{AN,a}R_{AN,b}-\delta_{ab}R_{AN}^{2}% }{R_{AN}^{5}}\end{split}$

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:

 $\eta=\frac{Q_{1}-Q_{2}}{Q_{3}}$ (10.69)

## 10.12.3.3 Electronic g-tensor

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

 $\hat{H}^{g-tensor}=\mu_{B}\mathbf{S}\cdot\mathbf{g}\cdot\mathbf{B},$ (10.70)

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

 $\mathbf{g}=g_{e}\mathbf{I}+\mathbf{g}^{rmc}+\mathbf{g}^{dso}+\mathbf{g}^{pso}.$ (10.71)

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

 $g^{rmc}_{pq}=-\frac{\alpha^{2}g_{e}}{2S}\delta_{pq}\sum_{\mu\nu}P^{\alpha-% \beta}_{\mu\nu}T_{\mu\nu}$ (10.72)

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

 $g^{pso}=\frac{1}{\alpha S}\sum_{N}\frac{\left\langle\Psi_{0}\left|h^{SO}\right% |\Psi_{N}\right\rangle\left\langle\Psi_{N}\left|h^{OZ}\right|\Psi_{0}\right% \rangle}{E_{N}-E_{0}}$ (10.73)

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

 $h^{OZ}=\mu_{B}\mathbf{L}\cdot\mathbf{B}$ (10.74)

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. , but with a computational approach following that used in the Q-Chem polarization code . At the moment the g-tensor is only implemented at the CCSD level.

## 10.12.3.4 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: $magnet TYPE: LOGICAL DEFAULT: FALSE OPTIONS: FALSE (or 0) Don’t calculate EFG tensors and nuclear quadrupole parameters. TRUE (or 1) Calculate EFG tensors and nuclear quadrupole parameters. RECOMMENDATION: None. For both hyperfine and EFG tensors, the results for all nuclei are automatically calculated. Calculation of g-tensor is activated by specifying the G_TENSOR keyword in the$rem section. Example 10.12.3.4 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 10.37  Calculating hyperfine and EFG tensors for the glycine 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


View output

Example 10.38  Calculating g-tensor for the water cation.

$molecule 1 2 O 0.00000000 0.00000000 0.13475163 H 0.00000000 -1.70748899 -1.06930309 H 0.00000000 1.70748899 -1.06930309$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


View output