10.11 NMR and Other Magnetic Properties 10.11.3 Linear-Scaling NMR Chemical Shift Calculations 10.12 Finite-Field Calculation of (Hyper)Polarizabilities

10.11.4 Additional Magnetic Field-Related Properties

(February 4, 2022)

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,

10.11.4.1 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}},

(10.68)

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}},

(10.69)

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

(10.70)

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

(10.71)

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.

10.11.4.2 Nuclear Quadrupole Interaction

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}},

(10.72)

	$\displaystyle Q_{ab}(N)$	$\displaystyle=\frac{\partial^{2}V_{eN}}{\partial X_{N,a}\partial X_{N,b}}+% \frac{\partial^{2}V_{NN}}{\partial X_{N,a}\partial X_{N,b}}$		(10.73)
		$\displaystyle\begin{split}\displaystyle=-\sum_{\mu\nu}P_{\mu\nu}^{\alpha+\beta% }\left\langle\chi_{\mu}\left\|\frac{3r_{N,a}r_{N,b}-\delta_{ab}r_{N}^{2}}{r_{N}% ^{5}}\right\|\chi_{\nu}\right\rangle\\ \displaystyle+\sum_{A\neq N}Z_{A}\frac{3R_{AN,a}R_{AN,b}-\delta_{ab}R_{AN}^{2}% }{R_{AN}^{5}}\end{split}$		(10.73)

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}}

(10.74)

10.11.4.3 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},

(10.75)

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}.

(10.76)

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}

(10.77)

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}}

(10.78)

where $h^{SO}$ is the spin-orbit coupling interaction where a spin-orbit mean-field approach³⁰² is used by default and $h^{OZ}$ the orbital Zeeman interaction

h^{OZ}=\mu_{B}\mathbf{L}\cdot\mathbf{B}

(10.79)

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.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.

10.11.4.4 Job Control and Examples

Only one keyword is necessary in the $rem section to activate the magnetic property module.

MAGNET

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 10.11.4.4 illustrates g-tensor calculation for water cation.

G_TENSOR

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 10.39 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 10.40 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