11.13 NMR and Other Magnetic Properties 11.13.2 Linear-Scaling NMR Chemical Shift Calculations 11.14 Finite-Field Calculation of (Hyper)Polarizabilities

11.13.3 Additional Magnetic Field-Related Properties

It is now possible to calculate certain open-shell magnetic field-related properties in Q-Chem. One is the hyperfine interaction (HFI) tensor, describing 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),

(11.73)

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

(11.74)

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

(11.75)

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

	$\displaystyle Q_{ab}(N)$	$\displaystyle=\frac{\partial^{2}V_{eN}}{\partial X_{N,a}\partial X_{N,b}}+% \frac{\partial^{2}V_{NN}}{\partial X_{N,a}\partial X_{N,b}}$		(11.76)
		$\displaystyle\begin{split}\displaystyle=-\sum_{\mu\nu}P_{\mu\nu}^{\alpha+\beta% }\left\langle\chi_{\mu}\left\|\frac{3r_{N,a}r_{N,b}-\delta_{ab}r_{N}^{2}}{r_{N}% ^{5}}\right\|\chi_{\nu}\right\rangle\\ \displaystyle+\sum_{A\neq N}Z_{A}\frac{3R_{AN,a}R_{AN,b}-\delta_{ab}R_{AN}^{2}% }{R_{AN}^{5}}\end{split}$		(11.76)

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}}

(11.77)

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.

11.13.3.1 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.

Example 11.31 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

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11.13.3 Additional Magnetic Field-Related Properties

11.13.3.1 Job Control and Examples