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

$$A^{\text{FC}}(N)=\frac{\alpha }{2}\frac{1}{S}\frac{8\pi }{3}{g}_e{g}_N{\mu }_N\sum _{\mu \nu }{P}_{\mu \nu }^{\alpha -\beta }⟨{\chi }_{\mu }|\delta ({𝐫}_N)|{\chi }_{\nu }⟩$$

(10.52)

and the spin-dipole (SD) contribution is

$$A_{ab}^{\text{SD}}(N)=\frac{\alpha }{2}\frac{1}{S}{g}_e{g}_N{\mu }_N\sum _{\mu \nu }{P}_{\mu \nu }^{\alpha -\beta }⟨{\chi }_{\mu }\left|\frac{3r_{N,a}{r}_{N,b}-{\delta }_{ab}{r}_N^2}{r_N^5}\right|{\chi }_{\nu }⟩$$

(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

	$Q_{ab}(N)$	$={\displaystyle \frac{{\partial }^2{V}_{eN}}{\partial X_{N,a}\partial X_{N,b}}}+{\displaystyle \frac{{\partial }^2{V}_{NN}}{\partial X_{N,a}\partial X_{N,b}}}$		(10.54)
		$\begin{array}{c}\hfill =-{\displaystyle \sum _{\mu \nu }}{P}_{\mu \nu }^{\alpha +\beta }⟨{\chi }_{\mu }\left|{\displaystyle \frac{3r_{N,a}{r}_{N,b}-{\delta }_{ab}{r}_N^2}{r_N^5}}\right|{\chi }_{\nu }⟩\\ \hfill +{\displaystyle \sum _{A\ne N}}{Z}_A{\displaystyle \frac{3R_{AN,a}{R}_{AN,b}-{\delta }_{ab}{R}_{AN}^2}{R_{AN}^5}}\end{array}$

for a nucleus $N$. Diagonalizing the tensor gives three principal values, ordered $|Q_1|\le |Q_2|\le |Q_3|$, 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

$$H^{g-tensor}={\mu }_B𝐒\cdot 𝐠\cdot 𝐁$$

(10.56)

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

$$𝐠=g_e𝐈+{𝐠}^{rmc}+{𝐠}^{dso}+{𝐠}^{pso}.$$

(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

$$g_{pq}^{rmc}=-\frac{{\alpha }^2{g}_e}{2S}{\delta }_{pq}\sum _{\mu \nu }{P}_{\mu \nu }^{\alpha -\beta }{T}_{\mu \nu }$$

(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

$$g^{pso}=\frac{1}{\alpha S}\sum _N\frac{⟨{\mathrm{\Psi }}_0\left|h^{SO}\right|{\mathrm{\Psi }}_N⟩⟨{\mathrm{\Psi }}_N\left|h^{OZ}\right|{\mathrm{\Psi }}_0⟩}{E_N-E_0}$$

(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

$$h^{OZ}={\mu }_B𝐋\cdot 𝐁$$

(10.60)

with $𝐋$ 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.

$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