10.10 NMR and Other Magnetic Properties 10.10.3 Linear-Scaling NMR Chemical Shift Calculations 10.11 Vibrational Circular Dichroism (VCD)

10.10.4 Additional Magnetic Field-Related Properties

(September 1, 2024)

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,

10.10.4.1 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}},

(10.90)

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}},

(10.91)

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

(10.92)

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

(10.93)

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.

Calculation of excited state (CIS/TDA-TDDFT) singlet-triplet hyperfine couplings are also available, but formatted differently than the ground state unrestricted counterpart. Excited state couplings are printed as contributions from FC and SD for each pair of excited states and for each spin operator $S_{x},S_{y},S_{z}$ . For this method, the nuclear spin states are averaged to get final coupling contributions ⁴² andd C. Climent S. R. May et al.
J. Phys. Chem. A
(2023), 127, pp. 3591–3597. Link .

10.10.4.2 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 nuclear quadrupole moment interacts with the local electric field gradient:

\hat{H}_{\text{NQI}}/h=\mathbf{\hat{I}}\cdot\mathbf{Q}\cdot\mathbf{\hat{I}},

(10.94)

	$\displaystyle Q_{ab}(N)$	$\displaystyle=\frac{\partial^{2}V_{eN}}{\partial X_{N,a}\partial X_{N,b}}+% \frac{\partial^{2}V_{NN}}{\partial X_{N,a}\partial X_{N,b}}$		(10.95)
		$\displaystyle\begin{split}\displaystyle=-\sum_{\mu\nu}P_{\mu\nu}^{\alpha+\beta% }\left\langle\chi_{\mu}\left\|\frac{3r_{N,a}r_{N,b}-\delta_{ab}r_{N}^{2}}{r_{N}% ^{5}}\right\|\chi_{\nu}\right\rangle\\ \displaystyle+\sum_{A\neq N}Z_{A}\frac{3R_{AN,a}R_{AN,b}-\delta_{ab}R_{AN}^{2}% }{R_{AN}^{5}}\end{split}$		(10.95)

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}}

(10.96)

10.10.4.3 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},

(10.97)

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}.

(10.98)

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}

(10.99)

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}}

(10.100)

where $h^{SO}$ is the spin-orbit coupling interaction where a spin-orbit mean-field approach ³⁴¹ Epifanovsky E. et al.
J. Chem. Phys.
(2015), 143, pp. 064102. Link is used by default and $h^{OZ}$ the orbital Zeeman interaction

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

(10.101)

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. ⁴⁰¹ Gauss J., Kállay M., Neese F.
J. Phys. Chem. A
(2009), 113, pp. 11541–11549. Link , but with a computational approach following that used in the Q-Chem polarization code ⁹¹⁹ Nanda K., Krylov A. I.
J. Chem. Phys.
(2016), 145, pp. 204116. Link . At the moment the g-tensor is only implemented at the CCSD level.

10.10.4.4 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.

HYPERFINE_FULL
       Activate calculation of excited state hyperfine couplings.
INPUT SECTION: $magnet
TYPE:
       LOGICAL
DEFAULT:
       FALSE
OPTIONS:
       False (or 0) Don’t calculate excited state hyperfine couplings. True (or 1) Calculate excited state hyperfine couplings.
RECOMMENDATION:
       None.

HYPERFINE_FULL_NROOTS
       Specify number of roots for excited state hyperfine calculation.
INPUT SECTION: $magnet
TYPE:
       INTEGER
DEFAULT:
       1
OPTIONS:
       n Calculate hyperfine couplings between lowest n CIS/TDA-TDDFT states.
RECOMMENDATION:
       None.

HYPERFINE_FULL_ROOT_OFFSET
       Specify offset for roots for excited state hyperfine calculation.
INPUT SECTION: $magnet
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       m Calculate hyperfine couplings between lowest n CIS/TDA-TDDFT states, starting at m ${}^{\text{th}}$ root.
RECOMMENDATION:
       None.

HYPERFINE_FULL_SINGLET_OTHER_SINGLET
       Specify whether to calculate couplings between singlet excited states.
INPUT SECTION: $magnet
TYPE:
       LOGICAL
DEFAULT:
       FALSE
OPTIONS:
       False (or 0) Don’t calculate excited state hyperfine couplings between singlets. True (or 1) Calculate excited state hyperfine couplings between singlets.
RECOMMENDATION:
       None.

HYPERFINE_FULL_TRIPLET_OTHER_TRIPLET
       Specify whether to calculate couplings between triplet excited states.
INPUT SECTION: $magnet
TYPE:
       LOGICAL
DEFAULT:
       FALSE
OPTIONS:
       False (or 0) Don’t calculate excited state hyperfine couplings between triplets. True (or 1) Calculate excited state hyperfine couplings between triplets.
RECOMMENDATION:
       None.

Calculation of g-tensor is activated by specifying the G_TENSOR keyword in the $rem section. Example 10.10.4.4 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 10.50 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
   MAGNET          = true
   INTEGRAL_SYMMETRY    = false
   POINT_GROUP_SYMMETRY = false
$end

$magnet
  hyperfine = true
  electric  = true
$end

Example 10.51 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
   NO_REORIENT     = true
   SCF_CONVERGENCE = 12
   CC_CONVERGENCE  = 12
   POINT_GROUP_SYMMETRY = false
$end

$gauge_origin
0.000000 0.000000  0.0172393
$end

Excited state HFC values are printed as

Nucleus: IDX
…
STATE1 / STATE2
FC: S_x_contribution S_y_contribution S_z_contribution
SD: S_x_contribution S_y_contribution S_z_contribution
…

Example 10.52 Calculating excited state hyperfine coupling for Pyrene-Dimethylaniline radical pair.

$molecule
    READ     pydma_xyz.txt
$end

$rem
    EXCHANGE = wb97x-d3
    BASIS = def2-svpd
    CIS_SINGLETS = true
    CIS_TRIPLETS = true
    CIS_MULLIKEN = true
    CIS_N_ROOTS = 15 ! 15 TDA-TDDFT roots for each spin state
    RPA = false
    magnet = true ! Turn on magnetman
    CC_PRINT_PREC = 12 ! Print precision affects precision of printed hfc
    sym_ignore = true
    cis_max_cycles = 500
    scf_max_cycles = 500
    SCF_CONVERGENCE = 10
    CIS_CONVERGENCE = 8
$end

$magnet
    hyperfine_full = true ! Activate excited state hfc calculation
    hyperfine_full_nroots = 30 ! Calculate all roots
    hyperfine_full_singlet_other_singlet = false ! Neglect singlet-singlet couplings
    hyperfine_full_triplet_other_triplet = false ! Neglect triplet-triplet couplings
$end

Search Results

10.10.4 Additional Magnetic Field-Related Properties

10.10.4.1 Hyperfine Interaction

10.10.4.2 Nuclear Quadrupole Interaction

10.10.4.3 Electronic g-tensor

10.10.4.4 Job Control and Examples