10.12 NMR and Other Magnetic Properties

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 nuclear quadrupole interaction from electric field gradient tensors, and

  • 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:

H^HFI/h=𝐒^𝐀𝐈^, (10.63)

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

Aabtot(N)=AabFC(N)δab+AabSD(N)+AabOZ/SOC, (10.64)

where the Fermi contact (FC) contribution is

AFC(N)=α21S8π3gegNμNμνPμνα-βχμ|δ(𝐫N)|χν (10.65)

and the spin-dipole (SD) contribution is

AabSD(N)=α21SgegNμNμνPμνα-βχμ|3rN,arN,b-δabrN2rN5|χν (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.

10.12.3.2 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:

H^NQI/h=𝐈^𝐐𝐈^, (10.67)
Qab(N) =2VeNXN,aXN,b+2VNNXN,aXN,b (10.68)
=-μνPμνα+βχμ|3rN,arN,b-δabrN2rN5|χν+ANZA3RAN,aRAN,b-δabRAN2RAN5

for a nucleus N. Diagonalizing the tensor gives three principal values, ordered |Q1||Q2||Q3|, which are components of the asymmetry parameter eta:

η=Q1-Q2Q3 (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

H^g-tensor=μB𝐒𝐠𝐁, (10.70)

where μB is the Bohr magneton, 𝐒 is the intrinsic molecular spin vector, and 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 𝐠rmc, diamagnetic spin-orbit coupling 𝐠dso and paramagnetic spin-orbit coupling 𝐠pso terms

𝐠=ge𝐈+𝐠rmc+𝐠dso+𝐠pso. (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 grmc is

gpqrmc=-α2ge2SδpqμνPμνα-βTμν (10.72)

with α as the fine-structure constant, Pα-β as spin density and T as kinetic energy integrals. The diamagnetic spin-orbit term gdso 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 gpso is a second-order term in the perturbation series but constitutes the main contribution to the g-tensor shift

gpso=1αSNΨ0|hSO|ΨNΨN|hOZ|Ψ0EN-E0 (10.73)

where hSO is the spin-orbit coupling interaction where a spin-orbit mean-field approach is used by default and hOZ the orbital Zeeman interaction

hOZ=μB𝐋𝐁 (10.74)

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. 343 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 787 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.

10.12.3.4 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 ρα-β(𝐫)ρα(𝐫)-ρβ(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