10.10 NMR and Other Magnetic Properties 10.10.1 Introduction 10.10.3 Linear-Scaling NMR Chemical Shift Calculations

10.10.2 NMR Chemical Shifts and J-Couplings

(September 1, 2024)

NMR calculations are available at both the Hartree-Fock and DFT levels of theory. ⁵¹² Helgaker T., Watson M., Handy N. C.
J. Chem. Phys.
(2000), 113, pp. 9402. Link ^, ¹²⁴⁵ Sychrovský V., Gräfenstein J., Cremer D.
J. Chem. Phys.
(2000), 113, pp. 3530. Link Q-Chem computes NMR chemical shielding tensors using gauge-including atomic orbitals ³¹⁵ Ditchfield R.
Mol. Phys.
(1974), 27, pp. 789. Link ^, ¹³⁸⁵ Wolinski K., Hinton J. F., Pulay P.
J. Am. Chem. Soc.
(1990), 112, pp. 8251. Link ^, ⁴⁹¹ Häser M. et al.
Theor. Chem. Acc.
(1992), 83, pp. 455. Link (GIAOs), an approach that has proven to reliable and accurate for many applications. ⁵¹¹ Helgaker T., Ruud M. Jaszuński K.
Chem. Rev.
(1990), 99, pp. 293. Link ^, ⁴⁰² Gauss J.
Ber. Bunsenges. Phys. Chem.
(1995), 99, pp. 1001. Link The shielding tensor $\bm{\sigma}$ is a second-order property that depends upon the external magnetic field, $\mathbf{B}$ , and the spin angular momentum $\bm{m}$ for a given nucleus:

\Delta E=-\bm{m}\bm{\cdot}(\mathbf{1}-\bm{\sigma})\bm{\cdot}\mathbf{B}\;.

(10.87)

Using analytical derivative techniques to evaluate $\bm{\sigma}$ , the components of this $3\times 3$ tensor are computed as

\sigma_{ij}=\sum_{\mu\nu}P_{\mu\nu}\left(\frac{\partial^{2}h_{\mu\nu}}{% \partial B_{i}\partial m_{j}}\right)+\sum_{\mu\nu}\frac{\partial P_{\mu\nu}}{% \partial B_{i}}\frac{\partial h_{\mu\nu}}{\partial m_{j}}

(10.88)

where $i,j\in\{x,y,z\}$ indicate Cartesian components. Note that there is a separate chemical shielding tensor for each $\bm{m}$ , that is, for each nucleus. To compute $\sigma_{ij}$ it is necessary to solve coupled-perturbed SCF (CPSCF) equations to obtain the perturbed densities $\partial P/\partial B_{i}$ , which can be accomplished using the MO-based “MOProp” module whose use is described below. (Use of the MOProp module to compute optical properties of molecules was discussed in Section 10.9.) Alternatively, a linear-scaling, density matrix-based CPSCF (D-CPSCF) formulation is available, ⁹⁴³ Ochsenfeld C., Kussmann J., Koziol F.
Angew. Chem.
(2004), 116, pp. 4585. Link ^, ⁷⁰³ Kussmann J., Ochsenfeld C.
J. Chem. Phys.
(2007), 127, pp. 204103. Link which is described in Section 10.10.3.

In addition to chemical shifts, indirect nuclear spin-spin coupling constants, also known as scalar couplings or $J$ -couplings, can be computed at the SCF level. The coupling tensor $\mathbf{J}^{AB}$ between atoms $A$ and $B$ is evaluated as the second derivative of the electronic energy with respect to the nuclear magnetic moments $\bm{m}$ :

\mathbf{J}^{AB}=\frac{\partial^{2}E}{\partial\bm{m}_{A}\partial\bm{m}_{B}}\;.

(10.89)

The indirect coupling tensor has five distinct contributions. The diamagnetic spin-orbit (DSO) contribution is calculated as an expectation value with the ground state wave function. The other contributions are the paramagnetic spin-orbit (PSO), spin-dipole (SD), Fermi contact (FC), and mixed SD/FC contributions. These terms require the electronic response of the systems to the perturbation due to the magnetic nuclei. Ten distinct CPSCF equations must be solved for each perturbing nucleus, which makes the calculation of $J$ -coupling constants more time-consuming than that of chemical shifts.

Some authors have recommended calculating only the Fermi contact contribution, ⁶⁵ Bally T., Rablen P. R.
J. Org. Chem
(2011), 76, pp. 4818. Link and skipping the other contributions, for $\rm{}^{1}H$ - ${}^{1}H$ coupling constants. For that purpose, Q-Chem allows the user to skip calculation of any of the four contributions: (FC, SD, PSO, or DSO. (The mixed SD/FC contributions is automatically calculated at no additional cost whenever both the SD and FC contributions are computed.) See Section 10.9.3 for details. Note that omitting any of the contributions cannot be rationalized from a theoretical point of view. Results from such calculations should be interpreted extremely cautiously.

Note: 1. Specialized basis sets are highly recommended in any $J$ -coupling calculation. The pcJ- $n$ basis set family ⁵⁹⁷ Jensen F.
J. Chem. Theory Comput.
(2006), 2, pp. 1360. Link has been added to the basis set library. 2. The Hartree-Fock level of theory is not suitable to obtain $J$ -coupling constants of any degree of reliability. Use GGA or hybrid density functionals instead.

10.10.2.1 NMR Job Control and Examples

This section describes the use of Q-Chem’s MO-based CPSCF code, which is contained in the “MOProp” module that is also responsible for computing electric properties. NMR chemical shifts are requested by setting MOPROP = 1, and $J$ -couplings by setting JOBTYPE = ISSC. The reader is referred to to Section 10.9.3 for additional job control variables associated with the MOProp module, as well as explanations of the ones that are invoked in the samples below. An alternative, ${\cal{O}}({N})$ density matrix-based implementation of NMR chemical shifts is also available and is described in Section 10.10.3. Setting JOBTYPE = NMR invokes the density-based code, not the MO-based code.

Example 10.45 MO-based NMR calculation.

$molecule
0  1
  H        0.00000        0.00000        0.00000
  C        1.10000        0.00000        0.00000
  F        1.52324        1.22917        0.00000
  F        1.52324       -0.61459        1.06450
  F        1.52324       -0.61459       -1.06450
$end

$rem
  METHOD             B3LYP
  BASIS              6-31G*
  MOPROP               1
  MOPROP_PERTNUM       0  ! do all perturbations at once
  MOPROP_CONV_1ST      7  ! sets the CPSCF convergence threshold
  MOPROP_DIIS_DIM_SS   4  ! no. of DIIS subspace vectors
  MOPROP_MAXITER_1ST 100  ! max iterations
  MOPROP_DIIS          5  ! turns on DIIS (=0 to turn off)
  MOPROP_DIIS_THRESH   1
  MOPROP_DIIS_SAVE     0
$end

In the following compound job, we show how to restart an NMR calculation should it exceed the maximum number of CPSCF iterations (specified with MOPROP_MAXITER_1ST, or should the calculation run out of time on a shared computer resource. Note that the first job is intentionally set up to exceed the maximum number of iterations, so will crash. However, the calculation is restarted and completed in the second job.

Example 10.46 Illustrates how to restart an NMR calculation. In this first job, we intentionally set the max number of iterations too small, to force premature end so that we can demonstrate restart capability in the 2nd job.

$molecule
0  1
  H        0.00000        0.00000        0.00000
  C        1.10000        0.00000        0.00000
  F        1.52324        1.22917        0.00000
  F        1.52324       -0.61459        1.06450
  F        1.52324       -0.61459       -1.06450
$end

$rem
   METHOD               B3LYP
   BASIS                6-31G*
   SCF_ALGORITHM        DIIS
   MOPROP               1
   MOPROP_MAXITER_1ST   10   ! too small, for demonstration only
   GUESS_PX             1
   MOPROP_DIIS_SAVE     0   ! don’t hang onto the subspace vectors
$end

@@@

$molecule
0  1
  H        0.00000        0.00000        0.00000
  C        1.10000        0.00000        0.00000
  F        1.52324        1.22917        0.00000
  F        1.52324       -0.61459        1.06450
  F        1.52324       -0.61459       -1.06450
$end

$rem
   METHOD               B3LYP
   BASIS                6-31G*
   SCF_GUESS            READ
   SKIP_SCFMAN          TRUE   ! no need to redo the SCF
   MOPROP               1
   MOPROP_RESTART       1
   MOPROP_MAXITER_1ST   100  ! more reasonable choice
   GUESS_PX             1
   MOPROP_DIIS_SAVE     0
$end

Example 10.47 $J$ -coupling calculation: water molecule with B3LYP/cc-pVDZ

$molecule
   0 1
   O
   H1 O OH
   H2 O OH H1 HOH

   OH  = 0.947
   HOH = 105.5
$end

$rem
   JOBTYPE           ISSC
   EXCHANGE          B3LYP
   BASIS             cc-pVDZ
   LIN_K             FALSE
   MOPROP_CONV_1ST   6
   INTEGRAL_SYMMETRY TRUE
$end

In the event that spin-spin couplings of only certain atom pairs are of interest, it is possible to limit the atom pairs for which the couplings are computed. Selection is done via the $spin-spin input section, which is zero-indexed. For example, the section

$spin-spin
0 1 5
$end

would compute couplings between all possible pairings of the first, second, and sixth atoms in the respective $molecule section: (1, 2), (1, 6), and (2, 6). If the $spin-spin section is not specified, couplings between all possible pairs of atoms in $molecule will be computed.

10.10.2.2 Nucleus-Independent Chemical Shifts: Probes of Aromaticity

Unambiguous theoretical estimates of degree of aromaticity are still on high demand. The NMR chemical shift methodology offers one unique probe of aromaticity based on one defining characteristics of an aromatic system: its ability to sustain a diatropic ring current. This leads to a response to an imposed external magnetic field with a strong (negative) shielding at the center of the ring. Schleyer et al. have employed this phenomenon to justify a new unique probe of aromaticity. ¹²⁸⁹ v. R. Schleyer P. et al.
J. Am. Chem. Soc.
(1996), 118, pp. 6317. Link They proposed the computed absolute magnetic shielding at ring centers (unweighted mean of the heavy-atoms ring coordinates) as a new aromaticity criterion, called nucleus-independent chemical shift (NICS). Aromatic rings show strong negative shielding at the ring center (negative NICS), while anti-aromatic systems reveal positive NICS at the ring center. As an example, a typical NICS value for benzene is about $-11.5$ ppm as estimated with Q-Chem at the Hartree-Fock/6-31G* level. The same NICS value for benzene was also reported in Ref. 1289 v. R. Schleyer P. et al.
J. Am. Chem. Soc.
(1996), 118, pp. 6317. Link . The calculated NICS value for furan of $-13.9$ ppm with Q-Chem is about the same as the value reported for furan in Ref. 1289 v. R. Schleyer P. et al.
J. Am. Chem. Soc.
(1996), 118, pp. 6317. Link . Below is one input example of how to the NICS of furan with Q-Chem, using the ghost atom option. The ghost atom is placed at the center of the furan ring, and the basis set assigned to it within the basis mix option must be the basis used for hydrogen atom.

Example 10.48 Calculation of the NMR NICS probe of furan, HF/6-31G* level. Note the ghost atom at the center of the ring.

$molecule
   0 1
   C    -0.69480       -0.62270       -0.00550
   C     0.72110       -0.63490        0.00300
   C     1.11490        0.68300        0.00750
   O     0.03140        1.50200        0.00230
   C    -1.06600        0.70180       -0.00560
   H     2.07530        1.17930        0.01410
   H     1.37470       -1.49560        0.00550
   H    -1.36310       -1.47200       -0.01090
   H    -2.01770        1.21450       -0.01040
   @H    0.02132        0.32584        0.00034
$end

$rem
   JOBTYPE               NMR
   METHOD                HF
   BASIS                 6-31G*
   PURCAR                111
   SCF_CONVERGENCE       7
   NO_REORIENT           1
   POINT_GROUP_SYMMETRY  FALSE
$end

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10.10.2 NMR Chemical Shifts and J-Couplings

10.10.2.1 NMR Job Control and Examples

10.10.2.2 Nucleus-Independent Chemical Shifts: Probes of Aromaticity