NMR calculations are available at both the Hartree-Fock and DFT levels of
theory.^{362, 895} Q-Chem computes NMR chemical
shielding tensors using gauge-including atomic
orbitals^{230, 1001, 346} (GIAOs), an
approach that has proven to reliable and accurate for many
applications.^{361, 287} 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:

$$\mathrm{\Delta}E=-\bm{m}\mathbf{\cdot}(\mathrm{\U0001d7cf}-\bm{\sigma})\mathbf{\cdot}\mathbf{B}.$$ | (11.69) |

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

$${\sigma}_{ij}=\sum _{\mu \nu}{P}_{\mu \nu}\left(\frac{{\partial}^{2}{h}_{\mu \nu}}{\partial {B}_{i}\partial {m}_{j}}\right)+\sum _{\mu \nu}\frac{\partial {P}_{\mu \nu}}{\partial {B}_{i}}\frac{\partial {h}_{\mu \nu}}{\partial {m}_{j}}$$ | (11.70) |

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 11.12.) Alternatively, a
linear-scaling, density matrix-based CPSCF (D-CPSCF) formulation is
available,^{677, 510} which is described in
Section 11.13.2.

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 $\bm{m}$:

$${\mathbf{J}}^{AB}=\frac{{\partial}^{2}E}{\partial {\bm{m}}_{A}\partial {\bm{m}}_{B}}.$$ | (11.71) |

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,^{51} and skipping the other contributions, for ${}^{1}\mathrm{H}$-${}^{1}\mathrm{H}$ 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 11.12.2 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^{420} 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.

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 11.12.2 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, $\mathcal{O}(N)$ density matrix-based implementation of NMR chemical shifts is also available and is described in Section 11.13.2. Setting JOBTYPE = NMR invokes the density-based code, not the MO-based code.

$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.

$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

$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 SYMMETRY TRUE MOPROP_CONV_1ST 6 $end

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.^{928} 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. 928. 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. 928. 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.

$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 GH 0.02132 0.32584 0.00034 $end $rem JOBTYPE NMR METHOD HF BASIS mixed SCF_ALGORITHM DIIS PURCAR 111 SEPARATE_JK 0 LIN_K 0 CFMM_ORDER 15 GRAIN 1 CFMM_PRINT 2 CFMMSTAT 1 PRINT_PATH_TIME 1 LINK_MAXSHELL_NUMBER 1 SKIP_SCFMAN 0 IGUESS core SCF_CONVERGENCE 7 ITHRSH 10 IPRINT 23 D_SCF_CONVGUIDE 0 D_SCF_METRIC 2 D_SCF_STORAGE 50 D_SCF_RESTART 0 PRINT_PATH_TIME 1 SYM_IGNORE 1 NO_REORIENT 1 $end $basis C 1 6-31G* **** C 2 6-31G* **** C 3 6-31G* **** O 4 6-31G* **** C 5 6-31G* **** H 6 6-31G* **** H 7 6-31G* **** H 8 6-31G* **** H 9 6-31G* **** H 10 6-31G* **** $end