10.13 NMR Shielding Tensors

NMR spectroscopy is a powerful technique to yield important information on molecular systems in chemistry and biochemistry. Since there is no direct relationship between the measured NMR signals and structural properties, the necessity for a reliable method to predict NMR chemical shifts arises. Examples for such assignments are numerous, for example, assignments of solid-state spectra [678, 679]. The implementation within Q-Chem uses gauge-including atomic orbitals (GIAOs) [680, 681, 682] to calculate the NMR chemical shielding tensors. This scheme has been proven to be reliable an accurate for many applications [683].

The shielding tensor, $\sigma$ , is a second-order property depending on the external magnetic field, $B$ , and the nuclear magnetic spin momentum, $m_ k$ , of nucleus $k$ :

$\begin{equation} \Delta E=-m_ j (1-\sigma )B \end{equation}$

(10.62)

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

$\begin{equation} \sigma _{ij} =\sum \limits _{\mu \nu } {P_{\mu \nu } } \frac{\partial ^2h_{\mu \nu } }{\partial B_ i \partial m_{j,k} }+\sum \limits _{\mu \nu } {\frac{\partial P_{\mu \nu } }{\partial B_ i }} \frac{\partial h_{\mu \nu } }{\partial m_{j,k} } \end{equation}$

(10.63)

where $i$ and $j$ represent are Cartesian components.

To solve for the necessary perturbed densities, $\partial P/\partial B_{x,y,z}$ , a new CPSCF method based on a density matrix based formulation [684, 685] is used. This formulation is related to a density matrix based CPSCF (D-CPSCF) formulation employed for the computation of vibrational frequencies [686]. Alternatively, an MO-based CPSCF calculation of shielding tensors can be chosen by the variable MOPROP. Features of the NMR package include:

Restricted HF-GIAO and KS-DFT-GIAO NMR chemical shifts calculations
LinK/CFMM support to evaluate Coulomb- and exchange-like matrices
Density matrix-based coupled-perturbed SCF (D-CPSCF)
DIIS acceleration
Support of basis sets up to $d$ functions
Support of LSDA/GGA/Hybrid XC functionals

10.13.1 Job Control

The JOBTYPE must be set to NMR to request the NMR chemical shifts.

D_CPSCF_PERTNUM

Specifies whether to do the perturbations one at a time, or all together.

TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

0

Perturbed densities to be calculated all together.

1

Perturbed densities to be calculated one at a time.

RECOMMENDATION:

None

D_SCF_CONV_1

Sets the convergence criterion for the level-1 iterations. This preconditions the density for the level-2 calculation, and does not include any two-electron integrals.

TYPE:

INTEGER

DEFAULT:

4

corresponding to a threshold of $10^{-4}$ .

OPTIONS:

$n<10$

Sets convergence threshold to $10^{-n}$ .

RECOMMENDATION:

The criterion for level-1 convergence must be less than or equal to the level-2 criterion, otherwise the D-CPSCF will not converge.

D_SCF_CONV_2

Sets the convergence criterion for the level-2 iterations.

TYPE:

INTEGER

DEFAULT:

4

Corresponding to a threshold of $10^{-4}$ .

OPTIONS:

$n<10$

Sets convergence threshold to $10^{-n}$ .

RECOMMENDATION:

None

D_SCF_MAX_1

Sets the maximum number of level-1 iterations.

TYPE:

INTEGER

DEFAULT:

100

OPTIONS:

$n$

User defined.

RECOMMENDATION:

Use the default.

D_SCF_MAX_2

Sets the maximum number of level-2 iterations.

TYPE:

INTEGER

DEFAULT:

30

OPTIONS:

$n$ User defined.

RECOMMENDATION:

Use the default.

D_SCF_DIIS

Specifies the number of matrices to use in the DIIS extrapolation in the D-CPSCF.

TYPE:

INTEGER

DEFAULT:

11

OPTIONS:

$n$

$n$ = 0 specifies no DIIS extrapolation is to be used.

RECOMMENDATION:

Use the default.

10.13.2 Using NMR Shielding Constants as an Efficient Probe 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 [687]. 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 Hartree-Fock/6-31G* level. The same NICS value for benzene was also reported in Ref. vonSchleyer:1996. 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. vonSchleyer:1996. 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 10.253 Calculation of the NMR NICS probe of furan with Hartree-Fock/6-31G* with Q-Chem.

$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 ! ghost at the ring center
$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*
****

0	Perturbed densities to be calculated all together.
1	Perturbed densities to be calculated one at a time.