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Nuclear magnetic shielding

 NMR spectrum!shielding tensor Cartesian $ \epsilon\tau$ component of the nuclear magnetic shielding tensor $ \sigma$$ _K$

$\displaystyle \sigma_{K,\epsilon\tau} = \left.\frac{\partial^2 {\cal E}^{el} \!...
...}\, \partial B_{\tau}}\right\vert _{\mbox{{\boldmath$m$}}_K = 0, {{\bf B}} = 0}$ (6.3)

nuclear magnetic moment $ \ m $$ _K =
\hbar \gamma_K {\bf I}_K$, where $ \gamma_K$ and $ {\bf I}_K$ are the gyromagnetic ratio and spin of nucleus $ K$, $ {{\bf B}}$ - external magnetic field, $ {\cal E} \!\left({\mbox {\boldmath $m$}}_K,{{\bf B}} \right)$ - energy. When the direct Zeeman interaction between the nucleus and the field is added we have

$\displaystyle \left.\frac{\partial^2 {\cal E} \!\left({\mbox{\boldmath$m$}}_K,{...
...\boldmath$m$}}_K = 0, {{\bf B}} = 0} = \mbox{\boldmath$\sigma$}_{K} - {\bf {1}}$ (6.4)

Calculation type
average value $ < H^{K,B}>$ 1.Average!$ <$ $ H^{K,B}$ $ >$ and linear response, $ \langle\langle $ $ {H^{K,{\rm PSO}}}$; l $ \rangle\rangle $ 2.Linear!$ <<$ $ {H^{K,{\rm PSO}}}$; l$ >>$

Symmetry

The nuclear shielding is a tensor of rank 2. The full rank-2 tensor can be written as a sum of three tensors of rank 0, 1 and 2, respectively

$\displaystyle \mbox{\boldmath$\sigma$}$$\displaystyle = \sigma_{iso}\left[\begin{array}{ccc} 1 & 0 & 0\\  0 & 1 & 0\\  ...
..._{xz}\\  d_{yx} & d_{yy} & d_{yz}\\  d_{zx} & d_{zy} & d_{zz}\end{array}\right]$ (6.5)

where the isotropic shielding $ \sigma_{iso}$ is

$\displaystyle \sigma_{iso} = \frac{1}{3}\left(\sigma_{xx} + \sigma_{yy} + \sigma_{zz}\right)$ (6.6)

the antisymmetry parameters are

$\displaystyle \sigma^{A}_{\mu\nu} = \frac{1}{2}\left(\sigma_{\mu\nu}-\sigma_{\nu\mu}\right)$ (6.7)

and finally the parameters

$\displaystyle d_{\mu\nu} = \frac{1}{2}\left(\sigma_{\mu\nu} + \sigma_{\nu\mu} - 2\sigma_{iso}\right)$ (6.8)

The rank-1 tensor is antisymmetric, and thus contains only three distinct elements, whereas the rank-2 tensor is symmetric and, with zero trace, has five independent elements. These definitions are discussed in ref [70]. Recently, a new standardized definition that relates the tensor components to the experimentally measured quantities has been proposed [71].

In most cases, the shielding tensors are expressed in the principal axis system, which is the coordinate system for which the rank-2 tensor is diagonal. In this case, the rank-1 tensor (antisymmetric part) is disregarded and eq 6.5 can be reduced to

$\displaystyle \mbox{\boldmath$\sigma$}$$\displaystyle ^{PAS} = \sigma_{iso}\left[\begin{array}{ccc} 1 & 0 & 0\\  0 & 1 ...
...{PAS} & 0 & 0\\  0 & d_{yy}^{PAS} & 0\\  0 & 0 & d_{zz}^{PAS}\end{array}\right]$ (6.9)

The usual definitions of the anisotropy and asymmetry of the shielding are

$\displaystyle \Delta\sigma = \sigma_{33}^{PAS} - \frac{1}{2}(\sigma_{11}^{PAS} + \sigma_{22}^{PAS})$ (6.10)

and

$\displaystyle \eta = \frac{(\sigma_{22}^{PAS} - \sigma_{11}^{PAS})} {(\sigma_{33}^{PAS} - \sigma_{iso})}$ (6.11)

where it is assumed that $ \sigma_{33}^{PAS} \geq \sigma_{22}^{PAS} \geq \sigma_{11}^{PAS}$. Other quantities sometimes used are the span, $ \sigma_{33}^{PAS} - \sigma_{11}^{PAS}$ and the skew, $ 3(\sigma_{iso} - \sigma_{22}^{PAS} )/
(\sigma_{33}^{PAS} - \sigma_{11}^{PAS})$  [71].

The $ S$ and $ A$ parameters are defined as

$\displaystyle S^2$ $\displaystyle =$ $\displaystyle (\Delta\sigma)^2(1 + \frac{1}{3} \eta^2)$ (6.12)
$\displaystyle A^2$ $\displaystyle =$ $\displaystyle (\sigma^{A}_{12})^2
+ (\sigma^{A}_{13})^2
+ (\sigma^{A}_{23})^2$ (6.13)

Comments

The antisymmetric part does not affect the standard NMR spectrum, and that is why the rank-1 tensor is disregarded above.

The partition of the shielding into diamagnetic and paramagnetic contributions depends on the approach used. For perturbation-independent basis sets it depends on the chosen gauge origin.

The paramagnetic contribution to the shielding obtained for the gauge origin at the shielded nucleus is related to the spin-rotation constant, see eqs. 6.326.33.

Units
dimensionless, expressed in ppm (parts per million),

to obtain ppm the computed values are multiplied by 10 $ ^6 \times \alpha_{fs}^2\cong $ 53.25136

             Shielding tensors in symmetry coordinates (ppm)

  Shielding constant:     42.1100 ppm
  Anisotropy:           -173.0362 ppm
  Asymmetry:                .1242

  S parameter:           173.4807 ppm
  A parameter:             6.9190 ppm

  Total shielding tensor (ppm):

                  Bx             By             Bz

  C1   x    92.57339868     9.80120030      .00000000
  C1   y    -4.03684899   -73.19735078      .00000000
  C1   z      .00000000      .00000000   106.95399445

...
  Antisymmetric and traceless symmetric parts (ppm):

               Bx         By         Bz             Bx         By         Bz

  C1   x      .0000     6.9190      .0000        50.4634     2.8822      .0000
  C1   y    -6.9190      .0000      .0000         2.8822  -115.3074      .0000
  C1   z      .0000      .0000      .0000          .0000      .0000    64.8440

  Principal values and axes:

  C1    1    106.953994  =   42.11  +  64.84:      .000000   .000000  1.000000
  C1    2     92.623495  =   42.11  +  50.51:      .999849   .017379   .000000

                     ! Summary of chemical shieldings !
 Definitions from J.Mason, Solid state Nuc.Magn.Res. 2 (1993), 285
 @1atom   shielding       dia      para     skew      span     (aniso     asym)
 @1C1       42.1100  366.7367 -324.6267    -.8410  180.2014   97.2660    2.5580
 Definitions from Smith, Palke and Gerig, Concepts in Mag.Res. 4 (1992), 107
 @2atom   shielding       dia      para     aniso      asym        S        A
 @2C1       42.1100  366.7367 -324.6267 -173.0362     .1242  173.4807    6.9190


next up previous contents index
Next: Shielding polarizabilities Up: Nuclear-spin related properties Previous: NMR effective spin Hamiltonian   Contents   Index
Michal Jaszunski 2002-11-10