NMR spectrum!shielding tensor Cartesian component of the nuclear magnetic shielding tensor
(6.3) |
nuclear magnetic moment , where and are the gyromagnetic ratio and spin of nucleus , - external magnetic field, - energy. When the direct Zeeman interaction between the nucleus and the field is added we have
(6.4) |
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
(6.6) |
(6.7) |
(6.8) |
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
(6.9) |
(6.11) |
The and parameters are defined as
(6.12) | |||
(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. 6.32, 6.33.
Units
dimensionless, expressed in ppm (parts per million),
to obtain ppm the computed values are multiplied by 10 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