Chemical Shift Tensor ConventionsContents: [Online converter] [Chemical shift / shielding] [Chemical shift tensors] [Standard] [HerzfeldBerger] [Haeberlen] [References] 
It is recommended that the IUPAC conventions [1] are obeyed:
Shifts, commonly used in solution and solid state NMR studies, are thus positive to high frequency. Absolute shieldings are positive to low frequency, and are only accessible via theoretical calculations. The establishment of a correspondence between a chemical shift scale and a magnetic shielding scale is not a trivial task and requires both careful theoretical calculations and experimental measurements [2].
The nuclear magnetic shielding (absolute shielding) is the molecular electronic property. The chemical shift is a quantity that we experimentalists have defined and use because of our inability to directly measure the absolute magnetic shielding. This inability results from our inability to know the magnitude of the magnetic field to an accuracy on the order of parts per billion, independent of the resonance experiment [3].
The symbol σ should only be used for absolute shieldings. Often, however, authors use a "pseudo" shielding scale, where the "shielding" is obtained by simply reversing the sign of the chemical shift. In our opinion, this adds only to the confusion without providing any additional insight.
Depending on the local symmetry at the nuclear site, the magnitude of the chemical shift will vary as a function of the orientation of the molecule with respect to the external magnetic field. This orientation dependence of the chemical shift is referred to as chemical shift anisotropy (CSA). Mathematically, the chemical shift anisotropy is described by a secondrank tensor (a 3 by 3 matrix), which in the case of the symmetric part of the chemical shift (CS) tensor consists of six independent components. Generally, one is able to express the chemical shift tensor in a coordinate frame where all offdiagonal elements vanish. In this principal axis system, the chemical shift tensor is fully described by the three diagonal elements  the principal components  and the three eigenvectors or Euler angles describing the orientation of the principal axes with respect to an arbitrary frame. In addition, various combinations of the principal components (and their orientations) are in use to describe the chemical shift tensor. Some of these conventions will be described in the next sections.

In the standard convention, the principal components of the chemical shift tensor, (δ_{11}, δ_{22}, δ_{33}), are labeled according to the IUPAC rules [4]. They follow the high frequencypositive order. Thus, δ_{11} corresponds to the direction of least shielding, with the highest frequency, while δ_{33} corresponds to the direction of highest shielding, with the lowest frequency. The isotropic values, δ_{iso}, are the average values of the principal components, and correspond to the center of gravity of the line shape.
The term "standard convention" has been coined by me to distinguish it from the other conventions. Of course, other groups will call in turn their convention the standard. So far, NMR spectroscopists are still fighting about the best way to report tensors. I think that it doesn't really matter as long as people clearly state their definitions.
My programs use this convention!

In the HerzfeldBerger notation [5], a tensor is described by three parameters,
which are combinations of the principal components in the standard notation:
The isotropic value, i.e., the centre of gravity, is the average value of the principal components.
The span describes the maximum width of the powder pattern.
The skew of the tensor is a measure of the amount and orientation of the asymmetry of the
tensor. As indicated, κ is given by 3a/Ω.
Depending on the position of δ_{22} with respect to
δ_{iso}, the sign is either positive or negative. If δ_{22}
equals δ_{iso}, a and the skew are zero. In the
case of an axially symmetric tensor, δ_{22} equals
either δ_{11} or δ_{33} and a = Ω/3. Hence, the skew is ±1.
The parameter μ used with the HerzfeldBerger tables is related to the span of a tensor by:
μ = Ω*ν_{ref}/spinning rate
The parameter ρ used with the HerzfeldBerger tables corresponds to the skew of a tensor described here. The HerzfeldBerger convention is related to the Standard convention via:
δ_{22} = δ_{iso}+κ Ω/3
δ_{33} = (3δ_{iso}δ_{22}Ω)/2
δ_{11} = 3δ_{iso}δ_{22}δ_{33}
Note that the exact formulation of the span, Ω, contains the factor (1σ_{ref}) [3]:
Ω = (δ_{11}δ_{33})(1σ_{ref})

The HaeberlenMehringSpiess convention uses different combinations of the
principal components to describe the line shape [6]. This convention
requires that the principal components are ordered according to their separation from the
isotropic value.
The centre of gravity of the line shape is described by the isotropic value, which is the average
value of the principal components.
The anisotropy and reduced anisotropy describe the largest separation from the centre of
gravity. (The term reduced anisotropy is not used in the literature, but I introduce it here in
order to be able to distinguish between "Haeberlen's δ" and Δδ ;
the use of the symbol δ is somewhat unfortunate, because it is more commonly used to
represent chemical shifts.) The sign of the anisotropy indicates on which side of the isotropic value one can
find the largest separation.
The asymmetry parameter indicates by how much the line shape deviates from that of an
axially symmetric tensor. In the case of an axially symmetric tensor, a =
(δ_{yy}  δ_{xx}) will be zero and hence η = 0.
Unfortunately, there are also some conventions in which the order of the components δ _{yy} and δ_{xx} is interchanged, as well as other definitions of Δδ.
The HaeberlenMehring convention is related to the Standard convention via:
for δ > 0 (i.e. δ_{zz} = δ_{11})  for δ < 0 (i.e. δ_{zz} = δ_{33})  
δ_{11} = δ_{iso}+δ  δ_{33} = δ_{iso}+δ  
δ_{22} = δ_{iso}δ(1η)/2  δ_{22} = δ_{iso}δ(1η)/2  
δ_{33} = δ_{iso}δ(1+η)/2  δ_{11} = δ_{iso}δ(1+η)/2 
The program SIMPSON by M. Bak uses this convention!
[ Anorg. Chemie ]  [ Go Home ]  webm@ster  last modified: 01.04.2022