Subject: Chemical Shift and Shielding Calculations

From: [email-masked] (Mark J Forster )
Date: Tue, 28 Nov 1995 10:03:11 -0800

Subject: Chemical Shift and Shielding Calculations. 

Seiji Mori asked the following question:

>   Dear Sirs,
> 
>  I posted to CCL about the methods of NMR calculation a fey days ago , and
> had an other question.
> In G94, I tried to calculate the chemical shifts , there are magnetic
> shieldings
snip... 
> Would you please tell me the physical meaning of isotropic  and anitrosopy, 
> and which is an equivalent of the experimental chemical shift? It is better
> that 
> you show the references. 
> 
>     Sincerely yours,
>     Seiji Mori email:[email-masked]

The property calculated by an ab initio shielding scheme is typically a
nuclear magnetic shielding tensor in the form of a 3x3 matrix. In 
physical terms the elements of the tensor represent the shielding along
x,y and z when the applied magnetic field is along x,y and z in turn.
This matrix may then be separated into symmetric and antisymmetric 
components, when the former is diagonalised then one obtains the three
principal components of the shielding tensor denoted 

	sigma(1,1) sigma(2,2) and sigma(3,3) 

with the convention 

	sigma(3,3) >= sigma(2,2) >= sigma(1,1)

the isotropic part of the shielding tensor is the trace or average of
these three components and represents the value expected for an atom
in a molecule that tumbles rapidly and isotropically. 
	trace = ( sigma(1,1) + sigma(2,2) + sigma(3,3)) /3

In non isotropic
systems such as single crystals or oriented liquid crystals the shielding
takes on an orientational dependence. The shielding anisotropy is defined
as 
	anisotropy = sigma(3,3) - ( sigma(1,1)+sigma(2,2) )/2

and is a measure of the deviation of the shielding from spherical symmetry.
For heavier nuclei the shielding anisotropy can be large (eg  > 5000 ppm 
in 199Hg and 205Tl compounds) and this can often be a significant 
mechanism for nuclear spin relaxation.

In addition the shielding asymmetry may be defined as

	asymmetry = (sigma(2,2)-sigma(1,1) ) / (sigma(3,3) -trace)

An axially symmetric shielding tensor has a non zero anisotropy but a
zero value for the asymmetry.

The chemical shift measured by NMR is the difference between 
the shielding tensors for a nucleus in the compound under study and in the 
reference compound (typically TMS for 1H, 13C and 29Si). So to compare
ab initio computed shielding with experimental chemical shifts two 
calculations will need to be performed one for the molecule under study
and one for the reference compound. In addition experimental shifts are
often found to be sensitive to environmental conditions such as solvent
used, temperature, salt concentration etc. These are usually not taken 
into consideration in an ab initio calculation and so the comparison may
only be valid if the intrinsic shift variations across a range of compounds
far outweigh the environmental effects. 

No references to hand at the moment but I hope this helps.

Best Wishes       __________________________________________________________
Mark             /
         ___    /                     
        /  /|  /                     Mark J Forster
       /  / | /                     Biosym / Molecular Simulations. 
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