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Induced fields at proton sites from magnetic dipole model; relating values
to absolute NMR shifts and referenced chemical shifts.
Sankarampadi Aravamudhan
Department of Chemistry
North Eastern Hill University
Shillong 793003 Meghalaya
saravamudhan@hotmail.com
Currently the ab initio quantum mechanical calculations can yield, as a matter of routine,
Absolute NMR shift values in molecules. Absolute shifts are essentially the electronic shielding
values for nuclei measured by the shift in the NMR resonance with respect to the bare nucleus.
When a magnetic dipole model is used for calculating the intra molecular shielding at a proton,
the induced secondary fields at the proton site, due to electron circulations in the neighborhood
is calculated. Since point dipole approximation becomes inapplicable at closer proximities, the
field induced at the proton site by electron circulations around proton cannot be calculated
accurately. A summation method (instead of the well-known integration procedures to evaluate
demagnetization factors) is now available currently. This summation procedure also could
have a provision for improving the validity of the point dipole approximation at much closer
distances. However, this method relies on the susceptibility tensor values of the functional
group within which the proton is also located. Susceptibility tensor being the result of electron
(charge cloud) circulations over the region of the functional group, the region, thus, includes
also the charge cloud at the proton itself. The summation procedure, even though improves the
validity of point dipole approximation, the uniform value of the (a matter of homogeneity)
susceptibility tensor may not be a good approximation. Hence, placing a point dipole at the
electrical centre of gravity within that functional group may result in inaccuracies in the
calculated induced fields at the proton. When comparing the absolute shift values with the
corresponding shifts calculated entirely from magnetic dipole model the inaccuracies may
result in discrepancies.. This, can be considered as a situation indicating the necessity for a
method (link/slide-18) yet to be evolved) for chemical shift referencing when intra molecular
proton shielding is calculated by magnetic dipole model.
The electron charge clouds in the molecules undergo changes in their circulations when an
external magnetic field is present. The changes in the induced fields, thus arising due to the
external fields give rise to the different peak positions for the corresponding nucleus within
the molecule. Such differences in peak positions are measured from the NMR peak of a
reference compound, which are the experimentally measured chemical shifts. When the
differences in peak positions are measured from the resonance peak of bare nucleus (not from
the resonance line reference chemical compound), such values are referred to as the absolute
shielding constants. Shielding and chemical shifts are expressed in units of “parts per
million”- ppm.
The Shielding constants can thus be obtained from experimental NMR spectrum, or by a
suitable ab initio computational quantum chemical method. There are differences
encountered between the experimentally measured quantities (experimental artifacts) and the
theoretical abinitio calculation results (approximations in the methods). It so happens that the
theoretical values have to be subjected to a scaling which results in a better comparison with
the experiment. Such procedures remain, as unsatisfactory approaches only, and the effort to
improve the situations are encountered with arbitrariness at some stage or other without
convincing.
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The induced fields from electron circulations much far away from the location of the
Nucleus, can be subjected to a classical magnetic dipole model for evaluating the shielding
values. However applying such classical methods of calculations for electronic charge clouds
at the nucleus would not be possible because the point dipole approximation required for the
magnetic dipole model may not be upheld as a valid approximation. Hence, such calculations
must use ab initio quantum chemical calculations.
In general, when the data on magnetic susceptibility are available for any specific electron
charge circulation at a location, correspondingly a magnetic dipole moment can be calculated
that would be generated due to the presence of an external magnetic field. This dipole can be
placed at the electrical centre of gravity, or at the geometrical center of gravity depending on
the symmetry prevailing over the charge cloud region. This point dipole can in turn generate
secondary fields at nearby locations to enable a nearby magnetic dipole to interact.
Considering the discussions on the “Theory of Chemical Shifts1”, it is significant to note that
for the interpretation of chemical shifts, the theory can be developed in terms of an atomic
breakdown of diamagnetic currents. The total screening constant for atom A is then written as
in equation-1 as follows with the interpretation of the various terms being given below.
A = dAA + pAA + B(A) AB + A, ring ---- Equation-1
(1) dAA is the contribution to the secondary magnetic field at nucleus A due to the
diamagnetic Langevin-type currents on the atom A itself, which give the corresponding
susceptibility term dA
(2) pAA is the contribution due to the paramagnetic-type currents on atom A which
give the susceptibility term pAA.
(3) AB is the contribution to the screening of atom A by the atomic circulations on
atom B2.
(4) A, ring is the contribution to the screening due to the ring currents1 which cannot be
localized by any atom.
Thus, it is possible to envisage a break up of molecular susceptibility values in terms of
susceptibilities for each atom. An effort was earlier3 made to obtain intra molecular shielding
values for proton in benzene by composing a set of break up values (as was available in
earlier works of several authors in different contexts) with the criterion that such a composed
break up values should add up to the total molecular susceptibility value by appropriate
summing. The significance of the term 3 in equation 1 has been emphasized1, 2, 4 by earlier
authors and several authors more recently, on the interpretation of NMR results.
It is to be emphasized at the outset, that the calculation of shielding constants has been using
the classical magnetic-dipole model (using susceptibility values) mainly for distant
contributions to Shielding; at such distances, that the immediate surroundings of the nucleus
do not have much electron charge overlaps with the region from where the shielding
contribution arises. In a much familiar terms, intermolecular nature for the charge cloud
(relevant orbital) overlap, and use of such classical model was unthinkable for distances close
to bond-lengths within molecule, mainly because of the criteria for the validity of point
dipole approximation.
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However, it has been possible recently to evolve a method to calculate induced fields within
the magnetized material, the demagnetization factors, based on classical magnetic dipole
model5. This inherently had the advantage that the point dipole model can become valid at
much closer proximities to the charge circulation region than it was possible before. In
addition, the method seemed to have no limitation for the validity of point dipole
approximation2. This advantage is based on the fact it was possible to choose any distance in
absolute measure by appropriate fragmentation. It is conditionally necessary to maintain an
appropriate ratio (about 0.2-0.1) for the dimensions of the fragmented region of charge cloud
circulation (in terms of a diameter of a spherical volume) to the distance of location the
induced fields due to this circulation are calculated.
r
R
FIG. 1
r = radius of charge circulation
= > 0.2
R = distance of Nucleus from Dipole moment
FIG.1. Region of charge cloud, the current circulation and the location of nucleus where
the induced field is to be calculated
When the susceptibility is isotropic in region i, the following equation would be applicable:
------ Equation-2
The corresponding Tensor form of the Equation is:
---- Equation-3
From the schematic diagram FIG.2 below, it can be discerned that the Calculations using point dipole
model (if the intra molecular Shielding can be calculated) would become an alternate, independent
method to calculate the shielding constant, depending on the theoretical methods of evaluation and
experimental determination of Susceptibility values. If it is possible to compare the results from all the
three routes, by a mutual inference from one method for the other, it may be possible to arrive at a
consistent set of Susceptibility values, and a theoretical formalism with much less need for arbitrary
scaling.
Calculation of the intra molecular shielding would require considering the molecule as a complete set
of convenient fragments3. Each fragment must have a justifiable susceptibility tensor in such a way,
that the total molecular susceptibility tensor1, 6, 7 can be obtained by the permissible addition of the
fragment tensor values. Each fragment tensor would be characterized by the principal axis directions
in conformity with the local disposition of the symmetry elements if any. Thus, the principal axis
system [PAS] for all the fragment would not be the same, and each one of such PAS would have well
specified direction cosines with respect to the Molecular Principal Axis System. Transformations from
the local principal axis system (for the fragment) to the molecular PAS would be necessary to add all
the tensor elements, without inconsistencies, to obtain the molecular tensor values.
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Absolute Shielding
Values
Experimentally
measured values
Theoretical Values
by QM calculations
Calculations using
Magnetic Point-Dipole
model.
Magnetic Susceptibility
data for molecules,
functional groups, bonded
regions & atoms
FIG 2 The over view of the different methods of calculating (induced fields) Shielding Constant
Categorizing them from the point of view of Experimental and Theoretical approaches
Contribution to the proton shielding from each fragment can be calculated, and proper transformation
properties must be ensured to get the total shielding contribution.
a
c
b
FIG 3 (a) Structure of typical organic molecule (b) Indicating and demarcating the
Charge clouds on atoms and bonded regions and the extent of charge
delocalization (c) The local charge circulations possible within a demarcated region
as a appropriate fragment, and the contribution (calculable from equation-3) to
shielding at one of the protons.
Once a complete set of molecular fragments are identified, then the task is to find appropriate
susceptibility tensor values, which can be attributed to the fragments. Sometimes it would be possible
to arrive at a consensus on the tensor element values gathered from several different sources, and
succeed in calculating the intra molecular proton shielding in the molecules as above ( in Fig.3 ).
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However, justifying the values of the susceptibility tensors used could prove to be intriguing, in the
sense that for one reason it may be justifiable and for a different reasons it may not be so easily
convincing. It is probably this aspect that could be an aid in trying to find out the more about the
requirement of having to scale the shielding values obtained by QM theoretical calculations. In this
particular work, the above considerations would be further illustrated by actually calculating the intra
molecular shielding contribution to protons in benzene.
A complete set of 25 fragments of Benzene molecule could be realized, such that with their
susceptibility tensors in Local PAS, a total Molecular Tensor value can result by summing them in the
molecular reference frame.
Having the fragmented tensors (Fig.4 and Fig.5) and the total tensor (Fig.6), well disposed with their
respective principal directions, in the Benzene molecule, the task of ascertaining the validity of point
dipole approximation for evaluating the contribution from the fragments leads to the following initial
settings:
1. For all the 24 tensor fragments (Fig.) of the 25, the point dipole approximation would have a
reasonable validity with the exception being that C-H () contribution of the bond to which the Proton
is attached. which may be designated as the Proton 12? This disposition and standpoint for the
exception can be visualized as depicted in the following figure (Fig.7).
It becomes evident that the contribution of the 24 tensors (Fig.8) can be calculated by placing a
magnetic dipole moment at the centre of the respective region. On the other hand, for proton 12, the
contribution from the corresponding C-H bulk susceptibility has to be calculated by considering the
volume of charge cloud region for the C-H (). It may appear that the proton 12 is well within the
charge cloud of the C-H  bond. Therefore, the calculation is carried out by subdividing this C-H
volume into smaller volume elements, and then summing up the contributions from the smaller
volume elements to make up the contribution of the total of charges responsible (Fig.8)
FIG. 4
Each one of the fragmented Tensor value can be used with Equation-3 to result in a shielding value at
Proton 1, and the resultant total shielding tensor can be obtained. Thus obtained Total Shielding
tensor can be diagonalised, and the trace can be calculated when the result would be an absolute
shielding for proton 1 corresponding to the experimentally obtainable HR NMR benzene peak value,
referenced to TMS becomes the Benzene proton chemical shift. In FIG-6, the results are displayed
and taking into consideration the required justification for the fragmented susceptibility values, the
intra molecular shielding by this model seems to have resulted in reasonably comparable values. In
particular, the value of 7.4 ppm is more appealing. Subdividing the region into smaller volume
elements and summing up the contributions to shielding from the subdivided volume elements, that is
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the summation procedure, even though improves the validity of point dipole approximation, the
FIG 5a
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uniform value of the (a matter of homogeneity) susceptibility tensor may not be a good approximation.
Hence, placing a point dipole at the electrical centre of gravity within that functional group may result
in inaccuracies in the calculated induced fields at the proton (Fig. 8). When comparing the absolute
shift values with the corresponding shifts calculated entirely from magnetic dipole model the
inaccuracies may result in discrepancies.. This situation can be considered as an indication of a
necessity for evolving method for chemical shift referencing when intra molecular proton shielding is
calculated by magnetic dipole model.
FIG 5b
The actuality is a variation in the charge cloud distribution. The carbon core-electron-cloud has been

taken into consideration in the C atom local
susceptibility (one of the25 fragments). For the
remaining part of the region, there is a single susceptibility tensor assigned and for practical reasons
further fragmentation of the C-H region does not seem worth the while. On the other hand, the region
of C-H bond itself is only about 4 Aº 3 in volume within which the variation (charge cloud gradient) not
insignificant. Hence even while subdividing this volume into smaller volume elements, cannot have
susceptibility values proportional to the volume of the smaller volume elements that is implicit in the
procedure adapted for C-H region contribution at the H atom. This induced field thus calculated would
not lead to the Absolute shielding values to the required level accuracy and confidence.
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FIG 6
FIG 7: Fragments indicated by the possible shape of the
charge distribution demarcating each fragment from the
neighboring one. The shapes relevant for the  and π are
depicted.
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Image-01
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FIG 8.
FIG 9.
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Each one of the fragmented Tensor value can be used with Equation-2 to result in a shielding value at
Proton 1, and the resultant total shielding tensor can be obtained. Thus obtained Total Shielding
tensor can be diagonalised, and the trace can be calculated when the result would be an absolute
shielding for proton 1 corresponding to the experimentally obtainable HR NMR benzene peak value,
referenced to TMS becomes the Benzene proton chemical shift. In FIG-6, the results are displayed
and taking into consideration the required justification for the fragmented susceptibility values, the
intra molecular shielding by this model seems to have resulted in reasonably comparable values. In
particular, the value of 7.4 ppm is more appealing. Subdividing the region into smaller volume
elements and summing up the contributions to shielding from the subdivided volume elements, that is
the summation procedure, even though improves the validity of point dipole approximation, the
uniform value of the (a matter of homogeneity) susceptibility tensor may not be a good approximation.
Hence, placing a point dipole at the electrical centre of gravity within that functional group may result
in inaccuracies in the calculated induced fields at the proton (Fig. 8). When comparing the absolute
shift values with the corresponding shifts calculated entirely from magnetic dipole model the
inaccuracies may result in discrepancies.. This situation can be considered as an indication of a
necessity for evolving method for chemical shift referencing when intra molecular proton shielding is
calculated by magnetic dipole model.
FIG 10 a. Homogeneous charge cloud
distribution can result in a Susceptibility is a
homogeneous in the region.
The actuality is a variation in the charge cloud distribution. The carbon core-electron-cloud has been

taken into consideration in the C atom local
susceptibility (one of the25 fragments). For the
remaining part of the region, there is a single susceptibility tensor assigned and for practical reasons
further fragmentation of the C-H region does not seem worth the while. On the other hand, the region
of C-H bond itself is only about 4 Aº 3 in volume within which the variation (charge cloud gradient) not
insignificant. Hence even while subdividing this volume into smaller volume elements; the smaller
elements cannot have susceptibility values proportional to the volume of the elements that is implicit
in the procedure adapted for C-H region contribution at the H atom. This induced field thus calculated
would not lead to the Absolute shielding values to the required level accuracy and confidence.
Locally on the hydrogen atom, besides the participation of the electron in the C-H bonding, a part of
the electron would remain as the core Hydrogen atom charge cloud. Hence, the contribution from this
core should be calculated as the induced field due to the H atom local diamagnetic susceptibitity and
should be added to the induced field obtained by magnetic dipole approximation for the C-H region.
When the C-h bond is placed farther from the H atom at which Shielding is calculated, then, a point
dipole placed at the center of the region seems a reasonable for obtaining the shielding values at
distant protons. As it is it seems that the value of the C-H fragment used must be subjected much
closer examination of the regional variation of the charge densities, when it is a question of calculating
the C-H region contribution to the H atom bonded to that carbon. The current state of this calculation8
as described above is summarized in the following Fig 9 & 12.
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FIG 10 b.
C
H
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FIG 12
Image-02
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From image-02 above and Fig-12 it would be possible to make an assessment of the Shielding trends
and the Electron Densities for varying distance between the Hydrogen atoms.
Image-02a
Image-02b
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6 Valence
Electrons
1 Valence
Electron
H
H
C
In Benzene Molecule (By semi
empirical QM) on H atom partial
charge is + 0.1926 (electrons
belonging Hydrogen atom= 0.8074)
In Benzene Molecule (By semi
empirical QM) on C atom partial
charge is -0.1926 (6.1926 electrons
belong to C atom)
H
C
FIG 13
While forming benzene molecule 0.1926 electrons from H atom has been
transferred to C atom. These number of electrons values relate to point electron
charges. Charge density maps on basis of charge cloud / orbital overlap basis
would yield a picture somewhat similar to the description below. Contrasting
complementary colors chosen for partial charges of opposite sign.
H
C
FIG.14
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Image-04
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References:
1. The theory of Chemical Shifts, J.A.Pople, Discussions of the Faraday Society, Vol.34, page-7,
I-NUCLEAR MAGNETIC RESONANCEIN DIAMAGNETIC MATERIALS (1962).
Theory of the Chemical Shift in Aromatic Heterocycles, G.G. Hall, A.Hardisson, and L.M.Jackman,
Disc. Farad. Soc., Vol.34, page-15, (1962).
2. Pople, Proc. Roy. Soc. A, Vol.239, p-550 (1957). McConnel, J. Chem. Physics, Vol. 27, p-226
(1957), http://www.ugc-inno-nehu.com/UH_PMR_Biphenyl_ExptThery.pdf
3. http://saravamudhan.tripod.com/id2.html
http://aravamudhan-s.ucoz.com/amudhan20012000/ismar_ca98.html
4, “Nuclear Magnetic Resonance Spectroscopy”, by F.A.Bovey, Academic Press 1969.
5. http://nehuacin.tripod.com/pre_euromar_compilation/index.html
http://www.ugc-inno-nehu.com/isc2009nehu.html
http://ugc-inno-nehu.com/ISC2014/ISC2014-abstract-fullpaper-SA.doc
http://www.ugc-inno-nehu.com/cmdays2011/0_3_16Aug2011.ppt
http://www.ugc-inno-nehu.com/rscfd177/rcma2005-fp.pdf
6. P.W.Selwood, Magnetochemistry, Interscience Publishers, New York, 1943
7. W.H. Flygare, Chemical Reviews, Vol.74, p-682 (1974)
8. http://www.ugc-inno-nehu.com/rscfd177/draft-full-paper-SA-RSC-FD177.doc