04/21/23 01:49 PM

Calculating Intra-molecular Proton Shielding Tensors Using Magnetic Dipole model; Possible Procedures and

Prerequisites

S.AravamudhanDepartment of Chemistry, North Eastern Hill University,

Shillong 793022 Meghalaya; INDIA

http://saravamudhan.tripod.com/id2.html

http://nehuacin.tripod.com/id6.html

http://www.ugc-inno-nehu.com/isc2009nehu.html

saravamudhan@hotmail.com

Link: acceptance for Oral presentation O-5 from Sectional President Chemical Sciences, ISC2014

Link for Abstract & Fullpaper:https://www.nitrocloud.com/p/PM1TPz3rJmHXmmQ9S_sKZ_

February 07, 2014

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Click for Research interests & publications

Professor S ARAVAMUDHAN

+

Nucleus at the Centre of the circulating charge cloud

(under the influence of primary moment)

Representing circulating electron charge cloud and (defined direction of current flow) associated induced field (moment)

This nucleus can come under the influence of the

neighboring charge cloud due to the secondary

fields i

j

This Secondary field influence depends on the internuclear distance Rij and

an angle θij to be defined

Rij

θij

extiiH = Susceptibility

To ascertain the magnetic moment it is

necessary to be sure about the Susceptibility

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e

extiiH extii

H

extiiH

= Susceptibility

Homogeneous through out the sample

Inhomogeneous Homogeneous

Induced field within the specimenHomogeneous field distribution is amenable for further calculations more

easily. In these cases a magnetic moment can be placed as effective for the field inside the specimen.

Above was the case of macroscopic specimen-What about charge circulations within A MOLECULE??

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These are intra molecular charge clouds

circulating each unit ‘i’ may be described with a magnetic susceptibility and hence in presence of an external field each would be characterized by a induced magnetic dipole moment

i

extiiH

HHtotal

ii

iitotal

The total molecular magnetic susceptibility would be given by

i

itotal

Hence the molecule can be characterized by a total magnetic moment

i

itotal

Thus,

Having considered the intra molecular summations and break ups, the next step is to build up sums for the ensemble of molecules in the material medium 04/21/23 01:49 PM 5

PROCEDURE: STEPS IN SEQUENCEDelineate the specific regions of electron Currents

Locate the moments (proportional to susceptibility) generated at the appropriate Center within the region

Calculate the secondary field due to this moment at the proton site

Sum up contributions from all fragmented regions to make up the whole.

H

H

O

OO

O

O O

i = {χi . (1-3.COS2θ)}/(Ri)3

Tensor form

Isotropic susceptibility form

Demagnetization effectsShape dependent demagnetization factors

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Equations used for such calculation of secondary fields described in the previous slide

H

H

O

OO

O

O O

A Charge density map representation of this molecule

Representing the circulation of charges

The magnetic moments and the secondary field consequences would be considered in the next slide

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H

NOTE that this task of subdividing the benzene molecule into 25 smaller regions and appropriately subdividing the Susceptibility Tensor also has been accomplished in such a way that the divided values when added up results in the total value comparable to the experimental values.

The details of molecular fragments and the corresponding local fragmented susceptibility tensor values would be dealt with in the subsequent slides.04/21/23 01:49 PM 8

Moments placed

Center of C-C bonds - 6

Center of C-H bonds - 6

On C atoms – 6 Set-1

On C atoms – 6 Set-2

Center of ring -1

RESULTS IN THIS PRESENTATION ARE ON BENZENE – SHIELDING OF AROMATIC PROTON

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C-H bond distance=1.087 A⁰

C-C bond length= 1.4A⁰

Angle C-C-C =120⁰

Angle C-C-H= 120⁰

-2.05 10-6

-3.05 10-6

-3.05 10-6

χC-C (σ)

Set of 6 Centers

C

CH

-4.21 10-6

-3.41 10-6

-4.21 10-6

χC-H (σ)

Set of 6 Centers

C

H

χC(localized π contribution)

10.8 10-6

7.9 10-6

6.5 10-6 Set of 6 Centers

C

H χC (atomic, diamagnetic, Contribution) IsotropicC

-9.35 10-6

-9.35 10-

6

-9.35 10-

6

Set of 6 Centers

-33.0 10-6

0.0 10-6

0.0 10-6

χC(delocalized π contribution) At ring center

One set only

Thus, these are 25 subdivided tensors with each molecular fragment which when added return the whole molecule.

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?

The spheres closely placed leave voids and which is in actuality filled by material medium. Hence a better approximation would be to place a cube at the place of the sphere and this would amount to change in the material volume and the Susceptibility per unit volume has to be multiplied by volume of cube instead of sphere in the formula. Volume of Cube / sphere =1.91 ratio

6 C-C σ

5 C-H σ

6 C atom diamagnetic

6 C atom local π

1 ring center delocalized π

1.2 Aº

=0.1 Aº

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Contribution of this C-H bond should be calculated differently

For the contributions at the

proton numbered 12

Thus the entire region for the C-H sigma contribution can be filled with close-packing small spheres, whose dimensions are all of such small radius that the ratio distance to proton ‘R’ / radius ‘r’ can be 10 which is in conformity for the point dipole approximation to be valid for the content of each of the close packing spheres.

With 0.1 Aº radius of the inner cavity, the circumference would be 2.π. 0.1=(6.28* 0.1) =0.628 Aº. With an angle of 2.5º as equal interval between the radius vectors from proton, there would be 144 divisions and the division length would be 0.628/144 =0.00436 Aº. Entire length of the circumference of inner cavity can be close packed with exact number 144 spheres of radius 0.00436 Aº. The ratio 0.1 (R)/0.00436 (r) =22.9358 > the required ratio 10. The procedure of close packing would ensure that this ratio is held true for every one of the spheres. Thus the summing procedure (essentially based on magnetic dipole model) for the calculation of demagnetization factors of ellipsoidal material specimen can be well integrated with the source program for the intra molecular proton shielding of molecules at the appropriate groups when for that group the point dipole model becomes gross violation for realistic values to be the result.

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The results displayed till now:-

1. Feasibility of finding susceptibility (break-up) values for the molecular fragments which on proper addition result in the experimentally measured molecular susceptibility tensor. (Slide #7)

2. That these fragmented susceptibility values (Slide #8) of a molecule, may be representing the actual electron circulations in the fragmented groups and hence, a magnetic moment would be generated at the (electrical centre of gravity of the) functional group, when the molecule is placed in an external magnetic field. (Slide #6)

3. Then these induced magnetic moments can be, in turn, producing secondary magnetic fields within the molecular fragment. These induced secondary magnetic fields relate to the (chemical shifts) shielding tensors for the protons at the various locations within the molecule. (Slide #6)

4. Hence, the possibility of calculating such shielding tensors of protons in a molecule.

What remains to be considered?

5. When the proton is located within the regions of electron circulations, the point- dipole approximations may not be adequate for extending the magnetic dipole model.

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-4.2110-6

-3.4110-6

-4.2110-6

χC-H (σ)

Set of 6 Centers

C

H

-7.4135 x 10-5

- 9.1580 x 10-5

- 9.1580 x 10-5

1.2 Aº

=0.1 Aº

C

H

Molar Susceptibility Tensor

Volume Susceptibility Tensor

( Molar value / Avagadro number)=Molecular value

(Molecular value / Volume of one molecule) = Volume susceptibility value 12

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H

C

1.08 Aº

0.3 Aº

R1

R1 = 0.15 Aº

CN=R1/r1 =10.0

r1 = 0.15/10.0 = 0.015 Aº

r1= 0.015 Aº

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log

log

1 1

CCR

R

n

n

= 1+ [log (1.08/0.15) / log (11.0/9.0)] = 1+ [ 0.8573 / 0.0872 ] = 1+ 9.8318 = 10.8318 (4/3) x π x r1

3 = v1 = 1.4143e-5Aº 3

0.00001413 Aº 3 =1.4143 x 10-29 cm3

Benzene Mol wt = 6 x 12 + 1 x6 = 72 + 6 = 78C =12 ; H=1, C-H = 13 gms = 1 mole of C-H = wt of 6.023 x 1023 C-H units

Volume of Cylinder = π x r2 x l = 22/7 x 0.152 x 1.38 = 0.09759 Aº 3 = 0.09759 x 10-24 cm3

-4.21 x 10-6 cgs units per mole = -4.21 x 10-6 / 6.023 x 1023 = -0.6990 x 10-29 cgs per one C-H unit

-0.6990 x 10-29 cgs units is per 0.09759 Aº 3 = per 0.09759 x 10-24 cm3 = 9.759 x 10-26 cm3

Per unit volume = (-0.6990 x 10-29 cgs units)/ 0.09759 x 10-24 = - 7.16262 x 10-5 cgs units

-3.41 x - 7.16262 / -4.21 = - 5.8015 x 10-5 cgs units

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-2.8841 ppm

Such a calculation yielded a value for isotropic shielding contribution

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The (locally) diagonal Tensors (in their respective X”,Y”,Z” frames) of the various parts of Benzene are all to be transformed to a common Molecular axis system X,Y,Z. The

transformation matrices are obtained with the corresponding direction cosines.

Coordinats of C atomsMidpoints of Cn+1-Cn

dm originC1 =90⁰ 1.4 Aº 0.0000 Aº 1.4000 Aº 0.0000 Aº -0.6062, 1.0500, 0.0000 [C2-C1]C2 =150⁰ 1.4 Aº -1.2124 Aº 0.7000 Aº 0.0000 Aº -1.2124, 0.0000, 0.0000 [C3-C2]C3 =210⁰ 1.4 Aº -1.2124 Aº -0.7000 Aº 0.0000 Aº -0.6062,-1.0500, 0.0000 [C4-C3]C4 =270⁰ 1.4 Aº 0.0000 Aº -1.4000 Aº 0.0000 Aº 0.6062,-1.0500, 0.0000 [C5-C4]C5 =330⁰ 1.4 Aº 1.2124 Aº -0.7000 Aº 0.0000 Aº 1.2124, 0.0000, 0.0000 [C6-C5]C6 =390 ≡30⁰ ⁰ 1.4 Aº 1.2124 Aº 0.7000 Aº 0.0000 Aº 0.6062, 1.0500, 0.0000 [C1-C6]

Coordinates of C atoms

Midpoint of C-H, location of Dipole, DM originC1-H1 =90⁰ 1.9435 Aº 0.0000 Aº 1.9435 Aº 0.0000 Aº

C2 =150⁰ 1.9435 Aº -1.6831 Aº 0.9718 Aº 0.0000 Aº

C3 =210⁰ 1.9435 Aº -1.6831 Aº -0.9718 Aº 0.0000 Aº

C4 =270⁰ 1.9435 Aº 0.0000 Aº -1.9435 Aº 0.0000 Aº

C5 =330⁰ 1.9435 Aº 1.6831 Aº -0.9718 Aº 0.0000 Aº

C6 =390 ≡30⁰ ⁰1.9435 Aº 1.6831 Aº 0.9718 Aº 0.0000 Aº

Midpoint of C-H, location of Dipole, DM origin 1.4+0.5(1.087)=1.9435 Proton CoordinatesC1-H1 =90⁰ 2.4870 Aº 0.0000 Aº 2.4870 Aº 0.0000 Aº

C2 =150⁰ 2.4870 Aº -2.1538 Aº 1.2435 Aº 0.0000 Aº

C3 =210⁰ 2.4870 Aº -2.1538 Aº -1.2435 Aº 0.0000 Aº

C4 =270⁰ 2.4870 Aº 0.0000 Aº -2.4870 Aº 0.0000 Aº

C5 =330⁰ 2.4870 Aº 2.1538 Aº -1.2435 Aº 0.0000 Aº

C6 =390 ≡30⁰ ⁰ 2.4870 Aº 2.1538 Aº 1.2435 Aº 0.0000 Aº

Proton Coordinates 1.4+1.087=2.4870

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The Dipole model calculation results in values which are comparable break up values as from different contexts for point dipole approximation. Actual comparisons require the consideration of “Absoulte” Shifts and Chemical shifts referenced to TMS, and the values in δ & τ Scales -

GIAO 26.9047

MD 40.07 57.6603

GIAO 27.9521

MD 17.64 5.7837

GIAO 23.6179

MD 14.08 3.2237

ISOTROPIC

GIAO 26.1582

MD 23.93 22.22

From ab initio EthyleneContribution from the nearest carbon atom

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The Shielding tensor component values: In black fonts: Ab initio QM resultsIn blue fonts: Dipole model results with 22 fragments, and one C-H bond by filling the region with closed packed spheres. (slide#9 &10)

In brown fonts: The spheres closely placed leave voids and which is in actuality filled by material medium. Hence a better approximation would be to place a cube at the place of the sphere and this would amount to change in the material volume and the Susceptibility per unit volume has to be multiplied by volume of cube instead of sphere in the formula. Volume of Cube / shpere =1.91 ratio

1

2

3

4

5

6

7

8

9

10

11

( -0.67 ppm) (+2.77 ppm) ( -7.56 ppm)

(17.31)

(47.63)

(11.31)

22

MD Blue are with value (Quantum Chemical calculation ) for ethylene included.MD Green values are only 22 tensors ,without ethylene values

6- C-C ()

5- C-H ()

5- C- atom Localized π

5- C- atom diamagnetic

1- delocalized π

π

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The Dipole model calculation results in values which are comparable break up values as from different contexts for point dipole approximation. Actual comparisons require the consideration of “Absoulte” Shifts and Chemical shifts referenced to TMS, and the values in δ & τ Scales -

Accounting for the contribution of the C-H bond of proton 12

-1.82

As displayed earlier , a value of -2.8841 was obtained for C-H (σ) bond of proton 12 -2.8841 ppm

The localized atom circulations induces fields and this to be added to make it absolute shifts

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Localized circulation at atom

Circulation effects from adjacent atoms/groups

When Only Circulation effects from adjacent atoms/groups are calculated by a method, then

17.5 ppm 15.5 ppm

Hydrogen atom limit

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Calculated isotropic shielding/chemical shifts

24 susc tensors

--5.510 ppm

-2.785 ppm

-13.40 ppm

-11.93

19.33

-11.93

One C-H bond susc

-7.2317-2.8841

-10.1157

isotropic

isotropic -total

Abs shift= Abs shift H + 10.1157 ppm15.5 +10.1157= approx 25.6 ppm δ=7.4ppmAbs shift= Abs shift H + 7.1157 ppm15.5 + 7.2317 = approx 22.7 ppm δ=10.3ppm

Bare nucleus

0 33 ppmTMS reference

Increasing “Shielding” effect

33 ppm 0Increasing “Deshielding” effect ‘δ’

17.5 ppm 15.5 ppm

χ. H0 = -σ.H0

χ. H0 for field strength unity will numerically be the same but dimensions would be that of field “gauss”

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How well the results of Slide #16 compare with the experimental values of NMR shifts of benzene (isotropic neat liquid values) & the various aromatic proton shielding tensor values ( referenced to ‘0’ value of TMS ) obtained by experimental HR PMR studies on single crystal specimen?

The final report in the previous slide would have to be further elaborated to find out the validity of magnetic dipole model for such shielding tensor calculations as much as the quantitative demagnetization effects have been reported till now.

Now, that the possibility of comparing such magnetic model calculations of shielding tensors with experimental values and the values obtained by ab initio quantum chemical calculations could be found viable, this makes possible the various theoretical formalisms of quantum chemical approaches (applicable for calculating both, the susceptibility tensor & shielding tensor) to be assessed and in turn the method to improve the magnetic dipole model, which has the more convincing possibility of calculating without much computational effort, and tractable in terms of classically describable secondary fields and point dipoles.

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The significance of the results in slides # 15 to18 would be for a full presentation at a later time.

What was to be emphasized at this juncture in the evolution of this method is the Procedure (the method of calculation) and the possibility of comparison with QM results and experimental values. And, the factors to be considered during such comparison have been pointed out.

Calculating Intra-molecular Proton Shielding Tensors Using Magnetic Dipole model; Possible Procedures and

Prerequisites

Subject matter for this lecture:

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