04/20/23 1

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 February 07, 2014

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

This is a provisional presentation file

The actual presentation file for this lecture would also be made available for download as soon as the final version ready.

http://www.ugc-inno-nehu.com/isc2014-venue.ppt

Magnetic Dipole Model: Principle of the method for calculating induced field distributions:

Consider a specimen or a region of charge cloud, which has a homogeneous

magnetic (volume) susceptibility tensor, v with a well defined principal axis system-PAS.

The boundary of this region/specimen should be describable by mathematical equation(s).That is, a regular shape like, cube, cylinder, ellipsoid or a sphere

When the specimen is placed in an external magnetic field, a magnetic moment is induced and it would be depicted to have been located at the electrical centre of gravity (may coincide with the geometrical centre) of the specimen / region.

The method described here consists of fragmenting (subdividing) this total tensor (a total value) and distributing over the entire extent of the specimen/region.

04/20/23 2

‘R’ Is the distance of the magnetic moment from the point (where the induced field value is to be known). The distance is along the radial Vector specified by its Polar angle

Z-Axis; the Direction of Magnetic Field: H

Polar Angle θ

Susceptibility is ISOTROPIC

In these calculations, VOLUME Susceptibility is used. That is specified as χ cm3 A typical value for Organic molecules (Diamagnetic) can be a convenient value and -2.855 x 10 -7 units can be typical per cc. of the material

Commensurate with the Susceptibility the magnetic moment M would be in accordance

with the equation M = χ v x H

The equation for induced field based on a dipolar model would then be

σ = -2.855 x 10-7 x [(4/3) π r3] χv x (1- 3 cos2 θ) / R3

From the above equation it is obvious that along this radial vector with the specified polar angle if spherical volume elements of the material are placed such that they all have the ‘radius- r’ to ‘distance-R’ ratio the same, then every one of such sphere would contribute the same induced field at the specified point.

r is the radius of the spherical magnetized material specifically demarcated. (4/3) π r3 will be the

spherical volume of the material at a distance R contributing at the point of origin in the illustration on the left

IN THE NEXT SLIDE: The possibility of close packing of subdivided spheres of the specimen is considered ; and for which an equation could be derived .

To Calculate the induced field due to the magnetic moment at this site

04/20/23 3

The possibility of close packing of subdivided spheres of the specimen is considered ; and for which an equation could be derived .

04/20/23 4

The derivation

R1

Rn

Radial Vector defined by a polar angle θ w.r.to Z

Polarangle

R1 Rn

Z-Axis; The Direction of magnetic field

This circular base of the cone with apex angle equal to the polar angle θ,

has radius equal to ‘R sin θ’: See Textbox below

11

log

log

1 1

CCRR

n

n

Using the equation ‘n’ along the vector length is calculated, for the direction with polar angle θ. Which is ‘σ’ per spherical magnetic moment x number of such spheres ‘n’. σθ =σ x n. At the tip of the vector, there is circle along which magnetic moment have to be calculated. This circle has radius equal to ‘R sinθ’. The number of dipoles along the length of the circumference = 2 π R sinθ/2.r = π R/ r sinθ. Again, (R/ r) is a constant by earlier criteria.

The three dimensional perspective of this procedure

Equation for calculating the number of spheres, “number of dipole moments” , along the radial vector is as given below:

With “C= Ri / ri, i=1, n”

For a sphere of radius =0.25 units, and the polar angle changes at intervals of 2 .5˚

There will be 144 intervals. Circumference= 2π/4 so that the diameter of each sphere on the circumference = 0.0109028; radius = 0.0054514

C = R/r = 0.25 / 0.0054514 = 45.859779 [46.859779/44.859779] = 1.04458334

Log (1.0445834) = 0.0189431 (r/R) 3=1.0368218e-5 =0.000010368218

04/20/23 5

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

The molecule considered in this

presentation is BENZENEThe peripheral protons are all related by symmetry and hence calculating the shielding tensor of one of the ring Protons would enable the other proton values to be deduced by appropriate transformation.

Molecular, total Susceptibility value.

The method described here consists of fragmenting (subdividing) this total tensor (a total value) and distributing over the entire extent of the specimen/region.

This means the molecule would be subdivided appropriately into smaller regions; then correspondingly the Total Susceptibility also would be subdivided to be assigned to the smaller regions04/20/23 6

NOTE that this task of subdividing the benzene molecule into 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.

04/20/23 7

04/20/23 8

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.

04/20/23 9

?

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.

1.2 Aº

=0.1 Aº

04/20/23 10

-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

04/20/23 11

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

11

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

04/20/23 12

04/20/23 13

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 ⁰2.4870 Aº

2.1538 Aº 1.2435 Aº 0.0000 Aº

Proton Coordinates 1.4+1.087=2.4870

04/20/23 14

04/20/23 15

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.

2. That these fragmented susceptibility values 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.

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 #14)

4. 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.

04/20/23 16

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

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

04/20/23 17

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.

04/20/23 18

The significance of the results in slides # 16 &17 have been considered for a full presentation at a later time.

Hence this aspect was not included in the full paper submitted.

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 while such comparison would be unambiguous have been pointed out.

The actual effectiveness and consequential studies have been kept in abeyance for the present as pointed out earlier.

The actual presentation file for this lecture would also be made available for download as soon as the final version ready.

http://www.ugc-inno-nehu.com/isc2014-venue.ppt