10/28/20151 Calculating Intra-molecular Proton Shielding Tensors Using Magnetic Dipole model; Possible Procedures and Prerequisites S.Aravamudhan Department of Chemistry, North Eastern Hill University, Shillong Meghalaya; INDIA Link: acceptance for Oral presentation O-5 from Sectional Presidentacceptance for Oral presentation O-5 Chemical Sciences, ISC2014 February 07, 2014 Link for Abstract & Fullpaper:Abstract & Fullpaper _ 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.

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. 10/28/20152

‘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 cm 3 A typical value for Organic molecules (Diamagnetic) can be a convenient value and x 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 = x x [(4/3) π r 3 ] χ v x (1- 3 cos 2 θ) / R 3 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 r 3 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 10/28/20153

Figure i n R 3 = ‘R i ’ R 4 = ‘R i+1 ’ R i+1 = R i + r i + r i Eq.1 If R i / r i = C, C=constant for all ‘i’ values then, R i = C x r i and [R i+1 / R i ] = [r i+1 / r i ] From Eq.1 [R i+1 / r i+1 ] = [R i / r i+1 ] + [r i / r i+1 ] + 1 [R i+1 / r i+1 ] = [(C x r i ) / r i+1 ] + [r i / r i+1 ] + 1 [R i+1 / r i+1 ] = [ (C+1) { r i / r i+1 }] + 1 C = [ (C+1) { r i / r i+1 }] + 1 (C -1) = (C +1) [R i / R i+1 ] R i+1 = R i (C+1)/(C-1) R i = R i-1 (C+1)/(C-1) R 2 = R 1 (C+1)/(C-1) R 3 = R 2 (C+1)/(C-1) Therefore, R 3 = R 1 (C+1)/(C-1) (C+1)/(C-1) = R 1 {(C+1)/(C-1)} 2 R n = R 1 {(C+1)/(C-1)} n-1 Hence, R n / R 1 = {(C+1)/(C-1)} n-1 Log(R n / R 1 ) = n-1 [log{(C+1)/(C-1)}] ‘n’ = 1+ {log(R n / R 1 )}/ [log{(C+1)/(C-1)}] Radius vector ‘R’,’θ’ and ‘φ’ The possibility of close packing of subdivided spheres of the specimen is considered ; and for which an equation could be derived. 10/28/20154 The derivation

R1R1 RnRn Radial Vector defined by a polar angle θ w.r.to Z Polar angle R1R1 RnRn 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 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’. σ θ =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= R i / r i, 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 = ; radius = C = R/r = 0.25 / = [ / ] = Log ( ) = (r/R) 3 = e-5 = /28/20155

Calculating Intra-molecular Proton Shielding Tensors Using Magnetic Dipole model; Possible Procedures and Prerequisites The molecule considered in this presentation is BENZENE The 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 regions 10/28/20156

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. 10/28/20157

8 C-H bond distance=1.087 A ⁰ C-C bond length= 1.4A ⁰ Angle C-C-C =120 ⁰ Angle C-C-H= 120 ⁰ χ C-C (σ) Set of 6 Centers C CHCH χ C-H (σ) Set of 6 Centers C H χ C( localized π contribution) Set of 6 Centers C H χ C (atomic, diamagnetic, Contribution) Isotropic C Set of 6 Centers χ C( 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.

10/28/20159 ? 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 = Aº. Entire length of the circumference of inner cavity can be close packed with exact number 144 spheres of radius Aº. The ratio 0.1 (R)/ (r) = > 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

10/28/ χ C-H (σ) Set of 6 Centers C H x x  1.2 Aº =0.1 Aº C H

10/28/ H C 1.08 Aº 0.3 Aº R1R1 R 1 = 0.15 Aº CN=R 1 /r 1 =10.0 r 1 = 0.15/10.0 = Aº r 1 = Aº = 1+ [log (1.08/0.15) / log (11.0/9.0)] = 1+ [ / ] = = (4/3) x π x r 1 3 = v 1 = e-5Aº Aº 3 = x cm 3 Benzene Mol wt = 6 x x6 = = 78 C =12 ; H=1, C-H = 13 gms = 1 mole of C-H = wt of x C-H units Volume of Cylinder = π x r 2 x l = 22/7 x x 1.38 = Aº 3 = x cm x cgs units per mole = x / x = x cgs per one C-H unit x cgs units is per Aº 3 = per x cm 3 = x cm 3 Per unit volume = ( x cgs units)/ x = x cgs units x / = x cgs units

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10/28/ 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 atoms Midpoints of C n+1 -C n dm origin C1 =90⁰ 1.4 Aº Aº Aº Aº , , [C 2 -C 1 ] C2 =150⁰ 1.4 Aº Aº Aº Aº , , [C 3 -C 2 ] C3 =210⁰ 1.4 Aº Aº Aº Aº , , [C 4 -C 3 ] C4 =270⁰ 1.4 Aº Aº Aº Aº0.6062, , [C 5 -C 4 ] C5 =330⁰ 1.4 Aº Aº Aº Aº1.2124, , [C 6 -C 5 ] C6 =390⁰≡30⁰ 1.4 Aº Aº Aº Aº0.6062, , [C 1 -C 6 ] Coordinates of C atoms Midpoint of C-H, location of Dipole, DM origin C1-H1 =90⁰ Aº Aº Aº Aº C2 =150⁰ Aº Aº Aº Aº C3 =210⁰ Aº Aº Aº Aº C4 =270⁰ Aº Aº Aº Aº C5 =330⁰ Aº Aº Aº Aº C6 =390⁰≡30⁰ Aº Aº Aº Aº Midpoint of C-H, location of Dipole, DM origin (1.087)= Proton Coordinates C1-H1 =90⁰ Aº Aº Aº Aº C2 =150⁰ Aº Aº Aº Aº C3 =210⁰ Aº Aº Aº Aº C4 =270⁰ Aº Aº Aº Aº C5 =330⁰ Aº Aº Aº Aº C6 =390⁰≡30⁰ Aº Aº Aº Aº Proton Coordinates =2.4870

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10/28/ 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.

10/28/ GIAO MD GIAO MD GIAO MD ISOTROPIC GIAO MD The Shielding tensor component values: In black fonts: Ab initio QM results In 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

10/28/ 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.

10/28/ 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.