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S.Aravamudhan Department of Chemistry, North Eastern Hill University,

Calculating Intra-molecular Proton Shielding Tensors Using Magnetic Dipole model; Possible Procedures and Prerequisites. Link : acceptance for Oral presentation O-5 from Sectional President Chemical Sciences, ISC2014

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S.Aravamudhan Department of Chemistry, North Eastern Hill University,

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  1. Calculating Intra-molecular Proton Shielding Tensors Using Magnetic Dipole model; Possible Procedures and Prerequisites Link: acceptance for Oral presentation O-5from Sectional President Chemical Sciences, ISC2014 Link for Abstract & Fullpaper: https://www.nitrocloud.com/p/PM1TPz3rJmHXmmQ9S_sKZ_ S.Aravamudhan Department of Chemistry, North Eastern Hill University, Shillong 793022 Meghalaya; INDIA saravamudhan@hotmail.com http://saravamudhan.tripod.com/id2.html http://nehuacin.tripod.com/id6.html http://www.ugc-inno-nehu.com/isc2009nehu.html February 07, 2014

  2. Click for Research interests & publications Professor S ARAVAMUDHAN

  3. Rij To ascertain the magnetic moment it is necessary to be sure about the Susceptibility Representing circulating electron charge cloud and (defined direction of current flow) associated induced field(moment) + e This Secondary field influence depends on the internuclear distanceRij and an angleθij to be defined This nucleus can come under the influence of the neighboring charge cloud due to the secondary fields j θij i Nucleus at the Centre of the circulating charge cloud (under the influence of primary moment)  = Susceptibility

  4. Homogeneous through out the sample  = Susceptibility Inhomogeneous Homogeneous Induced field within the specimen Homogeneous 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?? NEXT SLIDE

  5. The total molecular magnetic susceptibility would be given by Hence the molecule can be characterized by a total magnetic moment PROCEDURE: STEPS IN SEQUENCE Delineate 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. 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 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

  6. H O O O O O O H Demagnetization effects Shape dependent demagnetization factors Equations used for such calculation of secondary fields described in the previous slide Tensor form Isotropic susceptibility form i={χi . (1-3.COS2θ)}/(Ri)3

  7. H O O O O O O H A Charge density map representation of this molecule H Representing the circulation of charges The magnetic moments and the secondary field consequences would be considered in the next slide

  8. 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. RESULTS IN THIS PRESENTATION ARE ON BENZENE – SHIELDING OF AROMATIC PROTON 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 The details of molecular fragments and the correspondinglocal fragmented susceptibility tensor values would be dealt with in the subsequent slides.

  9. -2.05 10-6 χC-C (σ) CH -3.05 10-6 C -3.05 10-6 Set of 6 Centers -3.41 10-6 χC-H (σ) 0.0 10-6 H χC(delocalizedπ contribution) At ring center -4.21 10-6 C One set only -4.21 10-6 Set of 6 Centers 0.0 10-6 -33.0 10-6 10.8 10-6 χC(localizedπ contribution) H C 7.9 10-6 Set of 6 Centers 6.5 10-6 -9.35 10-6 χC(atomic, diamagnetic, Contribution) Isotropic C -9.35 10-6 Set of 6 Centers -9.35 10-6 Thus, these are 25 subdivided tensors with each molecular fragment which when added return the whole molecule. C-H bond distance=1.087 A⁰ C-C bond length= 1.4A⁰ Angle C-C-C =120⁰ Angle C-C-H= 120⁰

  10. 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. For the contributions at the proton numbered 12 Contribution of this C-H bond should be calculated differently ? 6 C-C σ 5 C-H σ 6C atom diamagnetic 6C atom local π 1 ring center delocalized π 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 1.2 Aº =0.1 Aº

  11. 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.

  12. -7.4135 x 10-5 - 9.1580 x 10-5 - 9.1580 x 10-5 -3.4110-6 χC-H (σ) H -4.2110-6 C -4.2110-6 Set of 6 Centers H Molar Susceptibility Tensor C Volume Susceptibility Tensor 1.2 Aº =0.1 Aº ( Molar value / Avagadro number)=Molecular value (Molecular value / Volume of one molecule) = Volume susceptibility value

  13. 0.3 Aº R1 = 0.15 Aº H CN=R1/r1 =10.0 R1 r1 = 0.15/10.0 = 0.015 Aº 1.08 Aº r1= 0.015 Aº C = 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 r13 = 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 = 78 C =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 10-24cm3 -4.21 x 10-6cgs 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 Such a calculation yielded a value for isotropic shielding contribution -2.8841 ppm

  14. 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. Coordinates of C atoms Proton Coordinates 1.4+1.087=2.4870 Midpoint of C-H, location of Dipole, DM origin 1.4+0.5(1.087)=1.9435

  15. 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 & τScales -

  16. 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) GIAO 26.9047 MD 40.0757.6603 (47.63) MD Blue are with value (Quantum Chemical calculation ) for ethylene included. MD Green values are only 22 tensors ,without ethylene values From ab initio Ethylene Contribution from the nearest carbon atom ( -7.56 ppm) ( -0.67 ppm) (+2.77 ppm) 7 11 10 9 8 1  Accounting for the contribution of the C-H bond of proton 12 3 22 GIAO 27.9521 MD 17.64 5.7837 6- C-C () (17.31) 5- C-H () As displayed earlier , a value of -2.8841 was obtained for C-H (σ) bond of proton 12 2 5- C- atom diamagnetic 5 6 4 -2.8841 ppm π -1.82 5- C- atom Localized π ISOTROPIC GIAO 26.1582 MD 23.93 22.22 1- delocalized π GIAO 23.6179 MD 14.08 3.2237 (11.31) 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 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 & τScales -

  17. Hydrogen atom limit 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 The localized atom circulations induces fields and this to be added to make it absolute shifts

  18. Increasing “Deshielding” effect ‘δ’ 33 ppm 0 0 33 ppm Increasing “Shielding” effect Bare nucleus TMS reference χ. H0for field strength unity will numerically be the same but dimensions would be that of field “gauss” Calculated isotropic shielding/chemical shifts isotropic -2.785 ppm -7.2317 24 susc tensors 19.33 -2.8841 One C-H bond susc isotropic -total χ. H0 = -σ.H0 -10.1157 --5.510 ppm Abs shift= Abs shift H + 10.1157 ppm 15.5 +10.1157= approx 25.6 ppm δ=7.4ppm -11.93 -13.40 ppm Abs shift= Abs shift H + 7.1157 ppm 15.5 + 7.2317 = approx 22.7 ppm δ=10.3ppm -11.93 17.5 ppm 15.5 ppm

  19. 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.

  20. Subject matter for this lecture: Calculating Intra-molecular Proton Shielding Tensors Using Magnetic Dipole model; Possible Procedures and Prerequisites 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. Thank you

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