12/21/2015 8:53:32 PM 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 Link for Abstract & Fullpaper:Abstract & Fullpaper February 07,

12/21/2015 8:53:32 PM2 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 R ij and an angle θ ij to be defined R ij θ ij  = Susceptibility To ascertain the magnetic moment  it is necessary to be sure about the Susceptibility  12/21/2015 8:53:32 PM3 e

 = Susceptibility Homogeneous through out the sample 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 12/21/2015 8:53:32 PM4

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 The total molecular magnetic susceptibility would be given by Hence the molecule can be characterized by a total magnetic 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 12/21/2015 8:53:32 PM5 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.

H H O O O O O O  i = {χ i. (1-3.COS 2 θ)}/(R i ) 3 Tensor form Isotropic susceptibility form Demagnetization effects Shape dependent demagnetization factors 12/21/2015 8:53:32 PM6 Equations used for such calculation of secondary fields described in the previous slide

H H O O O 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 12/21/2015 8:53:32 PM7 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. 12/21/2015 8:53:32 PM8 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

12/21/2015 8:53:32 PM 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. 9

12/21/2015 8:53:32 PM ? 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 10 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 = 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.

12/21/2015 8:53:32 PM 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. 11

12/21/2015 8:53:32 PM χ C-H (σ) Set of 6 Centers C H x x  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

12/21/2015 8:53:32 PM 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 ppm Such a calculation yielded a value for isotropic shielding contribution

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12/21/2015 8:53:32 PM 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 =

12/21/2015 8:53:32 PM16 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 MD GIAO MD GIAO MD ISOTROPIC GIAO MD From ab initio Ethylene Contribution from the nearest carbon atom 12/21/2015 8:53:32 PM 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 ( ppm) (+2.77 ppm) ( 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 17 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 As displayed earlier, a value of was obtained for C-H ( σ) bond of proton ppm

The localized atom circulations induces fields and this to be added to make it absolute shifts 12/21/2015 8:53:32 PM18 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

12/21/2015 8:53:32 PM19 Calculated isotropic shielding/chemical shifts 24 susc tensors ppm ppm ppm One C-H bond susc isotropic isotropic -total Abs shift= Abs shift H ppm = approx 25.6 ppm δ=7.4ppm Abs shift= Abs shift H ppm = approx 22.7 ppm δ=10.3ppm Bare nucleus 0 33 ppm TMS reference Increasing “Shielding” effect 33 ppm 0 Increasing “Deshielding” effect ‘ δ’ 17.5 ppm 15.5 ppm χ. H 0 = -σ.H 0 χ. H 0 for field strength unity will numerically be the same but dimensions would be that of field “gauss”

12/21/2015 8:53:32 PM 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

12/21/2015 8:53:32 PM 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: 21