Hopf Fibration on Relationship Between Dihedral Angle θ<sub>HnHn+1</sub>[deg] and Vicinal Angle ϕ[deg], Angles Calculated from NMR Data with 3-Shere Approach and Java Script

Carmen-Irena Mitan; Emerich Bartha; Constantin Draghici; Miron-Teodor Caproiu; Petru Filip; Robert Michael Moriarty

doi:doi:10.11648/j.sjc.20221001.13

| Peer-Reviewed

Hopf Fibration on Relationship Between Dihedral Angle θ_HnHn+1[deg] and Vicinal Angle ϕ[deg], Angles Calculated from NMR Data with 3-Shere Approach and Java Script

Carmen-Irena Mitan, Emerich Bartha, Constantin Draghici, Miron-Teodor Caproiu, Petru Filip, Robert Michael Moriarty

Published in Science Journal of Chemistry (Volume 10, Issue 1)

Received: 30 January 2022 Accepted: 19 February 2022 Published: 28 February 2022

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

3-Sphere theory, a hypersphere in four dimensions, is applied for calculation dihedral angles with the right stereochemistry and sign in D, -L ribitol series from proton and carbon chemical shift (Δδ_XnXn+1[ppm], X = H, C) and vicinal coupling constant (³J_HnHn+1[Hz]) with Java Script. A method in three steps, easy to calculate by hand or with Java Script program: 1. prediction of the dihedral angle only from ³J_HnHn+1[Hz], 2. calculation the angle of set A with manifold equation (conic section, Villarceau circles) from chemical shift, 3. building of the seven sets unit or six sets units, from which is chose an angle almost equal with the predicted one having its stereochemistry and sign. Angles of set A and set B, relationships between vicinal angle and dihedral angle (X₀-X₁₅) are introduced instead of polar angle and azimuthal angle in spherical coordinates (eq. 1 versus eq. 3). Hopf coordinates, trigonometric equations, confirmed by algebraic equations are disclosed for all cis-ae/ea, trans-ee, trans-aa stereochemistry. Octonionic fibration S⁷→S¹⁵→S⁸ in R¹⁶, with real fibration S⁰ →S¹ →S¹ as unit, reassembles all possible stereochemistry gives by the HCCH fragment on two congruent disks, each centered on the perimeter of the other with equilateral triangles as vertices. Complex Hopf fibration in R⁴ ensuring the calculation of the dihedral angle from vicinal angle and vice versa, demonstrating the relationships between sets A, B, C.

Published in	Science Journal of Chemistry (Volume 10, Issue 1)
DOI	10.11648/j.sjc.20221001.13
Page(s)	21-31
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2022. Published by Science Publishing Group

Keywords

3-sphere, Hopf Fibration, Villarceau Circles, Dihedral Angle, Vicinal Angle, Vicinal Coupling Constant, Java Script

References

[1]	B. Coxon, Developments in the Karplus equation as they relate to the NMR coupling constants of carbohylates, Adv. Carb. Chem. Biochem (2009), 62, 17, doi.org./10.1016/50065-2318(09)00003-1.
[2]	http://en.wikipedia.org/wiki/X: X = a. Vesica_piscis, b. Hopf_coordinates, c. conical_surface, d. Euler_angles.
[3]	C.-I. Mitan, E. Bartha, P. Filip, C. Draghici, M. T. Caproiu, R. Moriarty, NMR data and 3-sphere approach on calculation dihedral angles of iminocyclitols with Java Script, ACS National Meeting, 5 – 30 April 2021, Live virtual events Macromolecular chemistry: the second century. ANYL ID: 3549263, oral presentation, morressier, 39 pag. Publisher: American Chemical Society, Washington; 10.1021/scimeetings.1c00922.
[4]	C.-I. Mitan, E. Bartha, C. Draghici, M. T. Caproiu, P. Filip, L. Tarko, R. Moriarty, Dihedral angles beyond unit with seven sets on 3-sphere approach, ACS VIRTUAL National Meeting, 16-20 August 2020, Moving chemistry from bench to market, ANYL 22, oral presentation, morressier, 32 pag. Publisher: American Chemical Society, Washington, D. C.; DOI.10.1021/scimeetings.Oc06576.
[5]	E. Bartha, C.-I. Mitan, C. Draghici, M. T. Caproiu, P. Filip, R. M. Moriarty, Program for prediction dihedral angle from vicinal coupling constant with 3-sphere approach, Rev. Roum. Chim (2021), 66, 178-183; DOI: 10.33224/rrch.2021.66.2.08 (Eng).
[6]	H. Hopf, Über die abbildungen der dreidimensionalen sphäre auf die kugelfläche, Mathematische Annalen (1931), 104, 637, doi: 10.1007/BFo1457962.
[7]	D. W. Lyons, An elementary introduction to the Hopf fibration, Mathematic Magazine (2003), 76, 87, doi: 10.230713219300, lyons@lvc.edu.
[8]	J. V. Waite, The Hopf fibration and encoding torus knots in light fields, University of Nevada, Las Vegas may 2016: http://digitalscholarship.unlv.edu/theses dissertations, http://dx.doi.org/10.34917/9112204.
[9]	L. Moroni, The toric sections: a simple introduction, arXiv: 1708.00803v2 [math.HO] 6 aug 2017.
[10]	H. Gluck, F. Warner, W. Ziller, The geometry of the hope fibrations, L’Enseignement Mathématique, (1986), 32, 173.
[11]	L. Ornea, M. Parton, P. Piccinni, V. Vulturescu, Spin (9) geometry of the octonionic hopf fibration, arXiV: 1208.0899v2 [math.DG] 8Jul 201; DOI: 10.1007/S00031.
[12]	http://ncatlab.org/nlab/show/Hopf_fibration.
[13]	M. Zamboj, Synthetic construction of the Hopf fibration in the double orthogonal projection of the 4-space, arXiV: 2003.09236v2 [math.HO] 6 Jan 2021; DOI 10.10931/jcde/qwabo18.
[14]	C. E. Durán, A. Mendoza, A. Rigas, Blakers-Massey elements and exotic diffeomorphisms of S6 and S14 via geodesics, Transactions of the American Mathematical Society (2004), 356, 5025; DOI:https://doi.org/10.1090/50002-9947-04-03469-5.
[15]	L. Garnier, H. Barki, S. Toufou, L. Puech, Computation of Yvon-Villarceau circles on Dupin cyclides and construction of circular edge right triangles on tori and Dupin cyclides, Computers and Mathematics with Applications (2014), 68, 1689; http://dx.doi.org/10.1016/j,camwa.2014.10.020.
[16]	M. Schrott, B. Odehnal, Ortho-Circles of Dupin Cyclides, Journal for Geometry and Graphics (2006), 10, 73; http://www.heldemabb.de/J66/J6610/jgg10.htm.
[17]	C.-I. Mitan, E. Bartha, P. Filip, C. Draghici, M. T. Caproiu, L. Tarko, R. Moriarty, Proton chemical shift and vicinal coupling constant ratio on conformational analysis of iminocyclitols with α-D ribose stereochemistry, ACS VIRTUAL National Meeting, 16-20 August 2020, Moving chemistry from bench to market, ANYL 48, poster; demo - morressier. Publisher: American Chemical Society, Washington, D. C.; DOI.10.1021/scimeetings.Oc06567.
[18]	R. M. Moriarty, C. I. Mitan, N. Branza-Nichita, K. R. Phares, D. Parrish, The exo-imino to endo-iminocyclitol rearrangement. A general route to five membered antiviral azasugars, Org. Lett. (2006), 8, 3465, doi.org/10.1021/ol061071r.
[19]	C.-I. Mitan, E. Bartha, P. Filip, C. Draghici, M. T. Caproiu, R. M. Moriarty, Calculation of the dihedral angles of 1-α,-β-R-5-O methanesulfonyl-2,3-O-isopropilidene-L-lyxose from NMR data with 3-sphere approach and Lambert-Wu R value, ACS National Meeting, Fall 2021 August 22-26, Atlanta, GA, Resilience of Chemistry, poster ID: 3582748, Carb Row 5 – Virtual.
[20]	C.-I. Mitan, E. Bartha, P. Filip, C. Draghici, M. T. Caproiu, R. Moriarty, Relationships between vicinal constant couplings ³J_HH, ¹H and ¹³C-chemical shifts and torsional angles, 257^th ACS National Meeting in Orlando, Florida, March 31- April 4, 2019, Chemistry for the New Frontiers, ANYL 392, morressier, 39 pag. Publisher: American Chemical Society, Washington, D. C.
[21]	https://mathworld.wolfram.com/VillarceauCircles.
[22]	H. A. Taha, M. R. Richard, T. L. Lowary, Conformational analysis of furanoside containing mono- and oligosaccharides, Chem. Rev. (2013), 113, 1851, doi.org/10.1021/cr300249c.
[23]	C.-I. Mitan, E. Bartha, A. Hartopeanu, C. Stavarache, C. Draghici, M. T. Caproiu, M. Manganu, I. Man, L. Tarko, F. Teodorescu, C. Deleanu, Anomeric ratio and dihedral angles of α-, β-iodo-methyl-D-ribofuranose, ACS VIRTUAL National Meeting, 16-20 August 2020, Moving chemistry from bench to market, ANYL 49, poster; demo - morressier. Publisher: American Chemical Society, Washington, D. C.; DOI.10.1021/scimeetings.Oc06576.
[24]	B. L. Stocker, E. M. Dangerfield, A. L. Win-Mason, G. W. Haslett, M. S. M. Timmer, Recent developments in the synthesis of pyrroline containing iminosugars, Eur. J. Org. Chem. (2010), 1615, Doi.org/10.1002/ejoc.201090021.
[25]	G. Perry, D. Miller, Beginning Programming in 24 hours, Third edition, SAMS Indianapolis, Indiana, 46240 USA.
[26]	W3schols.com, http://www.w3schools.com, HTML, CSS, Java Script.

Cite This Article

Plain Text BibTeX RIS

APA Style

Carmen-Irena Mitan, Emerich Bartha, Constantin Draghici, Miron-Teodor Caproiu, Petru Filip, et al. (2022). Hopf Fibration on Relationship Between Dihedral Angle θHnHn+1[deg] and Vicinal Angle ϕ[deg], Angles Calculated from NMR Data with 3-Shere Approach and Java Script. Science Journal of Chemistry, 10(1), 21-31. https://doi.org/10.11648/j.sjc.20221001.13

Copy | Download

ACS Style

Carmen-Irena Mitan; Emerich Bartha; Constantin Draghici; Miron-Teodor Caproiu; Petru Filip, et al. Hopf Fibration on Relationship Between Dihedral Angle θHnHn+1[deg] and Vicinal Angle ϕ[deg], Angles Calculated from NMR Data with 3-Shere Approach and Java Script. Sci. J. Chem. 2022, 10(1), 21-31. doi: 10.11648/j.sjc.20221001.13

Copy | Download

AMA Style

Carmen-Irena Mitan, Emerich Bartha, Constantin Draghici, Miron-Teodor Caproiu, Petru Filip, et al. Hopf Fibration on Relationship Between Dihedral Angle θHnHn+1[deg] and Vicinal Angle ϕ[deg], Angles Calculated from NMR Data with 3-Shere Approach and Java Script. Sci J Chem. 2022;10(1):21-31. doi: 10.11648/j.sjc.20221001.13

Copy | Download

@article{10.11648/j.sjc.20221001.13,
  author = {Carmen-Irena Mitan and Emerich Bartha and Constantin Draghici and Miron-Teodor Caproiu and Petru Filip and Robert Michael Moriarty},
  title = {Hopf Fibration on Relationship Between Dihedral Angle θHnHn+1[deg] and Vicinal Angle ϕ[deg], Angles Calculated from NMR Data with 3-Shere Approach and Java Script},
  journal = {Science Journal of Chemistry},
  volume = {10},
  number = {1},
  pages = {21-31},
  doi = {10.11648/j.sjc.20221001.13},
  url = {https://doi.org/10.11648/j.sjc.20221001.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjc.20221001.13},
  abstract = {3-Sphere theory, a hypersphere in four dimensions, is applied for calculation dihedral angles with the right stereochemistry and sign in D, -L ribitol series from proton and carbon chemical shift (ΔδXnXn+1[ppm], X = H, C) and vicinal coupling constant (3JHnHn+1[Hz]) with Java Script. A method in three steps, easy to calculate by hand or with Java Script program: 1. prediction of the dihedral angle only from 3JHnHn+1[Hz], 2. calculation the angle of set A with manifold equation (conic section, Villarceau circles) from chemical shift, 3. building of the seven sets unit or six sets units, from which is chose an angle almost equal with the predicted one having its stereochemistry and sign. Angles of set A and set B, relationships between vicinal angle and dihedral angle (X0-X15) are introduced instead of polar angle and azimuthal angle in spherical coordinates (eq. 1 versus eq. 3). Hopf coordinates, trigonometric equations, confirmed by algebraic equations are disclosed for all cis-ae/ea, trans-ee, trans-aa stereochemistry. Octonionic fibration S7→S15→S8 in R16, with real fibration S0 →S1 →S1 as unit, reassembles all possible stereochemistry gives by the HCCH fragment on two congruent disks, each centered on the perimeter of the other with equilateral triangles as vertices. Complex Hopf fibration in R4 ensuring the calculation of the dihedral angle from vicinal angle and vice versa, demonstrating the relationships between sets A, B, C.},
 year = {2022}
}

Copy | Download

TY - JOUR
T1 - Hopf Fibration on Relationship Between Dihedral Angle θHnHn+1[deg] and Vicinal Angle ϕ[deg], Angles Calculated from NMR Data with 3-Shere Approach and Java Script
AU - Carmen-Irena Mitan
AU - Emerich Bartha
AU - Constantin Draghici
AU - Miron-Teodor Caproiu
AU - Petru Filip
AU - Robert Michael Moriarty
Y1 - 2022/02/28
PY - 2022
N1 - https://doi.org/10.11648/j.sjc.20221001.13
DO - 10.11648/j.sjc.20221001.13
T2 - Science Journal of Chemistry
JF - Science Journal of Chemistry
JO - Science Journal of Chemistry
SP - 21
EP - 31
PB - Science Publishing Group
SN - 2330-099X
UR - https://doi.org/10.11648/j.sjc.20221001.13
AB - 3-Sphere theory, a hypersphere in four dimensions, is applied for calculation dihedral angles with the right stereochemistry and sign in D, -L ribitol series from proton and carbon chemical shift (ΔδXnXn+1[ppm], X = H, C) and vicinal coupling constant (3JHnHn+1[Hz]) with Java Script. A method in three steps, easy to calculate by hand or with Java Script program: 1. prediction of the dihedral angle only from 3JHnHn+1[Hz], 2. calculation the angle of set A with manifold equation (conic section, Villarceau circles) from chemical shift, 3. building of the seven sets unit or six sets units, from which is chose an angle almost equal with the predicted one having its stereochemistry and sign. Angles of set A and set B, relationships between vicinal angle and dihedral angle (X0-X15) are introduced instead of polar angle and azimuthal angle in spherical coordinates (eq. 1 versus eq. 3). Hopf coordinates, trigonometric equations, confirmed by algebraic equations are disclosed for all cis-ae/ea, trans-ee, trans-aa stereochemistry. Octonionic fibration S7→S15→S8 in R16, with real fibration S0 →S1 →S1 as unit, reassembles all possible stereochemistry gives by the HCCH fragment on two congruent disks, each centered on the perimeter of the other with equilateral triangles as vertices. Complex Hopf fibration in R4 ensuring the calculation of the dihedral angle from vicinal angle and vice versa, demonstrating the relationships between sets A, B, C.
VL - 10
IS - 1
ER -

Copy | Download

Author Information

Carmen-Irena Mitan

Institute of Organic Chemistry C.D. Nenitescu of Roumanian Academy, Bucharest, Romania
Emerich Bartha

Institute of Organic Chemistry C.D. Nenitescu of Roumanian Academy, Bucharest, Romania
Constantin Draghici

Institute of Organic Chemistry C.D. Nenitescu of Roumanian Academy, Bucharest, Romania
Miron-Teodor Caproiu

Institute of Organic Chemistry C.D. Nenitescu of Roumanian Academy, Bucharest, Romania
Petru Filip

Institute of Organic Chemistry C.D. Nenitescu of Roumanian Academy, Bucharest, Romania
Robert Michael Moriarty

Department of Chemistry, University of Illinois at Chicago, Chicago, the United States

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Carmen-Irena Mitan, Emerich Bartha, Constantin Draghici, Miron-Teodor Caproiu, Petru Filip, et al. (2022). Hopf Fibration on Relationship Between Dihedral Angle θHnHn+1[deg] and Vicinal Angle ϕ[deg], Angles Calculated from NMR Data with 3-Shere Approach and Java Script. Science Journal of Chemistry, 10(1), 21-31. https://doi.org/10.11648/j.sjc.20221001.13

Copy | Download

ACS Style

Carmen-Irena Mitan; Emerich Bartha; Constantin Draghici; Miron-Teodor Caproiu; Petru Filip, et al. Hopf Fibration on Relationship Between Dihedral Angle θHnHn+1[deg] and Vicinal Angle ϕ[deg], Angles Calculated from NMR Data with 3-Shere Approach and Java Script. Sci. J. Chem. 2022, 10(1), 21-31. doi: 10.11648/j.sjc.20221001.13

Copy | Download

AMA Style

Carmen-Irena Mitan, Emerich Bartha, Constantin Draghici, Miron-Teodor Caproiu, Petru Filip, et al. Hopf Fibration on Relationship Between Dihedral Angle θHnHn+1[deg] and Vicinal Angle ϕ[deg], Angles Calculated from NMR Data with 3-Shere Approach and Java Script. Sci J Chem. 2022;10(1):21-31. doi: 10.11648/j.sjc.20221001.13

Copy | Download

@article{10.11648/j.sjc.20221001.13,
  author = {Carmen-Irena Mitan and Emerich Bartha and Constantin Draghici and Miron-Teodor Caproiu and Petru Filip and Robert Michael Moriarty},
  title = {Hopf Fibration on Relationship Between Dihedral Angle θHnHn+1[deg] and Vicinal Angle ϕ[deg], Angles Calculated from NMR Data with 3-Shere Approach and Java Script},
  journal = {Science Journal of Chemistry},
  volume = {10},
  number = {1},
  pages = {21-31},
  doi = {10.11648/j.sjc.20221001.13},
  url = {https://doi.org/10.11648/j.sjc.20221001.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjc.20221001.13},
  abstract = {3-Sphere theory, a hypersphere in four dimensions, is applied for calculation dihedral angles with the right stereochemistry and sign in D, -L ribitol series from proton and carbon chemical shift (ΔδXnXn+1[ppm], X = H, C) and vicinal coupling constant (3JHnHn+1[Hz]) with Java Script. A method in three steps, easy to calculate by hand or with Java Script program: 1. prediction of the dihedral angle only from 3JHnHn+1[Hz], 2. calculation the angle of set A with manifold equation (conic section, Villarceau circles) from chemical shift, 3. building of the seven sets unit or six sets units, from which is chose an angle almost equal with the predicted one having its stereochemistry and sign. Angles of set A and set B, relationships between vicinal angle and dihedral angle (X0-X15) are introduced instead of polar angle and azimuthal angle in spherical coordinates (eq. 1 versus eq. 3). Hopf coordinates, trigonometric equations, confirmed by algebraic equations are disclosed for all cis-ae/ea, trans-ee, trans-aa stereochemistry. Octonionic fibration S7→S15→S8 in R16, with real fibration S0 →S1 →S1 as unit, reassembles all possible stereochemistry gives by the HCCH fragment on two congruent disks, each centered on the perimeter of the other with equilateral triangles as vertices. Complex Hopf fibration in R4 ensuring the calculation of the dihedral angle from vicinal angle and vice versa, demonstrating the relationships between sets A, B, C.},
 year = {2022}
}

Copy | Download