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

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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

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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), 2024. Published by Science Publishing Group

Keywords

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

References

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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

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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

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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

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@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}
}

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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 -

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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

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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

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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

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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

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@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}
}

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