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

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 (³<i>J</i>_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 ³<i>J</i>_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 <i>versus</i> eq. 3). Hopf coordinates, trigonometric equations, confirmed by algebraic equations are disclosed for all <i>cis</i>-ae/ea, <i>trans</i>-ee, <i>trans-aa</i> 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 <i>vice versa</i>, demonstrating the relationships between sets A, B, C.