Suchergebnisse

The wave character of the NMR data (chemical shift ΔδCn[ppm] – vicinal coupling constant 3JHH[Hz]) enable calculation the tandem dihedral-vicinal or tetrahedral-internal angles with 3-sphere approach. Golden ratio, invers of Fibonacci number, and manifold inversion torus to Dupin cyclide under 3-sphere units ensure the calculation of the tetrahedral angles of five membered ring in opposite with dihedral angles sin versus tan functions. The transformation from space/time to space under electric and magnetic rule gives dodecahedron and icosahedron values for calculation tetrahedral angles from polyhedron and hypersphere equations. Tetrahedral angles of five membered ring can be calculated from carbon chemical shift ΔδCn[ppm] with Java Script program and hypersphere equations.

Hypersphere equations enable calculation tetrahedral angles φCn[deg] of six membered ring from carbon chemical shift δCn[deg]. The tetrahedral angles calculated directly from eq. 2a,b are confirmed by the dihedral angles found in close relationship on the five sets unit (Ci and Ti, i = 1-5, C, T = U or S) having ϕ2X, ϕ2Y equal with θZ1, θZ2 were Z is A or B and X, Y are D, E or F,G, angle of seven sets unit, or on two units with six sets. A method which is expected in case of six membered ring to ensure calculation of the tetrahedral angles for all conformations, chair – boat – skew.

Dihedral angles are predicted from vicinal coupling constant 3JHH[Hz] with 3-sphere approach, sphere or torus trigonometric equations of circle 1, 2 and circle inversion 3-5 for all cis-, trans-ee, trans-aa stereochemistry. The existence of circle inversion was demonstrated with conformational analysis on five and six membered rings. The sign and the stereochemistry result from vicinal coupling constant under trigonometric equations confirmed by algebraic equations, Hopf and Lie algebra theory. 3-Sphere, a hypersphere in 4D enable for all stereochemistry calculation dihedral angles under magnetic wave (NMR data -3JHH[Hz]), and all isomers from only one vicinal coupling constant.

3-Sphere approach is applied on prediction dihedral angle θHnHn+1[deg] only from vicinal coupling constant 3JHnHn+1[Hz] with Java script, in comparation with angles calculated from the differences between two atoms of carbon chemical shift (ΔδCnCn+1[ppm]) and Karplus equations. The trigonometric equations 1, 2 ensuring the right sign along the D-, L series rule.