Manifold inversion on prediction dihedral angle from vicinal coupling constant with 3-sphere approach

Abstract

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 on 9-O-(10,11-di-O-benzyl-12,14-O-benzylidene-α-D-galactopyranosyl)-1-butyl-2,3-O-isopropylidene-1,4-dideoxy-1,4-imino-1-<I>N</I>-dehydro-L-ribitol

“…The phase angle of the pseudorotation was established with 3-Sphere dihedral angles θ HnHn+1 [deg] calculated only from vicinal coupling constant [1] 1, Figure 3). Torsional angles of trans-ee, trans-aa stereochemistry are calculated as already established using the rule of 3-sphere, [22] relationships between two sets angles in agreement with corresponding trigonometric equations between vicinal and dihedral angles.…”

“…Torsional angles of trans-ee, trans-aa stereochemistry are calculated as already established using the rule of 3-sphere, [22] relationships between two sets angles in agreement with corresponding trigonometric equations between vicinal and dihedral angles. Dihedral angles θ HnHn+1 [deg] of five and six membered ring [1] are calculated with eq. 1-5, torus (eq.…”

“…Dihedral angles of five and six membered ring θ HnHn+1 [deg] can be calculated only from vicinal coupling constant 3 J HH [Hz] [1] or with one of the manifold equations [2,3] from carbon and/or proton chemical shift δ[ppm], resulting dihedral angles with right stereochemistry and sign from torus or invers of torus (Dupin cyclide) equations. From manifold equations result dihedral angles with values almost equals with angles calculated from vicinal coupling constant 3 J HH [Hz] recorded experimentally.…”

Tetrahedral angles of six membered ring calculated from NMR data with 3-sphere approach

“…2) Dihedral angles (eqs. 3a-d) are angles [1] at intersection of two disks in close relationships with vicinal angles, angles result from vicinal coupling constant 3 JHH[Hz] under torus circle inversion to Dupin cyclide circle or viceversa. From both angles ϕ, θHnHn+1[deg] can be calculated 3 JHH[Hz] with eq.…”

“…Dihedral angles of five membered ring [1][2][3] are calculated to date from vicinal coupling constant [1,3] 3 JHH[Hz] with 3-sphere approach, relationships between dihedral θHnHn+1[deg] and vicinal angles ϕ[deg] ensuring the right sign, and proton or/and carbon chemical shift from manifold equations [2] , conic section or Villarceau circles ensuring the right cis/trans-ee, -aa stereochemistry. The wave character of NMR data fitting well with the 3-sphere theory, Hopf fibration confirmed by Lie algebra.…”