3-Spherical Conformational Insights into Iminocyclitols with 1-α-D-Ribose and 1-β-L-Ribose Stereochemistry Under Homotopic Behaviour of Nuclear Magnetic Resonance Data

Carmen-Irena Mitan; Emerich Bartha; Petru Filip; Valeriu Dragutan; Ileana Dragutan; Calin Deleanu; Constantin Draghici; Miron-Teodor Caproiu; Robert Michael Moriarty

doi:doi:10.11648/j.sjc.20261401.11

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3-Spherical Conformational Insights into Iminocyclitols with 1-α-D-Ribose and 1-β-L-Ribose Stereochemistry Under Homotopic Behaviour of Nuclear Magnetic Resonance Data

Carmen-Irena Mitan^*

, Emerich Bartha, Petru Filip

, Valeriu Dragutan, Ileana Dragutan, Calin Deleanu

, Constantin Draghici, Miron-Teodor Caproiu, Robert Michael Moriarty

Published in Science Journal of Chemistry (Volume 14, Issue 1)

Received: 22 December 2025 Accepted: 6 January 2026 Published: 2 February 2026

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Abstract

3-Sphere, a hypersphere in 4 dimensions approach, applied for calculating stereochemical parameters of iminocyclitol 1 – 5 with Hopf fibration and Lie algebra is described. Three angles have been considered, i.e. dihedral θ_HnHn+1 [deg] – tetrahedral φ_Cn [deg] – phase angle of the pseudorotation P [deg] calculated from NMR data, vicinal coupling constant ³J_HH [Hz] and carbon chemical shift δ_C [ppm]. This approach gave for 1-α-methyl-1,4-imino-1,4-dideoxy-D-ribitol 2 two conformers E₃ and ³T₂ having different dihedral θ_HnHn+1 [deg] and tetrahedral φ_Cn [deg] angles with same vicinal angles ϕ [deg]. Notably, phase angle of pseudorotation P [deg] placed the conformations on the south side for D-ribitols 1 - 3 and on the north side for L-ribitol 4, excepting trifluoroacetate salt of L-ribitol 5. The wave character of NMR data introduced few homotopic switches, the transformation from torus to inverse of torus, the relationship between angles of set A and set B, the transformation from Planck constants h to h-bar, along with the transformation from Joule to Calorie (J 4.1868 ⇆ J^-1 0.238). Two methods for calculation of tetrahedral angles φ_Cn [deg], energy-graph and Euler conic with two ways for representing the angles, polyhedron and unit models are analyzed. The conformational parameters, phase angle of the pseudorotation P [deg] established with VISION molecular model and exocyclic 3-Sphere dihedral angles θ_HnHn+1 [deg] relative to endocyclic torsional angles θ_n,n₊₁ [deg] from Altona-Sundaralingan model have been evaluated. In addition, the corresponding angle of deviation from planarity θ_m [deg] has been determined.

Published in	Science Journal of Chemistry (Volume 14, Issue 1)
DOI	10.11648/j.sjc.20261401.11
Page(s)	1-11
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), 2026. Published by Science Publishing Group

Keywords

3-Sphere, Dihedral Angles, Vicinal Angles, Vicinal Coupling Constant, Tetrahedral Angles, Phase Angle of the Pseudorotation, Angle of Deviation from Planarity, Conformational Analysis, Iminocyclitols

1. Introduction

3-Sphere dihedral angles θ_HnHn+1 [deg] on five membered ring can be calculated from vicinal coupling constant ³J_HH[Hz] and / or chemical shift δ [ppm] with Hopf fibration and Lie algebra protocol.

[1]

The Hopf fibration trigonometric model, in agreement with the algebraic equations, and the 3-Sphere representation of six dihedral angles with cis-, trans-ee and trans-aa stereochemistry ensure calculation of dihedral angles θ_HnHn+1 [deg] with right sign and stereochemistry in close relationship with vicinal angle ϕ [deg]. The angles are placed on two units with three sets of angles (A, B, C), with two sets (A, B) per unit from the trigonometric point of view and considering the relationships between the vicinal ϕ [deg] and dihedral angles θ_HnHn+1 [deg].

[1-3]

In the case of tetrahedral angles in close relationship with dihedral angles, seven sets of angles on one unit are preferred.

[1]

Moreover, Altona-Sundaralingan model

[4]

used endocyclic torsional angles θ_n,n+1 [deg] for conformational analysis: phase angle of pseudorotation P [deg] and angle of deviation from planarity θ_m [deg]. In addition, Karplus dihedral angles

[5]

, calculated from vicinal coupling constant ³J_HH [deg], can be transformed in endocyclic torsional angles with PSEUROT equations,

[6]

a program able to give the right sign and stereochemistry useful for simulation of phase angle of the pseudorotation P [deg] with Gausian09W.

[7]

Essentially, the 3-Sphere dihedral angles calculated from carbon and / or proton chemical shift

[8]

can be used on conformational analysis using exocyclic angles

[9]

on Altona-Sundaralingan model

[10]

. By contrast, the phase angle of pseudorotation P [deg] can be evaluated with VISION molecular model

[11]

. In the present work, 3-Sphere approach was designed for establishing the relationship between different dihedral θ_HnHn+1 [deg] and tetrahedral φ_Cn [deg] angles with the same vicinal angle ϕ [deg] for iminocyclitols 1 – 5

[12]

with equations (1-5).

P = tan-1 [(θ3,4 – θ1,2)/3.077xθ2,3] (1)

θm = θ2,3/cosP (2)

θ_H2H3 < 0:

P = - (180-Pcalc) (3)

P = 180 + P^calc(4)

Where P^calc [deg] – phase angle of the pseudoroation calculated with Altona-Sundaralingan Eq. (3) for P^calc positive and with eq. (3) for P^calc negative; θ_n,n+1 [deg] – endocyclic torsional angle, used in Table 5 also as θ_HnH,n+1 [deg] exocyclic torsional angle.

θ_m= θ_2,3/cosP^calc(5)

Where θ_m – angle of deviation from planarity [deg] calculated from P^calc for θ_2,3 < 0 [deg].

2. Results

The vicinal coupling constant ³J_HH [Hz] under 3-Sphere trigonometric equations

[1-3]

gives three possible dihedral angles θ_HnHn+1 [deg] for every stereochemistry

[1]

, considering vicinal coupling constant for cis, trans-ee stereochemistry with smaller values and for trans-aa stereochemistry with higher values

[13, 14]

As recently demonstrated, vicinal coupling constant ³J_HH [Hz] can be higher for cis, trans-ee and smaller for trans-aa

[2]

, thus the number of possible dihedral angles θ_HnHn+1 [deg] increased to six. Conformational characterization of iminocyclitols 1 – 5 with 1-α-D-ribitol and 1-β-L-ribitol stereochemistry (Figure 1)

[12]

presented in this paper was based on the relationship between dihedral θ_HnHn+1 [deg] – tetrahedral φ_Cn [deg] and phase angle of pseudorotation P [deg]. In case of 3-Sphere dihedral angles θ_HnHn+1 [deg], as found from trigonometric equations,

[1]

cis and corresponding trans-aa stereochemistry are under 180 [deg] rule and cis and trans-ee stereochemistry under 120 [deg] rule,

[15]

with trans-ee^3,2stereochemistry moving between two units, the unit U and unit S in 2D

[3]

. For iminocyclitols 1 - 5 proposed for conformational analysis the rule for calculating dihedral angles from second vicinal angle ϕ [deg]

[2]

gives one more dihedral angle for vicinal coupling constant of 3.9, 5.4, 5.2 [Hz] with cis stereochemistry. Iminocyclitol 2, 1-α-methyl-1,4-imino-1,4-dideoxy-D-ribitol was used extensively in this paper for conformational analysis, since the calculated 3-Sphere dihedral angles θ_HnHn+1 [deg] are difficult to simulate on VISION molecular model.

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Figure 1. Five membered ring iminocyclitols with α-D-ribitol (1-3) and β-L-ribitol (4, 5) stereochemistry.

In this context, the dihedral θ_HnHn+1 [deg] and tetrahedral φ_Cn [deg] angles calculated from carbon chemical shifts

[1, 16]

for 1-α-methyl-1,4-imino-1,4-dideoxy-D-ribitol 2

[12]

are chosen in pair of angles from seven sets units: i. e. dihedral θ_HnHn+1 [deg] – vicinal ϕ [deg] from sets A and B and tetrahedral φ [deg] - internal γ [deg] from sets D, E or F, G, as presented in Table 1 without opposite rule and in Table 2 using opposite rule, dihedral from unit build with sin function and tetrahedral from unit build from tan function, or viceversa. Units are constructed from one angle of set A calculated from sin and tan function from carbon chemical shift δ_Cn [ppm] and Eular – conic manifold.

[1]

In function of the phase angle of the pseudorotation P [deg], dihedral θ_HnHn+1 [deg] and tetrahedral φ_Cn [deg] angles

[16]

are chosen from units in opposite mode for few possible conformers (Table 2). As an important remark, the transformations from U to S or S to U, two units with three sets angles (C = U^A,B,C, S^A,B,C), replace better two units with seven sets angles (C = U^A-G, S^A-G) calculated from sin and tan function. Between sets A and B one first angle must be higher than 5 [deg] on unit U and smaller than 5 [deg] on unit S. Building six sets in two units for compound 1 from C₂ carbon chemical shift δ_C2 [ppm], it is obvious that the transformation from S to U can be made with angle on unit S of 6.85 [deg]. Alternatively, between six sets angles on two units, the equation is known

[16]

for calculation in opposite dihedral and tetrahedral angles from angle θ^S4 of set A or B with equation (6). In case of iminocyclitol 1, from the unit build with C₃ under tan function with first angle of 7.05056 [deg] the transformation U to S gives the unit U instead of S, and the transformation S to U gives unit U. For the conformation of iminocyclitol 1 the NMR data in this case are somewhat difficult to be analyzed in strictly opposite rule.

cos^-1sin{[60 – (θ^S4– 90)]/1.5} = φ_Cn(6)

Where θ^S4 [deg] – forth angle on set A or B of unit S, φ_Cn [deg] – tetrahedral angle of five membered ring.

The tetrahedral angles φ_Cn [deg] in relationship with the corresponding dihedral angles θ_HnHn+1 [deg] are selected to be considered as well (Table 2). Two aspects are evident: 1. The dihedral θ_HnHn+1 [deg] and tetrahedral angles φ_Cn [deg] are in opposite way on seven sets angles on unit U or S. That is also true on six sets angles on two units (U and S), or building more units U₂ or S₂ from U₁ or S₁; 2. Dihedral angles θ_H3H4 for D-ribose stereochemistry with trans-aa vicinal coupling constant 8.8 [Hz] is over rule, dihedral and tetrahedral angles are found on sets D and E or F, G. In this case, for units S_1, the values calculated from sin and tan functions for vicinal coupling constant, are smaller (8.72, 8.71 [Hz]), but in S₂ both values increased to 8.79, 8.76 [Hz] almost equal with the recorded vicinal coupling constant ³J_H3H4 8.8 [Hz].

In Table 3 the tetrahedral angles φ_Cn [deg] are calculated with energy-graph theory

[17, 18]

using Planck constant h and polyhedron equations (7-9) characteristic for icosahedron and dodecahedron, inversely of h giving h-bar with a factor of correction f 0.94767, and a transformation from Joule to Calorie (J 4.1868 ⇆ J^-1 0.238) under homotopic process

[19]

. Relative to building units, polyhedron model used first tetrahedral angle φ_Cn [deg] calculated from sin or cos functions on its relationship with corresponding internal angle γ [deg] under 180 [deg], and continuum transformation from internal γ [deg] (corresponding to θ³) in tetrahedral φ [deg] (corresponding to θ⁴) with polyhedron equation (7). In Table 4 are presented dihedral θ_HnHn+1 [deg] and tetrahedral φ_Cn [deg] angles

[17, 18]

calculated for E₃ and ³T₂ conformations with values comparable with angles

[16]

presented in Table 2.

tan(A/2) =X = 2sin^-1(B/2)(7)

2xtan^-1{2xsin [(180– θ)/2]} = φ ~ 100 [deg](8)

180 – tan^-1(1/{[(sin [(180– θ)/2]}/2) = φ ~ 106 [deg](9)

Where θ ~ 105, 104 [deg].

Table 1. Dihedral θ_HnHn+1 [deg]

[13, 14]

and tetrahedral φ_Cn [deg] angles

[16]

of 1-α-methyl-1,4-imino-1,4-dideoxy-D-ribitol 2

[12]

calculated from carbon chemical shift δ_Cn [ppm].

Entry	³J_HH [Hz]	ϕ [deg]	δ_Cn [ppm]	ϕ [deg]	²J_HH [Hz]	θ_HnHn+1^calc [deg]	φ_Cn^calc [deg]
1.	cis: 3.1	38.44	C₁: 57.4	38.458	3.10	S: 51.541 T: -31.879	S – C₁: 109.229 S – C₂: 100.770 T – C₁: 105.770 T – C₂: 104.229
2.	cis: 3.9	60.84	C₂: 71.5	60.046	3.87	S: 29.953 T: -33.406	S – C₁: 105.023 S – C₂: 104.976 T – C₁: 106.733 T – C₂: 103.266
3.	trans-aa: 8.8	77.44	C₃: 71.7	60.138	3.87	S: 29.861 T: -33.530	S - C₄: 105.06 S - C₃: 104.93 T - C₄: 106.76 T - C₃: 103.23
4.			C₄: 66.8	76.152 75.939	8.72 8.71	S: -135.84^S1T: -135.87^S1	S - C₄: 106.15 S - C₃: 103.84 T - C₄: 105.93 T - C₃: 104.06
4.				77.305 76.878	8.79 8.76	S: -135.708^S2T: -135.75^S2	S - C₄: 107.30 S - C₃: 102.69 T - C₄: 106.878 T - C₃: 103.121

a. D₂O δ [ppm] ¹³C-NMR, 75 [MHz], ¹H – NMR 400 [MHz].

The phase angle of the pseudorotation P [deg] was established using exocyclic 3-Sphere dihedral angles on VISION molecular model

[9]

and Altona - Sundaralingam map

[4]

. In Table 5 the phase angles of pseudorotation P [deg] calculated with Altona Sundaralingan model

[10]

from exocyclic 3-Sphere dihedral angles

[9]

are presented in comparison with endocyclic torsional angles, and corresponding angles of deviation from planarity θ_m [deg]. Endocyclic torsional angles (eq. (10, 11)) calculated with method 2

[7]

in function of trans or corresponding trans exocyclic dihedral angles and vicinal coupling constant (Table 6) gives on Altona – Sundaralingam equation (1) results comparable with conformation established on VISION molecular models.

θ_n,n+1= θ_HnHn+1^trans-aa/³J_HH(10)

θ_n,n+1= (θ_HnHn+1^trans-eex³J_HH)/2(11)

where θ_n,n+1torsional endocyclic angles [deg], θ_HnHn+1transexocyclic dihedral angles [deg].

3. Discussions

3.1. Conformational analysis: Phase Angle of the Pseudorotation P [deg], Angle of Deviation from Planarity θ_m [deg]

In the present study, the phase angles of the pseudorotation P [deg] are determined with VISION molecular models and 3-Sphere dihedral angles θ_HnHn+1 [deg], exocyclic dihedral angles

[9, 20]

calculated in this case directly from carbon chemical shift δ_Cn [ppm] (Tables 2, 4) in comparation with phase angles of the pseudorotation calculated with Altona equation (1) from exocyclic torsional angles

[9]

and endocyclic torsional angles θ_n,n+1 [deg] in case of iminocyclitol 2 (Table 5) and only from endocyclic torsional angles θ_n,n+1 [deg] for iminocyclitols 1, 3 - 5 (Table 6).

Endocyclic torsional angles θ_n,n+1 [deg] are calculated from 3-Sphere dihedral angles θ_Hn,Hn+1 [deg] with method 1: under 180/120 rule for cis stereochemistry and 120 rule for trans stereochemistry, method 2: endocyclic torsional angles are calculated from 3-Sphere dihedral angles and vicinal coupling constant conform with equations reported to date

[7]

. In both cases, the sign in D-ribitol series: positive trans-ee endocyclic torsional angle relative to negative dihedral angle θ_H3H4 [deg], and in L-ribitol series: negative trans-ee endocyclic torsional angle relative to positive dihedral angle θ_H3H4 [deg], as reported recently based on Gaussian09W analysis

[7]

on iminocyclitol 2. Phase angle of pseudprotation P [deg] calculated with endocyclic torsional angles θ_n,n+1 [deg] with methods 2 in case of iminocyclitol 1 is 15.28 [deg] with a conformation near to E³ comparative to method 1 with 7.63 [deg] at border line between ₂T³ and E³ (Table 5, entry 2). According to our first analysis, the phase angle of pseudorotation of -143.12 [deg] with ₃E – ⁴T₃ conformation

[9]

results for the combination of dihedral angles: I. 50.20, 29.67, -168.6 [deg] calculated with modified Altona model (eq. (1)) from exocyclic torsional angles

[13]

. Because the first combination of angles is too hard to simulate on molecular models, Sundaralingam theory

[21]

with two atoms out of plane (³²E) was considered. With both positive angles for cis stereochemistry, the trans-aa H₃H₄ angle negative from D-ribitol stereochemistry becomes smaller as 90 [deg]. Subsequently, the optimized method for calculation of the dihedral angles from trigonometric equations

[14]

point out other two combinations: II. ₃E 50.20, -40.98, -135.57 [deg], III. ³T₂: -52.53, 29.67, -168.36 [deg].

[9]

The opposite trigonometric relationship between dihedral and tetrahedral angles sin versus tan function for building units or vice versa reveals unexpected relationship between two combinations of angles (Table 2): I. 51.54, -33.46 or -33.53, -135.708 [deg]: S – T – T – S (Table 2, entry 1-4, Table 5, entry 1) and II. -31.87, 29.95 or 29.86, -135.757 [deg]: T – S – S – T (Table 2, entry 5 – 8, Table 5, entry 2), with E₃ (-162 [deg]) and E₂ (-18 [deg]) conformations analyzed with VISION molecular models, comparative to phase angle of pseudorotation P [deg] calculated with Altona-Sundaralingan eq. (1) from endocyclic 3-Sphere torsional angles: E₃ (-163.51 [deg], -153.8 [deg]) and ³T₂ - E³(7.63 [deg], 15.28 [deg]). In second case from exocyclic angles resulting a phase angle of the pseudorotation P [deg] of -7.65 [deg] with a conformation between E₂ – ³T₂. On VISION molecular model conformation E₃ can be simulated with a good balance between angle of deviation from planarity θ_m [deg] and 3-Sphere dihedral angle θ_H2H3 [deg] (Table 5, entry 1, 3 - 5), otherwise dihedral angles θ_H3H4 with trans-aa^5,2stereochemistry becomes smaller as 120 [deg], near to 90 [deg] characteristic for trans-ee stereochemistry with tendency to become ⁰T₄ conformation with P 72 [deg], behavior recorded also with Gaussian09W

[7, 22]

: S – S – S – T having ⁰E (90 [deg]) conformation. In Table 5, entry 1, 3 - 5, exocyclic dihedral angles (+) cis-θ_H1H2 – (-) cis-θ_H2H3 – (-) trans-aa^5,2θ_H3H4 with E₃ configuration have θ_H2H3 -33.46, -41.15, -15.75 [deg], and angle of deviation from planarity θ_m -27.67, -20.67, -44.85 [deg] increasing since dihedral angle θ_H2H3 [deg] decreased. Phase angle of pseudorotation calculated from exocyclic dihedral angles are different with 13 – 26 [deg] relative to endocyclic torsional angles (Table 5, entry 1, 3), higher difference (52 [deg]) resulting from smaller exocyclic angle θ_H2H4 [deg] (Table 5, entry 4).

Conformation E₂ – ³T₂ (P -18 - 0 [deg]) having exocyclic dihedral angles (-) cis-θ_H1H2 – (+) cis-θ_H2H3 – (-) trans-aa^5,2 θ_H3H4 simulated on molecular models are comparable with twist ³T₂ conformation calculated with Altona-Sundaralingan eq. (1) from endocyclic torsional angles θ_n,n+1 [deg], phase angle of pseudorotation P 7.63 [deg] and angle of deviation from planarity θ_m 30.31 [deg].

Isopropylidene protected (1) and unprotected (3 - 5) iminocyclitols

[12]

with α-D-ribose (1, 3) and β-L-ribose (4, 5) stereochemistry, bearing nonyl chain at C₁ and NH (3, 4) and trifloroacetate salt with decyl at C₁ (5), shown conformations as calculated with eq. (1) from endocyclic torsional angles θ_n,n+1 [deg], in south side for α-D-ribose stereochemistry and north side for β-L-ribose stereochemistry, excepting for β-L-stereochemistry trifuoroacetate salt iminocyclitol 5 (Table 6). Protected iminocyclitol 1 with methyl group at C₁ and unprotected 3 with nonyl at C₁ and NH, have the following conformation: twist ²T₃ (-175.5, -173.9 [deg] 1), twist ²T₃ conformation (-168.021 [deg] 3) or envelope E₃ conformation (-163.46 [deg] 3), since iminocyclitol 2 shown two conformations from E³ (15.28 [deg]) and ⁴T₃ – E₃ (-153.8 [deg]). Nonyl chain at C₁ and NH on iminocyclitols 3 and 4 with D and L-ribose stereochemistry having conformation ²T₃ (-168.021 [deg]) – E₃ (-163.46 [deg]) and³T₂ – E³(11.91 [deg]) – ³T₂ (1.312 [deg]) between south to north side of the 2D map. Phase angle of pseudorotation of trifloroacetate salt (5) with decyl chain at C₁ is placed in south side with ²T₃ – E₃ conformation (-174.01 [deg]). Conformation simulated on VISION molecular models are compatible with angles calculated from endocyclic torsional angles θ_n,n+1 [deg], excepting P 11.91 with dihedral angles -2.92, 19.83, 88.99 [deg] (4). In Table 6 are presented tetrahedral angles in opposite relationship with dihedral angles, sin versus tan function. Between D-ribose 3 and L-ribose 4 iminocyclitols the conformations are change from south to north because in case of 3 dihedral angles θ_H1H2 negative and θ_H2H3 positive on molecular models give for θ_H3H4 trans-aa stereochemistry instead of trans-ee.

3.2. Homotopic Behavior of NMR Data

The wave character of NMR data introduces few homotopic switches on relationship between vicinal – dihedral – tetrahydric angles, at list on phase angle of the pseudorotation P [deg]. One topological space under continuous deformation from one to other

[19]

, i.e. transformation from Planck constant h to h-bar, along with transformation from Joule to Calorie (J 4.1868 ⇆ J^-1 0.238), transformation from torus to invers of torus (sin versus tan functions), relationship between angles of set A and set B.

Relationships between vicinal ϕ – dihedral θ_HnHn+1– tetrahedral φ_Cnangles [deg] and at least phase angle of pseudorotation P [deg] are used for establishing the real combination between calculated dihedral angles θ_HnHn+1 [deg] from only one recorded vicinal coupling constant ³J_HH [Hz] with two possible vicinal angles ϕ [deg],

[2]

totally six dihedral angles θ_HnHn+1 [deg]. Six dihedral angles are calculated for vicinal coupling constant higher as 4.5 [Hz],

[2]

in this case for 3.9 [Hz] from carbon chemical shift C₄ result an angle of -15.939 [deg] with a vicinal coupling constant of 3.92 [Hz] from second rule

[2]

. The sign of trans-aa θ_H3H4 [deg] is restricted by D ribitol – stereochemistry, negative only from first rule

[14]

, since from second rule

[2]

result only positive angles. The stereochemistry of D-ribitol iminocyclitol 1 was established based on the method of synthesis chosen

[12]

and confirmed by H-H COSY, H-C HMQC NMR experiments (ind. 16 Supporting information).

Having in mind the tetrahedral angles of natural crystal pyrite of 106.6, 102.6, 102.6, 106.6, 121.6 [deg], until now all the calculated tetrahedral angles φ_Cn [deg] are recalculated with golden angle and golden ratio (Table 3), as relationships between angles of polyhedron and its pentagon unit

[17]

Moreover, the tetrahedral angles calculated Energy-graph approach

[17]

using inverse of h-Planck are almost equals with angles calculated with Euler-conic section (Tables 3 and 4 in comparation with Tables 1, 2)

[16]

, polyhedron versus unit for angles representation. The relationships between tetrahedral φ_Cn [deg] and dihedral θ_HnHn+1 [deg] angles as a function of phase angle of the pseudorotation P [deg] (Tables 1 and 2) shown for E₃ conformation smaller value for φ_C1,φ_C4 tetrahedral angles (105.77, 105.93 [deg]) and higher for φ_C2,φ_C3 tetrahedral angles (104.97 and 104.93 [deg]) (Table 4). In addition, for ³T₂ conformation (Table 5, entry 2) the value on first unit (109.22 [deg]) are equal with φ_C1 tetrahedral angle calculated with polyhedron equation 109.8 [deg]

[15]

from inverse of square energy in J/molx10^-6 (Table 4), relative to 106.91 [deg] (Tables 2, 3).

Notably, the conformation E₃ with the following succession of angles φ_Cn (θ_HnHn+1) [deg]: Euler conic 105.77 (51.54), 104.97 (-33.40), 104.93 (-33.53), 105.93 (-133.12) [deg] (Table 2) and Energy-graph 105.38 (50.39), 104.54 (-33.80), 104.18 (-33.71), 105.09 (-135.58) (Table 4), relatively hard to simulate on molecular models, and with values of tetrahedral angles unexpected, but result directly from building unit in case of Euler conic approach (Tables 1, 2), are confirmed by the angles calculated with Energy-graph approach using inverse of h-bar and polynomial equations (Tables 3 and 4). Furthermore, the tetrahedral angles can be transformed in values near to 100 and 106 [deg] with polyhedron equations (7, 8)

[16, 17]

as follows: 105.77 (106.78), 104.97 (101.22), 104.93 (101.24), 105.93 (106.75) [deg] (Table 2), near to theoretical values of five membered ring.

Table 2. Phase angle of the pseudorotation P [deg] established with VISION molecular models from dihedral θ_HnHn+1 [deg]

[2, 13, 14]

and tetrahedral φ_Cn [deg] angles

[16]

of 1-α-methyl-1,4-imino-1,4-dideoxy-D-ribitol 2

[12]

calculated from carbon chemical shift δ_Cn [ppm] and vicinal coupling constant ³J_HH [Hz].

Entry	C_n^a	θ_HnHn+1[deg]	³J_HH[Hz]	φ_Cn[deg]	C_n^c[deg]	φ_Cn^Td[deg]	Entry	θ_HnHn+1[deg]	³J_HH[Hz]	φ_Cn[deg]	C_n[deg]	Entry	θ_HnHn+1[deg]	³J_HH[Hz]	φ_Cn[deg]
1.	C₁	S: 51.54	3.10	T: 105.77	109.48^c	106.789	5.	T: -31.87	3.1	S: 109.22	106.91^c	9.	S: 51.54	3.1	T: 105.77
2.	C₂	T: -33.46	3.91	S: 104.97	100.015	101.223	6.	S: 29.953	3.87	T: 103.266	101.155^c	10.	StoU: 49.984	3.16	108.466^c, 110.57or 99.42^e
3.	C₃	T: -33.53	3.907	S: 104.93	101.040	101.249	7.	S: 29.861	3.87	T: 103.234	101.176^c	11.	UtoS: -41.179 S: 29.86	3.90 3.87	100.040^c, 110.02 or 99.97^e T: 103.23
4.	C₄^b	S: ϕ 76.15^S1 -135.844 ϕ 77.30^S2 -135.708^S2	8.72 8.79^S2	T: 105.93^U1A	S₂: 106.878	106.757	8.	T: ϕ 75.93^S1 -135.87^S1 ϕ 76.87^S1 -135.757^S2	8.71 8.76	S: 106.15	S₂: 107.3	12.	T: ϕ 75.93^S1 -135.87^S1 ϕ 76.87^S2 -135.757^S2 T: -15.93^H3H2	8.71 8.76 3.92	S: 106.15 S: 107.3, 111.34or 98.65^f S: 111.92 or 98.076^f
5.		E₃						E₂ - ³T₂					E₃

a. D₂O δ [ppm] ¹³C-NMR, 75 MHz, ¹H – NMR 400 MHz, b. relationships between vicinal angle and tetrahedral angle, c. six sets on two units, d. φ_Cn^T tetrahedral angles calculated from polynomial eq. (7-9), e. seven sets angles from unit U₁, f. seven sets angles on one unit from set D in case of S₁ and from set F in case of unit S₂.

Table 3. Tetrahedral angles

[17, 18]

calculated from carbon chemical shift δ_Cn [ppm] in relationship with dihedral θ_HnHn+1 [deg] and vicinal ϕ [deg] angles for iminocyclitol 2.

Entry	δ_Cn [ppm]^a	1/E, θ [deg]	1/E^1/2, θ [deg]	1/E², θ [deg]	³J_HH [Hz], ϕ [deg], θ_HnHn+1 [deg]
1.	C₁ 57.4	0.58203	0.76291	0.338767	1/E²: ³J_H1H2: 3.14 [Hz] ³J_H2H3= {[2x(109.80 – 90)]^1/2}/2 ϕ = 39.603 [deg] θ_H1H2= 50.396, -32.516 [deg]
		71.187	75.436	70.198
		74.998	78.185	74.613
		81.328	80.556	82.018
		98.671	99.443	97.981
		105.001	101.482	105.386
		108.812	104.563	109.801
		116.826		117.139 → 116.664 108.308 108.116
2.	C₂ 71.5	0.46725	0.68356	0.21833
		68.573	72.367 76.632	77.389 64.778 75.963	1/E^{1/2: 3}J_H2H3: 3.90 [Hz] θ_H2H3= 2x(104.545 – 90) = 29.09 [deg] ϕ = 60.91deg] θ_H2H3 29.44, -33.804 [deg] ³J_H1H2: 3.13 [Hz] θ_H1H2= 2x(115.400 - 90) = 50.8 [deg] ϕ = 39.2 [deg]
		73.975	80.523 75.454	77.356 72.464 76.823
		83.183	86.246 78.505	77.304 86.055 78.185
		96.816	99.476 101.494	102.643 93.944 101.814
		106.624	93.753 103.367	102.695 107.535 103.176
		111.427	104.545 104.545 (115.400)	102.611 115.222→104.036
		117.638 ϕ = 62.361	107.632
3.	C₃ 71.7	1/E 0.46595	1/E^1/20.68258	1/E²0.217077
		68.909	63.579 75.484	64.920 75.514	1/E^{1/2: 3}J_H2H3: 3.904 [Hz] θ_H2H3= 2x(104.515 – 90) = 29.03 [deg] ϕ = 60.97 [deg] θ_H2H3 29.03, -33.71 [deg]
		74.1083	72.443 76.643	72.521 76.973 76.654
		79.3703	86.096 78.487	85.943 76.363 78.468
		82.9385	93.903 101.512	94.056 77.934 101.531
		97.0614	99.527 103.356	104.485 102.064 103.343
		100.629	104.515→104.515, 115.390	107.471 103.020 104.485
		105.8916	107.556	115.079 → 103.636
		111.090	116.420
4.	C₄ 66.8	0.500134	0.707201	0.25013409
			70.536	61.03	1/E^2: ³J_H3H4= 8.85 [Hz] ϕ = 78.468 [deg] θ_H3H4= -168.22, -135.58 [deg] 1/E:³J_H2H3= 3.87 [Hz] ϕ = 60.017 [deg] θ_H2H3= 29.982deg]
		60.017	74.745	75.514 70.949
		70.536	81.781	76.654 81.495
		89.984	89.983	78.468 89.117
		90.015	90.016	101.531 90.882
		109.463	98.218	103.345 98.506
		119.982	105.254	104.485 105.093
			109.463	115.379 109.051
				118.97 116.0, 106.46
5.	NH 3.33	0.531585	0.729099	0.2825831
		115.354	93.6218, 115.630	114.3422
		115.354	111.117, 107.699	106.4122

a. D₂O δ_Cn [ppm], ¹H NMR 400 [MHz], ¹³C NMR 75 [MHz].

Table 4. Phase angle of the pseudorotation P [deg] established with VISION molecular models from dihedral θ_HnHn+1 [deg] and tetrahedral φ_Cn [deg] angles

[17, 18]

of 1-α-methyl-1,4-imino-1,4-dideoxy-D-ribitol 2

[12]

calculated from carbon chemical shift δ_Cn [ppm] and vicinal coupling constant ³J_HH [Hz].

Entry	C_φ^a	θ_HnHn+1 [deg]	³J_HH [Hz]	φ_Cn [deg]	φ_Cn^Tb [deg]	Entry	θ_HnHn+1 [deg]	³J_HH [Hz]	φ_Cn [deg]
1.	C₁	1/E²: 50.396	3.14	105.38	106.86	5.	1/E²: -32.516	3.14	109.801 (108.11)
2.	C₂	1/E^1/2: -33.804	3.90	104.54	101.49	6.	1/E^1/2: 29.44	3.89	103.36 (101.49)
3.	C₃	1/E^1/2: -33.71	3.90	104.18	101.72	7.	1/E^1/2: 28.36	3.92	103.23 (101.72)
4.	C₄	1/E²: -135.58	8.85	105.09	106.91	8.	1/E²: -135.58	8.85	109.051 (106.46)
5.		E₃					E₂ - ³T₂

a. D₂O.δ [ppm] ¹³C-NMR, 75 MHz, ¹H – NMR 400 MHz, b. φ_Cn^T tetrahedral angles calculated from polyhedron equations (6-8).

Table 5. Phase angle of the pseudorotation P [deg] and angle of deviation from planarity θ_m [deg] calculated with optimized Altona-Sundaralingan model (eq. (1-5)) for iminocyclitol 2.

Entry	Vision	θ_HnHn+1^a [deg]	P [deg]	θ_m [deg]	θ_n,n+1^b [deg] Met. 1	P [deg]	θ_m [deg]	θ_n,n+1^b [deg] Met. 2	P [deg]	θ_m [deg]
1.	~E₃	51.54 -33.46 -135.708	-137.052 ⁴T₃ – E⁴	-45.711	8.46 -26.54 -15.708	-163.514 E₃	-27.677	41.43 -37.57 -15.421	-153.809 ⁴T₃ – E₃	-41.872
2.	³E	-31.87 29.953 -135.75	-7.65 ³T₂ – E₂	30.222	-28.13 31.047 -15.757	7.634 ³T₂	31.313	-47.783 38.473 -15.426	15.287 ³E	39.884
3.	E₃	49.984 -41.155 -135.757	-143.347 ⁴T₃	-51.298	10.16 -18.845 -15.757	-156.036 E₃	-20.677	41.94 -35.601 -15.426	-152.359 ⁴T₃ - E₃	-29.026
4.	E₃	49.984 -15.939 -135.757	-117.494 ⁴E	-35.525	10.016 -44.061 -15.757	-169.236 E₃	-44.850	41.940 -42.066 -15.426	-155.975 ⁴T₃ - E₃	-46.056

a. exocyclic 3-Sphere dihedral angles under 180 r ule for trans-aa stereochemistry, b. endocyclic torsional angles under 120 rule. Met. 1: 180/120 rule, Met. 2: θ_n,n+1= f(θ_HnHn+1, ³J_HH)

[7]

Table 6. Phase angle of the pseudorotation P [deg] and angle of deviation from planarity θ_m [deg] calculated with optimized Altona-Sundaralingan model (eq. (1-5)) for iminocyclitol 1, 3-6.

Entry	Comp.	³J_HnHn+1^a [Hz]	δ_Cn [deg]	θ_HnHn+1 [deg]	φ_Cn [deg]	Vision	θ_HnHn+1 [deg]	θ_n,n+1^b [deg] Met. 1	P [deg]	θ_m [deg]	θ_HnHn+1 [deg]	θ_n,n+1^b[deg] Met. 2	P [deg]	θ_m [deg]
1.	1	4.1	55.8	20.224	106.305	²T₃	20.224 -25.311 -88.447	39.775 -34.68 31.553	P -175.594 ²T₃	-41.720	20.224 -25.311 -88.447	38.969 -28.646 27.555	P -172.621 ²T₃	-28.885
2.		5.4	83.5	-25.311 -34.688^b	101.5744		20.224 -25.311 -88.447	39.775 -34.68 31.553	P -175.594 ²T₃	-41.720	20.224 -25.311 -88.447	38.969 -28.646 27.555	P -172.621 ²T₃	-28.885
3.		5.4	84.3	-25.051 -34.946^b	101.474		20.224 -34.688 -88.447	39.775 -25.312 31.553	P -173.986 ²T₃	-25.452	20.224 -34.688 -88.447	38.969 -25.312 27.551	P -173.973 ²T₃	-25.431
4.		0	65.9	-88.447	106.4007		20.224 -34.688 -88.447	39.775 -25.312 31.553	P -173.986 ²T₃	-25.452	20.224 -34.688 -88.447	38.969 -25.312 27.551	P -173.973 ²T₃	-25.431
5.	3	4.8	63.7	-/+2.043 -25.913^b	106.862 (105.36^c)	²T₃	2.043 -18.732 -88.992	57.95 -41.268 31.008	P -168.021 ²T₃- E₃	-42.186	2.043 -18.732 -88.992	37.074 -31.013 22.336	P -171.22 ²T₃	-31.380
6.		5.2	72.5	19.833 -30.500^b -/+18.732	101.261 103.108 100.166	²T₃	2.043 -18.732 -88.992	57.95 -41.268 31.008	P -168.021 ²T₃- E₃	-42.186	2.043 -18.732 -88.992	37.074 -31.013 22.336	P -171.22 ²T₃	-31.380
6.		5.2	72.5	19.833 -30.500^b -/+18.732	101.261 103.108 100.166	²T₃	2.043 -30.5 -88.992	57.957 -29.5 31.008	P -163.464 E₃	-30.772	2.043 -30.5 -88.992	37.074 -28.75 22.3368	P -170.541 ²T₃	-29.146
7.		5.2	74.0	19.614 -31.155^b -/+18.587	101.415 102.877 100.385
8.		0	69.3	-88.992	106.372
9.	4	4.8	68.4	-/+2.923^S2 -/+2.438^S1 -26.16^δC2	107.438^S2 106.796 (105.73^c) 106.883 (105.250)^c	³T₂	-2.923 19.833 88.992	-57.077 40.167 -31.008	P11.91 ³T₂ – E³	41.050	-2.923 19.833 88.766	-36.891 30.801 -10.855	P15.36 ³T₂ – E³	31.942
9.		4.8	68.4	-/+2.923^S2 -/+2.438^S1 -26.16^δC2	107.438^S2 106.796 (105.73^c) 106.883 (105.250)^c	³T₂	-26.16 19.833 88.992	-33.84 40.168 -31.008	P 1.312 ³T₂	40.178	-26.16 19.833 89.766	-32.05 30.801 -10.855	P 12.605 ³T₂ – E³	31.562
10.		5.2	72.5	19.833 -30.500^b -/+18.738	101.261 103.108 100.166
11.		5.2	72.7	19.803 -30.589^b -/+18.717	101.282 103.076 100.196
12.		0	70.9	89.766	106.636
13.	5	2.8	63.3	59.422	107.165	E₃	59.422 -12.968 135.745	0.578 -47.032 15.745	P -174.017 ²T₃ - E₃	-47.289	59.422 -12.968 135.745	43.063 -46.397 15.425	P -169.043 ²T₃ - E₃	-47.289
14.		3.6	72.1	-13.171	104.839
15.		3.6	73.4	-12.968	104.558
16.		8.8	63.9	166.98 135.745^d	106.855 (105.404^c)

a. δ_Cn [ppm] 1, 3 - 5: CDCl₃, ¹H NMR 400 [MHz], ¹³C NMR 75 [MHz], b. second rule

[2]

; c. polyhedron equations (6-8), d. angle calculated from ϕ [deg]. Met. 1: 180/120 rule, Met. 2: θ_n,n+1= f(θ_HnHn+1, ³J_HH)

[7]

Angles calculated from polynomial equation (6) are comparable with angles fund by building units or transformation from U to S or S to U; i.e. 106.91, 101.15, 101.17, 106.87 [deg] (Table 2).

Tetrahedral angles calculated from carbon C₁ chemical shifts with eq. (8) of 106.78 [deg] are almost equal with the angle 106.917 [deg] calculated on unit U₂ from sin function. In case of C₂ and C₃ on unit U₂ results angles in opposite 100.01 or 100.04 [deg], relative to polynomial angles calculated with eq. (7) of 101.223 and 101.249 [deg], closed to angles results directly 101.15, 101, 17 [deg]. Fibonacci numbers Fⁿ, golden ratio φ and golden triangle giving angles of 103.6545 [deg] and 101.01 [deg] for tetrahedral angle (eq. (12-15)).

[18]

Fⁿ= φⁿ-𝛹ⁿ/(5)^1/2(12)

Where: φ = 1.6180339887 goden ratio, 𝛹 =-1/φ = -0.6180339887 invers of golden ratio.

γ/2 = cos^-1(φ/2) = π/5 = 36 [deg](13)

2xcos^-1(1/φ) = 103.6545 [deg](14)

φ/2 = cos^-1[(φ)^1/2/2) = 50.50 [deg](15)

Where: γ – internal angle [deg], φ tetrahedral angle [deg].

4. Conclusions

Conformational parameter, phase angle of the pseudorotation P [deg] established with 3-Sphere approach gives for 1-α-methyl-1,4-imino-1,4-dideoxy-D-ribitol 1 two conformers E₃ and ³T₂ having different dihedral θ_HnHn+1 [deg] and tetrahedral φ_Cn [deg] angles but same vicinal angles ϕ [deg]. From units are choose dihedral angles θ_HnHn+1 [deg] with values of the corresponding vicinal angles ϕ [deg] almost equals with recorded. The alternation of the trigonometric functions corresponding to the first set of units is remarkable for the phase angle of pseudorotation, ³T₂ – E₂: N (7.4 [deg]), T-S-S-T (conform with VISION molecular models instead of endocyclic model) and E₃: S (-163 [deg]), S-T-T-S (Table 2). Considerable differences on phase angle of the pseudorotation (Table 5) are calculated from exocyclic torsional angles cis-θ_Hn,Hn+1 [deg]

[9]

relative to endocyclic torsional angles θ_n,n+1 [deg] calculated from 3-Sphere dihedral angles, in last case resulting a good correlation with conformations simulated on VISION molecular models. Conformations of D-ribose stereochemistry 1 and 3 are placed on south side, twist ²T₃ (1: -175.5, -173.9 [deg]), twist ²T₃ conformation (3: -168.021 [deg]) or envelope E₃ conformation (3: -163.46 [deg]) relative to L-ribose 4 in north side (³T₂ – E³(11.91 [deg]) – ³T₂ (1.312 [deg])), excepting trifloroacetate salt with L-ribose stereochemistry 5 (²T₃ – E₃ conformation (-174.01 [deg])). The angle of deviation from planarity θ_m [deg] decreased once the 3-Sphere dihedral angle θ_H2H3 [deg] increased.

Noteworthy, endocyclic torsional angles θ_n,n+1 [deg] are calculated from 3-Sphere dihedral angles θ_Hn,Hn+1 [deg] with right sign selectively in function of the stereochemistry, D-ribitol versus L-ribitol and trans-aa versus trans-ee. Overall, homotopy approach on Planck constant was analyzed on relationship between dihedral θ_HnHn+1[deg] – tetrahedral φ_Cn[deg] angles, with losing 0.94767 at transformation from Planck h to Planck h-bar constants, in case of energy-graph approach (Tables 3, 4) as compared to Euler-conic approach (Tables 1, 2), two ways for transforming carbon chemical shift from ppm to J/molx10⁶ or Gauss with polyhedron model (Table 3) or unit (Table 2). Also, two ways to establish the relationships between dihedral θ_Hn,Hn+1 [deg] and tetrahedral φ_Cn [deg] angles, in function of the phase angle of the pseudorotation P [deg], were consistently employed.

Abbreviations

RMN Data

Nuclear Magnetic Resonance Data

Conflicts of Interest

The authors have not conflict of interest.

References

[1]	C. I. Mitan, E. Bartha, P. Filip, R. Moriarty, Nuclear Magnetic Resonance Spectroscopy - Recent Research and Applications, chapter book: Dihedral and tetrahedral angles of five and six membered ring calculated from NMR data with 3-Sphere approach, IntachOpen 2025; http://dx.doi.org/10.5772/intechopen.1009208
[2]	C.-I. Mitan, E. Bartha, P. Filip, C. Draghici, M.-T. Caproiu, R. M. Moriarty, ACS Spring, San Diego march 23 - 27, 2025 CARB 623, https://doi.org/10.1021/scimeetings.5c11051
[3]	C.-I. Mitan, P. Filip, E. Bartha, C. Draghici, M.-T. Caproiu, R. M. Moriarty, ACS San Diego march 23 - 27, 2025 Sci-Mix CARB - ANYL 632, https://doi.org/10.1021/scimeetings.5c11052
[4]	C. Altona, M. Sundaralingam, Conformational analysis of the sugar ring in nucleosides and nucleotides. A new description using concept of pseudorotation, J. Am. Chem. Soc. 1972, 94, 8205; https://doi.org/10.1021/ja00778a043
[5]	B. Coxon, Chapter 3 Developments in the Karplus equation as they relate to the NMR coupling constants of carbohydrates, Adv. Carb. Chem. Biochem. 2009, 62, 17; https://doi.org/10.1016/50065-2318(09)00003-1
[6]	J. B. Houseknecht, C. Altona, C. M. Hadad, T. Lawary, Conformational analysis of furanose rings with PSEUROT: Parametrization for rings possessing the arabino, lyxo, ribo, and xylo stereochemistry and application to arabinofuranosides, J. Org. Chem. 2002, 67, 4647; https://doi.org/10.1021/jo025635q
[7]	P. Filip, C. I. Mitan, E. Bartha, 3-Sphere tetrahedral angles and phase angle of the pseudorotation P [deg] of C2-CH3-α-D ribitol iminocyclitol, Science Journal of Chemistry 2024, 12(3), 54; SciencePG: http://doi.org/10.11648/j.sjc.20241203.12
[8]	C.-I. Mitan, E. Bartha, P. Filip, C. Draghici, M.-T. Caproiu, R. M. Moriarty, Java Script programs for calculation of dihedral angles with manifold equations, Science Journal of Chemistry 2024, 12(3), 42; SciencePG: https://doi.org/10.11648/j.sjc.20241203.11
[9]	C.-I. Mitan, E. Bartha, P. Filip, C. Draghici, M. T. Caproiu, R. M. Moriarty, Dihedral angles calculated with 3-sphere approach as integer in conformational analysis on D-, L- ribitol series, Rev. Roum. Chim. 2022, 66(21), 941, https://doi.org/10.33224/rrch.2021.66.12.07
[10]	C. Altona, M. Sundaralingam, Conformational analysis of the sugar ring in nucleosides and nucleotides. Improved method for the interpretation of proton magnetic resonance coupling constant, J. Am. Chem. Soc. 1972, 2333; https://doi.org/10.1021/ja00778a043
[11]	VISION molecular models, darling Models, Inc. P. O. Box 1818, Stow, Ohio 44224 U.S.A.
[12]	R. M. Moriarty, C. I. Mitan, N. Branza-Nichita, K. R. Phares, D. Parrish, exo-Imino to endo-iminocyclitol rearrangement. A general route to five membered antiviral azasugars, Org. Lett. 2006, 8, 3465; https://doi.org/10.1021/ol061071r
[13]	E. Bartha, C.-I. Mitan, C. Draghici, M.-T. Caproiu, P. Filip, L. Tarko, R. M. Moriarty, Program for prediction dihedral angle from vicinal coupling constant with 3-sphere approach, Rev. Roum. Chim. 2021, 66(2), 179; https://doi.org/10.33224/rrch.2021.66.2.08
[14]	C.-I. Mitan, E. Bartha, P. Filip, C. Draghici, M. T. Caproiu, R. M. Moriarty, Manifold inversion on prediction dihedral angles from vicinal coupling constant with 3-Sphere approach, Rev. Roum. Chim. 2023, 68(3-4), 185; https://doi.org/10.33224/rrch.2023.68.3-4.08
[15]	E. Bartha, C.-I. Mitan, P. Filip, 3-Sphere torsional angles and six membered ring conformation, Am. J. Quant. Chem. and Mol. Spec. 2023, 7(1), 9; https://doi.org/10.11648/j.ajqcms.20230701.12
[16]	C.-I. Mitan, E. Bartha, P. Filip, Relationship between tetrahedral and dihedral on hypersphere coordinates, Rev. Roum. Chim. 2023, 68(5-6), 261; https://doi.org/1033224/rrch.2023.68.5-6.09
[17]	C.-I. Mitan, E. Bartha, C. Draghici, M.-T. Caproiu, P. Filip, R. M. Moriarty, Tetrahedral angles of five membered ring iminocyclitols with ribitol stereochemistry beyond the dihedral angles, Rev. Roum. Chim. 2022, 67(3), 165; https://doi.org/10.33224/rrch.2022.67.3.04
[18]	C.-I. Mitan, E. Bartha, P. Filip, M.-T. Caproiu, C. Draghici, R. M. Moriarty, Graph flux intensity and electromagnetic wave on 3-Sphere approach, Science Journal of Chemistry 2023, 11(6), 212; https://doi.org/10.11648/j.sjc.20231106.12
[19]	Mathematical Institute, University of Oxford, Homotopy, http://people.maths.ox.ac.uk/nanda/cat/lecture02Hmotopy.pdf
[20]	C. M. Cerda-Carcía-Rojas, L. Gerardo-Zepeda, P. Joseph-Nathan, Tetrahedron Computer Methodology 1990, 3(2), 113; https://doi.org/10.1016/0898-5529(90)90113-M
[21]	E. Westhof, M. Sundaralingam, A method for the analysis of puckering disorder in five - membered rings: the relative mobilities of furanose and proline rings and their effects on polynucleotide and polypeptide backbone flexibility, J. Am. Chem. Soc. 1983, 105, 970; https://doi.org/10.1021/ja00342a054
[22]	M. J. Frisch et. al., Gausian09W C. 01, Gaussian Inc., 340 Quinnipioc St Bldg 40, Wallingford, CT 06492 USA; HYPERLINK " http://www.gaussian.com" www.gaussian.com

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Mitan, C., Bartha, E., Filip, P., Dragutan, V., Dragutan, I., et al. (2026). 3-Spherical Conformational Insights into Iminocyclitols with 1-α-D-Ribose and 1-β-L-Ribose Stereochemistry Under Homotopic Behaviour of Nuclear Magnetic Resonance Data. Science Journal of Chemistry, 14(1), 1-11. https://doi.org/10.11648/j.sjc.20261401.11

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Mitan, C.; Bartha, E.; Filip, P.; Dragutan, V.; Dragutan, I., et al. 3-Spherical Conformational Insights into Iminocyclitols with 1-α-D-Ribose and 1-β-L-Ribose Stereochemistry Under Homotopic Behaviour of Nuclear Magnetic Resonance Data. Sci. J. Chem. 2026, 14(1), 1-11. doi: 10.11648/j.sjc.20261401.11

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Mitan C, Bartha E, Filip P, Dragutan V, Dragutan I, et al. 3-Spherical Conformational Insights into Iminocyclitols with 1-α-D-Ribose and 1-β-L-Ribose Stereochemistry Under Homotopic Behaviour of Nuclear Magnetic Resonance Data. Sci J Chem. 2026;14(1):1-11. doi: 10.11648/j.sjc.20261401.11

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@article{10.11648/j.sjc.20261401.11,
  author = {Carmen-Irena Mitan and Emerich Bartha and Petru Filip and Valeriu Dragutan and Ileana Dragutan and Calin Deleanu and Constantin Draghici and Miron-Teodor Caproiu and Robert Michael Moriarty},
  title = {3-Spherical Conformational Insights into Iminocyclitols with 1-α-D-Ribose and 1-β-L-Ribose Stereochemistry Under Homotopic Behaviour of Nuclear Magnetic Resonance Data},
  journal = {Science Journal of Chemistry},
  volume = {14},
  number = {1},
  pages = {1-11},
  doi = {10.11648/j.sjc.20261401.11},
  url = {https://doi.org/10.11648/j.sjc.20261401.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjc.20261401.11},
  abstract = {3-Sphere, a hypersphere in 4 dimensions approach, applied for calculating stereochemical parameters of iminocyclitol 1 – 5 with Hopf fibration and Lie algebra is described. Three angles have been considered, i.e. dihedral θHnHn+1 [deg] – tetrahedral φCn [deg] – phase angle of the pseudorotation P [deg] calculated from NMR data, vicinal coupling constant 3JHH [Hz] and carbon chemical shift δC [ppm]. This approach gave for 1-α-methyl-1,4-imino-1,4-dideoxy-D-ribitol 2 two conformers E3 and 3T2 having different dihedral θHnHn+1 [deg] and tetrahedral φCn [deg] angles with same vicinal angles ϕ [deg]. Notably, phase angle of pseudorotation P [deg] placed the conformations on the south side for D-ribitols 1 - 3 and on the north side for L-ribitol 4, excepting trifluoroacetate salt of L-ribitol 5. The wave character of NMR data introduced few homotopic switches, the transformation from torus to inverse of torus, the relationship between angles of set A and set B, the transformation from Planck constants h to h-bar, along with the transformation from Joule to Calorie (J 4.1868 ⇆ J-1 0.238). Two methods for calculation of tetrahedral angles φCn [deg], energy-graph and Euler conic with two ways for representing the angles, polyhedron and unit models are analyzed. The conformational parameters, phase angle of the pseudorotation P [deg] established with VISION molecular model and exocyclic 3-Sphere dihedral angles θHnHn+1 [deg] relative to endocyclic torsional angles θn,n+1 [deg] from Altona-Sundaralingan model have been evaluated. In addition, the corresponding angle of deviation from planarity θm [deg] has been determined.},
 year = {2026}
}

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TY  - JOUR
T1  - 3-Spherical Conformational Insights into Iminocyclitols with 1-α-D-Ribose and 1-β-L-Ribose Stereochemistry Under Homotopic Behaviour of Nuclear Magnetic Resonance Data
AU  - Carmen-Irena Mitan
AU  - Emerich Bartha
AU  - Petru Filip
AU  - Valeriu Dragutan
AU  - Ileana Dragutan
AU  - Calin Deleanu
AU  - Constantin Draghici
AU  - Miron-Teodor Caproiu
AU  - Robert Michael Moriarty
Y1  - 2026/02/02
PY  - 2026
N1  - https://doi.org/10.11648/j.sjc.20261401.11
DO  - 10.11648/j.sjc.20261401.11
T2  - Science Journal of Chemistry
JF  - Science Journal of Chemistry
JO  - Science Journal of Chemistry
SP  - 1
EP  - 11
PB  - Science Publishing Group
SN  - 2330-099X
UR  - https://doi.org/10.11648/j.sjc.20261401.11
AB  - 3-Sphere, a hypersphere in 4 dimensions approach, applied for calculating stereochemical parameters of iminocyclitol 1 – 5 with Hopf fibration and Lie algebra is described. Three angles have been considered, i.e. dihedral θHnHn+1 [deg] – tetrahedral φCn [deg] – phase angle of the pseudorotation P [deg] calculated from NMR data, vicinal coupling constant 3JHH [Hz] and carbon chemical shift δC [ppm]. This approach gave for 1-α-methyl-1,4-imino-1,4-dideoxy-D-ribitol 2 two conformers E3 and 3T2 having different dihedral θHnHn+1 [deg] and tetrahedral φCn [deg] angles with same vicinal angles ϕ [deg]. Notably, phase angle of pseudorotation P [deg] placed the conformations on the south side for D-ribitols 1 - 3 and on the north side for L-ribitol 4, excepting trifluoroacetate salt of L-ribitol 5. The wave character of NMR data introduced few homotopic switches, the transformation from torus to inverse of torus, the relationship between angles of set A and set B, the transformation from Planck constants h to h-bar, along with the transformation from Joule to Calorie (J 4.1868 ⇆ J-1 0.238). Two methods for calculation of tetrahedral angles φCn [deg], energy-graph and Euler conic with two ways for representing the angles, polyhedron and unit models are analyzed. The conformational parameters, phase angle of the pseudorotation P [deg] established with VISION molecular model and exocyclic 3-Sphere dihedral angles θHnHn+1 [deg] relative to endocyclic torsional angles θn,n+1 [deg] from Altona-Sundaralingan model have been evaluated. In addition, the corresponding angle of deviation from planarity θm [deg] has been determined.
VL  - 14
IS  - 1
ER  -

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Author Information

Carmen-Irena Mitan

Department of Organic Chemistry, “C. D. Nenitescu” Institute of Organic and Supramolecular Chemistry of Roumanian Academy, Bucharest, Romania

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http://orcid.org/0000-0002-4790-3122
Emerich Bartha

Department of Organic Chemistry, “C. D. Nenitescu” Institute of Organic and Supramolecular Chemistry of Roumanian Academy, Bucharest, Romania
Petru Filip

Department of Organic Chemistry, “C. D. Nenitescu” Institute of Organic and Supramolecular Chemistry of Roumanian Academy, Bucharest, Romania

Contact Email

http://orcid.org/0000-0003-2076-7244
Valeriu Dragutan

Department of Organic Chemistry, “C. D. Nenitescu” Institute of Organic and Supramolecular Chemistry of Roumanian Academy, Bucharest, Romania

Contact Email
Ileana Dragutan

Department of Organic Chemistry, “C. D. Nenitescu” Institute of Organic and Supramolecular Chemistry of Roumanian Academy, Bucharest, Romania
Calin Deleanu

Department of Organic Chemistry, “C. D. Nenitescu” Institute of Organic and Supramolecular Chemistry of Roumanian Academy, Bucharest, Romania

http://orcid.org/0000-0001-8206-5227
Constantin Draghici

Department of Organic Chemistry, “C. D. Nenitescu” Institute of Organic and Supramolecular Chemistry of Roumanian Academy, Bucharest, Romania
Miron-Teodor Caproiu

Department of Organic Chemistry, “C. D. Nenitescu” Institute of Organic and Supramolecular Chemistry of Roumanian Academy, Bucharest, Romania
Robert Michael Moriarty

Department of Chemistry, University of Illinois at Chicago, Chicago, USA

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Figure 1

Figure 1. Five membered ring iminocyclitols with α-D-ribitol (1-3) and β-L-ribitol (4, 5) stereochemistry.

Table 1

Table 1. Dihedral θ_HnHn+1 [deg] [13, 14] and tetrahedral φ_Cn [deg] angles [16] of 1-α-methyl-1,4-imino-1,4-dideoxy-D-ribitol 2 [12] calculated from carbon chemical shift δ_Cn [ppm].
Table 2

Table 2. Phase angle of the pseudorotation P [deg] established with VISION molecular models from dihedral θ_HnHn+1 [deg] [2, 13, 14] and tetrahedral φ_Cn [deg] angles [16] of 1-α-methyl-1,4-imino-1,4-dideoxy-D-ribitol 2 [12] calculated from carbon chemical shift δ_Cn [ppm] and vicinal coupling constant ³J_HH [Hz].
Table 3

Table 3. Tetrahedral angles [17, 18] calculated from carbon chemical shift δ_Cn [ppm] in relationship with dihedral θ_HnHn+1 [deg] and vicinal ϕ [deg] angles for iminocyclitol 2.
Table 4

Table 4. Phase angle of the pseudorotation P [deg] established with VISION molecular models from dihedral θ_HnHn+1 [deg] and tetrahedral φ_Cn [deg] angles [17, 18] of 1-α-methyl-1,4-imino-1,4-dideoxy-D-ribitol 2 [12] calculated from carbon chemical shift δ_Cn [ppm] and vicinal coupling constant ³J_HH [Hz].
Table 5

Table 5. Phase angle of the pseudorotation P [deg] and angle of deviation from planarity θ_m [deg] calculated with optimized Altona-Sundaralingan model (eq. (1-5)) for iminocyclitol 2.
Table 6

Table 6. Phase angle of the pseudorotation P [deg] and angle of deviation from planarity θ_m [deg] calculated with optimized Altona-Sundaralingan model (eq. (1-5)) for iminocyclitol 1, 3-6.

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APA Style

Mitan, C., Bartha, E., Filip, P., Dragutan, V., Dragutan, I., et al. (2026). 3-Spherical Conformational Insights into Iminocyclitols with 1-α-D-Ribose and 1-β-L-Ribose Stereochemistry Under Homotopic Behaviour of Nuclear Magnetic Resonance Data. Science Journal of Chemistry, 14(1), 1-11. https://doi.org/10.11648/j.sjc.20261401.11

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ACS Style

Mitan, C.; Bartha, E.; Filip, P.; Dragutan, V.; Dragutan, I., et al. 3-Spherical Conformational Insights into Iminocyclitols with 1-α-D-Ribose and 1-β-L-Ribose Stereochemistry Under Homotopic Behaviour of Nuclear Magnetic Resonance Data. Sci. J. Chem. 2026, 14(1), 1-11. doi: 10.11648/j.sjc.20261401.11

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AMA Style

Mitan C, Bartha E, Filip P, Dragutan V, Dragutan I, et al. 3-Spherical Conformational Insights into Iminocyclitols with 1-α-D-Ribose and 1-β-L-Ribose Stereochemistry Under Homotopic Behaviour of Nuclear Magnetic Resonance Data. Sci J Chem. 2026;14(1):1-11. doi: 10.11648/j.sjc.20261401.11

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@article{10.11648/j.sjc.20261401.11,
  author = {Carmen-Irena Mitan and Emerich Bartha and Petru Filip and Valeriu Dragutan and Ileana Dragutan and Calin Deleanu and Constantin Draghici and Miron-Teodor Caproiu and Robert Michael Moriarty},
  title = {3-Spherical Conformational Insights into Iminocyclitols with 1-α-D-Ribose and 1-β-L-Ribose Stereochemistry Under Homotopic Behaviour of Nuclear Magnetic Resonance Data},
  journal = {Science Journal of Chemistry},
  volume = {14},
  number = {1},
  pages = {1-11},
  doi = {10.11648/j.sjc.20261401.11},
  url = {https://doi.org/10.11648/j.sjc.20261401.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjc.20261401.11},
  abstract = {3-Sphere, a hypersphere in 4 dimensions approach, applied for calculating stereochemical parameters of iminocyclitol 1 – 5 with Hopf fibration and Lie algebra is described. Three angles have been considered, i.e. dihedral θHnHn+1 [deg] – tetrahedral φCn [deg] – phase angle of the pseudorotation P [deg] calculated from NMR data, vicinal coupling constant 3JHH [Hz] and carbon chemical shift δC [ppm]. This approach gave for 1-α-methyl-1,4-imino-1,4-dideoxy-D-ribitol 2 two conformers E3 and 3T2 having different dihedral θHnHn+1 [deg] and tetrahedral φCn [deg] angles with same vicinal angles ϕ [deg]. Notably, phase angle of pseudorotation P [deg] placed the conformations on the south side for D-ribitols 1 - 3 and on the north side for L-ribitol 4, excepting trifluoroacetate salt of L-ribitol 5. The wave character of NMR data introduced few homotopic switches, the transformation from torus to inverse of torus, the relationship between angles of set A and set B, the transformation from Planck constants h to h-bar, along with the transformation from Joule to Calorie (J 4.1868 ⇆ J-1 0.238). Two methods for calculation of tetrahedral angles φCn [deg], energy-graph and Euler conic with two ways for representing the angles, polyhedron and unit models are analyzed. The conformational parameters, phase angle of the pseudorotation P [deg] established with VISION molecular model and exocyclic 3-Sphere dihedral angles θHnHn+1 [deg] relative to endocyclic torsional angles θn,n+1 [deg] from Altona-Sundaralingan model have been evaluated. In addition, the corresponding angle of deviation from planarity θm [deg] has been determined.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - 3-Spherical Conformational Insights into Iminocyclitols with 1-α-D-Ribose and 1-β-L-Ribose Stereochemistry Under Homotopic Behaviour of Nuclear Magnetic Resonance Data
AU  - Carmen-Irena Mitan
AU  - Emerich Bartha
AU  - Petru Filip
AU  - Valeriu Dragutan
AU  - Ileana Dragutan
AU  - Calin Deleanu
AU  - Constantin Draghici
AU  - Miron-Teodor Caproiu
AU  - Robert Michael Moriarty
Y1  - 2026/02/02
PY  - 2026
N1  - https://doi.org/10.11648/j.sjc.20261401.11
DO  - 10.11648/j.sjc.20261401.11
T2  - Science Journal of Chemistry
JF  - Science Journal of Chemistry
JO  - Science Journal of Chemistry
SP  - 1
EP  - 11
PB  - Science Publishing Group
SN  - 2330-099X
UR  - https://doi.org/10.11648/j.sjc.20261401.11
AB  - 3-Sphere, a hypersphere in 4 dimensions approach, applied for calculating stereochemical parameters of iminocyclitol 1 – 5 with Hopf fibration and Lie algebra is described. Three angles have been considered, i.e. dihedral θHnHn+1 [deg] – tetrahedral φCn [deg] – phase angle of the pseudorotation P [deg] calculated from NMR data, vicinal coupling constant 3JHH [Hz] and carbon chemical shift δC [ppm]. This approach gave for 1-α-methyl-1,4-imino-1,4-dideoxy-D-ribitol 2 two conformers E3 and 3T2 having different dihedral θHnHn+1 [deg] and tetrahedral φCn [deg] angles with same vicinal angles ϕ [deg]. Notably, phase angle of pseudorotation P [deg] placed the conformations on the south side for D-ribitols 1 - 3 and on the north side for L-ribitol 4, excepting trifluoroacetate salt of L-ribitol 5. The wave character of NMR data introduced few homotopic switches, the transformation from torus to inverse of torus, the relationship between angles of set A and set B, the transformation from Planck constants h to h-bar, along with the transformation from Joule to Calorie (J 4.1868 ⇆ J-1 0.238). Two methods for calculation of tetrahedral angles φCn [deg], energy-graph and Euler conic with two ways for representing the angles, polyhedron and unit models are analyzed. The conformational parameters, phase angle of the pseudorotation P [deg] established with VISION molecular model and exocyclic 3-Sphere dihedral angles θHnHn+1 [deg] relative to endocyclic torsional angles θn,n+1 [deg] from Altona-Sundaralingan model have been evaluated. In addition, the corresponding angle of deviation from planarity θm [deg] has been determined.
VL  - 14
IS  - 1
ER  -

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