Research Article | | Peer-Reviewed

Java Script Programs for Calculation of Dihedral Angles with Manifold Equations

Received: 1 May 2024     Accepted: 20 May 2024     Published: 3 June 2024
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Abstract

Java Script programs for calculation dihedral angles from NMR data with manifold equations of 3-Sphere approach: rectangle, Villarceau circles of cyclide (Torus – Dupin Cyclide), polar equations, Euler-Conic. Manifolds are curves or surface in higher dimension used for calculation of dihedral angles under wave character of NMR data, carbon and/or proton chemical shift δXn[ppm] and vicinal coupling constant 3JHnHn+1[Hz]. 3-Sphere approach for calculation of the dihedral angles from NMR data in four steps: 1. Prediction, or more exactly calculation of the dihedral angles from vicinal coupling constant with trigonometric equations, 2. Calculation of the dihedral angles from manifold equations; 3. Building units from angle calculated with one of the manifold equations; 4. Calculation the vicinal coupling constant of the manifold dihedral angle. In this paper are presented Java Script programs of step 2 and from step 3 only the Java Script program for calculation of seven sets angles. The bond distances lCnCn+1[A0] between two atoms of carbon are under different polar equations (i.e. limaçons or cardioid, rose or lemniscale), our expectation was to find different manifold equations for calculation the best angle, differences are smaller but can be find sometimes a preferred one for a vicinal coupling constant. 3-Sphere approach has the advantages of calculation from vicinal angle or/and chemical shift the dihedral angle, tetrahedral angle and the bond distance lCnCn+1[A0], with application on conformational and configurational analysis.

Published in Science Journal of Chemistry (Volume 12, Issue 3)
DOI 10.11648/j.sjc.20241203.11
Page(s) 42-53
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

Manifold, Reactangle, Villarceau Circles, Euler-Conic, Hopf Fibration, Lie Algebra, 3-Sphere, Dihedral Angles, Java Script

1. Introduction
Curve or surface generalized in higher dimensions are named manifold, are applied in classical mechanics, general relativity and quantum field theory using elements of algebra, topology, and analysis. A manifold can be locally Euclidean space, but also smooth (Riemannian manifold), complex (Kahler structure), algebraic (sphere, Riemannian sphere, torus – elliptic curves – Weierstrass elliptic functions). Sixteen points on the Poincaré sphere results from four points of polar angle and azimuthal angle in case of Poincaré rotator, a polarization modulator, which enable dynamic control over a polarization state by execution an arbitrary rotation on the Poincaré sphere, same with Poincaré sphere on organic stereochemistry. The Poincaré rotor is an SU(2) Lie group operator, with two U(1) operations to change the phase and the amplitude of the state, able to achieve distinguishable 4x4 = 16, 8x8 = 64, 10x10 = 100 states on the Poincaré sphere after modulating both the phase and amplitude. The molecular structure of Buckmmsterfullerene (C60) was drawn using the polarization states on the sphere. Geometrical consideration of Stokes parameters on the Poincaré sphere, relationship for parametrization of ellipse based on angle orientation and ellipticity. The wave function corresponds to a point on a surface of a unit sphere R in 4-dimensions, isomorphic to a complex unit sphere C in 2-dimension. A point on a surface of a hypersphere described a state of coherent photons in quantum wave function.
Lie N-algebra is a N-manifold having homological vector field. N-Sphere having (N+1) dimensional Euclidean Space. Topological Poincaré conjecture is a 4-manifold, 1 torus is a circle and a square flat torus is a 3-sphere S3. Poincaré group is a 10 dimensional Lie group of affine isometries of the Minkowski space. Minlowski space closely associated but different with Einstein’s theories of special relativity and general relativity, in the Minkowski space the Einstein Universe is approximated at the point of observation with conventional time and energies in the local flat space. Segal replaced the Minkowski space Mo with unique four dimension manifold M, in fact the Euclidean space E3 of Mo is replaced by a sphere S3 of small but nonvanishing curvature, a deformation of M into Mo, and SU(2,2) into the Poincaré group as the formation of the “flat” limit of “curved” picture.
3-Sphere approach for calculation of the dihedral and tetrahedral angles from NMR data, having as mathematical theories Hopf fibration and Lie algebara. Manifolds locally in Euclidean space or algebraic can be used for calculation dihedral angles from carbon and/or proton chemical shift or only from vicinal coupling constant 3JHnHn+1[Hz]. Dihedral angle can be represented on three concentric cons having six dihedral angle and two algebraic angles, 3xϕ1 and 2xϕ2, which after translation from 3D to 2D can be represented on two sets angles having as point of linkage halt of ϕ1 of sets A and B equal with first angle of set B and A. First angles of sets A and B are equals ϕ2 of sets D and F resulting other four sets angles, totally seven sets angles under U or S rule: a. first angles higher as 5[deg] unit U, b. first angles smaller as 5 [deg] unit S. On six sets angles from two units can be found dihedral angles with all stereochemistry, on 14 sets angles are found in close relationship tetrahedral and dihedral angles: a. seven sets starting with U unit and seven with S unit in case of tetrahedral angles φCn[deg] of five membered ring calculated from vicinal coupling constant 3JHnHn+1[Hz], b. under sin and tan function in case of tetrahedral angles φCn[deg] of five membered ring calculated from chemical shift ΔδCnCn+1[ppm]. Recorded NMR data are transformed from ppm in gauss and transformed in angle with manifold equations , then with calculated angle are builds units:
Cn= UnNi, SnNi, Tn= UnNi, SnNi,
with I = 1 – 6, n = 1 – 7, N = A, B, C, D, E, F.
Angle with values almost equals with angle calculated only from vicinal constant coupling will be dihedral angle, its relationship with tetrahedral angle can be determined with already published trigonometric equations or following the relationship on 14 sets angles on two units based on θ and ϕ.
2. Results
2.1. Manifolds Java Script Programs
2.1.1. Rectangle Approach
Rectangle approach with its geometrical variation: kite – trapezoid – antiparallelogram – irregular, with non-coplanar line – middle line – antirecatangle middle line, transformed in angles of skew circles (eq. 1-2) and middle circles (eq. 3-4) are used for building units and established the dihedral angle.
Skew circles:
θAn = 2x90x(ΔδHnHn+1 ΔδCnCn+1)2[deg](1)
cis-ea, -ae, trans-ee
θAn = 90x(ΔδHnHn+1 ΔδCnCn+1)2[deg](2)
trans-aa6,1
Middle circles:
θAn = 2x90x(ΔδHnHn+1+ ΔδCnCn+1)2[deg](3)
cis-ea, -ae, trans-ee
θAn = 90x(ΔδHnHn+1- ΔδCnCn+1)2[deg](4)
trans-aa6,1
where: θAn –angle of set A from unit U or S, ΔδHnHn+1, ΔδCnCn+1 – the differences between two consecutive protons and two atoms of carbon [ppm].
Figure 1. Java Script program ScienceRectangle.html for calculation of dihedral angles with rectangle eq. 1-4 from chemical shift δ[ppm].
As observation: instead of n used for cis and trans stereochemistry in case of Skew and Middle circles can be applied the rule used for calculation of the vicinal coupling constant 3JHnHn+1[Hz] from vicinal angle ϕ[deg] (eq. 5-8).
Trans:3JHH=ϕ(5)
Cis:3JHH=ϕ2(6)
θAn=A, A = f(ΔδHnHn+1,ΔδCnCn+1)(7)
θAn=A2, A = f(ΔδHnHn+1,ΔδCnCn+1)(8)
Java Script program presented at ACS spring 2021 for calculation of the dihedral angles from carbon and proton chemical shift δCn[ppm] with rectangle equations in 3 steps : 1. Prediction of de dihedral angles θHnHn+1[deg] only from vicinal coupling constants 3JHnHn+1[Hz] without torus inversion; 2. Calculation the first angle of set B or half ϕ1 of set A; 3. Building three sets angles for all equations (eq. 1-4), and choosing of the dihedral angle with values almost equals with angles calculated only from vicinal coupling constant. If the required dihedral angle is not found on unit with three sets angles can be used the program for transformation from U to S and viceversa, also can be calculated the vicinal coupling constant from the calculated dihedral angle.
In figure 1 is presented Java Script program ScienceRectangle.html for calculation of the dihedral angles from carbon and proton chemical shift with rectangle eq. 1-4, in fact only one angle of set A. Then with this angle are build units with seven sets angles and analyzed the stereochemistry and the sign of dihedral angle
2.1.2. Cyclide Approach
Figure 2. Torus (A) and Dumpin Cyclide (C).
A. Villarceau circles Cv1 and Cv2, or two sphere Sv1 and Sv2 result at intersection with a bitangent plane Pv, which meet all of the parallel circular cross-sections of the torus at the same angle (Figure 2).
θTNA= sin-1(ΔδHHΔδCC)(9)
θTNB= cos-1(ΔδHHΔδCC)(10)
θDNA= tan-1(ΔδHHΔδCC)(11)
θDNB= cotan-1(ΔδHHΔδCC)(12)
Where: ΔδHnHn+1, ΔδCnCn+1 – the differences between two consecutive protons and two atoms of carbon [ppm], θTNA, NB, θDNA, NB - Villarceau circles of torus and Dupin cyclide.
B. Dupin cyclide non-spherical algebraic surface of degree four, compatible with inversion of torus, having embedded all four characteristics circles of torus (Figure 2).
Torus equations for calculation one characteristic angle of first set in function of the physical transformation reveals that the proton and carbon chemical shifts transformed in gauss don’t fit into the Hopf fibration relationship between three sets angles, instead using the transformation on Hz the units can be built in line with Hopf fibration rule. A. Transformation in Hz :
θWnA= sin-1(ΔδHnHn+1H/ΔδCnCn+1C)(13)
Euler character: sinθWEA= cosθWEB(14)
B. Transformation in gauss:
θWnA= sin-1(ΔδHnHn+1HH/ΔδCnCn+1CC)(15)
Without Euler character: sinθWFA≠ cosθWFB(16)
Where: ΔδHnHn+1, ΔδCnCn+1 – the differences between two consecutive protons and two atoms of carbon [ppm], ωL – Larmor frequency [13C: 75MHz, 1H 400MHz], ν – frequency [Hz], δ – chemical shifts [ppm], γ – gyromagnetic ratio: 13C: γ = 10.71 [MHzxT-1] = 6.7[107xradxT-1xs-1], 1H: γ = 42.57 [MHzxT-1] = 26.7[107xradxT-1xs-1].
Figure 3. Java Script program ScienceVillarceau.html for calculation of dihedral angles with Cyclide approach from chemical shift δ[ppm] with eq. 9-12.
Figure 4. Java Script program SciencePOLAReq.html for calculation of dihedral angles with polar equations 17-20 from carbon chemical shift δ[ppm].
Java Script program ScienceVillarceau.html presented in Figure 3 for calculation of the dihedral angles from carbon and proton chemical shift with cyclide equations (eq. 9-12) start with introduction of the vicinal coupling constant, proton and carbon chemical shift, calculation one angle of set A with eq. 9, 11. A program for calculation of the dihedral angles only from torus Villarceau (eq. 9) circles containing also the step for building of the unit with seven set angle was published.
2.1.3. Polar Equations Approach
Polar equations without (eq. 17, 18) or under stereochemistry rule (eq. 19, 20) and homotopic approach (continues transformation from torus to rectangle, from π to deg) are proposed for calculation of dihedral angles θHnHn+1[deg] from the ratio differences in chemical shift between two protons or atoms of carbon and vicinal coupling constant 3JHnHn+1[Hz]. A ratio used in literature for transformation of the differences in chemical shift between two atoms of carbon from ppm in Hz at values higher as 10, and Δν = Jx(3)1/2 at smaller differences of two protons chemical shift, in other words from AB to AX coupling.
RmH= ΔδHnHn+1/3JHnHn+1[s](17)
RmC= ΔδCnCn+1/3JHnHn+1[s](18)
Where x = C, H, Rmx = 1/Rmx or Rm = 10xRmx
cis,trans-ee: (2xRmH)2= θx(19)
trans-aa: (RmH)2= θx(20)
4–ν1)/(ν3–ν2) = tan2θ(21)
1–ν3) = (ν2–ν4) = (Δν2+J2)1/2(22)
Where νn – n = 1 – 4, four pics of the two coupled protons.
Figure 5. Java Script program EulerConic.html for calculation of dihedral angles with Euler-Conic approach with eq. 23.
2.1.4. Euler-Conic Approach
One angle of first set can be calculated from the differences in chemical shift between two protons ΔδHnHn+1[ppm] or two atoms of carbons ΔδCnCn+1[ppm] with Euler-Conic equation 23.
θ = sin-1Rmx[deg](23)
Where Rmx = ΔδXnXn+1Hx4x10-3H [gauss], x = C, H; ΔδXnXn+1 – the differences between two consecutive protons and two atoms of carbon [ppm], ωL – Larmor frequency [13C: 75MHz, 1H 400MHz], ν – frequency [Hz], δ – chemical shifts [ppm], γ – gyromagnetic ratio.
Figure 6. Java Script program ScienceSevenUnit.html for building seven sets units.
Java Script program EulerConic.html for calculation of the dihedral angles from carbon or proton chemical shift with Euler-Conic eq. 23 is presented in Figure 5. Same with all the programs presented in this paper, is a short program only for calculation one angle of first set. With the angle calculated then can be used a program for building of the units with seven sets angles from sin or tan function, and used the transformation from U to S and S to U published already . A program for calculation of two seven sets units starting with half ϕ1 of set A namely AQ1 was presented at ASC spring 2021 (Step3unitAQ1.html). As reported by Silverstein, G. C. Bassler difference between chemical shift of two coupled protons leads to difference in system of equations , thus on the unit are found in function of the values of coupling constant different equation: the transformation from U to S through first angle of set θUB1 leading to third angles of set θSA3 with eq. 24.
θSB3= nx(3xθUB1), n = 1, 2(24)
Where: θUB1 – first angle of set B from unit U (θN1 > 5[deg]), θSB3 – third angle of set B from unit S (θN1 < 5[deg]), N = A, B, C, D, E, F, G.
2.2. Seven Sets Unit
Seven sets angles, three Venn diagrams A, B, C - D, E, A - F, G, B, with angles between 69 -70[deg] in case of unit U1, and with angles between 60 - 65[deg] in case of unit S1, or two units U1 and one S1 and vice-versa, can be calculated with Java Script program ScienceSevenUnit.html (Figure 6).
U1 = U1Ai, U1Bi, U1Ci, U1Di, U1Ei, U1Fi, U1Gi
Where: U1A1 > 5[deg], S1A1 < 5[deg], i = 1-6.
3. Discussion
The Java Script programs presented for calculation from chemical shift δ[ppm] and/or vicinal coupling constant 3JHnHn+1[Hz] enable a carefully analysis of dihedral angles θHnHn+1[deg] with corresponding sign and stereochemistry. Since bond distances lCnCn+1[A0] between two atoms of carbon are all under different polar equations (i.e. limaçons or cardioid, rose or lemniscale) our expectation was to find different equations for calculation the best angle, much more the differences between dihedral angles calculated from carbon and proton chemical shift and vicinal coupling constant with varies manifolds to give information about the limits of deviation from planarity.
Dihedral angles calculated with Rectangle approach (eq. 1-4) and Java Script ScienceRectangle.html program from carbon and proton chemical shift are published to date, and results are remarkable.
Eular-Conic approach (eq. 23) for calculation of the dihedral angles from carbon or proton chemical shift with Java Script program EulerConic.html is probable the preferred one. Because smaller differences are observed in case of iminocyclitols studied to date between unit builds from sin and tan functions, in Java Script program presented in Figure 5 are calculated only sin and cos functions, angles of set A and B. Units calculated from sin and tan functions are used for calculation of the dihedral θHnHn+1[deg] and tetrahedral angles φCn[deg] in opposite.
Villarceau approach (eq. 9-12) and Java Script program ScienceVillarceau.html (Figure 3) enable calculation of the dihedral angles θHnHn+1[deg] from the ratio proton/carbon chemicals shift δ[ppm] transformed in Hz, since gauss transformation don’t give the expected correlation between set A and set B, resulting two seven set units. Along the sin function in this paper was introduced the tan function, in attempt to find the almost equals values between recorded and calculated vicinal coupling constant. In case of isopropylidene protected iminocyclitol 1 (Figure 7) for a vicinal coupling constant of 5.4[Hz] was calculated from tan function a vicinal coupling constant of 5.4[Hz] relative to 5.37[Hz] calculated from sin function.
Trans-ee: θH3H4= tan-1(sin-ϕ) (25)
Trans-ee: θH3H4= sin-1(tan-ϕ) (26)
Dihedral angles are calculated with Java Script SciencePolareq.html (Figure 4) and polar equations approach eq. 17-20 from the differences between two protons chemical shift ΔδHnHn+1[ppm] or two carbons chemical shift ΔδCnCn+1[ppm] and vicinal coupling constant 3JHnHn+1[Hz] (Tables 1 and 2), using once the sin and cos function and second the homotopic approach . In second case are calculated angles with and without vicinal angle rule (θA, θXA[deg]). All the programs presented gives one or two angles of first and second sets angles A and B, angles used for building seven sets unit and chose the dihedral angle with value almost equal with dihedral angle θHnHn+1[deg] calculated only from vicinal coupling constant 3JHnHn+1[Hz]In accord with D-ribitol stereochemistry, iminocyclitols 1-3 (Figure 7) have negative dihedral angle θH3H4[deg], angle calculated with equations 25, 26.
Figure 7. Iminocyclitols 1-3 with C1-R-α-Dribitol stereochemistry .
Table 1. Dihedral angles of iminocyclitols 1-3 calculated from differences between chemical shift ΔδHn[Hz] of two protons and vicinal coupling constant 3JHnHn+1[Hz] with polar equations 17, 19, 20.

Entry

3JHHa [Hz]

Hn [ppm]

RmH [π]

θ[deg],3JHH[Hz]

θA [deg]

θ[deg], 3JHH[Hz] eq. 17

θXA [deg]

θ[deg], 3JHH[Hz] eq. 19, 20

1

4.1

3.08

4.49

4.38

3.22

0.3439

20.11, 4.18

2.907

3.439

21.93U1C1, 4.12

22.29U1C1, 4.11

33.82

47.30

22.35U2B1, 4.11

23.65U1E1, 4.07

2

5.4

0.0203

-28.83, 5.45

49.09

-24.54U1E1, 5.35

-

-

3

0

d, H3, 0.1

0.0862

-94.96S1B4, 1.1tan

-92.47S1E4, 0.79tan

0.203

0.086

0.862

-24.96U1G1, 5.36

-90.08S1B4, 0.146tan

-90.86S1B4, 0.46tan

0.165

0.029

2.972

-24.97U1G1, 5.36

-90.029S1B4, 0.086tan

-92.97S1B4, 0.86tan

4

3.1

3.71

4.16

4.26

3.58

0.1451

51.65U1A2, 3.09

6.888

1.451

51.88U2F2, 3.08

50.24U1D2, 3.15

189.82

8.428

50.172U1A2, 3.15

51.571U1A2, 3.09

5

3.9

0.0256

29.26S1E1, 3.89

39.00

0.256

25.2S1G1, 4.01

29.74S1B1, 3.88

-

0.262

-

29.73S1B1, 3.88

6

8.8

0.0772

-166.88U1F6, 8.78tan

12.941

0.772

-166.71U1A6, 8.77tan

-169.57U1B6, 8.92tan

167.47

0.597

-167.16U1A6, 8.8tan

-169.63U1B6, 8.93tan

7

4.8

3.11

4.02

4.12

2.89

0.1895

-2.78S1A1, 4.81

5.274

1.895

-1.64U1S1A1, 4.78

-1.89S1A1, 4.79

111.290

14.376

-3.87U1S1A1, 4.84

-1.86S1B1, 4.79

8

5.2

0.0192

-15.55U1G1, 5.13

-19.66U1A1, 5.23

51.999

0.192

-19.00U1G1, 5.22

-19.93U1A1, 5.24

-

0.1479

-

-19.95U1A1, 5.24

9

0

bs, 0.81

0.6585

-93.57S1A4, 0.94tan

0.658

6.585

-90.65S1B4, 0.4tan

-93.29SD1, 0.90tan

1.7346

173.46

-91.73S1B4, 0.65tan

-93.27SE4, 0.90tan

[a] δ[ppm] 1-CDCl3, 2-D20, 3- CDCl3: 13C 75 [MHz], 1H 400 [MHz]
3-Sphere approach for calculation of the dihedral angles from NMR data, under Hopf fibration and Lie algebra, follow few steps:
1. Calculation of the dihedral angles only from vicinal coupling constant with Java Script program PREDICTIEACS2022.html. The program of prediction is based on trigonometric equation and not on algebraic equations. The differences between trans-ee4,1 and trans-ee3,23 and θ4) can be made only with algebraic equations on very good recorded spectra, probably. The transformation from U to S is performed with trans-ee3,2 algebraic equation, almost compatible with tan function, a general rule for calculation only the trans-ee3,2 angle, but from trigonometric equations, and algebraic equation for trans-ee4,1, first angle of set A or B, is too hard to made the differences between the two trans-ee angles. From eq. 25, 26 results cisSA1 angles with negative sign giving trans-ee4,1 dihedral angles in accord with D-ribitol stereochemistry.
2. Calculation one angle of set A with manifold equations from carbon and/or proton chemical shift.
3. Building of seven sets angles of unit U and S, and choosing the dihedral angle with values almost equals with calculated dihedral angles from vicinal coupling constant. The number of unit U and S can be increased through set C until the angles of first unit are almost equals with last unit, at list can be extracted and analyzed the differences results from chemical shift for many dihedral angles with values almost equals with dihedral angles calculated only from vicinal coupling constant.
4. Calculation of the vicinal coupling constant 3JHnHn+1[Hz] of the manifold dihedral angles.
Table 2. Dihedral angles of iminocyclitols 1-3 calculated from differences between chemical shift of two carbon ΔδCn[ppm] and vicinal coupling constant 3JHnHn+1[Hz] with polar equations 18, 19, 20.

Entry

3JHHa [Hz]

Cn [ppm]

RmC [π]

θ[deg],3JHH[Hz]

θA [deg]

θ[deg], 3JHH[Hz] eq. 18

θXA [deg]

θ[deg], 3JHH[Hz] eq. 19, 20

1

4.1

55.8

83.5

84.3

65.9

0.1480

21.48U1B1, 4.13

6.756

1.480

23.48U1B1, 4.13

20.98U1C1, 4.15

182.57

8.763

21.71U1C1, 4.13

21.23U1B1, 4.14

2

5.4

0.1481

-25.74S1E1, 5.37

6.75

1.481

-26.62S1E1, 5.39

-28.51S1B1, 5.44

182.25

8.779

-27.75S1B1, 5.42

-25.61S1E1, 5.37

3

0

d, H3

0.0054

-90.308S1B4, 0.277 tan

0.0054

0.0543

-90.005S1B4, 0.036 tan

-90.053S1B4, 0.115 tan

0.0001

0.0118

-90.0001S1B4, 0.005 tan

-90.0118S1B4, 0.053 tan

4

3.1

57.4

71.5

71.7

66.8

0.2198

51.35U1F2, 3.108

(-31.98, -53.10tan)

4.548

2.198

50.90U1C2, 3.12

49.26U1B2, 3.19

82.751

19.335

52.75U1B2, 3.05

50.33U1D2, 3.14

5

3.9

0.0512

28.53S1E1, 3.92

19.499

0.512

28.49S1A1, 3.92

29.48S1B1, 3.88

-

1.0519

-

28.94S1B1, 3.90

6

8.8

0.5568

-168.49U1F6, 8.87tan

1.7959

5.5681

-169.21U1F6, 8.91tan

-167.49U1F6, 8.81tan

3.225

31.004

-168.71U1F6, 8.88tan

-169.49U1F6, 8.92tan

7

4.8

63.7

72.5

74.0

69.3

0.5454

-3.055S1B1, 4.82

(-44.95, 3.06tan)

1.8333

5.4545

-1.833S1A1, 4.79

(-44.98, 3.06tan)

-2.727S1D1, 4.81

(-44.96, 2.73tan)

13.444

119.008

-

-1.983S1C1, 4.79

(-44.98, 1.98tan)

8

5.2

0.2884

-16.76U1A1, 5.16

(-43.75, 17.53tan)

3.4666

2.8846

-18.84U1A1, 5.21

(-43.42, 19.95tan)

-19.03U1A1, 5.22

(-43.38, 20.18tan)

48.0711

33.2840

-18.07U1B1, 5.19

(-43.55, 19.04tan)

-18.90U1A1, 5.21

(-43.41, 20.02tan)

9

0

bs, 0.81

0.1723

-90.182S1B4, 0.213 tan

5.8024

1.7234

-95.772S1B4, 1.20 tan

-91.724S1B4, 0.656 tan

134.674

11.8804

-91.953S1CB4, 0.698 tan

-95.668S1CB4, 1.187 tan

[a] δ[ppm] 1-CDCl3, 2-D20, 3- CDCl3: 13C 75 [MHz], 1H 400 [MHz].
4. Conclusions
3-Sphere approach, a method which enables calculation of the dihedral angles, tetrahedral angles, bond lengths, from chemical shift δ[ppm] and vicinal coupling constant 3JHnHn+1[Hz], was analyzed in this paper from the manifold point of view in attempt to find the best solution for building a program for calculation of the conformation and configuration of five and six membered ring, based on spatial representation.
Five Java Script programs for calculation dihedral angles from NMR data, carbon and/or proton chemical shift δ[ppm] along vicinal coupling constant 3JHnHn+1[Hz], with manifold equations of 3-Sphere approach are presented in Fig. 1, 3-6: rectangle, Villarceau circles of cyclide (Torus – Dupin Cyclide), polar equations, Euler-Conic, and unit of seven sets angles.
Abbreviations

RMN

Nuclear Magnetic Resonance

Author Contributions
Carmen-Irena Mitan: Conceptualization, Methodology, Writing – original draft, Writing – review & editing
Emerich Bartha: Conceptualization, Metodology
Petru Filip: Methodology
Miron-Teodor Caproiu: Formal Analysis
Constantin Draghici: Methodology
Robert Michael Moriarty: Supervision
Conflicts of Interest
The authors declare no conflicts of interest.
References
[1] Loring W. Tu, Manifolds, An introduction to Manifolds, Universititxt, Springer Link,
[2] euclidean space: Riemannian manifold, Kahler structure, algebraic: sphere, Riemannian sphere, torus – elliptic curves – Weierstrass elliptic functions.
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  • APA Style

    Mitan, C., Bartha, E., Filip, P., Draghici, C., Caproiu, M., et al. (2024). Java Script Programs for Calculation of Dihedral Angles with Manifold Equations. Science Journal of Chemistry, 12(3), 42-53. https://doi.org/10.11648/j.sjc.20241203.11

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    ACS Style

    Mitan, C.; Bartha, E.; Filip, P.; Draghici, C.; Caproiu, M., et al. Java Script Programs for Calculation of Dihedral Angles with Manifold Equations. Sci. J. Chem. 2024, 12(3), 42-53. doi: 10.11648/j.sjc.20241203.11

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    AMA Style

    Mitan C, Bartha E, Filip P, Draghici C, Caproiu M, et al. Java Script Programs for Calculation of Dihedral Angles with Manifold Equations. Sci J Chem. 2024;12(3):42-53. doi: 10.11648/j.sjc.20241203.11

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  • @article{10.11648/j.sjc.20241203.11,
      author = {Carmen-Irena Mitan and Emeric Bartha and Petru Filip and Constantin Draghici and Miron-Teodor Caproiu and Robert Michael Moriarty},
      title = {Java Script Programs for Calculation of Dihedral Angles with Manifold Equations
    },
      journal = {Science Journal of Chemistry},
      volume = {12},
      number = {3},
      pages = {42-53},
      doi = {10.11648/j.sjc.20241203.11},
      url = {https://doi.org/10.11648/j.sjc.20241203.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjc.20241203.11},
      abstract = {Java Script programs for calculation dihedral angles from NMR data with manifold equations of 3-Sphere approach: rectangle, Villarceau circles of cyclide (Torus – Dupin Cyclide), polar equations, Euler-Conic. Manifolds are curves or surface in higher dimension used for calculation of dihedral angles under wave character of NMR data, carbon and/or proton chemical shift δXn[ppm] and vicinal coupling constant 3JHnHn+1[Hz]. 3-Sphere approach for calculation of the dihedral angles from NMR data in four steps: 1. Prediction, or more exactly calculation of the dihedral angles from vicinal coupling constant with trigonometric equations, 2. Calculation of the dihedral angles from manifold equations; 3. Building units from angle calculated with one of the manifold equations; 4. Calculation the vicinal coupling constant of the manifold dihedral angle. In this paper are presented Java Script programs of step 2 and from step 3 only the Java Script program for calculation of seven sets angles. The bond distances lCnCn+1[A0] between two atoms of carbon are under different polar equations (i.e. limaçons or cardioid, rose or lemniscale), our expectation was to find different manifold equations for calculation the best angle, differences are smaller but can be find sometimes a preferred one for a vicinal coupling constant. 3-Sphere approach has the advantages of calculation from vicinal angle or/and chemical shift the dihedral angle, tetrahedral angle and the bond distance lCnCn+1[A0], with application on conformational and configurational analysis.
    },
     year = {2024}
    }
    

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  • TY  - JOUR
    T1  - Java Script Programs for Calculation of Dihedral Angles with Manifold Equations
    
    AU  - Carmen-Irena Mitan
    AU  - Emeric Bartha
    AU  - Petru Filip
    AU  - Constantin Draghici
    AU  - Miron-Teodor Caproiu
    AU  - Robert Michael Moriarty
    Y1  - 2024/06/03
    PY  - 2024
    N1  - https://doi.org/10.11648/j.sjc.20241203.11
    DO  - 10.11648/j.sjc.20241203.11
    T2  - Science Journal of Chemistry
    JF  - Science Journal of Chemistry
    JO  - Science Journal of Chemistry
    SP  - 42
    EP  - 53
    PB  - Science Publishing Group
    SN  - 2330-099X
    UR  - https://doi.org/10.11648/j.sjc.20241203.11
    AB  - Java Script programs for calculation dihedral angles from NMR data with manifold equations of 3-Sphere approach: rectangle, Villarceau circles of cyclide (Torus – Dupin Cyclide), polar equations, Euler-Conic. Manifolds are curves or surface in higher dimension used for calculation of dihedral angles under wave character of NMR data, carbon and/or proton chemical shift δXn[ppm] and vicinal coupling constant 3JHnHn+1[Hz]. 3-Sphere approach for calculation of the dihedral angles from NMR data in four steps: 1. Prediction, or more exactly calculation of the dihedral angles from vicinal coupling constant with trigonometric equations, 2. Calculation of the dihedral angles from manifold equations; 3. Building units from angle calculated with one of the manifold equations; 4. Calculation the vicinal coupling constant of the manifold dihedral angle. In this paper are presented Java Script programs of step 2 and from step 3 only the Java Script program for calculation of seven sets angles. The bond distances lCnCn+1[A0] between two atoms of carbon are under different polar equations (i.e. limaçons or cardioid, rose or lemniscale), our expectation was to find different manifold equations for calculation the best angle, differences are smaller but can be find sometimes a preferred one for a vicinal coupling constant. 3-Sphere approach has the advantages of calculation from vicinal angle or/and chemical shift the dihedral angle, tetrahedral angle and the bond distance lCnCn+1[A0], with application on conformational and configurational analysis.
    
    VL  - 12
    IS  - 3
    ER  - 

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Author Information
  • Organic Chemistry, Institute of Organic and Supramolecular Chemistry “C. D. Nenitescu”, Bucharest, Romania

  • Organic Chemistry, Institute of Organic and Supramolecular Chemistry “C. D. Nenitescu”, Bucharest, Romania

  • Organic Chemistry, Institute of Organic and Supramolecular Chemistry “C. D. Nenitescu”, Bucharest, Romania

  • Organic Chemistry, Institute of Organic and Supramolecular Chemistry “C. D. Nenitescu”, Bucharest, Romania

  • Organic Chemistry, Institute of Organic and Supramolecular Chemistry “C. D. Nenitescu”, Bucharest, Romania

  • Organic Chemistry, University of Illinois at Chicago, Chicago, USA