Java Script Programs for Calculation of Dihedral Angles with Manifold Equations

Carmen-Irena Mitan; Emeric Bartha; Petru Filip; Constantin Draghici; Miron-Teodor Caproiu; Robert Michael Moriarty

doi:doi:10.11648/j.sjc.20241203.11

Research Article |

| Peer-Reviewed

Java Script Programs for Calculation of Dihedral Angles with Manifold Equations

Carmen-Irena Mitan^*

, Emeric Bartha

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

Published in Science Journal of Chemistry (Volume 12, Issue 3)

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 δ_Xn[ppm] and vicinal coupling constant ³J_HnHn+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 l_CnCn+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 l_CnCn+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.

[1]

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).

[2]

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.

[3]

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.

[4]

Lie N-algebra is a N-manifold having homological vector field.

[5]

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 M_o with unique four dimension manifold M, in fact the Euclidean space E³ of M_o is replaced by a sphere S³ of small but nonvanishing curvature, a deformation of M into M_o, and SU(2,2) into the Poincaré group as the formation of the “flat” limit of “curved” picture.

[6]

3-Sphere approach for calculation of the dihedral

[7]

and tetrahedral angles

[8-10]

from NMR data, having as mathematical theories Hopf fibration and Lie algebara.

[11]

Manifolds

[11]

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 ³J_HnHn+1[Hz]. Dihedral angle can be represented on three concentric cons having six dihedral angle and two algebraic angles, 3xϕ₁and 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.

[11]

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 φ_Cn[deg] of five membered ring calculated from vicinal coupling constant ³J_HnHn+1[Hz], b. under sin and tan function in case of tetrahedral angles φ_Cn[deg] of five membered ring calculated from chemical shift Δδ_CnCn+1[ppm].

[8]

Recorded NMR data are transformed from ppm in gauss and transformed in angle with manifold equations

[11]

, then with calculated angle are builds units:

C_n= U_n^Ni, S_n^Ni, T_n= U_n^Ni, S_n^Ni,

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

[8-10]

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

[12]

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:

θ^{An} = 2 x \sqrt{\frac{90 x ({Δδ}_{HnHn + 1} x {Δδ}_{CnCn + 1})}{2}}

[deg](1)

cis-ea, -ae, trans-ee

θ^{An} = \sqrt{\frac{90 x ({Δδ}_{HnHn + 1} x {Δδ}_{CnCn + 1})}{2}}

[deg](2)

trans-aa^6,1

Middle circles:

θ^{An} = 2 x \sqrt{\frac{90 x ({Δδ}_{HnHn + 1} + {Δδ}_{CnCn + 1})}{2}}

[deg](3)

cis-ea, -ae, trans-ee

θ^{An} = \sqrt{\frac{90 x ({Δδ}_{HnHn + 1} - {Δδ}_{CnCn + 1})}{2}}

[deg](4)

trans-aa^6,1

where: θ^An –angle of set A from unit U or S, Δδ_HnHn+1, Δδ_CnCn+1 – the differences between two consecutive protons and two atoms of carbon [ppm].

Download: Download full-size image

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 ³J_HnHn+1[Hz] from vicinal angle ϕ[deg] (eq. 5-8).

Trans:³J_HH=

\sqrt{ϕ}

(5)

Cis:³J_HH=

\frac{\sqrt{ϕ}}{2}

(6)

θ^An=

\sqrt{A}

, A = f(Δδ_HnHn+1,Δδ_CnCn+1)(7)

θ^An=

\frac{\sqrt{A}}{2}

, A = f(Δδ_HnHn+1,Δδ_CnCn+1)(8)

Java Script program presented at ACS spring 2021 for calculation of the dihedral angles from carbon and proton chemical shift δ_Cn[ppm] with rectangle equations in 3 steps

[13]

: 1. Prediction of de dihedral angles θ_HnHn+1[deg] only from vicinal coupling constants ³J_HnHn+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,

[11]

also can be calculated the vicinal coupling constant from the calculated dihedral angle.

[14]

In figure 1 is presented Java Script program ScienceRectangle.html for calculation of the dihedral angles from carbon and proton chemical shift

[11]

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

Download: Download full-size image

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).

[14, 15]

θ_T^NA= sin^-1(

\frac{Δδ HH}{Δδ CC}

)(9)

θ_T^NB= cos^-1(

\frac{Δδ HH}{Δδ CC}

)(10)

θ_D^NA= tan^-1(

\frac{Δδ HH}{Δδ CC}

)(11)

θ_D^NB= cotan^-1(

\frac{Δδ HH}{Δδ CC}

)(12)

Where: Δδ_HnHn+1, Δδ_CnCn+1 – the differences between two consecutive protons and two atoms of carbon [ppm], θ_T^{NA, NB}, θ_D^{NA, 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).

[14, 15]

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

[15]

θ_W^nA= sin^-1(Δδ_HnHn+1xω_H/Δδ_CnCn+1xω_C)(13)

Euler character: sinθ_WE^A= cosθ_WE^B(14)

B. Transformation in gauss:

θ_W^nA= sin^-1(Δδ_HnHn+1xω_H/γ_H/Δδ_CnCn+1xω_C/γ_C)(15)

Without Euler character: sinθ_WF^A≠ cosθ_WF^B(16)

Where: Δδ_HnHn+1, Δδ_CnCn+1 – the differences between two consecutive protons and two atoms of carbon [ppm], ω_L – 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].

Download: Download full-size image

Figure 3. Java Script program ScienceVillarceau.html for calculation of dihedral angles with Cyclide approach from chemical shift δ[ppm] with eq. 9-12.

Download: Download full-size image

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,

[11]

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.

[11]

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 θ_HnHn+1[deg] from the ratio differences in chemical shift between two protons or atoms of carbon and vicinal coupling constant ³J_HnHn+1[Hz].

[16]

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)^1/2at smaller differences of two protons chemical shift, in other words from AB to AX coupling.

[17]

R_mH= Δδ_HnHn+1/³J_HnHn+1[s](17)

R_mC= Δδ_CnCn+1/³J_HnHn+1[s](18)

Where x = C, H, R_mx = 1/R_mx or R_m = 10xR_mx

cis,trans-ee: (2xR_mH)²= θ_x(19)

trans-aa: (R_mH)²= θ_x(20)

(ν₄–ν₁)/(ν₃–ν₂) = tan²θ(21)

(ν₁–ν₃) = (ν₂–ν₄) = (Δν²+J²)^1/2(22)

Where ν_n – n = 1 – 4, four pics of the two coupled protons.

Download: Download full-size image

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 Δδ_HnHn+1[ppm] or two atoms of carbons Δδ_CnCn+1[ppm] with Euler-Conic equation 23.

[11]

θ = sin^-1R_mx[deg](23)

Where R_mx = Δδ_XnXn+1xω_Hx4x10^-3/γ_H[gauss], x = C, H; Δδ_XnXn+1 – the differences between two consecutive protons and two atoms of carbon [ppm], ω_L – Larmor frequency [¹³C: 75MHz, ¹H 400MHz], ν – frequency [Hz], δ – chemical shifts [ppm], γ – gyromagnetic ratio.

Download: Download full-size image

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

[11]

. 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

[17]

, 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 θ^UB1 leading to third angles of set θ^SA3 with eq. 24.

θ^SB3= nx(3xθ^UB1), n = 1, 2(24)

Where: θ^UB1 – first angle of set B from unit U (θ^N1 > 5[deg]), θ^SB3 – third angle of set B from unit S (θ^N1 < 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 ³J_HnHn+1[Hz] enable a carefully analysis of dihedral angles θ_HnHn+1[deg] with corresponding sign and stereochemistry. Since bond distances l_CnCn+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.

[12]

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.

[11]

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 θ_HnHn+1[deg] and tetrahedral angles φ_Cn[deg] in opposite.

[8]

Villarceau approach (eq. 9-12) and Java Script program ScienceVillarceau.html (Figure 3) enable calculation of the dihedral angles θ_HnHn+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

[11]

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

[18]

(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: θ_H3H4= tan^-1(sin-ϕ) (25)

Trans-ee: θ_H3H4= 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 Δδ_HnHn+1[ppm] or two carbons chemical shift Δδ_CnCn+1[ppm] and vicinal coupling constant ³J_HnHn+1[Hz] (Tables 1 and 2), using once the sin and cos function and second the homotopic approach

[19]

. In second case are calculated angles with and without vicinal angle rule (θ^A, θ_X^A[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 θ_HnHn+1[deg] calculated only from vicinal coupling constant ³J_HnHn+1[Hz]In accord with D-ribitol stereochemistry, iminocyclitols 1-3

[18]

(Figure 7) have negative dihedral angle θ_H3H4[deg], angle calculated with equations 25, 26.

Download: Download full-size image

Figure 7. Iminocyclitols 1-3 with C₁-R-α-Dribitol stereochemistry

[18]

Table 1. Dihedral angles of iminocyclitols 1-3 calculated from differences between chemical shift Δδ_Hn[Hz] of two protons and vicinal coupling constant ³J_HnHn+1[Hz] with polar equations 17, 19, 20.

Entry	³J_HH^a [Hz]	H_n [ppm]	R_mH [π]	θ[deg],³J_HH[Hz]	θ^A[deg]	θ[deg], ³J_HH[Hz] eq. 17	θ_X^A [deg]	θ[deg], ³J_HH[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	21.93^U1C1, 4.12 22.29^U1C1, 4.11	33.82 47.30	22.35^U2B1, 4.11 23.65^U1E1, 4.07
2	5.4		0.0203	-28.83, 5.45	49.09	-24.54^U1E1, 5.35	-	-
3	0 d, H₃, 0.1		0.0862	-94.96^S1B4, 1.1^tan -92.47^S1E4, 0.79^tan	0.203 0.086 0.862	-24.96^U1G1, 5.36 -90.08^S1B4, 0.146^tan -90.86^S1B4, 0.46^tan	0.165 0.029 2.972	-24.97^U1G1, 5.36 -90.029^S1B4, 0.086^tan -92.97^S1B4, 0.86^tan
4	3.1	3.71 4.16 4.26 3.58	0.1451	51.65^U1A2, 3.09	6.888 1.451	51.88^U2F2, 3.08 50.24^U1D2, 3.15	189.82 8.428	50.172^U1A2, 3.15 51.571^U1A2, 3.09
5	3.9		0.0256	29.26^S1E1, 3.89	39.00 0.256	25.2^S1G1, 4.01 29.74^S1B1, 3.88	- 0.262	- 29.73^S1B1, 3.88
6	8.8		0.0772	-166.88^U1F6, 8.78^tan	12.941 0.772	-166.71^U1A6, 8.77^tan -169.57^U1B6, 8.92^tan	167.47 0.597	-167.16^U1A6, 8.8^tan -169.63^U1B6, 8.93^tan
7	4.8	3.11 4.02 4.12 2.89	0.1895	-2.78^S1A1, 4.81	5.274 1.895	-1.64^U1S1A1, 4.78 -1.89^S1A1, 4.79	111.290 14.376	-3.87^U1S1A1, 4.84 -1.86^S1B1, 4.79
8	5.2		0.0192	-15.55^U1G1, 5.13 -19.66^U1A1, 5.23	51.999 0.192	-19.00^U1G1, 5.22 -19.93^U1A1, 5.24	- 0.1479	- -19.95^U1A1, 5.24
9	0 bs, 0.81		0.6585	-93.57^S1A4, 0.94^tan	0.658 6.585	-90.65^S1B4, 0.4^tan -93.29^SD1, 0.90^tan	1.7346 173.46	-91.73^S1B4, 0.65^tan -93.27^SE4, 0.90^tan

[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.

[7]

The program of prediction is based on trigonometric equation and not on algebraic equations. The differences between trans-ee^4,1and trans-ee^3,2 (θ³ 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-ee^3,2algebraic equation, almost compatible with tan function, a general rule for calculation only the trans-ee^3,2angle,

[11]

but from trigonometric equations, and algebraic equation for trans-ee^4,1, 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 cis-θ^SA1 angles with negative sign giving trans-ee^4,1dihedral 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 ³J_HnHn+1[Hz] of the manifold dihedral angles.

Table 2. Dihedral angles of iminocyclitols 1-3 calculated from differences between chemical shift of two carbon Δδ_Cn[ppm] and vicinal coupling constant ³J_HnHn+1[Hz] with polar equations 18, 19, 20.

Entry

³J_HH^a [Hz]

C_n [ppm]

R_mC [π]

θ[deg],³J_HH[Hz]

θ^A[deg]

θ[deg], ³J_HH[Hz] eq. 1 8

θ_X^A [deg]

θ[deg], ³J_HH[Hz] eq. 1 9, 20

4.1

55.8

83.5

84.3

65.9

0.1480

21.48^U1B1, 4.13

6.756

1.480

23.48^U1B1, 4.13

20.98^U1C1, 4.15

182.57

8.763

21.71^U1C1, 4.13

21.23^U1B1, 4.14

5.4

0.1481

-25.74^S1E1, 5.37

6.75

1.481

-26.62^S1E1, 5.39

-28.51^S1B1, 5.44

182.25

8.779

-27.75^S1B1, 5.42

-25.61^S1E1, 5.37

d, H₃

0.0054

-90.308^S1B4, 0.277^tan

0.0054

0.0543

-90.005^S1B4, 0.036^tan

-90.053^S1B4, 0.115^tan

0.0001

0.0118

-90.0001^S1B4, 0.005^tan

-90.0118^S1B4, 0.053^tan

3.1

57.4

71.5

71.7

66.8

0.2198

51.35^U1F2, 3.108

(-31.98, -53.10^tan)

4.548

2.198

50.90^U1C2, 3.12

49.26^U1B2, 3.19

82.751

19.335

52.75^U1B2, 3.05

50.33^U1D2, 3.14

3.9

0.0512

28.53^S1E1, 3.92

19.499

0.512

28.49^S1A1, 3.92

29.48^S1B1, 3.88

1.0519

28.94^S1B1, 3.90

8.8

0.5568

-168.49^U1^F6, 8.87^tan

1.7959

5.5681

-169.21^U1^F6, 8.91^tan

-167.49^U1^F6, 8.81^tan

3.225

31.004

-168.71^U1^F6, 8.88^tan

-169.49^U1^F6, 8.92^tan

4.8

63.7

72.5

74.0

69.3

0.5454

-3.055^S1B1, 4.82

(-44.95, 3.06^tan)

1.8333

5.4545

-1.833^S1A1, 4.79

(-44.98, 3.06^tan)

-2.727^S1D1, 4.81

(-44.96, 2.73^tan)

13.444

119.008

-1.983^S1C1, 4.79

(-44.98, 1.98^tan)

5.2

0.2884

-16.76^U1A1, 5.16

(-43.75, 17.53^tan)

3.4666

2.8846

-18.84^U1A1, 5.21

(-43.42, 19.95^tan)

-19.03^U1A1, 5.22

(-43.38, 20.18^tan)

48.0711

33.2840

-18.07^U1B1, 5.19

(-43.55, 19.04^tan)

-18.90^U1A1, 5.21

(-43.41, 20.02^tan)

bs, 0.81

0.1723

-90.182^S1B4, 0.213^tan

5.8024

1.7234

-95.772^S1B4, 1.20^tan

-91.724^S1B4, 0.656^tan

134.674

11.8804

-91.953^S1CB4, 0.698^tan

-95.668^S1CB4, 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 ³J_HnHn+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 ³J_HnHn+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

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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

Carmen-Irena Mitan

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

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

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

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http://orcid.org/0000-0003-2076-7244
Petru Filip

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

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Constantin Draghici

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

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Miron-Teodor Caproiu

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

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Robert Michael Moriarty

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

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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

VL - 12
IS - 3
ER -

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