3-Sphere Tetrahedral Angles and Phase Angle of the Pseudorotation P[deg] of C<sub>1</sub>-R-α-D Ribitol Iminocyclitol

Petru Filip; Carmen Irena Mitan; Emerich Bartha

doi:doi:10.11648/j.sjc.20241203.12

Research Article |

| Peer-Reviewed

3-Sphere Tetrahedral Angles and Phase Angle of the Pseudorotation P[deg] of C₁-R-α-D Ribitol Iminocyclitol

Petru Filip

, Carmen Irena Mitan^*

, Emerich Bartha

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

Received: 13 May 2024 Accepted: 30 May 2024 Published: 19 June 2024

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Abstract

3-Sphere dihedral angles θ_HnHn+1[deg] calculated from NMR data, from vicinal coupling constant ³J_HnHn+1[Hz] with right sign and stereochemistry, are used for simulation of the conformation of the five membered ring with VISION molecular models and GaussView05. For a vicinal angle ϕ[deg], angle result from vicinal coupling constant ³J_HnHn+1[Hz], result three possible dihedral angles with negative and positive sign. Different phase angles of the pseudorotation results from combination of dihedral angles (exocyclic angles) with positive and negative sign in case of cis stereochemistry, and only negative sign for trans stereochemistry, in accord with D-ribitol stereochemistry. The sign of the endocyclic trans-ee torsional angle is positive, relative to trans-aa and cis stereochemistry with same sign as exocyclic angle, as visualized on VISION molecular models. Tetrahedral angles φ_Cn[deg] in close relationship with dihedral angles θ_HnHn+1[deg] are calculated only from vicinal coupling constant ³J_HnHn+1[Hz] in attempt to corelate the change in conformation with tetrahedral values φ_Cn[deg] and bond lengths l[A⁰], once the iminocyclitol push out from planarity one or two atoms of carbon, and once again to confirm the method for calculation of tetrahedral angles of five membered ring, sin/tan versus sin/cos units.

Published in	Science Journal of Chemistry (Volume 12, Issue 3)
DOI	10.11648/j.sjc.20241203.12
Page(s)	54-62
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

3-Sphere, Dihedral Angle, Tetrahedral Angle, Vicinal Angle, Vicinal Coupling Constant, C₁-R-α-D-Ribitol, Conformational Analysis

1. Introduction

Conformational analysis on five and six membered rings with biological activity (i.e. antiviral activity) as a pseudorotation concept appears around 1947, thermodynamic studies on cyclopentene conformation performed by Kilpatrich, Pitzer si Spitzer

[1]

and Pitzer si Donath

[2]

. First spectroscopical evidence was obtained in 1965 on far infrared of tetrahydro furan

[3]

. Cremer and Pople

[4]

used this model in calculation of the conformation of carbasugar, and Altona

[5]

in vitamin D five membered ring or nucleic acid conformation.

Recently are published results based on 1H NMR, DFT and X-ray data for determination of the conformation of furanose substrates with restricted freedom of rotation on the C₃-C₄, C₂-C₃, and C₁-C₂ bonds

[6]

, known as antivirale, antibacteriene or anticancer. i.e. Molnupiravir approve by WHO (World Health Organization) in treatment of COVID-19

[7]

. Optimization of the geometry in case of N-(α-D-glucofuranurono-6, 3-lactone) and N-(methyl β-D-glucopyranuronate)-p-nitroanilines was realized with DFT (B3YPP/6-311+G++ method), and structures are simulated with MOLDEN program and all calculation with Gaussian 03.

[8]

Our aim is to state a general method for calculation tetrahedral angles φ_Cn[deg] with 3-Sphere approach in close relationship with dihedral angles θ_HnHn+1[deg], analyzing the phase angle of the pseudorotation. Three dihedral angles θ_HnHn+1[deg] are calculated for every vicinal coupling constant ³J_HnHn+1[deg], and only for trans θ_H3H4[deg] dihedral angle the sign is resticted by D-ribitol stereochemistry.

2. Conformational Analysis

Conformational analysis,

[9, 10]

phase angle of the pseudorotation P[deg] (eq. 1) and angle of deviation from planarity θ_m[deg] (eq. 2) calculated with 3-sphere dihedral angles θ_HnHn+1[deg] and Altona formalism are published already

[11]

P = \tan^{- 1} \frac{(θ_{H 2 H 4} + θ_{H 1 N}) - (θ_{H 1 H 2} + θ_{H 4 N})}{θ_{H 2 H 3} (\sin 72 + \sin 144)}

(1)

θm = \frac{θ 0}{cosP}

(2)

Where: P – pahase angle of pseudorotation [deg], θ_m – angle of deviation from planarity[deg], θ_HnHn+1– 3-Sphere dihedral angles[deg].

2.1. Endocyclic Torsional Angles

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Figure 1. Exocyclic (dihedral angles) and endocyclic torsional angles.

Endocyclic angle θ_endo[deg], an angle at intersection of l_Cn-1Cnand l_Cn+1Cn+2[A⁰] along the bond length l_CnCn+1[A⁰] (Figure 1), in case of polynomial equations in close relationship with corresponding exocyclic angle θ_exo[deg] gives the PSEUROT program,

[12]

endocyclic angles θ_endo[deg] are usefully on Altona model.

[11]

Endocyclic torsional angles (Figure 1) calculated with PSEUROT equations for D-ribose (Eq. 3-5)

[12]

θ_endoH1H2=0.9090x(θ_exoH1H2–3.3)(3)

θ_endoH2H3=0.9174x(θ_exoH2H3–0.2)(4)

θ_endoH3H4=0.9090x(θ_exoH3H4+124.9)(5)

Other posibility under reflexion for calculation of the endocyclic torsional angles θ_endo[deg] can be: 3-sphere dihedral angle θ_HnHn+1[deg] divided with vicinal coupling constant ³J_HnHn+1[Hz] (eq. 6) in case of cis, trans-aa stereochemistry or multiplaid with vicinal coupling constant ³J_HnHn+1[Hz] (eq. 7, 8) in case of trans-ee stereochemistry, in line with polar equations. That must be confirmed by NMR data and X-ray. As observation eq. 8 gives too smaller endocyclic angles betwwen 0.1 and 0.5[Hz] for n = 2, angles result sometimes from simulation with GaussView05. In case of trans-ee stereochemistry the exocyclic angle transformed in endocyclic torsional angle (eq. 16) with corresponding sign can be also contemplate. In this light for trans-aa stereochemistry also the 120 rule must be used (eq. 18). On molecular models with an angle of 120[deg] between H₁ - OH₁ and H₂ - OH₂ the ring of iminocyclitol is plane, any modification with required number of degrees between H₁ and H₂ must be proportional with endocyclic angle.

cis-ae,trans-aa: θ_endo= θ_HnHn+1/³J_HH(6)

trans-ee: θ_endo=(θ_HnHn+1^trans-eex³J_HH)/2(7)

trans-ee:θ_endo=(θ_n,n+1X³J_HH)/n(8)

Where θ_endo – endocyclic torsional angle [deg], θ_HnHn+1 – dihedral angle or exocyclic angle [deg], θ_n,n+1 – exocyclic torsional angle [deg].

3-Sphere dihedral angles are calculated (eq. 9-13) from vicinal angle (eq. 14) an angle calculated from vicinal coupling constant.

[13]

cis,trans-ee^3,2: sin^-1cosϕ= θ_HnHn+1(9)

trans-aa^{6,1 or 5,2},trans-ee^4,1: cos^-1sin(-ϕ) = θ_HnHn+1(10)

tan^-1sin(-ϕ) = θ_HnHn+1(11)

sin^-1[cot(-ϕ)] = θ_HnHn+1(12)

sin^-1[tan(-ϕ)] = θ_HnHn+1(13)

ϕ= (nx³J_HnHn+1)²(14)

Where: θ_HH[deg] – dihedral angle, ϕ[deg] – vicinal angle, ³J_HH[Hz] – vicinalcoupling constant, with cis, trans-ee: m = 2, trans-aa: m = 1.

The relationship between the 3-Sphere dihedral angles with cis and trans stereochemistry was established based on Hopf fibration trigonometric equations applaid on six angles with cis and trans-ee and trans-aa stereochemistry (eq. 15-17),

[14]

totaly in contradiction with Karplus 120 rule (eq. 18).

trans-aa^{6,1 or 5,2}:θ_HnHn+1^cis= 180 - θ_HnHn+1^trans(15)

trans-ee^{4,1 or 3,2}:θ_HnHn+1^cis= 120 - θ_HnHn+1^trans(16)

trans-ee^{4,1 or 3,2}:θ_HnHn+1^trans= -90 -/+ θ_HnHn+1^cis(17)

θ_n,n+1= 120 - θ_HnHn+1(18)

Where θ_HnHn+1 – 3-Sphere dihedral angle with cis, trans-ee and trans-aa stereochemistry, θ_endo – endocyclic torsional angle.

The main question, 120 rule is applaid in case of endocyclic torsional angles only for trans stereochemistry? In this case the torsional angle of trans-aa dihedral angle -167.13[deg] is -47.13[deg] instead -12.87[deg], realtive to -18.99[deg] result from eq. 6 (Table 2). In case of dihedral angle with cis stereochemistry of 51.56[deg] the torsional angle is 8.44[deg] under 120 rule (eq. 18), realtive to 16.63[deg] result from eq. 6. In case of cis stereochemistry PSEUROT program (eq. 3, 4) shown smaller differences, no more as 3[deg].

[12]

2.2. Relationships Between Phase Angle of the Pseudorotation and Other Physical Coordinates

The angle of deviation from planarity can be calculated from the phase angle of the pseudorotation P[deg] with Altona equation eq. 2 from θ_H2H3or with eq. 19, 20 from θ_H1H2 or θ_H3H4[deg].

[11]

θ_m= θ_H1H2/cos(P–144) [deg](19)

θ_m= θ_H3H4/cos(P+144)[deg](20)

The atom coordinate, the out-of plane vibrations Zj calculated with Levitt methods (eq. 21).

[15]

Zj = (2/5)^1/2xq_mxcos(P+jx144–90), j 0 – 4(21)

Where: Z_j out-of plane vibrations, q_m – amplitude [A⁰] calculated from endocyclic torsional angle θ_endo[deg], P – phase angle of the pseudorotation [deg].

Grabb – Harvey relationships (22-27) for calculation torsional constraints ν⁰[deg] from phase angle of the pseudorotation P[deg] and q_m[A⁰]

[16]

ν⁰= q_mcos(P+72)(22)

ν¹= -q_mcos(P+36)(23)

ν²= q_mcos(P)(24)

ν³= -q_mcos(P-36)(25)

ν⁴= q_mcos(P-72)(26)

ν⁰+ ν¹+ ν²+ ν³+ ν⁴= 0(27)

Bartenev – Kameneva – Lipanov relationship for calculation of the tetrahedral angles φ_i[deg] from phase angle of pseudorotation P[deg] (Eq. 28-35).

[17]

φ_i= A_i+ B_ixcos(2P–72i+ci)(28)

φ₀= 108.8 + 1.4cos(2P–0x75–10)(29)

φ₁= 105.2 + 0.8cos(2P–1x75+16)(30)

φ₁= 106.6 + cos(P)(31)

φ₂= 103.5 + 1.5cos(2P–2x75–36)(32)

φ₃= 102.9 + 1.6cos(2P–3x75+9)(33)

φ₄= 106.2 + 1.1cos(2P–4x75+8)(34)

φ₄= 105.3 - (P)(35)

The relationship between proton-proton torsional angle φ [deg] and the pseudorotational parameters P[deg] and q_m[A⁰] for ³J_{2, 3}[Hz] extracted from X-ray data (eq. 36)

[18]

³J_2,3(ribo): φ_2,3= 0.2 + 1.09q_mcosP(36)

2.3. Bond Lengths l_CnCn+1[A⁰]

The bond lengths

[19]

can be calculated with eq 37 using the theoretic carbon-carbon bond length distance (l₀ = 1.54[A⁰]) with results equals

[20]

with a method for calculation published by Maksic and Randic

[21]

. Recently, based on dihedral angle under wave theory the proton-proton bond lengths l_HnHn+1[A⁰]was calculated without theoretic bond distance (results presented elsewhere). Thus, probably the eq. 37 can be reconsidered, eq. 38 for calculation carbon-carbon bond length distance l_CnCn+1[A⁰] without the theoretic bond length giving comparable results.

l_CnCn+1= [l₀x1.57xcos^1/2(θ/m)]^1/2[A⁰](37)

l_CnCn+1= 1.57xcos^1/4(θ^cis/n)[A⁰](38)

Where l_CnCn+1- carbon-carbon bond length distance [A⁰], l₀ - theoretic carbon-carbon bond length distance [A⁰], n = 1 for cis^6,1 and trans-aa^6,1, trans-ee^4,1 – transformed in cis, n = 2 for cis^5,2 and trans-aa^5,2, trans-ee^3,2 – transformed in cis, alternatively n = 4 for trans-ee.

Meyer et. al reported a model for calculation the bond length from pseudorotation angle P[deg] (Eq. 34-36).

[22]

r_(C2-C3)= r_cc⁽⁰⁾+ r_cc⁽²⁾cos2P(39)

r_(C1-C2)= r_cc⁽⁰⁾[1 + ε_lpcos2P] + r_cc⁽²⁾sin2P(40)

r_(C3-C4)= r_cc⁽⁰⁾[1 + ε_lpcos2P] - r_cc⁽²⁾sin2P(41)

where r_CC⁽⁰⁾ = 1.537[A⁰], r_CC⁽²⁾ the C-C bond extension due to the repulsion of the eclipsed vicinal C-H bonds, εlp the effect of the lone electron pairs of the oxygen.

3. Materials and Methods

Gausian09W

[23]

, VISION molecular models

[24]

3-Sphere approach for calculation of the dihedral angles θ_HnHn+1[deg], tetrahedral angles φ_Cn[deg], and bond lengths l_CnCn+1[A⁰], using Hopf fibration and Lie algebra, trigonometric equations and algebraic equations.

[25]

4. Results

3-Sphere dihedral angles θ_HnHn+1[deg] calculated from vicinal coupling constant ³J_HnHn+1[Hz] are used for simulation of the phase angle of the pseudorotation

[10]

and analyzed the tetrahedral angles in close relationship with dihedral angles (Table 1) with Bartenev – Kameneva – Lipanov

[17]

tetrahedral angles φ[deg] calculated from phase angle of the pseudorotation (Table 2).

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Figure 2. Iminocyclitol 1 with C₁-methyl-α-D ribitol stereochemistry.

Table 1. Tetrahedral angles φ_Cn[deg] in close relationship with dihedral angles θ_HnHn+1[deg]and vicinal angles ϕ[deg] calculated from vicinal coupling constant ³J_HnHn+1[Hz].

Entry

³J_HH[Hz]^a ϕ[deg] θ_HnHn+1[deg]

φ_Cn = f (θ_HnHn+1)

3.1

38.44

51.56

21.56

38.44

81.56

98.44

141.56

158.44

43.12

8.439

8.439

51.56

68.44

111.56

128.44

171.56

16.879

21.560

16.87

43.12

76.88

103.12

136.88

163.12

33.75

13.12

10.78

49.22

70.78

109.22

130.78

169.22

21.56

19.22

19.22

40.78

79.22

100.78

139.22

160.78

38.44

10.78

4.21

55.78

64.22

115.78

124.22

175.78

8.439

25.78

25.78

34.22

85.78

94.22

145.78

154.22

51.56

4.21

sin^-1cos38.44 = 51.56 θ_H1H2

cos^-1sin(-38.43/2) = 109.22 φ_C1

3.1

38.44

-34.68

U to S

~ tan

25.31

34.68

85.32

94.68

145.32

154.68

50.63

4.68

4.68

55.31

64.68

115.32

124.68

175.32

9.36

25.31

9.36

50.63

69.36

110.64

129.36

170.64

18.72

20.63

12.65

47.34

72.65

107.34

132.65

167.34

25.31

17.34

17.34

42.65

77.34

102.65

137.34

162.65

34.68

12.65

2.34

57.65

62.34

117.65

122.34

177.65

4.68

27.65

27.65

32.34

87.65

92.34

147.65

152.34

55.30

2.34

cos^-1sin-0.5x{ [60 – (98.44 – 90)]/1.5} =

cos^-1sin (-34.37/2) = 107.18 φ_C1

3.1

38.44

-31.86

28.13

31.86

88.13

91.86

148.13

151.86

56.26

1.86

1.86

58.13

61.86

118.13

121.86

178.13

3.73

28.13

3.73

56.26

63.73

116.26

123.73

176.26

7.47

26.26

14.06

45.93

74.06

105.93

134.06

165.93

28.13

15.93

15.93

44.06

75.93

104.06

135.93

164.06

31.86

14.06

0.934

59.06

60.93

119.06

120.93

179.06

1.86

29.06

29.06

30.93

89.06

90.93

149.06

150.93

58.13

0.93

tan^-1sin (-38.44) = -31.86 θ_H1H2

cos^-1sin (-31.86/2) = 105.93 φ_C1

3.1

38.44

-52.53

7.46

52.53

67.46

112.53

127.46

172.53

14.92

22.53

22.53

37.46

82.53

97.46

142.53

157.46

45.07

7.46

14.92

45.07

74.92

105.07

134.92

165.07

29.85

15.07

3.73

56.26

63.73

116.26

123.73

176.26

7.46

26.26

26.26

33.73

86.26

93.73

146.26

153.73

52.53

3.73

11.26

48.73

71.26

108.73

131.26

168.73

22.53

18.73

18.73

41.26

78.73

101.26

138.73

161.26

37.46

11.26

sin^-1tan (-38.44) = -52.53 θ_H1H2

sin^-1cos (-52.53) = 37.46 θ_H1H2

cos^-1sin (-37.46/2) = 108.73 φ_C1

3.9

60.84

29.15

0.84

59.16

60.84

119.16

120.84

179.16

1.68

29.15

29.15

30.84

89.16

90.84

149.16

150.84

58.31

0.84

1.68

58.31

61.68

118.32

121.68

178.32

3.36

28.31

0.42

59.58

60.42

119.58

120.42

179.57

0.84

29.58

29.58

30.42

89.58

90.42

149.57

150.42

59.16

0.42

14.57

45.42

74.58

105.42

134.57

165.42

29.15

15.42

15.42

44.58

75.42

104.58

135.42

164.57

30.84

14.57

sin^-1cos60.84 = 29.15 θ_H2H3

cos^-1sin (-29.15/2) = 104.58 φ_C2

3.9

60.84

S to U

~ tan

19.72

40.28

79.72

100.28

139.72

160.28

39.44

10.28

10.28

49.72

70.28

109.72

130.28

169.72

20.56

19.72

20.56

39.44

80.56

99.44

140.56

159.44

41.12

9.43

9.86

50.14

69.86

110.14

129.86

170.14

19.72

20.14

39.86

80.14

99.86

140.14

159.86

40.28

9.86

5.14

54.86

65.14

114.86

125.14

174.86

10.28

24.86

35.14

84.86

95.14

144.86

155.14

49.72

5.14

cos^-1sin-0.5x{ [60 – (119.16 – 90)]/1.5} =

cos^-1sin (-20.56/2) = 100.28 φ_C2

3.9

60.84

-41.12

18.87

41.12

78.87

101.12

138.87

161.12

37.74

11.12

11.12

48.87

71.12

108.87

131.12

168.87

22.15

18.87

22.25

37.74

82.25

97.74

142.25

157.74

44.51

7.74

9.43

50.56

69.43

110.56

129.43

170.56

18.87

20.56

20.56

39.43

80.56

99.43

140.56

159.43

41.12

9.43

5.56

54.43

65.56

114.44

125.56

174.43

11.12

24.43

24.45

35.56

84.44

95.56

144.44

155.56

48.8

5.56

tan^-1sin (-60.84) = -41.12 θ_H2H3

I. cos^-1sin (-41.12/2) = 110.56 φ_C2

cos^-1sin (-18.87/2) = 99.43 φ_C2

II. cos^-1sin [- (41.12 – 30)] = 101.12 φ_C2

cos^-1sin [-(41.12 – 18.87)/2] = 101.12 φ_C2

3.9

60.84

-33.91

26.09

33.91

86.09

93.91

146.09

153.91

52.19

3.90

3.90

56.09

63.91

116.09

123.91

176.09

7.81

96.69

7.81

52.18

67.82

112.18

127.82

172.18

15.63

22.18

13.04

46.95

73.04

106.95

133.04

166.95

26.09

16.95

16.95

43.04

76.95

103.04

136.95

163.04

33.91

13.04

1.95

58.04

61.95

118.04

121.95

178.04

3.90

28.04

28.04

31.95

88.04

91.95

148.04

151.95

56.09

1.954

sin^-1[1/tan (-60.84)] =

-33.91 θ_H2H3

cos^-1sin (-26.09/2) = 103.04 φ_C2

8.8

77.44

-167.13

12.87

47.12

72.87

107.12

132.87

167.12

25.74

17.12

17.12

42.87

77.12

102.87

137.12

162.87

34.25

12.87

25.74

34.25

85.74

94.25

145.74

154.25

51.49

4.25

6.43

53.56

66.43

113.56

126.43

173.56

12.87

23.56

26.56

36.43

83.56

96.43

143.56

156.43

47.12

6.43

8.56

51.43

68.56

111.43

128.56

171.43

17.12

21.43

21.43

38.56

81.43

98.56

141.43

158.56

42.87

8.56

sin^-1(1/tan-ϕ) = -12.873;

-167.126 θ_H3H4

cos^-1sin-17.12 = 107.12 φ_C4

8.8

77.44

-135.7

15.70

44.3

75.7

104.3

135.7

164.3

31.4

14.29

14.29

45.7

74.3

105.7

134.3

165.7

28.59

15.70

28.59

31.40

88.6

91.4

148.6

151.4

57.19

1.40

7.84

52.15

67.85

112.15

127.85

172.15

15.69

22.15

22.15

37.84

82.15

97.85

142.15

157.85

44.30

7.84

7.15

52.84

67.15

112.85

127.15

172.85

14.30

22.85

22.85

37.15

82.85

97.15

142.85

157.15

45.6

7.15

tan^-1sin-77.44 = -44.306;

-135.7 θ_H3H4

cos^-1sin (-15.70) = 105.7 φ_C4

8.8

77.44

167.44

17.43

42.56

77.43

102.56

137.43

162.56

34.87

12.56

12.56

47.43

72.56

107.43

132.56

167.43

25.12

17.43

25.12

34.87

85.12

94.87

145.12

154.87

50.24

4.87

8.71

51.28

68.71

111.28

128.71

171.28

17.43

21.27

21.28

38.71

81.28

98.71

141.28

158.71

42.56

8.71

6.28

53.71

66.28

113.71

126.28

173.71

12.56

23.71

23.71

36.28

83.71

96.28

143.71

156.28

47.43

6.28

cos^-1sin-77.44 = 167.44 θ_H3H4

cos^-1sin (-17.43) = 107.435 φ_C4

^*[a] δ[ppm] 2-D₂0, ¹³C 75 [MHz], ¹H 400 [MHz].

Tetrahedral angles calculated only from vicinal coupling constant are presented in Table 1 for all possible dihedral angles with negative or positive sign. As reported recently, dihedral θ_HnHn+1[deg] and tetrahedral φ_Cn[deg] angles calculated from carbon chemical shift δ_Cn[deg] are in opposite from the trigonometric point of view, sin versus tan functions, for five membered ring

[26]

comparative to six membered ring

[27]

In this case, tetrahedral angles calculated only from vicinal coupling constant, from vicinal angle in close relationship with dihedral angle, results from trigonometric eq. 42, 43 and 9-14 on seven sets angles unit (Table 1), relationship between two pairs of angles A, B with D, E or F, G. For a dihedral angle of 51.56[deg] was calculated a tetrahedral angle of 109.22[deg] in seven set unit, a value somewhat unexpected, probably on corresponding polyhedron

[27]

an angle of six membered ring. The transformation from U to S gives an angle of 107.18[deg], but also two characteristic angles 34.68 and 55.31[deg] with a vicinal coupling constant of 3.71[Hz], if 55.31 will be considered vicinal angle ϕ[deg], or as results from eq. 9-14, other dihedral angle -34.68[deg] in close relationship with the vicinal angle ϕ 38.44[deg] results from a vicinal constant coupling of 3.1[Hz]. All the equations presented in Table 2 are in close relationships with seven sets angles on one unit.

cos^-1sin(-ϕ/2) = φ_Cn(42)

cos^-1sin(-θ_HnHn+1/2) = φ_Cn(43)

Where θ_HnHn+1 – dihedral angle [deg], ϕ – vicinal angle [deg], tetrahedral angle φ_Cn [deg].

The vicinal coupling constant of 3.9[Hz] giving for a dihedral angle θ_H2H3 of 29.15[deg] a tetrahedral angle φ_C2 of 104.58, and from transformation S to U a tetrahedral angle of 100.28[deg], for a dihedral angle of -41.12[deg] an angle of 99.43 and other angle on same pair of angles of 101.12[deg], at list for a dihedral angle of -33.91[deg] a tetrahedral angle of 103.04[deg]. In case of dihedral angles θ_H3H4 with trans-aa stereochemistry, tetrahedral angles φ_C4 are 107.42 for 167.44[deg], 105.7 for -135.7[deg], and 107.12 for -167.13[deg]. In comparation with tetrahedral angles calculated from chemical shift,

[26]

in this case tetrahedral angle φ_C2 can be calculated in first case from vicinal coupling constant ³J_H1H2 or ³J_H2H3, and tetrahedral angle φ_C3 from vicinal coupling constant ³J_H3H4. For example, on two sets angles are found two tetrahedral angles: φ_C4 and φ_C3 – 107.41, 102.56[deg].

Table 2. Conformation simulated with GaussView5.0 from 3-Sphere dihedral angles θ_HnHn+1[deg] (eq. 9-14) and corresponding endocyclic torsional angles calculated with eq. 6.

[a] δ[ppm] 2-D₂0, ¹³C 75 [MHz], ¹H 400 [MHz], [b] θ_m – angle of deviation from planarity calculated from 3-Sphere dihedral angle θ_HnHn+1[deg], [c] θ_m – angle of deviation from planarity calculated from endocyclic 3-sphere torsional angle θ_endo[deg], [d] θ_m – angle of deviation from planarity calculated from Altona torsional angle θ_endo[deg] (eq. 3, 5), [e] Zj – out of plane vibrations (eq. 21) calculated from endocyclic 3-sphere torsional angle θ_endo[deg] (eq. 6) and phase angle of the pseudorotation P[deg], [f] Zj – out of plane vibrations (eq. 21) calculated from endocyclic Altona’s torsional angle θ_endo[deg] (eq. 3-5) and phase angle of the pseudorotation P[deg], [g] eq. 28, 29-35.

Two conformations are simulated with GaussView05 in Table 2 using 3-sphere dihedral angles and endocyclic torsional angles calculated with eq. 6. Combination of two consecutive positive or negative dihedral angles are too hard to simulate with VISION molecular models and also with GaussView05. The angle of deviation from planarity θ_m[deg] with corresponding amplitude q[A⁰] are calculated from phase angle of the pseudorotation simulated with GaussView05. The out of plane vibration Z_i (eq. 21) was calculated from endocyclic torsional angle θ_endo[deg] calculated with eq. 6. The angle of deviation from planarity calculated from 3-sphere torsional angles and phase angle of the pseudorotation of 30.65[deg] is expected for Altona formalism, relative to angle for deviation from planarity calculated from endocyclic torsional angle θ_endo[deg] of 7.8[deg].

Endocyclic torsional angles calculated with Altona’s eq. 3-5 are used for simulation of the conformation of iminocyclitol 1 (Table 1, entry 3), resulting E₂ conformation with θ_m 29.91[deg].

5. Discussion

3-Sphere dihedral angles θ_HnHn+1[deg] are used recentry on Altona model as exocyclic angles, and the phase angle of the pseudorotation P[deg] was established with VISION molecular models. The phase angle of the pseudorotation between ¹T₂ and ₂E with P = -24.12[deg] results from Altona model for angles -52.53, 29.67, -168.36[deg], and ₃E – ⁴T₃ with P = -143.12[deg] for angles 50.20, -40.98, -135.57[deg].

[10]

Exocyclic 3-Sphere torsional angles are considered angles under 180 rule in case of trans-aa and 120 rule in case of trans-ee.

[14]

As observation, on VISION molecular models endocyclic torsional angle θ_endo[deg] of D-ribitol stereochemistry have same sign as exocyclic 3-Sphere dihedral angles θ_exo[deg] in case of cis, trans-aa and opposite sign in case of trans-ee stereochemistry, but don’t forgot that VISION molecular models have carbon-carbon bond length and tetrahedral angles frozen.

Conformational results obtained with VISION molecular models are compared with GaussView. 05 program for prediction and visualization of the conformation in Table 2, in first case almost same (entry 1) ₂E and in second case different (entry 2) E⁰.

Download: Download full-size image

Figure 3. Simulated Conformation of iminocyclitol 1 with GaussView5.0 (Table 2).

Remarkably in case of iminocyclitol 1, the trans-aa dihedral angles of -135.62[deg] result from eq. 12 is preferred to -167.44[deg] result from eq. 13. on simulation of the conformation with GaussView5.0. Positive dihedral angles of 51.56[deg] and 29.15[deg] for vicinal coupling constants of 3.1 and 3.9[Hz] is too hard to established even with VISION molecular models.

Between the exocyclic dihedral angles and endocyclic torsional angles and implicit tetrahedral angles is almost impossible to obtain accurate correlation with calculated angles. GaussView5.0 program for prediction and visualization don’t corelate tetrahedral angle with dihedral angles as shown in Table 2, but enable a very good visualization of the conformation. Attempts to optimized the conformation with semiempirical PM6 method giving dihedral angles with other vicinal coupling constant ³J_HnHn+1[deg] relative to recorded vicinal coupling constant ³J_HnHn+1^exp[deg].

Bartenev – Kameneva – Lipanov tetrahedral angle φ[deg] (eq. 29),

[17]

particularized for all tetrahedral angles (eq. 29-35) are calculated from phase angles of pseudorotation P[deg] simulated with GaussView05 in Table 2. The atom coordinate, the out-of plane vibrations Zj are calculated with Levitt methods (eq. 21).

[15]

6. Conclusions

Tetrahedral angles are calculated only from vicinal coupling constant in attempt to corelate the theier relationships with dihedral angles – vicinal angle and vicinal coupling constant. In comparation with tetrahedral angles calculated from carbon chemical shift, tetrahedral angles calculated from vicinal coupling constant are not calculated in opposite with dihedral angles.

The phase angle of the pseudoroataion was simulated with GaussView05 using frozen 3-Sphere dihedral angles and endocyclic torsional angle, but tetrahedral angles are not corelated with dihedral angles in light of 3-Sphere approach. A program for simulation of the endocyclic torsional angles, dihedral angles, carbon-carbon bond lenghts, and tetrahedral angles in function of the vicinal coupling constant or carbon chemical shift is under work.

Abbreviations

NMR	Nuclear Magnetic Resonance

Author Contributions

Petru Filip: Methodology

Carmen-Irena Mitan: Methodology, Writing

Emerich Bartha: Methodology

Conflicts of Interest

The authors declare no conflicts of interest.

References

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APA Style

Filip, P., Mitan, C. I., Bartha, E. (2024). 3-Sphere Tetrahedral Angles and Phase Angle of the Pseudorotation P[deg] of C1-R-α-D Ribitol Iminocyclitol. Science Journal of Chemistry, 12(3), 54-62. https://doi.org/10.11648/j.sjc.20241203.12

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ACS Style

Filip, P.; Mitan, C. I.; Bartha, E. 3-Sphere Tetrahedral Angles and Phase Angle of the Pseudorotation P[deg] of C1-R-α-D Ribitol Iminocyclitol. Sci. J. Chem. 2024, 12(3), 54-62. doi: 10.11648/j.sjc.20241203.12

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AMA Style

Filip P, Mitan CI, Bartha E. 3-Sphere Tetrahedral Angles and Phase Angle of the Pseudorotation P[deg] of C1-R-α-D Ribitol Iminocyclitol. Sci J Chem. 2024;12(3):54-62. doi: 10.11648/j.sjc.20241203.12

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@article{10.11648/j.sjc.20241203.12,
  author = {Petru Filip and Carmen Irena Mitan and Emerich Bartha},
  title = {3-Sphere Tetrahedral Angles and Phase Angle of the Pseudorotation P[deg] of C1-R-α-D Ribitol Iminocyclitol
},
  journal = {Science Journal of Chemistry},
  volume = {12},
  number = {3},
  pages = {54-62},
  doi = {10.11648/j.sjc.20241203.12},
  url = {https://doi.org/10.11648/j.sjc.20241203.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjc.20241203.12},
  abstract = {3-Sphere dihedral angles θHnHn+1[deg] calculated from NMR data, from vicinal coupling constant 3JHnHn+1[Hz] with right sign and stereochemistry, are used for simulation of the conformation of the five membered ring with VISION molecular models and GaussView05. For a vicinal angle ϕ[deg], angle result from vicinal coupling constant 3JHnHn+1[Hz], result three possible dihedral angles with negative and positive sign. Different phase angles of the pseudorotation results from combination of dihedral angles (exocyclic angles) with positive and negative sign in case of cis stereochemistry, and only negative sign for trans stereochemistry, in accord with D-ribitol stereochemistry. The sign of the endocyclic trans-ee torsional angle is positive, relative to trans-aa and cis stereochemistry with same sign as exocyclic angle, as visualized on VISION molecular models. Tetrahedral angles φCn[deg] in close relationship with dihedral angles θHnHn+1[deg] are calculated only from vicinal coupling constant 3JHnHn+1[Hz] in attempt to corelate the change in conformation with tetrahedral values φCn[deg] and bond lengths l[A0], once the iminocyclitol push out from planarity one or two atoms of carbon, and once again to confirm the method for calculation of tetrahedral angles of five membered ring, sin/tan versus sin/cos units.
},
 year = {2024}
}

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TY  - JOUR
T1  - 3-Sphere Tetrahedral Angles and Phase Angle of the Pseudorotation P[deg] of C1-R-α-D Ribitol Iminocyclitol

AU  - Petru Filip
AU  - Carmen Irena Mitan
AU  - Emerich Bartha
Y1  - 2024/06/19
PY  - 2024
N1  - https://doi.org/10.11648/j.sjc.20241203.12
DO  - 10.11648/j.sjc.20241203.12
T2  - Science Journal of Chemistry
JF  - Science Journal of Chemistry
JO  - Science Journal of Chemistry
SP  - 54
EP  - 62
PB  - Science Publishing Group
SN  - 2330-099X
UR  - https://doi.org/10.11648/j.sjc.20241203.12
AB  - 3-Sphere dihedral angles θHnHn+1[deg] calculated from NMR data, from vicinal coupling constant 3JHnHn+1[Hz] with right sign and stereochemistry, are used for simulation of the conformation of the five membered ring with VISION molecular models and GaussView05. For a vicinal angle ϕ[deg], angle result from vicinal coupling constant 3JHnHn+1[Hz], result three possible dihedral angles with negative and positive sign. Different phase angles of the pseudorotation results from combination of dihedral angles (exocyclic angles) with positive and negative sign in case of cis stereochemistry, and only negative sign for trans stereochemistry, in accord with D-ribitol stereochemistry. The sign of the endocyclic trans-ee torsional angle is positive, relative to trans-aa and cis stereochemistry with same sign as exocyclic angle, as visualized on VISION molecular models. Tetrahedral angles φCn[deg] in close relationship with dihedral angles θHnHn+1[deg] are calculated only from vicinal coupling constant 3JHnHn+1[Hz] in attempt to corelate the change in conformation with tetrahedral values φCn[deg] and bond lengths l[A0], once the iminocyclitol push out from planarity one or two atoms of carbon, and once again to confirm the method for calculation of tetrahedral angles of five membered ring, sin/tan versus sin/cos units.

VL  - 12
IS  - 3
ER  -

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Author Information

Petru Filip

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

Contact Email

http://orcid.org/0000-0003-2076-7244
Carmen Irena Mitan

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

Contact Email

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

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

Contact Email

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Table 1

Table 1. Tetrahedral angles φ_Cn[deg] in close relationship with dihedral angles θ_HnHn+1[deg]and vicinal angles ϕ[deg] calculated from vicinal coupling constant ³J_HnHn+1[Hz].
Table 2

Table 2. Conformation simulated with GaussView5.0 from 3-Sphere dihedral angles θ_HnHn+1[deg] (eq. 9-14) and corresponding endocyclic torsional angles calculated with eq. 6.

Plain Text BibTeX RIS

APA Style

Filip, P., Mitan, C. I., Bartha, E. (2024). 3-Sphere Tetrahedral Angles and Phase Angle of the Pseudorotation P[deg] of C1-R-α-D Ribitol Iminocyclitol. Science Journal of Chemistry, 12(3), 54-62. https://doi.org/10.11648/j.sjc.20241203.12

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ACS Style

Filip, P.; Mitan, C. I.; Bartha, E. 3-Sphere Tetrahedral Angles and Phase Angle of the Pseudorotation P[deg] of C1-R-α-D Ribitol Iminocyclitol. Sci. J. Chem. 2024, 12(3), 54-62. doi: 10.11648/j.sjc.20241203.12

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AMA Style

Filip P, Mitan CI, Bartha E. 3-Sphere Tetrahedral Angles and Phase Angle of the Pseudorotation P[deg] of C1-R-α-D Ribitol Iminocyclitol. Sci J Chem. 2024;12(3):54-62. doi: 10.11648/j.sjc.20241203.12

Copy | Download

@article{10.11648/j.sjc.20241203.12,
  author = {Petru Filip and Carmen Irena Mitan and Emerich Bartha},
  title = {3-Sphere Tetrahedral Angles and Phase Angle of the Pseudorotation P[deg] of C1-R-α-D Ribitol Iminocyclitol
},
  journal = {Science Journal of Chemistry},
  volume = {12},
  number = {3},
  pages = {54-62},
  doi = {10.11648/j.sjc.20241203.12},
  url = {https://doi.org/10.11648/j.sjc.20241203.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjc.20241203.12},
  abstract = {3-Sphere dihedral angles θHnHn+1[deg] calculated from NMR data, from vicinal coupling constant 3JHnHn+1[Hz] with right sign and stereochemistry, are used for simulation of the conformation of the five membered ring with VISION molecular models and GaussView05. For a vicinal angle ϕ[deg], angle result from vicinal coupling constant 3JHnHn+1[Hz], result three possible dihedral angles with negative and positive sign. Different phase angles of the pseudorotation results from combination of dihedral angles (exocyclic angles) with positive and negative sign in case of cis stereochemistry, and only negative sign for trans stereochemistry, in accord with D-ribitol stereochemistry. The sign of the endocyclic trans-ee torsional angle is positive, relative to trans-aa and cis stereochemistry with same sign as exocyclic angle, as visualized on VISION molecular models. Tetrahedral angles φCn[deg] in close relationship with dihedral angles θHnHn+1[deg] are calculated only from vicinal coupling constant 3JHnHn+1[Hz] in attempt to corelate the change in conformation with tetrahedral values φCn[deg] and bond lengths l[A0], once the iminocyclitol push out from planarity one or two atoms of carbon, and once again to confirm the method for calculation of tetrahedral angles of five membered ring, sin/tan versus sin/cos units.
},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - 3-Sphere Tetrahedral Angles and Phase Angle of the Pseudorotation P[deg] of C1-R-α-D Ribitol Iminocyclitol

AU  - Petru Filip
AU  - Carmen Irena Mitan
AU  - Emerich Bartha
Y1  - 2024/06/19
PY  - 2024
N1  - https://doi.org/10.11648/j.sjc.20241203.12
DO  - 10.11648/j.sjc.20241203.12
T2  - Science Journal of Chemistry
JF  - Science Journal of Chemistry
JO  - Science Journal of Chemistry
SP  - 54
EP  - 62
PB  - Science Publishing Group
SN  - 2330-099X
UR  - https://doi.org/10.11648/j.sjc.20241203.12
AB  - 3-Sphere dihedral angles θHnHn+1[deg] calculated from NMR data, from vicinal coupling constant 3JHnHn+1[Hz] with right sign and stereochemistry, are used for simulation of the conformation of the five membered ring with VISION molecular models and GaussView05. For a vicinal angle ϕ[deg], angle result from vicinal coupling constant 3JHnHn+1[Hz], result three possible dihedral angles with negative and positive sign. Different phase angles of the pseudorotation results from combination of dihedral angles (exocyclic angles) with positive and negative sign in case of cis stereochemistry, and only negative sign for trans stereochemistry, in accord with D-ribitol stereochemistry. The sign of the endocyclic trans-ee torsional angle is positive, relative to trans-aa and cis stereochemistry with same sign as exocyclic angle, as visualized on VISION molecular models. Tetrahedral angles φCn[deg] in close relationship with dihedral angles θHnHn+1[deg] are calculated only from vicinal coupling constant 3JHnHn+1[Hz] in attempt to corelate the change in conformation with tetrahedral values φCn[deg] and bond lengths l[A0], once the iminocyclitol push out from planarity one or two atoms of carbon, and once again to confirm the method for calculation of tetrahedral angles of five membered ring, sin/tan versus sin/cos units.

VL  - 12
IS  - 3
ER  -

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