'Pseudo Positions of Symmetry': a Continuum Between Classical

Home , Fundamental domain, Glide reflection

'Pseudo Positions of Symmetry':

A Continuum Between Classical General

and Special Positions in Crystals

Thesis submitted in partial fulfillment of the requirements for the degree of “DOCTOR OF PHILOSOPHY”

Avital Steinberg

Submitted to the Senate of Ben-Gurion University of the Negev

February, 2010

Beer-Sheva

'Pseudo Positions of Symmetry':

A Continuum Between Classical General

and Special Positions in Crystals

Thesis submitted in partial fulfillment of the requirements for the degree of “DOCTOR OF PHILOSOPHY”

Avital Steinberg

Submitted to the Senate of Ben-Gurion University of the Negev

Approved by the advisor – Prof. Robert Glaser

Approved by the Dean of the Kreitman School of Advanced Graduate Studies

February, 2011

Beer-Sheva

This work was carried out under the supervision of Professor Robert Glaser In the Department of Chemistry Faculty of Natural Sciences

Acknowledgments:

I owe my deepest gratitude to my advisor, Prof. Robert Glaser, for everything he has taught me and for giving me fascinating projects which we both enjoyed very much.

I would like to thank Prof. David Avnir and all of his group members, especially Dr. Dina Einot-Yogev, Dr. Shahar Keinan, Chaim Dryzun and Amir Zayit for their help and support.

Thanks to Dr. Itzhak Ergaz, for synthesizing the adamantane and nefopam derivatives, and for very fruitful scientific discussions.

I thank my friends and colleagues Anna Kogan, Dr. Erez Boukobza, Noa Zamstein and Aliza Levkovich for their help and support.

I thank my dear parents, brother and sister.

This thesis is dedicated to my parents, Shoshana and Saul Steinberg

Table of Contents Abstract

1. Introduction 1

1A. Review of Some General Concepts in Symmetry, Periodic Arrays, and Crystallography 1

1B. Introduction for Research Subproject 1: The Use of the Continuous Symmetry Measure (CSM) to Study the Desymmetrization of Molecules with Platonic Solid Geometries Upon Entering a Crystal Lattice) 11

1.B.1 The Avnir Continuous Summetry Measure (CSM) 17

1.B.2 An In-Depth Overview of Avnir's CSM Methodology 20

1.B.3 Estimating the Errors for the CSM Values 24

1.B.4 CSM Computation Times 24

1.B.5 Calculation of Symmetry Measures with Respect to a Specific Permutation 25

1C. Introduction for Research Subproject 2: Symmetry Measure Continuum Studies on Pseudo Symmetry Between Molecules in Crystals Containing Multiple Molecules in the Asymmetric Unit Leading to a Subgroup-Supergroup Relationship. The Investigation of Intermolecular Forces Involved in Pseudo-Symmetrical Arrangements within a Crystal Lattice. 30

1.C.1 Analysis of Pseudo-Symmetry and Packing in Crystal Lattices Which

Are Related to a Pseudo P21/c Supergroup and Which Contain Two Molecules in the Asymmetric Unit. 32

1D. Introduction for Research Subproject 3: CSM Studies on the Orthorhombic

Pn21a Crystals of Diphenhydramine HCl (14), and its Relationship to the Bb21m Supergroup, and the Determination of the Forces Responsible for the Unusual Pseudo-Mirror Conformation of the Ammonium Cation. 39

2. Research Goals 43

2A. The Desymmetrization of Symmetrical Molecules Upon Entering a Crystal Lattice 43

2B. Studies of Intermolecular Pseudo Symmetry in Crystals Having Multiple Molecules in the Asymmetric Unit 44

2C. Studies on Crystals With Molecules Occupying Pseudo Special Positions of Symmetry 45

2D. Development of New Methodologies to Measure the Deviation from Screw Rotation and Glide Reflection Space Group Symmetries 46

3. Experimental 48

3A. Avnir CSM Programs for Point Group Pseudo Symmetry Operations 48

3B. Calculation of Centroids Within a Molecule 48

3C. CSD Searches 48

3D. Geometry Optimizations 48 3E. Graphics Programs 49 3F. Mathematical Calculation Methods Used for Space Group Pseudo Symmetry Operations 49

3F.1 Calculation of Centroids Between Pseudo Symmetry Related Atoms in Molecules A and B of the Asymmetric Unit 49

3F.2 Calculation of the xmean, ymean, and zmean Coordinates 50

3F.3 The Relocation of a ‘Best Pseudo Inversion Point’ from the Location of the Ideal Inversion Point 50

3F.4 Relocation of the ‘Best Pseudo Screw Displacement Axis’, Compared to the Location of the Ideal Screw Displacement Axis (a Numerical Example is Given) 51

3F.5 Relocation of the ‘Best Pseudo Glide Reflection Plane’ Compared to the Location of the Ideal Glide Plane (a Numerical Example is Given) 54

3F.6 The RmS(G) Pseudosymmetry Tool where G = Ci, , Cn, and Sn. 57

3F.7 Location of Statistically Determined Pseudo Positions of Symmetry. 58

3F.8 The RmS(translation) Pseudosymmetry Tool. 58

3F.9 The RmS(glide) and RmS(screw) Pseudosymmetry Tools. 59

3F.10 Calculation of the New RmS(P21/c) Tool 60

3F.11 The New RmS Fidelity(P21/x) Tool – an Additional Method which was

Developed in Order to Measure Deviations from Symmetry in Crystals 60

4. Results and Discussion 62

4A. The Role of the Environment Upon Molecular Structure: CSM Studies on the Crystalline State Desymmetrization of High Symmetry Platonic Solid Geometry Molecules. 62

4A.1 CSM Studies on Dodecahedrane, a Platonic Solids Geometry Molecule Incapable of Expressing its Five-Fold Axis of Symmetry in the Solid-State 62

4A.2 Discussion of the CSM Studies of Crystalline Dodecahedrane 67

4B. CSM Studies on the Distortion of Crystalline Molecules with Distant Three- Fold Rotors on an Adamantane Rigid Skeleton. The Crystalline State Stereochemistry of N-Methylated Adamantane 1,3-Diammonuim Salts Residing at Special and General Positions of Symmetry in a Crystal Lattice. 68

4B.1 Discussion of the CSM Studies of Three-Fold Rotors Affixed to an Adamantane Skeleton 74

4C. CSM Studies on the Distortion of Crystalline Platonic Solid Geometry Molecules with Three-Fold Rotors 76

4C.1 CSM Studies on the Distortion of Crystalline Molecules with Distant Three-Fold Rotors on a Triakis-Tetrahedron Geometry Skeleton. 76

4C.1a 2,4,6,8 – Tetra - tert. -Butyl - 1,3,5,7 – Tetraphospha - 2,4,6,8-Tetrasilapentacyclooctane (20) 76

4C.1b 2,4,6,8-Tetra-tert.-Butyl-1,3,5,7-Tetraphosphacubane 85

4C.2 CSM Studies on the Distortion of Crystalline Molecules with Spatially Close Three-Fold Rotors on an Octasilacubane Skeleton 88

4C.2a CSM Studies on Crystalline Octa-tert.-Butyl-Octasilacubane 88

4C.2b Superlattice Packing Arrangement in the Octa-tert.-Butyl- Octasilacubane Rhombohedral R32 Crystal. 110

4C.3 CSM Studies on the Distortion of Crystalline Molecules with Spatially

Close Three-Fold Rotors on Tetrahedrane and Tetrasilatetrahedrane

Skeletons. 110

4C.3a CSM Studies on Crystalline Tetra-tert.-Butyl-Tetrahedrane 110

4C.3b CSM studies on Tetrakis(trimethylsilyl)-Tetrahedrane 116

4C.4 CSM Studies on the Distortion of Crystalline Molecules with Very Large, Spatially Close Three-Fold Rotors on a Tetrasilatetrahedrane Skeleton. 124

4C.5 Discussions on the CSM Studies of the Distortion of Crystalline Molecules with Distant or Close Threefold Rotors 137

4D. The Continuum of RmS Symmetry Measures of P21/c Family ‘Supergroup

Character’ in Lower Symmetry P21, P–1, and Pn Space Group Crystals Containing Multiple Molecules in the Asymmetric Unit. 142

4D.1 RmS Symmetry Measures of Monoclinic P21 Chiral Space Group Crystals of (±)-(1RS,3SR,4RS)-1-Phenyl-cis-3,4-Butano-3,4,5,6-Tetrahydro-1H- 2,5-Benzoxazocine Hydrochloride, 11. 142

4D.1a Location of the pseudo positions of inversion in the N-

desmethyl-cis-3,4-butano-nefopam HCl P21 chiral crystal 147

4D.1b Development of the New RmS(i) Symmetry Measure Tool for Comparison of Inversion Pseudo Symmetry with Bona Fide Symmetry in the

N-Desmethyl-cis-3,4-Butano-Nefopam HCl P21 Chiral Crystal 151

4D.1c Relocation of Pseudo Special Positions of Inversion From the Special Positions for Bona Fide Inversion Symmetry 155

4D.1d RmS(21), Comparison of 21 Screw Displacement Pseudo Symmetry with Bona Fide Symmetry in the N-desmethyl-cis-3,4-Butano-

Nefopam HCl P21 Chiral Crystal 156

4D.1e Location of the Pseudo Special Positions of n-Glide Reflection

in the N-Desmethyl-cis-3,4-Butano-Nefopam HCl P21 Chiral Crystal 157 4D.1f RmS(n-glide), Comparison of n-Glide Reflection Pseudo Symmetry with Bona Fide Symmetry in the N-Desmethyl-cis-3,4-

Butano-Nefopam HCl P21 Chiral Crystal 158

4D.1g Relocation of the Pseudo Positions of Glide-Reflection from the Special Positions for Bona Fide Glide-Reflection Symmetry 160

4D.1h The RmS(P21/c) Value for the N-Desmethyl-cis-3,4-Butano-

Nefopam HCl P21 Chiral Crystal 160

4D.1i RmS(P21/n) Calculated for Just the Pseudo Enantiotopic Skeletons of Crystalline 11 161

4D.1j RmsFidelity(P21/n), a Check of the CSM RmS(P21/n) Method 162

4D.2 RmS Symmetry Measures of Triclinic P–1 Achiral Space Group Crystals of (±)-1-Phenyl-6-Cyano-1,3,4,5,6,7-Hexahydro-2,6- Benzoxazonine, 12. 163

4D.2a Location of the Pseudo Positions of 21 Screw-Rotation in the 2,6- Homonefopam-6-CN Triclinic P–1 Achiral Space Group Crystal 169

4D.2b RmS(21) - Quantification of 21 Screw-Rotation Pseudo Symmetry in the 2,6-Homonefopam-6-CN Triclinic P–1 Achiral Space Group Crystal 170

4D.2c Location of the Pseudo Positions of a-Glide Reflection in the 2,6-Homonefopam-6-CN Triclinic P–1 Achiral Space Group Crystal 171

4D.2d RmS(a-glide), Comparison of a-Glide Reflection Pseudo Symmetry in the 2,6-Homonefopam-6-CN Crystal with Bona Fide Symmetry 171

4D.2e Relocation of the Pseudo Special Positions of 21 Screw

Displacement Axes Versus the Special Positioned Ideal 21 Screw Axes in the 2,6-Homonefopam-6-CN Triclinic P–1 Achiral Space Group Crystal 172

4D.2f Relocation of the Pseudo Positions of a-Glide Reflection Planes in the 2,6-Homonefopam-6-CN Triclinic P–1 Achiral Space Group Crystal 173

4D.2g The RmS(P21/a) Value for the 2,6-Homonefopam-6-CN Crystal 173

4D.2h RmsFidelity(P21/n), a Check of the RmS(P21/n) Symmetry Measure Method 174

4D.3 RmS Symmetry Measures of Monoclinic Pn Achiral Space Group Crystals of (±) - (2RS,3SR,4RS) – tert. Butyl 3 – Hydroxyl – 4 – Phenyl – 2 – (p - Toluenesulfonylamino) Pentanoate Dichloromethane Solvate, 176

4D.3b RmS(i) Comparison of Inversion Pseudo Symmetry with Bona Fide Symmetry in the (±)-tert.-Butyl 3 – Hydroxyl – 4 – Phenyl – 2 -(p- Toluenesulfonylamino)Pentanoate Dichloromethane Solvate Monoclinic Pn Achiral Space Group Crystal. 184

4D.3c Reocation of Pseudo Positions of Inversion from the Special Positions for Bona Fide Inversion Symmetry 185

4D.3d Location of the Pseudo Positions of 21 Screw-Rotation Axes in the (±)-tert.-Butyl 3 – Hydroxyl – 4 – Phenyl – 2 -(p- Toluenesulfonylamino)Pentanoate Dichloromethane Solvate Monoclinic Pn Achiral Space Group Crystal. 185

4D.3f Relocation of the Pseudo Special Positions of 21 Screw- Rotation

Axes Versus the Special Positioned Ideal 21 Screw Axes in the (±)-tert.-Butyl 3 – Hydroxyl – 4 – Phenyl – 2 – (p-Toluenesulfonylamino)Pentanoate Dichloromethane Solvate Monoclinic

Pn Achiral Space Group Crystal. 187

4D.3g The RmS(P21/n) Value in the (±)-tert.-Butyl 3 – Hydroxyl – 4 – Phenyl – 2 -(p-Toluenesulfonylamino)Pentanoate Dichloromethane Solvate

Monoclinic Pn Achiral Space Group Crystal. 188

4D.3h RmsFidelity(P21/n), a Check of the CSM RmS(P21/n) Method 189

4D.4 The ‘Pseudo Positions of Symmetry’ Concept 190

4D.5 Discussion of the RmS Symmetry Measure Continuum of P21/c Family

‘Supergroup Character’ in Lower Symmetry P21, P–1, and Pn Space Group Crystals Containing Multiple Molecules in the Asymmetric Unit. 191

4D.5a The New RmS Symmetry Measure Tools of RmS(Ci), RmS(C2),

RmS(), RmS(translation), RmS(relocation-Ci), RmS(relocation-screw),

RmS(relocation-plane), RmsFidelity(Ci), RmsFidelity(21), and RmsFidelity(glide) 191

4D.5b Discussion of Stereochemistry and RmS Symmetry Measure

Results of Monoclinic P21 Chiral Space Group Crystals of (±)- (1RS,3SR,4RS)-1-Phenyl-cis-3,4-Butano-3,4,5,6-Tetrahydro-1H-2,5- Benzoxazocine Hydrochloride, 11. 192

4D.5c Advantages of Pseudo-Symmetry in the N-Desmethyl-cis-3,4-

Butano-Nefopam HCl P21 Chiral Crystal 194

4D.5d The Solid-State Cp/mas NMR Spectrum of N-Desmethyl-cis-

3,4-Butano-Nefopam HCl P21 Monoclinic Chiral Crystals 197

4D.5e Maximization of Aromatic…Aromatic Interactions and Packing via Distortion from Ideal (P21/n) Symmetry 198

4D.5f Discussion of Stereochemistry and RmS Symmetry Measure Results for Triclinic P–1 Achiral Space Group Crystals of (±)-1-Phenyl-6- Cyano-1,3,4,5,6,7-Hexahydro-2,6-Benzoxazonine, 12. 201

4D.5g Discussion of Stereochemistry and RmS Symmetry Measure Results for Monoclinic Pn Achiral Space Group Crystals of (±)- (2RS,3SR,4RS)-tert.-Butyl 3 – Hydroxyl – 4 – Phenyl – 2 -(p- Toluenesulfonylamino)Pentanoate Dichloromethane Solvate, 13 203

4D.5h Comparison of RmS(P21/c) Family Symmetry Measures for the Three Crystals Studied in this Section of the Thesis 204

4D.5i Comparison of RmsFidelity(P21/c) Symmetry Measures as a

Check of the CSM RmS(P21/n) Method 204

4D.5j Comparison of RmS Symmetry Measures According to the Type of Pseudo Symmetry Operation 205

4D.5k Does the Largest Distortion in a P21/c Family Supergroup Crystal Arise from Symmetry Deviations or From Relocation? 206

4E. Stereochemistry and Bb21m Supergroup RmS Symmetry Measures of

Orthorhombic Pn21a Crystals of Diphenhydramine HCl, an Antihistaminic Drug. 207

4E.1 CSM S(σ) of the Pseudo Mirror Plane in the Bb21m Supergroup 213

4E.2 RmS(21) of the Pseudo 21 Screw Axes in the Bb21m Supergroup 214

4E.3 RmS(b-Glide) of the Pseudo b-Glides in the Bb21m Supergroup 216

4E.4 Conclusions Concerning the Pseudo Cs Symmetry of Diphenhydramine

HCl in the Pn21a Space Group Crystal and its Bb21m Supergroup 218

5. Conclusions 220

6. References 223

7. Appendix: The Avnir Algorithms for Finding the Nearest Perfectly Symmetric Object with Respect to the Required Symmetry. 7A. The Folding-Unfolding Algorithm (Numerical Approximation) 7B. The Analytical Solution

8. Appendix - .cif Files (CDROM)

An electronic format of the .cif files was saved on a CDROM. The .cif files were named according to the molecular structure numbers . The .cif files can be opened, viewed and rotated with Mercury, which is freely available on the web: http://www.ccdc.cam.ac.uk/free_services/mercury/downloads/ Figure index

Figure Page Molecule Subject Number

1 2 - The two chiral molecules are related by a symmetry operation of the Second Kind

2 3 - The two chiral molecules are related by a symmetry operation of the first kind

3 3 - Staggered conformation molecule with C2 point group symmetry

4 4 - Staggered conformation molecule with Ci point group symmetry

5A 5 - A solvated symmetrical molecule may lose some or all of its symmetry elements upon crystallization due to packing constraints:

Change to a new C1 symmetry conformation upon crystallization at a general position.

5B 5 - Retention of conformation upon crystallization but object’s bona fide inversion center has become pseudoinversion cluster of points

6 6 - An extended array produced from one unit cell by translation

7 8 - Asymmetric unit of one full object (Z' = 1) Figure Page Molecule Subject

8 9 - Asymmetric unit of one-half and object (Z' = 0.5)

9 12 - Duality relationships in the Platonic solids

10A 15 - Latent octahedral symmetry in the icosahedron

10B 15 - Latent octahedral symmetry in the dodacahedron

11 17 - A cube consists of two tetrahedra of the same size

12 17 - A continuous distortion of the cube can lead to a series of triakis-tetrahedra

13 18 - The nearest symmetric structure can be found by using Avnir's CSM algorithms

14 19 - The deviation from C3 symmetry is measured by

comparing the structure to the nearest perfectly C3- symmetric structure.

15 21 - The folding-unfolding algorithm

16 22 - The analytical algorithm

17 23 - A permutation matrix determines the pairs of vertices which will be compared to each other.

18 26 - A method for defining a specific permutation

19A 27 2 The Td symmetric conformation for 20

19B 27 2 The T symmetric conformation for 20

Figure Page Molecule Subject

19C 29 2 Stabilizing proton-proton interactions for the Td symmetric conformation of tetrakis-tert-butyl- tetrahedrane

19D 29 2 There are more stabilizing proton-proton interactions for the T symmetric conformation of tetrakis-tert-butyl- tetrahedrane

20 40 14 A ball and stick representation of the (Ph)2CH– 'open butterfly wing' and pseudo-mirror conformation of 14 Solid-state cpmas 13C NMR spectrum 21 41 14

Latent octahedral symmetry in Th symmetry crystal 22 63 4

The four ideal C3 axes and coincidental four S6 axes in 23 63 4 Th symmetry crystalline 4.

24 64 4 A mirror plane in Th symmetry crystalline 4.

25 64 4 C2 axes are preserved in Th symmetry crystalline 4

26 64 4 The center of inversion was preserved in Th symmetry crystalline 4.

27 65 4 A pseudo C3 axis in Th symmetry crystalline 4.

28 66 4 A pseudo C2 axis in Th symmetry crystalline 4.

29 66 4 A pseudo mirror plane in Th symmetry crystalline 4.

30 67 4 A pseudo C5 axis in Th symmetry crystalline 4 Figure Page Molecule Subject

31 70 17 Adamantane-1,3-diammonium dication color coded

according to liquid state C2v symmetry equivalence.

32A 72 18A Pseudo C2 symmetric molecule 18A.

32B 72 20B Pseudo mirror symmetric molecule 20B.

33A 78 20A The color coded S4 symmetric achiral molecule

33B 79 20B The color coded T symmetric chiral molecule

34A 80 20A S4 symmetry triakis-tetrahedron 20A occupying special

positions of C2 rotational and S4 rotatory-reflection

symmetry in the P43n crystal.

34B 81 20A S4 symmetry triakis-tetrahedron 20A occupying general

positions of pseudo-C2 rotational and pseudo-S4

rotatory-reflection symmetry in the P43n crystal.

34C 81 20A S4 symmetry triakis-tetrahedron 20A occupying a

general position of pseudo-C3 rotational symmetry in the

P43n crystal.

34D 82 20A S4 symmetry triakis-tetrahedron 20A occupying a general position of pseudo-mirror reflection symmetry in

the P43n crystal.

35A 83 20B T symmetry triakis-tetrahedron 20B occupying a special

position of C2 rotational symmetry in the P43n crystal.

35B 84 20B T symmetry triakis-tetrahedron occupying a special position of C3 symmetry Figure Page Molecule Subject

35C 84 20B T symmetry triakis-tetrahedron 20B occupying a general position of pseudo mirror reflection symmetry in the

P43n crystal.

35D 84 20B T symmetry triakis-tetrahedron 20B occupying a general

position of pseudo S4 rotary-reflection symmetry in the

P43n crystal.

36 86 21 Numbering diagram

37A 94 22A D3 symmetry silacubane 22A with pseudo C4 & pseudo

C3 axes in the R32 crystal

37B 94 22A D3 symmetry silacubame 22A with pseudo C2 axes in the R32 crystal

37C 95 22A D3 symmetry silacubane

37D 95 22A D3 symmetry silacubane 22A with a pseudo inversion point in the R32 crystal

38 99 22A,22B. Arrangement of six subsections containing fifteen molecules in the R32 unit cell color 22C The schematic R32 unit cell of crystalline 22. 39 101 22

40 102 22 Stack of six schematic asymmetric units

41 104 22 Two shallow chiral depression

42 105 22 Two shallow chiral depression Figure Page Molecule Subject Number

43 107 22 A pseudo inversion 'best point' between (+)-twist

conformation D3 (-)-twist conformation C3

44 108 22 Layers of dense packed silacubanes separated by two layers of sparsely packed schematic units

45 113 - Closing the Ccore–Ccore–Ccore angle makes adjacent tert.- butyl substituents much more crowded in cubanes than in tetrahedranes and they are expected to be even more crowded in dodecahedranes.

46 113 23 Numbering diagram for carbon atoms in tetra-tert.- butyl-tetrahedrane 23 (CUCZUV).

… 47 116 23 (top) H H Distances in a C—C fragment from a Td symmetric conformational model with ideally staggered rotors on both ends. (bottom) H…H Distances in a C— C fragment from a T symmetric conformational model with same sense twisted rotors on both ends.

48 118 24A,24B Arrangement of the three molecules in the asymmetric unit. ,24C

49 119 24A,24B,24 Numbering diagram for carbon atoms in C1 symmetry C tetrakis(trimethylsilyl)-tetrahedrane 24A-C (OHABEE).

50 126 25A The C3 symmetric structure of crystalline 25A.

51 127 25A Three-fold rotationally symmetric –Si(t-Bu)3 rotor (red) in 25A having homotopic –9.06° twist angles Figure Page Molecule Subject Number

52 129 25A Numbering diagram for atoms in C3 symmetry tetrakis(tri-tert.-butylsilyl)-tetrahedrane

53 130 25B Numbering diagram for carbon atoms in C3 symmetry tetrakis(tri-tert.-butylsilyl)-tetrahedrane

54 133 25A A pseudo C3 axis in 25A.

55 133 25A A pseudo C2 axis in 25A.

56 133 25A A pseudo  -plane in 25A.

57 134 25A Another pseudo  -plane in 25A.

58 134 25A A pseudo S4-axis in 25A.

59 136 25C Numbering diagram for atoms in the C1 symmetric tetrakis(tri-tert.-butylsilyl)-tetrahedrane 25C (NAJRAR).

60 138 - Increase in deviation from symmetry of the Second Kind, e.g. S(), as a function of the total number of twisted (non-staggered conformation) tert.-butyl rotors in a molecule.

61 140 25A Close contacts between an edge fragment model of a carbon tetrahedrane analogue of 25A constructed using molecular graphics.

62 143 11 The differences in origin between the group and supergroup

Figure Page Molecule Subject Number

63 145 11 Illustration of the four positions which are equivalent in

the ideal P21/n unit cell.

64 147 11 Pseudo inversion position 1

65 148 11 Pseudo inversion position 2

66 149 11 Pseudo inversion position 3

67 149 11 Pseudo inversion position 4

68 156 11 21 screw rotation axis

69 157 11 Pseudo n-glide reflection planes

70 165 12 P21/c cell diagram

71 166 12 Illustration of the four positions which are equivalent in

the ideal P21/a unit cell. Bona fide inversion center of crystalline 12 relates (R)-A 72 167 12 (pink) with (S)-A (red), and (R)-B (black) with (S)-B (grey).

73 168 12 Pseudo 21 screw axes Pseudo a-glide-reflection planes 74 168 12

Diagram of symmetry operations in the P21/n cell 75 178 13 The four positions which are equivalent in the ideal 76 179 13 P21/n unit cell are shown here.

77 180 13 The bona fide glide reflection plane. Figure Page Molecule Subject

78 181 13 Four symmetry nonequivalent pseudo positions of inversion.

79 182 13 Pseudo 21 screw axes 1

80 193 11 An aromatic-aromatic coulombic interaction

81 193 11 Network of edge to face aromatic interactions

Changing the Hbenzhydryl–Cbenzhydryl–Cipsophenyl–Cortho 82 194 11 torsion angle by +59.2° from its original –42.8° value

As a result of changing the H–Cbenzhydryl–CipsoPhenyl–Cortho 83 195 11 torsion angle by +59.2° to +16.4°, the phenyl ring now

eclipses the Cbenzhydryl—Hbenzhydryl bond (green), but two aromatic…aromatic interactions are lost.

84 196 11 Another result of having hypothetical bona fide inversion symmetry

85 197 11 Solid state CPMAS NMR 13C-spectrum

86 198 11 Unit cell depicting lateral shear

87 199 11 Pseudo n-glide reflection plane

88 200 11 Pseudo n-glide reflection plane (z component)

89 202 12 Coulombic interactions in the asymmetric unit

90 203 13 Hydrogen bonding interactions in the asymmetric unit

91 207 14 The orthorhombic Pn21a space group (Z = 4) unit cell Figure Page Molecule Subject

92 208 14 and 'Open butterfly wing' conformation BEXHPA

93 208 14 Symmetry operations in the Bb21m space group viewed from the ac-plane

94 209 14 Symmetry operations in the Bb21m space group viewed from the bc-plane

95 209 14 Symmetry operations in the Bb21m space group viewed from the ab-plane

21 Screw-rotation operations in the Bb21m space group. 96 210 14

… – 97 212 14 Weak Cortho —H Cl bonds

98 215 14 Pseudo 21 screw axes

99 217 14 A pseudo b-glide-reflection plane

100 219 14 A Bb21m supergroup unit cell with

Table index

Table Page Molecule Subject Number

1 13 - Faces, vertices and edges of convex Platonic solids

2 16 5-9 Geometric parameters for compounds 5-9

3 73 15-17, CSM values for adamantane derivatives.

18A,18B,

19A,19B

4 80 20A CSM values for 20A

5 83 20B CSM values for 20B

6 87 21 CSM values for 21

7A 90 22A CSM values for 22A

7B 91 22B CSM values for 22B

7C 92 22C CSM values for 22C

8 97 22A,22B,22C S(average_O) and S(average_Oh) values

9 115 23 CSM values for 23

10 117 22A,22B,22C Pseudoinversion symmetry between molecules in the asymmetric unit of 22

11A 120 24A CSM values for the C1 symmetric molecule 24A

Table Page Molecue Subject

11B 121 24B CSM values for 24B

11C 122 24C CSM values for 24C

12A 131 25A CSM values for 25A

12B 132 25B CSM values for 25B

12C 137 25C CSM values for 25C

13 135 25C The Si–Si–C(quat)–C(quat) torsion angle for rotors in 25C

14 143 11 The four molecules in the unit cell

15 146 11 Pseudosymmetry related molecules

16A 150 11 Best pseudo inversion coordinates with phenyls

16B 151 11 Best pseudo inversion coordinates without phenyls

17 152 11 S(Ci) values

18 153 11 S(Ci) values for the phenyls and distances between phenyl centroids

19 162 11 Total deviation from the P21/x point group symmetries for 11

20 164 12 The four molecules in the unit cell

21 175 12 Total deviation from the P21/x point group symmetries for 12

22 177 13 The four molecules in the unit cell of 13 23 183 13 The four best inversion centers

24 184 13 S(Ci) values

25 189 13 Total deviation from the P21/x point group symmetries for 13

Pseudo Positions of Symmetry: a Continuum Between Classic General and Special Positions in Crystals

Abstract

Most molecules which have high point group symmetry lose some or all of their symmetry elements in a crystalline environment. Edge bonded polyhedra with Platonic Solid geometries are made from highly symmetrical building blocks. Solution state NMR spectra of such compounds show degeneracies which correspond to ideal Platonic Solid point group symmetries (by dynamic conformational averaging). Since there are less degrees of freedom in the solid state, molecules are less likely to express ideal symmetries within a crystalline environment. Only a molecule which resides on a special position of a certain type, may show bona fide symmetry of that type in the solid state. Addition of large rotors should cause a molecule to lose the symmetry operations of the Second Kind to a large extent, while preserving those of the First Kind to a large extent. Avnir's Continuous Symmetry Measures will be used in order to measure the deviations from ideal Platonic solid symmetries. An initial hypothesis was that as the rotors increase in size, the molecules would twist in a way which would preserve the symmetry operations of the First Kind and not those of the Second Kind.

According to an early computational study by Hounshell and Mislow, the ground state (global minimum calculated geometry) of tetrakis-tert-butyl tetrahedrane has T symmetry and is thus chiral.

Recently, Balci and Schleyer using computations performed with higher levels of theory (ab-intio restricted Hartree-Fock HF/6-31G* and density functional theory B3LYP/6- 31G*), have also shown that tetrakis-tert-butyl-tetrahedrane is more stable when it exhibits T symmetry. In these computations, the T symmetry geometry is 0.5-2.0 kcal/mol lower in energy than that with Td symmetry.

Some of the crystals examined here exhibited multiple molecules in the asymmetric unit. This provided multiple views of the same molecule in a crystalline environment. Experimental results were compared to gas phase calculation results for the Platonic solids. In most of the cases, the solid state conformation was the same as the one predicted by gas phase calculation (the conformations were the same, but not identical). An interesting case in which only two out of three conformations of molecules in the asymmetric unit were similar to the one predicted by gas phase calculations, led to an additional research project. In this project, the packing of crystals with multiple molecules in the asymmetric units was examined.

According to the literature, edge-to-face interactions and hydrogen bonds play an important role in generating multiple molecules within an asymmetric unit. Indeed, many of the molecules in the asymmetric unit in the crystals investigated in the prototype study interacted via edge-to-face and hydrogen bonding forces. Electrostatic long range interactions were found to play an important role in crystal packing, and as predicted by the literature, they could be responsible for the pseudo mirror conformation of crystalline diphenhydramine HCl.

The overall aim of the first PhD research subproject was to utilize the existing classical continuous symmetry measure algorithms developed by David Avnir and his research group to investigate the effect of environment upon molecular structure. In the first part of the project these algorithms were used to study the small, but measurable, degree of structural distortion when solvated molecules having convex Platonic solid geometries were resident within the confines of a crystal lattice. Molecules with Platonic solid geometries were chosen, since it is interesting to find out which of the many symmetry elements would be preserved, and to what extent. These algorithms were also used to study the degree of distortion to the Platonic solid geometry core which results in a desymmetrization when the molecule’s vertices were substituted with three- fold rotors such as tert.-butyl groups. It was of interest to investigate how the size of a spacer (adamantane or Platonic solid geometry skeletons) influenced the twist sense of rotors. Other structural questions were the effect of rotor size on the desymmetrization of rotor substituted Platonic solid geometry molecules. What is the effect of local site symmetry on rotor symmetry (C3 symmetrical versus asymmetric tert.-butyl rotors) and how does this effect the total CSM value of the molecule? What was the effect on the CSM component and total values when partially desymmetrized rotor substituted Platonic solid geometry crystal arrays were compared with completely asymmetric pseudopolymorphs? Many crystalline molecules have a bona-fide symmetry in the array but actually exhibit a higher pseudosymmetry geometry. In such a situation, would the total CSM values for a higher pseudosymmetry in pseudopolymorphic crystals of the same molecule reflect occupancies of different local site symmetries (i.e. positions of special versus general symmetry in the lattice)?

An important goal of the second subproject was to use the algorithms to investigate pseudo-symmetrical relationships in crystals containing multiple molecules in the asymmetric unit. The emphasis for the latter part of the dissertation project was the quantification of the distortion from bona-fide symmetry between molecules rather than within the molecule itself. When there is no pseudosymmetry, the bona fide space group descriptor testifies to the highest symmetry that the crystalline array can express. However, since pseudosymmetry is a continuum with varying degrees of distortion from ideality, then there should be numerous examples when an auspicious packing in the array can be aided by pseudosymmetry between the multiple molecules within an asymmetric unit. The combination of bona fide operations that the crystal has gives its space group. The combination of bona fide operations with pseudosymmetry operations yields an arrangement of the symmetry elements which resembles the symmetry of the crystal's bona fide supergroup. One of the main goals of this thesis was to develop a way to quantify a crystal's deviation from bona fide supergroup symmetry. The distortion from a particular supergroup symmetry would result from the individual contributions of the various pseudosymmetry operations of the set, and their total would report the degree of ‘pseudo supergroup character’ for that crystal. The packing in crystals belonging to the ‘P21/c family’ of space groups [P21/x where x come from either a-, c-, or n-glides] must be auspicious since ca. 37% of all organic molecules crystallize in this monoclinic space group. As a prototype of the method, methods would be developed in order to investigate lower symmetry P21, P–1, Pc or Pn space groups with two molecules in the asymmetric unit. These particular lower symmetry space groups were chosen since they were all related to the same ubiquitous P21/c family of space groups.

New methods were to be used to investigate crystals with both intermolecular and intramolecular pseudosymmetry relationships for the molecules constituting the array. Could quantifying the distortion from an ideal symmetry assist in understanding the constraints which could result in the formation of an unusual crystalline-state conformation? To achieve this goal it was decided to investigate orthorhombic crystals of Pn21a diphenhydramine hydrochloride in which the ammonium cation exhibited an unusual low synclinal (pseudo synperiplanar) 37.5° N–C–C–O torsion angle. The crystals could be related to a Bb21m supergroup as a direct result of this pseudo synperiplanar torsion angle. Was the ‘open butterfly wing’ conformation of the Ph2CH- moiety (as opposed to more usual orthogonal and helical twist) influencing the lattice packing or vice-versa?

In order to meet the above stated goals of using a method to study the pseudosymmetry between molecules in a crystal lattice, it was clear that novel tools would have to be developed. The existing Avnir methodologies were designed to investigate pseudosymmetry within an object, i.e. they were produced for ‘point group’ symmetry operations. Point group symmetry operations leave at least one point invariant in space. The Avnir CSM algorithms were not available to handle symmetry operations found only in space groups, i.e. translation, screw displacement (rotatory translation), and glide reflection (reflection-translation). The classical Avnir algorithms available for point group symmetry operations (rotation, reflection, inversion, and rotatory-reflection) would also have to be modified for use in space groups. The reason for the modification is that they incorporate a size normalization function whose purpose is to measure a particular symmetry in objects irregardless of their scale size. In this manner the Avnir algorithm would provide the identical CSM value for two objects whose only difference was their relative scale size. This size normalization function would be incorrect if the distances between molecules varied either within the same unit cell or in unit cells of different crystals. Since the distances between molecules in different crystals will always be non- identical by symmetry argument, it is clear that the size normalization function would have to be removed.

In addition, a technique for measuring pseudosymmetry in space groups would have to include the dislocation of a pseudosymmetry element defined by a statistical average of points vis-à-vis the special position of the bona fide element in the crystal lattice. All the

[xmean,ymean,zmean] points between bona fide symmetry related atom pairs coincide with special positions in the lattice which are either a single unique point of inversion, or reside on a unique axis of rotation, or define a unique plane of reflection. However, the

[xmean,ymean,zmean] points between pseudosymmetry related atom pairs all define general positions of symmetry in the lattice. As such, they must be treated as either statistical averages of pair-wise points of inversion (a so called ‘best point of inversion’ which now has associated with it an estimated standard of deviation), or they statistically define a ‘best axis of rotation' or a ‘best plane of reflection’. The crystal's space group lacks certain symmetry element, which exist in the bona fide supergroup. However, there may be some degree of pseudosymmetry with regards to these symmetry elements, and the degree of deviation from ideal symmetry can be measured. In addition to that, the dislocation of the best point, axis, or plane compared to the ideal location in the bona fide supergroup must also be taken into account. Only by these new yet-to-be-developed methods could we expect to have a quantitative measure of the difference between bona- fide symmetry and pseudosymmetry in those interesting lower symmetry crystals which mimic the ideal supergroup arrangement to a certain extent.

Throughout all the parts of the research investigation there was a common theme which was to utilize symmetry measures to investigate what happens to intra- or intermolecular symmetrical relationships when the molecular environment does not constrain a particular symmetry to exist in its ideal form.

New non-normalized rmS symmetry measure tools of rmS(i), rmS(C2), and rmS() had to be developed during the course of this research after the classical Avnir S(i) value was found to provide increased values as the distance between two pseudo inversion symmetry related phenyl rings decreased in crystals of N-desmethyl-cis-3,4-butano- nefopam HCl (11). In addition, a new rmS(translation) tool was developed together with new rmS symmetry measure tools of rmS(relocation-i), rmS(relocation-screw), and rmS(relocation-plane) in order to quantify pseudosymmetry in crystallographic unit cells. Their numerical values were sensical and their magnitudes were in agreement with the new rmS symmetry measure tools of rmsFidelity(i), rmsFidelity(21), and rmsFidelity(glide) indices that were developed as an additional method of measuring deviations from symmetry in crystals.

Such tools are very useful, since crystals can gradually desymmetrize from a bona fide higher space group symmetry to a lower symmetry one having multiple molecules in the asymmetric unit and ‘pseudo special positions of symmetry’. These ‘pseudo positions of symmetry’ represent a continuum between classical ‘general’ and ‘special positions of symmetry’ in crystals. Group theory deals with symmetry which either exists or does not exist. In this case, we have seen that there is a need for a continuous form of group theory which predicts how a deviation from one symmetry operation would effect the deviation from another symmetry operation. We have shown in this thesis that symmetry elements continue to be correlated with each other even when one of them shows a large deviation from perfect symmetry.

We have seen many examples in which a bona fide element of symmetry and a pseudo element of symmetry generate a third pseudosymmetry element. For example, when there are two molecules in the asymmetric unit and there is a crystallographic 21 screw axis symmetry together with pseudo inversion symmetry, then pseudo glide symmetry will also exist in that crystal. An interesting future research project in mathematics could further examine how group theory can be extended to cases in which there is deviation from a perfect symmetry. How does the deviation from one kind of symmetry effect the deviations from other symmetries? The use of ‘fuzzy logic’ would be needed in order to examine this aspect of group theory. We have shown an interesting example of how this ‘fuzzy logic’ was essential in understanding why Diphenhydramine HCl (14) exhibited a unique pseudo mirror conformation.

Desiraju's statement that: “high Z' structures may teach us something about the mechanism of crystallization… the answer to this question might not be found in the domain of chemistry but rather in mathematics”, is indeed verified by this research project.

Pseudo Positions of Symmetry: a Continuum Between Classic General and Special Positions in Crystals

1. Introduction

1A. Review of Some General Concepts in Symmetry, Periodic Arrays, and Crystallography. Most molecules which have high point group symmetry lose some or all of their symmetry elements in a crystalline environment. It is still very difficult to predict how a molecule will pack in a particular crystallographic lattice. However, correlations between molecular point group symmetries and space group symmetries were found.1 It is useful to begin by a brief review of some of the terminology and concepts used in this research project. If an object is said to be composed of a pattern of points, then an isometry exists between two objects if the interpoint distance matrix of one is identical to that of the other. Symmetry is an isometric relationship that enables one or more rotation, reflection, inversion, rotation-reflection, pure translation, rotation-translation (screw rotation) and reflection-translation (glide- reflection) operators to exchange subunits within a single object, or to exchange whole objects within a doubly or triply periodic array, while bringing the exchanged moieties into coincidence or self coincidence. Mislow2 defines symmetry as “that property of a body by which that body can be brought from an initial position in space to another, indistinguishable position by means of a movement (or movements such that, after the movement has been carried out, every point of the body is coincident with an equivalent (or perhaps the same) point of the body in its original orientation.” Symmetry is described by a mathematical group composed of a mathematical operation that acts on the elements of a mathematical set. The elements of the mathematical set (not to be confused with ‘symmetry elements’ described below) are all the symmetry operators (mathematical transforms or operations) acting on the object and must include identity.

The mathematical operation is ‘successive application’ or ‘multiplication’, i.e. first perform operation no. 1, then perform operation no. 2, and the result will be operation no. 3 which must also be a member of the same set. Point group symmetry enables subunit exchange by any of the first five transformations listed above so that at least one point remains invariant in space. Space group symmetry refers to doubly or triply periodic arrays and utilizes all seven mathematical transforms listed above plus identity. Mathematical transforms for the three space group operations involving translation (translation, screw- rotation or rotatory-translation, and glide-reflection or reflection-translation) leave no point invariant in space. The symmetry of molecules is defined in terms of symmetry elements and symmetry operations.3 A symmetry element is an imaginary geometrical entity such as a line, plane, or point about which the symmetry operation can be performed.3 Therefore, symmetry equivalence poses a structural constraint (i.e. an isometry and a special orientation and location- an ideality). Figure 1 shows two molecules which are related by a symmetry operation of the Second Kind (reflection). They are non-superimposable isometric structures and have opposite handedness. Symmetry operations of the Second Kind are those of reflection, inversion, rotatory-reflection, and glide-reflection.

Figure 1: The two chiral molecules are related by a symmetry operation of the Second Kind, and therefore have opposite handedness.

Figure 2 shows two molecules which are related by a symmetry operation of the First Kind. They are superimposable isometric structures and have the same handedness. Symmetry operations of the First Kind are those of identity, rotation, translation, and screw-rotation (rotation-translation).

Figure 2: The two chiral molecules are related by a symmetry operation of the first kind, and are therefore superimposable.

Figure 3 shows a staggered conformation molecule with C2 point group symmetry.

The molecule has two homotopic halves which are superimposable by a C2 rotation

operation. The C2 point group provides a structural constraint which gives rise to homotopic degeneracies in corresponding bond lengths, bond angles and torsion angles. The torsion angles have equal signs and equal magnitudes, since a C2 operation is a symmetry operation of the First Kind. For example, the two blue Cl– C–C–H torsion angles in Figure 3 are conrotatory (same directionality, anticlockwise magenta arrows) and equal to –60°.

H D Cl

D Cl H

Figure 3: Staggered conformation molecule with C2 point group symmetry.

Figure 4 shows a staggered conformation molecule with Ci point group symmetry. The molecule has two enantiotopic halves which are interconvertable by an inversion operation. The Ci point group provides a structural constraint which gives rise to enantiotopic degeneracies in corresponding bond lengths, bond angles and torsion angles. The torsion angles have opposite signs and equal magnitudes, since an inversion operation is a symmetry operation of the second kind. For example, the two blue colored Cl–C–C–H torsion angles in Figure 4 are disrotatory (opposite directionality) and equal to 60° in magnitude but have opposite signs.

Cl H D

D H Cl

Figure 4: Staggered conformation molecule with Ci point group symmetry.

All bonding parameter constraints (bond lengths, bond angles, torsion angles) in two or more molecular subunits OR between molecules in an extended array are removed when these objects are NOT related by a symmetry transform. The molecular subunits in a solvated molecule which were related by a symmetry transform in solution state, are not necessarily related by symmetry transforms in a solid state crystal lattice. They may be desymmetrized to a large or small extent upon crystallization since their environments are different. (see Figure 5)

Figure 5: A solvated symmetrical molecule may lose some or all of its symmetry elements upon crystallization due to packing constraints. (a) Change to a new C1 symmetry conformation upon crystallization at a general position. (b) Retention of conformation upon crystallization but object’s bona fide inversion center has become pseudo-inversion cluster of points [i.e. non-crystallographic symmetry].

Figure 5 represents a solvated three-dimensional achiral object (molecule) with a red center of inversion symmetry. Crystallization changes its environment. It might crystallize in a space group also having inversion as one of the elements of the set of symmetry operations. The fact that the space group is centrosymmetric, does not guarantee that the molecule's inversion symmetry will be preserved. If packing efficiency warrants it, then the crystal’s inversion center might be between molecules, and then the molecule will loose its inversion symmetry when confined in the crystal. Figure 5a shows a descernable conformational change upon crystallization and Figure 5b shows that retention of conformation upon crystallization, but the bona fide center of inversion in the solvated molecule has now become a pseudo-inversion cluster of points [i.e. non-crystallographic symmetry]. As is well known, the smallest part of a crystal that will reproduce the entire crystalline extended array by using only the translation operation is denoted as the ‘unit cell’.4 Figure 6 shows an example of a three dimensional unit cell, and an array of 4 unit cells which is obtained by translation of the repeat unit.

Figure 6: An extended array produced from one unit cell by translation.

“An asymmetric unit of a space group is a (simply connected) smallest part of space from which, by application of all symmetry operations of the space group, the whole of space is filled exactly.5 This implies that mirror planes and rotation axes [if present] must form boundary planes and boundary edges of the asymmetric unit.5 A twofold rotation axis may bisect a boundary plane.5 Centers of inversion must either form vertices of the asymmetric unit or be located at the midpoints of boundary planes or boundary edges.5

For glide planes and screw axes these simple restrictions do not hold.5 An asymmetric unit contains all the information necessary for the complete description of the crystal structure.5 In mathematics, an asymmetric unit is called a 'fundamental region' or 'fundamental domain'.5 ", ‘General position of symmetry’ "A set of symmetrically equivalent points, i.e. a 'crystallographic orbit', is said to be in 'general position', if each of its points is left invariant only by the identity operation but by no other symmetry operation of the space group.5 Each space group has only one general position.5 The coordinate triplets of a general position (which always start with x,y,z) can also be interpreted as a short-hand form of the matrix representation of the symmetry operations of the space group."5 We may denote the identity and translation operations as ‘trivial’ since they must be a member of the set of symmetry operations for every space group. The use of the term ‘trivial’ does not imply that they are insignificant or unimportant. Instead, the second meaning of ‘trivial’ is being used which means ‘common place’ or ‘expected’. Obviously, no special position may exist in the asymmetric unit (not including its boundaries). The letter Z denotes the number of molecules in the unit cell, and Z' is the number of molecules in the asymmetric unit.4 The ratio of Z/Z’ gives the number of equivalent general positions, in other words, the multiplicity of the general positions. If an atom resides at a crystallographic symmetry element, then one says that it is occupying a ‘special position of symmetry’ in the lattice.4 Such atoms are transformed into themselves by the particular symmetry operation at that site. Therefore, their multiplicity is lower than that of a general position.4

Figure 7 shows an asymmetric unit which is comprised of a three dimensional object. It is an asymmetric object and it occupies a general position of symmetry (Z’ = 1). For the purpose of illustration, a unit cell will be generated by applying an inversion symmetry transformation on the object. We note that there are now two objects in the unit cell (Z = 2). The number of equivalent general positions for each object is two (Z/Z’ = 2).

Figure 7: Asymmetric unit of one full object (Z’ = 1). Two objects in the unit cell (Z = 2) generated by applying the inversion symmetry operation of space group. Two equivalent positions per object (Z/Z’ = 2).

Figure 8 shows an example of a unit cell which is made up of an achiral object. The asymmetric unit is one half of the unit cell (and one half of the object). The object is occupying a special position of inversion symmetry. For an object to occupy a special position of symmetry in the crystal, the space group must have the same symmetry operation as that found in the object, so that the two symmetry elements can coincide providing that packing will be efficient. By definition, the inversion center must lie on the boundaries of the asymmetric unit and not inside it, since the asymmetric unit may not contain symmetry elements besides identity.

Since the letter Z denotes the number of complete objects in the unit cell and Z' is the number of objects in the asymmetric unit, then Z = 1 object and Z' = 0.5. The number of equivalent positions is Z/Z’ = 2.

Figure 8: Asymmetric unit of one-half an object (Z’ = 0.5). One object in the unit cell (Z = 1) generated by applying the inversion symmetry operation of space group. Two equivalent positions per object (Z/Z’ = 2).

Glaser used M. C. Escher’s periodic drawings to illustrate the concepts of ‘special’ and ‘general’ positions.6 As noted before, a molecule must reside in a special position in the crystal lattice in order to maintain the symmetry it had in another environment.6 For example, a solvated molecule with mirror symmetry in the solution state as its sole symmetry element, must occupy a crystallographic mirror plane in order to conserve its reflection symmetry in the crystalline environment. If efficient packing of space should obligate Nature to use a space group without a mirror reflection operation, then the molecule will be desymmetrized completely to an asymmetric geometry upon entering the crystal lattice.6 This is a symmetry argument but it does not tell us the degree of desymmetrization.

In covalent solids, covalent bonds in a definite spatial orientation link the atoms in a network extending through the crystal.7 Molecular solids, on the other hand, consist of separate molecules linked together by non-covalent electrostatic interactions, such as strong or weak hydrogen bonds8, aromatic – aromatic interactions9, and van der Waals forces7. My thesis will focus on the symmetry and conformations of organic molecules within a crystalline environment. Such crystals are called 'molecular solids'7 or 'molecular crystals'10 This thesis will focus on the way molecules are packed within such crystals.

Brock and Dunitz11 have stated that centrosymmetric molecules often occupy special positions of inversion symmetry upon packing. These authors have also stated that

crystals with space groups having C3 axes are not usually observed unless these axes are located within molecules of the same appropriate symmetry. In other words, a

crystal whose space group has a C3 axis as one of its symmetry elements will not

usually be found unless a molecule with a C3 axis point group symmetry element occupies a special position of three-fold rotational symmetry in the lattice (so that the

space group’s C3 axis and the molecule’s C3 axis both coincide). According to a statistical search of the Cambridge Structural Database (CSD) of x-ray crystallographically determined molecular structures, 95.2% of all centrosymmetric point group molecules (i.e. those with inversion centers) crystallized in centrosymmetric space groups, and 88.4% of the centrosymmetric molecules occupied special positions of inversion symmetry in the lattice.1 A search of the CSD showed that 76% of the molecules with more than five atoms which crystallized in space groups having symmetry elements of both sigma-planes and C2 axes had ‘high symmetry point groups’.1 The authors defined a ‘high symmetry point group’ in a 1 special ad-hoc way as “one which was not Ci, Cs, C2, or C1”. This non-standard definition of a high symmetry point group differs greatly from the more accepted one which states that high symmetry point groups are those that have more than one Cn rotation axis that is of order three or greater.

On the other hand, 84.1% of the molecules which crystallized in the completely

asymmetric P1 space group were themselves also completely asymmetric (i.e. they

had themselves C1 point group symmetry) while only 7.2% of the molecules which

crystallized in the completely asymmetric P1 space group exhibited high point group symmetry (defined in the non-standard manner of the authors) in other environments.1 In other words, on the basis of statistical studies, ‘high symmetry’ point group molecules preferred to pack in ‘high symmetry’ space group crystals, while asymmetric molecules preferred to pack in asymmetric space group crystals.1 According to a database survey of molecular and crystallographic symmetry, space groups with inversion centers, screw axes and glide planes that lead to 'bump-to- hollow' [large-close-to-small] molecular interactions are preferred over 'poor' symmetry operators like the mirror planes, which lead to 'bump-to-bump' [large-close- to-large] interactions.12 Molecules that possess an inversion center retain this symmetry element in more than 80% of the cases.12 Similarly high degrees of 12 retention were found for S4 and S6. Rotation axes were retained by molecules in approximately 50% of the cases and mirror planes were usually retained in less than 30% of the cases.12 Molecular elements that promote dense structures are retained in crystal structures, whereas molecular-symmetry elements that are not conductive to densely packed structures are lost.

1B. Introduction for Research Subproject 1: The Use of the Continuous Symmetry Measure (CSM) to Study the Desymmetrization of Molecules With Platonic Solid Geometries Upon Entering a Crystal Lattice). The Platonic convex solids are totally isotropic.12 This means that when viewed either edge-on, or vertex-first, the geometrical representation does not depend on the particular edge or vertex that was chosen.13 The platonic solids are the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron.13 Plato associated each of the four classical elements (earth, air, fire and water) with a regular solid.13

Fire was associated with the tetrahedron, Earth with the cube, Air with the octahedron, and Water with the icosahedron.13 The Heavens (‘ether’) were associated with the dodecahedron. Duality is a relationship between pairs of platonic solids.14 The operation demonstrating duality can be viewed as connecting the midpoints of ‘hedra’ (faces) so that they become the vertices of a second Platonic solid.14

The cube is the dual of the octahedron, the dodecahedron is the dual of the icosahedron, and the tetrahedron is the dual of itself.14 Figure 9 shows the duality relationships of the Platonic solids. Table 1 lists the numbers of faces, vertices and edges for each Platonic solid polyhedron,14 and their corresponding point group symmetries.15

Figure 9: Duality relationships in the Platonic solids.

Table 1: Faces, vertices, and edges of convex Platonic Solids. Platonic solids Faces vertices edges Point group symmetry

tetrahedron 4 4 6 Td

cube 6 8 12 Oh

octahedron 8 6 12 Oh

dodecahedron 12 20 30 Ih

icosahedron 20 12 30 Ih

It would be interesting to understand how molecules composed of such isotropic building blocks would distort upon being packed into a crystal lattice, and how rotatable substituents (e.g. tert.-butyl groups) attached to rigid cores would desymmetrize the Platonic solid skeleton. In molecules exhibiting edge-bonded polyhedra geometries, the edges correspond to chemical bonds.16 According to 17 calculations in the gas phase, tetrahedrane (1) has Td symmetry, tetrakis-tert.-butyl- 18 17 tetrahedrane (2) has T symmetry, and cubane (3) has Oh symmetry .

Dodecahedrane (4) was shown to have a single 13C and 1H NMR peak [66.93 ppm and

3.38 ppm] in the respective solution-state spectra, and therefore has Ih symmetry in solution.19

The aim of this particular thesis sub-project is to understand how molecules exhibiting such convex Platonic solid geometries in non-crystalline environments distort upon crystallization, and how substituents with three-fold axes ligated to the vertices desymmetrize these molecules when confined within the crystal lattice.

Alvarez and coworkers20 have elegantly shown that molecules with icosahedral symmetry were found to crystallize in cubic or nearly cubic crystal space group systems, which is probably related to their “latent octahedral symmetry”. Two examples of such latent octahedral symmetries in icosahedral Platonic solids are shown in Figure 10A,B. In Figure 10A, eight equilateral triangular faces of an icosahedron related by symmetry planes were selected, and a point was placed in each.20 As a result, a perfect cube was constructed20 The dodecahedron is the “dual” of the icosahedron.20 “Dual’ means that the twenty face centers in the icosahedron are exchanged into the twenty vertices of the dodecahedron, while the twelve face centers in the dodecahedron become the twelve vertices of the icosahedron.20 Therefore, there must be latent octahedral symmetry in the dodecahedron too, as can be seen in Figure 10B.

Figure 10A: Latent octahedral symmetry in the icosahedron.

Figure 10B: Latent octahedral symmetry in the dodecahedron.

Alvarez and coworkers16 have presented a definition of the minimum distortion paths between two polyhedra. Examples of these distortion paths are the spread path leading from the tetrahedron to the square, the Berry pseudorotation that interconverts a square-pyramid with a trigonal-bipyramid, and the Bailar twist for the interconversion of the octahedron with the trigonal-prism.16 The minimal distortion path leading from tetrahedricity to planarity was found to be preferred by copper complexes.21 Most of the transition metal hexacoordinated complexes were found to be (nearly) aligned along the Bailar twist, in which the initial octahedral structure is 22 gradually twisted into an achiral D3h structure. The intermediate structures 22 belonged to the chiral D3 point group.

Kost and coworkers23 have shown that tert-butyl substitution of pentacoordinate bischelate complexes of silicon (5-9) (which had a trigonal bipyramidal geometry before substitution) pushed the geometry towards the other end of the reaction coordinate, along the Berry pseudorotation path, as can be seen in Table 2.

Table 2: Geometric parameters for compounds 5-9.

compound 5 6 7 8 9 drawing of the N N o o first Si X Si X o coordination o N N sphere

X H Me CH2– tert.-Butyl tert.-Butyl (dimer) angle N–Si–N 171.2° 155.6° 146.9° 133.5° 121.5° angle O–Si–O 119.8° 134.3° 146.0° 152.2° 165.7° (C–Si–C) geometry trigonal ca. 35% square ca. 31% trigonal bipyramid partially pyramid partially bipyramid no. 1 distorted- distorted- no. 2 trigonal trigonal bipyramid bipyramid no. 1 no. 2

Another example by Alvarez and coworkers16 of a continuous distortion path is the one connecting the cube and the triakis-tetrahedron. As can be seen in Figure 11, a cube consists of two tetrahedra of the same size. The triakis-tetrahedron is built from two tetrahedra which differ in their size.

A gradual distortion from the cube to the triakis tetrahedron is illustrated in Figure 12. Tetrahedral heterocubanes often have symmetries which correspond to a minimal distortion path between a cube and a triakis – tetrahedron.16

Figure 11: A cube consists of two tetrahedra of the same size.

Figure 12: A continuous distortion of the cube can lead to a series of triakis- tetrahedra.

1B.1 The Avnir Continuous Symmetry Measure (CSM). The continuous symmetry measure (CSM) of a structure is a normalized root-mean- square distance function from the closest structure which has the desired symmetry, i.e., the nearest perfectly symmetric object with respect to the required symmetry.24,25 (see Figure 13)

A Finding the nearest 1 Â1 A1 Â1 c3 symmetric object Superimposition A A 2 2 Â3 Â2 Â2 Â3 A3 A3

Figure 13: The nearest symmetric structure can be found by using Avnir's CSM algorithms.

Given a distorted structure composed of N vertices (see Figure 14), the coordinate vectors of which are { Ak , k = 1, 2 ,…, N }, one searches for the vertex coordinate ˆ 25 vectors of the nearest perfectly G-symmetric object, { Ak , k = 1, 2 ,…, N }. Once at hand, the symmetry measure is defined as:

N ˆ 2   AA kk k1 S  min N 100 (1) 2  k  AA 0 k1

Where A0 is the coordinate vector of the center of mass of the investigated structure

1 N A0   Ak (2) N k 1

Ak - coordinates of the actual structure

Âk – coordinates of the nearest

ideal C3 symmetric structure

Figure 14: The root mean square distance deviation from ideal C3 symmetry is measured by superimposition of the atomic positions (depicted as blue colored balls) in the actual structure (with blue bonds) on the corresponding atoms (depicted as magenta colored balls) in the nearest ideally C3-symmetric structure (with magenta bonds). The yellow double head arrows are the distances between corresponding blue and magenta pairs of atoms, and in order to compute the CSM value, these distances are added up.

The CSM defined in Eq. 1 is independent of the position, orientation and size of the original structure.25 Equation 1 is general and allows one to evaluate the symmetry measure of any structure relative to any symmetry group G.25 The original structure is normalized to the Root Mean Square (RMS) distance from the center of mass of the structure (placed at the origin) to all vertices (the denominator in Eq. 1).25 The bounds are 100≥S≥0, where S(G) = integer-0 means that the structure has the desired ideal G symmetry.25 The symmetry measure increases as the structure departs from G symmetry. All S(G) values are on the same scale and are therefore comparable.25

This study uses CSM values with respect to symmetry operations in the particular high-symmetry tetrahedral, octahedral, or icosahedral point group, in order to characterize the distortion and understand how vertex substitution with three-fold rotors influences the molecular conformation in the crystal lattice.

1B.2 An In-Depth Overview of Avnir’s CSM Methodology. The nearest symmetric function must be searched for, and algorithms for finding it were created by David Avnir and his research group.26,27 A simple numerical algorithm was previously used in order to calculate the CSM values with respect to Cn operations when order n is greater than 2.26 This algorithm is called the ‘folding and 26 unfolding’ algorithm. Figure 15 illustrates the method for finding the nearest C3 symmetric triangle. The vectors' origin is the center of mass.26 The statistical ‘best plane’ is found by a least squares approximation.26,25,28

Folding:

0 A0 Ã0 0 Ã0 A is rotated A1 is rotated 2 Ã2 Ã0 The vectors Â1, Â2, Â3 counterclockwise counterclockwise A0 doesn’t Ã1 Ã1 are averaged move by 120˚ by 240˚

1 A1 1 A1 2 A2 A2 2 A2

Unfolding:

Â1 is Â2 is Â , Â , Â rotated Â0, Â2 rotated Â 0 1 2 clockwise clockwise 0 by 120˚ by 240˚

Â1 Â2 Â 1 Figure 15: The least squares numerical folding-unfolding algorithm was the first algorithm used for finding the nearest C3 symmetric structure.

A program based on an analytical algorithm was recently developed and has been used in my research for the calculation of the CSM values.25,28,29,30 This method is analogous to the folding and unfolding.26 While the folding and unfolding method is an approximation, the new algorithms provide exact analytical solutions.28

The analytical method for finding the nearest symmetric structure is illustrated in Figure 16 below.28

Figure 16: A new algorithm, developed by Avnir's research group, finds the nearest symmetric structure analytically.

A. All of the symmetry operations of the group are applied on the distorted 28 structure. (Qk is the set of coordinates for the distorted triangle) B. Three triangles are obtained – one for each symmetry operation.28

C. The coordinates of the three triangles are averaged, and the nearest C3 28 symmetric triangle is obtained (Qk' is the set of coordinates for this triangle). D. The nearest symmetric structure is superimposed on the distorted structure, and the CSM value is obtained.28 The mathematical form of the CSM equation in this case is:

2 1 N n 2 1 N n ' ' GS  min)( 2  k VQ ,ik  min 2  QRQ ,ikik  2Nnd k11i 2Nnd k11i N n 1 T ' 1 2 max  k iQRQ ,ik Nnd k 11i

28 Where d is the RMS size of the original structure, Qk. The distance function is

' 28 between two sets of nN points: nQk and V ,ik .

' V ,ik = , QPR kiki The subscript i stands for the symmetry operation index.28 In this case, the 28 symmetry element is C3. The index i can be 1,2 or 3 and it will be dropped for the sake of simplicity.28

As can be seen in Figure 17, Q1 should be compared to V3 , Q2 to V1 , and and

28 Q3 to V2 . In order to compare the correct points, a permutation matrix is used,

' ' and the points are rearranged into Vk , for which: Q1 is compared to V1 , Q2 is

' ' 28 compared to V2 , and Q3 is compared to V3 . A similar sequence of steps is taken for each of the group symmetry operations.28

Q1 V3

3 c3

Q2 V2 Q V1 3 Figure 17: A permutation matrix determines the pairs of vertices which will be compared to each other.

1B.3 Estimating the Errors for the CSM Values: The errors in the CSM values are estimated using the Error Estimation Program, which takes the number of unique reflections, the thermal factor and the CSM value into account as input, and returns a CSM range or an upper bound for the CSM values.30 When the CSM value is much larger than the errors of the atomic coordinates, the program gives a range of CSM values, which is in the 95% confidence interval.30 When the CSM value and the displacement errors have similar values, it is only possible to obtain the maximal CSM value.30 Therefore, the CSM results are sometimes expressed as ‘less than a certain value’ rather than as a number and error.30

1B.4 CSM Computation Times: The number of permutations grows with the number of vertices as N!.28 The new CSM programs can reduce this number by excluding forbidden permutations, such as permutations between different atom types, and permutations between atoms which differ in their first connectivity sphere.25 ,29

1B.5 Calculation of Symmetry Measures With Respect to a Specific Permutation: The CSM programs of Avnir and his research group initially did not have an explicit option for specifying the specific permutation that was desired. Without specifically specifying the desired symmetry permutation, most of the calculation times were very long since artificial intelligence was used in order to search for a permutation. Part of my research project was to develop a method for making significant reductions in the computation times by specifying the desired permutation. Since different atom types could be permutated in the older versions of Avnir's programs, random atom type symbols were used to specify symmetry equivalent positions.

A different atom symbol from the Periodic Table was used for each set of equivalent positions. As can be seen in Figure 18, graphically coloring one ball in methane by a unique color reduces the number of possible pseudo C3 axes. The mechanism for ‘coloring’ the atoms in the input file for the calculation was by specifying a different atom symbol for each symmetry set. It was decided to do this since the Avnir program treats atom types as simple chemically insignificant alphabetical descriptors ("colors") rather than treat them as actual chemical entities. The result was a marked increase in calculation speed, which was made possible by choosing which pairs of atoms would be permuted by the algorithm to give the particular symmetry operation.

Figure 18: In order for the program to work faster to find a specific C3 symmetry, the Td symmetry can be artificially lowered by coloring one of the atoms yellow whereby the green colored atoms define the specific C3 permutation.

In this research project, the ‘Continuous Symmetry Measures’ (CSM) of David Avnir (Hebrew University of Jerusalem) will be used in order to ascertain how many molecules with Platonic solid geometries distort upon residence in a crystalline environment, and how vertex substitution with three-fold rotors (e.g. tert.-butyl groups) influences the symmetry of the molecule. According to an early computational study by Hounshell and Mislow31, the ground state (global minimum calculated geometry) of tetrakis-tert-butyl tetrahedrane (2) has T symmetry and is thus chiral.31 All four tert.-butyl groups twist by about 14° and all 12 methyl groups twist by about 2-6°, all in the same direction from a staggered Td conformation. The calculations were done using a variety of force fields which were available in the 1970's.31 Recently, Balci and Schleyer18 using computations performed with higher levels of theory (ab-intio restricted Hartree-Fock HF/6-31G* and density functional theory B3LYP/6-31G*), have also shown that tetrakis-tert-butyl-tetrahedrane (2) is more stable when it exhibits T symmetry. In these computations, the T symmetry 18 geometry is 0.5-2.0 kcal/mol lower in energy than that with Td symmetry.

Figures 19A,B shows the achiral Td- and chiral T-symmetric conformations for tetrakis-tert-butyl-tetrahedrane (2), respectively.

H H H H H H H HH HH H H H H H H H H H H H H H H H H H H H H H H H H H

Figure 19A: The Td symmetric conformation for tetrakis-tert.-butyl- tetrahedrane (2).

H H H H H H H H H H H H H H H H H H H H H H H H H H H H HH H H H H H H

Figure 19B: The T symmetric conformation for tetrakis-tert.-butyl-tetrahedrane (2).

The driving forces of this distortion were attributed by Balci et al.18 to the relief of steric repulsion between methyl groups on different tert.-butyl groups. Twelve non- bonded H…H interactions are reduced from 2.25 Å to 2.43 Å.18 According to Matta et al.,32 the H…H interactions that were described by Balci and Schleyer18 as being ‘repulsive’, are in actuality stabilizing ‘Van der Waals’ interactions.

The reason for the stabilization is the fact that instead of 12 H…H interactions, 18 … 32 H H interactions are obtained, when the structure is twisted from Td to T symmetry. Figures 19C,D demonstrate how a twist generates additional interactions, when a pair of parallel interactions is substituted by three diagonal ones. Therefore, the loss of mirror symmetry yields stabilization of the structure. A Td to T geometry transformation also shortens the lengths of the tetrahedron core C—C bonds by 0.0014Å while the bonds between a core carbon and a tert.-butyl carbon are shortened 32 by 0.0020 Å. The remaining tert.-butyl Cquaternary—Cmethyl bond lengths all increase by less than 0.0004Å.32 The bonds which are shortened have a stabilizing effect and the ones which become longer have a destabilizing effect.32 Cohesive properties of crystals can be understood better, when favorable interactions such as ‘Van der Vaals’ H…H interactions are taken into account.32 This may be the reason why mirror planes are considered ‘poor’ symmetry operations in a crystalline environment: 'bump-to-bump' [large-close-to-large] interactions yield less interactions than 'bump- to-hollow' [large-close-to-small] interactions.

H H H H H H H HH HH H H H H H H H H H H H H H H H H H H H H H H H H H

Figure 19C: Stabilizing proton–proton interactions for the Td symmetric conformation of tetrakis-tert-butyl-tetrahedrane (2).

H H H H H H H H H H H H H H H H H H H H H H H H H H H H HH H H H H H H Figure 19D: There are more stabilizing proton–proton interactions for the T symmetric conformation of tetrakis-tert-butyl-tetrahedrane.

1C. Introduction for Research Subproject 2: Symmetry Measure Continuum Studies on Pseudo Symmetry Between Molecules in Crystals Containing Multiple Molecules in the Asymmetric Unit Leading to a Subgroup-Supergroup Relationship. The Investigation of Intermolecular Forces Involved in Pseudo- Symmetrical Arrangements Within a Crystal Lattice.

The symmetry of molecules may be affected by their environment. The degree of symmetry distortion of molecules residing in the crystal lattice was examined by the CSM method. A desire to understand this geometrical distortion prompted me to examine the packing arrangement in the crystal.

Often, the presence of multiple molecules within the asymmetric unit may be the key to explaining the particular symmetry of the molecules therein. Approximately 11% of all organic compounds in the CSD have more than one molecule in the asymmetric unit.33 One ponders why this occurs. One possibility is that these crystals are metastable, and a corresponding crystal with a single molecule in the asymmetric unit would always be more stable.33 However, the prevalence of Z' > 1 (i.e. multiple molecules in the asymmetric unit) in a subset of organometallic crystals was found to be less than average .34

Organometallic compounds are usually made by rapid cooling – a process which usually yields metastable crystals (kinetic), since the molecules are not allowed to reach their thermodynamic minimum geometries and the resulting crystals are also not thermodynamically stable.34

The multiple molecules in the asymmetric unit were found to have the strongest interactions in the crystal in 55.3% of the cases, and in 77.4% of the cases, these interactions were among the top three in strength.12 Classical hydrogen bonding interactions are considered strong, while aromatic C—H hydrogen interactions are considered weak.35

Two molecules in the asymmetric unit occur when it is difficult to simultaneously satisfy the criteria of close packing and bonding requirements.36 Edge-to-face aromatic-aromatic Coulombic interactions involving the +(H)→–(C) dipole in the aromatic H—C bonds often play a very important role in intermolecular arrangements found in the crystal lattice.37 These interactions were found to have an important stabilizing effect on protein structure.9 They are characterized by a distance of 4.5 to 7 Å between ring-centroids, and the dihedral angle between the rings is usually about 90˚.9 Another effect is exemplified by aliphatic polyamine salts forming strongly hydrogen-bonded networks.12 When a competing face-to-face aromatic-aromatic π- stacking interaction is introduced, structures with higher Z' values (more than one molecule in the asymmetric unit) are formed more readily.12

Crystals which have more than one molecule in the asymmetric unit have less space group operations.38 This is probably due to the fact that there are strong binding interactions between the molecules in the asymmetric unit.38 Out of 1850 crystals obtained from a search of the CSD, approximately 83% revealed some kind of pseudo-symmetry, 20% of the molecules had two-fold pseudo symmetry, 23% inversion, 30% translation, 57% screw and 60% glide pseudo symmetries.38 Gavezzotti listed molecules found in the CSD, which had pseudo symmetry of different types, up to a 0.5Å threshold in the RMS of their superimposition.38 A small percentage of the molecules which have pseudo inversion symmetry are chiral.

A simple mechanical mixture of two enantiomeric crystals in a 1:1 statistical ratio is called a ‘conglomerate’.39 About 10% of all racemic mixtures of organic compounds crystallize as conglomerates.40 A ‘racemic compound’ consists of crystals in which the enantiomers are present in a 1:1 ratio and reside at well defined positions in the unit cell.39 ‘Pseudo racemates’ are crystals that contain unequal amounts of the two enantiomers. About 90% of all organic compounds are racemic compounds.40 Crystals with two molecules in the asymmetric unit can be divided into four different subgroups:

1. The two molecules in the asymmetric unit have the same configuration and the same conformation (the conformations are the ‘same’, but not identical – since they are symmetry unrelated due to their residence in the asymmetric unit). The differences in the torsion angles are small, since if the differences were large, then the conformations would be different. The two molecules are related by pseudo symmetry of the First Kind. 2. The two molecules in the asymmetric unit have the same configuration and different conformations. 3. The two molecules in the asymmetric unit have opposite configurations and the same conformation. These molecules are related by pseudo symmetry of the second kind. 4. The two molecules in the asymmetric unit have opposite configurations and different conformations. This is the most interesting of all cases noted above, since it is relatively unusual. There is usually are substantial reasons for the existence of two different conformations in the asymmetric unit, and it is a challenge to understand their function.

Most of the crystals which have two molecules in the asymmetric unit belong to the first two cases, i.e. they consist of two molecules with the same handedness. Bishop and Scudder41 have estimated that only about 1% of all organic molecules have two molecules of opposite handedness in the asymmetric unit. They called such mechanical mixtures of chiral crystals 'false conglomerates' since the original racemic mixture is regenerated if one crystal is separated out and then dissolved.41

Sometimes multiple molecules in the asymmetric unit are arranged in an auspicious relationship which bears a close resemblance to a particular bona-fide symmetry. Subprojects 2-3 all exemplify another use of the Continuous Symmetry Measure to ascertain the degree of pseudo-symmetry in crystals containing multiple molecules in the asymmetric unit, and to gain insight into the forces responsible for both the pseudo-symmetry and the multiple molecular occupancy within an asymmetric environment.

1C.1 Analysis of pseudo-symmetry and packing in crystal lattices which are related to a pseudo P21/c supergroup and which contain two molecules in the asymmetric unit. Gavezzotti has noted that “with an asymmetry tolerance of 0.5 Å atom–1, 83% of the asymmetric crystal structures reveal some kind of symmetry beyond that of the assigned space group. In some cases the extra symmetry element(s) suggest a subgroup-supergroup relationship…”38

A search of the literature will show that many useful approaches have been developed to study pseudo symmetry in crystals - an important research topic in the crystallographic community. Gavezzotti developed an indexing method, which makes it possible to find crystals in the CCDC, which have more than one molecule in the asymmetric unit, and which are related by pseudo symmetry.38 His method makes it possible to search for a desired symmetry operation by machine intelligence and to select a threshold discrepancy limit from an ideal symmetry.38 It is based on two indices.38 The x,y, or z coordinates of corresponding pairs of atoms are added to each other for the Is+ index, while they are subtracted from each other for the Is– index.38 These indices make it possible to recognize the type of pseudo symmetry which exists between the two molecules in the asymmetric unit. For example, when there is pseudo inversion, the Is+ values for x,y and z are very low while the Is– value is high.38 For symmetry operations which involve translation, an additional number which quantifies the deviation from perfect translation is used.38 While Gavezzotti's method is very useful for identifying the types of pseudo symmetries present in a crystal, they are inconvenient to use as a symmetry measure, since three different indices have to be used for glide-reflection or screw-rotations. Our goal was to come up with a method which eventually yields a single number which quantifies the deviation from a certain type of symmetry element.

It had been shown in the literature that crystallographic software can be used in order to detect and investigate pseudo symmetry between multiple molecules in the asymmetric unit. But if the original crystallographer has published a structure where the existence of pseudo symmetry in the crystal packing was not recognized, then there are numerous cases where atom number labels affixed to a pseudo symmetrical pair are different and therefore do not provide any indication of the relationship. Since machine intelligence usually recognizes pseudo symmetrical pairs of atoms by their same atomic descriptor, this pseudo symmetrical relationship will not be found in a search. Fortunately, in cases where pseudo symmetry is now known to exist in a crystal, then software has been developed to reindex (if needed) the atom numbers in the cif file according to pseudo symmetry.42,43,44,45 A method to improve the pairing of atoms for molecules with two molecules in the asymmetric unit uses connectivity as the first stage, three dimensional fitting in the second stage, and a search for a mathematical operator (D) which interconverts sets of corresponding atom pairs as the third stage.46 Once the pairing of atoms has been completed, the molecules are then compared in terms of their conformational similarity and an operator D is compared to the ideal operator.46 Conformational similarity is examined in terms of root mean square deviation (RMSD) between pairs of atomic coordinates of two molecules, the torsion angle similarities between the two molecules, and similarity in bond lengths.46 Collins et al.46 state that one of the major drawbacks of using a simple RMSD comparison between the two molecules is the fact that it hinders differentiating between a general small deformation over the entire molecule and a situation in which the differences are mainly concentrated within a small portion of the molecule. Therefore, torsion angles and bond lengths are compared.46

The ‘PSEUDO’ program makes it possible to identify molecules which deviate from an ideal space group by a certain threshold value.47 The algorithm used in this program, applies the space group's symmetry operations on the atoms of the crystal's asymmetric unit.48 It looks for the ‘best match’ between atoms in the original space group and the transformed array of coordinates.48 The difference between the coordinates of each corresponding set of atoms is computed.48 The maximal difference between a corresponding set of atoms is used as an index, which makes it possible to recognize a set of crystals which are within a certain threshold from an ideal space group symmetry.48 This method gives a maximal deviation from the ideal symmetry.

But, the above method does not provide the average deviation from a particular space group symmetry. A method which uses differences in electron density in order to find the deviation from a certain supergroup's symmetry, is supposed to give an average, rather than maximal values.49 The program is called ‘DOPE’ and is available on the Bilbao Crystallographic Server.50,51 A .cif file (crystal information file) can be uploaded, and the program ‘DOPE’ gives a list of possible minimal supergroups. Unfortunately, the pseudo symmetry utility does not seem to work, and when the ‘pseudo symmetry check’ button is pressed, nothing happens.

The existence of multiple molecules in a crystal’s asymmetric unit provides an auspicious opportunity to study intermolecular pseudo symmetrical relationships. Three crystals having different space groups with two molecules in their asymmetric units, were chosen for a prototypical investigation of an extension of the CSM method to space group pseudo symmetry. The P21 and P–1 crystal structures chosen for this study were unpublished data of my research group, and data for the Pn crystal was 52 taken from the literature. The P21, P–1, and Pn space groups of these three crystals had lower bona-fide symmetry than that of a higher symmetry P21/c supergroup.

The higher symmetry P21/c space group is the most common achiral space group for organic crystals and has three non-trival symmetry operations: inversion, 21 screw- rotation and glide-reflection. Judging from its extremely high statistical frequency (36.6%. of all the structures in the CCDB)1 the combination of these three symmetry operations leads to efficient packing for a very wide variety of organic molecules. The arrangement of the two molecules in the asymmetric units of the lower symmetry space groups was such that each crystal had a different combination of one bona-fide operation (either inversion, or 21 screw-rotation or glide-reflection) plus two different pseudo symmetry operations. A P21 chiral crystal having two molecules of opposite handedness in the asymmetric unit is statistically relatively unusual. It exhibited true

21 screw-rotation and pseudo inversion plus pseudo glide relationships. A P–1 achiral crystal having two molecules in the asymmetric unit had ideal inversion symmetry and pseudo 21 screw-rotation and pseudo glide relationships. Finally, a Pn achiral crystal showed bona fide n-glide symmetry and pseudo inversion and 21 screw-rotation. The goal of this part of my dissertation was the quantification of deviation of pseudo symmetry operations in the pseudo P21/c supergroup using special CSM methods that were to be developed in order measure symmetry operations involving translation: i.e. screw-rotation and glide-reflection.

Cl +NH

CH3 10

Nefopam HCl (10) is a non-narcotic analgesic drug and a serotonin reuptake site blocker.53 As part of the synthesis of a large series of eight- and nine-membered ring analogues of 10 Ergaz54 reported the synthesis and crystallization of a racemic mixture of the secondary ammonium salt (±)-(1RS,3SR,4RS)-1-phenyl-cis-3,4-butano- 3,4,5,6-tetrahydro-1H-2,5-benzoxazocine hydrochloride (N-desmethyl-cis-3,4-butano- nefopam HCl, 11). At the time the work was performed, the crystals of 11 were considered to be special, since there were two molecules of opposite handedness in the asymmetric unit. Very recently, this type of special phenomenon has been assigned the name ‘false conglomerate’.41

+ NH2 Cl

The x-ray crystallographic structure determination was performed by Reuben Alfredo Toscano at the National Autonomous University of Mexico (UNAM) on a preliminary data set collected at Ben-Gurion University. The two molecules exhibited pseudo-inversion and pseudo-glide relationships in a chiral environment. A special goal of this research subproject is to understand why crystallization of a racemic mixture of 2 afforded a false conglomerate of P21 chiral crystals with two molecules of opposite handedness in the asymmetric unit, rather than a racemic compound achiral crystal with only one molecule in the asymmetric unit.

In such an achiral crystal the enantiomeric pair would be related by bona-fide inversion and glide symmetry. A packing analysis and the employment of Continuous Symmetry Measures were some of the techniques that were utilized in this quest.

The second crystal structure to be studied in this subsection of the dissertation was (±)-1-phenyl-6-cyano-1,3,4,5,6,7-hexahydro-2,6-benzoxazonine (2,6-homonefopam- 6-CN, 12). It was synthesized by Ganit Levi-Ruso55 and also by Itzhak Ergaz.54 It gave racemic compound crystals belonging to the achiral triclinic P-1 space group, and had two molecules in the asymmetric unit.

The third crystal structure investigated was that of (±)-(2RS,3SR,4RS)-tert-butyl 3- hydroxyl-4-phenyl-2-(p-toluenesulfonylamino)pentanoate dichloromethane solvate, CCDB REF code RIWMEP, 13. The crystals of 13 were described by Gorbitz, Kazmaier, and Grandel.52 They are racemic compound crystals belonging to the achiral monoclinic Pn space group, and have two molecules in the asymmetric unit.

1D. Introduction for Research Subproject 3: CSM Studies on the

Orthorhombic Pn21a Crystals of Diphenhydramine HCl (14), and its

Relationship to the Bb21m Supergroup, and the Determination of the Forces Responsible for the Unusual Pseudo-Mirror Conformation of the Ammonium Cation.

O Cl

+ NH CH3

H3C 14

Diphenhydramine HCl (14) is a first generation antihistaminic drug.53 Nefopam HCl (10) is an eight-membered closed-ring analogue of 14. Diphenhydramine was also found to have a dopamine reuptake site side effect.56 The dopamine reuptake site blocking capability of diphenhydramine derivatives may be useful for treating cocaine abuse.57,58 Diphenhydramine derivatives which are selective for the dopamine transporter site have been developed for this purpose.58,57 The x-ray crystallographic structure determination of the orthorhombic Pn21a achiral crystals of 14 was made by Haartmann-Moe and Glaser at the University of Bergen, Norway.53 The molecule exhibited an unusually small 37.5° N–C–C–O synclinal-type torsion angle and pseudo mirror symmetry.53 There is also an unsual ‘open butterfly wing’ type conformation for the phenyl rings in the diphenylmethyl [(Ph)2C] moiety (see Figure 20).

Figure 20: A ball and stick representation of the (Ph)2CH– ‘open butterfly wing’ and pseudo-mirror conformation of 14.

A more usual synclinal magnitude for the N–C–C–O torsion angle would make the molecule asymmetric in an environment incapable of permitting conformational interconversion. In such a condition, one phenyl ring and one methyl group are related to their corresponding partners as diastereotopic neighbors. In other words, one would expect to see anisochronous resonances for the two methyl carbons and a maximum of twelve aromatic carbon signals in a solid-state 13C NMR cpmas spectrum. The pseudo mirror symmetry of crystalline 14 affords an extremely simplified cpmas spectrum where the single ipso carbon ( 142.82) and single methyl ( 40.08) signals are readily assigned for the pseudo enantiotopic nuclei, see Figure 21.

diphenhydraHCl Diphenhydramine Hydrochloride (as is) 8000 vacptppm, spin-rate 11,500 Hz, o1 20,000 Hz, o2 1,250 Hz, sr -348.47 7500

N-methyl 7000

6500

ipso 6000 Diphenhydramine Hydrochloride crystalline n21m

5500

5000

4500

4000 OC CN 3500

benzhydrylic-C 3000

2500

2000

1500

spinning 1000 side band 500

-500

145 140 135 130 125 120 115 110 105 100 404550556065707580859095 f1 (ppm)

13 Figure 21: Solid-state cpmas C NMR spectrum of Pn21m crystalline 14 with readily assignable single aromatic-ipso carbon ( 142.82) and single N-methyl carbon ( 40.31) isochronous pseudo enantiotopic signals.

The purpose of this subproject was to investigate the packing conditions responsible for the unusually low 37.5° N–C–C–O torsion angle. The observation of a pseudo mirror was a ‘given’. CSM calculations were to be performed to see if the same extent of distortion applied to pseudo 21 screw-rotation and pseudo b-glide-reflection which would have to be present in a pseudo Bb21a supergroup. A packing analysis will be made to ascertain if pseudo mirror conformational packing caused the ‘open butterfly wing’ arrangement, or did an ‘open butterfly wing’ necessarily constrain the molecule to adopt a pseudo mirror?

2. Research goals

2A. The Desymmetrization of Symmetrical Molecules Upon Entering a Crystal Lattice The overall aim of the first PhD research subproject was to utilize the existing classical continuous symmetry measure algorithms developed by David Avnir26.27,24,25,28 and his research group to investigate the effect of environment upon molecular structure. In the first part of the project these algorithms were used to study the small, but measurable, degree of structural distortion when solvated molecules having convex Platonic solid geometries were resident within the confines of a crystal lattice. Molecules with Platonic solid geometries were chosen, since it is interesting to find out which of the many symmetry elements would be preserved, and to what extent. Desymmetrization is the process of systematic removal of symmetry 39 elements, starting from a higher symmetry group (such as Ih, Oh and Td). The Avnir algorithms26,27,24,25,28,30 were used to study the degree of distortion to the Platonic solid geometry core which results in a desymmetrization when the molecule’s vertices were substituted with three-fold rotors such as tert.-butyl groups. It was of interest to investigate how the size of a spacer (adamantane or Platonic solid geometry skeletons) influenced the twist sense of rotors. Other structural questions were the effect of rotor size on the desymmetrization of rotor substituted Platonic solid geometry molecules. What is the effect of local site symmetry on rotor symmetry [C3 symmetrical versus asymmetric tert.-butyl rotors] and how does this effect the total CSM value of the molecule? What was the effect on the CSM component and total values when partially desymmetrized rotor substituted Platonic solid geometry crystal arrays were compared with completely asymmetric pseudopolymorphs? Many crystalline molecules have a bona-fide symmetry in the array but actually exhibit a higher pseudo symmetry geometry. In such a situation, would the total CSM values for a higher pseudo symmetry in pseudopolymorphic crystals of the same molecule reflect occupancies of different local site symmetries (i.e. positions of special versus general symmetry in the lattice)?

2B. Studies of Intermolecular Pseudo Symmetry in Crystals Having Multiple Molecules in the Asymmetric Unit An important goal of the second subproject was to use the algorithms to investigate pseudo-symmetrical relationships in crystals containing multiple molecules in the asymmetric unit. The emphasis for the latter part of the dissertation project was the quantification of the distortion from bona-fide symmetry between molecules rather than within the molecule itself. When there is no pseudosymmetry, the bona fide space group descriptor testifies to the highest symmetry that the crystalline array can express. However, since pseudosymmetry is a continuum with varying degrees of distortion from an ideal symmetry, then there should be numerous examples when an auspicious packing in the array can be aided by pseudosymmetry between the multiple molecules within an asymmetric unit. The combination of bona fide operations that the crystal has gives its space group. The combination of bona fide special positions with pseudo positions of symmetry yields an arrangement of the pseudo and special position of symmetry, which resembles the arrangement of special positions of the crystal's bona fide supergroup. One of the main goals of this thesis was to develop a way to quantify a crystal's deviation from bona fide supergroup symmetry. The distortion from a particular supergroup symmetry would result from the individual contributions of the various pseudosymmetry operations of the set, and their total would report the degree of ‘supergroup character’ for that crystal. The packing in crystals belonging to the ‘P21/c family’ of space groups [P21/x where x come from either a-, c-, or n-glides] must be auspicious since ca. 37% of all organic molecules crystallize in this monoclinic space group.1 As a prototype of the method,

CSM methods would be developed to investigate lower symmetry P21, P–1, Pc or Pn space groups with two molecules in the asymmetric unit. These particular lower symmetry space groups were chosen since they were all related to the same ubiquitous P21/c family of space groups.

2C. Studies on Crystals With Molecules Occupying Pseudo Positions of Symmetry.

New methods were also to be used to investigate crystals with both intermolecular and intramolecular pseudo symmetry relationships for the molecules constituting the array. Could quantifying the distortion from an ideal symmetry assist in understanding the constraints which could result in the formation of an unusual crystalline-state conformation? To achieve this goal it was decided to investigate orthorhombic crystals of Pn21a diphenhydramine hydrochloride in which the ammonium cation exhibited an unusual low synclinal (pseudo synperiplanar) 37.5° N–C–C–O torsion angle. The crystals could be related to a Bb21m supergroup as a direct result of this pseudo synperiplanar torsion angle. Was the ‘open butterfly wing’ conformation of the Ph2CH- moiety (as opposed to more usual orthogonal and helical twist) influencing the lattice packing or vice-versa?

2D. Development of New Methodologies to Measure the Deviation from Screw- Rotation and Glide-reflection Space Group Symmetries In order to meet the above stated goals of using a method to study the pseudo symmetry between molecules in a crystal lattice, it was clear that novel tools would have to be developed. The existing Avnir methodologies were designed to investigate pseudo symmetry within an object, i.e. they were produced for ‘point group’ symmetry operations. Point group symmetry operations leave at least one point invariant in space. The Avnir CSM algorithms were not available to handle symmetry operations found only in space groups, i.e. translation, screw-rotation (rotation-translation), and glide-reflection (reflection-translation). The classical Avnir algorithms available for point group symmetry operations (rotation, reflection, inversion, and rotatory-reflection) would also have to be modified for use in space groups. The reason for the modification is that they incorporate a size normalization function whose purpose is to measure a particular symmetry in objects irregardless of their scale size.

In this manner the Avnir algorithm would provide the identical CSM value for two objects whose only difference was their relative scale size. This size normalization function would be incorrect if the distances between molecules varied either within the same unit cell or in unit cells of different crystals. Since the distances between molecules in different crystals will always be non-identical by symmetry argument, it is clear that the size normalization function would have to be removed.

In addition, a technique for measuring pseudo symmetry in space groups would have to include the relocation of a pseudo symmetry element defined by a statistical average of points vis-à-vis the special position of the bona fide element in the crystal lattice. All the [xmean,ymean,zmean] points between bona fide symmetry related atom pairs coincide with special positions in the lattice which are either a single unique point of inversion, or reside on a unique axis of rotation, or define a unique plane of reflection. However, the [xmean,ymean,zmean] points between pseudo symmetry related atom pairs all define general positions of symmetry in the lattice.

As such, they must be treated as either statistical averages of pair-wise points of inversion (a so called ‘best point of inversion’ which now has associated with it an estimated standard of deviation), or they statistically define a ‘best axis of rotation' or a ‘best plane of reflection’. The crystal's space group lacks certain symmetry element, which exist in the bona fide supergroup. However, there may be some degree of pseudosymmetry with regards to these symmetries, and the degree of deviation from ideal symmetry can be measured. In addition to that, the relocation of the best point, axis, or plane compared to the ideal location in the bona fide supergroup must also be taken into account. Only by these new yet-to-be-developed methods could we expect to have a quantitative measure of the difference between bona-fide symmetry and pseudo symmetry in those interesting lower symmetry crystals which mimic the ideal supergroup arrangement to a certain extent.

3. Experimental

3A. Avnir CSM Programs for Point Group Pseudo Symmetry Operations The CSM programs which were used in order to calculate the discrepancy from bona fide inversion S(Ci), reflection S(Cs), rotation S(Cn), and rotatory-reflection S(Sn), were provided through the generosity of Prof. David Avnir, Department of Organic Chemistry, Hebrew University of Jerusalem.29

3B. Calculation of Centroids Within a Molecule A centroid of a set of atoms is the center of mass of this set of atoms, i.e., its coordinates are the average coordinates of all the atoms in the set. In this project, the coordinates used were fractional coordinates from the CIF (crystal information file) file. For example, in order to calculate the x,y,z coordinates of a centroid in an aromatic ring, the respective x,y, and z coordinates of the six carbon atoms are individually averaged.

3C. CSD Searches Searches of the Cambridge Structural Database (also refered to as the Cambridge Crystallographic Database or CCDB) were conducted in order to find crystal structures with molecules that are edge-bonded Platonic polyhedra: tetrahedranes, tetrahedral cubanes, cubanes, dodecahedranes and icosahedranes.59

3D. Geometry Optimizations

Geometry optimizations were calculated the GaussianW0360 program using the B3LYP/6-31g* density functional theory (DFT) method. The GaussView 4.161 program was used to prepare an input file in which the molecule was constrained to exhibit a particular point group symmetry.

3E. Graphics Programs The Chem3D and ChemDraw versions 562,63 and 1264 programs were used for making the graphical illustrations of the molecules used in this dissertation. Coordinates from three dimensional models (arising from either x-ray crystallography or computer generated molecular modeling) were used as input for the Chem3D program. Two- dimensional projections (iconic drawings) of the actual 3D structure were produced using the ‘copy as a ChemDraw structure’ option of Chem3D. Iconic drawings were then graphically edited in ChemDraw with ‘spatial visual clues’ (e.g. bold-face for close bonds, grey colors for very distant bonds, larger font size for atom descriptors of close atoms, smaller font size for those of distant atoms, etc.)

3F. Mathematical Calculation Methods Used for Space Group Pseudo Symmetry Operations

3F.1 Calculation of Centroids Between Pseudo Symmetry Related Atoms in Molecules A and B of the Asymmetric Unit.

The centroid between a pair of pseudo symmetry related atoms in molecules A and B is defined by a point whose fractional coordinates are:

,, zyx

where:

 BxAx )()( x  2

 ByAy )()( y  2

 BzAz )()( z  2

The set of centroids for the n-pairs of pseudo symmetry related atoms generate an approximate point, or axis, or plane when the pseudo symmetry relationship is either that of inversion, screw-rotation, or glide-reflection, respectively.

3F.2 Calculation of the xmean, ymean, and zmean Coordinates

The xmean fractional coordinate for n pairs of pseudosymmetry related atoms in molecules A and B is the mean value of the set of coordinates for all pairs:

n  xi x  i1 mean n

The ymean and zmean coordinates are calculated in a similar fashion.

3F.3 The Relocation of a ‘Best Pseudo Inversion Point’ from the Location of the Ideal Inversion Point

The [xmean,ymean,zmean] fractional coordinates of the statistical average ‘best pseudo inversion point’ is calculated from all n-centroids in the set of pseudo inversion related atoms. The distance between the best pseudo inversion point and the ideal inversion point is found. This is the RmS(relocation-Ci) tool, which quantifies the Relocation of a Pseudo Inversion ‘Best Point’ Relative to that of the Ideal Inversion Center.

The distance between the best approximate inversion point and the ideal inversion point is found (see section 3F.3). This distance will be referred to as the rmS(relocation-Ci), since this is the difference between the ideal special position and the least squares approximation for the 'pseudo special position'.

3F.4 Relocation of the ‘Best Pseudo Screw-Rotation Axis’, Compared to the Location of the Ideal Screw-Rotation Axis (a Numerical Example is Given) The centroids of all screw-rotation related pairs of atoms, and a list of dummy atoms which show the origin and directions of a, b and c are converted to Cartesian coordinates, together with the coordinates of point P, which is on the ideal axis according to the following transformation equation65:

aˆ  ba  c  )cos()cos( xˆ     bˆ   )sin(0 cb   )cos()cos()cos( yˆ     )sin(   cˆ  00 c v zˆ      )sin(  

Where: v 2 2 2   )cos()cos()cos(2)(cos)(cos)(cos1

If the ideal screw-rotation axis is in the c direction, the c axis is later set to coincide with the z axis.

The Cartesian coordinates of point P are written down. The Cartesian coordinates of the centroids are saved in a text file and loaded as a matrix to matlab. They are used as input for a least squares Matlab modual66 which finds the best line and the centroid of the best line. It is possible to find the angle between the best line (L = Q12, which is the directional cosines vector) and the ideal line (L0, which is a unit vector in the z direction). The scalar product of the directional cosines of the ideal axis with the directional cosines of the best line, will give us the cosine of the angle between the two lines:

 LL cos  0 LL 0

In this case, cosθ = 0.9994

A measure of how far we are from a situation in which the best axis is perfectly parallel to c is:

1 – |cosθ|, which is 0.0006 in this case. If this value were zero, it would mean that the axes are perfectly parallel. If it were 1, then the axes would be orthogonal. The obtained value shows that the best axis is very close to being ideally parallel to the z axis. From now on, we can assume that the best axis is parallel to the z axis, and this is a justification for the validity of this assumption. The values of 1 - |cosθ| were very low (lower than 0.002) in all cases examined in this thesis, and therefore, the best axis was assumed to be parallel to the ideal axis.

If the Cartesian coordinates of P are: P = [-0.0220,-0.0280,2.0260]

For a crystal with the following unit cell parameters: a = 8.9104(10) Å, b = 10.4110(12) Å, c = 16.199(2) Å, α = 90.640(4)°, β = 91.266(5)°, γ = 99.057(5)°

The distance between this point and the best line will be the relocation. Q1 and Q2 are points on the best line. We will find the distance between the point P and the vector connecting Q1 with Q2.

Q1 = [0.0080,0.0033,-0.1319]

This is the centroid of the best axis, obtained from the Matlab modual67

The Matlab modual67 gives the direction cosine vector:

Q21  Q2 – Q1

Which is equal to: [0.0314, –0.0127, 0.9994] in this numerical example.

PQ1  P – Q1

Which is equal to: [–0.0300, –0.0313, 2.1579] in this example.

 PQQ D  21 21 Q21 And in this case, D = 0.0978 Å.

By taking the average atomic thermal factor as the experimental error, we obtain a relocation for the screw-rotation of:

D = 0.10(7) Å

This distance will be called the rmS(relocation-21), since it is the distance between the ideal special position in the bona fide supergroup and the least squares pseudosymmetry statistical ‘best’ position.

3F.5 Relocation of the ‘Best Pseudo Glide-reflection Plane’ Compared to the Location of the Ideal Glide Plane (a Numerical Example is Given) The point P is on the ideal glide plane: P: [0, 0, ¼]

The point on the ideal plane is converted to Cartesian coordinates according to the following transformation equation:65

Where: v 2 2 2   )cos()cos()cos(2)(cos)(cos)(cos1 together with the coordinates of the centroids of the glide related pairs of atoms, and a list of dummy atoms which show the origin and directions of a, b and c. If the ideal axis which is perpendicular to the plane is in the c direction, the c axis is later set to be the z axis.

The coordinates of the point P in Cartesian coordinates are written down:

P: [–4.034, –1.312, 2.025]

For a crystal with the following unit cell parameters: a = 8.9104(10) Å, b = 10.4110(12) Å, c = 16.199(2) Å, α = 90.640(4)°, β = 91.266(5)°, γ = 99.057(5)°.

The cartesian coordinates of the centroids are saved in a text file and loaded as a matrix with matlab. They are used as input for a least squares Matlab modual66 which finds the directional cosines of the normal to the best plane and the centroid of the best plane. It is possible to find the angle between the normal of the best plane

(N) and the normal of the ideal glide plane (N0). The normal of the ideal glide plane is parallel to the z axis. The cross product of the directional cosines of the normal to the best plane with the directional cosines of the normal of the ideal plane, will give us the cosine of the angle between the two lines:  NN cos  0 NN 0 A measure of how far away we are from a situation in which the best glide plane is perfectly parallel with the ideal glide plane is:

1 – |cosθ|

The directional cosines of the normal to the best plane are: [0.0383, 0.0100, 0.9992]

And the cross product of the directional cosines of the normal to the best plane with the normal to the ideal glide plane is:66 cosθ = 0.9992

The following is a measure which can assume values from 0 to 1. A value of 0 means that the planes are perfectly parallel to each other, and a value of 1 means that they are orthogonal to each other. In this case the value is:

1 – |cosθ| = 0.0008 From now on, the best plane will be treated as parallel to the ideal glide plane. Since the values obtained for all crystals in this thesis were smaller than 0.002, the best plane treated as being parallel to the ideal plane.

The formula of the best plane is: 0.0383x + 0.0100y + 0.9992z – 1.8412 = 0

In order to find the distance between the point P[x1,y1,z1] (which is on the ideal glide plane) to the best plane, the following formula was used:

And the relocation from the ideal glide special position is: D = 0.01(1) Å

The distance between the ideal and best pseudo glide planes will be called the rmS(relocation-glide), since it is the distance between the ideal special position in the bona fide supergroup and the least squares pseudosymmetry statistical ‘best’ position.

3F.6 The RmS(G) Pseudosymmetry Tool where G = Ci, , Cn, and Sn. CSM calculations were performed at the website of the Avnir group at the Hebrew University of Jerusalem (http://www.csm.huji.ac.il/new/?cmd=symmetry). Input file formats and a tutorial are provided at the site. These calculations include the deviation from ideal G point group symmetries of G = i, , Cn, and Sn. The fractional coordinates of an ensemble consisting of G pseudosymmetry related n-pairs of atoms within two molecules/or fragments were converted to Cartesian space and then arranged in an MDL (*.mol) input file. Another input file specifies the permutation for the particular Avnir CSM program. The coordinates of the theoretical ‘nearest ideal geometrical structure’ having a bona fide G symmetry relationship were calculated from the input coordinates using the appropriate Avnir CSM algorithm.29 These coordinates and the size normalized CSM S(G) value appear in the ‘Results’29 output file and may be saved. Then, atoms in the ‘nearest ideal G symmetry ensemble of geometrical structures’ were superimposed upon corresponding atoms in the actual ensemble of a pseudosymmetry related pair of molecules/or fragments to give the RMS interatomic distance. This is an average deviation from ideal G symmetry which is defined as a crystallographic G pseudosymmetry measure called ‘the non-size normalized rmS(G) tool’.

3F.7 Location of Statistically Determined Pseudo positions of symmetry. The midpoints (i.e. average coordinates)  ,, zyx  between corresponding n-pairs of atoms within the approximately enantiomeric fused-ring skeletons of the (+)-A and (–)-B molecules generate a cluster of points (or an approximately linear string of points- not applicable for crystalline 1) or an almost planar array of points, if the fragments are related by pseudo-inversion (or pseudo-21 screw displacement), or pseudo-glide reflection, respectively. The xmean value for all the midpoints between n-pairs of pseudosymmetry related atoms in molecules A and B is:

n  xi x  i1 mean n

The point defined by the [xmean,ymean,zmean] fractional coordinates is the statistical ‘best point’, and its estimated standard deviation (esd) is the precision of this value. A statistically calculated ‘best line’ or ‘best plane’ can be calculated from the mid-point ,, zyx coordinates between corresponding n-pairs of atoms in pseudo-screw and pseudo-glide arrays of points using moduals developed for MatLab68,66,67.

3F.8 The RmS(translation) Pseudosymmetry Tool. The xi-translation or zi- translation of an atom in molecule (–)-B into one in molecule (+)-A can be calculated from the difference in their respective xi or zi fractional coordinates, e.g. Δxi = xi(A) – xi(B). The difference between an actual x-translation (Δxi) minus the fractional number ½ value for a bona-fide glide reflection (or 21-screw rotation) is calculated for each of the n-pairs of atoms in the ensemble. The RMS value for the set of n- differences is the deviation from an ideal x-translation (in fractional coordinates), rmS(x-translation)coordinate:

n 2  xi  )5.0( (xrmS  ntranslatio )  i1 coordinate n

The rmS(x-translation)coordinate value was multiplied by the a-axis cell length to give the relevant crystallographic translation pseudosymmetry measure, rmS(x-translation) now in Å.

3F.9 The RmS(glide) and RmS(screw) Pseudosymmetry Tools. The xcoordinate- translation (or ycoordinate-translation, not applicable for 1) or zcoordinate-translation is the mean value of the Δxi (or Δyi) or Δzi fractional coordinate distances between the n- pairs of atoms involved in a translation of molecule (–)-B to molecule (+)-A. For example:

n xi x  ntranslatio  i1  coordinate n

If the directional cosine values of the , , and  angles which the ‘best line’67 or the ‘normal to the best plane’66 make with the positive directions of the coordinate axes show that these lines are parallel to a cell face, then one is justified in treating the pseudo screw or pseudo-glide reflection as separate translation and rotation/or reflection components, respectively. If statistically insufficient midpoints are present in one unit cell to calculate a ‘best line’ or ‘best plane’ that is parallel to a cell face, then midpoints from adjacent unit cells should be added to the set. For a pseudo n- glide in crystalline 1, a ‘dummy’ molecule (–)-B was created from molecule (–)-B by transforming its atomic coordinates by the respective xmean and zmean values so that it was opposite to molecule (+)-A. Now, from an input file of the Cartesian coordinates of the ensemble of ‘dummy’-B and actual (+)-A molecules, the coordinates of the closest theoretical geometrical structure with bona fide reflection symmetry was calculated with the Avnir Continuous Symmetry Measure S(Cs) program.

The non-distance dependent deviation from the true reflection symmetry in the

‘dummy’ (–)-B plus actual (+)-A ensemble, [rmS(Cs)glide], was calculated in a similar manner to rmS(Cs) described above. If a pseudo-21 screw relationship exists within the ensemble of ‘dummy’ and real molecules, then rmS(C2)screw is calculated by analogy to rmS(C2) also described above. The rmS(glide) or rmS(screw) is the summation of their two components: rmS(Cs)glide + rmS(translation)glide or the rmS(C2)screw + rmS(translation)screw, respectively, and their values are in Å.

3F.10 Calculation of the New RmS(P21/c) Tool

RmS(P21/c) = rmS(P21/c-symmetry) + rmS(P21/c-relocation),

where rmS(P21/c-symmetry) = rmS(inversion) + rmS(21) + rmS(glide), and rmS(P21/c-relocation) = rmS(relocation-inversion) + rmS(relocation-screw) + rmS(relocation-glide).

3F.11 The New RmsFidelity(P21/x) Tool – an Additional Method Which Was Developed in Order to Measure Deviations from Symmetry in Crystals

This is done by first reindexing the fractional coordinate data from each crystal to generate a new unit cell whose origin is crystallographically in accord with that acceptable for the appropriate P21/x (where x = c,a,or n) supergroup. One of the two molecules in the asymmetric unit is taken as the reference molecule [e.g. molecule- A]. A ‘dummy-A’ molecule is then generated using one of the symmetry operations listed for the P21/x (where x = c,a,or n) supergroup. If the parent molecule-A was related to molecule B by a bona-fide symmetry operation of the first kind (or of the second kind if the handedness of ‘dummy-A’ is inverted), then corresponding atoms in new ‘dummy-A’ would superimpose ideally on molecule-B with an RMS = integer-0, and there would have been only one molecule in the asymmetric unit.

However, by definition, both A and B reside in the asymmetric unit so they must be different, i.e. they can only be related by pseudo symmetry, i.e., the RMS of superimposition ≠ integer-0. The rms difference between corresponding atoms in molecules ‘dummy A’ and B is a new index of fidelity (rmsFidelity) of their pseudo symmetry which includes components from both the symmetry distortion as well as from the relocation. The smaller the rmsFidelity(G) value, the lower the degree of symmetry distortion and/or relocation which influences the RMS of superimposition. In other words, the smaller rmsFidelity(G) is, the higher the degree of fidelity between ‘dummy-A’ and B molecules.

4. Results and Discussion

4A. The Role of the Environment Upon Molecular Structure: CSM Studies on the Crystalline State Desymmetrization of High Symmetry Platonic Solid Geometry Molecules. Solution-state 1H and 13C-NMR spectroscopy are very useful techniques for the determination of the symmetry of solvated species, as evidenced by degenerate chemical shift signals from homotopic and enantiotopic nuclei. Since there is free Brownian motion in the liquid state, a time averaged species is observed in solution state.

4A.1 CSM Studies on Dodecahedrane, a Platonic Solids Geometry Molecule Incapable of Expressing its Five-Fold Axis of Symmetry in the Solid-State. 1 Dodecahedrane (4) (C20H20), due to its solution-state Ih symmetry, shows one H signal (3.38 ppm) and one 13C signal (66.93 ppm) in the respective NMR spectra. This means that there are 19 degenerate frequencies carbons and 19 degenerate frequencies for hydrogens for the solvated molecule.19 However, when such a molecule crystallizes, it is expected to lose its five-fold axes (C5) of symmetry, since a C5 rotation operation is not allowed in a triply periodic crystal lattice (an interesting exception is in quasi-periodic crystals, in which C5 symmetry is allowed, and therefore, icosahedral symmetry is made possible69). One cannot tile a floor with regular pentagonal tiles since they do not completely tessellate (i.e. you will always see some of the floor underneath). The CSM S(C5) values for crystalline dodecahedrane are therefore obliged to be different from integer zero, and according to X-ray crystallographic structure analysis70 the symmetry of the molecule indeed changed from Ih symmetry having five-fold axes to Th symmetry (a subgroup of Ih) without five-fold axes of rotation. Figure 22 shows a crystal-state desymmetrized molecule illustrated with a characteristic color given to each of the two sets of symmetry equivalent atoms: twelve atoms (colored green), occupy special positions of σh symmetry, and eight atoms (colored white) occupy special positions of three- fold rotational symmetry.

The latent octahedral symmetry of dodecahedrane was described by Alvarez20 and can be seen in Figure 22, since the eight white carbon atoms form an ideal octahedron. The symmetry operations for dodecahedrane which were preserved upon entering the crystal lattice are those of the Th point group (see the four ideal C3 and S6 operations in Figure 23). Figure 24 illustrates the σh symmetry which was preserved. Additional symmetry operations which were preserved were C2 (Figure 25) and inversion (Figure 26).

Figure 22: The latent octahedral symmetry in Th symmetry crystalline 4 is shown by the eight white symmetry color-coded set of atoms.

Figure 23: The four ideal C3 axes and coincidental four S6 axes in Th symmetry crystalline 4.

Figure 24: A mirror plane in Th symmetry crystalline 4.

Figure 25: C2 axes are preserved in Th symmetry crystalline 4.

Figure 26: The center of inversion was preserved in Th symmetry crystalline 4.

The degree of geometry distortion from ideal Ih solution-state symmetry to crystal- state Th symmetry is small since the molecule is rigid, and thus symmetry operations which were present in the Ih point group but not in the Th point group now become, pseudo symmetry operations.

Figure 27 shows that the green colored atoms do not sit on special positions of C3 symmetry. Pseudo C3 axes pass through the green atoms, and the the deviation from bona-fide C3 symmetry is exemplified by a S(C3) CSM small but significantly non- integer value of 0.00019(2). It is significant since this small degree of molecular distortion from true C3 symmetry is what enables the crystallographer to observe two carbon atoms and two protons in the asymmetric unit rather than only one of each type. Since we expect the dodecahedrane rigid molecular framework to undergo only very small changes in geometry upon entering the crystal lattice, we can say that the 0.00019(2) value is trivial since it is expected or commonplace when the five-fold axes become forbidden in the lattice.

Figure 27: A pseudo C3 axis in Th symmetry crystalline 4.

Similarly, there are pseudo C2 axes (shown in Figure 28) and their S(C2) values are also small in magnitude [0.00029(3)]. Likewise, the pseudo σ planes also show small

S(Cs) = 0.00029(3) deviations from ideal Cs symmetry (shown in Figure 29).

Figure 28: A pseudo C2 axis in Th symmetry crystalline 4.

Figure 29: A pseudo mirror plane in Th symmetry crystalline 4.

Finally, the crystal lattice’s incompatibility with the dodecahedrane’s C5 axes (see

Figure 30) is illustrated by a non-integer zero small value of S(C5) = 0.00039(3). In all the cases shown above these minor skeletal distortions for rigid dodecahedrane in the crystal lattice are significant but trivial since we expect them to happen to only a small extent. Clearly, we should not expect dodecahedrane to undergo distortion to a great extent when it enters the crystal lattice.

Figure 30: A pseudo C5 axis in Th symmetry crystalline 4.

4A.2 Discussion of the CSM Studies of Crystalline Dodecahedrane

In spite of the fact that the rigid dodecahedrane molecule cannot occupy a special position of five-fold rotational symmetry in a crystal (since this is not allowed by symmetry rules for a triply periodic crystal lattice), we are not surprised to observe that the CSM values for distortion are very small indeed. These very small CSM values are significant since they show that ideal Ih symmetry has indeed been lost when the molecule enters the crystal lattice. If they were not significant, then the crystallographer would not have been able to observe two different carbon atoms and two different protons in accord with Th symmetry. However, as noted before, they are ‘trivial’ since they are expected to occur due to the rules of symmetry but the distortion cannot be large since the molecular skeleton is rigid. Crystalline 4 exhibits a bona-fide Th symmetry conformation when resident within the confines of the crystal lattice, but its pseudo symmetry is actually higher (pseudo Ih symmetry) since all of the CSM values for the suboperations of the Ih point group are low.

Four multi N-methylated crystal-state derivatives of adamantane-1,3-diamine 2HCl (15) were analyzed in terms of their CSM values: N,N,N'N'-tetramethyl-1,3- diaminoadamantane dihydrochloride monohydrate (16), ,N,N,N,N',N',N'-hexamethyl- 1,3-diaminoadamantane diodide monohydrate (17), N,N,N,N',N',N'-hexamethyl-1,3- diaminoadamantane diodide monohydrate within a curcurbit[8]uril clathrate complex (18), CCDB REF code SAXKEI), and 5,7,N,N,N,N’,N’,N’-Octamethyl-1,3- diaminoadamantane dibromide dihydrate (19).71,72 The adamantane bis– quaternaryammonium dication cucurbit[8]uril clathrate clathrate (18) was prepared in the literature in order to mimic the biological process of molecular recognition.71 The adamantane bis–quaternaryammonium dication was found to be complementary to the interior of the cyclic host in both size and shape.71 All the diammonium salts occupied general positions of symmetry within the crystal lattice.72 The +2 N,N,N,N’,N’,N’-hexamethyl-adamantane-1,3-diamonium dication (C16H32N2) (17) 1 13 has C2v symmetry in solution state (see Figure 31) according to H and C-NMR spectroscopy.71

Figure 31 shows six color-coded sets of symmetry equivalent atoms for the C2v symmetry adamantane-1,3-diammonium dication based upon degeneracies found in 71 the solution-state NMR spectra. From group theory, a C2v symmetry molecule has two -planes intersecting at 90° and as a result it also has a C2 axis. As a result of these symmetry elements the adamantane core (skeleton) is composed of five differently colored sets of symmetry equivalent atoms. Figure 31 shows that an additional set is composed of the N-methyl carbons, which undergo fast topomerization about the C—N bond in the solution state, and therefore give rise to a single weighted time-averaged chemical shift.71

Figure 31: Adamantane-1,3-diammonium dication color coded according to liquid state C2v symmetry equivalence.

The 5,7,N,N,N,N’,N’,N’-octamethyl-1,3-diaminoadamantane dibromide dihydrate had 71 72 two molecules (19A,B) in the asymmetric unit. , The two C2v reflection planes were differentiated by assigning a σ|| descriptor to the N(1)–C(1)–C(2)–C(3)–N(2) 72 plane and a σ┴ descriptor to the perpendicular plane. The CSM values of the adamantane-1,3-diammonium salts are listed in Table 3.72 The three non-complexed bisquaternary ammonium dications (17,19A,B) in the limited series studied exhibited the lowest deviation from bona fide symmetry for S(σ┴) while that for S(C2) was 2- 7.5 times higher. The clathrate crystal comprising a quaternary ammonium dication within a curcurbit[8]uril ring had two complexes in the asymmetric unit (18A,B).

The lowest deviation from bona fide symmetry for one of the two complexed bisquaternary ammonium dications (18B) was also S(σ┴) while that for S(C2) was 12 times higher. In this molecule the C(2)–C(1)–N(1)–Cmethyl and C(2)–C(3)–N(2)–

Cmethyl torsion angles are respectively –165.9° and +166.2°. The two pseudo + enantiotopic – N(Me)3 rotors clearly exhibit opposite twist sense in pseudo Cs (σ┴) symmetry molecule 18B (see Figure 32B). For the second complexed bisquaternary ammonium dication (18A) the lowest deviation from bona fide symmetry was S(C2) while that for S(σ┴) was 7.5 times higher. In this molecule the pseudo homotopic

C(2)–C(1)–N(1)–Cmethyl and C(2)–C(3)–N(2)–Cmethyl torsion angles are respectively

–166.6° and –169.0°, and both rotors obviously exhibit the same twist sense. It should be noted that one of the –N(Me)3 rotors is within the host cage while the other one is more exposed in the two complexes. However, the twist sense for the more internal rotor differs in each of the two complexes, and there does not seem to be a difference in the degree of twist between the internal versus the exposed partner.



Figure 32A: Pseudo C2 symmetric molecule 18A.



Figure 32B: Pseudo mirror symmetric molecule 20B.

Table 3: S(C2), S(σ┴), and S(σ||) values for the full molecule or for the adamantane C10 skeletal core or for the external fragment composed of the two trimethylammonium external rotors

Full Molecules: S(C2) S(σ┴) S(σ||) Bisprimary diammonium dication (15) < 0.0003 < 0.0005 < 0.0004 Bistertiary diammonium dication (16) 0.0727(3) 0.0023(2) 0.0749(3) Uncomplexed bisquaternary 0.046(5) 0.022(4) 0.0665(5) diammonuym dication full molecule (17) Complexed bisquaternary dication (18A) 0.048(4) 0.36(1) 0.37(2) Complexed bisquaternary dication (18B) 0.39(2) 0.032(4) 0.37(2) Uncomplexed bisquaternary dication 5,7- 0.091(3) 0.0121(1) 0.081(3) diMethyl (19A) Uncomplexed bisquaternary dication 5,7- 0.148(3) 0.030(2) 0.121(3) diMethyl (19A) External Fragments: Bisprimary diammonium dication (15) 0.0041(2) 0.0472(1) 0.0169(3) Bistertiary diammonium dication (16) 0.0343(3) 0.0039(1) 0.038(1) Uncomplexed bisquaternary 0.0288(2) 0.0335(2) 0.0606(2) diammonium dication (17) Complexed bisquaternary dication (18A) 0.023(1) 0.545(1) 0.546(1) Complexed bisquaternary dication (18B) 0.14(1) 0.020(4) 0.14(2) Uncomplexed bisquaternary 0.0087(1) 0.152(4) 0.1600(4) dication 5,7-diMethyl (19A) Uncomplexed bisquaternary dication 5,7- 0.148(3) 0.030(2) 0.121(3) diMethyl (19B) Core Fragments: Bisprimary diammonium dication (15) 0.0004(1) 0.0012(1) 0.0008(1) Bistertiary diammonium dication (16) 0.0003(1) < 0.0001 0.0003(1) Uncomplexed bisquaternary diammonium 0.0039(1) 0.0040(1) 0.0050(1) dication (17) Complexed bisquaternary dication (18A) 0.066(1) 0.0389(3) 0.0602(4) Complexed bisquaternary dication (18B) 0.08(1) 0.040(5) 0.06(1) Uncomplexed bisquternary dication 5,7- 0.0008(1) 0.0014(1) 0.0014(1) diMethyl (19A) Uncomplexed bisquternary dication 5,7- 0.0013(1) 0.0018(1) 0.0015(1) diMethyl (19B)

All of the adamantane-1,3-diammonium salts occupied general positions of symmetry in the crystal lattice and thus their solution-state C2V symmetry was desymmetrized down to C1 point group symmetry. Bottom line: while these salts exhibited bona-fide asymmetric structures, their solid-state pseudo symmetries were actually higher

(either pseudo Cs or pseudo C2 symmetry).

4B.1 Discussion of the CSM Studies of Three-Fold Rotors Affixed to an Adamantane Skeleton

The relatively large distance between the two rotors is probably behind the findings that about half of the crystalline molecules 15-19 in this limited series under investigation had pseudo C2 symmetry (markedly lower S(C2) & higher S(σ┴), while the other half showed pseudo Cs(σ┴) symmetry (higher S(C2) & markedly lower

S(σ┴). If one –N(Me)3 rotor has a particular twist sense, then the distant second rotor has an equal probability to twist either with the same or with the opposite sense due to the relatively far inter-rotor relationship. From this one can conclude that the two –

N(Me)3 rotors are isolated from each other, i.e. they do not interact through space. Perfect staggering about sp3—sp3 single bonds is a constraint in the crystal state. Relaxation of this constraint affords a twist sense to the three-fold rotor. Therefore, in such a limited statistical sample, it is not a surprise that none of the diammonium cations exhibited a pseudo Cs(σ||) structure went resident at a general position of symmetry within the lattice since this would have obligated more ideal C(2)–C(1)–

N(1)–Cmethyl and C(2)–C(3)–N(2) –Cmethyl torsion angles of about 180° each which is a double constraint since two angles are involved.

Some additional comments can be made about the data in Table 3. In general, all of the CSM indices were the lowest for the bisprimary diammonium dication 15 compared to all other (larger) dications in the series studied. The relatively low degree of discrepancy (distortion) from ideal C2v symmetry for the small dication 15 shows that loss of symmetry constraints upon occupation of a general position has not changed the overall structure of the small rigid molecule (as expected). Dication 15 is the smallest of the series since it does not have three-fold rotors (the protons on the ammonium nitrogen are usually put in by the crystallographer using ideal geometrical factors). All the other dications except for the bistertiary dication 16 have larger –

NMe3 rotors ligated to the adamantane skeleton, and each rotor can act independently of the other in terms of the degree of twist. Another point is that CSM indices for the ten carbon adamantane skeletal core fragment were always lower than those for the external two –NMe3 rotors. This is also readily explained by the rigidity of the tricyclic skeletal core versus the much greater flexibility for rotations of the three-fold rotors about the C—N+ bonds. From these observations one can conclude that occupancy of a general position for the adamantane-1,3-bisquaternary diammonium dications frees the unit from symmetry constraints regarding the disposition of the rotors to the skeletal center as well as the relative relationship between the rotors themselves.

4C. CSM Studies on the Distortion of Crystalline Platonic Solid Geometry Molecules with Three-Fold Rotors.

4C.1 CSM Studies on the Distortion of Crystalline Molecules with Distant Three-Fold Rotors on a Triakis-Tetrahedron Geometry Skeleton.

4C.1a 2,4,6,8 – Tetra - tert. -Butyl - 1,3,5,7 – Tetraphospha - 2,4,6,8-Tetrasilapentacyclooctane (20),

2,4,6,8-Tetra-tert.-butyl-1,3,5,7-tetraphospha-2,4,6,8-tetrasilapentacyclooctane (20),

C16H36P4Si4, has alternating P and tert-butyl substituted-Si atoms. In the literature (CCDB REF code SENLAY) the triakis-tetrahedron cubane-like geometry molecules 73 are reported to crystallize in an achiral P43n space group crystal. The unit cell contains three different molecules in a ratio of 2:6 respectively: an S4 (achiral 20A) diastereomer with an achiral meso-molecular conformation and a dl-pair of enantiomers (chiral 20B & 20B-bar), each with an enantiomeric T molecular conformation.73 Since the space group is achiral and one of the molecules in the asymmetric unit exhibits a chiral conformation (20B) while that of the other is achiral (20A), we expect that the (20B-bar) enantiomer must also present in this crystal.

The alternating substitution pattern of the skeleton lowers the cubane symmetry from

Oh to Td, which is the highest possible symmetry which such a tetrasubstituted molecule can have. Each of the two solid-state molecules has point group symmetries which are subgroups of Td. Had the tert.-butyl groups been placed adjacent to each other, then molecular mirror planes would result in sterically unfavorable 'bump-to-bump' arrangements, and one would have expected the molecule to lose its mirror symmetry. Since the bulky substituents are present only on every other vertex, then a Td-like arrangement could still happen. Figures 33A,B show the torsion angle which deviates from 180° marked in yellow. The arrows in

Figure 33A show two pairs of opposite twist sense antiperiplanar P–Si–Cquat–Cmethyl torsion angles for S4 symmetric crystalline molecule 20A although one tert.-butyl group is hidden from view in the illustration. All four tert.-butyl groups in 20A are asymmetric. Their methyl carbons are diastereotopic [not symmetry related but with the same constitution (connectivity)] when resident in the crystal lattice, and therefore, each of the three methyl carbon atoms bears a different color. There are three different antiperiplanar values for the P–Si–Cquat–Cmethyl torsion angles within a tert.-butyl group: 174.89°, 177.00°, and 175.70° (average twist angle +4(1)°). Within crystalline 20A there is a second tert.-butyl group that is homotopic [related by C2 rotation, a symmetry operation of the First Kind] and its three different antiperiplanar values for the P–Si–Cquat–Cmethyl torsion angles are identical to those given above.

There are two other tert.-butyl groups that are enantiotopic [related S4 rotary- reflection, a symmetry operation of the Second Kind]. The three antiperiplanar values for the P–Si–Cquat–Cmethyl torsion angles have the same magnitude but opposite signs to those noted above, i.e. –174.89°, –177.00°, and –175.70° (average twist angle –4(1)°).

All four tert.-butyl groups in T symmetry crystalline 20B have three-fold rotational symmetry. Therefore, all four tert.-butyl groups (and all three methyl carbons in each group) represent two sets of homotopic symmetry equivalent subunits. Therefore, the methyl carbon atoms have been assigned the same color and there are three degenerate antiperiplanar P–Si–Cquat–Cmethyl torsion angles: –173.23°, –173.23°, and –173.23° (twist angle –6.77°). The arrows in Figure 33B show that the antiperiplanar P–Si–Cquat–Cmethyl torsion angles in all four tert.-butyl groups have the same twist sense for the T symmetric molecule 20B although one tert.-butyl group is hidden from view in the illustration. In spite of the fact that the twist has the same tropicity (direction) it is highly unlikely that the twist sense is correlated since the distance between adjacent three-fold rotors is too large for intramolecular steric interactions.

Figure 33A: The color coded S4 symmetric achiral molecule 20A has two pairs of opposite twist sense yellow colored antiperiplanar P–Si–Cquat–Cmethyl torsion angles although one tert.-butyl group is hidden from view in the illustration.

Figure 33B: The color coded T symmetric chiral molecule 20B has four yellow antiperiplanar P–Si–Cquat–Cmethyl torsion angles with the same twist sense although one tert.-butyl group is hidden from view in the illustration.

The atoms in Figures 33A & 33B were color coded according to symmetry equivalence, i.e., symmetry equivalent atoms were assigned the same color. CSM deviations from all symmetry subtypes comprising the Td point group were evaluated for the S4 symmetry molecule (20B in Figure 33A). The results are listed in Table 4.

The largest CSM value was S(C3), which means that the molecule lost more of its ‘C3- character' than of any of the other types of symmetries. The three-fold rotors on the exterior were desymmetrized to the largest extent. The core skeleton of the molecule did not distort to a significant extent, which is reasonable for a relatively rigid entity.

There is a symmetry constraint on 20A since it occupies a special position of S4 rotatory-reflection symmetry in the achiral P43n crystal. This symmetry constraint requires P–Si–Cquat–Cmethyl torsion angles which are related by an S4 operation to have equal magnitudes and opposite signs. The deviation from C3 symmetry is large since there are two pairs of opposite twist sense in 20A. Figures 34A-D illustrate bona- fide and pseudo sub-symmetry operations for 20A in the P43n crystal.

Table 4. CSM values for the bona-fide S4 symmetric molecule (20A)

Axis type S(C2) full S(C2) core S(C2) external No. of molecule fragment rotors degenerate elements pseudo 0.041(3) 0.0000 0.046(5) 2 bona-fide 0.0000 0.0000 0.0000 1

Axis type S(C3) full S(C3) core S(C3) external molecule fragment rotors pseudo 0.11(1) < 0.004 0.13(1) 4

Type of S(σd) full S(σd) core S(σd) external mirror plane molecule fragment rotors not through 0.086(6) < 0.004 0.096(8) 4 the C2 axis through the 0.041(4) 0.0000 0.046(5) 2 C2 axis

Axis type S(S4) full S(S4) core S(S4) external molecule fragment rotors pseudo 0.107(6) < 0.004 0.12(4) 2 bona-fide 0.0000 0.0000 0.0000 1

Figure 34A: S4 symmetry triakis-tetrahedron 20A occupying special positions of

C2 rotational and S4 rotatory-reflection symmetry in the P43n crystal.

Figure 34B: S4 symmetry triakis-tetrahedron 20A occupying general positions

of pseudo-C2 rotational and pseudo-S4 rotatory-reflection symmetry in the P43n crystal.

Figure 34C: S4 symmetry triakis-tetrahedron 20A occupying a general position of pseudo-C3 rotational symmetry in the P43n crystal.

Figure 34D: S4 symmetry triakis-tetrahedron 20A occupying a general position of pseudo-mirror reflection symmetry in the P43n crystal.

On the other hand, the second molecule in the asymmetric unit (20B) exhibits T- symmetry since all of its symmetry operations of the First Kind (C3 & C2 axes) were preserved while the four identically twisted three-fold rotors effectively removed all symmetry operations of the Second Kind, see Figures 35A,B. In the enantiomeric pair of T symmetry molecules 20B, the four P–Si–C(quat)–C(methyl) torsion angles are either all +173.2°or –173.2°, see Figure 33B. Thus, if a Td triakis-tetrahedron skeleton with four three-fold rotors is crystallized in a space group lacking a mirror reflection operation, then each of the rotors will have a twist sense which will be either the same or different from that of its adjacent neighbor. If they are the same, then the highest symmetry conformation can be T in the solid-state. If they are different, then the highest symmetry conformation can be S4 in the crystal.

The CSM values for the chiral T symmetry molecule 20B are listed in Table 5. Not surprisingly, deviations from symmetries of the Second Kind are rather large in this case, e.g. –0.118 and 0.12 for the respective Cs mirror and S4 symmetries, see Figures 35A-D.

Table 5. CSM values for the bona-fide T symmetric molecule (20B)

Axis type S(C2) full S(C2) core S(C2) external No. of molecule fragment rotors degenerate elements bona-fide 0.0000 0.0000 0.0000 3

Axis type S(C3) full S(C3) core S(C3) external molecule fragment rotors bona-fide 0.0000 0.0000 0.0000 4

Type of S(σd) full S(σd) core S(σd) external mirror plane molecule fragment rotors pseudo 0.118(6) 0.0000 0.13(5) 6

Axis type S(S4) full S(S4) core S(S4) external molecule fragment rotors pseudo 0.12(4) 0.0000 0.13(5) 3

Figure 35A: T symmetry triakis-tetrahedron 20B occupying a special position of

C2 rotational symmetry in the P43n crystal.

Figure 35B: T symmetry triakis-tetrahedron 20B occupying a special position of

C3 rotational symmetry in the P43n crystal.

Figure 35C: T symmetry triakis-tetrahedron 20B occupying a general position of pseudo mirror reflection symmetry in the P43n crystal.

Figure 35D: T symmetry triakis-tetrahedron 20B occupying a general position of pseudo S4 rotary-reflection symmetry in the P43n crystal.

As can be seen from this triakis-tetrahedron example, a Td cubane molecule which loses its mirror symmetry, cannot simultaneously exhibit both perfect C3 and perfect

S4 symmetries. The deviation from bona-fide C3 symmetry operations for molecule

20A was of the same magnitude as the deviation from a S4 symmetry operation for the T symmetric molecule 20B. Since preservation of mirror planes upon crystallization is known to be sterically unfavorable for packing, many molecules tend to lose their mirror symmetry as a consequence of entering a lattice. In this case, the loss of mirror symmetry meant that the triakis-tetrahedron molecule would be still be able to preserve its C3 or S4 symmetry since they involve skewed rotors. Since the rotors were distant, they could skew in different ways, even if they were in the same crystal.

4C.1b 2,4,6,8-Tetra-tert.-Butyl-1,3,5,7-Tetraphosphacubane (21)

t-Bu

t-Bu P P

P t-Bu t-Bu

As was noted above, the triakis-tetrahedron molecules 20A,B occupied special positions of symmetry in the unit cell, and thus it was of interest to see the effect of general positional occupancy on the CSM values. 2,4,6,8-Tetra-tert.-butyl-1,3,5,7- tetraphosphacubane (21) is an analogue of 20 which has alternating P and tert-butyl substituted-C atoms. In the literature (CCDB REF code VAVYAS) the triakis- tetrahedron cubane-like geometry molecules are reported to crystallize in an achiral 74 P21/c space group crystal and they asymmetric (C1 symmetry).

Its CSM values are listed in Table 6. The numbering diagram needed to interpret the results in Table 6 is given as Figure 36. The 179.4(4)° average of 12 antiperiplanar P–C(cubane)–C(quat)–C(methyl) torsion angles in crystalline 21 is quite close to (but not equal) to 180°. Thus, crystalline 21 has a pseudo Td conformation since its desymmetrization from the time-averaged solution-state conformation is so small.74 Therefore, the low CSM values show that the fact that crystalline 21 occupied a general position of symmetry in the lattice was not enough to generate larger than trivial distortions of its molecular geometry.

Figure 36: Pseudo Td symmetry triakis-tetrahedron 21 numbering diagram.

Table 6. CSM values for the pseudo Td symmetric molecule (21)

Symmetry S(C2) full S(C2) core S(C2) external related pair molecule fragment rotors 1,2 < 0.0014 < 0.00064 < 0.0014 1,3 < 0.0007 < 0.0008 < 0.0007 1,4 < 0.0018 < 0.0007 < 0.0018

Symmetry S(C3) full S(C3) core S(C3) external related set molecule fragment rotors 5,7,8 0.007(3) < 0.002 0.006(2) 5,6,8 0.006(3) < 0.002 0.005(2) 5,6,7 0.006(3) < 0.0012 0.005(2) 6,7,8 0.007(3) < 0.002 0.006(2)

Symmetry S(σd) full S(σd) core S(σd) external related pair molecule fragment rotors 1,2 < 0.003 < 0.0006 < 0.0022 1,3 < 0.006 < 0.0011 < 0.006 1,4 < 0.0016 < 0.0009 < 0.0016 2,3 < 0.003 < 0.0009 < 0.003 2,4 < 0.006 < 0.0011 < 0.0063 3,4 < 0.004 < 0.0006 < 0.004

S4 axis cuts S(S4) full S(S4) core S(S4) external the bonds molecule fragment rotors 1-2, 3-4 < 0.0036 < 0.0009 < 0.0036 1-3,2-4 0.008(3) < 0.0017 0.006(2) 1-4,2-3 < 0.0036 < 0.0009 < 0.0036

4C.2 CSM studies on the Distortion of Crystalline Molecules with Spatially Close Three-Fold Rotors on an Octasilacubane Skeleton.

4C.2a CSM Studies on Crystalline Octa-tert.-Butyl-Octasilacubane (22)

The skeleton of octa-tert.-butyl-octasilacubane (22), Si8C32H72, is an octa-silicon analogue of cubane in which a tert.-butyl three-fold rotor is ligated to each vertex.75 The first crystal structure of 22 (CCDB REF code YISJUF) was reported to crystallize in the trigonal R32 chiral space group with one molecule in the asymmetric unit.76 The molecule occupied a special position of three-fold rotational symmetry but was disordered causing a high 9.96% R factor.76 A polymorph was later reported (CCDB REF code YISJUF01) which crystallized also in a trigonal R32 chiral space group.75 It refined to a better 4.9% R factor and there were two one-third molecules plus one one-sixth of another molecule in the asymmetric unit. This Z’ = 0.83 value arises since all three molecules occupy different special positions of C3 three-fold rotational symmetry, and one of the molecules (22A) also occupies three special positions of C2 two-fold rotational symmetry. Therefore, the unit cell contains one palindromic molecule with bona-fide D3 symmetry (22A, closest to the unit cell’s origin) and two non-palindromic molecules with bona-fide C3 symmetry (22B and 22C, respectively ‘intermediate-to’ and ‘farthest-from’ the origin).75 It is obvious that these three molecules must be diastereomers since they all reside in the asymmetric unit. The desymmetrization of these molecules in the chiral crystal lattice will be analyzed in detail.

t-Bu t-Bu t-Bu t-Bu Si Si Si Si

Si Si t-Bu Si Si t-Bu

t-Bu t-Bu

29 When crystals of 22 are dissolved in ortho-xylene-d10 the Si NMR spectrum measured for the resulting solution shows a single peak at 13.042 ppm.75 Also the 1H NMR spectrum shows a single peak at 1.483 ppm, while the 13C{1H} NMR spectrum contains two peaks, one for the tert.-butyl quaternary carbons at 32.786 ppm 75 and one for the methyl carbons at 25.983. Therefore, 22 has either a static Oh symmetry conformation at a slow conformational exchange kinetic regime or a weighted time-averaged Oh symmetry structure consisting of a fast kinetic regime equilibrium of lower symmetry conformations in the liquid state (probably an enantiomerization of O-symmetry conformers). There is no information about the temperature dependence of the NMR spectra for 22 which might enable us to differentiate between these two possibilities.

The CSM values for crystalline 22A-C molecules are listed in Tables 7A-C, respectively.

Table 7A. CSM values for the bona-fide D3 symmetric molecule (22A).

Axis type S(C2) full S(C2) core S(C2) external No. of molecule fragment rotors degenerate elements pseudo 0.0203(1) 0.0052(2) 0.021(2) 3 pseudo C2 0.0204(1) 0.0050(2) 0.021(2) 3 coincides with C4 bona-fide 0.0000 0.0000 0.0000 3

Axis type S(C3) full S(C3) core S(C3) external molecule fragment rotors pseudo 0.0271(2) 0.0067(2) 0.0281(2) 3

bona-fide 0.0000 0.0000 0.0000 1

Axis type S(C4) full S(C4) core S(C4) external molecule fragment rotors pseudo 0.0305(2) 0.0075(2) 0.0317(2) 3

Type of S(σh) full S(σh) core S(σh) external mirror plane molecule fragment rotors pseudo 0.384(1) 0.0045(2) 0.403(1) 3

Type of S(σd) full S(σd) core S(σd) external mirror plane molecule fragment rotors pseudo 0.384(1) 0.0046(2) 0.403(1) 3 pseudo 0.392(1) 0.0013(1) 0.412(1) 3

S(Ci) full S(Ci) core S(Ci) external molecule fragment rotors pseudo 0.392(1) 0.0014(1) 0.412(1) 1

Axis type S(S4) full S(S4) core S(S4) external molecule fragment rotors pseudo 0.394(1) 0.0071(2) 0.414(1) 3

Axis type S(S6) full S(S6) core S(S6) external molecule fragment rotors pseudo 0.3925(5) 0.0013(1) 0.412(1) 1 along C3 axis pseudo 0.4002(5) 0.0068(2) 0.420(1) 3

Table 7B. CSM values for the bona-fide C3 symmetric molecule (22B).

Axis type S(C2) full S(C2) core S(C2) external No. of molecule fragment rotors degenerate elements pseudo 0.0124(1) 0.0005(1) 0.0130(1) 3 pseudo 0.0104(1) 0.00019(3) 0.0109(1) 3 pseudo C2 0.0055(1) 0.00030(5) 0.0057(1) 3 coincides with C4

Axis type S(C3) full S(C3) core S(C3) external molecule fragment rotors pseudo 0.0073(1) 0.00049(5) 0.0076(1) 3 bona-fide 0.0000 0.0000 0.0000 1

Axis type S(C4) full S(C4) core S(C4) external molecule fragment rotors pseudo 0.0152(1) 0.0007(1) 0.0159(1) 3

Type of S(σh) full S(σh) core S(σh) external mirror plane molecule fragment rotors pseudo 0.2360(3) 0.0005(1) 0.248(1) 3

Type of S(σd) full S(σd) core S(σd) external mirror plane molecule fragment rotors pseudo 0.2326(3) 0.00029(4) 0.2444(4) 3 pseudo 0.2356(3) 0.0000 0.2477(4) 3

S(Ci) full S(Ci) core S(Ci) external molecule fragment rotors pseudo 0.2383(3) 0.00029(4) 0.2505(3) 1

Axis type S(S4) full S(S4) core S(S4) external molecule fragment rotors pseudo 0.2353(3) 0.00049(5) 0.2473(3) 3

Axis type S(S6) full S(S6) core S(S6) external molecule fragment rotors pseudo 0.2384(3) 0.00029(4) 0.2505(3) 1 along C3 axis pseudo 0.2405(3) 0.0007(1) 0.2527(4) 3

Table 7C. CSM values for the bona-fide C3 symmetric molecule (22C).

Axis type S(C2) full S(C2) core S(C2) external No. of molecule fragment rotors degenerate elements pseudo 0.0244(2) 0.0004(1) 0.0254(2) 3 pseudo 0.0281(2) 0.0010(1) 0.0294(2) 3 pseudo C2 0.0117(1) 0.0004(1) 0.0121(1) 3 coincides with C4

Axis type S(C3) full S(C3) core S(C3) external molecule fragment rotors pseudo 0.0156(1) 0.0007(1) 0.0161(1) 3 bona-fide 0.0000 0.0000 0.0000 1

Axis type S(C4) full S(C4) core S(C4) external molecule fragment rotors pseudo 0.0340(1) 0.0012(1) 0.0355(2) 3

Type of S(σh) full S(σh) core S(σh) external mirror plane molecule fragment rotors pseudo 0.0486(2) 0.0010(1) 0.0510(1) 3

Type of S(σd) full S(σd) core S(σd) external mirror plane molecule fragment rotors pseudo 0.0386(2) 0.0005(1) 0.0404(2) 3 pseudo 0.0449(2) 0.0000 0.0470(2) 3

S(Ci) full S(Ci) core S(Ci) external molecule fragment rotors pseudo 0.0535(2) 0.0005(1) 0.0535(2) 1

Axis type S(S4) full S(S4) core S(S4) external molecule fragment rotors pseudo 0.0444(2) 0.0007(1) 0.0465(2) 3

Axis type S(S6) full S(S6) core S(S6) external molecule fragment rotors pseudo 0.0535(2) 0.0005(1) 0.0562(2) 1 along C3 axis pseudo 0.0581(2) 0.0011(1) 0.0608(2) 3

There are asymmetric and C3 symmetric tert.-butyl groups in crystalline 22A-C similar to the example of triakis-tetrahedron 22A,B discussed above. Only C3 symmetric tert.-butyl groups contain methyl carbons with the same color in Figures

37A-D. The integer-0 values in Table 7A calculated for S(C3) and for S(C2) are expected for D3 symmetry conformation 22A. The molecule also has low 0.0203-

0.0305 values for S(Cn), where n = 2,3, and 4 for pseudo Cn rotation axes. Much lower values were found for the silacubane skeleton (none higher than 0.008) in accord with its rigid nature. We can interpret our results as showing that the distortion of the full molecule from bona-fide Cn rotational symmetries arises from the external rotor fragments since their CSM values are very similar to those calculated for the full molecule. On the other hand, the distortion of the full molecule from bona-fide symmetries of the Second Kind is more than ten times higher (0.38-0.40) than distortions from those of the First Kind. Similarly, the distortion of the C3 symmetry conformation 22B full molecule from bona-fide symmetries of the Second Kind is also much higher (0.235-0.241) than distortions from those of the First Kind

(0.006-0.015). However, the situation is different for the second C3 symmetry conformation 22C full molecule since the distortions from bona-fide symmetries of the First Kind (0.012-0.034) were almost similar in magnitude to those distortions from bona-fide symmetries of the Second Kind (0.0390-0.058)!

Figure 37A: D3 symmetry silacubane 22A with pseudo C4 & pseudo C3 axes in the R32 crystal.

Figure 37B: D3 symmetry silacubane 22A with pseudo C2 axes in the R32 crystal.

Figure 37C: D3 symmetry silacubane 22A with pseudo h (top left) & d (top right and bottom) planes in the R32 crystal.

Figure 37D: D3 symmetry silacubane 22A with a pseudo inversion point in the R32 crystal.

There are two sets of tert.-butyl groups in D3 symmetry conformation 22A. Six homotopic asymmetric groups contain diastereotopic methyl carbons and give rise to three unique antiperiplanar Si–Si–Cquat–Cmethyl torsion angles: 171.01°, 167.28°, and

177.51°. Two homotopic C3 symmetric groups contain homotopic methyl carbons and give rise to only one antiperiplanar Si–Si–Cquat–Cmethyl torsion angle: 162.12°. All four angles have the same plus sign, and the average twist for these four angles is

+13(4)°. There are also two sets of tert.-butyl groups in C3 symmetry conformation 22B. Six homotopic asymmetric groups contain diastereotopic methyl carbons and give rise to six unique antiperiplanar Si–Si–Cquat–Cmethyl torsion angles: –171.48°, –

160.76°, –169.95°, –171.01°, –167.58°, and –169.38°. Two homotopic C3 symmetric groups contain homotopic methyl carbons and give rise to two unique antiperiplanar

Si–Si–Cquat–Cmethyl torsion angle: –170.96° and –167.67°. All eight angles have the same minus sign and their average twist is –11(4)°. In addition, there are also two sets of tert.-butyl groups in the C3 symmetry conformation 22C partner. Six homotopic asymmetric groups contain diastereotopic methyl carbons and give rise to six unique antiperiplanar Si–Si–Cquat–Cmethyl torsion angles: 176.17°, 177.87°,

179.23°, 178.55°, 173.82°, and 176.55°. Two homotopic C3 symmetric groups contain homotopic methyl carbons and give rise to two unique antiperiplanar Si–Si–

Cquat–Cmethyl torsion angle: 178.63° and 170.55°. All eight angles have the same plus sign and their relatively low average twist is +4(3)°.

The six CSM values of symmetry operations of the First Kind may be averaged to give a S(average_O) value which shows their deviation from 'O character'. Similarly, the thirteen CSM values of the First and Second Kinds may also be averaged to give a

S(average_Oh) value for their deviation from 'Oh character'. When this is done we get the following results:

Table 8: 'O-symmetry character' and 'Oh-symmetry character' S-values for crystalline molecules (22A), (22B) and (22C) :

Molecule 'O character' 'Oh character'

Molecule (22A) 0.0164 0.2182

Molecule (22B) 0.0085 0.1224

Molecule (22C) 0.0190 0.0350

From these values one can clearly see that molecules 22A,B form a set that appears to be different than 22C. The ratio of ('Oh character')/('O character') = 13.3 for 22A and 14.4 for 22B, while it is 1.8 for 22C.

On the basis of First and Second Kind CSM values for 22A-C and the degree of rotor twist sense one may state the following:

1. Crystalline 22A has a bona-fide D3 symmetry relatively high +13(4)°-twist sense chiral conformation when resident within the confines of the crystal lattice, but its pseudo symmetry is actually higher (pseudo O symmetry (+)-twist sense chiral conformation) since all of the CSM values for the suboperations of the O point group are low.

2. Crystalline 22B has a bona-fide C3 symmetry relatively high –11(4)°-twist sense chiral conformation when resident within the confines of the crystal lattice, but its pseudo symmetry is also actually higher (a pseudo O symmetry (–)-twist sense chiral conformation) since all of the CSM values for the suboperations of the O point group are low.

3. Crystalline 22C has a bona-fide C3 symmetry relatively low +4(3)°-twist sense chiral conformation when resident within the confines of the crystal lattice, but its pseudo symmetry is actually higher (a pseudo Oh symmetry achiral conformation) since all of the CSM values for the suboperations of the Oh point group are low. 4. The distortions of symmetry operations of the First Kind are always lower than those of the Second Kind since all eight tert.-butyl groups twist with the same sense (i.e. the same sign) in each of crystalline molecules 22A-C. This is due to the close placement of the rotors on the octasilacubane skeleton which results in a correlated sense of twist.

DFT calculations in the B3LYP/6-31g* level of theory were performed in order to ascertain the most stable gas phase conformation for 22. The molecule was constrained in either a Oh, O, D3 or C3 point group conformation followed by geometry optimization. The O symmetry conformation of 22 was most stable in the gas phase. The relative energies of the Oh, C3 and D3 conformation model vis-à-vis that for the O symmetry model were respectively 3.642, 0.453 and 0.231 kcal/mol less stable. The average twist for the tert.-butyl rotor [based on antiperiplanar Si–Si–

Cquat–Cmethyl torsion angles] for the calculated D3 and C3 symmetry conformation models was respectively 11.8(7)° and 11.9(4)°. These values compare nicely with the 13(4)° and 11(4)° absolute average twist values experimentally determined for D3 symmetry crystalline 22A and C3 symmetry crystalline 22B. It is apparent that the 22A,B pair can be considered to be related as pseudo O symmetry conformations of opposite handedness (based on their CSM values listed in Tables 7A,B). Assuming this to be correct, then the same 13(4)° and –11(4)° average twist values of the pseudo O symmetry structures are also consistent with the 11.5° average twist value for the DFT bona-fide O symmetry conformational model.

The gas phase geometry optimized low relative energies for D3 and C3 symmetry conformations compared to the O symmetry global minimum in the limited series is in good accord with their experimental observation in the crystalline state. A detailed analysis of the packing of 22A-C in the R32 space group crystal was undertaken in order to investigate why one of the molecules exhibited a pseudo Oh symmetry conformation. This finding was a bit surprising since a gas phase bona-fide Oh symmetry conformation molecule is relatively quite high in energy. An analysis of the packing was performed since a pseudo Oh conformer is found in the crystal.

4C.2b Superlattice Packing Arrangement in the Octa-tert.-Butyl- Octasilacubane Rhombohedral R32 Crystal.

0 a

Figure 38: Arrangement of six subsections containing fifteen molecules in the R32 unit cell color coded as 22A (red), 22B (blue), and 22C (green), where the a- axis is labeled on the bottom right.

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Very little has been written in the literature report75 concerning the stereochemistry of the molecules in the asymmetric unit. The magnitude and direction of the screw sense was not described for the three molecules in the asymmetric unit. They were described as having “highly symmetrical tert.-butyl groups” and were given the 75 numerical descriptors 1-3. The authors gave descriptors of ‘1’ for what the D3 22A molecule described here, ‘2’ for C3 22C, and ‘3’ for C3 22B. They noted that the molecules are stacked in the unit cell, and that all occupied a three-fold axis in the crystal.75 Molecule 1 also occupied a two-fold axis of the crystal.75 A black and white illustration was provided similar to the color coded structure in Figure 38. The centroids of each silacubane are coplanar in this figure. The stacking order for fifteen molecules was described as being ‘132231322313223’ but no rationale was provided for this observation other than the statement that “the superlattice structure was caused primarily by the difference in the symmetry of the substituents”.75

The packing diagram for crystalline 22A-C was inspected with the Mercury program77 as part of the CSM study of these molecules. It was apparent that the upper and lower ab planes of the unit cell contained two fused equilateral triangles. The R32 unit cell is schematically depicted in Figure 39. It may be split by a plane (grey) encompassing the short diagonals of the top and bottom faces. The rhombohedral cell is thus divided into two fused equilateral triangular prisms with equal 12.232(3) Å a and b axes sides and a long c axis length of 125.186(4) Å. The c axis and the three parallel cell edges and the dashed axes (magenta) through each equilateral triangle’s centroid are all three-fold axes of rotational symmetry. The plane encompassing the long diagonals in the top and bottom faces contains four C3 axes spaced 7.06 Å apart: the two dashed axes and also the edge axes perpendicular to points [1,0,0] and to [0,1,0] (respectively marked a and b in blue). One may consider this plane to contain an imaginary ‘reference’ line (green) which intersects the bottom face at an angle of 49.6°. Within this plane additional ‘reference’ lines are constructed parallel to the first green line at perpendicular distances of 16.1 Å (magenta double-headed arrow).

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Green reference lines intersect the parallel-to-c axes defined by points [1,0,z] or [0,1,z]. These intersecting points define layers in which octa-tert.-butyl- octasilacubanes occupy all four vertices of rhombohedral planes parallel to the top and bottom of the unit cell. Silacubane analogues also occupy each of the two magenta color points on a green reference line.

Figure 39: The schematic R32 unit cell of crystalline 22. Octa-tert.-butyl- cubane analogues 22A-C only occupy magenta circles on imaginary green construction lines and vertices of the fused equivilateral trianglar (rhombohedral) planes.

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Figure 40: Stack of six schematic asymmetric units in plane through points [1,0,0], [0,1,0], [1,0,1], and [0,1,1] encompassing the two dashed axes of three-fold rotational symmetry (magenta) that pass through the centroids of the fused equilateral triangles. Zig-zag arrangement on imaginary green construction lines of (+)-twisted pseudo O symmetry 22A (red circle), (–)-twisted pseudo O

symmetry 22B (blue circle), pseudo Oh symmetry 22C (green circle) units, and C2 axes (black fused arcs).

The asymmetric unit is composed of one-third molecule of (–)-twisted pseudo O

symmetry conformation (bona fide C3 symmetry 22B, blue circles in Figure 40), one-

third molecule of pseudo Oh symmetry conformation (bona fide C3 symmetry 22C, green circles), and one-sixth molecule of (+)-twisted pseudo O symmetry

conformation (bona fide D3 symmetry 22A, red circles). Thus, the ratio of these molecules in the unit cell is 1:1:0.5.

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Figure 40 illustrates the green zig-zag imaginary construction line on which color coded circles reside. Two-fold axes of rotation (black fused arcs) are located between 2.5 molecule asymmetric unit segments of the green line. The axes either pass through (+)-twisted pseudo O symmetry conformation red circles (D3 symmetry

22A) or between two adjacent pseudo Oh symmetry green circles (22C). The periodicity of the asymmetric unit plus the C2 axes generates an arrangement where pseudo O symmetry molecules of the same twist sense are never adjacent to each other. The ordering of adjacent silacubanes repeats itself on the seventh vertical stack of zig-zag asymmetric units, and this accounts for the long 125.186(4) Å c-axis length. Figure 40 is similar to the stacking diagram in the literature with the exception that two-fold axes of symmetry have been added, and the molecules have become color coded circles for simplicity.

The ab planes at z = 0, 1/3, 2/3, or 1 all have 1/6 of a D3 symmetry 22A molecule occupying a vertex of each equilateral triangle for a one half-molecule, see Figure 41.

As noted above, each vertex is on a three-fold axis. Three mutually perpendicular C2 axes reside in these ab planes and generate the second half of the 22A molecules. The axes intersect the midpoints of a set of parallel edges in the silacubanes. It was noted that the fused equilateral triangles formed a layer that was occupied either by all 22A, all 22B or all 22C units. The layer extends laterally throughout the entire crystal. It is a densely populated layer since each molecule occupies a three-fold axis and is thus surrounded by three lateral homotopic neighbors. This three-dimensional arrangement was not readily apparent from the ‘linear’ relationship of coplanar silacubane skeleton centroids graphically shown in the literature or in Figures 38 and 40. There is a shallow chiral depression (green) above and below the equilateral triangle’s centroid. A single silacubane unit occupies the space either above or below the depression (but not at both sides simultaneously). This arrangement is similar to a ‘host-guest’ type ensemble.

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Figure 41: Two of four empty shallow chiral depressions (green) within a layer of four (+)-twisted pseudo O-symmetry silacubane analogues (D3-22A, red). Two other shallow chiral depressions (each on a different opposite side of the layer) have not been illustrated.

Occupancy by (–)-twisted pseudo O-symmetry guests (C3-22B) above or below a shallow chiral depression (green) between equilateral triangular ensembles of diastereomeric (+)-twisted pseudo O-symmetry silacubane analogues (D3-22A) is illustrated in Figure 42. Carbon atoms of the C3-22B three-fold symmetrical tert.- butyl group pointing into the ‘depression’ and those of a three-fold symmetrical rotor pointing upward in one of the three adjacent lateral D3-22A are colored green in Figure 42. These antiparallel tert.-butyl groups reside in a common ca. 0.75 Å thick layer (black dashed lines in Figure 43) since methyl carbons at the top periphery of the host depression are 4.53 Å above the ab plane at z = 0, while the methyl carbons of the guest are only 3.78 Å above the same level. This common region of antiparallel tert.-butyl groups extends laterally over the entire crystal. Molecular arrangements of the type depicted in Figure 42 were not mentioned or reported before.

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Since these arrangements seemed to play a role in the superlattice architecture, it is interesting to try to unravel as much of the superlattice architecture as possible.

Figure 42: Two shallow chiral depressions (green) within a dense packed layer of (+)-twisted pseudo O-symmetry silacubane analogues (D3-22A, red) with a dense packed layer of (–)-twisted pseudo O-symmetry ‘guests’ (C3-22B, blue) on both sides.

Occupancy of a special position of three-fold rotational symmetry by the –11(4)° average twist pseudo O-symmetry guest (C3-22B) enables it to have a pseudo inversion relationship with each of its three +13(4)° average twist pseudo O- symmetry hosts (D3-22A). This pseudo inversion relationship may be the cause of the auspicious steric fit, although the subtle details of the intermoleculear interactions require proton coordinates which were not reported in the literature. We propose that these interactions are a type of ‘chiral discrimination’ since they involve the interaction between diastereomers that are related as ‘pseudo enantiomers’, while interactions between homomeric chiral molecules are absent in this chiral crystal (see sections below).

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A modification of Avnir’s algorithm for pseudo inversion was developed as part of this research so that the distortion from an ideal symmetry between molecules would be independent of their intermolecular distance. The Avnir algorithms were developed to access distortion from an ideal symmetry within an object. This discrepancy index for bona-fide inversion was given the descriptor ‘rmS(Ci)’ to distinguish it from the size independent Avnir ‘S(Ci)’. Its development and use is described in detail in the last project of the dissertation and in the Experimental section. At this stage we will just state that the rmS(Ci) is determined by calculating the theoretical ‘nearest ideal geometrical form’ having a bona fide inversion symmetry relationship which is generated from the input coordinates by using the Avnir ‘inversion’ algorithm. These coordinates appear in the csm.exe53 output file. Then the ‘best inversion related geometrical pair’ is superimposed upon corresponding atoms in the actual pseudo symmetry 22A,B pair of molecules. The rmS(Ci) value is a relatively low 0.203(2) Å and the coordinates of the ‘best point of inversion’ (magenta) between oppositely handed 22A,B units in the .cif file asymmetric unit are [0.167(14),0.333(14),0.0332(2)], see Figure 43. To put this

0.203(2) Å value in a proper perspective, an rmS(Ci) calculation was also performed for the imperfect arrangement between –11(4)° average twist 22B and +4(3)°average twist 22C. The rmS(Ci) was now a much higher 2.4316(2) Å.

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As noted above, since the 22B ‘guest’ is surrounded by three equivalent 22A ‘hosts’, there are three pseudo inversion ‘best points’. A 22A,B pair having an inversion relationship is illustrated in Figure 44. There is a different location in the cell where 22A,B pairs show pseudo inversion and this will be discussed later on.

Figure 43: A pseudo inversion ‘best point’ (magenta) between pseudo

‘enantiomeric’ (+)-twist conformation D3 22A (red) and (–)-twist conformation

C3 22B (blue) diastereomers. Three-fold symmetry antiparallel tert.-butyl groups (green) in adjacent molecules delineate a ca. 0.75 Å thick layer extending laterally over the entire crystal.

What appears to be the result of a chiral discrimination is also seen both in the right and left sections of Figure 44. A bona-fide Oh symmetry conformation would have an ideally staggered arrangement for each of its tert.-butyl rotors vis-à-vis the silacubane skeleton rather than a ca. 12° twist. The 22C tert.-butyl group’s pseudo staggered (non-twist) arrangement enables it to interact with potential packing partners without constraints of chiral discrimination, i.e. the pseudo Oh symmetry conformation 22C rotors do not seem to discriminate between their adjacent neighbors by excluding either pseudo Oh symmetry conformation rotors, nor between nearby (+)- or (–)-twist sense pseudo O symmetry rotors.

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Figure 44: Layers of dense packed silacubanes separated by two layers of sparsely packed schematic units. A. (right drawing) Three types of ‘host-guest’ type arrangements of a sparse packed layer on top and on bottom of a dense packed layer in the R32 crystal: dense layers of (+)-twisted pseudo O symmetry 22A (red circles) with sparse layers of (–)-twisted pseudo O symmetry 22B (blue circles) on both sides; dense layers of pseudo Oh symmetry 22C (green circles) with sparse layers of either (–)-

twisted pseudo O symmetry 22B (blue circles) or pseudo Oh symmetry 22C (green circles) on adjacent sides, and dense layers of (–)-twisted pseudo O symmetry 22B (blue circles) with sparse layers of either (+)-twisted pseudo O

symmetry 22A (red circles) or pseudo Oh symmetry 22C (green circles) on adjacent sides. B. (left drawing) The different color codes of adjacent sparse density layers of octa-tert.-butyl-silacubanes show that same twist-sense handedness (+)…(+) or (–)…(–) units are never adjacent to each other in the R32 crystal.

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Stacks of dense packed layers (so called ‘hosts’) separated by two sparsely packed layers (so called ‘guests’) are schematically depicted in Figure 44. The ‘guest’ layers may be considered to be ‘sparsely packed’ since the molecule therein occupies a three-fold axes but has no adjacent neighbor on the same level. The bottom host- guest ensemble is identical to the upper one in the stack and these have already been illustrated in Figure 42 as an iconic drawing. The second host from the bottom

(green balls) is a rhombus of silacubanes 22C. While the bona-fide C3 symmetry structure 22C has a +4(3)° average twist sense, it is so low that it can be considered to represent pseudo staggered tert.-butyl rotor conformation (as compared to the much larger twists for 22A,B). Thus, the units in the dense packed ‘host’ layer can be thought of as having a pseudo achiral Oh conformation. The shallow depressions within the fused equilateral ensembles of four 22C units may also be considered to be pseudo achiral. Inspection of the stacks of six layers with top and bottom neighbors shows that there seems to be no difficulty for pseudo achiral depressions to host either another pseudo achiral 22C or a (–)-twist sense pseudo O symmetry conformation chiral guest (22B) on the other side. Similarly, layers of four (–)-twist conformation pseudo O symmetry units also are capable of hosting either opposite (+)-twist conformation pseudo O symmetry conformers or the pseudo achiral Oh conformation partner. What are absent are sparse packed layers of ‘guests’ that have the same handedness as that of the adjacent dense packed ‘host’ layer. Therefore, this perhaps can be called a ‘chiral discrimination’.

For emphasis, the left part of Figure 44 shows circles that are color coded only for units located in adjacent sparse packed layer stacks. It is seen that a second type of pseudo inversion relationship is found between the pseudo ‘enantiomerically’ twisted silacubanes between the sparse packed layers near z = 1/3 and 2/3. Similar to the case of adjacent dense and sparse packed layers in the right side of the figure, two adjacent sparse packed layers also never contain molecules of the same handedness. Therefore, in conclusion, the packing arrangement of twisted and almost non-twisted units in the R32 crystal seems to be based on the ‘principle’ that opposite handed twist sense molecules fit next to each other in an auspicious manner.

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Perhaps the pseudo Oh symmetry units act to space the pseudo O symmetry units in such a manner that homochiral interactions are prevented. Bottom line: one can provide a rationalization of the packing within the R32 crystal that is based on the absence of the same twist sense in and between adjacent layers of units. Finally, the pseudo O symmetry character of 22A,B and the pseudo Oh character of 22C has never been commented upon, to the best of my knowledge.

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4C.3 CSM Studies on the Distortion of Crystalline Molecules with Spatially Close Three-Fold Rotors on Tetrahedrane and Tetrasilatetrahedrane Skeletons.

4C.3a CSM Studies on Crystalline Tetra-tert.-Butyl-Tetrahedrane (23)

Tetra-tert.-butyl-tetrahedrane (23, CCDB REF code CUCZUV) crystallizes in the 78 P63/m space group, and occupies a special position of mirror symmetry. Due to the large thermal ellipsoids, it is likely that the structure has disorder and may be a dynamic or static average of two T symmetric structures.78 Two of its rotors are asymmetric and two are Cs symmetric. Tetra-tert.-butyl-tetrahedrane (23) and octa tert.-butyl-octasilacubane (23) have the same tert.-butyl rotor units. However, 23 is pseudo Td symmetric with either bona fide ideal staggered or pseudo staggered rotor geometries vis-à-vis the Platonic solid skeleton.78 Our DFT calculations have shown that the Oh conformation for 22 was 3.642 kcal higher in energy than for the O symmetry analogue. We rationalized that the existence of a pseudo Oh conformation unit (22C) in the YISJUF crystal lattice was in the role of a spacer molecule which enabled an adjacent pseudo O symmetry silacubane molecule to only interact with a molecule of opposite rotor twist sense. One ponders why pseudo O symmetry crystalline 22A,B has large deviations from symmetries of the Second Kind while 78 crystalline 23 is pseudo Td symmetric?

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The answer is geometric since a tetrahedrane skeleton permits adjacent tert.-butyl groups to be angled farther away from each other than does the cubane skeleton.

This is shown by a 145° CtBuquat–Ccore–Ccore angle versus a 125° CtBuquat–Sicore–Sicore angle. It results from the fact that as the Ccore–Ccore–Ccore internuclear angle in a Platonic solid becomes increasingly more distorted from ideal tetrahedrality as the skeleton changes from dodecahedrane to cubane and on to tetrahedrane. The reason for this is that the atomic orbitals of the core have to have increased p-character (compared to p = integer 3 in ideally tetrahedral sp3) in order to build the skeletal 3+ 3+ sp —sp ‘banana’ bonds. As a result, there is higher s-character from the Ccore partner in the CtBuquat—Ccore bond, and the CtBuquat–Ccore–Ccore angle opens up. We can predict that the CtBuquat–Ccore–Ccore angle for Ih dodecahedrane will be closer to the accepted ideal tetrahedral value than the corresponding value for a Td tetrahedrane. A corollary is that crowding between adjacent tert.-butyl rotors will increase as one goes from tetrahedrane to cubane to dodecahedrane, see Figure 45. Therefore, adjacent tert.-butyl groups are more sterically demanding when ligated to a cubane skeleton than to a tetrahedrane skeleton. Of course, the Sicore—Sicore bonds are longer than the Ccore—Ccore bonds when we are comparing the stereochemistry of 22 to 23, but the respective CtBuquat–Xcore–Xcore angular relationships remain the same. One may even speculate that the longer Sicore—Sicore bonds enabled 22 to be synthesized since the synthesis of octa-tert.-butyl-dodecahedrane with a carbon skeleton is expected to be a major challenge in organic synthesis.

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Figure 45: Closing the Ccore–Ccore–Ccore angle makes adjacent tert.-butyl substituents much more crowded in cubanes than in tetrahedranes and they are expected to be even more crowded in dodecahedranes.

Figure 46: Numbering diagram for carbon atoms in tetra-tert.-butyl- tetrahedrane 23 (CUCZUV).

The CSM values for 23 are listed in Table 9. A numbering diagram of carbons referred to in Table 8 is presented in Figure 46. As can be seen in Table 9, all of the CSM values are lower than 0.0046. Figure 47 shows a fragment of molecule 23 consisting of one side of the tetrahedrane and two tert.-Bu groups. The top fragment in the figure is from a Td symmetric model with ideally staggered rotors, while the lower fragment is from a T symmetric unit with rotors twisted in the same sense.

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The distance between adjacent ideally staggered tert.-butyl protons is not markedly smaller than that in the twisted conformation, and as a result, both conformational models are low energy (the energy difference between them is only 0.5 Kcal/mol).18 t Due to the open 145° C(Bu quaternary)–C(core)–C(core) angle, the skeleton can readily accommodate two adjacent ideally staggered tert.-butyl groups. Thus, crystalline 23 has a bona-fide Cs symmetry achiral conformation when resident within the confines of the crystal lattice, but its pseudo symmetry is actually higher (pseudo Td symmetry) since all of the CSM values for the suboperations of the Td point group are low.

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Table 9. CSM values for the bona-fide Cs symmetric molecule (23).

Symmetry S(C2) full S(C2) core S(C2) external related pair molecule fragment rotors of atoms 1,2 0.0031(2) < 0.0003 0.0031(1) 1,3 0.0031(2) < 0.0003 0.0031(1) 1,14 0.0031(2) 0.0000 0.0032(1)

Symmetry S(C3) full S(C3) core S(C3) external related set of molecule fragment rotors atoms 2,3,14 0.0034(2) < 0.0007 0.0035(1) 1,3,14 0.0021(2) < 0.0007 0.0021(1) 1,2,14 0.0042(2) < 0.0006 0.0043(1) 1,2,3 0.0034(2) < 0.0007 0.0035(1)

Symmetry S(σd) full S(σd) core S(σd) external related pair molecule fragment rotors of atoms 1,2 0.0031(2) < 0.0004 0.0016(1) 1,3 0.0016(2) < 0.0006 0.0032(1) 1,14 0.0000 0.0000 0.0000 2,3 0.0031(2) 0.0000 0.0032(1) 2,14 0.0031(2) < 0.0004 0.0016(1) 3,14 0.0016(2) < 0.0006 0.0032(1)

The bonds S4 S(S4) full S(S4) core S(S4) external axis passes molecule fragment rotors through 2-14,1-3 0.0041(3) < 0.0006 0.0041(1) 1-2,3-14 0.0041(3) < 0.0004 0.0041(1) 1-14,2-3 0.0046(3) < 0.0002 0.0046(1)

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2.318 Å

145.0°

1.486 Å

2.54 Å

2.53 Å

2.54 Å

… Figure 47: (top) H H Distances in a C—C fragment from a Td symmetric conformational model with ideally staggered rotors on both ends. (bottom) H…H Distances in a C—C fragment from a T symmetric conformational model with same sense twisted rotors on both ends.

4C.3b CSM studies on Tetrakis(trimethylsilyl)-Tetrahedrane

Si Si Si

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Tetrakis(trimethylsilyl)-tetrahedrane (CCDB REF code OHABEE) crystallizes in the triclinic P-1 space group with three full molecules in the asymmetric unit (24A- C).79 The arrangement of the three molecules in the asymmetric unit is depicted in Figure 48. By visual inspection one can see that the right and left molecules (24A,C) are related by a horizontal translation thereby keeping their orientation the same. The middle molecule (24B) is rotated by ca. 90° vis-à-vis its two lateral neighbors. A pseudo inversion relationship exists between pairs of orthogonally oriented neighbors 24A&B and between 24B&C, but not between parallel neighbors 24A&C (see Table 10)

Table 10: Pseudoinversion symmetry rmS(Ci) and S(Ci) values between pairs of molecules in the asymmetric unit

Pseudo rmS(Ci) S(Ci) inversion pair

24A&B 0.211(2) 0.112(2)

24B&C 0.189(2) 0.118(2)

24A&C 1.979(2) 3.895(2)

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Si Si

Si Si Si Si

Si Si

Figure 48: Arrangement of the three molecules in the asymmetric unit: 24A (red, left), 24B (blue, middle), and 24C (red, right). Pseudo centers of inversion (green) exist between 24A,B and between 24B,C but not between 24A,C.

Molecules 24A-C are very similar to 23 with the obvious differences being that the –

Si(CH3)3 rotors are slightly larger than –C(CH3)3 analogues and also farther away from the skeletal core [Ccore—Si 1.82 Å and Si—Cmethyl 1.86 Å versus Ccore—Cquat

1.49 Å and Ccore—Cmethyl 1.52 Å]. The geometries of 24A-C are very similar to each 79 other, and all of them have C1 symmetry since they occupy general positions. As can be seen from the CSM values for Td suboperations listed in Tables 11A-C, the fact that solution-state Td symmetry 24A-C molecules have been desymmetrized down to complete asymmetry (C1 point group) when resident within the confines of the crystal lattice does have any relationship their degree of distortion from bona-fide

Td symmetry. A numbering diagram of carbons referred to in Tables 11A-C is presented in Figure 49.

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Figure 49: Numbering diagram for carbon atoms in C1 symmetry tetrakis(trimethylsilyl)-tetrahedrane 24A-C (OHABEE).

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Table 11A. CSM values for the C1 symmetric molecule (24A).

Symmetry S(C2) full S(C2) core S(C2) external related pair molecule fragment rotors of atoms 5,6 0.047(2) < 0.0008 0.047(2) 5,7 0.046(2) < 0.0011 0.045(2) 5,8 0.014(1) < 0.001 0.013(1)

Symmetry S(C3) full S(C3) core S(C3) external related set of molecule fragment rotors atoms 6,7,8 0.056(2) < 0.0022 0.057(2) 5,7,8 0.064(1) < 0.0034 0.064(2) 5,6,8 0.035(2) < 0.0031 0.0346(2) 1,2,3 0.055(2) < 0.0033 0.055(2)

Symmetry S(σd) full S(σd) core S(σd) external related pair molecule fragment rotors of atoms 5,6 0.073(2) < 0.0014 0.074(2) 5,7 0.042(2) < 0.0014 0.042(2) 5,8 0.065(2) < 0.0022 0.066(2) 6,7 0.065(2) < 0.0022 0.065(2) 6,8 0.064(2) < 0.0008 0.065(2) 7,8 0.048(2) < 0.001 0.049(2)

The bonds S4 S(S4) full S(S4) core S(S4) external axis passes molecule fragment rotors through 5-8,6-7 0.033(2) < 0.004 0.032(2) 5-7,6-8 0.073(2) < 0.0017 0.073(2) 5-6,7-8 0.084(2) < 0.002 0.084(2)

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Table 11B. CSM values for the C1 symmetric molecule (24B).

Symmetry S(C2) full S(C2) core S(C2) external related pair molecule fragment rotors of atoms 5,6 0.019(1) < 0.0008 0.018(2) 5,7 0.016(1) < 0.0011 0.016(2) 5,8 0.016(2) < 0.001 0.017(1)

Symmetry S(C3) full S(C3) core S(C3) external related set of molecule fragment rotors atoms 6,7,8 0.0010(3) < 0.0009 0.0011(4) 5,7,8 0.022(2) < 0.0008 0.022(2) 5,6,8 0.024(2) < 0.002 0.023(2) 1,2,3 0.024(1) < 0.002 0.023(2)

Symmetry S(σd) full S(σd) core S(σd) external related pair molecule fragment rotors of atoms 5,6 0.0126(5) 0.0000 0.012(1) 5,7 0.014(1) < 0.0018 0.013(1) 5,8 0.011(1) < 0.0019 0.010(1) 6,7 0.017(2) < 0.0008 0.018(2) 6,8 0.018(1) < 0.0008 0.018(1) 7,8 0.017(1) 0.0000 0.017(2)

The bonds S4 S(S4) full S(S4) core S(S4) external axis passes molecule fragment rotors through 5-8,6-7 0.022(1) < 0.0026 0.021(2) 5-7,6-8 0.020(1) < 0.0024 0.019(1) 5-6,7-8 0.022(2) < 0.0024 0.021(2)

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Table 11C. CSM values for the C1 symmetric molecule (24C).

Symmetry S(C2) full S(C2) core S(C2) external related pair molecule fragment rotors of atoms 5,6 0.015(1) < 0.0025 0.015(1) 5,7 0.017(1) < 0.0018 0.019(1) 5,8 0.025(2) < 0.0027 0.025(2)

Symmetry S(C3) full S(C3) core S(C3) external related set of molecule fragment rotors atoms 6,7,8 0.042(2) < 0.0034 0.043(2) 5,7,8 0.021(2) < 0.0008 0.022(2) 5,6,8 0.035(2) < 0.0032 0.035(2) 1,2,3 0.043(2) < 0.0031 0.043(2)

Symmetry S(σd) full S(σd) core S(σd) external related pair molecule fragment rotors of atoms 5,6 0.036(2) < 0.0028 0.036(2) 5,7 0.036(2) < 0.0002 0.036(2) 5,8 0.030(1) 0.0000 0.030(1) 6,7 0.009(1) < 0.0026 0.009(1) 6,8 0.028(1) < 0.0006 0.028(1) 7,8 0.044(2) < 0.0007 0.045(2)

The bonds S4 S(S4) full S(S4) core S(S4) external axis passes molecule fragment rotors through 5-8,6-7 0.033(1) < 0.0035 0.032(1) 5-7,6-8 0.030(2) < 0.0037 0.030(2) 5-6,7-8 0.035(2) < 0.0037 0.035(2)

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All the crystalline state asymmetric molecules 24A-C when resident within the confines of the crystal lattice exhibit very low CSM values for the suboperations of the Td point group, but their pseudo symmetry is actually higher (pseudo Td symmetry). A symmetry descriptor such as ‘asymmetric’ is based on idealities. However, it brings to mind visions of distortion and chirality when in actuality the structure may only be very slightly distorted from a higher symmetry. Thus, while the crystalline molecules are described as ‘asymmetric’, their pseudo symmetry might even be an achiral higher pseudo symmetry.

The CSM method can quantify the amount of Td symmetry in a particular crystalline molecule by an analogy to its use with YISJUF. If all of the CSM values for the Td suboperations exhibited by bona fide Cs 23 and asymmetric 24A-C are summed up and averaged for each molecule, one can rank the molecules in terms of their distortion from ideal Td symmetry. This was done and the 'Td character' values were:

0.0030 for Cs 23, 0.0173 for C1 24B, 0.0299 for C1 24C, and 0.0540 for C1 24A. It is readily apparent that the three molecules in the OHABEE asymmetric unit differ in terms of their 'Td character'. In the next two subprojects of this thesis I will use new algorithms whose development was based on the original CSM.

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Two constitutional isomeric models were calculated: one with a carbon core, and four

–Si(CH3)3 substituents, and the other with a silicon core and four –C(CH3)3 groups. While in fairness these models are not diastereomers it was felt that one could gain some insight into their relative stabilities. The B3LYP/6-31g* energy of a molecule with a carbon core, attached to four Si(CH3)3 groups is 80.3 kcal/mol lower than a molecule with a Si core attached to four C(CH3)3 groups. This might be due to the fact that a better overlap can be formed between carbon atoms, which are second row elements, compared to those of Si atoms, which are third row elements. Therefore, a carbon cage with Si substituents seems to be preferred over a Si core with carbon substituents.

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4C.4 CSM Studies on the Distortion of Crystalline Molecules with Very Large, Spatially Close Three-Fold Rotors on a Tetrasilatetrahedrane Skeleton.

Tetrakis(tri-tert.-butylsilyl)-tetrasilatetrahedrane (25, CCDB REF code PEMNEA), (t-

Bu3Si)4Si4, is a tetrahedrane with a Si core and very large –Si(t-Bu)3 rotors 80 80 [Si8C48H108]. It crystallized in the trigonal P213 space group. The asymmetric unit contains two coaxial molecules 25A,B on the same special position of three-fold rotational symmetry.80 The asymmetric unit also contains one bis(tri-t-butylsilane) dimer and one benzene-d6 solvent.80 A pseudopolymorph (CCDB REF code NAJRAR) of tetrakis(tri-tert.-butylsilyl)-tetrasilatetrahedrane 25C crystallizes in the 81 monoclinic P21/c space group together with benzene-d6 solvent.

t-Bu t-Bu Si t-Bu

t-Bu Si t-Bu t-Bu Si Si Si Si Si t-Bu t-Bu t-Bu t-Bu Si t-Bu t-Bu

Multinuclear solution state NMR80 studies of 25 are consistent with a structure having symmetry equivalent t-Bu3Si, Sicore, Cquat, Cmethyl, and Hmethyl nuclei. A peak at 32.16 ppm (methyl) and one at 24.68 (quat) were measured in the13C {1H} NMR 1 (C6D6) spectrum. One peak at 1.36 ppm was present in the H NMR spectrum. A 29 peak at 53.07 ppm (t-Bu3Si) and one at 38.89 ppm (Si4) were found in the Si NMR spectrum. No mention was made of variable temperature NMR studies, so one does not know if the liquid state structure arises from stable or rapidly inconverting T symmetry enantiomers. Judging by the very large bulk of the rotor units, it is not very reasonable that the molecule exhibits Td symmetry in solution.

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Occupancy of a special position of three-fold rotational symmetry shows that the high symmetry structure of 25A,B has been desymmetrized to C3 symmetry in the P213 crystal, while 25C occupies a general position of symmetry in the P21/c crystal and has become asymmetric. The structure of crystalline 25A is depicted in Figure 50.

Figure 50: The C3 symmetric structure of crystalline 25A.

There are two types of –Si(t-Bu)3 rotors in 25A – a symmetric and an asymmetric rotor. One of the four –Si(t-Bu)3 rotors occupies the C3 axis and has 3 homotopic t- Bu groups (green).

There are three homotopic –170.94° Si–Si–Si–C(quat) torsion angles (red) on the C3 symmetric –Si(t-Bu)3 rotor of 25A. One of three homotopic –9.06° twist angles based on the above mentioned torsion angle in 25A is shown in Figure 51. Each of the asymmetric –Si(t-Bu)3 rotors has three diastereotopic Si–Si–Si–Cquat torsion angles: –166.35°, –170.09°, and –172.09°. Therefore, the average twist of the four –

Si(t-Bu)3 rotors is –10(2)° for 25A.

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Similarly, all of the torsion angles for molecule 25B have a negative sign, and the average twist of the four –Si(t-Bu)3 rotors is –9.2(7)° for 25B. Thus, both 25A,B each have correlated rotor twisting since all twist angles have the same sign (twist sense) and approximately the same magnitudes. The two chiral molecules (25A,B) in the asymmetric unit happen to have the same handedness, i.e. they are chiral proximate homomers.

t-Bu t-Bu

Si t-Bu

t-Bu Si t-Bu Si t-Bu t-Bu t-Bu Si Si Si Si t-Bu t-Bu

t-Bu t-Bu

Figure 51: Three-fold rotationally symmetric –Si(t-Bu)3 rotor (red) in 25A having homotopic –9.06° twist angles (other rotors are asymmetric).

The CSM values for suboperations of the Td point group are listed in Tables 12A,B for silatetrahedranes 25A,B. A numbering diagram of carbons referred to in Tables 12A,B is presented in Figures 52,53, respectively. Inspection of Tables 12A,B show that all CSM values for 25A of the First Kind are low (0.0012-0.0017) while those for the Second Kind are very high (1.1000). Similar magnitude corresponding values of 0.018-0.025 and 1.17-1.18 are seen for 25B.

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The average of the CSM values for the First Kind provides us with a 'T character' value of 0.0012, while the average for all the suboperations of Td (i.e. all the CSMs for both the First and Second Kind) gives a 'Td character' value of 0.762. The 'T character' and 'Td character' values are 0.0018 and 0.606 respectively for the crystalline 25B partner. Therefore, while crystalline 25A,B both have bona-fide C3 symmetry chiral conformations when resident within the confines of the crystal lattice, their pseudo symmetry is actually higher (a pseudo T symmetry (–)-twist sense chiral conformation). It is not surprising to observe that the CSM values for operations of both the First and Second Kind are all low in magnitude for the rigid core of four silicon atoms. The CSM values of the fragment which consists of rotors only, are the same as those of the entire molecule, which is also in accord with their size and twist. Figures 54-58 illustrate pseudo symmetry operations for color coded crystalline 25A. The relatively large rotor size enables the viewer to visually discern the absence of bona-fide symmetry in these figures.

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Figure 52: Numbering diagram for atoms in C3 symmetry tetrakis(tri-tert.- butylsilyl)-tetrahedrane 25A (PEMNEA).

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Figure 53: Numbering diagram for carbon atoms in C3 symmetry tetrakis(tri- tert.-butylsilyl)-tetrahedrane 25B (PEMNEA).

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Table 12A. CSM values for C3 symmetric molecule (25A).

Symmetry S(C2) full S(C2) core S(C2) external related pair molecule fragment rotors of atoms 2,3 0.0012(3) < 0.0026 0.0012(3) 2,21 0.0012(3) < 0.0026 0.0012(3) 2,22 0.0012(3) < 0.0036 0.0012(3)

Symmetry S(C3) full S(C3) core S(C3) external related set of molecule fragment rotors atoms 3,21,22 0.0000 < 0.0028 0.0000 2,21,22 0.0017(3) < 0.0053 0.0017(3) 2,3,22 0.0017(3) < 0.0053 0.0017(3) 2,3,21 0.0017(3) < 0.0036 0.0017(3)

Symmetry S(σd) full S(σd) core S(σd) external related pair molecule fragment rotors of atoms 2,3 1.10(1) < 0.0028 1.10(1) 2,21 1.10(1) < 0.0030 1.10(1) 2,22 1.10(1) < 0.0037 1.10(1) 3,21 1.10(1) 0.0000 1.10(1) 3,22 1.10(1) < 0.0027 1.10(1) 21,22 1.10(1) < 0.0025 1.10(1)

The bonds S4 S(S4) full S(S4) core S(S4) external axis passes molecule fragment rotors through 2-3,21-22 1.10(1) < 0.0047 1.10(1) 2-21,3-22 1.10(1) < 0.0047 1.10(1) 2-22,3-21 1.10(1) < 0.0047 1.10(1)

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Table 12B. CSM values for C3 symmetric molecule (25B).

Symmetry S(C2) full S(C2) core S(C2) external related pair molecule fragment rotors of atoms 2,3 0.018(2) < 0.006 0.018(1) 2,21 0.018(2) < 0.006 0.018(1) 2,22 0.018(2) < 0.006 0.018(1)

Symmetry S(C3) full S(C3) core S(C3) external related set of molecule fragment rotors atoms 3,21,22 0.025(2) < 0.008 0.024(1) 2,3,22 0.025(2) < 0.008 0.024(1) 2,3,21 0.025(2) < 0.008 0.024(1) 3,21,22 0.0000 0.0000 0.0000

Symmetry S(σd) full S(σd) core S(σd) external related pair molecule fragment rotors of atoms 2,3 1.17(1) < 0.006 1.17(1) 2,21 1.17(1) < 0.006 1.17(1) 2,22 1.17(1) < 0.006 1.17(1) 3,21 1.18(1) 0.0000 1.19(1) 3,22 1.18(1) 0.0000 1.19(1) 21,22 1.18(1) 0.0000 1.19(1)

The bonds S4 S(S4) full S(S4) core S(S4) external axis passes molecule fragment rotors through 2-3,21-22 1.17(2) < 0.008 1.18(1) 2-21,3-22 1.17(2) < 0.008 1.18(1) 2-22,3-21 1.17(2) < 0.008 1.18(1)

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Figure 54: A pseudo C3 axis in 25A.

Figure 55: A pseudo C2 axis in 25A.

Figure 56: A pseudo  -plane in 25A.

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Figure 57: Another pseudo  -plane in 25A.

Figure 58: A pseudo S4-axis in 25A.

Does occupancy of a general position of symmetry in the unit cell markedly effect the CSM values of tetrakis(tri-tert.-butyl-silyl)-tetrasilatetrahedrane (25C) when it crystallizes without crystallographic symmetry structural constraints in the monoclinic 81 P21/c space group? It is fortunate that the C1 symmetry silatetrahedrane 25C can be compared with the C3 symmetry 25A,B molecules discussed above. Not surprisingly, the C1 symmetric 25C also shows a correlated twist sense for its large rotors. All of the rotors are asymmetric and all of the Si–Si–Si–C(quat) torsion angles are diastereotopic. (see Table 13)

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Table 13: The Si–Si–Si–C(quat) torsion angle values for different rotors in crystalline molecule (25C)

Si–Si–Si–C(quat)

Rotor 1 –170.66° –170.65° –171.10°

Rotor 2 –167.93° –169.32° –168.32°

Rotor 3 –163.91° –162.98° –162.05°

Rotor 4 –165.40° –167.46° –167.73°

The average twist angle of these diverse values is –13(3)° which compares nicely with the –10(2)° and –9.2(7)° twist values for 25A,B. The CSM values for 25C are listed in Table 12C, and the numbering diagram is depicted in Figure 57. The 'T character' value is 0.0249 and the 'Td character' value is 1.381 for the asymmetric 25C, compared to 'T character' values of 0.0012 and 0.0018 for the C3 symmetric 25A,B. Thus, it appears that the imposition of crystallographic three-fold rotational symmetry on 25A,B has given them slightly higher ‘T symmetry character’. However, the inherent forces acting on the adjacent –Si(t–Bu)3 large rotors has resulted in a correlation of their twists irrespective of the crystal environment of 25A-C. Finally, the pseudo T symmetry character of 25A-C has never been commented upon to the best of our knowledge.

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Figure 59: Numbering diagram for atoms in the C1 symmetric tetrakis(tri-tert.- butylsilyl)-tetrahedrane 25C (NAJRAR).

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Table 12C. CSM values for the C1 symmetric molecule (25C).

Symmetry S(C2) full S(C2) core S(C2) external related pair molecule fragment rotors of atoms 1,2 0.048(2) < 0.001 0.049(1) 1,3 0.0185(5) < 0.001 0.019(1) 1,4 0.0344(5) < 0.001 0.0345(5)

Symmetry S(C3) full S(C3) core S(C3) external related set of molecule fragment rotors atoms 2,3,4 0.016(1) < 0.0044 0.016(3) 1,3,4 0.031(1) < 0.0044 0.031(1) 1,2,4 0.052(1) < 0.0044 0.052(1) 1,2,3 0.049(1) < 0.0044 0.049(1)

Symmetry S(σd) full S(σd) core S(σd) external related pair molecule fragment rotors of atoms 1,2 2.23(1) < 0.0012 1.635(5) 1,3 2.52(1) < 0.0012 1.660(1) 1,4 2.57(1) < 0.0006 1.65(2) 2,3 2.50(1) 0.0000 1.656(1) 2,4 3.18(1) < 0.0015 2.035(5) 3,4 3.15(1) < 0.0012 1.665(5)

The bonds S4 S(S4) full S(S4) core S(S4) external axis passes molecule fragment rotors through 1-2,3-4 1.66(1) < 0.002 1.67(1) 1-3,2-4 1.64(1) < 0.002 1.65(1) 1-4,2-3 2.35(1) < 0.001 2.37(1)

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4C.5 Discussions on the CSM studies of the distortion of crystalline molecules with distant or close three-fold rotors:

The molecules that had rotors which were close to each other, showed correlated twist sense, and those that were far away from each other, did not. There is no quantitative correlation of the CSM values for symmetry operations of the First Kind with the degree of twist-correlation between rotors. As can be seen in Figure 60, deviations from symmetry of the Second Kind increase with the total number of twisted (non- staggered conformation) tert.-butyl rotors in a molecule.

Figure 60: Increase in deviation from symmetry of the Second Kind, e.g. S(), as a function of the total number of twisted (non-staggered conformation) tert.- butyl rotors in a molecule.

Since symmetry is a structural constraint which generates mirror image copies of molecule subunits, one of the primary purposes of this subsection of the research was to test the sensitivity of the CSM measurement to desymmetrization when these constraints are removed upon occupancy of general positions of symmetry in a crystal lattice. In other words, what happens to a solution - state symmetrical molecule when it may not express its symmetry since the only site symmetry operations are the trivial ones of identity and translation? The CSM was used in order to quantify the effect of desymmetrization on molecules whose structural distortion was difficult to ascertain by mere visual inspection of its molecular graphic (e.g. crystalline dodecahedrane, 4) as well molecules whose distortions were more easily recognized

(e.g. tetrasilatetrahedrane with its large –Si(t–Bu)3 rotors, 25).

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The CSM values ranged varied from 0.0004 for the inability of crystalline dodecahedrane to express bona fide five-fold rotational symmetry to 3.18 for the inability of adjacent large Si(t–Bu)3 rotors of 25 to exhibit bona fide staggered conformations. The use of three-fold rotors proved to be a useful device to probe the general concept of ‘conformation’. Occupancy of a general position of symmetry by 25C resulted in all of its rotors being asymmetric. Occupancy of a special position of

C3 symmetry resulted in having one symmetric rotor and three asymmetric ones in

25A,B. S(C3) was seen to be sensitive to these subtle structural differences since its mean value for the four rotors was 0.036(17) for 25C, and 0.017(2) and 0.025(2) for

25A and 25B, respectively. Despite the relatively higher S(C3) value for 25C relative to those of 25A,B, the value is still low enough for the three-fold rotor rotation axes in

25C to be described as pseudo C3 – axes. The expression of a solution-state molecule’s ideal symmetry in a crystalline molecule is usually not the norm due to packing considerations which favor either occupancy of a general position or desymmetrization to a lower symmetry. The use of the CSM proved to be a useful method in assigning conformational descriptors to crystalline molecules by affixing the word ‘pseudo’ to the conformational symmetry. Three molecules which visually appeared to be geometrically similar (e.g. 24A-C) were shown to exhibit two different conformational geometries by use of their CSM values. Pseudo symmetry probed by the CSM method is a useful tool to assign conformational descriptors since the concept of ‘conformation’ is a similarity and not an identity.

Some other structural conclusions can be made. How crucial was the existence of long bonds [2.33 Å Sicore—Sicore, 2.37(2) Å Sicore—Sirotor and 1.94(2) Å Sirotor—Cquat] to the successful realization of synthesizing a 25A molecule having such bulky rotors? In other words, what are the chances that a carbon tetrahedrane skeleton could be synthesized with large –Si(t–Bu)3 rotors appended to it? Probably quite low! The corresponding distance for a Ccore—Ccore tetrahedrane bond is 1.486(2) Å and as a first approximation a 1.94(2) Å Sirotor—Cquat bond length can be utilized for a Sirotor—Ccore bond.

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An edge fragment model of a carbon tetrahedrane skeleton analogue of 25A can be constructed from the x-ray structure using molecular graphics and the two new bond lengths, see Figure 61. Inspection of the extremely close H…H contacts of ca. 1.0 Å in the model strongly predicts a failure for such a synthesis. However, one can envision long 2.38 Å Si(core)—Si(core) bonds on an octasilacubane skeletal frame.

What is the chance of an octasilacubane being synthesized with large –Si(t–Bu)3 rotors? Again, probably low, since the 144° Si(rotor)–Si(core)–Si(core) angle in 25A would have to close to ca. 125° for a octasilacubane analogue. A similar molecular graphics model shows even closer H…H contacts for this new model than those shown in Figure 61

Figure 61: Close contacts between an edge fragment model of a carbon tetrahedrane analogue of 25A constructed using molecular graphics.

Using a variety of force fields, Mislow predicted that tetra-tert-butyl tetrahedrane would be T symmetric in its ground state.31 In an earlier article, he showed that the rotation of the tert-butyl groups in tri-tert-butyl silane should be correlated.31 The achiral pseudo Td symmetry crystal geometries of 23 and later 24 may be a surprise to those who expect to ‘see’ a crystallographic ‘ball and stick model of Mislow’s31 predicted T structure for these molecules.

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A possible explanation is that crystal packing forces have overcome the slightly higher energy of the Td conformer. The crystallographic validation of the ‘twisted rotor’ prediction finally came in 1996 when silicon skeleton Platonic solid geometry molecules (crystalline 22 and 25, respectively) were synthesized with large rotors. 25A, 25B and 25C had pseudo T symmetry, as expected by Mislow's prediction. 22A and 22B had pseudo O conformations, which were also compatible with Mislow's predication. 22C had pseudo Oh symmetry, and this conformation could only be explained by a detailed analysis of the packing of 22. The analysis of the packing arrangement revealed that there are interesting host guest chiral recognition interactions, which are made possible by the existence of the almost achiral, pseudo O symmetric 22C.

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4D. The Continuum of RmS Symmetry Measures of P21/c Family ‘Supergroup

Character’ in Lower Symmetry P21, P–1, and Pn Space Group Crystals Containing Multiple Molecules in the Asymmetric Unit.

It is obvious that the ‘P21/c family’ of space groups includes P21/c, P21/a, and P21/n as members. In this Results and Discussion section we will use the term ‘P21/c family’ in a general sense when referring to its general traits, and we will reserve the actual P21/c, P21/a, and P21/n descriptor for specific situations when it is necessary.

4D.1 RmS Symmetry Measures of Monoclinic P21 Chiral Space Group Crystals of (±)-(1RS,3SR,4RS)-1-Phenyl-cis-3,4-Butano-3,4,5,6-Tetrahydro-1H-2,5- Benzoxazocine Hydrochloride, 11.

+ NH2 Cl

(±)-(1RS,3SR,4RS)-1-phenyl-cis-3,4-butano-3,4,5,6-tetrahydro-1H-2,5- benzoxazocine hydrochloride (N-desmethyl-cis-3,4-butano-nefopam HCl, 11) was synthesized by Itzhak Ergaz.54 It gave a ‘false conglomerate’ of chiral crystals belonging to the chiral monoclinic P21 space group with a = 10.224(2) Å, b = 13.969(2) Å, c = 12.724(2) Å,  = 98.996(2)°, V = 1794.88 Å3, and Z = 4 which shows that there are 2 molecules in the asymmetric unit and four molecules in the unit cell.

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In order to unequivocally define these four molecules, we will state their symmetry transforms in the .cif (crystal information file) and will arbitrarily choose the C(1) atom in the molecule to give its coordinates and atom-number (#) from the Mercury program list of atoms derived for a complete unit cell.77 (Table 14)

Table 14: The four molecules in the unit cell of (11). Atom numbers are from the Mercury77 program and are found upon choosing ‘display’, ‘more information’, and then ‘atom list’.

Molecule Symmetry coordinates of C(1), a Atom transform representative atom number (1R,3S,4R)-A [x,y,z] [0.9310, 0.16396, 0.07255] # 3 (1S,3R,4S)-B [x,y,z] [0.6377, 0.84018, 0.42829] # 97 (1R,3S,4R)-A [1–x, ½+y,1–z] [0.0690, 0.66396, 0.92745] # 49 (1S,3R,4S)-B [1–x, –½+y,1–z] [0.3623, 0.34018, 0.57171] # 143

The crystal packing arrangement shows bona fide 21 screw-rotation, and pseudo inversion plus pseudo n-glide symmetry. The pseudo positions of symmetry are correlated with a P21/n supergroup arrangement. A simple translation along the glide plane is performed in order to be able to compare the chiral P21 crystal to the achiral

P21/c analogue with unique axis b, cell choice 2 from the international tables. The setting is called P21/n, since it has an n-glide.

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Figure 62 shows the differences in origin chosen by the crystallographer for the P21 unit cell versus that of a P21/n unit cell is presented below:

a 0

0' a' b =1/4 b =3/4

Figure 62: The differences in origin chosen by the crystallographer for the P21 unit cell (top rectangle) versus that of a P21/n unit cell (bottom rectangle).

If the unit cell would have had P21/n symmetry, all four molecules in the unit cell would have been symmetry equivalent. Since the crystal has P21 symmetry and pseudo P21/n symmetry, there are four molecules in the unit cell, and two in the asymmetric unit. If the molecules were color coded according to symmetry equivalence, two colors would be needed – one for each symmetry equivalent set.

Figure 63 shows a diagram of the four equivalent positions in the ideal P21/n cell.

Since our crystal has pseudo P21/n symmetry, there are two sets of symmetry equivalent positions. Positions which belong to the same set were given the same color. The pseudo inversion and pseudo glide operations interconvert yellow and white positions (Figure 63) which are not symmetry equivalent.

They are pseudo equivalent, and it is possible to check how different these positions are, by evaluating their deviation from ideal symmetry.

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0’ a’

½+ , ½ - ,- c’

Figure 63: Illustration of the four positions which are equivalent in the ideal

P21/n unit cell. They are color coded according to symmetry equivalence. Since our crystal is pseudo P21/n, there are two pairs of symmetry equivalent positions. A position which is marked by an empty circle has opposite handedness to a circle which has a comma inside it. A plus sign means ‘+y’, a minus sign means ‘–y’, ½ + means ‘½ + z,’ and ½ – means ‘½ – z’.

The crystal has a set of four pseudo inversion centers that are parallel to the ac-plane. (see Table 15)

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Table 15: Pseudo-inversion and pseudo-glide symmetry relationships between molecules in crystalline (11) Type of Pair of molecules related by this Figure pseudosymmetry type of pseudosymmetry inversion (1R,3S,4R)-A[x,y,z] and 64 (1S,3R,4S)-B[x,y,z] inversion (1R,3S,4R)-A [2–x,y+½,1–z] and 65 (1S,3R,4S)-B [1–x,y–½,1–z]. inversion (1R,3S,4R)-A [1–x,y+½,1–z] and 66 (1S,3R,4S)-B [1–x,y–½,1–z] inversion (1R,3S,4R)-A [1-x, ½+y,1-z] with 67 (1S,3R,4S)-B [1-x,- ½+y,1-z] glide translation (1R,3S,4R)-A[x,y,z] with 69 (1S,3R,4S)-B [1-x,- ½+y,1-z] glide translation (1R,3S,4R)-A [1-x, ½+y,1-z] with 69 (1S,3R,4S)-B[x,y,z])

The four pseudo inversion centers are not equivalent. It also has two pseudo n-glide- reflection planes that cut the z-axis. (Table 15) Since the same bona fide 21 screw axis converts (1R,3S,4R)-A[x,y,z] with (1R,3S,4R)-A [1-x, ½+y,1-z] as well as (1S,3R,4S)-B[x,y,z] with (1S,3R,4S)-B [1-x,- ½+y,1-z], we will consider only the single axis which goes through the cell at the ideal points [½,0,½] and [½,1,½].

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4D.1a Location of the pseudo positions of inversion in the N-desmethyl-cis-3,4- butano-nefopam HCl P21 chiral crystal There are four pseudo positions of inversion between the two pseudo n-glide planes in the unit cell of 11, and they are shown below in Figures 64-67). The coordinates of the statistical ‘best pseudo inversion point’ are the [xmean,ymean,zmean] of the cluster of

[xaverage,yaverage,zaverage] points calculated for all the pairs of atoms having a pseudo inversion relationship. It is a pseudo position of inversion since it is the statistical mean of the cluster and has an estimated standard deviation (as opposed to a bona fide inversion center which has no esd) plus it is dislocated from the bona fide point of inversion position. Figure 64 depicts a pseudo position of inversion between (1R,3S,4R)-A [x,y,z] and (1S,3R,4S)-B [x,y,z] showing a hydrogen-bonding ring pattern. Figure 65 shows a pseudo position of inversion between (1R,3S,4R)-A [x– 1,y,z] and (1S,3R,4S)-B [x,y,z]. Figure 66 illustrates a pseudo position of inversion between (1R,3S,4R)-A [2–x,y+½,1–z] and (1S,3R,4S)-B [1–x,y–½,1–z] and Figure 67 illustrates a pseudo position of inversion between (1R,3S,4R)-A [1–x,y+½,1–z] and (1S,3R,4S)-B [1–x,y–½,1–z] showing an edge-to-face phenyl…phenyl Coulombic interaction.

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Figure 64: The pseudo position of inversion between (1R,3S,4R)-A [x,y,z] and (1S,3R,4S)-B [x,y,z] showing a hydrogen-bonding ring pattern.

Figure 65: The pseudo position of inversion between (1R,3S,4R)-A [x–1,y,z] and (1S,3R,4S)-B [x,y,z].

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Figure 66: The pseudo position of inversion between (1R,3S,4R)-A [2–x,y+½,1– z] and (1S,3R,4S)-B [1–x,y–½,1–z].

Figure 67: The pseudo position of inversion between (1R,3S,4R)-A [1–x,y+½,1– z] and (1S,3R,4S)-B [1–x,y–½,1–z] showing an edge-to-face phenyl…phenyl Coulombic interaction.

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The xmean, ymean, and zmean coordinates describing the location of the pseudo inversion statistical ‘best point’ (the pseudo position of inversion) between the atoms in (1R,3S,4R)-A[x,y,z] and corresponding atoms in (1S,3R,4S)-B[x,y,z] were calculated by the method described in the Experimental methods section. The ‘best pseudo inversion point’ coordinates listed below where calculated with the coordinates of the N-desmethyl-cis-3,4-butano-nefopam skeleton and those for the six phenyl carbon atoms.

Table 16A: Coordinates of ‘best inversion points’– (molecules include phenyl rings) of crystalline (11)

Pseudo inversion pair Coordinates of ‘Best inversion’ point (1R,3S,4R)-A [x,y,z] and [0.779(3),0.501(2),0.250(4)] (1S,3R,4S)-B [x,y,z] (1R,3S,4R)-A [x–1,y,z] and [0.721(3),0.501 (2),0.750 (4)] (1S,3R,4S)-B [x,y,z] (1R,3S,4R)-A [2–x,y+½,1–z] and [0.279 (3),0.501 (2),0.250(4)] (1S,3R,4S)-B [1–x,y–½,1–z] (1R,3S,4R)-A [1–x,y+½,1–z] and [0.221(3),0.501 (2),0.750 (4)] (1S,3R,4S)-B [1–x,y–½,1–z]

The pseudo positions of inversion listed below where calculated for the N-desmethyl- cis-3,4-butano-nefopam pseudo enantiotopic skeletons only. Since the phenyl rings in an (1R,3S,4R)-A[x,y,z]/(1S,3R,4S)-B[x,y,z] pair are twisted differently, the coordinates below are more meaningful than those given above.

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Table 16B: Coordinates of ‘best inversion points’ of crystalline (11) - without phenyls

Pseudo inversion pair Coordinates of ‘Best inversion’ point (1R,3S,4R)-A [x,y,z] and [0.780 (3),0.501 (2),0.250(5)] (1S,3R,4S)-B [x,y,z] (1R,3S,4R)-A [x–1,y,z] and [0.720 (2),0.501(2),0.750 (5)] (1S,3R,4S)-B [x,y,z] (1R,3S,4R)-A [2–x,y+½,1–z] and [0.280(2),0.501(2),0.250(5)] (1S,3R,4S)-B [1–x,y–½,1–z] (1R,3S,4R)-A [1–x,y+½,1–z] and [0.220(2),0.501(2),0.750 (5)] (1S,3R,4S)-B [1–x,y–½,1–z]

4D.1b Development of the New RmS(Ci) Symmetry Measure Tool for Comparison of Inversion Pseudo Symmetry with Bona Fide Symmetry in the N-

Desmethyl-cis-3,4-Butano-Nefopam HCl P21 Chiral Crystal In considering the pseudo symmetry relationships of molecules or their fragments it is obvious that their distortion from ideal symmetry must be determined. All the pseudo symmetry related pairs of atoms in (1R,3S,4R)-A[x,y,z] and (1S,3R,4S)- B[x,y,z] (and symmetry-related molecules) were utilized to calculate Avnir’s Continuous Symmetry Measure (CSM) algorithm29 for inversion symmetry distortion. The S(Ci) values differed according to which pseudo-inversion center was used:

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Table 17: S(Ci) values for the pseudoinversion related pairs of molecules in crystal (11).

Pseudosymmetry pair Figure S(Ci) S(Ci) (all (no phenyl atoms) no Cl) (1R,3S,4R)-A [x,y,z] and 64 0.19(1) 0.040(1) (1S,3R,4S)-B [x,y,z] (1R,3S,4R)-A [x–1,y,z] and 65 0.46(1) 0.026(1) (1S,3R,4S)-B [x,y,z] 1R,3S,4R)-A [2–x,y+½,1–z] and 66 0.53(1) 0.027(1) (1S,3R,4S)-B [1–x,y–½,1–z] (1R,3S,4R)-A [1–x,y+½,1–z] and 67 0.78(1) 0.020(1) (1S,3R,4S)-B [1–x,y–½,1–z]

The reason for the differences in the S(Ci) values could not be clearly understood until

S(Ci) was calculated for the relationship between only the two phenyl rings. Since the most asymmetry between (1R,3S,4R)-A[x,y,z] and (1S,3R,4S)-B[x,y,z] molecules was in the twist of the phenyl rings, it was reasoned that S(Ci) values should be decreased for an ‘all atom’ calculation versus one that was ‘phenyls only’. Clearly, in a 23 atom ‘all atom’ calculation, the asymmetry contribution from the six phenyl carbons and chloride anion is ‘diluted’ upon being averaged with the better pseudo symmetry contribution from the 16 atoms of the N-desmethyl-cis-3,4-butano-nefopam skeleton.

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… Table 18: S(Ci) values vs. Phcentroid Phcentroid distance in crystal (11). … Pseudosymmetry pair Figure S(Ci) Phcentroid Phcentroid (phenyls distance only) (1R,3S,4R)-A [x,y,z] and 64 0.32(1) 15.724 Å (1S,3R,4S)-B [x,y,z] (1R,3S,4R)-A [x–1,y,z] and 65 0.29(1) 16.596 Å (1S,3R,4S)-B [x,y,z] (1R,3S,4R)-A [2–x,y+½,1–z] and 66 1.37(1) 7.184 Å (1S,3R,4S)-B [1–x,y–½,1–z] (1R,3S,4R)-A [1–x,y+½,1–z] and 67 2.64(1) 4.838 Å (1S,3R,4S)-B [1–x,y–½,1–z]

As noted in the introduction, the Avnir continuous symmetry measure (CSM) of a structure is a normalized root-mean-square distance function from the closest structure which has the desired symmetry, i.e. the nearest perfectly symmetric object with respect to the required symmetry. For chemistry, the ‘object’ investigated may be a full molecule, fractions of a molecule, or ensembles of molecules. The algorithms incorporate a size normalization function since they were designed to afford the same S-value deviation from an ideal symmetry for each of two objects whose sole geometrical difference was just their relative spatial scale, i.e. one was either enlarged or reduced in size vis-à-vis the other. Thus, a size normalization function based on ‘centers of mass’ is utilized in the algorithms so that the calculation for one of two differently scaled objects would afford the same S-value when it was performed for the other. The four pseudo positions of inversion are between differently placed molecules within the P21 unit cell of crystalline 11. Four different values of S(Ci) (ranging from 0.29 to 2.64) were initially obtained for the pseudo enantiotopic skeletons. Initially, the ‘centers of mass’ of the (1R,3S,4R)-A[x,y,z] or (1S,3R,4S)-B[x,y,z] molecules could not be readily visualized by mere inspection.

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However, this was not the case when S(Ci) was calculated for an ensemble of just two phenyl rings. As a result, it became obvious that the S(Ci) values were inversely … proportional to the Phcentroid Phcentroid distance. Since the twist difference of the phenyl rings was the most striking difference between (1R,3S,4R)-A[x,y,z] and (1S,3R,4S)-B[x,y,z] pseudo-inversion related molecules, it was apparent from the

S(Ci) results that as these dissimilar phenyl moieties became spatially closer, then the degree of distortion from bona fide inversion symmetry increased due to the size normalization function. Therefore, the instead of using the CSM values given in the input file, the RMSD of superimposition of the initial input object's coordinates on the corresponding coordinates of the nearest perfectly inversion symmetric object was calculated. These two objects were obtained from the CSM program's output file.

The resulting values will be called 'rmS(Ci) values'. The CSM values do not have units because of the normalization. The rmS(Ci) values are in Å units, since they are not normalized. Now, the same rmS(Ci) 0.088(1) Å value was obtained for the pair of pseudo-enantiotopic skeletons irrespective of their relative distance in the ensemble. If phenyls were added to the skeletons, the value increased to 0.26(1) Å.

A goal of this subproject was to use symmetry measures to rank different lower symmetry space group crystals that share a particular supergroup. For example, if the appropriate pseudo positions of symmetry are present in P21, P–1, or Pc (Pn) crystals having two molecules in the asymmetric unit, then the packing in these crystals might show the ubiquitous P21/c family supergroup spatial arrangement. It was expected that when comparing various crystals, there should be a range of distortion in the pseudo symmetry relationships vis-à-vis those of the ideal supergroup. Since pseudo symmetry-related molecules will be differently spaced in different crystals, it is the rmS(Ci) values that have to be compared, and not the S(Ci)-values.

Using this technique, an rmS(Ci) = 0.46(1) Å was calculated for the distortion from bona fide inversion symmetry between just the two diastereotopically twisted phenyl rings. This is more than five times the rmS(Ci) value of 0.088(1) Å found for just the pseudo enantiotopic skeletons.

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4D.1c Relocation of Pseudo Positions of Inversion From the Special Positions for Bona Fide Inversion Symmetry In investigating pseudo symmetry relationships between molecules one must consider the relocation of the best least squares pseudo symmetry element vis-à-vis the special position. The fact that the esd values of the coordinates defining the above points are not integer-0 is to be expected since we are dealing with pseudo symmetry. The larger the esd-values of the coordinates defining the points, the less accurate is the location of that point. In a bona fide P21/n cell such as depicted above in Figure 62 the four special positions of inversion symmetry are located at: [¼,½,¼] & [¼,½,¾] & [¾,½,¼] & [¾,½,¾]. These four points are not equivalent. Clearly, the pseudo inversion ‘best points’ in crystalline 11 are ‘general positions of symmetry’. General positions exhibit only the bona fide trivial operations of identity and translation. In order to measure the spatial relocation of these general positions from their ideal counterparts, the fractional coordinates of atoms and special positions of inversion symmetry were first converted into Cartesian coordinates. The Cartesian-space distances between the ideal inversion center and the best pseudo inversion points were found. (the calculation method is decribed in detail in the method section) The distance is: rmS(relocation-Ci) = 0.30(6) Å for the four pseudo inversion points calculated for only the N-desmethyl-cis-3,4-butano-nefopam skeletons and it is the same value when calculated for the skeletons plus the phenyls but without the chlorides.

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4D.1d RmS(21), Comparison of 21 Screw-Rotation Pseudo Symmetry with Bona

Fide Symmetry in the N-desmethyl-cis-3,4-Butano-Nefopam HCl P21 Chiral Crystal

There is nothing to compare, since the 21 screw-rotation transformation is bona fide in space group P21, see Figure 68. Therefore, S(21) = rmS(21) = rmS(relocation-21) = integer-zero in this case.

Figure 68. Bona fide 21-axis in magenta relating (1R,3S,4R)-A[x,y,z] (black colors) with (1S,3R,4S)-B[x,y,z] (red colors)

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4D.1e Location of the Pseudo Positions of n-Glide-reflection in the N-

Desmethyl-cis-3,4-Butano-Nefopam HCl P21 Chiral Crystal

Figure 69: Two pseudo n-glide-reflection planes (red) relating (1R,3S,4R)- A[x,y,z] with (1S,3R,4S)-B [1-x, ½+y,1-z] on right and (1R,3S,4R)-A [1-x, ½+y,1- z] with (1S,3R,4S)-B[x,y,z] on left. Both pseudo n-glide planes in this crystal are symmetry equivalent and both are parallel to the ac-plane. They are illustrated in Figure 69. One pseudo plane relates atoms in (1R,3S,4R)-A[x,y,z] with corresponding atoms in (1S,3R,4S)-B [1-x,-

½+y,1-z] and cuts the y-axis at ymean = 0.251(3) for just the N-desmethyl-cis-3,4- butano-nefopam skeletons, where ymean was calculated in the same manner as described for the pseudo inversion points. The second pseudo n-glide plane relates between corresponding atoms in (1R,3S,4R)-A [1-x, ½+y,1-z] and (1S,3R,4S)-

B[x,y,z] and cuts the y-axis at ymean = 0.751(3) for just the pseudo enantiotopic skeletons. The fact that the esd value for ymean is not integer-0 is to be expected since we are dealing with pseudo symmetry. The larger is the esd of the points defining the pseudo-plane, the more this set of points is non-planar.

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4D.1f RmS(n-glide), Comparison of n-Glide-Reflection Pseudo Symmetry with

Bona Fide Symmetry in the N-Desmethyl-cis-3,4-Butano-Nefopam HCl P21 Chiral Crystal An n-glide-reflection symmetry operation is composed of two symmetry transform components: mirror-reflection and two translations: one by ½ of the unit’s a-axis cell length & one by ½ of the unit cell’s c-axis length. The two components can be separated, since the reflection leaves x and z invariant, while the translation leaves y invariant. If the pseudo n-glide-reflection plane relates atoms in (1R,3S,4R)-A[x,y,z] with corresponding atoms in (1S,3R,4S)-B [1-x,- ½+y,1-z], then x-translationmean = 0.563(8) (calculated without the phenyl carbons and the chloride anions). Since it is a n-glide, z-translationmean = –0.507(10) which is calculated by analogy. Once again, the fact that the both the x-translationmean and the z-translationmean are not the fractional number ½ nor are their esd values integer-0 is to be expected since we are dealing with pseudo symmetry. The RMS difference between the actual x-translation of the atoms in (1S,3R,4S)-B [1-x,- ½+y,1-z] relative to corresponding atoms in (1R,3S,4R)-A[x,y,z] compared to the fractional number ½ z-translation for a bona fide n-glide-reflection was calculated according to the method in the Experimental section as rmS(x-translation)coordinate = 0.0594. It must be multiplied by the 10.224 Å a-axis length to afford an rmS(x-translation) = 0.6076 Å. In a similar manner, the rmS(z- translation)coordinate = 0.0125 must be multiplied by the 12.724 Å c-axis length to give rmS(z-translation) = 0.1595 Å. Since it is an n-glide, the two values are averaged to give the rmS(n-translation) = 0.38(6) Å for just the N-desmethyl-cis-3,4-butano- nefopam skeletons. The rmS(n-translation) = 0.75(6) Å increased when it was calculated for the skeletons plus the phenyls but without the anions.

To calculate the pseudo n-glide-reflection’s mirror symmetry component, the x- and z-coordinates of (1S,3R,4S)-B [1-x,- ½+y,1-z] atoms were first transformed by the respective x-translationmean and z-translationmean values noted above. In this manner a dummy (1S,3R,4S)-B [1-x,- ½+y,1-z] molecule was generated next to (1R,3S,4R)- A[x,y,z].

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Avnir’s Continuous Symmetry Measure (CSM) algorithm29 for mirror-reflection gave average S(Cs) = 0.61(5) (for skeletons plus the phenyls but without the chlorides), which decreased to S(Cs) 0.07(1) for just the N-desmethyl-cis-3,4-butano- nefopam skeletons. When the size normalization function was removed, the average value for both planes is rmS(Cs) = 0.91(1) Å (with only the phenyl carbons being calculated), which decreased to rmS(Cs) = 0.26(1) Å (for the N-desmethyl-cis-3,4- butano-nefopam skeletons plus the phenyl carbons but without the anions), which then further decreased to rmS(Cs) = 0.076(1) Å for just the pseudo enantiotopic skeletons.

Therefore, the CSM for the pseudo n-glide-reflection (skeletons plus the phenyls but without the anions) is: rmS(Cs) + rmS(n-translation) = 0.26(1) Å + 0.75(6) Å = rmS(n-glide) = 1.01(6) Å

The CSM for the pseudo n-glide-reflection is reduced for just the N-desmethyl-cis-

3,4-butano-nefopam skeletons: rmS(Cs) + rmS(n-translation) = 0.076(1) Å + 0.38(6) Å = rmsS(n-glide) = 0.46(6) Å. This is the only important value since it pertains to the deviation from ideal n-glide-reflection symmetry between the pseudo-enantiotopic skeletons.

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4D.1g Relocation of the Pseudo Positions of Glide-Reflection from the Special Positions for Bona Fide Glide-Reflection Symmetry

The 21 screw-rotation related pseudo n-glide planes are not located at the special positions of y = ¼ and y = ¾ as they are in P21/n. This non-ideality of locations gives rise to a ‘relocation’ of the y-coordinate values defining the pseudo-planes versus those of their bona fide counterparts. The fractional coordinates of (1R,3S,4R)-A[x,y,z] and pseudo σ-reflection related ‘dummy’ (1S,3R,4S)-B [1-x,- ½+y,1-z] plus those of the ideal ac –plane cutting the b axis at y = ¼ were converted to Cartesian coordinates. The pseudo position of glide-reflection ‘best plane’ between the pseudo reflection related molecules was then calculated (without the phenyl carbons nor the chloride anions). The perpendicular distance between this ‘best plane’ and an ideal plane (at the Cartesian-space equivalent of y = ¼) was calculated to be rmS(relocation-glide) = 0.02(2) Å for just the N-desmethyl-cis-3,4- butano-nefopam skeletons, and 0.01(1) Å when it was calculated for the skeletons with the phenyls but without the anions.

4d.1h The RmS(P21/c) Value for the N-Desmethyl-cis-3,4-Butano-Nefopam HCl

P21 Chiral Crystal

rmS(P21/n) = rmS(Ci) + rmS(21) + rmS(n-glide) + rmS(relocation-Ci) + rmS(relocation-21) + rmS(relocation-glide).

The rmS(C2), rmsS(y-translation) and rmS(relocation-21) values are all integer-0 since 21 screw-rotation is a bona fide operation in this crystal.

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4d.1i RmS(P21/n) Calculated for Just the Pseudo Enantiotopic Skeletons of Crystalline 11: rmS(Ci) + rmS(n-glide) = rmS(Ci) + rmS(Cs) + rmS(n-translation) = 0.088(1) Å +

0.076(1) Å + 0.38(6) Å = rmS(symmetry-P21/n) = 0.54(6) Å and, rmS(relocation-Ci) + rmS(relocation-glide) = 0.30(6) Å + 0.02(2) Å = rmS(relocation-P21/n) = 0.32(6) Å therefore, rmS(symmetry-P21/n) + rmS(relocation-P21/n) = 0.54(6) Å + 0.32(6) Å = rmS(P21/n) = 0.86(6) Å

This can be compared to a much larger rmS(P21/n) = 1.58(8) Å value for the skeletons plus the phenyls but without the anions. This can be subdivided into rmS(symmetry-P21/n) = 1.27 Å and rmS(relocation-P21/n) = 0.31(6) Å contributions. From this it is readily observable that the rmS(symmetry-P21/n) distortion increases by a factor of 2.35 when the conformationally different phenyl rings are entered into the calculations while rmS(relocation-P21/n) remains the same.

The three components of rmS(symmetry-P21/n) are 0.26(1) Å + 0.26(1) Å + 0.75(6)

Å for rmS(Ci), rmS(Cs), and rmS(n-translation), respectively. Therefore, the largest contributor to rmS(symmetry-P21/n) is the rmS(n-translation) component which is also the case when just the pseudo enantiotopic skeletons are considered. The two components of rmS(relocation-P21/n) are 0.30(6) Å + 0.01 Å for rmS(relocation-Ci) and rmS(n-glide), respectively. Thus, the largest contributor to rmS(relocation-

P21/n) is rmS(relocation-Ci) which again is also the case when just the pseudo enantiotopic skeletons are considered.

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4D.1j RmsFidelity(P21/n), a Check of the CSM RmS(P21/n) Method This root mean squared difference (RMSD) superimposition method is explained in section 1C of the Introduction and also in the Methodology sections. The single value of its rmsFidelity(Ci) or rmsFidelity(glide) or rmsFidelity(P21/n) index contains simultaneous contributions from both symmetry deviation and relocation factors.

Table 19: rmStot(G) is the total deviation from a certain type of symmetry, including the non normalized CSM value for the superimposition of A and B on the nearest symmetric object, the relocation and the translation (for glides and 21 screw rotations). The rms difference between corresponding atoms in molecules B and ‘dummy A’ is a new index of fidelity (rmsFidelity) of their pseudo symmetry which includes components from both the symmetry distortion as well as from the relocation. The rmStot(G) and rmSFidelity(G) were calculated for crystalline (11).

Ci Glide 21 Screw P21/n translation rotation rmSFidelity(G) including phenyls 0.79(6) Å 0.79(6) Å 0 Å 1.58(6) Å rmSFidelity(G) not including 0.63(6) Å 0.66(6) Å 0 Å 1.29(6) Å phenyls

rmStot(G) including phenyls 0.56(6) Å 1.02(6) Å 0 Å 1.58(6) Å

rmStot(G) not including phenyls 0.39(6) Å 0.48(6) Å 0 Å 0.87(6) Å

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4D.2 RmS Symmetry Measures of Triclinic P–1 Achiral Space Group Crystals of (±)-1-Phenyl-6-Cyano-1,3,4,5,6,7-Hexahydro-2,6-Benzoxazonine, 12.

(±)-1-phenyl-6-cyano-1,3,4,5,6,7-hexahydro-2,6-benzoxazonine (2,6-homonefopam- 6-CN, 12) was synthesized by Ganit Levi-Ruso55 and also by Itzhak Ergaz.54 It gave racemic compound crystals belonging to the achiral triclinic P-1 space group with a = 8.910(1) Å, b = 10.411(1) Å, c = 16.199(2) Å,  = 90.640(4)°,  = 91.266(5)°,  = 99.057(5)°, V = 1483.46 Å3, and Z = 4 which shows that there are 2 molecules in the asymmetric unit (e.g., (R)-A and (R)-B) so that there are four molecules ((R)-A, (S)-A, (R)-B & (S)-B) in the unit cell. In order to unequivocally define these four molecules, we will state their symmetry transforms in the .cif (crystal information file) and will arbitrarily choose the C(1) atom in the molecule to give its coordinates and atom-number (#) from the Mercury77 program list of atoms derived for a complete unit cell. (Table 20)

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Table 20: The four molecules in the unit cell of crystalline (12). Atom numbers are from the Mercury77 program and are found upon choosing ‘display’, ‘more information’, and then ‘atom list’.

Molecule Symmetry Coordinates of C(1), a Atom transform representative atom number (R)-A [x,y,z] [0.3796,0.1415,0.0559] #1 (S)-A [1–x,1–y,1–z] [0.6204,0.8585,0.9441] #40 (R)-B [x,y,z] [0.1196,0.8610,0.5573] #79

(S)-B [1–x,1–y,1–z] [0.8804,0.1390,0.4427] #118

The crystal packing arrangement shows bona fide inversion and pseudo 21 screw- rotation plus pseudo a-glide symmetry. There is a P21/c supergroup arrangement where the unique axis is c, cell choice 1. It is called P21/a, since it has an a-glide. A diagram of a P21/a unit cell is presented below in Figure 70:

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¼ b 0

Figure 70: Diagram of a P21/c unit cell showing bona fide inversion centers (hollow circles) and pseudo two-fold screw axes, color coded according to symmetry equivalence.

If the unit cell would have had P21/a symmetry, all four molecules in the unit cell would have been symmetry equivalent. Figure 71 shows a diagram of the four equivalent positions in the ideal P21/a cell. Since our crystal has pseudo P21/a symmetry, there are two sets of symmetry equivalent positions. Positions which belong to the same set were given the same color. The pseudo 21 screw and pseudo glide operations interconvert yellow and white positions (Figure 71) which are not symmetry equivalent.

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They are pseudo equivalent, and it is possible to check how different these positions are, by evaluating their deviation from ideal symmetry.

b 0 + ½+

½- ,- a

Figure 71: Illustration of the four positions which are equivalent in the ideal

P21/a unit cell. These positions are color coded according to symmetry equivalence. Since our crystal is pseudo P21/a, there are two pairs of symmetry equivalent positions. A position which is marked by an empty circle has opposite handedness to a circle which has a comma inside it. A plus sign means ‘+z’, a minus sign means ‘–z’, ‘½ +’ means ‘½ + z’, and ‘½ –’ means ‘½ – z’.

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The bona fide special position of inversion is illustrated in Figure 72.

Figure 72: Bona fide inversion center of crystalline 12 relates (R)-A (pink) with (S)-A (red), and (R)-B (black) with (S)-B (grey).

Two pseudo 21 screw axes are parallel to the c-axis and are symmetry related by the center of inversion. One converts (R)-A [x,y,z] to (R)-B [x,y,z] and the other converts (S)-B [1–x,1–y,1–z] to (S)-A [1–x,1–y,1–z]. These will be described as pseudo 21 screw axes 1 and 1-bar, respectively, see Figure 73. There is a different pair of pseudo 21 screw axes (2 and 2-bar) that are parallel to the c-axis and are also inversion symmetry related. Axis 2 converts (R)-A [x,y,z] to (R)-B [x,y-1,z] and axis 2-bar converts (S)-B [1–x, -y,1–z] to (S)-A [1–x,1-y,1–z]. It also has two inversion symmetry equivalent pseudo a-glide-reflection planes that cut the z-axis [one relates (R)-A with (S)-B while the other relates (R)-B with (S)-A] (see Figure 74).

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Figure 73: Close upper turquoise pseudo 21 screw axis 1 relates (R)-A [x,y,z] (pink) to (R)-B [x,y,z] (red) and symmetry equivalent close lower turquoise pseudo 21 screw axis 1-bar relates (S)-B [1–x,1–y,1–z] (grey) with (S)-A [1–x,1– y,1–z] (black). Far upper and lower magenta dashed pseudo 21 screw axes are 2 and 2-bar, respectively.

Figure 74: Left pseudo a-glide-reflection plane relates (S)-A [1–x,1–y,1–z] (lower structure) with (R)-B [x,y,z] (upper structure) and right inversion equivalent plane relates (S)-B [1–x,1–y,1–z] (lower structure) with (R)-A [x,y,z] (upper structure).

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4D.2a Location of the Pseudo Positions of 21 Screw-Rotation in the 2,6- Homonefopam-6-CN Triclinic P–1 Achiral Space Group Crystal

Four pseudo 21 screw axes (1, 1-bar, 2 and 2-bar) are parallel to the c-axis. The xmean and ymean values for the defining points of these pseudo positions of 21 screw-rotation axes were calculated in the same manner as described for N-desmethyl-cis-3,4- butano-nefopam HCl above. Since the pseudo 21 screw axes are parallel to the c axis to a good approximation, the centroids of each pseudosymmetry related pair of atoms fall approximately on an axis which is parallel to the c axis. Therefore, the centroids of each pseudosymmetry related pair of atoms all have approximately constant x and y values, which can be averaged. These average values will be referred to as Xmean and Ymean, in order to describe the location of each pseudo 21 screw axis. For axis 1, which interconverts (R)-A [x,y,z] with (R)-B [x,y,z], the

Xmean = 0.254(4) and Ymean = 0.503(2). For its symmetry equivalent

[1–x,1–y,1–z] partner, 1-bar: (S)-B with (S)-A, Xmean= 0.746(4) and

Ymean = 0.746(4). The location of axis 2: (R)-A [x,y,z] and (R)-B [x,y-1,z] can be defined by Xmean = 0.254(4) and Ymean = 0.003(2). Finally, the location of axis 2-bar:

(S)-B [1–x,-y,1–z] with (S)-A [1–x,1–y,1–z] can be defined by Xmean = 0.746(4) and

Ymean = –0.003(2). The fact that the esd values of the xmean and ymean coordinates defining the above axes are not integer-0 is to be expected since we are dealing with pseudo symmetry. The larger the esd-values of the two defining points, the less accurate is the location of the pseudo-axis.

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4D.2b RmS(21) - Quantification of 21 Screw-Rotation Pseudo Symmetry in the 2,6-Homonefopam-6-CN Triclinic P–1 Achiral Space Group Crystal

The bona fide 21 screw-rotation symmetry operation is composed of two symmetry transform components: C2-rotation and translation (in this case, a translation by ½ of the c-axis unit cell length). The two components can be separated, since the translation leaves x and y invariant, while the C2 rotation leaves z invariant. If the pseudo 21 screw axis 1 relates atoms in (R)-A with corresponding atoms in (R)-B, and since (R)-B is more distant on the c-axis from the origin than is (R)-A, then z- translationmean = 0.501(4) which was calculated in the same manner as for the n- glide in crystalline 11. It is the same value for both axes 1 & 2. The fact that z- translationmean is not the fractional number ½ nor is its esd value integer-0 is to be expected since we are dealing with pseudo symmetry. The distortion from ideal z- translation of ½ for both axes is rmS(z-translation)coordinate = 0.0039 and must be multiplied by the 16.199 Å c-axis length to afford rmS(z-translation) = 0.06(6) Å.

To calculate the pseudo 21 screw-rotation’s C2 rotation component, the z-coordinates of (R)-B atoms were transformed by the z-translationmean value noted above to 29 generate a dummy (R)-B molecule next to (R)-A. Avnir’s CSM algorithm for C2- rotation gave average S(C2) = 0.0046(6) for the two axes. The non-normalized rmS(C2) = 0.028(1) Å for both axes in crystalline 2,6-homonefopam- 6-CN.

Therefore, the rmS(21) for the pseudo 21 screw-rotation is: rmS(C2) + rmS(z-translation) = 0.028(1) Å + 0.06(7) Å = rmS(21) = 0.09(7) Å

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4D.2c Location of the Pseudo Positions of a-Glide-Reflection in the 2,6- Homonefopam-6-CN Triclinic P–1 Achiral Space Group Crystal Both pseudo a-glide planes in this crystal are symmetry equivalent and both are parallel to the ac-plane. One relates atoms in (R)-A with corresponding atoms in

(S)-B and cuts the z-axis at zmean = 0.250(2) Å, where zmean was calculated in the same manner as described crystalline 11. The second pseudo a-glide plane relates atoms in (R)-B with corresponding atoms in (S)-A and cuts the z-axis at zmean = 0.750(2) Å. The fact that the esd value for z is not integer-0 is to be expected since we are dealing with pseudo symmetry. The larger is the esd of the points defining the pseudo-plane, the less planar is the set of points.

4D.2d RmS(a-glide), Comparison of a-Glide-Reflection Pseudo Symmetry in the 2,6-Homonefopam-6-CN Crystal with Bona Fide Symmetry An a-glide-reflection symmetry operation is composed of two symmetry transform components: mirror-reflection and a translation by ½ of the unit cell’s a-axis length. Since the translation leaves y and z invariant and the reflection leaves x invariant, the two components can be separated. If the pseudo a-glide-reflection plane relates atoms in (R)-A with corresponding atoms in (S)-B’, and since (S)-B’ is more distant on the a-axis from the origin than is R-A, then x-translationmean = 0.492(8) Å which is calculated in the same manner as for crystalline 11. Again, the fact that the x- translationmean is not the fractional number ½ nor is its esd value integer-0 is to be expected since we are dealing with pseudo symmetry. The distortion from ideal x- translation of ½ is rmS(x-translation)coordinate = 0.0111 which must multiplied by the 8.9104 Å a-axis length to afford rmS(x-translation) = 0.10(7) Å.

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To calculate the pseudo a-glide-reflection’s mirror component, the x-coordinates of

(S)-B’ atoms were first transformed by the x-translationmean value calculated above. In this manner a dummy (S)-B molecule was generated next to (R)-A. Avnir’s CSM 29 algorithm for mirror-reflection gave a value of S(Cs) = 0.0346(1). The non-normalized rmS(Cs) = 0.052(1) Å.

Therefore, the rmS(a-glide) is: rmS(Cs) + rmS(x-translation) = 0.052(1) Å + 0.10(7) Å = rmS(a-glide) = 0.15(7) Å

4D.2e Relocation of the Pseudo Positions of 21 Screw-Rotation Axes Versus the

Special Positioned Ideal 21 Screw Axes in the 2,6-Homonefopam-6-CN Triclinic P–1 Achiral Space Group Crystal

The points defining the two pseudo 21 screw axes 1 or 2 are not located at the special positions of [¼,½,0] & [¼,½,1] or [¼,0,0] & [¼,0,1], respectively, as they are in P21/a with the appropriate origin. This non-ideality of locations gives rise to a ‘relocation’ of the points defining the pseudo-axes versus those of their bona fide counterparts.

The fractional coordinates of (R)-A and pseudo C2 axis 1 related ‘dummy’ (R)-B, as well as for the molecules of pseudo axis 2, plus the points defining the ideal axes were all converted to Cartesian coordinates. A vector was defined by the ‘best axis’ calculated from the pseudo C2 related molecules on either axis 1 or 2. A second vector was calculated from the ideal 21 screw axis defining points. The distance between the pseudo position of 21 screw-rotation ‘best axis’ and ideal axis vectors is average rmS(relocation-21) = 0.10(7) Å.

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4D.2f Relocation of the Pseudo Positions of a-Glide-Reflection Planes in the 2,6- Homonefopam-6-CN Triclinic P–1 Achiral Space Group Crystal The inversion related pseudo a-glide planes are not located at the special positions of y = ¼ and y = ¾ as they are in P21/a. This non-ideality of locations gives rise to a ‘relocation’ of the y-coordinate values defining the pseudo-planes versus those of their bona fide counterparts. The fractional coordinates of (R)-A and pseudo σ- reflection related ‘dummy’ (S)-B’ molecule plus those of the ideal ac-plane cutting the b-axis at y = ¼ were converted to Cartesian coordinates. The pseudo position of a-glide-reflection ‘best plane’ between the pseudo reflection related molecules was then calculated. The perpendicular distance between this ‘best plane’ and the ideal plane was calculated to be rmS(relocation-glide) = 0.01(1) Å.

4D.2g The RmS(P21/a) Value for the 2,6-Homonefopam-6-CN Crystal

The rmS(P21/a) value is the summation of all the non-trivial rmS(symmetry-P21/a)- values, i.e. rmS(Ci) + rmS(21) + rmS(a-glide) plus the summation of all the components of rmS(relocation-P21/a), i.e. rmS(relocation-Ci) + rmS(relocation-21) + rmS(relocation-glide). The rmS(Ci) and rmS(relocation-Ci) values are integer-0 since inversion symmetry is a bona fide operation in this crystal.

Then, rmS(21) + rmsS(a-glide) = rmS(C2) + rmS(z-translation) + rmS(Cs) + rmS(x- translation) = 0.028(1) Å + 0.06(7) Å + 0.0520(1) Å + 0.10(7) Å = rmS(symmetry-

P21/a) = 0.24(7) Å and, rmS(relocation-21) + rmS(relocation-glide) = 0.10(7) Å + 0.01(1) Å = rmS(relocation-P21/a) = 0.11(7) Å therefore, rmS(symmetry-P21/a) + rmS(relocation-P21/a) = 0.24(7) Å + 0.11(7) Å = rmS(P21/a) = 0.35(7) Å

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4D2.h RmsFidelity(P21/n), a Check of the RmS(P21/n) Symmetry Measure Method This root mean squared difference (RMSD) superimposition method is explained in the section C of the Introduction and also in the Methodology sections. The single value of its rmsFidelity(Ci) or rmsFidelity(glide) or rmsFidelity(P21/n) index contains simultaneous contributions from both symmetry deviation and relocation factors.

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Table 21: rmStot(G) is the total deviation from a certain type of symmetry, including the non normalized CSM value for the superimposition of A and B on

the nearest symmetric object, the relocation and the translation (for glides and 21 screw rotations). The rms difference between corresponding atoms in molecules B and ‘dummy A’ is a new index of fidelity (rmsFidelity) of their pseudo symmetry which includes components from both the symmetry distortion as well as from the relocation. The rmStot(G) and rmSFidelity(G) were calculated for crystalline (12).

Ci Glide translation 21 Screw rotation P21/a rmSFidelity(G) 0 Å 0.13(7) Å 0.13(7) Å 0.26(7) Å

rmStot(G) 0 Å 0.16(7) Å 0.19(7) Å 0.35(7) Å

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The crystals of 13 were described by Gorbitz, Kazmaier, and Grandel.52 They are racemic compound crystals belonging to the achiral monoclinic Pn space group with a = 14.4318(14) Å, b = 10.2646(10) Å, c = 17.997(2) Å,  = 103.689(1)°, V = 2590.29 Å3, and Z = 4, which shows that there are 2 molecules in the asymmetric unit so that there are four molecules in the unit cell.52 In order to unequivocally define these four molecules, we will state their symmetry transforms in the .cif (crystal information file) and will arbitrarily choose the S(1) atom in the molecule to give its coordinates and atom-number (#) from the Mercury77 program list of atoms derived for a complete unit cell.

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Table 22: The four molecules in the unit cell of crystalline (13). Atom numbers are from the Mercury77 program and are found upon choosing ‘display’, ‘more information’, and then ‘atom list’.

Molecule Symmetry Coordinates of S(1), a Atom transform representative atom number (2R,3S,4R)-A [x,y,z] [0.61275,0.93941,0.36480] #2 (2S,3R,4S)-A [–½+x,1–y,½+z] [0.37896,0.59202,0.64513] #117 (2S,3R,4S)-A [–½+x,1–y,½+z] [0.11275,0.06059,0.86480] #59 (2R,3S,4R)-B [½+x,1–y,–½+z] [0.87896,0.40798,0.14513] #175

The crystal packing arrangement shows bona fide n-glide-reflection and pseudo inversion symmetry plus pseudo 21 screw-rotation, and packs in a P21/n supergroup arrangement where the unique axis is b (cell choice 2 from the international tables).

The setting is called P21/n, since it has an n-glide. The diagram for this setting is shown in Figure 75.

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¼ a 0

Figure 75: Diagram of symmetry operations in the P21/n cell. The pseudo positions of symmetry are colored according to their symmetry equivalence.

If the unit cell would have had P21/n symmetry, all four molecules in the unit cell would have been symmetry equivalent. Figure 76 shows a diagram of the four equivalent positions in the ideal P21/n cell. Since our crystal has pseudo P21/n symmetry, there are two sets of symmetry equivalent positions. Positions which belong to the same set were given the same color. The pseudo inversion and pseudo

21 screw operations interconvert yellow and white positions (Figure 76) which are not symmetry equivalent.

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They are pseudo equivalent, and it is possible to check how different their positions are, by evaluating the deviation from ideal symmetry.

a 0

+ ½+ , ½- ,- c

Figure 76: The four positions which are equivalent in the ideal P21/n unit cell are shown here. They are color coded according to symmetry equivalence.

Since our crystal is pseudo P21/n, there are two pairs of symmetry equivalent positions. A position which is marked by an empty circle has opposite handedness to a circle which has a comma inside it. A plus sign means ‘+y’, a minus sign means ‘–y’, ‘½ +’ means ‘½ + y’, and ‘½ –‘ means ‘½ – y’.

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Since our crystal has Pn symmetry, this means that it has bona fide crystallographic n- glide-reflection planes (see Figure 77).

Figure 77: The bona fide n-glide-reflection plane at y = 0.5 in Pn crystal of 13, by which (2R,3S,4R)-A[x,y,z] (red) is related to (2S,3R,4S)-A[-½+x,1-y,½+z] (pink), and (2S,3R,4S)-B[x,y,z] (grey) is related to (2R,3S,4R)-B[½+x,1-y,-½+z] (black).

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It has four pseudo inversion centers between the n-glide planes. These relate (2S,3R,4S)-B [x,y,z] either with (2R,3S,4R)-A [x,y,z] (or its (2R,3S,4R)-A [x,y,z+1], (2R,3S,4R)-A [x–1,y,z], or (2R,3S,4R)-A [x–1,y,1+z] symmetry equivalent partner), see Figure 78. These four pseudo inversion centers are not equivalent.

Figure 78: Four symmetry nonequivalent pseudo positions of inversion ‘best centers of the cluster’ between magenta dashed line bona fide y = ½ and 1 n-glide planes. The close-left-top blue pseudo inversion position relates (2S,3R,4S)- B[x,y,z] (black) with (2R,3S,4R)-A[x,y,z] (grey). The other blue pseudo inversion positions involve (2S,3R,4S)-B[x,y,z] with (2R,3S,4R)-A [x,y,z+1], (2R,3S,4R)-A [x–1,y,z], or (2R,3S,4R)-A [x–1,y,1+z] symmetry equivalent partners which do not appear in the drawing. The close-right-top light blue pseudo inversion position relates (2R,3S,4R)-B (red) with (2S,3R,4S)-A (pink).

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There are two pseudo 21 screw axes parallel to the b-axis and they are symmetry related by the n-glide-reflection. One converts (2R,3S,4R)-A [x,y,z] to (2R,3S,4R)-B [½+x,1–y,–½+z] and the other converts (2S,3R,4S)-A [–½+x,1–y,½+z] with

(2S,3R,4S)-B [x,y,z]. These will be described as pseudo positions of 21 screw- rotation axes 1 and 1-bar, respectively, see Figure 79. There is a different pair of 21 screw axes (2 and 2-bar) that are parallel to the b-axis and are also n-glide-reflection symmetry related. Axis 2 converts (2R,3S,4R)-A [x,y,z] to (2R,3S,4R)-B [½-x,1–y,– ½+z] and axis 2-bar converts (2S,3R,4S)-A [½+x,1–y,½+z] to (2S,3R,4S)-B [x,y,z].

Figure 79: Close upper turquoise pseudo 21 screw axis 1 relates (2R,3S,4R)-A [x,y,z] (pink) with (2R,3S,4R)-B [½+x,1–y,-½+z] (red) and symmetry equivalent far lower turquoise pseudo 21 screw axis 1-bar relates (2S,3R,4S)-B [x,y,z] (black) with (2S,3R,4S)-A [–½+x, 1–y,½+z] (grey). Far upper and close lower magenta dashed pseudo 21 screw axes are 2 and 2-bar, respectively.

In all the following symmetry analyses of this crystal, the dichloromethane solvent molecules were excluded since one of them was disordered.

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4D.3a Location of the Pseudo Positions of Inversion in the (±)-tert.-Butyl 3 – Hydroxyl – 4 – Phenyl – 2 -(p-Toluenesulfonylamino)Pentanoate Dichloromethane Solvate Monoclinic Pn Achiral Space Group Crystal. There are four pseudo positions of inversion located between the n-glide planes in this crystal. The same (2R,3S,4R)-A [x,y,z] and (2S,3R,4S)-B [x,y,z] pair of molecules that are involved in hydrogen-bonding (see above) are also related by pseudo inversion symmetry. The xmean, ymean and zmean coordinates defining each of the four pseudo positions of inversion can be seen in Table 23. They were calculated in the same manner as that described above for crystalline 11.

Table 23: The four best inversion centers between molecules of crystalline (13).

Molecules Related by Pseudo- Coordinates of ‘Best inversion’ inversion symmetry point 2R,3S,4R)-A [x,y,z] and [0.500(4),0.763(11),0.500(5)] (2S,3R,4S)-B [x,y,z] (2S,3R,4S)-B [x,y,z] and [0.500(4),0.763(11),1.000(5)] (2R,3S,4R)-A [x,y,z+1] (2S,3R,4S)-B [x,y,z] and [0.000(4),0.763(11),0.500(5)] (2R,3S,4R)-A [x–1,y,z] (2S,3R,4S)-B [x,y,z] and [0.000(4),0.763(11),1.000(5)] (2R,3S,4R)-A [x–1,y,1+z]

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4D.3b RmS(Ci) Comparison of Inversion Pseudo Symmetry with Bona Fide Symmetry in the (±)-tert.-Butyl 3 – Hydroxyl – 4 – Phenyl – 2 -(p- Toluenesulfonylamino)Pentanoate Dichloromethane Solvate Monoclinic Pn Achiral Space Group Crystal.

The classical size normalized Avnir S(Ci) was calculated by the same method as used for crystalline 11 and its values apprear in Table 24. The numerical spread of these values results from the size normalization function as was also observed for S(Ci) values calculated for crystalline 1. The non-normalized rmS(Ci) = 0.159(1) Å for all four pseudo inversion ‘best points’.

Table 24: S(Ci) values for the four pseudoinversion pairs between molecules of crystalline (13).

Pseudoinversion pair S(Ci) 2R,3S,4R)-A [x,y,z] and 0.1011(1) (2S,3R,4S)-B [x,y,z] (2S,3R,4S)-B [x,y,z] and 0.0524(1) (2R,3S,4R)-A [x,y,z+1] (2S,3R,4S)-B [x,y,z] and 0.0430(1) (2R,3S,4R)-A [x–1,y,z] (2S,3R,4S)-B [x,y,z] and 0.0430(1) (2R,3S,4R)-A [x–1,y,1+z]

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4D.3c Relocation of Pseudo Positions of Inversion from the Special Positions for Bona Fide Inversion Symmetry Once again, the fact that the esd values of the coordinates defining the above points are not integer-0 is to be expected since we are dealing with pseudo-symmetry. The larger the esd-values of the coordinates defining the points, the less accurate is the location of that point. In a bona fide P21/n cell such as depicted above in Figure 75, the four special positions of inversion symmetry are located at: [¼,½,¼], [¼,½,¾],

[¾,½,¼] and [¾,½,¾]. Due to a different choice of the origin in the literature Pn unit cell under analysis, the four ideal points are at: [0,¾,½], [0,¾,1], [½, ¾,½] and [½,¾,1]. In order to measure the spatial relocation of the general positioned ‘best points’ from their ideal counterparts, the distances between the ideal inversion point and the best pseudo inversion points were calculated. An rmS(relocation-Ci) of 0.13(4) Å was obtained for the four pseudo positions of inversion.

4D.3d Location of the Pseudo Positions of 21 Screw-Rotation Axes in the (±)- tert.-Butyl 3 – Hydroxyl – 4 – Phenyl – 2 -(p-Toluenesulfonylamino)Pentanoate Dichloromethane Solvate Monoclinic Pn Achiral Space Group Crystal.

Four pseudo 21 screw axes (1, 1-bar, 2 and 2-bar) are parallel to the b-axis and are symmetry related by the n-glide-reflection plane. The Xmean and Zmean values, were calculated in the same manner as described for crystalline 11 above. For Axis 1:

(2R,3S,4R)-A [x,y,z] and (2R,3S,4R)-B [½+x,1–y,–½+z] , Xmean = 0.750(3) and Zmean = 0.250(3). Its symmetry equivalent partner, 1-bar, relates between (2S,3R,4S)-A [–

½+x,1–y,½+z] and (2S,3R,4S)-B [x,y,z], and has an Xmean value of 0.250(3) and a

Zmean value of 0.750(3). Axis 2: relates between (2R,3S,4R)-A [x,y,z] and

(2R,3S,4R)-B [½-x,1–y,–½+z], and has a Xmean value of 0.250(3), and a Zmean value of 0.250(3). Finally, axis 2-bar, which relates between: (2S,3R,4S)-A [½+x,1–y,½+z] and (2S,3R,4S)-B [x,y,z], has a Xmean value of 0.750(3) and a Zmean value of 0.750(3).

The fact that the esd values of the Xmean and Zmean coordinates defining the above axes are not integer-0 is to be expected since we are dealing with pseudo-symmetry. The larger the esd-values of the two defining points, the less accurate is the location of the pseudo-axis.

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4D.3e RmS(21) Quantification of 21 Screw-Rotation Pseudo Symmetry in the (±)-tert.-Butyl 3 – Hydroxyl – 4 – Phenyl – 2 -(p- Toluenesulfonylamino)Pentanoate Dichloromethane Solvate Monoclinic Pn Achiral Space Group Crystal. The calculations in this section were all performed according to the method described for the pseudo 21 screw axes in crystalline homonefopam-6-CN (12). In the solvate adduct crystal, the pseudo 21 screw-rotation symmetry operation is composed of two components: pseudo C2 rotation and pseudo translation by ½ of the b-axis unit cell length. (the components can be separated, since the translation leaves x and z invariant, while the rotation leaves y invariant) For both axes 1 & 2, the same y- translationmean = 0.52(2) value was found when calculated by analogy to translations in the two preceding crystals. The fact that y-translationmean is not the fractional number ½ nor is its esd value integer-0 is to be expected since we are dealing with pseudo-symmetry. The distortion from ideal y-translation of ½ for both axes is rmS(y-translation)coordinate = 0.0335 and must be multiplied by the 10.2646 Å b-axis 29 length to afford rmS(y-translation) = 0.344(4) Å. Avnir’s CSM algorithm for C2- rotation gave average S(C2) = 0.0444(1) for the two axes. The non-normalized average rmS(C2) values are 0.114(13) Å for the two axes.

Therefore, the rmS(21) for the pseudo 21 screw-rotation is: rmS(C2) + rmS(z-translation) = 0.114(13) Å + 0.345(4) Å = rmS(21) = 0.46(1) Å

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4D.3f Relocation of the Pseudo Positions of 21 Screw-Rotation Axes Versus the

Special Positioned Ideal 21 Screw Axes in the (±)-tert.-Butyl 3 – Hydroxyl – 4 – Phenyl – 2 -(p-Toluenesulfonylamino)Pentanoate Dichloromethane Solvate

Monoclinic Pn Achiral Space Group Crystal.

The points defining the two pseudo 21 screw axes 1 or 2 are not located at the corresponding special positions of [¾,0,¼] & [¾,1,¼] or [¼,0,¼] & [¼,0,¼], respectively, as they are in P21/n with the appropriate origin. This non-ideality of locations gives rise to a ‘relocation’ of the points defining the pseudo-axes versus those of their bona fide counterparts. The fractional coordinates of 2R,3S,4R)-A

[x,y,z] and pseudo C2 axis 1 related ‘dummy’ (2R,3S,4R)-B [½+x,1–y,–½+z], as well as for the molecules of pseudo axis 2, plus the points defining the ideal axes were all converted to Cartesian coordinates. A vector was defined by the ‘best axis’ calculated from the pseudo C2 related molecules on either axis 1 or 2. A second vector was calculated from the ideal 21 screw axis defining points. The distance between the 21 screw ‘best axis’ and ideal axis vectors is rmS(relocation-21) = 0.017(4) Å for both axes.

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4D.3g The RmS(P21/n) Value in the (±)-tert.-Butyl 3 – Hydroxyl – 4 – Phenyl – 2

-(p-Toluenesulfonylamino)Pentanoate Dichloromethane Solvate Monoclinic Pn Achiral Space Group Crystal. rmS(P21/n) =rmS(Ci) + rmS(21) + rmS(n-glide) + rmS(relocation-Ci) + rmS(relocation-21) + rmS(relocation-glide).

The rmS(Cs), rmS(x-translation), rmS(z-translation) and rmS(relocation-Ci) values are all integer-0 since n-glide-reflection symmetry is a bona fide operation in this crystal.

Then, rmS(Ci) + rmS(21) = rmS(Ci) + rmS(C2) + rmS(y-translation) = 0.159(1) Å +

0.114(13) Å + 0.345(4) Å = rmS(symmetry-P21/n) = 0.62(1) Å and rmS(relocation-Ci) + rmS(relocation-21) = 0.13(4) Å + 0.017(4) Å = rmS(relocation-

P21/n) = 0.15(4) Å therefore, rmS(symmetry-P21/n) + rmS(relocation-P21/n) = 0.62(1) Å + 0.15(4) Å = rmS(P21/a) = 0.77(4) Å

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Table 25: rmStot(G) is the total deviation from a certain type of symmetry, including the non normalized CSM value for the superimposition of A and B on

Ci Glide 21 Screw rotation P21/n translation rmSFidelity (G) 0.41(4) Å 0 Å 0.41(4) Å 0.82(6) Å

rmStot(G) 0.29(4) Å 0 Å 0.48(1) Å 0.77(6) Å

4D.3h RmsFidelity(P21/n), a Check of the CSM RmS(P21/n) Method This root mean squared difference (RMSD) superimposition method is explained in section C of the Introduction and also in the Methodology sections. The single value

of its rmsFidelity(Ci) or rmsFidelity(glide) or rmsFidelity(P21/n) index contains simultaneous contributions from both symmetry deviation and relocation factors.

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4D.4 The ‘Pseudo Positions of Symmetry’ Concept In Experimental section 3F.1 it was noted that the set of centroids defined by n pairs of pseudo symmetry related atoms generate a cluster of points. When there is pseudo inversion, the cluster of points can be averaged in order to obtain a best point. (see

3F.2 and 3F.3) When there is a pseudo 21 screw-rotation operation, the cluster of points forms an approximate axis. The least squares 21 screw axis can be found, and this is defined as the best axis. (see 3F.4) When there is a pseudo glide operation, the cluster of points forms an approximate plane. The least squares plane can be found, and this is defined as the best glide plane. (see 3F.5) These best points, lines or planes should be considered to be ‘pseudo positions of symmetry’. They differ from bona fide ‘special positions of symmetry’ since they are statistical mean-points, mean- axes, or mean-planes and are all located at general positions of symmetry within the lattice. All of the clusters of n centroids which form a best point, line or plane have estimated standard deviations (esd). The space group defines the special position’s specific locations in the unit cell, and their fractional coordinates do not carry an estimated standard deviation since ideality is involved. The concept of ‘pseudo positions of symmetry’ represents a continuum involving both the relocation of the pseudo symmetry element from the bona fide position plus the magnitude of the defining point’s esd. In part 4D of this dissertation the ‘pseudo positions of symmetry’ were between molecules. In the last part of this thesis (4E) a crystal will be investigated where there were conformational consequences due to the fact that the molecule itself ‘occupied’ a ‘pseudo position of symmetry’.

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4D.5 Discussion of the RmS Symmetry Measure Continuum of P21/c Family

‘Supergroup Character’ in Lower Symmetry P21, P–1, and Pn Space Group Crystals Containing Multiple Molecules in the Asymmetric Unit.

In Results sections 4D.1-3, three crystals which had P21/c family supergroup pseudo symmetry were quantitatively analyzed in terms of the symmetry and relocation of a particular pseudo symmetry operation’s contribution to the total value for a space group. Each one of the crystals had a different space group (P21, P–1, and Pn) which was a subgroup of the larger P21/c family supergroup. From group theory we know that when a true inversion center and a true screw axis exist, then a true glide plane must also exist. It has been shown that a third pseudo symmetry element was generated when there was a combination of one crystallographic and one pseudo symmetry element. For instance, the combination of a true 21 screw axis and a pseudo position of inversion generated a pseudo glide-reflection in crystalline 11.

4D.5a The New RmS Symmetry Measure Tools of RmS(Ci), RmS(C2), RmS(Cs),

RmS(translation), RmS(relocation-Ci), RmS(relocation-screw), RmS(relocation- plane), RmsFidelity(Ci), RmsFidelity(21), and RmsFidelity(glide)

New non-normalized rmS symmetry measure tools of rmS(Ci), rmS(C2), and rmS(Cs) had to be developed during the course of this research since the magnitude of the classical Avnir S(Ci) value was found to be inversely proportional to the distance between two pseudo inversion symmetry related phenyl rings crystalline 11. In addition, a new rmS(translation) tool was developed together with new rmS symmetry measure tools of rmS(relocation-Ci), rmS(relocation-screw), and rmS(relocation-plane) in order to quantify pseudo symmetry in crystallographic unit cells. Their numerical values were sensical and their magnitudes were in agreement with the second rmS symmetry measure tools of rmsFidelity(Ci), rmsFidelity(21), and rmsFidelity(glide) indices which were developed here as a simple tool for measuring the overall deviation from each type of pseudo symmetry operation.

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4D.5b Discussion of Stereochemistry and RmS Symmetry Measure Results of

Monoclinic P21 Chiral Space Group Crystals of (±)-(1RS,3SR,4RS)-1-Phenyl-cis- 3,4-Butano-3,4,5,6-Tetrahydro-1H-2,5-Benzoxazocine Hydrochloride, 11. Both molecules (1R,3S,4R)-A and (1S,3R,4S)-B show the same boat-boat conformation for the eight-membered ring as shown by inverting one of the molecules and then measuring a 0.051 Å root mean squared (RMS) difference for the superimposition of corresponding non-hydrogen atoms (with the exception of those in the phenyl ring and the Cl anions). There are significant differences (59.2°) in the phenyl-ring twist as shown by a 0.492 Å RMS of superimposition of all non-hydrogen atoms. The phenyl rings in the two molecules (1R,3S,4R)-A and (1S,3R,4S)-B greatly differ in the twist of their phenyl rings. The (1R,3S,4R)-A phenyl ring almost eclipses the Cbenzhydryl―CipsoBenzo bond: +20.1° Cortho–CipsoPhenyl–Cbenzhydrylic–CipsoBenzo torsion angle, while the (1S,3R,4S)-B phenyl ring almost eclipses the the

Cbenzhydryl―Hbenzhydryl bond: –16.5° Cortho’– CipsoPhenyl–Cbenzhydryl–Hbenzhydryl torsion angle.

There are two types of attractive forces between the (1R,3S,4R)-A and (R)-B 2 molecules in the asymmetric unit. The first is a R 4 ring hydrogen-bonding pattern graph set involving two hydrogen-bond acceptors and four donors in an eight- membered ring cycle, see Figure 80. The second are chains of edge-to-face aromatic-aromatic interactions within the crystal and within the asymmetric unit (see Figure 81). There is an edge-to-face benzo…phenyl Coulombic interaction between the partially negatively charged (1S,3R,4S)-B [x,y,z] benzoface centroid and

(1S,3R,4S)-B’ [1–x,y–½,1–z] phenyledge partially positively charged aromatic protons (– C→H +) with a 4.880 Å centroid…centroid distance. There is a second edge-to- face phenyl…phenyl Coulombic interaction between the (1S,3R,4S)-B’ [1–x,y–½,1–z] – + phenyledge partially positively charged aromatic protons ( C→H  ) and the partially negatively charged (1R,3S,4R)-A’ [1–x,y+½,1–z] phenylface centroid with a 4.839 Å centroid…centroid distance.

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Finally, there is another edge-to-face phenyl…benzo Coulombic interaction between the (1R,3S,4R)-A’ [1–x,y+½,1–z] phenyledge partially positively charged aromatic protons (– C→H +) and the partially negatively charged (1R,3S,4R)-A’’ [X–1,y,1+z] … phenylface centroid with a 4.745 Å centroid centroid distance.

Figure 80: An aromatic aromatic Coulombic interaction.

Figure 81: Network of edge-to-face aromatic-aromatic Coulombic interactions in the P21 unit cell of 1 (black dashed line through cell center is the bona fide 21 screw axis).

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4D.5c Advantages of Pseudo-Symmetry in the N-Desmethyl-cis-3,4-Butano-

Nefopam HCl P21 Chiral Crystal One can rationalize the advantages to having only a pseudo-symmetrical relationship between the two molecules of opposite handedness having different phenyl ring twists in the asymmetric unit. Changing the Hbenzhydryl–Cbenzhydryl–CipsoPhenyl–Cortho torsion angle by 59.2° from –42.8° (see Figure 82) to +16.4° (see Figure 83) so that all molecules now have the same phenyl ring twist, should be a relatively low energy process in an isolated molecule. But, there are a number of negative consequences to this change. First of all, two out of the three aromatic…aromatic edge-to-face interactions are lost. In their place is a face-to-face arrangement of the two phenyls, but they are displaced sideways by about 4.7 Å from each other so that - stacking interactions are expected to very weak, if they exist at all. Secondly, there are now bad close contacts of 1.97-1.99 Å involving aromatic protons, and there is a very bad 1.33 Å close contact involving an aromatic proton and the alkyl proton H(4eq) common to both the 6- and 8-membered rings (see Figure 84).

Figure 82: Changing the Hbenzhydryl–Cbenzhydryl–Cipsophenyl–Cortho torsion angle by +59.2° from its original –42.8° value in which the phenyl ring eclipsed the

Cbenzhydryl—CipsoBenzo bond (green).

195

Figure 83: As a result of changing the H–Cbenzhydryl–CipsoPhenyl–Cortho torsion angle by +59.2° to +16.4°, the phenyl ring now eclipses the more usual … Cbenzhydryl—Hbenzhydryl bond (green), but two aromatic aromatic interactions are lost.

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Figure 84: Another result of having all phenyl rings having the same

conformation in which they eclipse the Cbenzhydryl—Hbenzhydryl bond is that there are bad close contacts between aromatic protons (blue) on the two close rings and an aromatic proton (green) is very close to an alkyl proton H(4eq) (green) common to both the 6- and 8-membered rings.

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4D.5d The Solid-State Cp/mas NMR Spectrum of N-Desmethyl-cis-3,4-Butano-

Nefopam HCl P21 Monoclinic Chiral Crystals

IE-189B_cp (1SR,3RS,4SR)-N-desmethyl Nefopam cis-3,4-butano HCl salt 100

IE-189B (all cis) 13C Seltics 86.22 84.52 79.53 50.16 48.51 28.10 25.74 22.98 18.92 17.02 spin-rate 5.0 kHz; o2 -4275.0 Hz; sr -1502.91 Hz

-10

101520253035404550556065707580859095100105110115120125130135140145150 f1 (ppm) Figure 85: The solid-state cp/mas 13C NMR spectrum of N-desmethyl-cis-3,4- butano-nefopam HCl P21 monoclinic chiral crystals. The solid-state cp/mas 13C NMR (125.72 MHz, spin-rate 5.0 KHz)) spectrum of N- desmethyl-cis-3,4-butano-nefopam HCl P21 monoclinic chiral crystals was measured using the SELTICS pulse program which removes spinning side bands, see Figure 85. By running an NQS spectrum which shows quaternary and methyl nuclei since they are not effectively relaxed by a dipolar mechanism, and also a CPPI (methylene carbon only) spectrum, the two peaks at 86.2 and 84.5 ppm were assigned to the benzhydrylic C(1) nuclei of the (1R,3S,4R)-A and (1S,3R,4S)-B molecules. A GIAO (Gauge-Independent Atomic Orbital) calculation60, DFT B3LYP/6-311g+(2d,p)) based on crystal geometry as invariant input (single point) gave calculated shielding constants which were then converted to calculated chemical shifts referenced to TMS calculated values61. The calculated chemical shift values for the benzhydrylic C(1) nuclei were 82.1 and 80.1 ppm for the phenyl eclipsing the C—H bond and the phenyl eclipsing the C—C bond, respectively.

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4D.5e Maximization of Aromatic…Aromatic Interactions and Packing via

Distortion from Ideal (P21/n) Symmetry

Figure 86: Unit cell depicting a-axis lateral shear relocation in the b = 0.501(3) plane of the pseudo inversion centers between (1R,3S,4R)-A & (1S,3R,4S)-B (labeled 1) and (1R,3S,4R)-A’ & (1S,3R,4S)-B’ (labeled 2).

The blow-ups in Figure 86 depict the a-axis relocation in the b = 0.501(3) plane of statistically determined pseudo positions of inversion (solid red-circles) between

(1R,3S,4R)-A & (1S,3R,4S)-B (labeled 1) and also between 21 screw axis related (1R,3S,4R)-A’ & (1S,3R,4S)-B’ (labeled 2) versus their nearest ideal points (solid green-circles). The respective x or y or z fractional coordinate differences of the pseudo inversion position between (1R,3S,4R)-A & (1S,3R,4S)-B minus corresponding coordinates of the special positions of the ideal partner are 0.030(2), 0.001(3), and 0.000(6) and between (1R,3S,4R)-A’ & (1S,3R,4S)-B’ they are – 0.030(2), 0.001(3), and 0.000(6). The distortion from ideality of the x coordinate is 30 times that measured for either y or z.

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As a result this relocation, the set of (1R,3S,4R)-A & (1S,3R,4S)-B molecules & pseudo inversion position 1 residing within the c = 0 to c = ½ half-portion of the cell and the set of (1R,3S,4R)-A’ & (1S,3R,4S)-B’ molecules & inversion center 2 within the c = ½ to c = 1 half-portion of the cell have been moved parallel to the a-axis in opposite directions from each other and are related by a net lateral shear relocation of x = 0.060(2) fractional coordinate.

Figure 87: Unit cell with bold-face magenta pseudo n-glide-reflection plane at b = 0.251(3) between molecules (1S,3R,4S)-B’ & (1R,3S,4R)-A exhibits a pseudo rmS(x-translation)coordinate = 0.059 distortion (magenta hollow-arrows between the red solid-circle and its partially obscured green solid-circle ideal position on the a-axis).

Figure 87 shows that the same 0.060(2) a-axis net lateral shear relocation between pseudo inversion centers 1 and 2 is manifested in an rmS(x-translation)coordinate = 0.059 fractional coordinate distortion on the a-axis (hollow magenta arrows between pseudo x-translationmean red solid-circle and partially obscured green solid-circle ideal position of x = fractional ½). The x-translation is depicted as a solid magenta arrow on the a-axis from the origin towards the red solid-circle.

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Figure 88: Unit cell with bold-face magenta pseudo n-glide-reflection plane at b = 0.251(3) between molecules (1S,3R,4S)-B’ & (1R,3S,4R)-A exhibits a pseudo rmS(z-translation)coordinate = 0.013 distortion (magenta hollow-arrows between the red solid-circle and its partially obscured green solid-circle ideal position on the c-axis).

Figure 88 shows the smaller rmS(z-translation)coordinate = 0.013 fractional coordinate distortion on the c-axis (hollow magenta arrows between pseudo z-translationmean red solid-circle and partially obscured green solid-circle ideal position of z = fractional ½). The z-translation is depicted as a solid magenta arrow on the c-axis from the red solid-circle towards the origin. The x-translationmean and z-translationmean values are respectively 0.563(8) and –0.51(1). As with the pseudo inversion ‘best points’, the translation distortion from ideality on the a-axis is larger than that on the c-axis. Since we are dealing with a (pseudo) symmetry group, the distortions on the a-axis exhibited by the pseudo inversion centers and the x-translations are obviously concurrent and of course interrelated. Figures 86-88 illustrate the interpretation that the a-axis realignment of the aromatic rings occurred primarily to maximize their edge-to-face Coulombic interactions (depicted by blue dashed-lines) and the overall packing within crystalline 11.

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Application of rmS quantitative methodology to the symmetry components of space groups has enabled an estimation of the rmS(P21/n) supergroup character of the monoclinic P21 crystals of N-desmethyl-cis-3,4-butano-nefopam HCl. This value (and those of its component parts) will be compared with those of other crystals in the following sections of this thesis.

4D.5f Discussion of stereochemistry and rmS symmetry measure results for triclinic P–1 achiral space group crystals of (±)-1-phenyl-6-cyano-1,3,4,5,6,7- hexahydro-2,6-benzoxazonine, 12.

Both (R)-A and (R)-B molecules in the asymmetric unit have the same skew-chair- boat type 2 conformation of the nine-membered ring and the superimposition of all non-hydrogen atoms give an RMS difference of only 0.040 Å.82 The RMS difference is reduced to 0.017 Å if the phenyl-ortho,meta, & para carbons are removed from the superimposition, which shows that there are only minor differences (4.46°) in the degree of phenyl-ring twist. Both phenyl rings are almost eclipsing the

Cbenzhydryl―CipsoBenzo bond: –33.4°, –28.9° (R)-A and (R)-B Cortho–CipsoPhenyl–Cbenzhydryl–

CipsoBenzo torsion angles, respectively. There are two types of attractive forces between the (R)-A and (R)-B molecules in the asymmetric unit. There is an edge-to- face phenyl-phenyl Coulombic interaction between a (S)-A phenyledge partially positively charged aromatic proton (– C→H +) and the partially negatively charged … (R)-B phenylface centroid (5.184 Å centroid centroid distance).

202

In addition, there is another Coulombic interaction between the partial positively charged nitrogen in the N–CN  + N=C=N: – (R)-B cyanamide moiety and the partially negatively charged (S)-A benzo centroid (4.183 Å centroid…N+ distance), see Figure 89.

Figure 89: Coulombic interactions in the asymmetric unit consisting of (S)-A & (R)-B (from left to right).

The rmS symmetry measure methodology now makes it possible to compare the

0.35(7) Å rmS(P21/a) P21/a supergroup character present in triclinic P–1 crystals of

2,6-homonefopam-6-CN with the larger 0.86(6) rmS(P21/n) value for the pseudo- enantiotopic skeletons in N-desmethyl-cis-3,4-butano-nefopam HCl (11) P21 chiral crystals. A major contributor to this difference in ranking is the less distorted pseudo glide-reflection translation component in the 2,6-homonefopam-6-CN crystal compared to that in the crystal of 11.

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4D.5g Discussion of Stereochemistry and RmS Symmetry Measure Results for Monoclinic Pn Achiral Space Group Crystals of (±)-(2RS,3SR,4RS)-tert.-Butyl 3 – Hydroxyl – 4 – Phenyl – 2 -(p-Toluenesulfonylamino)Pentanoate Dichloromethane Solvate, 13

Both (2R,3R,4R)-A and (2R,3R,4R)-B molecules in the asymmetric unit have the same conformation since the superimposition of all non-hydrogen atoms gave an RMS difference of 0.129 Å.82 The superimposition RMS difference is only reduced to 0.113 Å if the ortho,meta,& para carbons of the two phenyls are removed from the superimposition, which shows that the respective phenyl rings are twisted in the same manner in both molecules. The sole hydrogen-bonds in the crystal are between (2R,3R,4R)-B [x,y,z] and (2S,3S,4S)-A [x,y,z] molecules which can designate an asymmetric unit. The hydroxyl group oxygen and hydrogen are involved as hydrogen-bond acceptors and donors to the respective N–H proton and carboxyl oxygen atoms of the partner, see Figure 90.

Figure 90: Hydrogen bonding interactions between (2R,3S,4R)-A [x,y,z] and (2S,3R,4S)-B [x,y,z] which comprise the asymmetric unit.

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4D.5h Comparison of RmS(P21/c) Family Symmetry Measures for the Three Crystals Studied in this Section of the Thesis.

rmS(P21/a) = 0.35(7) Å for cyanamide 12 [P–1 triclinic achiral crystal] rmS(P21/n) = 0.77(4) Å for sulfonamide 13 [Pn monoclinic achiral crystal] rmS(P21/n) = 1.58(8) Å for secondary amine 11 [P21 monoclinic chiral crystal], pseudo-enantiotopic skeletons plus phenyls rmS(P21/n) = 0.86(6) Å Å for secondary amine 11 [P21 monoclinic chiral crystal], for pseudo-enantiotopic skeletons only

From this detailed analysis of pseudo symmetry deviations from ideal symmetry, the conclusion is that the 2,6-homonefopam-6-CN triclinic P–1 crystal with two molecules in the asymmetric unit is the closest to the ideal P21/c family supergroup character. Even after removing the obvious structural differences involving phenyl groups and the anions, the N-desmethyl-cis-3,4-butano-nefopam HCl P21 monoclinic chiral crystal is the farthest away from ideal P21/c family supergroup character.

4D.5i Comparison of RmsFidelity(P21/c) Symmetry Measures as a Check of the

CSM RmS(P21/n) Method

rmsFidelity(P21/a) = 0.26(7) Å for cyanamide 12 [P–1 triclinic achiral crystal] rmsFidelity(P21/n) = 0.82(6) Å for sulfonamide 13 [Pn monoclinic achiral crystal] rmsFidelity(P21/n) = 1.58(6) Å for secondary amine 11 [P21 monoclinic chiral crystal], pseudo-enantiotopic skeletons plus phenyls rmsFidelity(P21/n) = 1.29(6) Å for secondary amine 11 [P21 monoclinic chiral crystal], for pseudo-enantiotopic skeletons only

The results from this second pseudo symmetry quantification method are in accord with those of the rmS symmetry measures discussed above. Therefore, the rank of highest degree of P21/c character is still the 2,6-homonefopam-5-CN [P–1 triclinic achiral crystal].

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4D.5j Comparison of RmS Symmetry Measures According to the Type of Pseudo Symmetry Operation

Comparison of rmS(Ci) values: the pseudo inversion character of 11 of the pseudo enantiotopic skeletons is superior to that found in the sulfonamide 13 (rmS(Ci)

0.088(1) Å versus 0.159(1) Å, respectively). When one considers the rmS(Ci) = 0.26(1) Å value for the complete ammonium cation (skeletons plus phenyl rings) then the ordering is now reversed. However, the relocation of the pseudo positions of inversion from the ideal inversion positions was greater for the pseudo enantiotopic skeletons of 11 than for 13: rmS(relocation-Ci) 0.30(6) Å for 11 versus 0.017 Å for 13.

Comparison of rmS(n-glide) values: the pseudo n-glide character of the enantiotopic skeletons in 11 was inferior to the a-glide character of cyanamide 12 [0.563(8) a- length x-translationmean component & –0.51(1) c-length z-translationmean component of the n –translation versus a 0.492(8) a-length x-translationmean, respectively]. In addition, the relocation of the n-glide from its ideal position in 11 was larger than that for the a-glide in the cyanamide 12 [rmS(relocation-glide) 0.02(2) Å versus 0.01(1) Å]. Finally, the distortion from ideal mirror symmetry for the pseudo enantiotopic skeletons in 11 was slightly improved versus that of the cyanamide 12 [rmS(Cs) 0.076(1) Å versus 0.052 Å, respectively].

Comparison of rmS(21) values: Both the rmS(C2) 0.028(1) Å & rmS(z-translation) 0.06(7) Å in cyanamide 12 are considerably smaller than the values in the sulfonamide 13 0.114(13) Å & 0.345(4) Å, respectively.

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4D.4k. Does the Largest Distortion in a P21/c Family Supergroup Crystal Arise from Symmetry Deviations or From Relocation?

In this limited series of P21/c family supergroup crystals it is interesting to ascertain which factors contribute to the largest distortions from true symmetry. In the case of cyanamide 12 [P–1 triclinic achiral crystal] it is rmS(relocation-21) = 0.10(7) Å & rmS(x-translation) = 0.10(7) Å; for sulfonamide 13 [Pn monoclinic achiral crystal] it is rmS(n-translation) = 0.345(4) Å & rmS(Ci) = 0.16(1) Å; and for the pseudo enantiotopic skeletons of N-desmethyl-cis-3,4-butano-nefopam HCl [P21 monoclinic chiral crystal], rmS(n-translation) = 0.38(6) Å & rmS(relocation-Ci) = 0.30(6) Å. Therefore, in some cases both symmetry deviations and relocation seem to make large contributions to distortion in a P21/c supergroup crystal. However, the statistical sample for comparison is really too low to reach significant general conclusions. It is interesting that in the limited number of crystals studied, the largest symmetry deviations came from the non-ideal ½ translation distances in all three crystals. But, once again, caution is called for since there are an insufficient number of samples to jump to a general conclusion.

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4E Stereochemistry and Bb21m Supergroup RmS Symmetry Measures of

Orthorhombic Pn21a Crystals of Diphenhydramine HCl, an Antihistaminic Drug.

O Cl

+ NH CH3

H3C 14 Diphenhydramine HCl (14, CCDB REF code JEMJOA) packs in the orthorhombic 53 Pn21a space group, Z =4. The crystal’s four bona fide glide planes (red) and 21 screw axes (green) are shown in Figure 91.

Figure 91: The orthorhombic Pn21a space group (Z = 4) unit cell of diphenylhydramine HCl (14) with its four bona fide glide-reflection planes (red) and 21 screw axes (green).

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Figure 92: ‘Open butterfly wing’ conformation common to both (P)-14 (on the right) and for the diphenhydramine derivative BEXHPA (on the left) with respective +37.5° and +55.6° N–C–C–O torsion angles (red).

Glaser and Maartmann-Moe reported that the pseudo mirror symmetry of 14, pseudo

21 screw rotation (see right-hand side of Figure 92) plus the bona fide Pn21a space group special positions of symmetry resulted in a Bb21m supergroup relationship for 53 this crystal. Figures 93-96 illustrate the symmetry operations in the Bb21m space group.

Bona fide n-glide plane Pseudo b-glide plane

Bona fide a-glide plane Pseudo 21 screw axis

Pseudo mirror plane c

0 a

Figure 93: Symmetry operations in the Bb21m space group viewed from ac- plane.

209

Figure 94: Symmetry operations in the Bb21m space group viewed from bc- plane

Figure 95: Symmetry operations in the Bb21m space group viewed from ab- plane.

210

+ ½+ ,

½+ , +

½+, +

, + ½+

Figure 96: 21 Screw-rotation operations in the Bb21m space group.

The unusually small ±37.5° N–C–C–O synclinal-type torsion angle assisted in the exhibition of pseudo mirror symmetry for the molecule.53 The presence of an ‘open butterfly wing’ conformation for the diphenylmethane moiety (as opposed to orthogonal or helical twisted rings) also contributes to pseudo mirror symmetry. It was of interest to search the CCDB to ascertain if pseudo mirror conformational packing causes an ‘open butterfly wing’ arrangement, or does an ‘open butterfly wing’ necessarily constrain the molecule to adopt a pseudo mirror? A graphical

‘Ph2CHOCH2CH2N–’ fragment search afforded entries with more usual synclinal magnitude torsion angles. The average N–C–C–O torsion angle of eight crystals found with this fragment was ±68° with a standard deviation of 11° .83,84,85,86,87,58,88,89 86 87 The conformation of the Ph2CH– moiety varied from orthogonal, , to helical twist47,87,89 to ‘open butterfly wing84,85,88,89 showing that the aromatic rings in crystalline 14 can express varied solid-state conformational arrangements that are not correlated with the N–C–C–O torsion angle.

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The right-hand structure in Figure 92 illustrates that the chloride anion lies in very close proximity to the dashed blue line pseudo mirror (150° N+–H…Cl– angle and 3.00

Å N+…Cl– distance). An ‘(M), minus’ or ‘(P), plus’ sign of the N–C–C–O torsion angle will be used as a descriptor for the chiral conformation of 14. Chloride anions can form weak hydrogen atoms with C—H atoms,8 and most of the C—H hydrogen bonds found in the literature (72%) have a D—H…A angle of 150º or higher.90 The

‘open butterfly wing’ conformation enables an Cortho—H bond on each ring to point directly towards the nearby anion [pro-R ring (P)-14 left ring in Figure 92): 171° C– H…Cl– angle and 2.75 Å H…Cl– distance and pro-S ring: 173° C–H…Cl– angle and

3.00 Å H…Cl– distance]. The left-hand structure in Figure 92 also illustrates two intramolecular edge-to-face aromatic…aromatic interactions in a purin-8-yl quaternary ammonium salt analogue (CCDB REF code BEXHPA)84 related to (P)-14.

BEXHPA also exhibits an ‘open butterfly wing’ conformation for Ph2CH– but its +55.6° N–C–C–O torsion angle represents a more usual synclinal magnitude. The pro-R ring (BEXHPA left ring in Figure 92) has a 168° C–H…centroid angle and a

2.82 Å H…centroid distance and the pro-S ring has a 159° C–H…centroid angle and a

3.34 Å H…centroid distance. Although distances are long, electrostatic forces depend on a factor of r–1, rather than the short range (r–6) dependence of dispersion forces.91 Weak hydrogen bonds such as these have been discussed by Desiraju and Steiner (and references therein). Other symmetry equivalent aromatic rings also participate in this type of interaction with the chloride anion, see Figure 97. These long range electrostatic forces might be considered important, since they have been known to 92 … – occasionally cause a conformational change. Multiple weak Caromatic–H Cl bonds around the chloride anion occupying a pseudo position of mirror reflection may pay part of the energy cost of closing the N–C–C–O torsion angle to 37.5°. The ability of pseudo Cs symmetry 14 to utilize the Bb21m supergroup packing arrangement probably also contributes to stabilizing its unusual pseudo mirror geometry.93

212

… – Figure 97: Multiple weak Cortho—H Cl bonds (green) from (M)-14 (light blue) and from symmetry equivalent neighboring molecules (magenta).

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4E.1 CSM S(Cs) of the Pseudo Mirror Plane in the Bb21m Supergroup

Tools for pseudo symmetry quantification were not available at the time of the original publication of the crystal structure of 14.53 Now that these techniques are available, one can investigate the pseudo symmetry operations present in such a

Bb21m supergroup arrangement. The pseudo symmetry relationship for crystalline 14 is different from the examples studied earlier since there are both intermolecular and intramolecular pseudo symmetry relationships in the crystal. There are two (P)- 14 molecules in the center of the unit cell that are perpendicular to the c-axis. When the pro-R and pro-S halves of the molecule are used to generate a mirror reflection permutation file (including those non-hydrogen atoms that are permuted into themselves due to their closeness to the pseudo plane) the molecule has a CSM S(Cs) value of 0.16(1). This testifies to a reasonably low distortion from the ideal. The pseudo position of mirror reflection ‘best plane’ for the closer-to-the-origin (½- x,½+y,½+z) (P)-14 molecule is defined by points [0,0,0.491(7)], [0,1,0.491(7)], [1,0,0.491(7)], and [1,1, 0.491(7)]. It is interesting to note that since the pseudo plane is only ‘close-to’ but ‘not-on’ the mid-point z = ½ ab-plane, the bona fide 21 screw generates a parallel ‘best plane’ on the other side of the mid-point z = ½ ab- plane for the farther-from-the-origin (½+x,y,½–z) (P)-14 molecule. This second plane is defined by points [0,0,0.509(7)], [0,1,0.509(7)], [1,0,0.509(7)], and [1,1, 0.509(7)]. The mean of these two pseudo mirrors is the z = ½ ab-plane which would have been an ideal mirror if the symmetry was bona fide Bb21m. Each one of the two planes is dislocated from the ideal by the same z = 0.009 increment. This suggests that there is a continuum which relates Pn21a with Bb21m since the pseudo symmetry of the two closely spaced pseudo mirror planes for crystalline 14 in Pn21m represents a gradual departure from the single mirror in Bb21m.

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4E.2 RmS(21) of the Pseudo 21 Screw Axes in the Bb21m Supergroup

The Bb21m space group would have had Z = 8 if the asymmetric unit had one molecule occupying a general position of symmetry. Occupancy of a ‘pseudo position of mirror reflection’ in Pn21a (Z = 4) enables a Bb21m (Z = 4) supergroup relationship for crystalline 14. This means that one-half of a pseudo mirror symmetry molecule may be considered to be a ‘pseudo asymmetric unit’. For example, pseudo 21 screw axes (magenta) relate the (x,y,1+z) pro-S half of (M)-14 (red) with the (½-x,½+y,½+z) pro-S half of (P)-14 (pink) as well as the (½+x,y,½–z) pro-S half of (P)-14 (dark blue) with the (1–x, ½+y,–z) pro-S half of (M)-14 (light blue), see Figure 98. While it is proper to relate two pro-R (or two pro-S) halves via a symmetry operation of the First Kind, it is only a pseudo symmetry operation since these two halves reside in molecules of opposite (M,P)-handedness. For the first pseudo 21 screw axis, the centroids of pseudosymmetry related pairs of atoms forms a cluster of points. The Xmean value for this cluster of points is 0.250(9) and the Zmean value is 0.741(8). For the second axis, Xmean = 0.750(9), and Zmean = 0.259(8).

Symmetry equivalent molecules are used generate the two additional pseudo 21 screw axes (green). One of them has a Xmean value of 0.250(9) and a Zmean value of 0.241(8).

The other one has a Xmean value of 0.750(9) and a Zmean value of 0.741(9). The y- translationmean for pairs of pseudo 21 related atoms in two full molecules is 0.50(2). This is not the ideal fractional ½ value since one is dealing with a pseudo symmetry translation exhibiting a non-integer-0 value for the esd of y-translation. The RMS fractional coordinate discrepancy from an ideal fractional ½ value of y-translation between pairs of atoms is rmS(y-translation)coordinate = 0.02(1). When this is multiplied by the 10.761 Å b-axis length, the rmS(y-translation) = 0.22(1) Å. The deviation from the C2 rotation component of the pseudo 21 screw operation is shown by S(C2) = 0.10(1) and rmS(C2) = 0.16(1) Å.

215

Figure 98: Pseudo 21 screw axes (magenta) relate the pro-S half of (M)-14 (red) with the pro-S half of (P)-14 (pink) and the pro-S half of (P)-14 (blue) with the pro-S half of (M)-14 (light blue) while two other pseudo 21 screw axes (green) relate symmetry related molecules.

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4E.3 RmS(b-Glide) of the Pseudo b-Glides in the Bb21m Supergroup

A pseudo b-glide-reflection plane (magenta) at x = ca. 0.5 relates the (½+x,y,½–z) pro-S half of (P)-14 (red) with the (½-x,½+y,½+z) pro-R half of (P)-14 (pink) as well as the (x,y,1+z) pro-S half of (M)-14 (blue) with the (1–x, ½+y,–z) pro-R half of (M)- 14 (light blue), see Figure 99. While it is proper to relate pro-R and pro-S halves via a symmetry operation of the Second Kind, it is only a pseudo symmetry operation since these two halves reside in molecules of identical (M,M)- or (P,P)-handedness. This pseudo b-glide is defined by points [0.500(9),0,0], [0.500(9),1,0], [0.500(9),0,1], and [0.500(9),1,1]. Additional planes are near x = 0 & 1. The y-translationmean for pairs of pseudo b-glide related atoms in two full molecules is 0.50(2). Once again, this is not the fractional ½ value since one is dealing with a pseudo symmetry translation exhibiting a non-integer-0 value for the esd of y-translation. The RMS fractional coordinate discrepancy from a fractional ½value of y-translation between pairs of atoms is rmS(y-translation)coordinate = 0.02(1). When this is multiplied by the 10.761 Å b-axis length, the rmS(y-translation) = 0.22(1) Å. The normalized and non normalized distortion in the rotation component of the pseudo b-glide operation is shown by S(Cs) = 0.13(1) and rmS(Cs) = 0.21(1) Å.

217

Figure 99: A pseudo b-glide-reflection plane at x = ca. 0.5 (magenta) relates the (½+x,y,½–z) pro-S half of (P)-14 (red) with the (½-x,½+y,½+z) pro-R half of (P)- 14 (pink) as well as the (x,y,1+z) pro-S half of (M)-14 (blue) with the (1–x, ½+y,– z) pro-R half of (M)-14 (light blue).

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4E.4 Conclusions Concerning the Pseudo Cs Symmetry of Diphenhydramine

HCl in the Pn21a Space Group Crystal and its Bb21m Supergroup

The symmetry deviations from the Bb21m supergroup symmetry are: the S(C2) value is 0.16(1); the 21 screw-rotation is devided to its components: rmS(C2) = 0.16(1) Å & rmS(y-translation) = 0.22(1) Å; and the b-glide components are: rmS(Cs) = 0.21(1) Å & rmS(y-translation) 0.22(1) Å. Conclusion: the observation that the above pseudo symmetry operations have low CSM values is indeed consistent with a Bb21m supergroup symmetry unit cell. The ability of Pn21a space group crystalline 14 molecules to compress their ca. 60° N–C–C–O synclinal-type torsion angle to a smaller pseudo synperiplanar ±37.5° value via placement of the chloride anion on a

‘pseudo position of mirror symmetry’, together with an ‘open butterfly wing’ Ph2CH– moiety conformation, was enabled by (or occurred concurrently with) the formation of N—H…Cl– and multiple C—H…Cl– hydrogen bonds. As a result of these packing arrangements, crystalline 14 ammonium cation also occupies a ‘pseudo position of mirror symmetry’ and this enables pseudo Cs molecular symmetry in a Bb21m supergroup whereby reducing Z from eight full formula units to eight pseudo halves.

Figure 100 now shows the Bb21m supergroup unit cell with its full complement of bona fide glide planes (red) and 21 screw axes (green) together with its vertical pseudo mirror and horizontal b-glide planes (magenta) and its pseudo 21 screw axes (blue).

219

This finding strengthens our argument that the ‘open butterfly wing’ conformation is

a result of 14 being able to pack in the Bb21m supergroup crystalline array.

Figure 100: A Bb21m supergroup unit cell with its full complement of bona fide

glide planes (red) and 21 screw axes (green) together with its vertical pseudo mirror and horizontal b-glide planes (magenta) and its pseudo 21 screw axes (blue).

220

5. Conclusions:

New non-normalized rmS symmetry measure tools of rmS(Ci), rmS(C2), and rmS(Cs) had to be developed during the course of this research after the classical Avnir S(Ci) value was found to provide increased values as the distance between two pseudo inversion symmetry related phenyl rings decreased in crystals of N-desmethyl-cis-3,4- butano-nefopam HCl (11). In addition, a new rmS(translation) tool was developed together with new rmS symmetry measure tools of rmS(relocation-Ci), rmS(relocation-screw), and rmS(relocation-plane) in order to quantify pseudosymmetry in crystallographic unit cells. Their numerical values were sensical and their magnitudes were in agreement with the new rmS symmetry measure tools of rmsFidelity(Ci), rmsFidelity(21), and rmsFidelity(glide) indexes that were developed as an additional method of measuring deviations from symmetry in crystals.

Molecules exhibiting very high symmetry in the solution state (by dynamic conformational averaging) often exhibited lower symmetry when resident in crystalline environments. Addition of large rotors should cause a molecule to lose the symmetry operations of the Second Kind to a large extent, while preserving those of the First Kind to a large extent. An initial hypothesis was that as the rotors increase in size, the molecules would twist in a way which would preserve the symmetry operations of the First Kind and not those of the Second Kind. This was indeed the case. However, there was no clear quantitative correlation found between the substituent’s size and the CSM values of the First Kind. A qualitative correlation was found between the number of twistable rotors in the molecule and the CSM values of the Second Kind. The greater the number of twistable rotors the more the chirality was increased in the molecule within the crystal lattice.

221

According to calculations in the gas phase, 22 is expected to have a chiral pseudo O conformation. 22A and 22B had a pseudo O conformation. 22C had an almost achiral pseudo Oh conformation. The packing of the crystal was examined, in order to understand why a conformation like this exists. The pseudo Oh symmetry units seem act to space the pseudo O symmetry units in such a manner which prevents homochiral interactions. This phenomenon is called 'molecular recognition'. The CSM tools were used in order to determine the molecular conformation within the crystal. The existence of a pseudo Oh conformation which couldn't be explained by examining the stereochemistry of the molecule, showed the need to examine the crystal packing arrangement.

CSM studies of crystals focused mainly on network solids.94,95 The CSM values of units such as an atom surrounded by the first and second coordination spheres were often examined.95,94 The methods developed here make it possible to analyze aspects which are unique to molecular crystals. For example, the existence of two different molecules (A and B) in the asymmetric unit raises the question: how do the conformational differences between the two molecules effect the packing? Within the crystal, there are different arrangements of A with respect to B. What types of pseudosymmetry relationships can exist between A and B? Do certain molecular recognition processes influence the packing arrangements of A and B molecules within the crystal? Molecule A may assemble in various ways with respect to molecule B in the crystal. For example, pseudo P21/x crystals have four best pseudo inversion centers, each one relating a different set of A and B molecules. Analysis of organic molecular crystals in terms of their chirality is now possible, since the deviations from pseudoinversion and pseudo glide symmetries can be now be measured. Therefore, it will be possible to check if molecular organic crystals with more rotatable bonds are more chiral.

222

According to the literature, edge-to-face interactions and hydrogen bonds may play an important role in generating multiple molecules within an asymmetric unit. Indeed, many of the molecules in the asymmetric unit in the crystals investigated in my prototype study interacted via edge-to-face and hydrogen-bonding forces. Electrostatic long range interactions were found to play an important role in crystal packing, and as predicted by the literature, they could be responsible for the pseudo mirror conformation of crystalline diphenhydramine HCl (14).

Crystals can gradually desymmetrize from a bona fide higher space group symmetry to a lower symmetry one having multiple molecules in the asymmetric unit and ‘pseudo positions of symmetry’. These ‘pseudo positions of symmetry’ represent a continuum between classical ‘general’ and ‘special positions of symmetry’ in crystals. Group theory deals with symmetry which either exists or does not exist. In this case, we have seen that there is a need for a continuous form of group theory which predicts how a deviation from one bona fide crystallographic symmetry would effect the deviation from another crystallographic bona fide symmetry. We have shown in this thesis that it many cases, mimicry of higher order space groups can arise in molecular crystals containing multiple molecules in the asymmetric unit when the group elements of the proximate higher order space group are those of bona fide symmetry and pseudo-symmetry.

223

We have seen many examples in which a crystallographic element of symmetry and a 'pseudo special position of symmetry' generate a third 'pseudo special position'. For example, when there are two molecules in the asymmetric unit and there is a crystallographic 21 screw axis symmetry together with pseudo inversion symmetry, then pseudo glide symmetry will also exist in that crystal. An interesting future research project in mathematics could further examine how group theory can be extended to cases in which there is deviation from a perfect symmetry. How does the deviation from one kind of symmetry effect the deviations from other symmetries? The use of ‘fuzzy logic’ would be needed in order to examine this aspect of group theory. I have shown an interesting example of how this ‘fuzzy logic’ was essential in understanding why Diphenhydramine HCl (14) exhibited a unique pseudo mirror conformation.

224

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7. Appendix: The Avnir Algorithms for Finding the Nearest Perfectly Symmetric Object with Respect to the Required Symmetry.21,22,19,20,23

7A. The Folding-Unfolding Algorithm (Numerical Approximation)22 With this method, a shape with the required symmetry is constructed. In this case, 22 a D3 symmetry hexagon will be built: (Figure 101) 22 1. Rotate point P1 by 2/3 radians to create point P3. (first step in Figure 101)

2. Rotate point P3 by 2/3 radians to create point P5. (second step in Figure 101)22

3. Reflect points P1, P3 and P5 to obtain points P2, P4 and P6. (third step in Figure 101)22

4. A D3 symmetry collection of 6 points is constructed. (Fourth step in Figure 101)22

P1 P1 P1 1 2

C3 P3 P3

4 P2 P1 P2 P1

P3 P6 P3 P 6 P 4 P5 P4 P 5 Figure 101

The process of obtaining a symmetric shape by applying a set of operations gi on a point is called unfolding.22 The inverse procedure of unfolding is folding: points –1 22 P2 to P6 are folded onto point P1 by applying the inverse operations gi . All 22 points coallesce into a single point, P1. However, a hexagon which is not 22 perfectly D3 symmetric would give a cluster of points rather than a single point.

The Folding-Unfolding method will be explained for the basic case, in which the 22 number of points (np) equals the number of elements in the symmetry group (ng). The folding –unfolding method is performed as follows:

1. The centroid of the pseudo D3 symmetric object is set to be at the origin, and scale the object so that the maximal distance from the origin to any of the vertices is 1. 22 2. Translate the symmetry group so that all operations are about the origin (i.e., all rotations are about the origin and all reflection lines or planes pass through the origin). Select an ordering of the operations of the desired symmetry group 22 that follows the connectivity of the Pi vertices. –1 3. Fold the vertices P1...P6 by applying to each Pi the symmetry operation gi . A

~ 22 cluster of Pi points is obtained. ~ ˆ 22 4. Average the points Pi , obtaining the average point P1 :

ng ng ˆ 1 1 1 ~ P1  i Pg i   Pi ng i1 ng i1 ˆ 5. Unfold the average point P1 by applying on it each of the gi operations and ˆ obtaining Pi : ˆˆ  ii PgP 1 i = 1,..., ng 22 A D3 symmetric shape is obtained.

The Folding-Unfolding method was proved mathematically.22 Some definitions and Lemmas are required for the mathematical proof:

Orbits: The orbit of x under a group G is the set {gx  g  G }.22 x and y belong to the same orbit if y = gx for some g  G.22 Given a finite group G and given an ordering of its elements, g1, g2,…, gm, then the orbit under G of a point x in

Euclidean space is x1, x2,…, xm such that xi = gix for i = 1,.., m. If g1 = e (the 22 identity element of G) then xi = gix1 for i = 1,..., m. For example, P2, P6 and P4 belong to the same orbit, and another orbit is P3, P1, P5 (Figure 101).

Lemma 1. The centroid of an orbit of finite point-symmetry group G is invariant under G.22

For example, if the points belonging to the C3 orbit: P2, P6 and P4 (Figure 101) would be averaged, the centroid would coincide with the C3 rotation axis, and would therefore be invariant under a C3 rotation.

Lemma 2. If x is a general point with respect to G, then all points in the orbit of x 22 22 are general points . Furthermore for g1, g2  G, g1  g2  g1x  gxx. Thus if x is a general point, its orbit contains N(G) different points (N(G) is the 22 number of elements in group G). For example, in the C3 orbit: P2, P6, and P4, there are three general points and three C3 group elements. (Figure 101)

Lemma 3. If the orbit of x has a point in common with the orbit of y under G, then the two orbits are equal.22 For any x  X, the group Gx = {g  G  gx = x} is called the isotropy subgroup of G at x and it contains all elements of group G that leave x invariant.22 If x is a general point, its isotropy subgroup contains a single element of G – the identity element, i.e. Gx = {e}.22 For example, only the identity operation leaves the general point P2 invariant. (Figure 101)

Lemma 4. If G is finite, the number of different points in the orbit containing x is N(G) / N(Gx).22 For example, the number of different points in the orbit containing

P2 (Figure 101) is 3. N(C3) is the number of elements in C3, which is 3. The number of symmetry elements of C3 which leave P2 invariant is 1 (the identity operation). N(G) / N(Gx) is indeed equal to 3.

Proof of the Folding Method: The purpose is to prove that the folding method finds the closest symmetric set of points.22 Given a finite point-symmetry G centered at the origin and an ordering of its m elements {g1 = e,…gm} and given m general points P1,…,Pm, find m points ˆ ˆ ˆ ˆ 1,..., PP m and find a rotation matrix R and translation vector w such that 1,..., PP m

form an ordered orbit under G' (where G' is the symmetry group G rotated by R and translated by w) and brings the following expression to a minimum:

m ˆ 2  i  PP i i1 ˆ Since G has a fixed point at the origin and G' has the centroid of orbit Pi as a ˆ fixed point (see Lemma 1), we have that w is the centroid of orbit Pi :

m 1 ˆ w   Pi m i1 ˆ ˆ The points 1,..., PP m form an orbit of G', thus the following must be satisfied:

' ˆˆ t ˆ ii 1  i ( 1 )  wwPRgRPgP , i = 1,...,m

Where gi is the matrix representation of the i-th symmetry element of G' and is 22 equal to the i-th symmetry element gi of G rotated by R and translated by w. The following equation must be minimized:

m 2 m m ˆ t ˆ t ˆ 1 ˆ PP ii  i ( i  j ( 1 () wwwPRgRP   Pi ) i1 i1 m i1

22 Where  and i for i = 1,..., m are the Lagrange multipliers. Equating the derivatives to zero gives:

m ˆ ( i PP i  0) i1 1 m w   Pj m j 1 ˆ ˆ This means that the centroid of P1,…, Pm coincides with the centroid of 1,..., PP m . t Noting that R giR for i =1,.., m are isometries and are distance preserving:

m m ' t ˆ t t ˆ  i ( ii )   i ( PPRgRPPg ii  0) i1 i1

m m ˆ t t t t 1  i i   i RwgRRPgRmwPm i1 i1

m ˆ 1 t t 1 wP   i i  wPRgR )( m i1 The geometric interpretation of this equation is the folding method, thus showing that the folding method results in the G-symmetric set of points closest to a given set.22

7B. The Analytical Solution:20,23

Figure 102

The Folding-Unfolding method made it possible to find the nearest G symmetric object. The new method, which makes it possible to find an analytical solution for the CSM value (rather than a numerical approximation) uses a new approach. The symmetry operations of G are applied on the input structure (Figure 102a).23 The resulting structures are shown in Figure 102b. V’k,i is a G symmetric object (Figure 102c) which was obtained by using all of the symmetry operations of point group G.23 The similarities between this new method and the Folding- Unfolding methods will be shown. The CSM value calculated with the Folding-Unfolding method will be referred to as SF-U(G) and the CSM value calculated with the new method will be called S(G): 23 ˆ Qk (Figure 102a) is the original structure. Qk is the nearest G-symmetric structure, found by the Folding-Unfolding method.23 The CSM value can be calculated using the Folding-Unfolding method:

N 1 ˆ 2 UF GS  min)( 2   QQ kk Nd k 1 It is possible to show that both methods for calculating the CSM value give the 23 ˆ same results. For this purpose, bk will be defined as: kkk  ,..,1, nkQQb . min ˆ Qk is the nearest G-symmetric structure, found by the Folding-Unfolding method.23

23 ak,i will be defined as  ,..,1,' niVQa . V ' are the structures , kkik ,i min ,ik obtained in Figure 102b, after applying the symmetry operations of point group G.23

Using ak,i and bk, the expressions for SF-U(G) and the analytical S(G) will be written:

N N 1 ˆ 2 1 2 UF GS  min)( 2  QQ kk  2  bk Nd k1 nd k1

N n n n 2 1 ' 2 1 GS  min)( 2  k VQ ,ik  22 a ,ik 2Nnd k 11i 2 dn i11k

Center a Of mass k,i

Figure 103

From Figure 103, it is easily seen that ak,i = bk – bi It was shown that:

N 1 2 UF GS )(  2  bk nd k1 And:

1 n n 2 GS )(  22 a ,ik 2 dn i11k

This equation may be written in terms of bk and bi:

n n 2 n n 2 n n 2 1 1 1 2 GS )(  22 a ,ik  22  bb ik  22 ( k i bbbb ik )2  2 dn i11k 2 dn i11k 2 dn i11k 1 n 2 1 n n 1 n 2 b bb b 2 i  22 i k  2  i nd i1 dn i k 11 nd i1

It was shown that S(G) is equal to SF-U(G), which means that the new CSM method gives the same values as the old Folding-Unfolding method.23

פורמט אלקטרוני של קובצי ה - cif. נשמר ב - CDROM -. קובצי ה cif. נקראו על שם מספר המבנה המולקולרי. ניתן לפתות אותם, לצפות בהם בתלת מימד וגם לסובב אותם בעזרת התוכנה מרקורי, הניתנת להורדה מהאתר: http://www.ccdc.cam.ac.uk/free_services/mercury/downloads/

האלה, וניתן למדוד את מידת הסטייה מסימטריה אידיאלית. בנוסף לכך, ההזזה מהנקודה הטובה ביותר, הציר הטוב ביותר או המישור הטוב ביותר תילקח גם היא בחשבון. רק בעזרת השיטות החדשות שפותחו, ניתן היה למדוד באופן כמותי את ההבדל בין הסימטריה האידיאלית לפסאודו-סימטריה בגבישים בעלי סימטריה הנמוכה מסימטריית ה - supergroup האידיאלית, אך בעלי סימטריה הדומה במידה מסויימת למבנה ה - supergroup.

החוט המקשר בין כל חלקי עבודת המחקר הוא השימוש במדדי סימטריה על מנת לחקור את מה שקורה כאשר הסביבה המולקולרית היא כזו, שלא מחייבת סימטריה מסוג מסויים.

כלים למדידת סימטריה רציפה הנקראים (rmS(C2) , rmS(i -, ו (rmS(σ פותחו במהלך המחקר לאחר שמדד הסימטריה הקלאסי של אבניר נתן ערכים העולים כפונקציה של ירידת המרחק בין שני פנילים בגבישים של 11. בנוסף, מדד (rmS(translation פותח, יחד עם (rmS(relocation-i , (rmS(relocation-screw -, ו (rmS(relocation-plane , על מנת לכמת את הפסאודסימטריה ביחידות התא של גבישים. הערכים המספריים שלהם היו הגיוניים וסדר הגודל שלהם תאם את מדדי הסימטריה

החדשים שפותחו – (rmSFidelity(i), rmSFidelity(21 ו - (rmSFidelity(glide, כשיטה נוספת למדידת סטיות מסימטריה בגבישים.

כלים כאלה הם שימושיים מאוד, מכיוון שגבישים יכולים להתרחק באופן הדרגתי מסימטריה אידיאלית אמיתי לסימטריה יותר נמוכה כשיש כמה מולקולות ביחידה האסימטרית ו'עמדות פסאודו סימטריה'. עמדות הפסאודו סימטריה מייצגות רצף שבין עמדות מיוחדות ועמדות כלליות בגבישים. תורת החבורות עוסקת בסימטריה שקיימת או אינה קיימת במקרה זה, ראינו שיש צורך בתורת חבורות רציפה, שמנבאת כיצד סטייה מסימטריה מסוג מסויים תשפיע על הסטייה מסימטריה מסוג אחר. הראינו בעבודת הדוקטורט הזאת שאלמנטי סימטריה ממשיכים להיות קשורים זה לזה גם כשאחד מהם מראה סטייה גדולה מסימטריה מושלמת.

ראינו דוגמאות רבות שבהן אלמנט סימטריה אידיאלי ואלמנט פסאודו סימטריה מייצרים אלמנט סימטריה

שלישי. לדוגמא, כשיש שתי מולקולות ביחידה האסימטרית ויש בורג 21 יחד עם פסאודו אינברסיטה, אז יש לחפש גם פסאודו סימטריית glide . פרוייקט מחקר עתידי מעניין תחום המתימטיקה יכול לבחון כיצד ניתן להרחיב את תורת החבורות למקרים שבהם יש סטייה מסימטריה אידיאלית. כיצד ההתרחקות מסימטריה מסוג מסויים משפיע על סימטריות אחרות? השימוש בלוגיקה מעורפלת דרוש על מנת לבחון את ההיבט הזה של תורת החבורות. הראינו דוגמא מעניינת לצורה שבה לוגיקה מעורפלת היא חיונית על מנת להבין למה 14 הראתה קונפורמציית פסאודו מראה מיוחדת.

Desiraju כתב ש: "מבנים בעלי 'Z גבוה יכולים ללמד אותנו משהו לגבי מנגנון הגיבוש... התשובה לשאלה הזאת אולי נמצאת בתחום המתימטיקה ולא בתחום הכימיה", וזה דבר שאכן בא לידי ביטוי בפרוייקט המחקר הזה. המרחבית המונוקלינית הזאת. שיטות חדשות יוצגו על מנת למדוד סימטריה בגבישים בעלי

החבורות המרחביות P21, P-1, Pc או Pn בעלות שתי מולקולות ביחידה האסימטרית. החבורות המרחביות האלה נבחרו מכיוון שכולן קשורות למשפחת החבורות המרחביות

המיוחדת בשכיחותה הגבוהה, P21/c .

שיטות חדשות פותחו על מנת לשמש בחקר גבישים בעלי יחסי פאודו-סימטריה פנים וחוץ- מולקולריים. האם כימות העיוות מסימטריה אידיאלית יעזור להסביר את קיומן של קונפורמציות מיוחדות? על מנת להשיג את המטרה הזאת, הוחלט לחקור גבישים אורתורומביים

של diphenhydramine HCl בעלי סימטריית Pn21a, שבהם קטיון האמוניום היה בעל זוית פיתול N-C-C-O יוצאת דופן של 37.5º. אפשר להסביר את הזוית הזאת ולהראות את הקשר

שלה ל -supergroup של Bb21m. האם קונפורמציית 'כנף הפרפר הפתוחה' של יחידת ה - -

-Ph2CH (לעומת קונפורמציה שגרתית יותר – למשל אורתוגונלית או עם עיוות סלילי) משפיעה על האריזה של הגביש, או שההיפך הוא הנכון?

על מנת להשיג את המטרות שפורטו, לצורך חקירת הפסאודו-סימטריה בין מולקולות בגביש, היה ברור שיש צורך בפיתוח שיטות חדשות. המדדים הקיימים של אבניר פותחו על מנת למדוד פסאודו-סימטריה בתור אובייקט, כלומר, הן פותחו לסימטריה של חבורות נקודתיות. פעולות סימטריה נקודתיות משאירות לפחות נקודה אחת במקומה. האלגוריתמים של אבניר לא טיפלו בפעולות סימטריה שנמצאות רק בחבורות מרחביות, למשל טרנסלציה, בורג או glide .

האלגוריתמים הקלאסיים של אבניר ניתנים לשימוש עבור סיבוב, שיקוף, אינברסיה, ו - Sn , ויש לשנותם על מנת להשתמש בהם לפעולות במרחב. סיבה נוספת שמצריכה את שינוי השיטות, היא שמדדי הסימטריה כוללים נרמול לגודל, במטרה למדוד סימטריה ללא תלות בגודל האובייקט. בצורה כזו, האלגוריתם של אבניר יכול לתת את אותו מדד סימטריה עבור שני אובייקטים דומים, שההבדל היחידי ביניהם הוא בגודלם. פונקציית הנרמול לגודל תיתן תוצאות שגויות כאשר האובייקט מורכב משתי מולקולות, מכיוון שאז גודל האובייקט יושפע מהמרחק בין שתי המולקולות.

בנוסף, שיטה למדידת הפסאודו סימטריה בחבורות מרחביות צריכה לכלול גם את ההזזה של אלמנט הפסאודו-סימטריה שהוגדר ע"י מיצוע של נקודות, לעומת המיקום של העמדה המיוחדת

של אלמנט הסימטריה בגביש. כל הנקודות [xmean, ymean, zmean] בין זוגות אטומים שיש ביניהם פעולת סימטריה אידיאלית, יתלכדו עם העמדה המיוחדת בגביש, שהיא נקודה, קו או

מישור. אבל ה - [xmean,ymean,zmean] בין זוגות אטומים שיש ביניהם אלמנט פסאודו -סימטריה - מגדירים עמדה כללית בגביש. ככאלה, צריך להתייחס אליהם כמגדירי מרכז האינברסיה הטוב ביותר, ציר הבורג הטוב ביותר, או מישור השיקוף (כולל glide ) הטוב ביותר. החבורה המרחבית של הגביש לא כוללת אלמנטי סימטריה מסויימים, שקיימים ב - supergroup האידיאלי. למרות זאת, יכולה להיות מידה מסויימת של פסאודו-סימטריה עבור האלמנטים

על פי הספרות המקצועית, קשרי מימן ואינטראקציות edge to face מהווים גורם משמעותי ליצירת מבנים בעלי כמה מולקולת ביחידה האסימטרית. אכן, מולקולות רבות בגבישים שנחקרו כאן הראו אינטראקציות מהסוג הזה. נמצא שגם אינטראקציות אלקטרוסטטיות לטווח ארוך חשובות לתהליך אריזת הגביש, וכפי שמנבאת הספרות המקצועית, הכוחות האלה יכולים להסביר את קונפורמציית הפסאודו -HClמראה ineשל diphenhydram . המטרה הכוללת של תת הפרוייקט הראשון בדוקטורט היתה השימוש במדדי הסימטריה הרציפים הקלאסיים שפותחו ע"י אבניר וקבוצת המחקר שלו על מנת לחקור את השפעת הסביבה על מבנה המולקולה. בחלק הראשון של הפרוייקט, האלגוריתמים של אבניר שימשו לחקירת שינויים קטנים, אך מדידים, שעברו המולקולות כתוצאה מהימצאות במצב גבישי, לעומת מצבן בתמיסה. זה היה מעניין לבדוק עד כמה גודל השלד של המולקולה השפיע על כיווניות העיוות של המתמירים. שאלות מבניות אחרות כללו את השפעת גודל המתמיר על הדה-סימטריזציה של מולקולות בעלות מבנים של מוצקים פלטוניים. מה ההשפעה של השהייה

בעמדה מיוחדת על סימטריית המתמיר (סימטריית C3 לעומת אסימטריה של המתמירים) וכיצד זה משפיע על מדד הסימטריה הכולל של המולקולה? מה היתה ההשפעה על מדדי הסימטריה של חלקי המולקולה וכן של המולקולה כולה כשמתמירים ששימרו אלמנטי סימטריה מסויימים במלואם הושוו לפולימורפים שהיו לגמרי אסימטריים? מולקולות רבות משמרות סימטריה קריסטלוגרפית מסוג מסויים במלואה, לעומת גבישים שהם פסאודו-פולימורפיים, שבהם המולקולות אינן משמרות אף סימטריה. האם וכיצד תבוא תופעה זו לידי ביטוי במדדי הסימטריה הרציפים?

מטרה חשובה של תת הפרוייקט השני היתה השימוש באלגוריתמים על מנת לחקור יחסי פסאודוסימטריה בגבישים המכילים כמה מולקולות ביחידה האסימטרית. הדגש בתת הפרוייקט השני של עבודת הדוקטורט היה על מדידות העיוות מסימטריה אידיאלית בין מולקולות, ולא בתוך המולקולה. כשאין פסאודו סימטריה, מבנה הגביש תואם את הסימטריה המקסימלית שיכולה לבוא לידי ביטוי בצורת האריזה. אבל מכיוון שסימטריה היא תכונה רציפה, יכולות להיות מידות שונות של עיוות מאידיאליות, ויכולות להיות דוגמאות רבות למקרים שבהם הפסאודו סימטריה בין המולקולות ביחידה האסימטרית יכולה לסייע ליציבות של המבנה הגבישי. השילוב של פעולות הסימטריה שיש לגביש נותן את החבורה המרחבית שלו. השילוב בין פעולות סימטריה אידיאליות לכאלה שסוטות מאידיאליות יוצר מבנה שמזכיר את המבנה של ה - supergroup . אחת המטרות העיקריות של המחקר הזה היתה לפתח דרך למדידת הסטייה של גביש מסימטריה אידיאלית של supergroup . הסטייה מסימטריית ה - - supergroup נובעת מהסטייה מרכיבי הסימטריה השונים. האריזה בגבישים השייכים למשפחת

ה - 'P21/c ' של החבורות המרחביות [P21/x -, כאשר ה x מייצג סימטריית glide מסוג a, c או n ] חייבת להיות יעילה, מכיוון שכ - 37% מהגבישים האורגניים מתגבשים בחבורה

'עמדות פסאודו סימטריה':

הרצף שבין עמדות כלליות לעמדות מיוחדות בגבישים

תקציר:

רוב המולקולות בעלות סימטריה גבוהה מאבדות חלק מאלמנטי הסימטריה שלהן או את כולם בסביבת הגביש. מולקולות בעלות מבנים של מוצקים פלטוניים עשויות מאבני בניין בעלות סימטריה גבוהה. מולקולות כאלה מראות ניוונים המתאימים לסימטריה אידיאלית של מוצקים פלטוניים בתהודה מגנטית גרעינית עבור תמיסות (ע"י מיצוע הקונפורמציות). מכיוון שיש פחות דרגות חופש במצב מוצק, סביר להניח שמולקולות הנמצאות בסביבה גבישית יראו סימטריה נמוכה יותר. רק מולקולה הנמצאת בעמדה מיוחדת בעלת סימטריה מסוג מסויים, יכולה לבטא את אותה הסימטריה במצב גבישי. הוספת מתמירים גדולים אמורה לגרום למולקולה לאבד את פעולות הסימטריה מהסוג השני במידה רבה, ולשמר, במידה רבה, את פעולות הסימטריה מהסוג הראשון. מדדי הסימטריה הרציפים של אבניר ישמשו למדידת הסטייה מסימטריה אידיאלית. ההשערה הראשונית היתה, כי ככל שהמתמירים יגדלו, כך המולקולת יתעוותו באופן סלקטיבי, המשמר את אלמנטי הסימטריה מהסוג הראשון על חשבון אלה מהסוג השני.

על פי מחקר מוקדם שבוצע ע"י Mislow ו - Hounshell, קונפורמציית מצב היסוד (גיאומטריית המינימום הגלובלי) של tetrakis-tert-butyl tetrahedrane היא כיראלית, והסימטריה שלה היא .T

לאחרונה, Balchi ו-Schleyer ביצעו חישובים ברמות תיאוריה גבוהות יותר (*HF/6-31G ו - *B3LYP/6-31G ), וגם הם הראו ש - -tetrakis-tert-butyl tetrahedrane הוא יציב יותר כאשר הסימטריה שלו היא T . בחישובים האלה, גיאומטריית T T

בנמוכה - kcal/mol 0.5-2.0 מסימטריית Td .

בחלק מהגבישים שנחקרו כאן, היתה יותר ממולקולה אחת ביחידה האסימטרית. זה נתן הזדמנות לראות כמה מופעים של אותה המולקולה בסביבת הגביש. התוצאות הניסיוניות הושוו למבנים שניבאו סימולציות בפאזה הגזית, עבור מולקולות בעלות גיאומטריה הדומה לזו של המוצקים הפלטוניים. ברוב המקרים, הקונפורמציה היתה דומה לזו שניבאו חישובים בפאזה הגזית (הקונפורמציות היו דומות, אך לא זהות). מקרה מעניין שבו רק שתיים מתוך שלוש מולקולות היו דומות למבנה שניבאו החישובים, הוביל לפרוייקט מחקר חדש. בפרוייקט הזה, אופן האריזה של גבישים בעלי כמה מולקולות ביחידה האסימטריה נחקר.

התיזה מוקדשת להורי, שושנה ושאול שטיינברג.

תודות

אני מודה מקרב לב למנחה שלי, פרופ' רוברט גלזר, על כל מה שלימד אותי, ועל הפרויקטים המרתקים, ששנינו נהנינו לחקור.

תודה לפרופ' דוד אבניר וכל חברי קבוצתו, ובמיוחד לד"ר דינה יוגב עינות, ד"ר שחר קינן, חיים דריזון ואמיר זית על העזרה והתמיכה.

תודה לד"ר יצחק ארגז, שסינתז את נגזרות האדמנטן והנפופם, ועל הדיונים המדעיים הפוריים מאוד שניהלתי איתו.

לחברי הקרובים, שהם גם עמיתים – אנה קוגן, ד"ר ארז בוקובזה, נועה זמשטיין ועליזה ברקוביץ' תודה על העזרה והתמיכה.

תודה להורי, אחי ואחותי.

העבודה נעשתה בהדרכת

פרופסור רוברט גלזר

במחלקה לכימיה

בפקולטה למדעי הטבע

'עמדות פסאודו סימטריה':

הרצף שבין עמדות כלליות לעמדות מיוחדות בגבישים

מחקר לשם מילוי חלקי של הדרישות לקבלת תואר "דוקטור לפילוסופיה"

מאת

אביטל שטיינברג

הוגש לסינאט אוניברסיטת בן גוריון בנגב

אישור המנחה ______

אישור דיקן בית הספר ללימודי מחקר מתקדמים ע"ש קרייטמן ______

אדר, תש"ע פברואר, 2010

באר שבע

'עמדות פסאודו סימטריה':

הרצף שבין עמדות כלליות לעמדות מיוחדות בגבישים

מחקר לשם מילוי חלקי של הדרישות לקבלת תואר "דוקטור לפילוסופיה"

מאת

אביטל שטיינברג

הוגש לסינאט אוניברסיטת בן גוריון בנגב

אדר, תש"ע פברואר, 2010

באר-שבע