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Ab initio, second-order, Meller-Plesset (MP2) perturbation theory calculations of the equilibrium geometries, harrnonic vibrational frequencies, relative stabilities, dipole moments, and static dipole polarizabilities are reported for 87 different 6n-electron monocycles containing boron and nitrogen. These include al1 17 azaborinines (cornmonly called azaborines) isosteric to benzene, 26 azaborinines isosteric to pyridine, 16 azaboroles, and 28 oxazaboroles.

The rnost stable isomers have as many as possible consecutive BHNH groups and, where applicable, contain the substructure XBHNH, where X = N, NH, or O is the base- ring heteroatom. Planar conformations are stable minima for dl but 15 five-membered rings and one six-membered ring. Lower level calculations are unreliable in predicting which molecules are planar. Good agreement is found with the available electron diffraction and X-ray structures of substituted rings. The ratio of MP2 to Hartree-Fock

(HF) hannonic frequencies is found to Vary around an average of 0.94.

The polarizabilities, along with earlier results for the azoles, oxazoles, and azines, constitute a uniform quality data set for 120 heteroaromatic rings. Additive atom and bond polarizability modeIs which are accurate to within a few percent are constructed for the

104 planar molecules. The presence of boron causes scatter of the polarizabilities of isomers; hence the additive models of polarizability are less accurate than if only heterocycles containing C, N, and O are included.

The relative aromaticities of azines, azoles, oxazoles, and thiazoles are analyzed using polarizabilities and bond orders. The MP2 polarizability anisotropy and the

iii anisotropy of the n-polarizability-calculated through a combination of uncoupled

Hartree-Fock and electron correlated data-predict quite different scales of arornaticity.

HF bond orders combined with the accurate MP2 geometries, an alternative to the popular

Gordy bond orders derived from assorted experimental geometries, are used in order to improve previously proposed geometry-based aromaticity indexes-the Harrnonic

Oscillator Mode1 of Aromaticity, Pozharskii, Ring Current, and Bird Indexes. Acknowledgements

My motivation to study science stems from my desire to think God's thoughts after him, to discover the bountiful creation in al1 its diversity, to bnng order out of chaos. The first and biggest thanks go to my wife, Chen Yu-Chu. Her tender encouragement helped me to overcome every obstacle on the path to my Ph.D. But more than that, she has made Our life together a wondehl adventure. Thanks to my little Rosalie Shinwei. Thanks to Dr. Thakkar, who motivated me to be careful, thoughtful, and productive. Thanks for generous financial support to Dr. Thakkar, NSERC for a Postgraduate Scholarship, the Department of Chemistry for three PhysicaVTheoretical Awards, and the School of Graduate Studies for conference travel. Thanks to Room 304 colleagues: Drs. Bündgen, Das, Hoffmeyer, Hu, Kassimi, Lin, Ms. Steeves, and visitors Drs. Koga and Sharma, al1 of whom contributed to a congenial, work-oriented atmosphere. Their example forrns much of my experience: ZL for leaving first, REH for endurance, PJB for doing charrning calculations first and reporting them later, HCH for being systematic, VJS for being a chemist, AKD for combined pride and humility, BKS for efficacious workmanship, TK for tenaciously pursuing furtdamentals. Most importantly, thanks to Dr. N. El-Bakali Kassimi, who taught me much about science and about Iife. Thanks to al1 the librarians of UNB for a11 their help. Thanks to Cornputer Services Department personnel for services rendered. In particular, thanks to Mr. Brian Kaye for cornputer time on his workstation. Thanks to al1 the secretaries of the Department of Chemistry. Thanks to Mr. Dan Drummond for mass spectrometry and other assistance. Thanks to al1 Department professors, including members of my Advisory Cornmittee: Dr. Fritz Grein for being a dynarnic, living example of a theoretical chemist to contrast with my own boss; Dr. Allan Adam, for his enthusiastic partiipation in Our Department; and Dr. Dave Magee for being a friendly, encouraging professor. Thanks to Dr. Jack Passmore for many things, including recommending this department. Thanks to Dr. Merrill Edwards for his kind words of encouragement along my joumey . Thanks to rny fellow students for giving me a bit of themselves through sharing their experiences. May life be long and worthy in this world and beyond! Thanks to my brothers, Daniel J. and Alan. V. W. for helping to create an atmosphere conducive to love of life, from my first memory till now. Finally, thanks to my parents, Mrs. Nan (Enns) Doerksen and Dr. Daniel W. Doerksen. Without their love for each other and for me, 1 would not be standing here today. Table of Contents

O. Front Matter Title Page . i Dedication . ii Abstract . . iii Acknowledgements . v Table of Contents . vi List of Tables xi List of Figures . . xiv

1. Introduction

1.1 Overview of Introduction

1.2 Molecules Studied in this Thesis 1 .S. 1 Motivation for molecule choice . 1.2.1.1 Motivation for work on BN-containing heterocycles 1.2.1.2 Overview for the series 1.2.1.3 Details of the pattern of molecufes . 1 -2.2Nomenclature 1.2.2.1 General comments 1.2.2.2 Options and recommended narnes . 1.2.3 Previous work 1.2.3.1 Molecules not containing boron 1.2.3.2 Molecules containing boron . 1.2.3.2.1 Summary of review references . 1.2.3-2.2 Summary of moIecules previously synthesized

1.3 Properties 1.3.1 Geometries . 1.3.1.1 Experimental . 1.3.1.2 Calculated 1.3.1.2.1 Levels used 1.3.1.2.2 Choice of and accuracy of MP216-3 lG(d) . 1.3.1.2.3 Methods used for deteknining nonplanar geometries . 1.3.1.3 Comparison between experirnental and calculated geometries 1.3.2 Bond orders 1.3.3 Energies . 1.3.3.1 Ground state and zero-point . 1.3.3.2 Relative stabilities 1.3.3.3 Orbital energies . 1.3.4 Frequencies . 1.3.5 Dipole moments .

1.4 Polarizabilities . 1.4.1 Definition . 1.4.2 Practical uses of polarizability data 1.4.3 Experimental polarizabiIities 1 -4.4Theoretical polarizabilities . 1.4.4.1 Uncoupled Hartree-Fock 1.4.4.2 Coupled Hartree-Fock . 1.4.4.3Finite-field 1.4.4.4 Other . 1.4.4.5 Basis sets, and selection of ours 1.4.5 Theory vs experiment 1.4.6 Available polarizabilities of monocycles . 1.4.7 Additive models of polarizability . 1.4.7.1 Motivation for making additive models 1.4.7.2Explanation of Our procedure . 1.4.7.3 Other additive models of polarizability 1.4.7.4 Comparison with other models

1.5 Aromaticity . 1.5.1 Definition . 1.5.2 Brief introduction to criteria not included in this work . 1 S.2.1 Energetic 1 S.2.2 Magnetic 1.5.3 Summary of work of others on quantifying aromaticity .

1.6 Overview of Thesis . 1.6.1 OveraIl purpose . 1.6.2 Summary of thesis . 1.6.3 My contribution to work with CO-authors . 1.6.4 Publication details of chapters .

1.7 List of References

2. Azaborinines: Structures, Vibrational Frequencies, and Polarizabilities 2.1 Introduction . 2.2 Computational Methods 2.3 Equilibrium Geometries 2.3.1 Results 2.3.2 Comparison with Previous Calculations 2.3.3 Comparison with Experiment . 2.3.4 Geometrical Trends 2.4 Harmonic Vibrational Frequencies .

vii 2.5 Relative Stabilities . 2.6 Dipole Moments 2.7 Polarizabilities . 2.7.1 Results 2.7.2 Comparison with Previous Work . 2.7.3 Trends 2.7.4 Observations on Methodology . 2.7.5 Additive Atom Polarizability Models 2.7.6 Additive Bond Polarizability Models 2.8 References and Notes .

3. Heteroaromatic Azaboracycles: Structures and Vibrational Frequencies 3.1 Introduction . 3.2 Computational Methods 3.3 Equilibrium Geometries 3.3.1 Conformations 3.3.2 Comparison with Previous Work . 3.3.3 Trends in Bond Lengths and Angles 3.4 Harmonic Vibrational Frequencies . 3SRelativeStabilities . 3.6 Dipole Moments 3.6.1 Results 3.6.2 Comparison with Previous Work . 3.7 Concluding Remarks . 3.8 References and Notes .

4. Electron-Correlated Polarizabilities of 70 Heteroaromatic Azaboracycles 4.1 Introduction . 4.2 Computational Methods 4.3 Polarizabilities . 4.3.1 Results 4.3.2 Trends 4.3.3 Observations on Methodology . 4.3.4 Additive Atom Polarizability Models . 4.3.5 Additive Connection Polarizability Models 4.4 Discussion 4.5 References and Notes .

5. Polarizability- and Bond-Order-Based Quantitative Aromaticity Indexes Applied to Comrnon Heterocycles: Azines, Azoles, Oxazoles, and Thiazoles . . 155 5. 1 Introduction . . 156 5.2 Methods . 159 5.2.1 Computations . 159 5.2.2 Bond-Order-Based Aromaticity Indexes . . 160

viii 5.2.3 Angle-Based Aromaticity Indexes . 5.2.4 Polarizability-Based Indexes 5.2.5 HOMO-LUMO-Energy-B ased Indexes 5.3 Results . 5.4 Discussion 5.4.1 Geometry . 5.4.2 Bond Orders 5.4.3 Bond-Order-Based Indexes 5.4.4 Bond-Length-Based HOM& Index 5.4.5 Angle-Based Indexes 5.4.6 Polarizability-Based Indexes 5.4.7 Hardness . 5.5 Guideposts for Accuracy of Aromaticity Indexes 5.5.1Aza-substitution . 5.5.2 Heteroatom Type . 5.5.3 Ring Size . 5.6 Conclusion 5.7 References and Notes .

6. Conclusion . 6.1 Main Results of Thesis 6.2 Possible Follow-up Work . 6.3 Cornparison with Relevant Recent Results 6.3.1 Geornetry of s-triazine . 6.3.2 Results for polarizabilities . 6.4 References

Appendixes .

Al. Polarizabilities of Heteroaromatic Molecules: Azines Revisited Al.1 Introduction . A1.2 Methods and Definitions . A 1.3 Equilibrium Geometries . A1.4 Dipole Moments . A 1.5 Polarizabilities A 1.6 Polarizability Models A 1.7 References .

A2. The C Basis Set .

A3. Specifics on Making Scale Structural Diagrams A3.1 Getting started A3.2 Scale drawings of planar molecules . A3.3 Scale drawings of nonplanar molecules . A3.4 Concluding remarks . A3.5 References .

A4. Relevant Hantzsch-Widman Names for Molecules . Basic mles . Unsaturation . Indicated hydrogen . Locant numbering . References .

AS. Supplementary Material for Chapter Azaborinines: Structures, Vibrational Frequencies, and Polarizabilities . 248

A6. Supplementary Material for Chapter 3, Heteroaromatic Azaboracycles: Structures and Vibrational Frequencies . . 256

Vita . List of Tables

Table 1.1 Units and Conversion Factors .

Table 1.2 Names, Number of Isorners, and Number of Atom Groups for Five-membered Rings . 7

Table 1.3 Names, Number of Isomers, and Number of Atom Groups for Six-membered Rings . 7

Table 2.1 Trends in MP216-3 1G(d) Geometries of 15 Planar Azaborinines . 72

Table 2.2 Relative Stabilities, Dipole Moments and Angles, Polarizabilities and Angles, n-Fractions, and Reciprocd Hardnesses . . 76

Table 2.3 Parameters and Errors of Additive Atom Models of Polarizability . 86

Table 2.4 Parameters and Errors of Additive Connection Models of Polarizability 86

Table 3.1 Ring Bond Lengths and Angles For Al1 Diazaborinines and Triazaborinines . 106

Table 3.2 Ring Bond Lengths and Angles For Al1 Azaboroles and Oxazaboroles 108

Table 3.3 Relative Stabilities, Dipole Moments and Angles . 122

Table 4.1 MP2 Polarizabilities and Angles, UCHF n-fractions, and Reciprocal Hardnesses . . 138

Table 4.2 Percent Difference, for Key Properties, for Planar Relative to Nonplanar Geometry, for 3 Oxazaboroles . 145

Table 4.3 Parameters and Emors of Additive Atom Models of Polarizability . 149

Table 4.4 Parameters and Errors of Additive Connection Models of Polarizability 149

Table 5.1 Gordy and SCF/C Bond Orders for Benzene and the Twelve Azines . 166

Table 5.2 Gordy and SCFIC Bond Orders for the Azoles, Oxazoles, and Thiazoles 167

Table 5.3 Bond-Order-Based Aromaticity Scales with Gordy Bond Orders and with SCF Bond Orders, and Scales Based on Bond Lengths and Angles . . 169 Table 5.4 Aromaticity Scales Based on Polarizability; Also, Hardnesses and Absolute Values of the Chemical Potential . . 171

Table 5.5 Uncoupled Hartree-Fock Polarizabilities of the Thiazoles with the Cl Basis at their Optimized Geometries . . 173

Table A 1.1 Cornparison of MP2/6-3 1G* Geornetries with Others . 220

Table A1.2 MP2IC Dipole Moments and Polarizabilities at the MP2/6-31G* Geometries . . 222

Table A1.3 Parameters and Errors of Various Polarizability Models . . 229

TabIe AS. 1 The C Basis Set . . 237

Table AS. 1 Calculated MP216-31G(d) Geometries for C2, and C2, Azaborinines 249

Table A5.2 Calculated MP216-31G(d) Geometries for Cs Azaborinines . . 250

Table A53 Harmonic Vibrational Frequencies for C, and Cab Azaborinines . 252

Table A5.4 Hannonic Vibrational Frequencies for CsPzaborinines . 254

Table A6.1 XH Bond Lengths and XYH Angles for Al1 Diazaborinines and Triazaborinines 257

Table A6.2 XH Bond Lengths and XYH Angles for Al1 Azaboroles . . 259

Table A6.3 XH Bond Lengths and XYW Angles for Al1 Oxazaboroles . . 260

Table A6.4 Dihedral Angles, Stabilities witb respect to Planar Conformation En,, and Dipole Moment Angles for All Nonplanar Five-Membered Rings . 262

Table A6.5 Hannonic Vibrational Frequencies and Infrared Intensities for Diazaborinines . 263

Table A6.6 Harmonic Vibrational Frequencies and Infrared Intensities for Cs Triazadiborinines . . 265

Table A6.7 Harmonic Vibrational Frequencies and Infrared Intensities for Triazadiborinines . . 267

Table A6.8 Harmonic Vibrational Frequencies and Infrared Intensities for diazaboroles and Triazadiboroles . . 269

xii Table A6.9 Harmonic Vibrational Frequencies and Infrared Intensities for Cs Triazaboroles, Oxazaboroles, and Oxadiazadiborole . 271

Table A6.10 Harmonic Vibrational Frequencies and Infrared Intensities for Non-Cs Triazaboroles, Oxazaborole, and Oxadiazadiboroles . 272

Table A6.11 Harmonic Vibrational Frequencies and Infrared Intensities for the C, Tetrazaborole and Oxadiazaboroles . . 273

Table A6.12 Harmonic Vibrational Frequencies and Infrared Intensities for Tetrazaboroles and Oxadiazaboroles . . 274

Table A6.13 Harmonic Vibrational Frequencies and Infrared Intensities for Oxatriazaboroles . 275

xiii List of Figures

Figure 1.1 Fundamental aromatic 6n-electron structures . . 44

Figure 2.1 Azaborinines at their optimized geometry . 64

Figure 2.2 Distribution of ratios for harmonic vibrational frequencies of the azaborinines . . 74

Figure 2.3 Stabilities of the azaborinines relative to the most stable isomer . 75

Figure 2.4 Percent differences between the smallest principal component of the polarizability computed by a given method and its counterpart calculated at the CHF level in basis C . . 83

Figure 2.5 Percent differences between the mean polarizability computed by a given method and its counterpart calculated at the CHF level in basis C . 84

Figure 3.1 The diazaborinines at their optirnized geometry . 95

Figure 3.2 The triazadiborinines at their optimized geometry . . 96

Figure 3.3 The azaboroles at their optimized geometry . . 97 Figure 3.4 The oxazaboroles and oxadiazaboroles at their optimized geometry . 98

Figure 3.5 The oxatriazaboroles and oxadiazadiboroles at their optimized geometry 99

Figure 3.6 Relative stability of nonplanar vs. planar conformations . 102

Figure 3.7 Al1 nonplanar five-membered rings at their optimized geometry , 104

Figure 3.8 King bond lengths for 87 molecules . , 114

Figure 3.9 Ratios ri = oi(MP2)/wi(HF)versus MP2 frequency for al1 harrnonic vibrational frequencies of 83 molecules . . 116

Figure 3.10 Stabilities of the azaborinines and diazaborinines, each relative to their most stable isomer . . 119

Figure 3.11 Stabilities of the diazaboroles, triazaboroles, tetrazaboroles, and triazadiboroles, each relative to their most stable isomer . . 120

xiv Figure 3.12 Stabilities of the oxazaboroles, oxadiazaboroles, oxatriazaboroles, and oxadiazadiboroles, each relative to their most stable isomer . 121

Figure 4.1 One molecule out of each set of isomers for which polarizability calculations are reported . . 134

Figure 4.2 One molecule out of each set of five-membered ring isomers for which polarizability calculations were previously reported . 135

Figure 4.3 Trends in polarizability changes upon substituting N for CH, for 120 molecules . . 142

Figure 4.4 Trends in polarizability changes upon substituting O for MH, for 120 molecules . . 143

Figure 4.5 Trends in polarizability changes upon substituting BN for CC, for 120 molecules . 144

Figure 4.6 Ratios of CHFAJCHF and MP2KHF for out-of-plane polarizabilities 146

Figure 4.7 Ratios of CHF/UCHF and MP2ICHF for the mean polarizability . 147

Figure 5.1 Benzene, pyridine, pyrrole, furan, and thiophene . 157

Figure 5.2 The Bird index 1, for al1 42 molecules calculated using Gordy bond orders, at selected experimental and at AM1 geometnes compared with at the MP2 geometries . 174

Figure 5.3 The percent difference between the Gordy and SCF bond orders, for al1 ring bonds in the 42 molecules, both at the MI2 geornetries . . 177

Figure 5.4 Correlation between POZPC and HOM%, both calculated with Gordy bond orders, at the MF2 geometries . 179

Figure 5.5 Correlation between POZPC and HOM%, both calculated with SCF bond orders, at the MF2 geometries . 180

Figure 5.6 Correlation between POZPC and IA, both calculated with SCF bond orders, at the MP2 geornetries . . 181

Figure 5.7 Correlation between POZPC, calculated with SCF bond orders, and Ag, both at the MP2 geometries . 186 Figure 5.8 Correlation between MP2 Aa and the reciprùcal of the absolute value of the chernical potential, both at the MF2 geometries . . 190

Figure 6.1 Bais set and electron correlation effects on the C-N bond length of s-triazine . 204

Figure 6.2 Polarizability of furan . . 207

Figure 6.3 Average polarizabilities for heteroaromatic rings . 208

Figure 6.4 Anisotropic poIarizabilities for heteroaromatic rings . . 209

Figure Al.l The azines at their optirnized geometry . 218

Figure A1.2 Percent differences between the mean polarizability computed by a given method and its counterpart calculated at the CHFK level . . 224

Figure A1.3 Percent differences between the polarizability anisotropy Ala computed by a given method and its counterpart cdculated at the CHFIC level . 225

xvi Chapter 1. Introduction 1.1 Overview of Introduction

Zn this introduction, 1 will explain briefly the important ideas that are necessary to understand the body of the thesis, introduce definitions and nomenclature, and give background material to help demonstrate the value of the research.

The initial topic is the molecules which were studied. Next, various properties which were calculated for those molecules will be introduced: polarizabilities, geometries, bond orders, energies, vibrational frequencies, and dipole moments. Some cornrnents on quantifying aromaticity and an outline of the rest of the thesis wiIl conclude the introduction. The introductory material found in each later chapter may, in places, overlap with or supplement the content of Chapter 1.

Table 1.1 shows the units which are used in this thesis and conversion factors to

SI and other popular units.

1.2 Molecules Studied in this Thesis

This thesis is exclusively about heteroaromatic monocycles with no substituents.

Chapters 2 to 4 are on boron-nitrogen-containing rings. The fifth chapter contains more common heterocycles. Al1 sets of molecules are connected by two common threads: aza- substitution and the inclusion of al1 ring isomers.

1.2.1 Motivation for molecule choice. My original motivation was to find a collection of heteroaromatic molecuIes that would be suitable subjects for a study of trends in polarizability. Before the work of this thesis, Dr. Thakkar's research group was already interested in issues such as how well the polarizability of an aromatic molecule Table 1.1 Vbits ami Conversion Factor~.~ Physical Quantity Symbol Base Conversion Other unitsb FactorC Units length x Pm 1.88973 0.01 enersyd En kcal/mol 1.59360 4.33641 6.94770 electric field F a.u. 5.14221 electric dipole moment p D 0.393427 a.u.

3 -33561 ~o-~OC m polarizabilitye a a.u. 1.64878 l~-~~F m2 O. 148185 IL3 vibrational frequencyf O cm-' 29.9792 GHz 4.55633 10-~a-u.

a Fundamental physical constants from 1986 least-squares adjustment (Cohen and Taylor 1994). Cf. Lide (1994, 1-24, 1-34) and McWeeny (1973). Base units are those used most frequently in this thesis. Base unit is equivalent to conversion factor times other unit. Molar energy (E,) or energy (E),with E, = NAE. Polarizability (a) or polarizability volume (a,), with a = 4.Jreo%. Wavenumber (a),fxequency (V) , or energy (El , related by E = hV = hm. cm be represented as an additive function of subunits of that molecule. This ied to the early work on the azines (Archibong and Thakkar 1994), azoles (El-Bakali Kassimi et al.

1995) and oxazoles (El-Bakali Kassimi et al. 1996). There were severd possible ways to branch out. Some were taken by others-a move from CNO-containing monocycles to bicycles (El-Bakali Kassimi and Thakkar 1996; Hinchliffe and Machado 1993, 1994,

1995) or to rings including third-period atoms (El-Bakali Kassimi and Lin 1998).

1 chose to study isoelectronic series of molecu1es containing B and N. Other researchers have considered a set of molecules that is not isoelectronic, such as C,H4X, with X varying across the periodic table (Hinchliffe and Sosciin 1995). Another approach with much potential is to study the effects of substituents on heteroaromatic rings. Some 4

attempts have been made already, for exarnple on pyridine oxides (Lazzeretti et al. 1993;

Berthier et al. 1992). Several workers have studied polymer chains for which the

monomer unit is a heterocyclic ring (e.g., Luo et al. 1998; Champagne et al. 1994).

A philosophical issue is whether to study just experimentdly-known molecules.

To study al1 molecules of a series, whether or not there is experimental data with which

to compare, can be justified in various ways: research should explore new territory. Also,

if unknown and known molecules are studied together, then insight cm be gained into the

reasons for the relative instability of some. Finally, the list of experimentally observed

molecules is slowly increasing: only recently has the unsubstituted 1,2,3-triazine been

studied (Neunhoeffer et al. 1985); the first triazadiborinine was announced in 1995

(Klofkorn et al.).

Al1 the molecules of this thesis are of such a size that calculating their

polarizability accurately is currently difficult, but possible. Whether they are large,

medium, or small is a matter of perspective. Suffice it to Say that they are at the upper limit of size for the goal of performing calculations on a large series of them at the same meaningfully accurate level.

1.2.1.1 Motivation for work on BN-containing heterocycles. Why do people study these molecules? The attraction is that they are similar to carbocycles, yet have important differences in key properties. B and N are each quite different from C,but together they can replace two C's, yielding an isoelectronic product. In practice, the end result is very different from benzene, though "aromatic" still is a valid description of some of the BN-heterocycles, particularly when C is mixed in. 5

Sometimes heteroaromatic azaboracycles have been studied for possible therapeutic

effects (Niedenzu 1975; Yale et al. 1962). Some derivatives have been shown to be

effective antibacterial agents (Hogenauer and Woisetschlager 198 1). Benefits could result

from making BN-analogs of biological molecules, such as in the work of Groziak et al.

(1994). Polymers containing B, N and other elements have been synthesized and studied

(Gates and Manners 1997). Others have studied the BN-counterparts of fullerenes (Bardo et al. 1995).

There has been much effort exerted to make BN-containing rings. The literature is scattered. This thesis contains systematic ab initio calculations of the properties of a large set of such molecules and an assessrnent of the similarities and differences among the molecules. The results may prove useful to a future generation of chemists as they try to make and study new molecules.

1.2.1.2 Overview for the series. There are two contrasting groups of molecules. One consists of the most cornrnon heterocycles of chemistry, such as pyridine, pyrrole, and furan. These are the base units of heterocyclic chemistry, which itself includes about half of al1 known chemical compounds (Katritzky 1985). The other, containing the majority of molecules in this thesis, is a group of rare or as-yet- undiscovered heterocycles. This includes some molecules which are relatively unstable because of an unusual preponderance of adjacent N's or B's.

What ties these disparate groups together? The molecules are in isoelectronic series: either 42-electron six-membered rings or 36-electron five-membered rings. They 6 are generated by a regular procedure of rearranging a few atoms. They contain second- penod atoms each bonded to two other ring atoms and at most one H.

1.2.1.3 Details of the pattern of molecules. The 120 molecules containing

BH, CH, NH, N, and O fit into regular patterns. The common rule governing my choice of molecules is that a BH and an NH unit are always selected together. The azoles, oxazoles, and azines can each be considered as being derived from the parent pyrrole, furan, and benzene, respectively, by aza-substitution-replacing CH by N, an isoelectronic substitution. Similarly, the BN-containing molecules are each generated from one of the other molecules by replacing two CH's with one BH and one NH.

Table 1.2 shows the pattern for the five-membered rings. Each is made from four items selected from {BH, CH, NH,NI, plus one NH for pyrroIe, azoles, and azaboroles, or one O for furan, oxazoles, and oxazaboroIes. The azoles and oxazoles al1 match: each family contains five unique molecular forrnulae, with 1, 2, 4, 2, and 1 ring isomer(s).

They are al1 included in Chapter S. The BN-containing molecules derived from azoles and those derived from oxazoles do not match, because in the azaboroles there are always at least two NH's which (in terms of isomer-generation) are indistinguishable. The 16 azaboroles and 28 oxazaboroles are al1 studied in Chapters 3 and 4.

Table 1.3 lists six-mernbered rings. Each molecule is made from six items selected from (BH, CH, NH, N). For each pairing in Table 1.3, the left molecules are those with more CH's. Benzene and the 12 azines are al1 studied in Chapter 5 (and Appendix 1). The

17 azaborinines, diazadiborinines, and triazatriborinines are the focus of Chapter 2. The

10 diazaborinines and 16 triazadiborinines are included in Chapters 3 and 4. They fit Table 3.2 Names, Number of Isomers, and Number of Atom Groups for Five-membered Rings. Azoles and Derivatives Oxazoles and Derivatives --

Name No. N CH ~NH~BH n~ ~CH~NH ~BH n~ no No. Name pyrrole 1 4100 40001 1furan pentazole 1 0104 00041 1 oxatetrazole diazoles 2 3101 30011 2oxazoles tetrazoles 2 1103 10031 2oxatriazoles triazadiboroles 2 0320 02201 4 oxadiazadiboroles triazoles 4 2102 20021 4oxadiazoles diazaboroles 4 2210 21101 6 oxazaboroles tetrazaboroles 4 0212 O1121 6 oxatriazaboroles triazaboroles 6 1211 11111 12 oxadiazaboroles

Table 1.3 Names, Ehrmber of Isomers, and Number of Atm Groups for Six-membered Rings.

Name No- N CH %II n~~ n~ CH ~NH n~~ n~ No* Name benzene 1 6 O O O O O O 6 1 hexazine pyr idine 1 5 O O 1 1 O O 5 1 pentazine diazines 3 4 O O 2 2 O O 4 3 tetrazines azaborinines 3 4 1 1 O O 1 1 4 3 pentazaborinines triazines 3 3 O O 3 O 3 3 O 3 triazatxiborinines diazaborinines 10 3 1 1 1 1 1 1 3 10 tetrazaborinines diazadiborinines 11 2 2 2 O O 2 2 2 11 tetrazadiborinines triazaborinines 16 2 1 1 2 1 2 2 1 16 triazadiborinines

4 8 together because they each have one pyridinic nitrogen. The other azaborinines rernain to be studied: from the diazines, 16 triazaborinines and 1 1 tetrazaborinines can be formed, from the triazines, 10 tetrazaborinines, and from the tetrazines, 3 pentazaborinines.

1.2.2 Nomenclature. Chemistry has an interesting history of naming molecules and compounds. Often narnes have been chosen for substances before the substances were properly identified. Sometimes the name has no connection to the nature or properties of the substance being named.

1.2.2.1 General comments. The first important narning problem for the molecules in this thesis is the naming of the overall collection-heteroarornatic. The prefix refers to the presence of non-carbon atoms in addition to carbon atoms in the ring.

Aromaticity is an old concept, originally associated with benzene, that has its origin in the aroma of benzene and similar compounds. Kekulé (1865) devised the theory of the structure of benzene which helped explain the unique properties of aromatic compounds.

Hückel's 4n + 2 ruIe States that planar monocycles which contain 4n + 2 n-electrons in the ring, where n is a whole number, will be relatively stable (Streitweiser 1961). Al1 of the molecules in this thesis have six n-electrons. The concept of aromaticity has moved beyond the realm of nomenclature, and so will be discussed further in Section 1.5.

That the molecules al1 fit into the class of "heteroaromatic" has implications for proper naming. Some names for the boron-nitrogen-containing rings were designed to address this issue. Dewar (1964) proposed that the names of such rings should include not only a reference to boron and nitrogen, but also an extra designator that the rnolecule was 9

aromatic, aro. Thus, the simplest one should be called borazarobenzene. This style of

name was commonly used by some workers, but now is rare.

1.2.2.2 Options and recommended names. There are two major sources of

names for molecules in chemistry, the International Union of Pure and Applied Chemistry

(IUPAC) and the American Chemical Society's Chemical Abstracts Index Guide (1996).

There are three kinds of names for molecules: common or trivial names, systematic

names, and names which contain a mixture of common and systematic parts. Common

names are unpredictable. Commonly-used trivial names are easy to remember, sometimes

fairly short, and occasionally have a connection to a property of the molecuie being

named. However, now that there are millions of known molecules, it is hopelessly inefficient to give each a unique common name. On the other hand, systematic names have traditionally been coined on the basis of some already existing cornrnon name.

The third option is the one followed in the IUPAC-recommended (Powell 1983) extended Hantzsch-Widman systern: it uses names that are a mixture of cornrnon roots and systematic modifications and additions. There is a set of rules, themselves subject to modification from time to time, specifying prefixes, suffixes, ordering, and numbering.

The main rules relevant to BN-containing molecules are listed in endnote 8 of Chapter

2. A more complete explanation of important nomenclature rules is given in Appendix 4.

The chernical literature is full of cases of deliberate or unintentional breaking of the Hantzsch-Widman rules. Even Chemical Abstracts does not follow them completely.

The American Chemical Society has a compromise perspective: they suggest that 10

systematic narnes always be given, even if the authors then prefer to use cornrnon or other

names.

The nomenclature of inorganic chemistry (Leigh 1990) has rules for hornogeneous

monocycles (with a repeating unit in the ring), such as: always include the prefix cyclo-;

use the same prefixes for atoms as is standard, but a different order (e.g. bora- before

aza-); use suffix -une unless there are explicit unsaturations, in which case change to -ene

or -yne, as is done for aliphatic hydrocarbons. In this system, pentazole is called

cyclopentazadiene, and borazine would be cyclotriborazane. In any case, for

inhomogeneous rings, the Hantzsch-Widrnan system is still recommended (Leigh). 1chose

to follow the Hantzsch-Widman system (Powell 1983) exclusively.

1.2.3 Previous work. The important previous work is al1 mentioned in the

individual chapters. This section contains an overview of reference literature. Many more

experiments and calculations have been done on the molecules without boron than on the azaboracycles. 1 sumrnarize work on the former molecules only briefly, since they are well known.

1A3.l Molecules not containing boron. The moles, oxazoles, and azines are the fundamental unsaturated heterocycles. Regular review articles appear that cover heterocyclic chemistry. A good one-volume summary has been given by Katritzky (1985); a detailed account has recently appeared (Katritzky et al. 1996) which updates the previous edition (Katritzky and Rees 1984).

The commonness of the heterocycles is roughly inversely praportional to their degree of aza-substitution. Those containing O, 1, 2, or 3 pyridinic N's, as well as 11 pentazoles, pentazines, and tetrazines are al1 well-known. The books cited above give references for experimental work on their synthesis, function, and properties. The ones with no experimental data available are oxatetrazole and hexazine. Recent papers on polarizabilities (El-Bakaii Kassirni et al. 1995, 1996; cf. Appendix 1) cite relevant experiments and calculations on the properties of the molecules.

1.2.3.2 Molecules containing boron. The important dates to note are for the discovery of borazine (Stock and Pohland 1926), and the synthesis of the first BNC ring

(Ulmschneider and Goubeau 1957). Stock did extensive research on the hydrides of boron.

Much work has been done on organic boron compounds (Brown 1972). There is a vast literature on borazines, but much less work has been done on mixed B, N, C, and/or O rings (Morris 1982; Housecroft 1995).

1.2.3.2.1 Summary of review references. An overview of borazaromatic molecules is given by Pelter and Smith (1979). There are many annual reviews which include heteroaromatic BN-compounds. Annual Reports on Progress in Chernistry, Section

A Inorganic Chernistry (e.g., Beckett 1993) and its predecessors have coverage al1 the way back to the 1950s. The most useful annual reviews 1 consulted are the "Boron Annual

Survey Covering the Year 19--" in the Journal of Organometallic Chemistry and "Group

III: Boron" in Organometallic Chernistry (A Specialist Periodical Report). The former was written between 1972 and 1984 by Niedenzu (e.g., Niedenzu 1984), who himself worked on certain BN-aromatics. The latter flourished under the authorship of Gupta (e.g.,

1977) and of Morris (e.g., 1984) but because of heavy pressure to cut the number of pages finally becarne too bare-bones to be of any interest. 12

Niedenzu and Dawson (1965) have al1 the necessary references regarding which rings had been made up to that time. Their book also discusses nomenclature and the nature of the B-N bond. A symposium sumqfrom the same time has much important work by many of the key BN-workers of the tirne, such as Dewar (1964); it includes

Hoffmann (1964b) on extended Hückel work, though an article of his (Hoffmann 1964a) contains more that is relevant to the set of 87 BN-containing molecules. Niedenzu (1972,

1975) also reviewed the next 10 years of synthesis. Fritsch (1977) gave a good account of the extensive work from the Dewar group on attempts to make novel heterocycles, and discussed the aromaticity of the azaboracycles. Much interesting detail can be found in

Haiduc's book (1970).

Related articles irregularly appear in Advances in Organometallic Chemistry, such as Seibert (1980) on the important potential use of azaborarnatics as transition metal ligands, and in Advances in Inorganic Chemistry, e.g., Paetzold (1987) on iminoboranes, which cm be used in heterocyclic synthesis.

1.2.3.2.2 Surnmary of molecules previously synthesized. There are three motivations for knowing which molecules have been synthesized. If a molecule has been made, that implies that it is to some extent interesting. Caiculations can be checked if corresponding experimental work has been done on the molecule. Knowledge that some molecules have not been synthesized may prompt new initiatives to try to prepare them.

Of the 87 BN-rings, only one, borazine, has been isolated; 12 have been made with ring substituents, ranging from small (methyl) to large; five others have appeared in a molecule with a fused ring. The detailed references and names are listed in Chapters 2 13

and 3. The fused rings cm be expected to distort the properties of the base ring.

Substituents also can have a large effect on the properties that are described in this thesis.

Fundamenta1 effects such as whether or not the ring will be planar cannot be answered

based on experimental data of substituted rings, since certain substituents help to stabilize

a ring. Thus, it will be a major undertaking to make new unsubstituted rings.

1.3 Properties

This section gives background information for the basic properties (other than

poiarizabilities, treated below in Section 1.4) which were calculated for the 87 BN-

containing molecules: geometries, energies, frequencies, and dipole moments. For the

azoles, oxazoles, mines, and thiazoles, bond orders are included in Chapter 5, and

introduced in this section, whereas their geometries, energies, and dipole moments were calculated previously (El-Bakali Kassimi et al. 1995, 1996; Doerksen and Thakkar 1996;

El-Bakali Kassimi and Lin 1998).

1.3.1 Geometries. A geometry is a precise specification of the relative position of al1 the atomic centers in a rnolecule. This can be expressed with coordinates of various kinds, the rnost common of which are "intemal," Le., bond lengths, bond angles, and dihedral angles. An alternative is Cartesian coordinates.

Though much chemistry can be done without detailed knowledge of the geometry of the molecules involved, the geometry is important for quantitative understanding.

Accurate geometries can confimi structure and give clues about bond strength and steric 14

factors. For the theorist, a good geometry is necessary input for further calculations of

other properties of a rnolecule.

1.3.1.1 Experimental. The early history of attempts to use spectroscopy to

determine molecular geometry is described briefiy by Lide (1975). The first bond lengths

were reported between 1914 and 1920, just after the development of Bohr's atomic

theory. The early work was in the IR region, vibration-rotation and pure rotational spectra.

The development of microwave spectroscopy after 1945 allowed much better resolution,

including the detecting of isotopic differences (Lide).

By observing rotational spectra, it is possible to determine a molecule's moments

of inertia. Each molecule has at most three unique moments of inertia; thus, beyond

diatomics and other high symmetry species it is necessary to use the subtle changes in the

moments upon isotopic substitution to locate atoms one-by-one (Kraitchman 1953).

Microwave spectroscopy is limited to molecules with permanent dipole moments.

For large molecules, the number of isotopes to be studied becomes large. Nevertheless, the microwave spectra of many five- and six-membered rings have been determined. The method works well for unsubstituted rings, because they have relatively few atoms. Bulky substituents pose a formidable banier to microwave structural determination.

Electron diffraction (ED) has the advantage that it can be used for molecules such as borazine with a vanishing dipole moment.

X-ray crystalIography is an alternative method for structure determination (Dunitz

1979). Unlike rotational spectroscopy and ED which treat an isolated gas-phase molecule,

X-ray is for solid-state structures. Thus, intermolecular forces-stacking interactions, 15 hydrogen bonding, and steric repulsions-can play important roles in the final structure.

Most available geometries of azabora-rings are from X-ray diffraction of heavily substituted variants.

1.3.1.2 Calculated. Theoretical methods can focus directly on determining the geometry of an isolated, non-vibrating molecule. The advent of computer codes with built in optimization techniques using analytical gradients and Hessians for higher levels of electron correlation has made it straightforward, in principle, to determine the minimum geometry (Urban et al. 1987). For most of the molecuies in this thesis, there were no available geometries, experimental or calculated. We decided to calculate them, using the same theoretical level in order to have consistent geometries for al1 molecules. Systematic errors should be approximately the sarne across the series. Since previous work is limited, the geornetries themselves are of interest for this novel class of BN-containing molecules.

1.3.1.2.1 Levels used. In Chapters 2 and 3, there are three levels of geometry.

The lowest, modified neglect of differential overlap (MNDO), was used to find starting- point geometries for the higher level calculations. Hartree-Fock (HF) geometries are often used for molecules of the size of those in this study. HF does not include the correlation of electrons having opposite spins. Mgller-Plesset second-order (MP2) is the first correction to HF in perturbation theory. MP2 gives a large percent of the correlation gap, without extrerne computational demands (Dykstra 1988).

The HF geometries were closer than MNDO to MP2. Nevertheless, the errors in

HF were significant-e.g., on average 1.3 pm and 0.6" for the molecules in Chapter 2- showing that it was important to include electron correlation. 16

1.3.1.2.2 Choice of and accuracy of MP2/6-31G(d). The 6-3 1G(d) basis set

is split-valence with polarization functions on second-period atoms (Hariharan and Pople

1973). In the 1980s, extensive tests were perfonned on the MP2/6-3 lG(d) method; it was

found to be accurate within 1% for many molecules containing second-period atoms

(Hehre et al. 1986). This does not mean that MP2/6-31G(d) is the best possible

combination of basis set and correlation, nor that MP2 or 6-31G(d) are individually

outstanding, though they each have their merits. But it was found that increasing the

nurnber of basis functions, e.g. to MP216-3 1 lG(d), shortens bond lengths, while improving

the treatment of electron correlation to MP4/6-3 lG(d) gives bond lengths that are too long

(Hehre et al. 1986). Thus MP2/6-3 lG(d) is a compromise solution that, on average, works

very well.

1.3.1.2.3 Methods used for determining nonplanar geometries. For

flexible molecules, it is important to use many different starting geometries in order to

explore the potential energy surface. However, in this thesis Our concern was only for the

ring isomers. A good starting point was the planar (non-minimum) geometry, with each

dihedral angle set to be slightly different from planar.

MP2 harmonic vibrational frequencies were calculated to verify pianarity (al1 positive frequencies) or nonplanarity. First, only exceptionai cases were treated at the

MP2 level-the molecules whose MNDO andor HF planar structures were non-minima.

For molecules which were planar with MNDO or HF, it was assumed that the (similar)

MP2 geometry would be planar minima as well. Later, that assumption was verified by calculating MP2 frequencies for al1 MP2 structures. 17

1.3.1.3 Comparison between experimental and calculated geometries.

Structures can be classified as equilibrium (r,), effective (r,), substitution (r,), or average

(r,), arnong others (Lide 1975). In this notation, r represents any geometric parameter, or in general the geometry. Ab initio calculations give re, for the isolated, non-vibrating molecule. Rotational spectroscopy directly yields ro structures for the ground vibrational state. Different combinations of data can give substantially different r, structures. Precise extrapolation to r, is possible once vibrationally excited States have been studied, but this is easy only for simple molecules. It is preferable to have r, structures, which are determined by systematically comparing a molecule's spectra before and after individual isotopic substitutions have been made. No exact relation has been found between r, and r, bond lengths, though usually re < r, iro (Lide). As for r,, the averaging is done over vibrational motion using a harmonic potential function, often based on ab initio data.

ED geometries are termed r,, for a temperature-dependent average position of al1 atoms including molecular vibrations, and r,, which constrains atoms to the midpoint of off bond-axis vibrations (Lide 1975).

There are no MW geornetries available for the BN-containing molecules. ED and

X-ray geometries are fairly close to MP2/6-31G(d) (cf. Chapters 2 and 3). Accurate MW geometries have been detennined for 16 azoles, oxazoles, azines, and thiazoles (cf. references in El-Bakali Kassimi et al. 1995, 1996; Doerksen and Thakkar 1996; and El-

Bakali Kassimi and Lin 1998). The agreement of these with MP2/6-3 1G(d) is on average

0.7 pm for bond lengths and 0.4" for bond angles, suggesting that the geometries for the azabora-compounds are also reliable. 18

1.3.2 Bond orders. For heteroaromatic molecules, one would expect al1 the ring

bonds to be between single or double, i.e., a bond order between 1 and 2. The bond order

cm be defined in ab initio theory and has been given several empirical definitions. The

appropriateness of any particular definition of bond order should not be assumed.

Gordy (1 947) plotted l/(bond length12 against bond order for each type of bond

(say, X-Y), and obtained a straight line. Hence, he could take the dope and intercept as

empirical parameters to predict the bond order from any X-Y bond length. This type of

correlation is appropriate to bonds between second-period atoms but fails for bonds of

lower-period atoms, because for the latter the proportion of bonding- to nonbonding- electrons is relatively small (McWeeny 1979).

Sannigrahi (1992) has reviewed the theoretical calculation of bond orders by electrmic structure methods. Mayer (1986) promoted the following closed-shell

Hartree-Fock definition of the bond order between two atoms A and B:

in which P, the density matrix, and S, the overlap matrix, are calculated from the molecular orbitals.

The Gordy philosophy (and that of other sirnilar empirical methods such as

Pauling [1947] bond numbers) is that the atom type and bond length completely determine the bond order. If the C-C bond length is 139 pm then the Gordy bond order is always 1.77 (Gordy 1947), no matter what molecule that bond is in. By contrast, for a given bond length (and pair of atoms) there could be several values for the ab initio bond order, depending on detailed features of the particular molecule. 1.3.3 Energies. There are several important energies in this thesis.

1.3.3.1 Ground state and zero-point. The total energy of a molecule is the

sum of the kinetic energy of electrons and nuclei, the electron-electron and nucleus-

nucleus repulsion, and the electron-nucleus attraction. The electronic energy, which leaves

out the nuclear kinetic energy, is the energy that is calculated for any fixed arrangement

of the nuclei. At the equilibrium geometry of a molecule, the electronic energy is a

minimum. That is the ground state energy (McWeeny 1979).

The zero-point vibrational energy (ZPE) is calculated as half the sum of the

harmonic vibrational frequencies.

Finite calculations produce only an approximate ground state energy. An improved

basis set would give lower energies. The MP2 energies are dways lower than HF. What

is significant is to compare the energies of isomers, al1 calculated with precisely the same

method.

1.3.3.2 Relative stabilities. The potential energy surface-describing the

electronic energy as a function of geometry-of a given set of 6-12 atoms, as in this

work, will have several local minima corresponding to the equilibrium geometries of

different isomers (not al1 of them rings), and local maxima for transition States.

The relative stability of an isomer is the difference between its energy and that of the lowest energy isorner. We included the ZPE for comparing relative stabilities of

isomers. The other important kind of relative stability is the energetic difference between conformations of the same isomer, in particular, the difference between planar and nonplanar conformations. In a sense, this is the same problem: the difference in energy 20 between two points on the potential energy surface. Often this involves compacing a transition state to a minimum structure; therefore, we report these differences uncorrected for ZP vibrations.

The relative stabilities will not necessarily be accurately determined with a particular level of calculation. In Chapters 2 and 3, three methods were used to estirnate the relative stabilities. In each case, the energies were obtained at the equilibrium geometry of that method. Other workers determine relative stabilities using a more accurate calculated energy at a previously deterrnined geometry (either experimental or calculated with a lower level of theory). The MP2 calculated stabilities are at a reasonable level, but of course could be made more accurate in the future. Kranz and Clark (1992) noted the trends did not change much when using singles and doubles quadratic configuration-interaction (QCISD) versus MP2, for 3 azaborinines.

The absolute ZPE's are larger than the relative stabilities in many cases-e.g., for molecuIes of Chapter 3 they range between 25 and 60 kcal/mol. However, the variance in ZPE arnong a group of isomers is srnall (on the order of 2 kcallmol).

1.3.3.3 Orbital energies. The rnolecular orbital (MO) energies obtained from a Hartree-Fock calculation can be used to approximate ionization potentials. The highest occupied MO-lowest unoccupied MO (HOMO-LUMO) gap should be smaller for more polarizable molecules and larger for less polarizable ones. The relationship holds only approximately, with many exceptions.

The hardness, q, and chexnical potential, X, can be approximated using the HOMO and LUMO energies (Pearson 1993), as: e~~~~ - E~~~~ &LUMO + e~~~~ "l= ; x= 2 2

The LUMO energies are al1 very close to O, whereas the HOMO energies Vary over a wider range. Thus, the variance of the HOMO energy with the static pohizability, a,was almost the same as that of q with a,or of x with a.

1.3.4 Frequencies. Harmonic vibrational frequencies are calculated in order to verify that a geometry is a true minimum and to obtain the ZPE. Also, the frequencies cm prove helpful for identifying a molecule or its derivatives in the lab. The frequencies of the 87 azaboracycles are presented in Chapters 2 and 3.

A non-linear molecule with N atoms will have 3N - 6 vibrations (plus 3 rotations and 3 translations). The vibrations are harmonic to a first approximation (Herzberg 1945).

The intensity of IR vibration is largest for those vibrations that most affect the dipole moment. The intensity of Raman vibration shows how much the polarizability will change if the atoms are moved according to that particuIar vibration.

The most important differences between experimental and calculated frequencies are that the vibrations probed by experiment are anhamonic and at a finite temperature, whereas the calculated frequencies are harmonic and for a rnolecule at O K. An example of anharmonicity is that cornpressing a bond is more difficult than stretching it. Vibrations can be pictured with software such as Molwin (1993-94). Some are easy to describe and to assign to known categories, e.g., XH bond stretches, whereas others are heavily mixed.

Much work has been done to check how well a particular method and basis set will predict vibrational frequencies. Early reports on the methods used in this thesis, 22

HW6-31G(d) and MP2/6-31G(d), are given in Hehre et al. (1986). More recent results,

for more than 1000 frequencies in 100 molecules containing second- and third-period

atoms are summarized by Scott and Radom (1996) (cf. Chapters 2 and 3).

Calculating geometries depends on accurately finding a stationary point, whereas

cakulating frequencies depends also on describing the potential energy surface in the

neighborhood of the stationary point (cf. Urban et al. 1987), a more difficult task.

1.3.6 Dipole moments. The dipole moment is one possible measure of charge

distribution in a neutral rnolecule.

Dipole moments can be detennined from measurements of the dielectric constant

of a medium, E, using the relation from Debye (1929):

with p the dipole moment, a the static polarizability, V,,, the molar volume, NA

Avogadro's number, k the Boltzmann constant, and T the absolute temperature. The term containing p2 is called the orientation polarization, refemng to the distortion of the dipole moment orientation in the electric field (Bogaard and Orr 1975). Note that the sign of the dipole moment cannot be found with this method.

Proper care must be taken to allow for the other types of polarization in eq 3.

Scaife and Laubengayer (1966) found that some experimentalists had ignored atomic polarization (vibrational polarizability) contributions for certain symmetrically tri- substituted borazines. Inciuding such terms reduced otherwise anomolously non-zero dipole moments to zero. Presumably this applies also for the dipole moment of borazine 23

itself, which at times has been assigned a value greater than 0.5 D (Watanabe and Kubo

1960) (cf. Chapter 2).

A dipole moment cm be calculated from the expectation value of r, the position

vector; or as the derivative of the energy of a molecule perturbed by a weak field,

deterrnined analytically or by the finite-field method (cf. Section 1.4.4.3, below). We

applied the last method to obtain HF and MP2 dipole moments, using a double-zeta basis

set with two sets of polarization functions (cf. Section 1.4.4.5, below). For the molecules

of Chapter 5, the MP2 dipole moments were usually higher than those from microwave

experiment and agreed within 7% on average (El-Bakali Kassimi et al. 1995, 1996;

Doerksen and Thakkar 1996; El-Bakali Kassimi and Lin 1998). No experimental dipofe

moments are available for the unsubstituted azaboracycles (other than borazine).

The dipole moment of RBNR species cm give insight into the polarity of the B-N

bond. Paetzold (1987) suggests that such dipole moments are small like that of CO (about

0.1 D), but non-zero. This makes iminoboranes and CO more reactive than alkynes and

N2, respectively. Paetzold (1987) noted that "a difference between CO and RBNR is the direction of the dipole; in CO, the more electronegative O bears the small positive charge... [C-=O+]," versus WN-R.

1.4 Polarizabilities

Polarizability, a measure of what change occurs in a molecule placed in a weak electric field, is difficult to determine either by theory or experiment. In this section, 1 24

will define polarizability, suggest some practical uses, discuss experimental and theoretical

rneans to obtain polarizability data, and introduce additive models of polarizability.

There is a vast literature on the polarizability of atoms and of molecules, which

is being added to regularly. Some recent compilations are available of experimental

(Miller 1994) and caiculated (Hasanein 1993) polarizabilities. In the following, 1 give

examples and references primarily for heteroaromatic molecules, for which relatively little

work is available.

1.4.1 Definition. The energy E of a neutral molecule in a weak, homogeneous,

electric field F can be written:

where Eo, p, and a are the energy, dipole moment vector, and dipole polarizability tensor,

respectiveiy, of the free moIecule. The summatisns are for the Cartesian coordinates, x,

y, and z. Higher terrns in the expansion would involve the hyperpolarizabilities, while a

non-linear field would require inclusion of terrns in the quadrupole moment and

quadrupole (hyper)polarizabilities, and so on (Buckingham 1967).

The polarizability can dso be obtained from the dipole induced by such an

extemal field. The difference between field-dependent and permanent dipole, Ap, can be expanded as:

Api = a,Fi + (5) i where i and j again can be x, y, or z (Buckingham 1967). In either eq 4 or eq 5, the polarizability cm be found by taking the appropriate derivative with respect to the field. 25

If a molecule is asymmetric, then a field applied dong one Cartesian direction

could induce a non-zero dipole moment component in any of three directions. Thus in

general the polarizability is a syrnmetric 3 x 3 tensor. The eigenvalues of that tensor, a,

I a, 5 a,, are the principal polarizabilities; they can be combined to forrn the average

polarizability (here called a) and various anisotropies (listed in detail in Chapter 2).

Buckingham (1967) has a convenient table showing how many components of a, will be

unique and non-zero for each moIecular syrnmetry group.

1.4.2 Practical uses of polarizability data. Many physical phenornena depend

on polarizability. Miller (1994) lists appropriate formulae. The dielectric constant, index

of refraction, and Rayleigh scattering cross-section of a substance can be expressed as

functions of the frequency-dependent polarizability. Ion mobility in a gas, the Langevin

capture cross-section, and van der Waals interactions are also polarizabiiity-dependent

(Miller 1994; Miller and Bederson 1977).

The equations governing a molecule's interaction with an electric or magnetic field, or gradient, involve combinations of the anisotropic polarizability, the quadrupole moment, and magnetic quantities such as the magnetizability anisotropy (Bogaard and Orr

1975). Thus if the poIarizability is known, it can be used to help obtain the other quantities (cf., e.g., Dennis and Ritchie 1993).

Recent work on force-fields for proteins has tried various ways to include polarizabilities in order to mode1 charge transfer (Jorgensen and McDonald 1998). The accurate description of solvent effects requires correct treatment of polarization (Gao

1997). 26

1.4.3 Experimental polarizabilities. That polarizabilities play an important role

in a variety of experiments suggests there should be several ways to determine them.

Lorentz (1880) and Lorenz (1880) found the following relationship between the

average polarizability, a,and the index of refraction, n (equal to the ratio of the speed

of Iight in a vacuum to the speed of light in that medium):

where rois the permittivity of free space, V, is the molar volume, and NA is Avogadro's number; eq 6 also defines the molar refraction, R, (Bogaard and OIT 1975). Both a and n depend on the frequency, a,of light shone on the substance, and so must be carefully extrapolated to obtain static values.

Refractivities could be measured more than 100 years ago (cf. the account by Le

Fèvre 1965). Many more-accurate measurements of the rnolar refraction were obtained in the 1930's and after (Andrews et al. 1966; Stuart 1967; Le Fèvre 1965). These often were performed for liquids, which should give different refraction than the corresponding gases.

The Debye relation (eq 3, Section 1.3.5, above) cm be used to obtain the static polarizability from the dielectric constant. For non-polar molecules, p = O, so that the second terrn on the right of eq 3 is zero; for polar molecules, measurements at several temperatures will yield the polarizability and the dipole moment (Bogaard and Orr 1975).

When a bearn of light hits a substance, refraction occurs dong the path of incoming light. But light is also scattered perpendicular to the incoming beam-Rayleigh (1899) scattering at the incorning frequency and Raman scattering at other frequencies due to perturbation by molecular transitions. Stnitt (1918) found that the scattered light is depolarized by anisotropic molecules. The depolarization ratio, p (Ull),is:

3 (&al2 P = (7) 4(4a12 + 45 a2 in which the molecular anisotropic polarizability in the principal axis system is given by:

Applying an electric field to a molecule causes a shift in its rotational spectra, the

Stark effect (Buckingham 1972). With strong fields-in order to reach high values of the rotational angular momentum-the anisotropic polarizability can be deterrnined (Bogaard and Orr 1975).

Many physical effects involve not only the polarizability but also the hyperpolarizability components. An example is the electro-optic Kerr effect (Bogaard and

Orr 1975), discovered in 1875 (Kerr). A sample is placed in a static electric field. Then

Iinearly polarized light is shone through, suppose in the z-direction. The refractive index will change depending on whether the axis of polarization is parallel or perpendicular to the electric field direction. The difference (11 - I),calIed birefringence, leads to an equation for the first Kerr virial coefficient, A,: which includes temperature dependence; the average and components of the frequency-

dependent, a(@, and static, a, polarizabilities; the dipole moment, p; plus particular

components of the first, p, and second, y, hyperpolarizability. Depending on the particuiar

rnolecule's symmetry, using eq 9, various types of experimental data can be combined to

obtain the polarizabilities or other desired properties (Bogaard and Orr 1975). The

magneto-optical Cotton-Mouton effect (Buckingham and Pople 1956; Battaglia and

Ritchie 1977a) is a corresponding magnetic birefringence which also can be used in

conjunction with other experiments to yield polarizability data.

1.4.4 Theoretical polarizabilities. Early methods for calculating polarizabilities

and results are surnrnarized by Miller and Bederson (1977). In the last 10 years, the

number of published calculations of molecular polarizabilities has increased rapidly. Every

month, many new papers appear. In this section, 1 present several ways to calculate

polarizabilities, focussing on those used in this thesis.

1.4.4.1 Uncoupled Hartree-Fock. The uncoupled Hartree-Fock (UCHF) polarizability is calculated using the molecular orbitals (MO) of the unperturbed molecule

(Dalgarno 1962). It is insightful to divide the polarizability into parts: direct for the change in electron density produced by non-interacting electrons; and induced for the changes caused by the interacting electrons (Grant and Pickup 1990, 1992). Cornpared to the unperturbed molecule, in the presence of a weak extenid field the MO'S would adjust. 29

The adjustment of each MO would affect how the other MO's adjust. UCHF neglects the induced polarization: it does not allow the MO'S to adjust in the presence of the applied field.

The sum-over-states formula for the UCHF polarizability is:

j E occ a E vir where the Greek subscripts each can be any of the Cartesian coordinates, x, y, or z; r,, or r,, are components of the position vector; and the unperturbed MO'S, @, and their energies,

E, are labelled j for occupied and a for virtual.

The inner sum in eq 10 can be performed first; the intermediate values for each j can be collected as individual MO polarizabilities. Our program (Thakkar and Doerksen,

UCHF), lists these. For molecules with n-MO'S, such as those of this thesis, the UCHF program returns the n-poIarizability. The UCHF polarizabilities show that the core MO's are not very polarizable, in contrast to the valence orbitais. The denominator of eq 10 will be the smallest for HOMO to LUMO excitations, so those can contribute the most to the

UCHF polarizability .

The UCHF method gives a qualitative picture of the polarizability and requires just an additional =1% of computation time after the HF wavefunction has been obtained. As with other ab initio methods, results are basis-set dependent.

1.4.4.2 Coupled Hartree-Fock. Hartree-Fock (HF) polarizabilities can be calculated using the coupled HY (CHF) method. CHF uses an iterative procedure to solve 30

coupled Fock equations containing the perturbed MO'S (cf. a review by Pickup [1992]).

It takes typically 2-3 times longer than an ordinary HF energy calculation.

1.4.4.3 Finite-field. In the finite-field (FF) method (Cohen and Roothaan 1965),

derivatives of the energy are calculated approximately using finite differences. For

example, if 2 fields, +Fx and -Fx, are separately applied, where the magnitude Fx is

suitably chosen, then odd-order terrns in the field in eq 4 will cancel, yielding (to third

order):

2 E(0) - E(Fx) - E(- F,) axx =

If the dipole moment component in the direction, Say x, of the applied field is zero by

symmetry, then fields of only one sign need be applied, since then E(+F,) = E(-F,). For

a particular polarizability component, several different fields can be applied, one by one; then al1 the data is least-squares fitted to a polynomial in F.

Two options in the FF method are the number of fields, Np to apply and the number of terms, Np, of eq 4 (or of eq 5) to include. If NI > Np then the extra data help to compensate for the uncertainty in the energies (El-Bakali Kassimi and Thakkar 1994).

We use a least-squares fitting program (Thakkar et al., POL20I) to calculate a particular polarizability component several times, using data from al1 fields but with increasing Np, and check that the results are converged. If not, then data from more fields are added.

It is important to choose fields carefully to avoid two extremes: If fields are too weak then it will not be possible to attain a with enough significant figures, because then

E(F) - E(0) will not be sufficiently different from zero. If they are too strong, then the 31

influence of higher (hyperpolarizability) terms in the expansion of eq 4 will introduce

error into a. The overall FF computational time is proportional to the number of fields.

The beauty of the FF method is that the only ab initio calculations which need to

be perforrned are to determine the energy in the presence of a weak static electric field.

This requires only trivial modifications to the standard ab initio programs used to obtain

the zero-field energy. No special code to obtain derivatives of the energy is needed for

the FF method. Thus it has been possible to apply the method to many correlated

approaches. The FF method is not used for a(o).

For Hartree-Fock, we calculated FF polarizabilities from the energy (eq 4) and

frorn the dipole moment (eq S), and checked that the results agreed. This only requires

a set of tightly converged HF calculations in the presence of a field. The computation

time for HF/FF is longer than for CHF. However, we used the FF method for MP2

polarizabilities, and thus obtained the field-dependent HF energies as well without

additional calculation.

There are two practical considerations that make applying FF in the MEmethod

different from in HF. Conventional computer codes require convergence criteria to be set

for how accurately the HF energy will be detennined. The MP2 energy has no such criterion, because it is not calculated iteratively. However, if the HF energy is accurate

to, Say, 8 decimal pIaces, then the MP2 correction to the energy will be less accurate than that. So one must keep in mind what accuracy will be required for the MP2 energy when

one is setting the accuracy on the HF/SCF procedure. 32

Secondly, the MP2 correction to the polarizability is usually a small fraction of the total MP2 polarizability (which is HF + MP2). Thus stronger finite fields must be applied for MP2 than for HF in order to obtain the MP2 correction with sufficient accuracy.

1.4.4.4 Other. There are a few important alternative methods. Semiempirical methods have been used for calculating polarizabilities (Hasanein 1993). They are more approximate than HF. Certain such methods involve scaling to experimental polarizabilities. There is still some merit in using the methods for very large molecules; but there is no assurance that they will give a reasonable answer. An example is calculations on azines which gave average polarizabilities close to MP2 for pyrrole, but were in error by 24% for four other azoles (Waite and Papadopoulos 1990).

Density functional theory in various forrns has also been used for calculating polarizabilities (Sim et al. 1992; Matsuzawa and Dixon 1994). In the rare cases when careful cornparisons were made, MP2 gave as good or better results.

At the other extreme are methods that include a better treatment of electron correlation than that of MP2 perturbation theory. In the past five years, the improved efficiency of computers and the development of computer codes has made such methods more readiIy accessible.

The obvious route to improvement is to move to higher levels of perturbation theory, MP2, MP3, MP4. SDQMP4 (including single, double, and quadruple substitutions) was found to reduce by about 2% the polarizability of certain azines, azoles, and thiazoles

(compared to MP2) (Archibong and Thakkar 1994; El-Bakali Kassirni et al. 1995; El- 33

Bakali Kassimi and Lin 1998). However, effects on azaboracycles are hard to predict.

Coupled cluster (CC) methods (Bartlett 1995) could be the next logical progression after

MP2. A recent example of CC applied to furan (Christiansen et al. 1997) is discussed in

the conclusion. But for more accurately correlated methods, it is usually best also to use

a larger basis set. So, in all, that would require a much more time-consuming calculation.

Other promising methods are polarization propagator methods. Pickup (1992) gave

a thorough theoretical introduction to propagator and other techniques for computing

molecular properties.

1.4.4.5 Basis sets, and selection of ours. The selection of a suitable bais

set is of fundamental importance for accurate ab initio calculations. A basis set that is good for calculating one kind of property is not necessarily good for some other kind

(Davidson and Feller 1986). Computational time increases rapidly with basis set size. For exarnple, the MP2 calculation time is proportional to p,for N basis functions. Thus, N must be kept as Iow as possible for a given calculation.

The bais set should be particularly selected to give an accurate description of the polarizability. The core electrons are relatively unpolarizable; the valence electrons are the ones which must be described accurately (Helgaker and Taylor 1995). A double-zeta substrate with one set of polarization functions is a minimum starting point. Extensive testing has shown that diffuse functions of al1 symmetry and polarization functions improve the calculated polarizabilities. Much larger basis sets, including several functions of higher angular momentum, have been found to improve calculated polarizabilities for smaller molecules, such as HCl (Maroulis 1998). In special cases, a smaller basis rnight 34

provide a reasonably accurate polarizability, such as for the component dong the axis of

a linear chain molecule (Perpete et al. 1997).

For the heterocycles of Chapter 5, the trends were clear: a second set of

polarization functions improves the MP2 average polarizabilities by about 6%, but a third

set of d's is relatively unimportant (El-Bakali Kassimi et al. 1995; El-Bakali Kassirni and

Lin 1998). Sirnilarly, the effect of a set of f-functions was found to be small (El-Bakali

Kassimi and Lin 1998). Other further increases of bais set are not expected to have a

large effect for molecules of that type.

In this work, the same size of basis set was used for each second-period atom. The

[Ss3p2d/3s2p] basis, called "C" (El-Bakali Kassimi et al. 1995, 1996) has 24 functions

for each atom from B to 0, and 9 for each H. (It is detailed in Appendix 2.) Thus, e.g.,

oxatetrazole requires just 120 functions, whereas borazine needs 198, so a single borazine

MP2 calculation will be about 12 times longer (about 6 hours on a modern workstation).

The exponents of the Gaussian functions Vary fairly regularIy, with the functions on B

most diffuse and on O tightest.

Other basis sets give variable results compared to the C bais set. CHF

polarizabilities with the larger 6-3 1G+(3d,3p) basis set, not designed for polarizabilities,

were 3% to 5% lower than CHFK (Hinchliffe and Sosciin 1993, 1994). The Spackman

(1989) basis has only one set of d functions; with it, polarizabilities of azines were too

low (Archibong and Thakkar 1994). The Sadlej (1988) basis is quite similar to C, and has yielded average polarizabilities within 1% where comparison has been made (cf. discussion in Chapter 2 and Appendix 1). 35

1.4.5 Theory vs experiment. Experimentai or caiculated polarizabilities cm include various factors to be noted: frequency dependence, solvent effects, intermolecular interactions, vibrational effects. The polarizabilities calculated in this thesis are al1 for zero frequency, non-vibrating, isolated molecules.

Medium effects are important if an experiment is performed on a dilute solution or on a sample dissolved in a solvent. The intermolecular interactions will change the average polarizability relatively less than the anisotropy. The anisotropic polarizability can be much lower in a solvent. Also, it varies greatly with the solvent used (e.g., Battaglia and Ritchie 1977a).

Frequency-dependent polarizabilities can be adjusted to obtain static polarizabilities. This is difficult, so sometimes extra assumptions are made such as that the static and dynamic polarizabilities are equal (e.g., Dennis and Ritchie 1993) or that components of the polarizability follow a pattern found for a similar molecule (e.g.,

Battaglia and Ritchie 1977b; Calderbank et al. 198 1).

Polarizabilities can be calculated systernatically for many molecules, even those that are not isolable. But calculations are always approximate and finite. Experimental polarizabilities require correct conditions which are often difficult to prepare. Each molecule must be treated differently; some must be studied in solution, are sensitive to air, or are available in the gas-phase over only a narrow range. Carefully interpreted experimental polarizabilities in a few select cases can provide a guideline for testing the accuracy of calculated polarizabilities. 36

1A.6 Available polarizabilities of monocycles. Miller (1994) lists many

molecules for which the experimental polarizability is available. Included are the

following monocycles: benzene and substituted benzenes, such as nitro-, methyl-, ethyl-,

or halo-, and the heteroaromatic pyridine, two diazines, and thiophene.

Much of that experimental polarizability data is from liquid moiar refractivity

measurements, which retum only the average polarizability, but which can be combined

with additive subunit polarizabilities (cf. Section 1.4.7, below) to generate approximate

anisotropic information.

Ritchie and CO-workershave measured polarizability anisotropies in solution for

benzene and 5 azines (Battaglia and Ritchie 1977a, 1977b; Blanch et al. 1991), furan and

thiophene (Dennis et al. 1983), pyrrole, pyrazole, and imidazole (Calderbank et al. 198 l),

and borazine (Dennis and Ritchie 1993). Recently, they have obtained gas-phase results

for bemene (Gentle and Ritchie 1989; cf., also, Alms et al. 1975), furan and thiophene

(Coonan et al. 1992) (plus for some unrelated molecules). Heitz et al. (1991) obtained the

anisotropic polarizability of s-tetrazine through Stark shifts. These are the only molecules from among the 129 of this thesis for which gas-phase polarizabilities have been measured.

Systematic calculated polarizabilities for the molecules of Chapter 5 have been reported elsewhere (El-Bakali Kassirni et al. 1995, 1996; Doerksen and Thakkar f 996;

El-Bakali Kassirni and Lin 1998), and any prior calculations are summarized therein. 1 will discuss more recent calculations in the conclusion. 37

Only for borazine, of the 87 B-containing molecules in this thesis, has there been

any experimental or theoretical polarizability data reported with which to compare. Hough

et al. (1955) detemined the average polarizability from liquid molar refractisn data and

Dennis and Ritchie (1993) obtained the magnitude of the polarizability anisotropy from

the molar Kerr constant of a dilute solution of borazine in cyclohexane. CHF

polarizabiIities have been published by Fowler and Steiner (1997), with a [8s6p2d/6s2pJ

basis and by Lazzeretti et al. (1991) using the Sadlej (1988) basis; Archibong and

Thakkar (1994) obtained Spackrnan (1989) basis MP2 average and anisotropic polarizabilities (cf. Chapter 2).

1.4.7 Additive models of polarizability. Additive models are attractive as supplements to accurate experimental or calculated polarizabilities. This section describes our procedure for developing additive polarizability models, gives a short history of the approaches that others have taken, and compares the various methods. Note that the original additive models developed in Dr. Thakkar's research group were published earlier

(El-Bakali Kassimi et al. 1995, 1996). More recent work appears in this thesis in Chapters

2 and 4 and Appendix 1.

1.4.7.1 Motivation for making additive models. Additive polarizability models can have various uses: to help explain why polarizabilities vary as they do from molecule to molecule; or, to find to what extent the polarizability of a moiecule is an additive function of constituent elements. Also, the subunit polarizabilities which are obtained can be used to predict the polarizabilities of other molecules. 38

Often, workers choose to use model polarizabilities because the whole-molecule polarizabilities are unavailable. Atomic polarizabilities have been used to model the dielectric properties of a protein (Simonson et al. 1991). Cao and Li (1998) developed a rnolecular polarizability effect index to model solubility of aliphatic molecules based on group polarizabilities. One use of bond polarizabilities is to help in the prediction of the components of the polarizability tensor for molecules for which only partial experimental poIarizability data are available (Aroney et al. 1974).

1.4.7.2 Explanation of Our procedure. Our models were designed to test the regularity of such patterns as the regular decrease in polarizability with aza-substitution found in the azines. The usual additive models might be less effective for heteroaromatic molecules-because of conjugation-than for, Say, hydrocarbons. Refined models could be used to describe those non-additive effects.

Our models al1 begin with accurately calculated polarizabilities. For a set of molecules, these can form the array a(ij),where i signifies a particular molecule and j refers to the polarizability component. An M-parameter model (for M an integer), is constructed in the forrn:

The subscript O designates a particular value for i and for j. The rk's are molecular parameters. The coefficients ck, which are the same for al1 molecules in the data set, are obtained by least-squares fitting to the calculated MP2 polarizabilities. For each chosen set of T's, even for a new M, a completely new set of c's is deterrnined. 39

The T's fall into four categories: number of atoms of a particular type; number of connections (bonds without regard to bond order) of a particular type; molecular length or area; or a property related to the HOMO-LUMO gap. The former two categories depend only on the stoichiometry or connectivity of a molecule. The latter two are obtained from the MP2/6-3 1G(d) geometries and HFK HOMO and LUMO orbital energies, respectively.

This type of model is not primarily predictive; rather, its purpose is to explain trends in the accurate polarizabilities. The models could, however, be used to predict the polarizabilities of other similar molecules.

1.4.7.3 Other additive models of polarizability. The history of additive models of polarizability begins in the last century. At that time, the rnodels were for molecular refractions which are proportional to polarizabilities. Le Fèvre (1965) points out that Brühl (1880) was the first to analyze rnolecular refractions as sums of atom refractions. The "atom" units also included groups such as CH2. The goal was to explain the bulk property (refraction) as a function of molecular composition and structure. BrÜhI noted that this type of model worked better if atoms could have different refractivities depending on their environment. For exarnple, the carbonyl O refractivity was 50% more than that of the hydroxyl O.

The atorn refractions were used to help determine the structure of unknowns, e.g., to distinguish between geometric isomers. Non-additive effects were classified and analyzed (Le Fèvre 1965). 40

Later, the large body of refraction data was used to develop sets of bond

refractions (Denbigh and Vickery 1949; Vogel et al. 1952), which could predict whole-

molecule refractions within a few percent.

Refraction data were limited to average effects. Additive polarizability models, by

contrast, could include anisotropic data from Kerr and other experiments. Le Fèvre (1965) traces the development of modeIs that predicted the principal polarizabilities or anisotropic bond polarizabilities-which split bond polarizabilities into longitudinal, transverse, and vertical components.

More recently, a variety of additive polarizability models have been developed through fitting to experimental polarizabilities. Models by Applequist (Applequist et al.

1972; Shanker and Applequist 1996), Miller (Miller 1990a, 1990b; Miller and Savchik

1979), and Bulgarevich and CO-workers(Movshovich et al. 1989) are some of the most well-known. Other examples are those based on Ünsold polarizabilities (Mulder et al.

1979), or on uniforrnly scaled semiempirical polarizabilities (Raabe et al. 1997; Metzger

198 1). Stout and Dykstra (1995, 1998) have developed anisotropic additive atom models fitted to ab initio polarizabilities.

1.4.7.4 Comparison with other rnodels. A few of the previous models have been applied to heteroaromatic molecules. Mulder et al. (1979), studying azines, found it worthwhile to split bond polarizabilities into G and n parts, assurning only a single

Kekulé structure. They tested the predictive ability of their bond polarizabilities on bicyclics. Nowever, the polarizabilities to which they fitted their modeIs were not very 41

accurate (cf. Appendix 1). Our models could be fitted to UCHF cr- and K-polarizabilities,

though the UCHF polarizabilities are aIso rather inaccurate compared to MPS.

Movshovich et al. (1989) used anisotropic bond polarizabilities obtained from

experimental frequency-dependent polarizabilities of a few molecules to model the

average and anisotropic polarizabilities of 26 other moIecules, including 4 azines and

azoles. The agreement with experirnent was fairly good for the average polarizability and

poor for the anisotropy.

Miller developed two models. One, for the average polarizability, was based on

additive hybrid contributions (Miller 1990a; Miller and Savchik 1979); for 400 molecules,

the average errors were 2-3%, with a standard deviation of 6%. Another, for polarizability components, positioned point dipoles at each atomic site in a molecule (Miller 1WOb).

The latter was an extension to Applequist's atom dipole interaction mode1 (Applequist et al. 1972). Errors averaged only 6% for the individual components. Miller (1990b) supposed that the average errors in the experimental polarizability data he used are about

3% for the average polarizability and 6% for the components.

Stout and Dykstra modelled the polarizabilities of 35 small molecules containing

H, C, N, 0, and F, fitting atomic polarizability components (on atomic centers) to accurate CHF (1995) and MP2 (1998) data at assumed geometries. They allowed different parameters for different atornic hybridizations. The average MP2 polarizabilities could be reproduced with an error of ~2%for a 12-parameter mode1. The model for components resulted in larger errors. 42

Our models used the simplifying assumption that al1 bonds X-Y depend only on

the atom-type, X and Y, not on bond order. Considering that al1 the molecules are

heteroaromatic monocycles, the hybridization of atoms in different locations should not

Vary significantly. Separate fits were made to the average and to the anisotropic

polarizabilities; the latter are more difficult to mode1 accurately.

1.5 Aromaticity

In this section, 1 will introduce the concept of aromaticity, and describe the

attempts which have been made to quantify it. The topic of aromaticity is central to

organic and heterocyclic chemistry. Aromatic compounds are those for which substitution

reactions are favored over addition (Erlenmeyer 1866). The idea is that aromatic

molecuies will have an extra stability compared to nonaromatic and antiaromatic ones.

But attempts to define aromaticity precisely and to quantify it have met with great difficulty. Here 1 will discuss the definition and two centrai criteria of aromaticity: energetic and magnetic. Structural and polarizability criteria are the subject of Chapter 5.

Quantifying a concept often means trying to understand it better. Standard molecules are readily classified as aromatic or not. The difficulty is with finding an index which holds for al1 types of molecules (Pozharskii 1985), for example, three-dimensional aromatic molecules such as Cs,or closo-B,,H,," (Schleyer and Jiao 1996). Also, more fundamental questions such as the effect of aza-substitution on aromaticity have not been completely resolved. 43

Several books have dealt with aromaticity, but they are not totally satisfactory.

Garratt (1986) is superficid, particularly on quantifying aromaticity. Minkin et al. (1994) have extensively reviewed quantitative indexes and the many different types of aromaticity. However, the book does not satisfactody resolve important issues and contains some errors.

1.5.1 Definition. Minkin et al. (1994) define aromaticity as "cyclic electron

(bond) delocalization" effects. They especially want to avoid the idea that conjugation is uniquely associated with aromaticity (since noncyclic molecules can also be conjugated).

Pozharskii (1985) notes three fundamental aromatic 6.n-electron structures, containing 36,42, and 48 electrons, respectively, and the corresponding hetero-compounds that can be derived from them, depicted in Figure 1.1. In the structures, X (pyrrolic) must contribute two n-electrons, Y (pyridinic) one n-electron, and Z (borepinic) just an empty p-orbital. Typical examples are X = {NH, O, PH, S); Y = {N, P, O+);and Z = {BH,

AIH, CH').

Schleyer and Jiao (1996) suggest that magnetic properties are the only ones uniquely tied to aromaticity, though in most cases other criteria can also be applied to distinguish aromatic and antiaromatic species.

1.5.2 Brief introduction to criteria not included in this work. Here my focus is on quantitative criteria. Note, though, that even the qualitative ranking of molecules is stiI1 unclear in many cases.

1.5.2.1 Energetic. The energetic criterion of aromaticity is based on the idea that aromatic molecules have added energetic stability (resonance energy) compared to some Figure 1.1 Fundamental aromatic 6n-electron structures (after Pozharskii [1985])

reference molecule. Many such indexes have been proposed, often differing primarily in the choice of reference, such as a conjugated polyene, Kekulé structure, saturated ring, or combinations (Glukhovtsev 1997).

Schaad and Hess (1982) found that important energetic measures developed in the

1970's were fundamentally similar, and suggested that those were the first energetic criteria to be qualitatively successful. They noted that resonance energies must be used per electron, or n-electron (REPE), or carbon-since otherwise they would just increase with molecular size. REPE is obtained from the difference between the n-energy and the sum of bond energies. Other schemes are Hückel, Dewar, and topological resonance 45

energies. The various proposed resonance energy definitions in general give the same

results (Minkin et al. 1994).

Resonance energies have been defined based on the enthalpy of isodesmic or

homodesmotic equations (Minkin et al. 1994). The choice of reactants and products

affects the resulting energy.

One difficulty with the energetic criterion is that other factors such as strain

energy may force some aromatic molecules to be nonplanar (Schleyer and Jiao 1996).

Thus, the most stable isomers rnight not be the most aromatic (Subrarnanian et al. 1996).

1.5.2.2 Magnetic. Special magnetic properties have long been associated with

aromaticity. A recent brief review is available (Schleyer and Jiao 1996). Applying a

magnetic field to delocaiizable electrons creates a ring current (Pauling 1936), which in

turn affects chemicai shifts. The enhanced effect found in aromatic molecules results in

a diamagnetic susceptibility exaltation (A) (Dauben et al. 1971). As with resonance

energies, the exaltation must be defined with respect to a reference structure. Thus

different reference structures can give different A. Magnetic effects can be measured or calculated. Again, results from experiments or theory often differ.

The main problem with A as an aromaticity index is that it increases with ring size

(Schleyer et al. 1996). Benson and Flygare (1970) used the in-plane minus out-of-plane anisotropic rnagnetic susceptibility as an aromaticity index. An alternative rneasure is the proton chernical shift (Pople 1956). Protons outside a ring show a slight downfield shift in aromatic rings compared to corresponding nonaromatic rings. For species having intemal protons, the difference in shift is much more noticeable. However, few species 46

have protons inside the ring (Schleyer and Jiao 1996). Thus, Schleyer et al. (1996)

proposed the nucleus-independent chernical shift (NICS), which is measured at ring

center. Fowler and Steiner (1997) use maps of the calculated ring current density to assess

a molecule's aromaticity.

1.5.3 Summary of work of others on quantifying aromaticity. The main

research of the last decade has been by the following, with their CO-workers:Katritzky

et al. (1996), Jug (Behrens et al. 1994), Schleyer (Zywietz et al. 1998), Bird (1998), and

Krygowski et al. (1998) (where in each case 1 have cited their most recent paper on the

topic; those contain references to previous work).

A fundarnentally important question is, "How does one relate the different criteria of aromaticity?" From the previous work, there is some validity to the idea that the different criteria are complementary. Katritzky et al. (1989) used principal component analysis to assess the predictions of energetic, structural, and magnetic criteria. They conciuded that aromaticity is two-dimensional-i.e., that the indexes describe somewhat different phenornena-with energetic and structural indexes related, but magnetic indexes separate. Jug and Koster (1991) came to a similar conclusion of two-dimensionality, but found energetic orthogonal to geornetric criteria, with magnetic ones in the middle. Bird

(1996) and Schleyer et al. (1995) disagreed, instead finding that the three types of criteria are positiveIy correlated. Krygowski et al. (1995) pointed out that choosing different indexes from the three categories can lead to contradictory results. Subramanian et al.

(1996) contend that energetic and other criteria predict aromaticity differently for certain fused rings because the former indexes are unreliable for those molecules. More careful 47

analysis of definitions and of implementations of indexes must occur before it can be said

that aromaticity has been successfully quantified.

1.6 Overview of Thesis

To conclude this introduction, here I reiterate the purpose of this research, list the

contents of the individual chapters, discuss my role in joint projects included in this

thesis, and give information regarding publication of thesis portions.

1.6.1 Overall purpose. This thesis is an attempt to deterrnine accurate

polarizabilities and other properties for a large set of molecules and to analyze the trends

in the resulting data, particularly in terrns of additive models of polarizability and scales

of aromaticity. The molecules are BN-containing and other heteroaromatic rings.

1.6.2 Summary of thesis. There are four chapters in the body of the thesis.

Chapters 2 to 4 tell the story of 87 five- and six-membered rings containing B and N. For each of 87 molecules, there was a standard procedure: calculate the planar geometry; verify that it is a minimum structure; if not, look for a nearby nonplanar geometry; calculate vibrational frequencies, dipole moment, and polarizability at the final geornetry.

For most of the properties, three levels of calculation are used and compared, MNDO,

HF, and MP2; UCHF, CHF, and MP2 for polarizabilities. Trends are analyzed: relative stabilities, and patterns in the other properties.

Chapter 2 describes the BN-containing heterocycles which have been studied the most, the 6H-azaborinines. Previous theoretical work helped motivate the choice of these mo1ecules for further study and helped make it worthwhile to consider them separately. 48

Chapter 3 is about the geometries, relative stabilities, harmonic vibrational

frequencies, and dipole moments of 70 other azaboracycles. No previous systematic study

of the 70 nor of any of the subsets of isomers has been published, despite the fact that

experimental data have been reported for 12 of the 70.

Chapter 4 focusses on îhe polarizabilities of the 70 BN molecules that were treated

in Chapter 3. The polarizabilities of the 70 were combined with previously available data

from the 17 azaborinines, as well as for the azines, moles, and oxazoles, to make additive

models of polarizability. The changes upon substitution of N for CH, O for NH, and BN

for CC are noted.

Chapter 5 attempts to quantify the aromaticity of the non-boron-containing

molecules. Previously calculated geometries and polarizabilities are analyzed to see what

they predict for the aromaticity of the molecules. Indexes based on bond order deviations,

and on anisotropic polarizabilities are implemented and compared.

The conclusion completes this thesis. Appendixes foIIow, containing related work on the geometry and polarizabilities of the azines, basis set specification, notes on scale drawings and on nomenclature, and supplementary material for Chapters 2 and 3.

1.6.3 My contribution to work with CO-authors.AII of my research has been done in cooperation with my supervisor, Professor A.J. Thakkar. He has arranged for some of my thesis work to be done in cooperation with others in his research group.

Chapter 5 has other CO-authors. I took a lead role in the work on quantifying aromaticity, including perforrning the calculation of UCHF polarizabilities for thiazoles, and wavefunctions for azines; calculating al1 aromaticity indexes from bond order and polarizability data; analyzing results; and preparing Chapter 5 in draft manuscript form.

The literature search was done together with VJS. The other work was done by the other

authors. VJS also assisted with some of the oxazaborole calculations for Chapter 3.

1.6.4 Publication details of chapters. Chapter 2 has been published

(Doerksen and Thakkar 1998a). Chapter 3 has been submitted for publication (Doerksen

and Thakkar 1998b). Appendix A has been published (Doerksen and Thakkar 1996). The

three are given here in their published (or submitted) form. The scientific content of

Chapters 4 and 5 will be submitted for publication in future, and so they are presented

here in draft manuscript format.

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Waite, J.; Papadopoulos, M. G. 1990 Dependence of the polarizability, a, and hyperpolarizabilities, P and y, of a series of nitrogen heterocyclics on their molecular structure. A comparative study. J. Phys. Chem. 94, 1755.

Watanabe, H.; Kubo, M. 1960 The dipole moments and structure of borazole and its derivatives. J. Am. Chem. Soc. 82, 2428.

Yale, H. L.; Bergeim, F. H.; Sowinski, F. A.; Bernstein, J.; Fried, J. 1962 New boron heterocycles. 5,6- and 7-membered systems containing nitrogen, oxygen and sulfur. J. Am. Chem. Soc. 84, 68.

Zywietz, T. K.; Jiao H. J.; Schleyer, P. v. R.; Demeijere, A. 1998 Aromaticity and antiaromaticity in oligocyclic annelated 5-membered ring-systems. J. Org. Chem. 63, 3417. Chapter 2. Azaborinines: Structures, Vibrational Frequencies, and ~olarizabilities*

Robert J. Doerksen and Ajit J. Thakkar

* Published in Journal of Physical ChemLFtry A 1998, 102, 4679. The Abstract and Acknowledgements have been removed for this thesis. 2.1 Introduction

Borazine has been studied extensively, both experimentally and theoretically, because it is isoelectronic and isostructural with benzene. Borazine, B3N3H6, cm be obtained conceptually from benzene by replacing each CC group by an isoelectronic BN group. Similarly, one can generate 16 other CC2,BnNnH6azaborinines from benzene by replacing n pairs of carbon atoms by n nitrogen and n boron atoms. Three monoazamonoborinines, 11 diazadiborinines, and three triazatriborinines are formed when n = 1, 2, and 3, respectively. Figure 2.1 shows al1 17 azaborinines.

A note on nomenclature is merited because the azaborinines are often given différent names. Sometimes the terrn borazarobenzenes is used to emphasize that these

17 isoelectronic molecules al1 have six z-electrons and so are potentially aromatic.

Borazine is the common name' for 1,3,5,2,4,6-triazatriborinine,but does not follow the

IUPAC convention regarding order of prefixes for heteroatom~.~The ending "borine" is used frequently,' but the correct IUPAC stem for "bora" is Strict adherence to the IUPAC convention2 requires the addition of a prefix specifying the location of the hydrogens bonded to atoms other than carbon as in 1,2,3,4-tetrahydro- 1,2,3,4- diazadiborinine. We use the more compact "azaborinine" form (eg. 1,2,3,4- diazadiborinine) since there is a hydrogen bonded to each ring atom in al1 the molecules studied here.

Of the 17 molecules we consider, only borazine (17) has been isolated in unsubstituted form.' There have been experiments to determine its geometry$5 its vibrational spectra: its dipole moment? and its polarizability in solution.89 Only indirect Figure 2.1 Azaborinines at their MP2/6-31G(d) optimized geornetry (except MPU6-3 lG(d,p)) (with Cssyrnmetry or as listed in parentheses): 1 ,~azaborinineil), 1,3- azaborinine (2), 14-azaborinine (3) (C,,), 1,2,3,4-diazadiborinine (4), 1,S,3,5- diazadiborinine (5), 1,2,3,6-diazadiborinine (6) (C2,), 1,2,4,5-diazadiborinine (7) (C,,), 1,3,2,4-diazadiborinine (8), 1,3,2,5-diazadiborinine (9) (C,,),1,4,2,3-diazadiborinine (10) (C,,),1,4,2,5diazadiborinine (11) (C2h), 1,4,2,6-diazadiborinine (12) (C,,), l,5,2,3- diazadiborinine (13), 1,5,2,4-diazadiborinine (14) (C2,),1,2,3,4,5,6-triazatriborinine (15) (two views are given of this nonplanar molecule, one perpendicular to the Cs plane, and the other parailel), 1,2,4,3,5,6-triazatriborinine (16), and borazine or 1,3,5,2,4,6- triazatriborinine (17) (qh).MP2K dipole moments are shown to scale, with the arrow head pointing to the negative end. The solid line is the I,-axis of inertia, and the dotted line is the 3-axis of polarizability (nonunique and therefore not shown for borazine). 65

experimental work has been done on other azaborinines. There are crystd structures of

6 and 8 with ring substit~ents,~~*~~and of 3 and 1 with fused ~ewar''detected

but could not isolate 1; however, many of its derivatives have been studied.' Molecules

with fused rings containing an azaborinine unit with the pattern of 10 have been

synthesized. l6

There have been relatively more theoretical calculations on the azaborinines but

most of them focus on borazine. The relative stabilities of ail 17 azaborinines were

studied at the extended Hückel level by ~offmann.l7 Semiempirical geometries have been

published for 15 a~aborinines,'~~~but electron correlated, ab initio geometries have been

reported for only se~en.~l-~~Calculations of other properties are available only for

borazine. Ab initio calculations have been reported for the vibrational frequencies2' and

the static dipole polarizability2k26 of borazine.

We report systematic, electron correlated, ab initio calculations of the equilibrium

geometries, vibrational frequencies, relative stabilities, dipole moments, and static dipole polarizabilities of al1 seventeen azaborinines. Planar conformations are found to be stable minima for 16 of the 17 azaborinines. Additive and other simple models are developed for the dipole polarizability of 49 planar rings including 16 azaborinines and 33 heteroaromatic molecules that we have studied previously.25*27-29

2.2 Computational Methods

Equilibrium geometries and harrnonic vibrational frequencies were computed for al1 17 azaborinines using three different methods. Prelirninary calculations were made 66 with the serniernpirical modified neglect of differential overlap (MNDO) rn~del.~'

Subsequently we calculated ab initio Hartree-Fock (HF) and second-order Mgller-Plesset

(MP2) perturbation the01-y~~"~geometries and frequencies using the 6-31G(d) and

6-31G(d,p) split-valence plus polarization basis sets.34 Al1 these calculations were made

with GAUSSIAN-~O.~~The harmonic frequencies were used to verify that al1 Our

optimized geometries are true minima.

Dipole moments and static polarizabilities were calculated by the finite-field

rneth~d,~~using HF and valence MP2 field-dependent energies. Finite-field HF

polarizabilities are equivalent to coupled Hartree-Fock (CHF) ones." The details of the

techniques used were the same as in Our previous ~ork.~~-~'Hence, only a concise

summary is given here. Since the 6-31G(d,p) basis set is inadequate for polarizabilities,

we used a [5s3p2d/3s2pJ basis set, denoted C in ref 27, of contracted Gaussian-type functions (GTF) for the finite-field calculations. It consists of a double zeta substrate3' augmented by diffuse s- and p-type GTF, a set of p- and d-type GTF optimized for polarizabi~ities~~and another set of p- and d-type GTF optimized for electron c~rrelation.~~Complete details of this bais set can be found in ref 27 except the exponent of the diffuse d-GTF on boron which was taken to be 0.15. We use uncoupled

Hartree-Fock (UCHF) polarizabilitie~~~*~~to estimate the relative contribution of the K- electrons. Full details are given in ref 27.

2.3 Equilibrium Geometries

2.3.1 Results. It is not obvious how many of the 17 azaborinines have planar 67 geometries. At the MNDO level, we found 12 azaborinines to be planar. Five molecules were more stable as nonplanar species: 16 and 5 by about 0.2 kcal/mol, 7 and 4 by 3-4 kcaVmol, and 15 by 23 kcal/mol, each compared to stationary, saddle points on the planar potential energy surface. At the HF/6-31G(d) level, 16 is a planar molecule and planar conformers of 7 and 5 are within 0.0001 kcal/mol of the stable nonplanar forms. Only 15

(22 kcal/mol) and 4 (0.6 kcal/mol) are significantly more stable in nonplanar conformations. Next, at the MP2/6-31G(d) level, 15 and 4 were the only remaining nonplanar species, by 13 and 0.0001 kcaymol, respectively. Finally, at the

MP2/6-3 lG(d,p) level, the planar form of 4 was a true minimum, but 15 was still more stable as a nonplanar Cs molecule by a substantial 12 kcaVmol. It is conceivable but unlikely that 15 would be found to be planar at an even higher level of theory. We adopted the MP2/6-31G(d) geometry for al1 molecules except 4, for which we chose the

MP2/6-31G(d,p) planar geometry. Thus we have planar geometries for 16 of the 17 azaborinines.

A few methodological observations are of interest. There are significant differences between our MNDO, HF, and MP2 geometries. Let us compare the methods using the average 6, and maximum 6, values of the absolute difference of bond lengths and angles from their MP2/6-3 1G(d) values for al1 planar conformations of the azaborinines. We find that MNDO bond lengths and angles respectively deviate from MP216-31G(d) values by an average of 6, = 1.8 pm and 6, = 1.3O, and a maximum of 6, = 7.6 pm and 6, = 5.8'.

The HF/6-3 lG(d) results are better than the MNDO ones; the average deviations of bond lengths and angles are reduced to 1.3 pm and 0.6", respectively. But the maximum 68 differences from MP2 are still large: the worst case HF bond length, MC6 in 5, is too long by 4.9 pm, and the worst bond angle, HBN in 15, is too large by 2.7".

For three molecules, 4, 5, and 15, we can compare Our MP216-31G(d) and

MP2/6-3 1G(d,p) geometries. The main difference is that al1 MP216-3 1G(d,p) XH bonds are predictably shorter by an average of 0.75 pm, 0.58 pm, and 0.54 pm for X = B, N, and C, respectively. This is similar to the results of Boese et al.5 who found that

MP2/6-31+G(d,p) BH and NH bond lengths in borazine (17) differ from their

MP2/6-31G(d) counterparts by 0.8 pm and 0.4 pm, respectively. Ramondo et aL2' found that at the SCF level, bond lengths and angles of borazine changed by no more than 0.3 pm and 0.2" upon enlargement of the 6-3 lG(d) basis set.

Figure 2.1 shows al1 the azaborinines to scale at our best geometries. Notice that

15 is in a boat conformation; we have drawn it from two perspectives, one showing a plane perpendicular to the Cs plane, and the other showing a plane parallel to the Cs plane. Tables A5.1 and A5.2 list Our best calculated geometries for dl the azaborinines except borazine. We do not list our geometry for borazine (17) because an identical one can be found in Ramondo et al.'s work.*' The MP2/6-31G(d) geometries for the azaborinines can be expected31J2 to be accurate to within 1%.

2.3.2 Cornparison with Previous Calculations. Our MNDO geornetries are in reasonable agreement with previous ~ork'~'~at the same level. However, unlike us,

Massey and ~oellne8~did not find a minimum ring geometry for 7; their optimizations al1 ended with a chah conformation. Their incorrect result could be due to their rather 69

inappropriate choice of starting guess geometries such as a regular hexagon with al1 bond

lengths equal to 80 pm.

Our W16-31G(d) geometry for borazine is in perfect agreement with Ramondo

et But our HFl6-31G(d) bond lengths have noticeable discrepancies of 0.3 pm or

greater with Kranz and ~lark's~~results for 1, 2, and 3. In particular, their N1C2 bond

lengthZ2for 3 is too long by 0.9 pm. This suggests that theiZ2 HF geometries were not

completely converged. Kar et al.'s HF/6-3 lG(d) geornetry23of 8 is in error by 1.4 pm for

the BC bond length (and the adjacent angles are also wrong), as can be inferred even by

comparing with their own HFl6-3 1G+(d,p) geometries.

Our MP216-31G(d) geometries agree virtually exactly with previously published

ones for seven a~aborinines:~'-~~1,2,3,8, 11,14, and 17. Our MP216-31G(d) geometries

for six of these molecules are included in Tables A5.1 and A5.2 because Kranz and

~lark~~reported only ring bond lengths for 1, 2, and 3, and Kar et al.23 reported only ring

bond lengths and angles for 1, 8, 11, and 14.

2.3.3 Corn parison with Experiment. Experimental gas-phase geometries are available only for borazine (17). Harshbarger et al? measured the borazine geometry by electron diffraction but they could not determine whether it was planar D,, or nonplanar

C,. ~andolt-~ornstein~'increased Harshbarger et al.'s error estimates by a factor of two.

The MP2/6-31G(d) geometry is in fairly good agreement with the electron diffraction geometry. The smdl residual differences, primarily due to vibrational effects, have been analyzed by Ramondo et al?' More recently, Boese et al.; using assumed values for the

XH bond lengths, obtained a C2X-ray crystal structure for borazine with mean values of 70

142.9 -c 0.1 pm for the BN bond length, 117.1 2 0.1" for LNBN, and 122.9 0.1" for

LBNB. Remarkably, their crystal structure agrees almost exactly with the D3,,

MP2/6-3 1G(d) structure.

There are X-ray crystal structures for substituted derivatives of the planar

azaborinines 6 and 8. Siebert et al.'' obtained a crystal structure of 4,s-diethyl-3,6-

dimethyl- 1,2,3,6-diazadiborinine (a derivative of 6) with a planar ring having N-N, N-B ,

B-C, and C-C bond lengths of 139.1, 138.7, 156.1, and 137.1 pm, respectively, al1 with

an estimated uncertainty of t0.4 pm. The largest difference between their bond lengths

and our MP216-31G(d) values is 2.7 pm for B-C. Our C-C bond length is the same as

theirs but, contrary to their result, we find the N-B length to be greater than N-N. Their

ring angIes, LNNB = 123.2", LNBC = 117.2", and LBCC = 119S0, differ by less than

1" from Our values. Schreyer et al." deterrnined the X-ray structure of 1,3,2,4- diazadiborinine (8) with six bulky ligands: a tert-butyl group on each N, a mettiylester group on each C, and a methyl group on each B. Their ring is a boat shape with B2 and

CS at the two ends although their bond lengths suggest some cyclic delocalization." Their bond lengths are longer than our calculated ones by an average of 1.6 pm. Their N-B bond lengths differ from ours by more than 2 pm, whereas their bond lengths for the other haIf of the ring are quite similar to ours. Their two ring angles at N are ~118"or more than 5" sma1ler than ours, their prow and stem angles are larger by ==3",whiIe the other two ring angles are within 1" of the MP2/6-31G(d) values.

There are X-ray crystal structures of polycyclic derivatives of the azaborinines 1 and 3. Bel'skii et al.I2 determined the crystal structure of 2,7,9-trimethyl-IO-phenyl-9-aza- 71

10-boraanthracene which contains 3 as its central ring. Their molecule has C,symmetry with the phenyl ring rotated 49" with respect to the prirnary plane. Their six central ring angIes differ by an average of only 0.8"from our MP2 values for 3. Their N-C bonds are longer than ours, and their C-C bonds are 4 pm longer than ours presumably because of the fused rings. Kranz et a1.13 measured the crystal structure of a different substituted heteroanthracene with the azaborinine 3 as the central ring and 1-methyl and 4-rnesityl substituents. The central ring was slightly twisted.13 Their X-ray bond lengths had radier large error estimates of ~1.5to 1.8 Pm. They compared their X-ray structure with their own MP2/6-3 lG(d) calculation" for 3. The C-C bonds common to two rings are again

4.4 pm longer than the calculated ones for 3. Harris et al.14 found the crystal structure of

10-hydroxy-10,9-borazarophenanthrenewhich has the azaborinine 1 as the central ring.

They gave the geometry for one of two different structures contained in the unit cell, but they did not specify how close the central ring is to planarity. Their CNB angle is the same as our MP2/6-3 1G(d) value for 1, but their N-B bond length is 1.9 pm shorter and their NBC angle is 2.3" larger than ours.

The overall agreement of the X-ray crystal structures with our calculated

MP2/6-3 1G(d) ones is as good as could be expected given the differences created by the substituents and by the packing forces in the solid-state.

2.3.4 Geometrical Trends. Table 2.1 presents overall trends for the

MP216-3 lG(d) bond lengths and angles of the 15 azaborinines (al1 but 4 and 15) that are planar at that level of calculation. Each type of XH bond is almost constant with a range no greater than 1 pm. Ring bonds are significantly more variable; the range for BN bonds TABLE 2.1: Trends in MP2/6-3iG(d) Geometzies of 15 Planar Azaborininesa . bond number average maximum minimum BH 29 119.7 120.1 119.1 CH 32 108.8 109.1 108.3 NH 29 101.5 102.0 101.1 BB 4 166.3 168.8 164.4 BC 22 150.5 153.4 147.8 BN 28 143.2 148.5 140.2 CC 10 137.8 141.7 135.4 CN 22 136.0 138.6 132.7 NN 4 136.3 139.5 133.4 angle at

a Al1 azaborinines except 4 and 15. Bond lengths in picometers and bond angles in degrees. is more than 8 pm. It is remarkable that the ring angle at nitrogen is always greater than the ideal sp2 value of 120" and the angle at boron is always less than 120". By contrast, the angle at carbon is close to 120' but can be on either side of it. Compared to the carbons in benzene, a boron will be further out, while a nitrogen will be a slightly smaller distance in toward the center of the molecule.

The scaie drawing in Figure 2.1 shows that the azaborinines differ in size and shape. A measure of size is the area A enclosed by the ring. It ranges from 0.5 18 to 0.552 nm2 for the planar azaborinines as compared with 0.506 nm2 for benzene. The n = 1 azaborinines with a single N and B have the smallest A; next smallest is 8, noticeably smaller than the other diazadiborinines. The three molecules that have two adjacent B atorns and two adjacent N atorns have the highest A; 16 is the largest. Another size 73

measure is the span L, the longest distance between a pair of atoms in the molecule. L

ranges between 505 and 534 pm for the azaborinines as compared with 496 prn for

benzene. Because the BH bond length is longer than the CH or NH bond length, L can

be expected to be, and indeed is, longest for the azaborinines (6, 9, 11, 16) with a pair

of BH's on opposite sides of the ring. Azaborinines 3, 17, and 14, with N's and B's

directly opposite each other, have the smallest L.

2.4 Harmonic Vibrational Frequencies

Tables A5.3 and A5.4 list Our MPU6-3 lG(d) harmonic vibrationai frequenciesoi

for the azaborinines at Our best geometries; the frequencies are MP2/6-31G(d,p) for 4.

Frequencies for borazine (17) are not listed because MP216-3 lG(d) values were reported

earlier;21 Ramondo et al?' found that they correspond reasonably well to the gas-phase

IR and liquid Raman ~~ectra.~Tables A53 and A5.4 show that the lowest frequency of the planar azaborinines ranges from a minimum of 86 cm-' for 4 through 283 cm" for borazine to a maximum of 38 1 cm-' for 3.

HF harmonic frequencies are sometimes scaled by a factor close to 0.90 to obtain values that are presumed to be more accurate. Figure 2.2 is a histogram of the ratio

= w.(Mp2)/w.(m) in the 6-3 1G(d) basis for 16 azaborinines; 4 was excluded due to its 'i 'i i nonplanarity in the 6-31G(d) basis. Note that ri varies around an average of 0.95 but can exceed 1. Thus a simple scaling is moderately accurate in an isoelectronic set of molecules like the azaborinines. One occurrence at 1.82

Ratio of MP2 ta HF frequency

Figure 2.2 Distribution of ratios ri =wi(MP2)/wi(HJ?) for dl harmonic vibrational frequencies of the azaborinines (excluding 1,2,3,4-diazadiborinine) with the 6-3 1 G(d) basis. The height of a bar labelled 0.95 is the number of al1 ratios such that 0.945 c ri I 0.955. Each molecule's frequencies are paired by order of increasing frequency in each symmetry class. Figure 2.3 Stabilities of the azaborinines relative to the most stable isorner for each n.

2.5 Relative Stabilities

Figure 2.3 shows MNDO, HF, and MP2 energies relative to the most stable isorner

of a given molecular formula, for planar conformations only. The relative energies include

zero point energy (ZPE) corrections. Table 2.2 lists our MP2/6-31G(d) relative stabilities.

Figure 2.3 shows that HF differs from MP2 in many places for the relative stability of

the 11 n = 2 isomers, whereas MNDO only puts 9 out of place. The energy gaps for the n = 3 isomers are much larger than for the n = 1 isomers. Al1 three methods predict that TABLE 2.2 : Relative Stabilitiesa E,, Digole ~oments~*'and Angles, b'd ~olarizabilities~~~and mglest, b*d n-E'ractionsf f, and Reciarocal ~ardnesses~~~~~ no. Es

a MP2/6-31G(d)//MP2/6-31G(d) with respect to lowest energy isomer. In kcal/mol. MP2/C. In debyes. In degrees . In atomic units. ' Of UCHF polarizabilities. In percent. In C basis . ' Al1 except stabilitieç at the geometries of Figure 2.1. 77

the most stable n = 1, 2, 3 isomers are 1,2-azaborinine (1)- 1,3,2,4-diazadiborinine (a),

and borazine (17), respectively.

Hoffmann reported calculation~'~for al1 17 azaborinines using the extended

Hückel (EH) model with idealized planar geometries and standard bond lengths. The EH

model gets the three most stable isomers correct but not the second most stable

diazadiborinine; it is 12, not 14. Massey and ~oellner'~**~compared the MNDO A& of

the n = 1 and 2 isorners at their MNDO geometries. We found slightly different energies

cornpared to their results, in addition to their error in the planarity of 7 discussed in

section 3.2.

Kar et exarnined the relative stability of various isomers of azabora

derivatives of benzene and naphthalene. They assumed that the most stable isomers would

be those with consecutive, alternating B and N pairs, so they considered only three

diazadiborinines, 8, 11, and 14, and found energies identical to ours. However, 12, which

they left out, is more stable than 11 or 14. It is conceivable that this omission may render

incorrect their concl~sions~~about the most stable azaboranaphthalene isomers as well.

Kranz and ~1a-k~~also considered relative stabilities of the n = 1 isomers. Our

MP216-31G(d) energies agree with theirs. They gave as their best result

QCISDl6-3 1G(d)l/MP2/6-3 1G(d) energies corrected b y the HFl6-3 1G(d)//IIF/6-3 1G(d)

ZPE. Al1 methods show 1 to be the most stable and 2 the least stable; also, 2 remains fairly constantly about 23 kcaVmol above 1. Adding the MP2/6-31G(d) ZPE correction to Our MP2/6-31G(d) energies reduces the gap between 2 and 3 from 5.3 to 5.2 kcal/mol. 78

Kranz and Clark's QCISD cal~ulations~~led to a considerably larger gap of 9.9 kcal/mol; it is reduced to 9.6 kcal/mol if Our MP2 ZPE is used.

2.6 Dipole Moments

Table 2.2 lists the magnitude of Our MP2/C dipole moments and the angleea between the dipole moment and the principal axis of inertia 1,. The dipole moments are depicted to scale in Figure 2.1, which also shows the inertiaI axis 1,. The dipole moments of 11 and 17 vanish by symrnetry. In the other azaborinines, the dipole moments range from a modest 1.6 D in 15 to a large 7.6 D in 7. The dipole moment vectors of the azaborinines with C2,symmetry necessarily coincide with the syrnmetry axis. The dipole moments of the azaborinines with Cssymmetry lie in the symmetry plane, which is the molecular plane for al1 but 15. 1, is perpendicular to the symrnetry plane for 15, and thus

8, is 90";for a complete specification of the dipole moment orientation in 15, it is necessary to add that the angle between the dipole moment and I,, is 8.7'. We did not find any previous calculations or measurements of these dipole moments other than a confirmation7 that p = O for borazine.

Molecular dipole moments can be interpreted as vector sums of bond dipoles. This allows us to draw some conclusions from Figure 2.1. The BH group is more negative than the CH (cf. 10 and 12) and NH groups (cf. 3, 7, 14, 15, and 16). Sirnilady CH is more negative than NH (cf. 6 and 9). The molecule 7 with the largest dipole moment has two

B's on one side and two N's on the other. The azaborinines with the next three Iargest dipole moments (13, 5, and 4) al1 differ from 7 by a single exchange of a pair of atoms 79

that leads to a slightly more balanced distribution of charge. The C-B bond dipole = 1.1

D from 12, and the N-C bond dipole = 1.3 D from 9. Structure 1 suggests that the N-B

bond dipole is about 2 D whereas rnolecule 14 suggests about 2.8 D. Thus a bond dipole

mode1 would not be quantitative. Roughly speaking, pN4 + k-B= pW.

2.7 Polarizabilities

2.7.1 Results. Polarizabilities are important because they determine long range intermolecular induction and dispersion forces, various cross sections, and phenornena such as collision-induced spectral line ~hifts~'*~~.It is most useful to report quantities that are invariant to the choice of coordinate system. Familiar polarizability invariants can be constructed from the eigenvalues of the polarizability tensor al I q Iq. The most cornrnon invariant is the mean polarizability:

The difference between the mean in-plane and out-of-plane components is an invariant:

The plane is the molecular plane except for 15 in which it is the symmetry plane. An invariant related to the Kerr effect is:

The in-plane (defined as above) anisotropy is a more intuitive invariant: In this work we use atomic uni& for polarizabilities; one atomic unit of polarizability = 4ng0a0 =

1.648 78 x 10-~'F m2 in SI units.

Table 2.2 lists our MP2K polarizability eigenvalues and invariants for the azaborinines. The orientation of the principal 3-axis of greatest polarizability is shown in

Figure 2.1 and specified in Table 2.2 by the angle s between the 3-axis and the inertial axis 1,. For 15, the 3-axis is in the syrnrnetry plane but 1, is not; the angle between the

3-axis and Ib is 2.4O (Xb runs between and y, if they al1 are considered to pass through one point). Borazine is a symmetric top, and its 3-axis cm be placed anywhere in the molecular plane.

2.7.2 Comparison with Previous Work. Experimental and previously calculated polarizabilities are available only for borazine. The average polarizability was detemined from molar refraction data8 to be 59.7, which is about 5% below Our MP2K result. Dennis and ~itchie~obtained the magnitude of the polarizability anisotropy A2a from the molar Kerr constant of a dilute solution of borazine in cyclohexane measured at 632.8 nm. ~he~~made the assumption that the static anisotropy is approximately the sarne as that at opticd frequencies. Not unexpectedly, their result of 17.6 I 1.0 is lower than our free molecule value by 33%. Their planned gas-phase experimentsg should give much closer agreement with our anisotropy, as was the case for benzene (for which our cdculated value2g was 6% Iow). 81

Our CHF/C polarizability components for borazine differ only slightly from the

CHF values computed at the HW6-31G geometfl by Lazzeretti et with the Sadlej

(S) basis set? Archibong and Thakkar's best hybrid values2' for borazine, at a planar

geometry taken from Harshbarger et al.: differ from Our MP2/C results by only 0.96%

and 1.4% for a and A,a, respectively. Recently, Fowler and ~teine?~published

CHF/[8sGp2d6~2p]//HF/6-31G(d,p)polarizabilities for borazine. Their results are 2.6%

and 3.6% lower than our CHFK values for a and A,a,respectively, because of small

differences in the geometry used and deficiencies in their marginally bigger basis set.

Presumably using the UCHF method, they found borazine's x-polarizability to be more

isotropie than we did; they found A* = -0.77 to be compared with our UCHF/C value

of -2.58.

2.7.3 Trends. Table 2.2 reveals that the mean polarizability ranges from 62.7 for

borazine (17) to 78.3 for 13 as compared with 69.3 for ben~ene.~'Unlike the azoles,

oxazoles, and azines, the azaborinines do not show a systematic trend in the mean

polarizability with the number of heteroatoms. The mean polarizabilities of the n = 2

azaborinines range from 67.1 to 78.3. Generally, the n = 3 azaborinines have a lower

mean polarizability than the others. The three azaborinines with the highest mean

polarizability, 13,7, and 5, also have the largest dipole moments. Note that for each fixed n, the molecules with the smallest mean polarizability are the most stable isomers: 1, 8,

and 17 for n = 1, 2, and 3, respectively. The polarizability anisotropies A,a and^,^ range from 42 and 43 in 13 to 20 in nonplanar 15 to be compared with 36 for benzene." 82

Figure 2.1 shows the principal 3-axis of greatest polarizability in the azaborinines.

It tends to pass close to as many of the borons as possible because they are more

poiarizable than the carbon and nitrogen atoms. The inertiai 1, axis tries to pass through

as many of the heaviest atoms as possible. Thus the 1, and 3-axis are often nearly

perpendicular to each other. Deviations from this overall trend are due to more subtle

factors.

Although UCHF polarizabilities are not very accurate, they have interpretative

value because they can be partitioned uniquely into contributions from each of the

occupied rnolecular orbitals (MOs). Table 2.2 includes the UCHFK parallel and perpendicular rc-fractions f (x,~)= a(X,n)/a(X), x = 11,~for the planar azaborinines. The n-electron contribution to the mean polarizability varies from 61% to 38%. The contribution of the highest occupied MO (HOMO) (a n-MO for al1 the planar azaborinines) to the UCHF mean polarizability varies from a large 47% to 16%.

2.7.4 Observations on Methodology. Figure 2.4 shows percent differences

6a,(X) = (a,(X) - a,(CHF/C)) x 1OO/(a,(CHF/C)) between out-of-plane polarizabilities obtained by various methods and their CFWC counterparts; Figure 2.5 shows these differences for the mean polarizability.

Electron correlation increases the polarizability of second-period atoms from the right-hand side of the periodic table but decreases the polarizability of second-period atoms from the left-hand ~ide.4~Thus in Our previous work on azoles, oxazoles, and azines, we have usually observed cxMm > +HF. However, the presence of the borons in the azaborinines leads to MP2 polarizabilities that are sometimes higher and sometimes 16 ExPt- * Perpendicular polarizability

Figure 2.4 Percent differences between the smallest principal component of the polarizability computed by a given method and its counterpart calculated at the CHF level in bais C. The molecules are numbered as in Figure 2.1. MPZIA is a small basis set calcuIation from ref 25. lower than the CHF ones; see Figures 2.4 and 2.5. The absolute differences between

MP2lC and CHFIC average to 7, 13, and 15% for a, Ala, and A2a, respectively.

Correlation effects are largest for molecule 11: 16, 32, and 43% for a, A,u, and A,a, respectively.

The correlation effects on the polarizabilities do not have a simple trend with n.

However, close examination of the CHF and MP2 polarizabilities reveals that electron Mean polarizability MP- Hybrid

Figure 2.5 Percent differences between the mean polarizability computed by a given method and its counterpart calculated at the CHF level in basis C. The molecuIes are numbered as in Figure 2.1. ME/A is a srnall basis set calculation from ref 25. correlation irons out some of the extremes of the CHF level, so that the MP2 polarizabilities of the azaborinines vary over a narrower range than do the CHF ones.

The difference between CHF and UCHF polarizabilities can be interpreted4' as an inductive contribution that in turn is made up of a positive self-interaction terrn and a negative back polarization term. The UCHFIC values of il, a2,o;, A,a and are always lower than their CHF counterparts by an average of 9, 13, 13, 34, and 34%, respectively, whereas the UCHFK alis higher than its CHFK value by an average of

3%. Thus, unlike the cases of the azoles, oxazoles, and mines, there is a uniformly 85

positive inductive contribution dominated by the self-interaction term for al1 the

azaborinines. UCHF is most different from CHI? for the azaborinines with no carbons.

A disproportionately large UCHF HOMO polarizability is an indicator that the

correlation correction will be relatively large. This cm be rationalized by noting that a

small HOMO-LUMO gap can lead to both these features. Interestingly, those azaborinines

with a high (low) UCHF n;-fraction have negative (positive) correlation corrections to the

polarizability .

2.7.5 Additive Atom Polarizability Modefs. The polarizabilities of Table 2.2,

together with those for the azoles," ~xazoles,~~and a~ines,~' constitute a set of uniformly

good quality polarizabilities for 50 heteroaromatic molecuIes. This data set will be used

to examine the utility and lirnits of simple models for these polarizabilities. An additive

atom mode1 of polarizability applicable to the azaborinines, azoles, oxazoles, and azines

is

a - b, n, + bc nc + b, nN + bo no + bHnH (5)

in which niis the number of atoms of type i in a molecule, and bi can be regarded as the polarizability of an atom of type i in a planar heteroaromatic molecule.. Linear regression of our MPZ/C polarizabilities for the 16 planar azaborinines and 33 o~azoles,~~ and a~ines~~leads to the bi parameters shown in Table 2.3. Notice that the bi are smaller than the corresponding free atom polarizabilities46indicating that bonding has lowered the polarizability. The values of b,, bc, b,, and bo are very close to those obtained previously2g for the 33 azoles, oxazoles. and azines. Table 2.3 shows that the average absolute error 6, of Mode1 5 is 2.4% but the maximum error is 8, = 17%. By contrast, TABLE 2.3: parametersa and Errors of Additive tom ~odels~of ~olarizability - a 401 5 5a Sb 5

43 16 19 -22.2 14

a In atomic units. Mode1 numbers refer to the text.

TABLE 2.4: aram met ers^ and Errors of Additive Connection ~odels~of Polarizability - a A1a 6 6a 6 6a

CI 20.36 17.7 11 13

C2 16.13 13.9 8.5 10

c3 11.43 10.1 5.77 6.6

C4 10.48 9.3 4.58 5

Cs 9.15 7.8 4.76 5.59 7 5.9 3.32 4

C7 6.7 4.96 1.86 3

Ca 5.01 3.85 1.9 2.6

c9 O 1.53 O -0.94 6,(%) 1.4 0.95 4.7 3.9 &(%) 5.0 3.8 17 12 a In atornic units. Mode1 numbers refer to the text. 87 if the azaborinines are excluded, then the additive atom mode12' has an average error of only 1%. This is so because structural isomers of azaborinines have significantly more varied polarizabilities than do the structural isomers of azines, azoles, and oxazoles. The error in Model 5 can be considered a measure of nonadditive effects.

As in Our earlier ~ork,~~~~~one- and two-parameter improvements to Model 5 are obtained by adding 4 /q (Model 5a) and b /q + bAA 312 (Model 5b), respectively, where 'l rl = ( cLUMO- ~~~~~)/2is an approximation to the molecular hardness, and A is the ring area so that A" is an approximate molecular volume. Table 2.3 shows that these tems improve the fit but destroy the physical interpretation of the atomic terrns.

Model 5 can be applied to the anisotropies A,a and A2a with approximately half the accuracy (see Table 2.3) obtained for a. Models Sa and Sb do not significantly improve upon Model 5 for the polarizability anisotropies. Using Model 5 for &,a is tantamount to using different in- and out-of-plane "atornic polarizabilities" bi. It works for 4a as weii because A2a .c A,a in these molecules.

2.7.6 Additive Bond Polarizability Models. The additive connections model"

(ACM) expresses the polarizability as a linear combination of nij, the number of connections (i-e., bonds without regard to bond order) between atoms of type i and j. For our 49 molecules there are 11 distinct nij. However, they are not linearly independent because there are four stoichiometric constraints. Thus the ACM for the 49 azoles, oxazoles, azines, and planar azaborinines has eight tems which we choose as follows: 88

The parameters obtained by linear regression for Mode1 6 and Model 6a (which includes

a hardness term) are listed in Table 2.4. The eight-parameter Model 6 is not as efficient

as the seven-parameter Model 5b for the mean polarizability. The nine-parameter Model

6a is Our most accurate mode1 with average errors of 0.95 and 3.9% for a and A,a,

respectively.

The models presented above al1 exclude nonplanar 1,2,3,4,5,6-triazatriborinine

(15); the errors of al1 our models increase perceptibly if molecule 15 is included in the

data set. For example, the errors for Model 6 applied to A,a increase from 6, = 4.7% to

6.5%, and from 6, = 17 to 47%. We are continuing attempts to develop simple

polarizability models that are more accurate, particularly for anisotropies, and applicable

to a larger set of heteroaromatic molecules.

Supporting Information: Appendix 5 contains MP2/6-3 1G(d) geometries and

harmonic vibrational frequencies of the azaborinines excluding borazine; the results are

MP2/6-3 1G(d,p) for 4.

2.8 References and Notes

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1683.

Boese, R.; Maulitz, A. H.; Stellberg, P. C'hem. Ber. 1994, 127, 1887.

Niedenzu, K.; Sawodny, W.; Watanabe, H.; Dawson, J. W.; Totani, T.; Weber,

W. Inorg. Chem. 1967, 6, 1453.

Scaife, C. W. J.; Laubengayer, A. W. Inorg. Chem. 1966, 5, 1950; Watanabe,

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Hough, W. V.; Schaeffer, G. W.; Dzums, M.; Stewart, A. C. J. Am. Chem. Soc.

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Dennis, G. R; Ritchie, G. L. D. J. Phys. Chem. 1993, 97, 8403.

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Schreyer, P.; Paetzold, P.; Boese, R. Chem. Ber. 1988, 121, 195.

Bel' skii, V. K.; Nesterova, S. V.; Reikhsfel'd, V. O. Zh. Strukt. Khim. 1987,28,

186.

Kranz, M.; Hampel, F.; CIark, T. J. Chem. Soc., Chern. Commun. 1992, 1247.

Harris, K. D. M.; Kariuki, B. M.; Lambropoulos, C.; PhiIp, D.; Robinson, J. M.

A. Tetrahedron 1997, 53, 8599.

Davies, K. M.; Dewar, M. J. S.; Rona, P. J. Am. Chem. Soc. 1967, 89, 6294.

Noth, H.; Fritz, P.; Meister, W. Angew. Chem. 1961, 73, 762. Hoffmann, R. J. Chem. Phys. 1964, 40, 2474.

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Massey, S. T.; Zoellner, R. W. Int. J. Quantum Chem. 1991, 39, 787.

Massey, S. T.; Zoellner, R. W. Inorg. Chem. 1991, 30, 1063.

Rarnondo, F.; Portalone, G.; Bencivenni, L. THEOCHEM 1991, 236, 29.

Kranz, M.; Clark, T. J. Org. Chem. 1992, 57, 5492.

Kar, T; Elmore, D. E.; Scheiner, S. THEOCHEM 1997, 392, 65.

Lazzeretti, P.; Tossell, J. A. THEOCHEM 1991, 236, 403.

Archibong, E. F.; Thakkar, A. J. Mol. Phys. 1994, 81, 557.

Fowler, P. W.; Steiner, E. J. Phys Chem A 1997, 101, 1409.

El-Bakali Kassimi, N.; Doerksen, R. J.; Thakkar, A. J. J. Phys. Chem. 1995, 99,

12790.

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100, 8752.

Doerksen, R. 3.; Thakkar, A. J. Int. J. Quantum Chem. 1996, 60, 421. In Table

II, some of the entries for 1,2-diazine are incorrect. The correct values are: a,

= 67.78, a = 58.99, ~,a= 31.77, ~,a= 31.92, and Ap = 3.12.

Dewar, M. J. S.; Thiel, W. J. Am. Chem. Soc. 1977, 99, 4899.

Bartlett, R. J. Annu. Rev. Phys. Chem. 1981, 32, 359.

Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular

Orbital Theory; Wiley: New York, 1986. 91

Dykstra, C. E. Ab Initio Calculation of the Structures and Properties of

Molecules; Elsevier: Amsterdam, 1988.

Hariharan, P. C.; Pople, J. A. Theor. Chim. Acta 1973, 28, 213.

Frisch, M. J.; Head-Gordon, M.; Trucks, G. W.; Foresman, J. B.; Schlegel, H.

B.; Raghavachari, K.; Robb, M. A.; Binkley, J. S.; Gonzalez, C.; Defrees, D. J.;

Fox, D. J.; Whiteside, R. A.; Seeger, R.; Melius, C. F.; Baker, J.; Martin, R. L.;

Kahn, L. R.; Stewart, J. J. P.; TopioI, S.; Pople, J. A. Gaussian 90, Revision J;

Gaussian, Inc.: Pittsburgh, PA, 1990.

Cohen, H. D.; Roothaan, C. C. J. J. Chem. Phys. 1965, 43, S34.

Thakkar, A. J.; Koga, T.; Saito, M.; Hoffmeyer, R. E. Int. J. Quantum Chem.

Syrnp. 1993, 27, 343.

Dunning, T. H., Jr.; J. Chem. Phys. 1989, 90, 1007.

Dalgamo, A. Adv. Phys. 1962, 11, 281.

Thakkar, A. J.; Doerksen, R. J. UCHF, unpublished.

Callomon, J. H.; Hirota, E.; Kuchitsu, K.; Lafferty, W. J.; Maki, A. G.; Pote, C.

S. "Structure data of free polyatomic molecules" Landolt-Bonzstein, New Series,

Group II Hellwege, K. H.; Hellwege, A. M.; eds. Springer: Berlin, 1976.

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Miller, T. M.; Bederson, B. Adv. At. Mol. Phys. 1977, 13, 1; 1988, 25, 37.

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3602. 92

(45) Sadlej, A. J. Collect. Czech. Chem. Commun. 1988,53, 1995; Theor. Chim. Acta

1991, 79, 123.

(46) Werner, H.-J.;Meyer, W. Phys. Rev. A 1976, 13, 13.

(47) Grant, A. J.; Pickup, B. T. Chem. Phys. ha 1990, 174, 523; J. Chem. Phys.

1992, 97, 3521. Chapter 3. Heteroaromatic Azaboracycles: Structures and Vibrational ~requencies*

Robert J. Doerksen and Ajit J. Thâkkar

* Submitted for publication, 1998. The Abstract and Acknowledgements have been removed for this thesis. 3.1 Introduction

Since the first synthesis,l in 1957, of a heteroaromatic molecule containing B, N,

and C, there has been great interest in making and understanding molecules that bridge

the gap between benzene and borazineO2Despite great difficulty in making the

unsubstituted heteroaromatic monocycles, the interest continues.""or example,

derivatives of one diazaborinine have demonstrated strong anti-bacterial activity and hence

the potential to be useful as d~-u~s.~*'

Figures 3.1-3.5 show the 70 molecules exarnined in this paper. They are al1 five-

and six-rnembered rings that meet the 4n + 2 n-electron criterion of aromaticity. The

nomenclature of heteroaromatic rings follows the extended Hantzsch-Widman system8

recommended by IUPAC. The complete names are given in the captions of Figures

3.1-3.5. Alternate names are included there for one member of each set of isomers.

The formula (CH),N, (NBH,), , with a +b+2c =6, describes al1 azines (c=O) and

azaborinines (d).We have previously studiedg the 17 azabonnines with b=O (the

monoazamonobor-, diazadibor-, and triazatriborinines), and use the numbering from that paperg for them (1-17). In this paper we consider the BN-analogs of pyridine: 10

diazaborinines (18-27) (b=c= 1) (Figure 3.l), and 16 triazadiborinines (28-43) (b= 1 ,c=2)

(Figure 3.2). Higher values of b yield other azaborinines.

The formula (CH),N ,(NBH,),X , a +b +2c =4, defines al1 azoles (c=O) and azaboroIes (CH)if X=NH, and describes al1 oxazoles @=O) and oxazaboroles (d)if

X-O. We consider al1 16 possible azaboroles: 4 diazaboroles (44-47), 6 triazaboroles

(48-531, 4 tetrazaboroles (54-57), and 2 triazadiboroles (58,59) (Figure 3.3). We also Figure 3.1 The diazaborinines at their MP2/6-31G(d) optimized geometry (al1 with Cs symmetry) : 1,3-dihydro-l,2,3-diazaborinine or 2,4,5,6-tetradehydro- 1,2,3-diazaborinane (18), 2,3-dihydro- 1,2,3-diazaborinine (19)' 1,4-dihydro-1'2'4-diazaborinine (20), 2,4- dihydro- l,2,4-diazaborinine (21), 1,2-dihydro- 1,3,2-diazaborinine(22)' 1'5-dihydro- 1,3,5- diazaborinine (23), 1,2-dihydro-l,4,2-diazaborinine (24), 2,4-dihydro- l,4,2-diazaborinine (25), 1,2-dihydro-1,5,2-diazaborinine (26), and 2,5-dihydro- l,5,2-diazaborinine (27). Locant numbering is counter-clockwise, starting at 6 o'clock. The long line is the principal axis of inertia. Dipole moments are shown to scale, with origin at center of mass and arrow head at negative end. Figure 3.2 The triazadiborinines at their MP2/6-31G(d) optimized geometry (with Cs symmetry or as listed in parentheses): 1,2,4,5-tetrahydro-1,2,3,4,5-triazadiborinine or 3,6- didehydro-1,2,3,4,5-tnazadiborinane (a), 1,3,4,5-tetrahydro- 1,2,3,4,5-triazadiborinine (29), 2,3,4,5-tetrahydro-1,2,3,4,5-triazadiborinine (30), 1,2,4,6-tetrahydro-1,2,3,4,6- triazadiborinine (31), 1,3,4,6-tetrahydro-1,2,3,4,6-triazadiborinine (32) (C,,), 1,2,3,5- tetrahydro-1,2,4,3,5-triazadiborinine (33), 1,3,4,5-tetrahydro-1,2,4,3,5-triazadiborhine (34), 2,3,4,5-tetrahydro-1,2,4,3,5-triazadiborinine (35), 1,2,3,4-tetrahydro-l,2,5,3,4- triazadiborinine (36), 1,3,4,5-tetrahydro-1,2,5,3,4-triazadiborinie (37), 2,3,4,5-tetrahydro- 1,2,5,3,4-triazadiborinine (38), 1,2,3,5-tetrahydro-1,3,4,2,5-triazadiborinine (39), 1,2,4,5- tetrahydro-1,3,4,2,5-triazadiborinine (40),2,3,4,5-tetrahydro- l,3,4,2,5-triazadiborinine (41), 1,2,3,4-tetrahydro-1,3,5,2,4-triazadiborinine (42), 1,2.4,5-tetrahydro-l,3,5,2,4- triazadiborinine (43) (Cz,,). Locant numbering is counter-clockwise,starting at 6 o'clock. The long line is the principal axis of inertia. Dipole moments are shown to scale, with origin at center of mas and arrow head at negative end. Figure 3.3 The azaboroles at their MP216-3 1G(d) optimized geometry (with Cssymmetry or as listed in parentheses): 2,3-dihydro-1 H- l,2,3-diazaborole or 4,5-didehydro- 1,2,3- diazaborolidine (44) (C,), 2,4-dihydro- 1 H -c2,4-diazaborole (45) (C,),2,3-dihydro- 1 H - 1,3,2-diazaborole (46) (CZv),2,4-dihydG- 1 H - 1,4,2-diazaborole (47), 2,4-dihydro- 1 H- l ,2,3,4-triazaborole or 3,s-didehydro- l ,2,3,4%iazaborolidine (48) (C,), 3,4-dihydro- l E- l ,2,3,4-triazaborole (49), 3,4-dihydro-2 H- 1,2,3,4-triazaborole (50) (CI), 2,3-dihydro- l H- 1,2,4,3-triazaborole (51) (Cl), 3,4-dihydG- l H - l ,2,4,3-triazaborole (52), 3,4-dihydro-2H- 1,2,4,3-triazaborole (53), 2,5-dihydro- 1H -tetGaborole or 1,2-didehydro-tetrazaborolidEe (54) (C,), 3,5-dihydro- 1H -tetrazaboroE (SS), 4,5-dihydro- 1 H -tetrazaboroIe (56) (C,,), 3,s-dihydro-2 H -tetrazabGole (57) (C,,), 2,3,4,5-tetrahydro-TH - 1,2,3,4,5-triazadiborole or 1,2,3,4,5-trzadiborolidine (58) (Cs, nonplanar) (note thatThere is an H above the central N), 2,3,4,5-tetrahydro- 1 H - 1,2,4,3,5-triazadiboroe or 1,2,4,3,5-triazadiborolidine (59) (C*). Locant numbering is Gunter-clockwise, starting at 6 o'clock. The long line is the principal axis of inertia. Dipole moments are shown to scale, with origin at center of mas and arrow head at negative end. Figure 3.4 The oxazaboroles and oxadiazaboroles at their MP216-31G(d) optimized geometry (with Cs symmetry or as listed in parentheses): 2,3-dihydro- l,2,3-oxazaborole or 4,5-didehydro- l,2,3-oxazaborolidine (60)' 2,4-dihydro- l,2,4-oxazaborole (61), 2,5- dihydro- 1,2,5-oxazaborole (62) (C,), 2,3-dihydro- 1,3,2-oxazaborole (63), 3,4-dihydro- 1,3,4-oxazaborole (64), 2,4-dihydro- l,4,2-oxazaborole (65), 2,4-dihydro- l,2,3,4- oxadiazaborole or 3,5-didehydro-1,2,3,4-oxadiazaboro1idine(66) (C,), 3,4-dihydro- 1,2,3,4- oxadiazaborole (67), 2,5-dihydro- 1,2,3,5-oxadiazaborole (68) (C,), 3,5-dihydro- 1,2,3,5- oxadiazaborole (69), 2,3-dihydro- l,2,4,3-oxadiazaborole (70), 3,4-dihydro- 1,2,4,3- oxadiazaborole (71), 2,3-dihydro-l,2,5,3-oxadiazaborole (72), 3,s-dihydro- l,2,5,3- oxadiazaborole (73), 2,3-dihydro-1,3,4,2-oxadiazaborole (74), 2,4-dihydro-1,3,4,2- oxadiazaborole (75), 2,3-dihydro- 1,3,5,2-oxadiazaborole (76), 2,5-dihydro- I,3,5,2- oxadiazaborole (77) (Cl). Locant numbering is counter-clockwise, starting at 6 o'clock. The long line is the principal axis of inertia. Dipole moments are shown to scale, with origin at center of mass and arrow head at negative end. Figure 3.5 The oxatriazaboroles and oxadiazadiboroles at their MP2/6-3 1G(d) optimized geornetry (with Cs symmetry or as listed in parentheses): 2,5-dihydro- 1,2,3,4,5- oxatriazaborole or 3,4-didehydro-1,2,3,4,5-oxatriazaborolidine(78) (C,), 3,5-dihydro- 1,2,3,4,5-oxatriazaborole (79), 4,5-dihydro- 1,2,3,4,5-oxatriazaborole @O), 2,4-dihydro- 1,2,3,5,4-oxatriazaborole (81), 3,4-dihydro- 1,2,3,5,4-oxatriazaborole(82), 4,5-dihydro- 1,2,3,5,4-oxatriazaborole(83), 2,3,4,5-tetrahydro- 1,2,3,4,5-oxadiazadiboroleor 1,2,3,4,5- oxatriazaborolidine (84) (Cl), 2,3,4,5-tetrahydro-1,2,4,3,5-oxadiazadiboroleor 1,2,4,3,5- oxatriazaborolidine (85), 2,3,4,5-tetrahydro-1,2,5,3,4-oxadiazadiborole or 1,2,5,3,4- oxatriazaborolidine (86) (C2,,), 2,3,4,5-tetrahydro- 1,3,4,2,5-oxadiazadiboroleor 1,3,4,2,5- oxatriazaborolidine (87) (C2). LOcant numbering is counter-clockwise, starting at 6 o'clock. The long line is the principal axis of inertia. DipoIe moments are shown to scale, with origin at center of mass and arrow head at negative end. 100 study al1 28 possible oxazaboroles: 6 plain oxazaboroles (60-65), 12 oxadiazaboroles

(66-77), 6 oxatriazaboroles (7&83), and 4 oxadiazadiboroles (84-87) (Figures 3.4 and

3.5).

Experimental information is available for 12 of the 70 molecules we study: four six-membered rings, four five-rnembered rings without 0, and four with O. Although none of the molecules have been synthesized in unsubstituted forrn, ten have been made with ring substituents: 19," 42: 46," 53,12 56,13 59,14 63," 76,16 85," 87,14 and two others with a fused ring: 2216 and 26.18 Experimental X-ray structures are available for five of the twelve, 19,4*1942; 46,3*2072L~6,~~ and 76,23924but there is gas-phase electron diffraction data in only one case, ~6.~~Detailed vibrational spectral analysis has been performed for derivatives of 56.13+26-29Infrared, and occasionally Raman, data have been reported for 19:~'~ 22:1-33 26," 46,3d4 53,12 59:' 63," and 76.243'*36Dipole moments have been measured only for variants of 19.'~

Calculations, largely semiempirical, have been reported for seven of the molecules.

A semiempirical geometry and dipole moment were calculated for dimethyl-56.38

Semiempirical geometry3' and themochemical and magnetic property40 calculations were made for 59. There have been calculations at assumed geometry to determine the molecular orbitals, energy, aromatic stabilization, and spectra of 4634 and ~6;~'~"the last study" featured Hartree-Fock level population analysis and semiempirical excited States.

Semiempirical charge densities and bond orders have been calculated2' for 76. Extended

Hückel relative stabilities have been rep~rted~~for three of the diazaborinines, 22,24, and

26. 101

We report systematic, electron-correlated, ab initio calculations for al1 70

molecules; we consider their geometries, energies and relative stabilities, harmonic

vibrational frequencies, and dipole moments.

3.2 Computational Methods

Equilibrium geometries and harmonic vibrational frequencies were computed for

al1 70 molecules using three different methods. Preliminary calculations were made with

the semiempirical Modified Neglect of Differential Overlap (MNDO) rn~del.'~~~~

Subsequently we calculated ab initio Hartree-Fock (HF) and second-order M~ller-Plesset

(MPZ) perturbation theory4"' geometries and harmonic frequencies using the 6-3 1G(d),

and in two cases 6-3 lG(d,p), split-valence plus polarization basis sed8A11 calculations

were made with GAUSSIAN-90.~' The harmonic frequencies were used to venfy that the

calculated geometries are true minima. Dipole moments were calculated by the finite-field

meth~d?~using HF and valence MP2 field-dependent energies obtained with a larger

[5s3p2d/3s2p] basis set denoted C in Our earlier work.'' Further technical details can be

found else~here??~1s52

3.3 Equilibrium Geometries

3.3.1 Conformations. The conformational search was restricted to ring isomers.

For each molecule, and for each level of calculation, the geometry was first deterrnined under the constraint of planarity. If the planar conformation was a true minimum, then no further search was made. If the planar conformation was not a true minimum structure 102

at a particular level of theory, then that conformation was used as a starting guess to find

any nearby minima. Al1 stationary points found at a particular level were used as starting

guesses for the next higher level of calculations.

Figure 3.6 Relative stability (in kcal/moI) of nonplanar vs. planar conformation for al1 cases in which the nonplanar conformation is more stable for at least one method. Molecule numbering as given in Figures 3.2-3.5.

Figure 3.6 displays the energy gap for 22 molecules in which planar conformations were higher in energy than nonplanar ones at one or more levels of calculation. It shows that the MNDO and HF models did not reliably predict planarity. Five molecules were nonplanar with MNDO but planar otherwise. Two molecules were planar with MP2 even >C 1O3

though both MNDO and HF predicted them to be nonplanar. The HF structure of 66 was

planar but the MNDO and MP2 ones were not.

In every case, the energy gap decreases in the order MNDO, HFl6-31G(d),

MP216-3 1G(d). A nonplanar fom of 1,2,3,4-tetrahydro- 1,2,3,4-diazadiborinine (4) w as

previously foundg to be more stable by 3, 0.6, and 0.0001 kcallmol at these three levels,

but the planar form was more stable with MP2/6-31G(d,p). Hence, we tried increasing

basis set size for the two molecules with the smallest MP2/6-31G(d) energy gap: 48 and

54. At the MP2/6-31G(d,p) level, the gaps shrank to 0.033 and 0.13 kcaVmol for 48 and

54, respectively; however, the gaps increased to 0.14 and 0.29 kcal/mol at the

MP2/6-31 lG(d,p) level. So we retain the MP2/6-3 1G(d) nonplanar structures for 48 and

54. Future investigations with higher leveIs of correlation and even larger basis sets may

give different results because the difference between planar and nonplanar conformations

is very small for some of the molecules.

The minimum structure is nonplanar for 15 of the 70 molecules at the

MP216-31G(d) level. Their nonplanarity is shown in Figure 3.7 made with the help of

ORTEP-3 for ~indows."There are two possible enantiomers for each nonplanar species;

one can be obtained from the other by taking the negative of al1 dihedral angles.

The 15 nonplanar molecules are dl five-membered rings. The nonplanar

conformation of 11 of these is necessarily of C,symrnetry. However, 45, 58, 59 and 87

have C2, symmetry in their planar conformations and hence could have either Cs or C2 nonplanar conformations. In fact, at al1 three levels of theory, the nonplanar, minimum- energy structure of 4559, and 87 is C, whereas the minimum-energy structure of 58 is Figure 7. Al1 nonplanar five-membered rings at their MP216-3 lG(d) optimized geometry. Molecule and locant numbering and symmetry as given in Figures 3.3-3.5. Only one of two enantiomers is shown for each molecule. 105

Cs but nonplanar. This can be explained by noting the tendency of NH to pyramidalize

but for BH to maintain a trigonal planar structure around it. For 58, there are three

adjacent NH's; for the Cs structure, the central NH bends out-of-plane in one direction

while its two neighbors go in the other direction. For 45, 59, and 87 there are only two

neighboring NH's; one bends up and the other down, yieIding the C2 structure.

Figures 3.1-3.5 show the molecules to scale at their MP2/6-3 lG(d) geometry. The

MP2/6-3lG(d) geometries for the azaborinines can be e~~ected~'.~~to be accurate to

within 1%. Tables 3.1 and 3.2 list the computed and relevant experimental ring bond

lengths and angIes for al1 the six- and five-membered rings, respectively. Tables

A6.1-A6.3 list computed XH bond lengths and XYIf bond angles. Table A6.4 lists

dihedral angles for the nonplanar species.

3.3.2 Cornparison with Previous Work. As in Our previous paperP the

MNDO and HF bond lengths and angles were quite different from the MP2 ones.

Calculated geometries have been published previously for only two of these 70 rnoIecules.

Maouche et aL's MNDO geometry3gof 59 is nearly the same as ours except for an NNB angle that is too large by 1.2". Our MNDO geometry of 56 differs substantially from an

MNDO geometry of the 1,4-dimethyl ~~ecies.'~

The only gas-phase geometry of a derivative of any of the 70 molecules of this paper is an electron diffraction (ED) geometry by Chang et aL2' of the 56-like 1,4- dimethyl-4,5-dihydro- 1 -H-tetrazaborole (called cyclotetrazenoborane by those authors).

They assumed a value for the BH bond length and the NCH angle. Considering al1 data, they preferred a planar conformation for the molecule, although the evidence was not TABLE 3.1: Ring Bond ~engths~and Anglesb For Al1 Diazaborinines and Triazaborinines; MP2/6-31Q(d) and This Work Except as Noted. dc W d a.. *am mrlri rirld

wme... ~dcm Wmm rlrlri TABLE 3.2: Ring Bond ~engths'and Anglesb For Al1 Azaboroles and Oxazaboroles; MP2/6-310(d) aed hie Work Except a6 Noted. a In picorneters. In degrees. X-ray, bis compound, ref 3. Substituted derivative; cf. text. X-ray, bis compound, ref 20. ' X-ray, ref 21. 1,3-diisopropyl-2-meth 1-. 1,3-diethyl-2-methyl-. 1,2,3-trimethyl- at -73OC. 1.2.3-trimethyl-. at -17I0C. * MP2/6-31G(d.p). 'MP2/6-311G(d,p). ' X-ray, ref 22. " Electron diffraction, ref 25. O X-ray, ref 23. X-ray, ref 24. 111 conclusive. Our ring bond lengths agree within 0.7 pm, which is about the sarne as the estimated experimental error. Our ring angles differ by 0.6 to 2A0, compared to their estimated errors of 0.6 to 1.0". Interestingly, the ED external ring angles are similar to ours, despite the presence of the methyl groups.

There are relatively more experirnental geometries of substituted derivatives obtained by X-ray (XR) diffraction. Brett et al." obtained an XR geometry of the 1,5- diisopropyl-4-phenyl- derivative of 56. They found the hetero-ring to be planar within

0.2". The average absolute differences, Sa, between their ring parameters and the MP2 ones are only 0.9 pm and 0.9" to be compared with experimental errors of 0.4 pm and

0.3", respectively. However, their N2=N3 bond was only 127.9 pm, noticeably shorter than our 129.6 pm and the 129.1 pm of Chang et al.25

Schrnid et aL2' determined the XR structure for three versions of 46, with 1,3- diisopropyl-2-methyl-, with 1,3-diethyl-2-methyl-, and with 1,2,3-trimethyl- substituents.

Comparing the three XR geometries shows that significant variance occurs just because of different substituents. It is especially valuable that XR structures were obtained at two temperatures, -73°C and - 17 1 OC, for the trimethyldiazaborole. Two sets of parameters were presented that differ insignificantly in angles but by as much as 2 pm in bond lengths. The lower temperature structure is the one much closer to Our unsubstituted MP2 structure; the difference 6, is only 0.4 pm for ring bond lengths and 6, = 1" for angles.

This agreement is noteworthy, especially considering that their structures are not C,,, but instead have pairs of parameters that differ by as much as 0.6 pm, presumably because of interrnolecular interactions in the crystal. 112

Weber et alm3obtained the XR structure of bis(2,3-dihydro- 1,3,2-diazabory1)oxane with t-butyl groups on each N, and Sawitzki et aLZ0determined the XR structure of a fused ring molecule containing 46. As could be expected, these structures differ a iittle more from our unsubstituted molecule than do the compounds of Schrnid et al. ,21 because of substituent effects. However, 6, is still only 1.3 pm and 2.7" for the Weber et al. c~rn~ound.~

XR structures have been reported for 4-methyl-2-phenyl-76 by ~a~er~~and for 2- phenyl-4-(4-rnethylpheny1)-76 by Mohrle et In each case the heterocycle is nearly planar. ~a~e?~reported rather large estimated errors of I pm and = 0.7". Our geometry differs from his by 6, = 1.3 pm and 1". Our MP2 structure agrees more closely with that of Mohrle et al.24with 6, = 0.7 pm and 0.8' for ring parameters. In each case, the largest differences are an O-N bond longer by =2 pm and a corresponding 1234 larger by =2" than the unsubstituted MP2 structure.

Klofkorn et al? made the first triazadiborinine, a derivative of 42, and obtained its X-ray structure. Their ring bond lengths are similar to our MP2 ones for the unsubstituted ring, with B-N bonds in the range of 14 1-145 pm, and one long and one short C-N bond. Also, they5 reported that the sum of 3 angles at the ring atoms other than N5 was close to 360". However, the sum of the 6 ring angles is only = 700°, not the expected 720°, suggesting significant deviation from planarity. Their ring bond angles are quite different from MP2, by as much as 10". Apart from the usual methodological differences, this could be caused by the nature of the substituents. The two N's have a t-butyl group whereas the two B's have pentafluorobenzyl attached. In the same paper: 113

the structure of a borazine (17) derivative was reported with the same substituent pattern.

In both structures: the ring angles at N's having the t-butyl substituent are much less than

120". By contrast, the angles in unsubstituted 17"+~~follow the pattern observed for the

MP2 geometry of al1 the azaborinines9 and the molecules in this paper, with the angle at

NH greater than 120'.

Two fused-ring 19's have been studied by XR diffraction. Comparing Aurivillius and Lofving's 7-hydroxy-6-methyl-7,6-borazarothieno[3,2-c~pyrdinegand Groziak et al.'s

1,2-dihydro- 1-hydroxy-2,3,1 -benzodiazadibone the experimental ring bonds are al1 longer by up to 3 pm, except for the C-N bond which is 3 pm shorter than the MP2 19 structure. The average difference 6, between ring bond lengths is 1.3 pm when comparing the two experimental structures, but is 1.7 pm when comparing either to our MP2 structure. Differences occur because of hydrogen bonding in the crystal, the slight nonplanarity of the rings, and the tendency for the fused ring to cause longer bonds in the hetero-ring.

3.3.3 Trends in Bond Lengths and Angles. We consider the trends in the

MP2 geometrical parameters of al1 87 BN-rings (1-87). Bond lengths are within the same range for five- and six-membered rings. Bond angles differ greatly between five- and six- membered rings-obviously because of the different number of angles.

Al1 ring bond lengths, XY, where X and Y are selected from B, C, N, FJ, and 0, are shown in Figure 3.8. It separates those N's that have an H attached (called N) from those that do not (catled N). Figure 3.8 shows that bonds between two pyridinic N's

(N-N) are always shorter than 132 pm, whereas N-N bonds between two pyrrolic N's are 114 always longer than 133 pm. For six-membered rings, the ring angle at N varies between

112.6" and 1 t 9.4', while at N the range is 120.9" to 131.1 O. The distinction is not quite

as sharp for five-membered rings, but again the angles at N tend to be larger.

-BB Range of bond lengths

H 50 bonds Bond

Figure 3.8 Ring bond lengths in picometers for 87 molecules (1-87),separated by atom types. N is reserved for pyrrolic nitrogen while N is used to refer to pyridinic nitrogen. At the best geometry, whether planar or nonplanar. Al1 MP2/6-3 lG(d), except for 4 which is MP2/6-3 1G(d,p).

Most of the ring bond lengths fa11 between typical single and double bond lengths because of n-electron delocalization. An extrerne case is 43; it contains both the longest and the shortest of al1 B-N bond lengths, different by 13 pm. Though they are slightly shorter for five-membered rings, al1 bond lengths to H lie within 1.2 pm of their average 115

values: 119.2 pm for BH, 108.9 pm for CH, and 101.5 pm for NH. For six-membered

rings, the angle at B is at most 120.2";the range of values is just slightly broader than

we found for the 17 azaborininesg alone. Angles involving the H's vary greatly. The bond

lengths of nonplanar species fit into the range of values found for planar species because

the nonplanar conformations are close to being planar.

Table A6.4 shows that the dihedral angles are farthest frorn planar when they

include the out-of-plane N-H bond, the most significant nonplanar feature. The

appropriate HNNH or HNCH dihedral angle is between 26" and 77" for the nonplanar molecules. The 6-31G(d7p) basis reduces the dihedral angles for 48 and 54, but

6-3 1lG(d,p) gives almost the same angles as 6-3 lG(d) does.

3.4 Harmonic Vibrational Frequencies

Tables A6.SA6.13 list MP2/6-3 lG(d) harrnonic vibrational frequencies and infrared (IR) intensities at the best MP2/6-31G(d) geometry for 55 planar and 15 nonplanar molecules. For each molecule, the frequencies are Iisted in increasing order, sorted by symmetry type, and labelled with a qualitative indicator of intensity (very weak to very strong). Al1 frequencies pertain to the more abundant "B isotope. Frequencies involving 'OB would be =1û-15 cm-' higher.

Figure 3.9 plots the ratio ri = oi(MP2)hi(HF) (frequencies are matched by symmetry in increasing order) versus MP2 frequency, for 83 of 87 BN-containing molecules that we have studied, including the a~aborinines;~four molecules are excluded because their HF and MP2 conformations are not of the same symmetry. We find that ri

117

is consistently 0.99 for B-H stretch (v) vibrations, 0.96 on average for v(C-H), and 0.94

for v(N-H). The ratio ri is more scattered for the lower frequency vibrations. The overall

average is 0.94, close to the ratio of 0.95 found by Scott and 13adoms6 for 122 molecules

using the sarne basis set.

Many of the vibrational modes are mixed. Some trends do occur, but often with

many exceptions in a set of isomers. The highest fundamental frequencies are X-H

stretches occumng around 3650, 3300, and 2800 cm-' for X=N,C, and B, respectively.

The next highest frequency is usually v(C-N), if such a bond is present; the B-H in-plane

deformation (6) is usually quite distinct, between 863 and 987 cm-'; the B-H out-of-plane

deformation (y), at =850 cm-', is usually but not always the highest of the X-H wags. The

lowest frequency sometimes involves the whole ring, but other times is y(N-H). Other

types of frequencies Vary widely over the series of molecules.

It is difficult to compare harmonic frequencies calculated for isolated molecules with experimental ones that are anharmonic, temperature-dependent, and usually deterrnined in solution. A simple device is to unifomily scale calculated frequencies to improve agreement with experiment. Scott and ~adorn'~suggested a scale factor of

0.9427 for MP2/6-3 lG(d) frequencies. They gave alternate factors that yield maximum agreement for low frequencies.

SeIected experimental lines have previously been assigned for some BN- heterocycles. The results are: v(N-H), 3240-3450;4*18*2450-31*36v(C-H), 3045;~' v(B-H),

2542-26~6;~~~~~~"~~v(C=N), 1599-1650;~*~'-~~*~~ v(B-N), 1340-1400;~~*~~ and v(N-O), 118 92d6 cd.The X-H stretch frequencies are about 5% higher than the MP2 values as expected.

Measured vibrational spectra have been assigned for the 1,4-diphenyl- l3 and 1,4- dimethy126derivatives of 56. For the latter, liquid and vapor IR and liquid Raman spectra were ~btained,~~to help assign al1 lines accurately. The spectra of 1-rneth~l-4-~hen~l-~~ and 1-~hen~l-4-chloro~hen~l-~~derivatives of 56 have also been reported. It was straightforward to match seven lines from the twelve or so frequencies of the

56-derivatives that did not explicitly depend on the substit~ents.~~*~~*~~*~~The experimental frequencies (from the dimethyl derivativeZ6or a range of values from several of the four derivatives previously s~died~~~~~~~~~~~) for v(B-H), v(N=N), ring breathing, 6(B-H), y(B-H), and the two Iowest out-of-plane ring folds (y) of A, and of 13, symmetry are 2648

2 12, 1363, 1095 t 4, 1057 I 12, 814 I 4, 554 and 522 cm", respectively. The corresponding MP2 frequencies are 2825, 13 13, 1097, 871, 849, 635 and 525 cm-', respectively. Agreement is reasonable except for 8(B-H), which however should be strongly affected by substituents at the two neighboring N's.

The relative energies, including zero-point corrections, of the 70 molecules are shown in Figures 3.10-3.12, with respect to the lowest energy isomer for each group. The

MP2 stabilities are listed in Table 3.3. Al1 the energies are for the lowest conformation that we found, planar or nonplanar. Stabilities relative to lowest energy isomer P

26 27 25 18 20 43 40 38 32 37 33 29 28 Molecule

Figure 3.10 Stabilities of the azaborinines and diazaborinines, each relative to their most stable isomer. For HF and MP2, the 6-31G(d) basis set was used.

Al1 methods agree on the most stable species of each type. But Figures 3.10-3.12 show that the lower levels of calculation do not correctly predict the order of some of the other molecules. MNDO is correct more often than HF. MNDO tends to underestimate, and HF to overestimate, the gap between isomers.

An -XBHNH- unit is always present in the lowest energy isomers: X=NN in the four most stable azaboroles, X=O in the four lowest energy isomers of the oxazaboroles, and X=N in the most stable isomer, 22, of the diazaborinines. The lowest energy pair of triazadiborinine tautomers each have an additional SN pair, Le., an -NBHNHBHNH-unit. 100 Stabilities relative to lowest energy isomer

Molecule

Figure 3.11 Stabilities of the diazaboroles, triazaboroles, tetrazaboroles, and triazadiboroles, each relative to their most stable isomer. For HF and MP2, the 6-3 1G(d) basis set was used.

~offrnan~~calculated the following order of extended Hückel energies for diazaborinines:

22 c 26 c 24. Those are, in fact, the three lowest energy isomers in the correct MP2 order.

Certain features are unsatisfactory predictors of stability. Al1 12 molecules for which there is experimental data have a -BHNH-unit but this alone is not a guarantee of greater relative stability. Two such units are found in 41, but it is relatively unstable because of its adjacent NH's. The oxazaboroles 60 and 63 each have a -BHNH-next to

0, but 63 is more stable by 59 kcal/mol because of the preferred OBN order. Tautomers 1 Stabilities relative to lowest energy isomer I

65 62 61 76 77 71 68 72 73 79 83 81 85 86 Molecule

Figure 3.12 Stabilities of the oxazaboroles, oxadiazaboroles, oxatriazaboroles, and oxadiazadiboroles, each relative to their most stable isomer. For HF and MP2, the 6-31G(d) basis set was used.

can be equally stable (e.g., 20 and 21, within 0.4 kcal/mol) or separated by as much as

66 kcal/mol (33 and 35). Each of the four most stable azaboroles, 46, 53, 56, and 59, is the isomer with smallest dipole moment, but this trend does not hold for the other types of molecules. The molecules that we found to be nonplanar are sometimes but not always the highest energy isomers.

In the highest energy species, adjacent pairs of BH7s or of NH's are found. For example, the two highest energy triazadiborinines, 28 and 30, have both of these features, whereas 32 is relatively more stable despite having an NNN pattern because the dehydro- TABLE 3.3 : Relative Stabilitiesa, Es, and Dipole ~oments~~'and ~ngles~*~.* NO. E, CI 0, No. E, P 8, 18 41.1 4.35 73.4 53 O 1.92 6.3 19 23.4 0.71 23.6 54 33.6 5.71 19.3 20 48.0 3.27 84.4 55 15.0 2.02 22.2 21 47.5 1.97 29.9 56 O 2.75 90 22 O 4.22 22.9 57 44.6 5.59 90

a MP2/6-3l$(d)//MP2C6-31G(d) with respect to lowest energy isomer. In kcal/mol. MP2/C. Ln debyes. In degrees. Al1 at the geometries of Figures 3.1-3.5. 123 position is at the middle N. In molecules without BH or NH pairs, there are other patterns. For the diazaborinines, the common structure for the three least stable isomers is -NNH-.The highest energy oxadiazaboroles and oxatriazaboroles contain -NONH- andlor ONN (with or without one H).

The ease with which a molecule can be synthesized can be related to the relative energetic stability of that molecule compared to its isomers. Comparing the list of 12 molecuIes which have been made (with substituents) with the relative stabilities in Table

3.3 shows that in general those 12 are the lowest energy isomers. The two exceptions are

19, which is one of three known diazaborinines, but is only the fifth most stable; and 76, which is onIy the second most stable oxadiazaborole. We predict derivatives of 74, the lowest energy isomer of the 12 oxadiazaboroles, and 80, similar to 56 and the most stable of the hitherto unobserved oxatriazaboroles, to be accessible synthetic targets.

3.6 Dipole Moments

3.6.1 Results. Table 3.3 lists the magnitude of the MP2IC dipole moments and the angle ea between the dipole moment vector and the principal axis of inertia Ia; the latter was calculated using the masses of the most abundant isotopes for each atom.

Figures 3.1-3.5 show the inertial axes 1, and the dipole moments; the latter are drawn to scale with the arrowhead at the negative end. The angle ea suffices to describe the orientation of the dipole moment for the 55 planar molecules because symmetry requires the dipole moment vector to be in the rnolecular plane. The angles e,, ec, between the 124 dipole moment and the inertial axes Ib, I,, are given in Table A6.4 to complete the specification for the nonplanar molecules.

The dipole moments vary in size from 8.33 D in 33 to under a debye in 19, 46,

60, 68 and 74. The dipole moments Vary greatly for isomers and tautomers. One can see in Figure 3.7 that each CImolecule has one N-H bond pyrarnidalized such that the H is out-of-plane in one direction and the N is displaced a smaller amount in the other direction. The plus-to-minus dipole moment is parallel to this N-H with the negative end toward the N. The "in-plane" electron distribution of 68 is so remarkably uniform that p is almost perpendicular to the "moIecular plane"; this is why the dipole moment of 68 is not shown in Figure 3.4.

3.6.2 Camparison with Previous Work. The dipole moments of 5-ethyl-2,3- dimethyl and 4-ethyl-2,3-dimethyl derivatives of 19 were rnea~ured~~to be 1.2 D and 1.3

D, respectively, in benzene at 25°C. The contributions made by the substituents to the dipole moment of 19 can be estimated by comparing dipole moments of various methylpyridines and ethylpyridines.37*57The methyl groups should counteract each other with a net addition of only ~0.05D to the dipole moment of 19. By analogy with the dipole moment difference between ethylpyridine and pyridine,57 the ethyl group in the 5- and 4-positions should add =0.3 D and ==0.4D respectively to the dipole moment of 19.

Subtracting these substituent contributions from Gronowitz and MaItesson's rneasurement~)~leads to an estimate of p = 0.85 I0.1 D for 19 in benzene to be compared with our MP2 value of 0.71 D for isolated 19. 125

Dewar and ~ou~hert~~'calculated the MNDO dipole moment of the 1,3-dimethyl

derivative of 56 to be 2.3 D. By cornparison, Our MNDO and MP2 results for 56 are 2.37

D and 2.75 D, respectively.

3.7 Concluding Remarks

The MP2 calculations of the equilibrium geometries, hannonic vibrational

frequencies, relative stabilities, and dipole moments have helped paint a broad-brush

picture of 70 azaboracycles. The most stable isomers have the substructure XBHNH,

where X = N, NH, or O is the base-ring heteroatom. Planar conformations are stable

minima for al1 but 15 five-membered rings.

Supporting Information: Appendix 6 contains additional MP216-3 1G(d)

geometric parameters, dipole moment angies, harrnonic vibrational frequencies, and

Infrared intensities of lû-87.

3.8 References and Notes

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(2) Morris, J. H. Boron in ring systems. In Comprehensive Organometallic

Chemistry; Wilkinson, G, Ed.; Pergamon: Oxford, 1982; Vol. 1, p. 3 1 1.

Housecroft, C. E. Compounds with three- or four-coordinate boron, emphasizing

cyclic systems. In Comprehensive Organometallic Chemistry II; Abel, E. W.,

Ed.; Elsevier: Oxford, 1995; Vol. 1, p. 129. 126

Weber, L; Dobbert, E.; Stammler, H.4.; Neumann, B.; Boese, R.; Blaser, D.

Chem. Ber./Recueil1997,130, 705.

Groziak, M. P.; Chen, L.; Yi, L.; Robinson, P. D. J. Ain. Chem. Soc. 1997, 119,

7817.

Klofkorn, C.; Schmidt, M.; Spaniol, T.; Wagner, T.; Costisor, O.; Paetzold, P.

Chem. Ber. 1995, 128, 1037.

Wendler, F.; BergIer, H.; Prutej, K.; Jungwirth, H.; Zisser, G.; Kuchler, K.;

Hogenauer, G. J. Biol. Chem. 1997, 272, 27091.

Baldock, C.; Rafferty, J. B.; Sedelnikova, S. E.; Baker, P. J.; Stuitje, A. R.;

SIabas, A. R.; Hawkes, T. R.; Rice, D. W. Nature 1996, 274, 2107.

Powell, W. H. Pure Appl. Chem. 1983, 55, 409. The most important rules are:

The prefixes for heteroatoms must go in the order oxa-, aza-, bora-. Saturation is defined in terrns of the normal valence of the heteroatoms, 3 for B and N, 2 for 0. For fully unsaturated B-containing six-membered rings, the stem is

-borinine, and for five-membered rings, -borole; saturated rings would be

-borinane and -borolidine (because N is present), respectively. Partially unsaturated rings can be named based on either saturation extreme, though the unsaturated is the preferred reference. We chose to name al1 such compounds with respect to the unsaturated base. The prefix hydro- is used to specify points where an unsaturation has been removed. The presence of pyrrolic N in azole derivatives requires the use of an indicated H. This affects only the 16 azaboroles. The lowest possible locants are assigned in the following priority 127 order: a. first heteroatom (O before N before B); b. al1 heteroatoms; c. indicated hydrogen; d. hydro- referrants. The literature cited below often used different narnes.

Doerksen, R. J.; Thakkar, A. J. J. Phys. Chem. A 1998, 102, 4679.

Namtvedt, J.; Gronowitz, S. Acta Chem. Scand. 1968, 22, 1373.

Merriam, J. S.; Niedenzu, K. J. Organornet. Chem. 1973, 51, Cl.

Paetzold, P. 1. 2. Anorg. Allg. Chem. 1963, 326, 64.

Greenwood, N. N.; Morris, J. H. J. Chem. Soc. 1965, 6205.

Noth, H.; Regnet, W. 2. Naturforsch., B: Chem. Sci. 1963, 18, 1138.

Letsinger, R. L.; Hamilton, S. B. J. Org. chim. 1960, 25, 592.

Yale, H. L.; Bergeim, F. H.; Sowinski, F. A.; Bernstein, 3.; Fried, J. J. Am.

Chem. Soc. 1962, 84, 68.

Kroner, J.; Nolle, D.; Noth, H; Winterstein, W. Chem. Ber. 1975, 108, 3807.

Dorokhov, V. A.; Boldyreva, O. G.; Bochkareva, M. N.; Mikhailov, B. M. Bull.

Acad. Sci. USSR, Div. Chem. Sci. 1979, 28, 163.

Aurivillius, B.; Lofving, 1. Acta Chem. Scand. 1974, 828, 989.

Sawitzki, G.; Einholz, W.; Haubold, W. 2. Naturforsch., B: Chern. Sci. 1988,18,

1179.

Schmid, G.; Polk, M.; Boese, R. Inorg. Chem. 1990, 29, 4421.

Brett, W. A.; Rademacher, P.; Boese, R. Acta Crystallogr., Sect. C: Cryst.

Struct. Commun. 1990, 46, 689.

Raper, E. S. Acta Crystallogr., Sect. B: Struct. Sei. 1978, 34, 3281. 128

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El Cerrito, CA, 1989; Vol. 3. Also, cf. VOIS.1 and 2. Chapter 4. Electron-Correlated Polarizabilities of 70 Heteroarornatic Azaboracycles

Robert J. Doerksen and Ajit J. Thakkar 4.1 Introduction

Polarizabilities are important properties of molecules, but they are difficult to

determine either by experiment or by calculation. The average and anisotropic

polarizabilities play a key role in physical phenornena such as scattering and van der

Waals interactions.'~~Polarizabilities are also in demand as parameters for force field

descriptions of proteins and liquids.3

There have been several approaches to modeling polarizabilities.M Often the focus

has been on additivity of the polarizabilities of certain sub-units of the molecule: of

atoms, or of atoms allowing for different valence, or of bonds, or of chernicaI groups.

Important questions are: how much does bonding affect the polarizability of the atoms,

and how widely will the polarizability of a bond vary with bond length or order? In other

words, to what extent are group polarizabilities transferrable?

Accurate polarizability models must be based on accurate polarizabilities, whether

experimental or calculated. For the molecules we have targeted-aromatic

heterocycles-there were only lirnited data available, either theoretical or experimental.

Thus, we have been developing an accurate database of calculated polarizabilities for testing whether the polarizabilities of conjugated ring molecules can be represented as ail additive function of components of a molecule and to probe the relationship between polarizability and aromaticity.

In this work we present ab înitio electron correlated polarizabilities for 70 heterocyclic molecules containing B, C, N, and in some cases There have been no previous studies of the polarizabilities of the 70 molecules. We have aiready reportedg the 133

geometries of the 70, and in that work we gave the full name of each of them, following

the IUPAC narning convention.1° We refer to the molecules mostly by nurnber as found

in the previous paper? Figure 4.1 depicts one molecule out of each set of isomers. There

are 10 diazaborinines (1&27), 16 triazadiborinines (2W3), 4 diazaboroles (44-47), 6

triazaboroles (48-53), 4 tetrazaboroles (54-57), 2 triazadiboroles (58,59), 6 oxazaboroles

(6û-65), 12 oxadiazaboroles (66-77), 6 oxatriazaboroles (7&83), and 4 oxadiazadiboroles

(84-87).

We aIso constructed models of polarizability for 120 molecules including those

we have studied previously (again, following the numbering order found in those papas):

the 17 azaborinines which have 6 hydrogens (1-17),11 benzene and the 12 azines (88-100)

(cf. Figures 2.1 and Al. 1 in this thesis),12 pyrrole and the 9 azoles (101-110),13 and furan

and the 9 oxazoles (111-120)14 (cf. Figure 4.2 for 101-120). We show that either atorn or bond-type polarizability models can predict the polarizabilities of al1 the moIecules within a few percent.

4.2 Computational Methods

Polarizabilities were calculated by the finite-field method,ls using Hartree-Fock

(HF) and valence second-order Mflller-Plesset (MP2) perturbation the~ry'"'~field- dependent energies. These calculations were made with GAUSSIAN-90. l9 Finite-field HF polarizabilities are equivalent to coupled Hartree-Fock (CHF)ones," so we will refer to them as CHF. We use uncoupled Hartree-Fock (UCHF)polarizabilities20~21 to estimate the relative contribution of the n-electrons. The details of the techniques used were the H H. C-B' 1 \ ,,MC, /NXH O 60 N-N

Figure 4.1 One molecule out of each set of isomers for which polarizability cakulations are reported: 1,3-dihydro-l,2,3-diazaborinine (lS), 1,2,4,5-tetrahydro-l,2,3,4,5- triazadiborinine (28), 2,3-dihydro- 1H - 1,2,3-diazaborole (44), 2,4-dihydro- 1 H - 1,2,3,4- triazaborole (48), 2,s-dihydro- 1 H -tetrazaborole (54), 2,3,4,5-tetrahydro- 1 ~3,2,3,4,5- triazadiborole (58)' 2,3-dihydro- 1,z3-oxazaborole (60), 2,4-dihydro-l,2,3,4-oxaaiazaborole (66), 2,5-dihydro- l,2,3,4,5-oxatriazaborole (78), and 2,3,4,5-tetrahydro- 1,2,3,4,5- oxadiazadiborole (84). Al1 are at their MP2/6-31G(d) optimized geometry. Locant numbering is counter-clockwise, starting at 6 o'clock. The number of isomers is listed beside each molecule's picture. H H 'c- d I \ 2 N

112

N-N

Figure 4.2 One molecule out of each set of five-membered ring isomers for which polarizability calculations were previously reported: pyrrole (101), pentazole (102)' pyrazole (103), 1,2,3-triazole (105), 1,2,3,4-tetrazole (log), furan (11l), isoxazole (112), 1,2,3-oxadiazole (114)' 1,2,3,4-oxatriazole (119), and oxatetrazole (120). AI1 are at their MP2/6-3 lG(d) optimized geometry. Locant numbering is counter-clockwise, starting at 6 o'clock. The number of isomers is listed beside each molecule's picture. 136

same as in Our previous ~ork."~~~*'~The basis set we used, denoted C in ref 13, was

[5s3p2d/3s2p]. It previously yielded accurate polarizabilities for the azoles,13 oxazoles,14

mines, l2 and azaborinines."

Al1 calculations for the 70 molecules of this work were at the MP216-31G(d)

equilibrium geometries we reported earlier? In that work: we calculated harmonic

vibrational frequencies to verify that of the 70 molecules, 15 are nonplanar, while for 55

of them the planar conformation is a true minimum.

4.3 Polarizabilities

4.3.1 Results. The eigenvalues of the polarizability tensor a, I: a, 5 a, can be

combined to forrn various invariants. We use the mean polarizability:

and three measures of anisotropy:

A3a = [(&,al2 - (A~a)2]1/2 = -43 (a, - g ) 2

In this work we use atomic units for polarizabilities; one atomic unit of polarizability

3 = 4neoao = 1.648 78 x 1041 F m2 in SI units. 137

Table 4.1 lists the MP2/C polarizabilities for the 70 molecules (1û-89) and the angle $, between the principal axis of polarizability and the principal inertial axis 1,.

The 1,-axis for each of the 70 molecules is depicted in Figures 1-5 of ref 9. The polarizabilities cover a wide range: the average polarizability is between 38 a.u., for 80, and 72 a.u., for 30.

Table 4.1 also includes the UCHWC parallel and perpendicdar n-fractions f (XJ) = =(X,n)/a(X), X = 11, for the planar azaborinines. The 6n-electron contribution to the mean polarizability varies from 35% to 58% with an average of 50%. The n- fractions are Iargest for 40 and 47 and smallest for 86.

4.3.2 Trends. Consider the trends in the polarizabilities. The most polarizable five-membered rings, 45 and 47, are almost as polarizable as the least polarizable six- rnembered ring, 42. In every set of isomers, the least polarizable isomer is also the most stable. If a set of isomers inchdes two carbons, then the isomers with the C's adjacent are the least polarizable. If a set of isomers includes two borons, then the isomers with the B's adjacent tend to be the most polarizable.

Aza-substitution-i.e., N replacing CH-in every case reduces the polarizability, because the polarizability of a nitrogen atom is smaller than that of a carbon and hydrogen atom pair. Among the 87 BN-containing molecules, there are 62 cases where one rnolecule differs from another by a single aza-substitution. The average effect is a reduction by 4.6, 5.8, and 6.5 a.u. for al,a, and ai, respectively. The average reduction is a little larger than that we found previously for the azoles,13 oxazoles, and mines; for those 33 molecules, 88-120, aza-substitution reduces a*, a, and a,,by 3.2, 4.5, and 5.2 TABLE 4.1: MP2 polarizabilitiesa and Angles, UCHF n-f ractions, and ~eciprocal~ardnesses~ in C, Symmeéry Except as ~oted. - No. a1 a2 a3 a Ala A2a A3a 4% f (LTc) f(lA 1-1

f a In a.u. In degxees. In percent. With the C basis at MP2/6-31G(d) geometry. Czv. Cl. C2. C, nonplanax . 141

a.u., respectively. Figure 4.3, for al1 120 molecules, shows that there is a fairIy broad

distribution of differences.

Each oxazaborole is formally derived from an azaborole by putting an O in place

of an NH. For the 28 such substitutions, the azaborole is always more polarizable than - the corresponding oxazaborole(s), by an average of 4.3, 5.6, 6.3, and 2.1 a.u. for a,, a,

a,,and A,a, respectively. The reduction is similar to that we found previously14for the

oxazoles compared to the azoles, for which O replacing NH reduces oi by 5.1 a.u. Figure

4.4 shows the narrow distribution of differences for the 38 cases.

Figure 4.5 shows that the corresponding BNHz for CzHz replacement sometimes

increases and sometimes decreases the polarizability. There are 136 such substitutions.

Note that each diazaborinine is formally derived from pyridine. The average

polarizabilities of the diazaborinines differ from that of pyridine by 13.04 to 4.05 a.u.

There are 55 forma1 substitutions into a diazaborinine to make a triazadiborinine, for

which E changes by between +4.2 and -6.9 a.u. Thus, no simple rule holds. However,

one gross feature is noteworthy: the first BN-for-CC substitution, and the second one in

the six-membered rings, tend to increase the polarizability; the second substitution in the

five-membered rings and the third substitution in the six-membered rings (making the

triazatriborinines) generally decreases the polarizability.

For al1 the above substitutions, the arrangement of'atorns in the former and latter

molecule is very important. For example, the three azole-to-oxazole substitutions which change the polarizability the most are for species containing CXC,where X is either NH or O.

A given set of isomers without B are almost equaily polarizable. By contrast, the range in polarizability for members of a set of isomers containing B can be quite great.

Consider the influence of atom arrangement on polarizability in the latter case. The most stable isomers are also the least polarizable. The key factor leading to stability was foundgvl'to be the presence of XBHNH,where X is the base-ring heteroatom, either NH,

N, or O. B's or N's placed side-by-side increase the polarizability and decrease the relative stability. To further describe the extreme types, it is interesting to note the

Mulliken charges, which we calculated using the 6-31G(d) basis set. A most-stable isomer, such as 42, has sharply altemating (+, then -, with large absolute value) charges going around the ring. A relatively unstable isomer, such as 28, has low Mulliken charges that do not alternate in a regular pattern.

We calculated the dipole moments and polarizabilities of three nonplanar molecules at their MP2/6-3 lG(d) planar (nonminimum) stationary point. Table 4.2 shows

TABLE 4.2: Percent Difference, for Key Properties, for Planar Relative to Nonplanar Geometry, for 3 ~xazaboroles.*

CHF MP2 CHF MP2 CHI? MP2

a Nonplanar dipole moments p and angles 8, £rom ref 9. that the dipole moments are as much as 20% larger for the planar conformation, whereas

E, A,a, and A,a, are larger by just 0.5% to 3%.

4.3.3 Observations on Methodology. Figure 4.6 shows ratios of CHFNCHF

and MP2/CHF for out-of-plane (al)polarizabilities. Figure 4.7 shows these ratios for the

mean polarizability. The Figures include not only lû-87, but also the azaborinines, 1-17,

and non-B molecules, 88-120 as well. Across the series of molecules, q,a,, A,a, and

A2a show a pattern like that of a, with most UCHF values lower than CHF and most

MP2 values higher than CHF but with one or two dozen cases where MP2 is substantially lower than CHF.The only component with an exceptional pattern is a,.

Perpendicular polarizability i MP2 I I mi- CHF I 1.1 m i .r m -111 '=m m -Am * =m I i -i Li i I =i i m i - - =i i I m i i- -= I =< i ni' O " -'gm = i Qi O m 2 mm O - O i- m - I i i I :='&a - 1 I œ - - - -a i 0 O 0' mYzOo' gBD O 0 0 &'O O O *= m P œ O 0 Ci I I po 0 0 Op~ooSpam % pa i m 'a -CHF - UCHF

1 ml20 Molecule

Figure 4.6 Ratios of CEFNCHF and MP2lCHF for out-of-plane polarizabilities, in bais C. The molecules are numbered as in Figures 4.1-4.2 and text. 1 - Average polarizability

Molecule

Figure 4.7 Ratios of CHFKJCHF and MP2/CHF for the mean polarizability, in basis C. The molecules are numbered as in Figures 4.1-4.2 and text.

The presence of the borons lead to MP2 polarizabilities that are sometirnes higher and sometimes lower than the CHF ones. The average absolute differences for a, A,a, and Ap, are 7%, 27%, and 15%, respectively, between UCHF/C and CHF/C, and 7%,

1 1 %, and 12%,respectively, between MP2lC and CHFIC. In the extremes, MP2lCHF can be between 0.83 and 1.12 for 5. The UCHF polarizabilities tend to be lower than CHFy except for al.

The principal polarizability (q)is the component with the largest differences:

MP2 is lower than CHF for 28 of the 87 B-containing molecules, by a percent difference 148

of 15% or more for 7, 11, 12, 28, 40, 66, and 67. The two triazadiborinines, 28 and 40,

have the same halves, NHCHBH and BHNNH. Sirnilarly, the two oxadiazaboroles, 66 and

67, each contain OCHBH and NNH. The uncorrelated methods have difficulty treating

the polarizability of those sub-units.

4.3.4 Additive Atom Polarizability Models. We constructed simple additive

polarizability models based on the MP2/C polarizabilities of the 120 moIecules from this

and Our previous work. An additive atom model13 (AAM) of polarizability applicable to

al1 120 molecules is

in which niis the number of atoms of type i in a molecule, and bi can be regarded as the polarizability of an atom of type i in a planar heteroaromatic molecule.

Linear regression of Our MP2K polarizabilities for the 55 planar molecules from this work and the 49 planar azaborinines," azoles," oxa~oles,~~and azines12 leads to the biparameters shown in Table 4.3. The values of bH,bo bN,and bo are very close to those obtained previouslylljust for the 49 azaborinines, azoles, oxazoles, and mines. The bi are smaller than the corresponding free atorn polarizabilities22 indicating that bonding has lowered the polarizability. Table 4.3 shows that for the average absolute error 6, of

Model 5 is 3% but the maximum error is 6, = 14%. For A,a,6, = 6% and 6, = 40%.

One- and two-parameter improvements to Model 5 are obtained by addingb,~

(Model 5a) and bA~+bq/q(Model 5b), respectively, where A is the ring area and

/2 is an approximation to the molecular hardness. Table 4.1 lists the q=(e LUMO- EHOMO) C basis set reciprocal hardnesses. Table 4.3 shows that these terms improve the fits but TABLE 4.3: Parameters and Errors of Additive Atom Models of Polarizability

TABLE 4.4: Parametere and Erroxs of Additive Connection Models of Polarizability

%C ncc n BN n~~ n~~

~NN

"CO n~~ nrl n~ 6a(%) 150 destroy the physical interpretation of the atomic terms. If A", an approximate rnolecular volume, is used in place of A for this set of molecules, then 6, is about O. 1% larger.

Interestingly, reducing the AAM can still give a reasonably good fit. Note that the coefficients of the H- and O-terms are rather small. In the a,AAM, the coefficient of nH is only -0.09; it is almost negligible. For the average polarizability, a model with O,

N, B, and C has 6, = 3.1% and the even more compact B, C, N model has 6, just 3.7%

(2.4% for ai).

4.3.5 Additive Connection Polarizability Models. The additive connections model13 (ACM) expresses the polarizability as a linear combination of nij-the number of connections (i.e., bonds without regard to bond order) between atoms of type i and j.

For Our 120 molecules there are 12 distinct ni,, only one more than for the previous set of 50 molecules we studied,ll the extra one being nBo. Again there are four stoichiometric constraints; thus the ACM for the 120 molecules has nine terms which we choose as follows:

a = c, n,, + c, nBc + cg nCC + c4 nBN + cgnCN + c6 nNN

(6) + + '7 'CO '8 n~~ + '9 'BO

The parameters obtained by linear regression for Models 6, 6a (which includes a hardness terrn, q-'), and 6b (which includes and a term) for 104 planar molecules are listed in Table 4.4. The nine-parameter Model 6 is not as efficient as the seven- parameter Model 5b for the rnean nor for the anisotropic polarizability. The 1 1-parameter

Model 6b is more accurate, with errors of 6, = 1.4% and 6, = 6.9% for a and 6, = 4.4% and 6, = 22% for A,a. 15 1

The ACM can be expanded by distinguishing between nitrogens bonded to H and those not bonded to H. The extended ACM (EACM) model in that case must have 14 parameters. The EACM for E gives 6, = 1.6% but 6, is still about 7%; for Ap,6, = 4% and 6, = 16%. Further extensions to the EACM using area or hardness terms did not provide substantial improvement. The same procedure does not irnprove on the AAM model because of linear dependencies.

A natural progression after atom- and bond-polarizabilities is to include 3-body terms. The patterns NNN and NCB tend to be found in more-polarizable isomers, while

XBHNH (see above) is found in less-polarizable rings. (For the azaborinines, 1-3 and

15-17, we used the number of BHNH's, the nearest equivalent index.) XBHNH provided slight irnprovement to the additive atom model 5b, reducing 6, by O. 1%. The other 3-body terms we tried were not helpful.

The errors of the various models increase noticeably if the nonplanar molecules,

1,2,3,4,5,6-triazatriborinine(15), 8 azaboroles, and 7 oxazaboroles, are included in the data set. For Models 5 and 6, 6, increases by 0.2% and 0.1% for 6,and by 0.8% and

1.1 % for A p,respectively .

4.4 Discussion

Apart from the additive atom and additive connections models (AAM and ACM), one can also consider a mode1 using only HOMO-LUMO- or Area-based properties. The relationship between polarizability and hardness has often been st~died.~~**~Note, however, that the model a = constant + bq/q (7) gives large errors, with 6, = 15% for for the set of 104 molecules and = 14% for al1

120 molecules. Replacing 1/77 with q in Mode1 7 still gives about the same average absolute error. Instead, either n, or A reduces S, to about 8%.

Consider the relative merits of the AAM and ACM. The AAM is simple, requiring only 5 parameters, and yields values for al1 atom types, whereas the ACM must have 9 parameters, and does not give values for the polarizability of 3 other connected pairs.

However, if the goal is to obtain the most accurate model possible, then the more complicated ACM is preferable, particularly in combination with the extra q- and A- dependent terms. Including a distinction between pyridinic and pyrrolic N gives an even more precise model.

The dominant phenomena at play in the polarizabilities of the 120 molecules are: i. N substituted for CH reduces the polarizability; ii. O substituted for NH reduces the polarizability; iii. inclusion of B causes scatter of the polarizabilities of isomers. The last factor causes the simple additive models of polarizability to be less accurate for azaboracycles than for the other heterocycles we have studied. We are continuing Our efforts to develop better models of polarizability using this database of accurate polarizabilities of 120 molecules. 4.5 References and Notes

Miller, T. M.; Bederson, B. Adv. At. Mol. Phys. 1977, 13, 1.

Miller, T. M.; Bederson, B. Adv. At. Mol. Phys. 1988, 25, 37.

Lobaugh, J.; Voth, G. A. J. Chem. Phys. 1997, 106, 2400.

Miller, K. J. J. Am. Chem. Soc. 1990, 112, 8533.

Stout, J. M.; Dykstra, C. E. J. Phys. Chem. A. 1998, 102, 1576.

Bader, R. W. F.; Keith, T. A.; Gough, K. M.; Laidig, K. E. Mol. Phys. 1992, 75,

1167.

Morris, J. H. Boron in ring systems. In Comprehensive Organometallic

Chemistry; Wilkinson, G.; Stone, F. G. A.; Abel, E. W., Eds.; Pergamon:

Oxford, 1982; Vol. 1, p. 31 1.

Housecroft, C. E. Compounds with three- or four-coordinate boron, emphasizing

cyclic systems. In Comprehensive Organornetallic Chemistry II; Abel, E. W.;

Stone, F. G. A.; Wilkinson, G., Eds.; Elsevier: Oxford, 1995; Vol. 1, p. 129.

Doerksen, IR. J.; Thakkar, A. J. Znorg. Chern. 1998, submitted.

Powell, W. H. Pure Appl. Chem. 1983, 55, 409.

Doerksen, R. J.; Thakkar, A. J. J. Phys. Chem. A 1998, 102,4679.

Doerksen, R. J.; Thakkar, A. J. Znt. J. Quantum Chem. 1996, 60, 42 1. In Table

II, some of the entries for 1,Zdiazine are incorrect. The correct values are: a,

= 67.78, a = 58.99, ~,a= 31.77, Ap= 31.92 and ~p = 3.12.

El-Bakali Kassimi, N.; Doerksen, R. J.; Thakkar, A. J. J. Phys. Chem. 1995, 99,

12790. 154

El-Bakali Kassimi, N.; Doerksen, R. J.; Thakkar, A. J. J. Phys. Chem. 1996,

100, 8752.

Cohen, H. D.; Roothaan, C. C. J. J. Chem. Phys. 1965,43, S34.

Bartlett, R. J. Annu. Rev. Phys. Chem. 1981, 32, 359.

Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular

Orbital Theory; Wiley: New York, 1986.

Dykstra, C. E. Ab Znitio Calculation of the Structures and Properties of

Molecules; Elsevier: Amsterdam, 198 8.

Frisch, M. J.; Head-Gordon, M.; Tnicks, G. W.; Foresman, J. B.; Schlegel, H.

B.; Raghavachari, K.; Robb, M. A.; BinMey, J. S.; Gonzalez, C.; Defrees, D. J.;

Fox, D. J.; Whiteside, R. A.; Seeger, R.; Melius, C. F.; Baker, J.; Martin, R. L.;

Kahn, L. R.; Stewart, J. J. P.; Topiol, S.; Pople, J. A. Gaussian 90, Revision J;

Gaussian, Inc.: Pittsburgh, PA, 1990.

Dalgarno, A. Adv. Phys. 1962, 11, 281.

Thakkar, A. J.; Doerksen, R. J. UCHF, unpublished.

Das, A. K.; Thakkar, A. J. J. Phys. B 1998, 31, 2215.

Roy, R. K.; Chandra, A. K.; Pal, S. THEOCHEM 1995, 331, 261.

Simonmanso, Y.; Fuentedba, P. J. Phys. Chem. A 1998, 102, 2029. Chapter 5. Polarizability- and Bond-Order-Based Quantitative Aromaticity Indexes Applied to Common Heterocycles: Azines, Azoles, Oxazoles, and Thiazoles

Robert J. Doerksen, Noureddin El-Bakali Kassimi, Valerie J. Steeves, and Ajit J. Thakkar 5.1 Introduction

Aromaticity is a fundamental and useful concept in cherni~try.~~There bas been much recent work on quantifying ar~maticity~~in order to be able to determine the relative aromaticity of any particular molecules, as well as to gain insight into the fundamental meaning of the term. However, aromaticity is to some extent ill-defined,24 which naturally leads to difficulties when attempts are made to quantify it. To date, many quantitative aromaticity indexes have been used, but they often disagree with each other in their prediction of the relative aromaticities of the molecules considered. Nevertheless, one can retain hope that it will be possible to quantify the concept of aromaticity. The indexes thus far devised have been able to provide some insight into various practical chemical problems such as the planarity of cyclopalladated nngsS or the choice of suitable heterocyclic bridges for tuning the hyperpolarizability of donor-acceptor chrorn~~hores.~

Various criteria of aromaticity have been proposed, and around these have developed corresponding quantitative indexes. The main criteria discussed have been categorized as energetic, geometric, and rnagr~etic.~~-"'~There is some evidence both

and against12i13the idea that the different criteria describe different types of aromaticity.

This work focusses on the aromaticity of single ring heterocycles. Recently, we have calculated accurate geometries and polarizabilities for 42 azines,'' azoles,16 oxa~oles,'~and thiazoles.18 We use the data to generate and compare aromaticity scales based on bond order, bond angle, hardness, and polarizability. Figure 5.1 depicts the molecules in our study. Benzene 12 Azines P

10 Azoles 10 Oxazoles 9 Thiazoles Q P

Figure 5.1 The molecules in this chapter are: Benzene; pyrrole; furan; thiophene; pyridine (shown to scale at their best optimized geometry), and the other molecules formally derived from these by aza-substitution: the 11 other azines; the 9 other azoles; the 9 other oxazoles; and 8 other thiazoles.

Geometric indexe^^^"^'^-^^ have been widely used as indicators of aromaticity. We examine several bond-order-based indexes of aromaticity to see if they predict the same results, and to find if the indexes Vary depending on the type of geometry and of bond order used in their detemination. ~ozharskii~suggested that a measure of the variance of ring angles might be helpful in describing aromaticity, especially for molecules with some symmetry. We make a first attempt to implement this type of aromaticity index.

In recent new work and surnmaries on quantifying ar~maticity,'~ polarizability criteria have been neglected. This is partly because little accurate polarizability data had 158

previously been available for cyclic moIecules (a reason specifically given in ref 9). Apart

from Our own ~ork,'~*"*~~there have been only a few other attempts to relate

polarizability to aromati~it~.""' In this work, we consider the polarizabilities of the 42

molecules and compare several polarizability-based indexes of arornaticity with each other

and with geometric indexes. We tried to be consistent in our calculation rnethod~'~-~*so

that our polarizabilities and geometries would be approximately of the same accuracy for

al1 molecules considered.

Previous work has found a con-elation between hardness and aromati~it~~~*~*or

between hardness and polarizability (cf., e-g., ref 36), so we also tested HOMO-LUMO

gap properties as aromaticity indexes.

Geometric aromaticity scales measure equalization of bonds, which should be a

measure of electron delocalization (note that this is delocalization as it is found in the ground state, equilibrium ge~metry);~polarizability scales measure delocalization by noting how easily the electrons can be polarized in a perturbing electric field, i.e., how free they are to move, when called upon to do so. The two cannot be assumed to measure the sarne thing.

When choosing which aromaticity index is the best, it is helpful to have some idea of what is the correct ordering (and relative spacing) of certain key molecules. There are plenty of questions which have not been completely resolved, even just for the molecules we consider. The main important ones we will discuss below are:

1. Does aza-substitution always reduce aromaticity or sometimes increase it? 159

2. What is the relative ordering of pyrrole, thiophene, furan (and hence, their respective groups, the azoles, thiazoles and oxazoles)?

3. What is the relative aromaticity of five-membered rings versus six-rnembered rings?

5.2 Methods

5.2.1 Computations. We have previously published15-17 or will publish el~ewhere'~the ab initio geometries used for al1 the molecules in this study. The geometries of the 12 azines and benzene,Is 10 azoles,16 10 oxa~oles,'~and 9 thiazoles18 were al1 calculated using full second-order Moller-Plesset (MP2) perturbation the~ry,)~~~~ using a split-valence plus polarization basis set (6-31G(d) for all, except 6-31G(d7p)for the thia~oles~~~~~).

We computed self-consistent-field (SCF)bond orders, PJSCF, using the statistical interpretation of ~a~er:'which has previously been applied in many cases.42The bais set consisted of a double-c ~ubstrate~~with [ldlp] polarization functi~ns;~for S, the corresponding references are refs 45 and 46. For cornparison, we dso computed empirical

Gordy bond orders:' NO, based on Our best calculated geometries, given by:

where R is a bond length and with parameters a and b different for each unique atom pair, X and Y, in an X-Y bond. We use Gordy's a's and 6'~~~which were supplemented by ~ird?~ 160

The MP2 (frozen-core) finite-field static dipole polarizabilities4849we analyze have been obtained previously by us.15-18For the a~oles,'~oxa~oles,'~ and azines,15 the basis set was double-< plus diffuse functions of s, p, and d symmetry plus poiarization

[5s3p2d/4s2p] (basis C of ref 17). For the thiazoles, the bais set for H was the same, a slightly different basis set of the same size was used for C and and the functions for S were 6s5p2d18*50(basis Cl, detailed in ref 18). In order to get a measure of the a- eiectron part of the polarizability, we use uncoupled Hartree-Fock (UCHF)'~ with the C basis (C 1 for thiazoles), previously calculated for azoles,16 oxazoles, l7 and azines,I5 and freshly calculatedS2 for thiazoles.

5.2.2 Bond-Order-Based Aromaticity Indexes. Early geometry-based aromaticity indexes were functions of bond lengths.lg For heterocycles, bond Iength will

Vary with atom type, so that is controlled for by using bond orders derived frorn the bond lengths. Several bond-order-based aromaticity indexes have been proposed.7*20*22-28We present results here for four such indexes which we calculated using both the Gordy and

SCF bond orders, al1 at our MP2 geometries. The indexes aII follow the rule that a lower value means the molecule is less aromatic.

The simplest, the Ring Current index (RCI) by ~ug,~~is given by the lowest bond order of al1 ring bonds, which is supposedly a measure of the magnitude of a ring

~urrent.~~The other three indexes al1 include some measure of the differences between the n ring bond orders, Ni, with increased equalization of bond orders irnplying increased aromaticity. Bird's IA uses a measure of variance V,, given by, which includes differences from the mean bond order, i.The index was put on a convenient O to 100 scale:

where vKrefers to the variance in the corresponding Kekule structure.23~ater,~~ the index was generalized so that it applies to both six- and five-membered rings by setting 1, =

I, = 1.235 x I,, where 1.235 is the ratio of the Hückel delocalization energy of the cyclopentadienyl anion to that of benzene." The Pozharskii index7 is merely an average of al1 possible differences of bond orders:

where the dividing prefix is the number of such combinations. It is made into a percentage index called POZPC = 100 x (1 - POZ/0.49), since 1 and 0.49 are the PO2 values for benzene and cyclopentadiene, respectively? We put the bond-order version of

Krygowski's harmonic oscillator mode1 of aromaticity (HOMA), called HO MA^,^' again ont0 a O to 100 scale, so that it is written:

HOMA, = 100 - where NB is the bond order for benzene given by a particular method. (In the original work, n bond orders were used, with NB = 213.~') 162

A major difference between the indexes is that the bond-order-difference factor is raised to the power of 2 in HOMAp, whereas only to a power of 1 for POZPC and 1,.

Also note that 1, and HOMA, use a different reference bond order: 1, uses the mean of bond orders of that particular molecule, while HOM% uses the bond order of the archetypal aromatic molecule, benzene.

Krygowski has not recently used the bond-order version of his index, instead having developed a new versionx of his bond-length-based HO MA^' (which he now calls just plain "HOMA"), including coverage of heteroat~ms~~using the empirical

Pauling bond n~rnbers.'~The method requires use of "typical" single and double bond lengths for each pair of atom-types, al1 given by Krygowski and ~~raiiski~~except for

N-S and N-S, which we chose to be 174 pm and 153 pm, respectively (cf. ref 54).

Heterobonds are transformed into CC-equivalent bond lengths, ri; r,, is the average of the ri's for a molecule and r,, is the ideal (benzene) bond ~ength.'~Krygowski and

~~rafiski~~also suggested dividing HOMA to obtain two sub-indexes which (following

Katritzky et they said measure different (orthogonal) types of aromaticity: EN for energetic, based on how the average bond length of a molecule compares to r,,,

(+ for r,, > r,, and - for the opposite) and GE0 for geometric, based on deviations from the average, n 163

where a is a scale factor to zero the index for benzene's Kekule structure.28 Then

HOMA, = 100 - EN - GEO. We calculated these indexes using Our MP2 geometries.

5.2.3 Angle-Based Aromaticity Indexes. A weakness often n~ted~*~*"in the bond-order-alternation-based indexes is that D,, molecules such as s-triazine are ranked unduly high (often they have the same index value as benzene), a ranking attributable to symmetry rather than to aromaticity. In such molecules, the ring bond lengths are al1 equal, but there are two different angles. ~ozharskii' suggested two ways around this problem: use a measure of aiternation of either charge or angle. Minkin et aL2 did not mention any such index having been implemented. So, we tried several angle-altemation indexes.

Indexes based on angle alternation require two parts: a measure of variation and a way to set the extreme values (i.e., relative scaling). Our indexes' relative scaling is based on giving benzene an aromaticity rating of 100 and a six-membered ring with alternating 110" and 130" angles, 0. The latter is only approximately the maximum distortion that is found in Our set of molecules, so that a few of them have index values that are negative. An angle index based on consecutive angle aiternation [similar to the bond alternation coefficient A BAC)^^] is:

with en+,= el, where 20 is the approximate maximum variance between any two angles.

A ~ozharskii-t~~e~index for angles is: where the factor 15/180 is included to get the above-mentioned APOZPC = O condition.

A HOMA-I~~~*'index is:

1 n ADIFFPC = 1O0 1 - C [Bavg- 8'1 1 Because of how they are designed, the indexes will not give the same results for

any given five-membered rings. For a ring with consecutive angles (in degrees) 108, 118,

98, 118, 98, APOZPC gives 0, but AVGNEXT gives 20 and ADIFFPC, 33.

5.2.4 Polarizability-Based Indexes. Previous ~ork~~*~~~~~~has suggested

several different polarizability-based aromaticity indexes: the in-plane a,,out-of-plane

a*, or anisotropic (in-plane minus out-of-plane) polarizability Aa, each either using the total (o+ n) or the x-polarizability. We use our MF2 total polarizabilities, whereas the rc-polarizabilities in-plane (6,)or out-of-plane (a?) are calculated as:

X %KHF TOTAL an = -TOTAL aMp2 au,,, and the n-polarizability anisotropy (AaR)is the difference between the two. We not only considered al1 these, but also tested the effect of including various normalization factors for the polarizabilities: the number of ring atoms, number of electrons, or the area enclosed by the ring atoms (calculated from our best geometries). (Sometimes the number 165 of n-electrons has also been considered, but al1 the molecules in this study have six n- electrons.)

5.2.5 HOMO-LUMO-Energy-Based Indexes. Finally, chemical hardness has often been suggested to be inversely correlated with either polarizability36 or arornati~it~."~*~~We determined the hardness using q = (E,,,, - EHoM0)/2 and the chemical potential using x = (ELUMO+ EHOM0)/2.Since al1 the ~'sare negative, for convenience, in the rest of the paper we refer to the absolute value of the electronegativity as X. The energies were taken from the SCF wave function generated with the polarizability basis sets, C and Cl. Al1 polarizabilities and energies are in atomic units.

5.3 Results

The Gordy and SCF bond orders are listed in Tables S. 1 (six-membered rings) and

5.2 (five-membered rings). The SCF and Gordy bond-order-based aromaticity indexes IA,

POZPC, and HOMA,,, and our calculateù HOMAd and EN are in Table 5.3, along with the three angle-based indexes. The polarizability indexes anl,Aa', a,,and Aa are listed in Table 5.4. SCF/C or Cl hardnesses and chemical potentials are also tabulated in Table

5.4. Table 5.5 contains the UCHFKl total and z- average, perpendicular, and parallel, poiarizabilities for the thiazoles.

5.4 Discussion

5.4.1 Geometry. To be able to make an accurate cornparison of aromaticities for a group of molecules, the geometries of al1 the molecules must be equally accurate. Large cn rl m O Ln. w. ww. . rl rl rl rl

mm 4 m CO. LI. Cow. . rl rt rl rl

m m m m COw. . 'Q. w. rl rl rl rl

a3 Qi 0 0 Vl. Ln. mw. . rl rl drl

Cr) rl rn -4' Lnw. Ln. Ln. rl rl rl rJ

m O 0 Q' ln. Ln. VI. Ln . FI rl rl rl

0 01 w m Ln. ln. Ln. d . rl rl rl 84

m m rl cn m. m. m. dc rl rl rlrl

m w u m m. Ln. m. -a'. rl rl 4 3-4

O m Co m Ln. -3' . (V. d . rlrl rl rl

Ln l- m ln m. m. -dm. . rl rl rl rl

Ln ln Ca m m. N. (V. m. rlrl rl rl

-rl .rld : s ii -rl k I.( JJ ti al al -4 -4d kk I? ?? Q'. Ln. m a' m n . . -5 CV N N N . . 5 rl rl rl TABLE 5.2: Gordy and SCF/C Bond Orders for the Azolea, Oxazoles, and ~hiazolea.'

Molecule 1-2 2-3 3-4 4-5 5-1 1-2 2-3 3-4 4-5 5-1 Pyrrole Pyrazole Irnidazole 1,2,3-Triazolea 1,2,4-~riazole~ 1,2,5-Triazolec 1,3,4-~riazole~ 1,2,3,4-Tetrazolee 1,2,3,5-~etrazole~ Pentazole Furan 1,2-Oxazoleg 1, 3-Oxazoleh 1,2,3-Oxadiazole 1,2,4-Oxadiazole 1,2,5-Oxadiazolei 1,3,4-Oxadiazole 1,2,3,4-Oxatriazole 1,2,3,5-Oxatriazole Oxatetrazole Thiophene t Bond labels refer to locant numbers in the ring (N-H, O, or S numbered first, (other) N1s nurnbered next), al1 at the MP2 geornetries. Common names are the following: a. 1H-1,2,3-triazole; b. 1H-1,2,4-triazole; c. 2H-1,2,3-triazole; b. 4H-1,2,4-triazole; e. 1H-tetrazole; f. 2H-tetrazole; g. isoxazole; h. oxazole; i. furazan; j. isothiazole; k. thiazole. TABLE 5.3: Bond-Order-Based Aromaticity Scales POZPC, 1,, with Gordy Bond Orders and with SCF Bond Orders, and Scales Based on Bond Leagths-HOM% and ADIFFPC, and AVGNEXT-for Al1 42 Molecules, Al1 at the MP2 Geometriee.

With Gordy bond With SCF bond Wi th bond With bond angles orders: orders : lengths Molecule POZPC IA HOMAp POZPC IA HOMAp HOMAd EN APOZ ADIFPC AVGNEX Benzene Pyridine 1,2-Diazine 1,3-Diazine 1,4-Diazine 1,2,3-Triazine 1,2,4-Triazine 1,3,5-Triazine 1,2,3,4-Tetrazine 1,2,3,5-Tetrazine 1,2,4,5-Tetrazine Pentazine Hexazine Pyrrole Pyrazole Imidazole 1,2,3-Triazole 1,2,4-Triazole 1,2,5-Triazole 1,3,4-Triazole 1,2,3,4-Tetrazole 1,2,3,5-Tetrazole Pentazole Furan 1,2-Oxazole 1,3-Oxazole 1,2,3-Oxadiazole 1,2,4-Oxadiazole 1,2,5-Oxadiazole 1,3,4-Oxadiazole 1,2,3,4-Oxatriazole 1,2,3,5-Oxatriazole Oxatetrazole Thiophene 1,2-Thiazole 1,3-Thiazole 1,2,3-Thiadiazole 1,2,4-Thiadiazole 1,2,5-Thiadiazole 1,3,4-Thiadiazole 1,2,3,4-Thiatriazole TABLE 5.4: Aromaticity Scales Based on Polarizability-a,, Aa, aXIfAaZ. Hardnesses, q, and Absolute Values of the Chemlcal Potential, X, for Al1 42 Molecules, Al1 with the C (except Cl for Thiazoles) Basis at the MP2 Geametries. MP2 Scaled SCF Molecule a Aa an' AaX "l IX f Benzene Pyridine l,2-Diazine 1,3-Diazine 1,4-Diazine 1,2,3-Triazine 1,2,4-Triazine 1,3,5-Triazine 1,2,3,4-Tetrazine 1,2,3,5-Tetrazine 1,2,4,5-Tetrazine Pentazine Hexazine Pyrrole Pyrazole Imidazole 1,2,3-Triazole 1,2,4-Triazole 1,2,5-Triazole 1,3,4-Triazole 1,2,3,4-Tetrazole 1,2,3,5-Tetrazole Pentazole Furan 1,2-Oxazole 1,3-Oxazole 1,2,3-0xadiazole 1,2,4-Oxadiazole 1,2,5-0xadiazole 1,3,4-Oxadiazole 1,2,3,4-Oxatriazole 1,2,3,5-Oxatriazole 40.85 16.57 19.06 6.21 0.251 0.205 Oxatetrazole 36.01 14.16 16.05 5.24 0.274 0.243 Thiophene 73.27 28.53 37.54 8.84 0.187 0.187 1,2-Thiazole 67.79 27.68 33.81 9.07 0.203 0.203 1,3-Thiazole 67.23 26.88 32.84 7.93 0.195 0.195 1,2,3-Thiadiazole 63.36 26.59 30.94 9.03 0.211 0.211 1,2,4-Thiadiazole 61.57 25.34 28.82 7.49 0.216 0.216 1,2,5-Thiadiazole 62.12 26.16 31.43 10.05 0.210 0.210 1,3,4-Thiadiazole 61.67 24.94 28.41 6.68 0.217 0.217 1,2,3,4-Thiatriazole 57.68 24.25 26.41 7.43 0.241 0.241 1,2,3,5-Thiatriazole 57.40 24.17 28.08 9.13 0.219 0.219 TABLE 5.5: Total and n- Uncoupled Hartree-Fock (UCHF) Polarizabilities of the Thiazoles with the Cl Ba~is.~ Molecule - a a, E* oi"l uni Thiophene 64.08 47.49 72.37 34.88 30.47 37.08 1,2-Thiazole 59.23 42.77 67.45 31.23 26.39 33.65 1,3-Thiazole 58.75 43.32 66.46 30.56 26.74 32.47 1,2,3-Thiadiazole 57.15 39.72 65.86 29.33 23.66 32.16 1,2,4-Thiadiazole 53.39 38.71 60.73 26.55 22.79 28.43 1,2,5-Thiadiazole 55.42 38.74 63.76 29.19 23.04 32.26 1,3,4-Thiadiazole 54.39 39.46 61.85 26.78 23.35 28.49 1,2,3,4-Thiatriazole 52.03 35.88 60.10 25.14 20.37 27.52 1,2,3,5-Thiatriazole 52.15 35.61 60.42 26.48 20.31 29.56 a At their MP2/6-31G(d,p) optirnized geometries .la differences in the bond-order-based indexes for a particular molecule can be caused by small geometrical changes. Bird I6 and I5 and Pozharskii indexes (al1 using Gordy bond orders) for 38 of the molecules at AM1 geometries were published previously by

Katritzky et We converted their Bird index values to I,~~and plotted them (as well as Bird's 1992 I~~~from experimental geometries) against our Gordy-bond-order 1, values at MP2 geometries. Figure 5.2 shows that the correlation is not good, and thus illustrates the large effect geometry can have on the value of the aromaticity index (not necessariIy on the aromaticity itself'). Pozharskii index resu~ts~~~~show a similar lack of correlation.

Katritzky et ale's 1996" contribution studied the effect of geometry on the resultant value of the IA or POZPC indexes. They term it a study of the effects of "the molecular envir~nment,"~~but the geometric indexes of course just depend on the geometry, not on the source of the geometry. It is already well known that gas phase, X- ray, and in-solvent geometries wilt be different. ~he~"discuss the geometry effects as I Expt. (Bird) +r AM1 (Katrittky)

20-! I 1 I I I I 1 1 I 20 30 40 50 60 70 80 90 100 110 1 At MP2 Geometries (this work)

Figure 5.2 The Bird index 1, for al1 42 molecules calculated using Gordy bond orders, at selected e~~erirnental~~and at AM^^^^^ geometries cornpared with at the MP2 geometries.

a direct indication of arornaticity change, or of how change in dipole moment and in the

indexes could affect the aromaticity. What they do not comment on much is the change

due to calculational method. They present AM 1 and HF/6-3 1G(d) geometries (including

a reaction field to mode1 solvent effects). So, for imidazole, for AMI, the gas

phase/dioxane/water progression for the Bird index 1, is 84, 85, 88; for HF, the result is

59, 60, 61 .55 But regardless of the smail increase in aromaticity in both cases, is not the most notable feature the large difference between AM1 and HF, for any phase? For pyrrole, the gap due to geornetry is a different size; and HF shows no medium effect, 175

whereas AM1 does. Katritzky et do not comment on this at dl. The main points,

then, are: a. medium affects aromaticity; b. calculational technique affects the predicted

values of aromaticity indexes (through the geometry) even more.

Katritzky et aLSs also discuss the large difference in bond-order-based aromaticity

indexes for 1,2,4-triazole, depending on whether its microwave (MW)~~or neutron

diffractions7 geometry is used. We have noted previously'6 that the MW geometry of

1,2,4-triazo1es6 is less reliable than our MPU6-3 lG(d) one. l6 Our Gordy-bond-order 1,

and POZPC values at the MP2/6-31G(d) geometry, 107 and 79, are fairly close to the

values Katritzky et UL" determined for the neutron diffraction geometry, 102 and 73. but

far from the values at the MW geometry, 77 and 41, respectively.

Given that geometry can affect the resulting bond-order-based aromaticity indexes,

it is important to note that some other workers have used a different approach than ours to geometry deterrnination. Consider the effects of geometry selection in the recent work of Krygowski et al. (cf., e.g., refs 25,26, and 28) who regularly use X-ray structures from the Cambridge Structural Database. They: a. place X-ray (or solid state) structures as the ideal; b. are often concerned with rings containing substituents (either in themselves or as a mode1 of the base ring); c. include structures obtained from rnany different experiments and labs (and even with different levels of assigned precision); d. sometimes use the average of several crystal structures with the sarne base unit but different substituents. Points a and b might affect cornparison of their results to others', while points c and d contribute to a lack of consistency in their results. 176

Others such as Bird (cf., e.g., refs 23, 29) just look for the best available geometry in the Iiterature for each molecule, even mixing gas phase experirnent, X-ray, molecules with substituents, and calculated geometries, without much analysis of the possible ramifications.

Similar to our approach, Schleyer et al. try to use consistent caIculated geometries for their magnetic-property-based aromaticity indexes; depending on the size of the molecule involved they have used either MP2 or B3LYP,usually with the 6-3 lG(d) basis set (cf., e-g., refs 3, 12, and 58).

Our geometries emulate the gas phase. They are calculated consistently for the whole set of molecules. One must be careful in comparing indexes for a molecule, when different geometries have been used for the index determinations. A small difference in geometry can change the value of the indexes for that molecule considerably.

5.4.2 Bond Orders. Figure 5.3 shows the percent difference between the Gordy and SCF bond orders, both at our best calculated geometries. The percent difference between Gordy and SCF bond orders gets larger, the shorter the bond length. The Gordy orders are most often considerably larger than the SCF, particularly for the six-membered rings. The sum of al1 Gordy bond orders averages 10.5 for six-membered rings and 7.7 for five-membered rings, compared to 8.6 and 6.9, respectively, for SCF bond orders. The

C-C bond orders are particularly different, with the Gordy values higher by 1540%.

The differences can be partially explained by remembering Gordy's source for the pararneters.47 For C-C bonds, first, he used reference data from bonds in hydrocarbons, not from heter0c~cles.4' Secondly, the reference bond lengths used were selected from -10 Bonds

Figure 5.3 The percent difference between the Gordy and SCF bond orders, for al1 ring bonds in the 42 molecules, both at the MP2 geometries. various pre-1940 e~~eriments.~~Thirdly, the bond orders were taken from Mulliken et aL7s 1941 work?' which included the effects of hyperconjugation (e.g., contribution of the hydrogens into the C-C bond order of benzene) to increase the bond order. This gave

(arnong others) for C-C in ethane not 1, but 1.1 18, for ethene not 2, but 2.124, and for benzene, 1.77P7 The C-C Gordy bond orders could evidently be normalized differently

(as mentioned by Mulliken et aLSg)to agree more with the SCF bond orders.

For other bond types, ~ord~~'used two ways to assign the bond orders: whole numbers were assumed for "pure bonds" in standard molecules, e.g., for N-N bonds, he 178 assigned bond orders of 1 and 2 for hydrazine and azomethane, respectively, which would lead to a difference compared to the C-C curve; and he computed fractional bond orders with an empirical re~ation.~'For those bond types for which there were no experimental lengths available for the pure bonds, he used the Schomaker-Stevenson rule61 which gives a bond length as a function of covalent radii and electronegativities of the participating atoms, and hence in combination with the Gordy relation obtained a and b.

By contrat, consider again Our bond orders. The geometry of each molecule was carefully calculated; and the bond orders were calculated from the wave function of the individual molecule, at that geometry, in each case. Thus, our bond orders should be superior to the Gordy ones used previously.

5.4.3 Bond-Order-Based Indexes. Compare the bond-order-based indexes we obtained using SCF bond orders with those we obtained using Gordy bond orders. Let 1 or J represent any of the four indexes; and let a superscript indicate whether SCF or

Gordy (G) bond orders were used. Then there are four types of cornparisons to make: (a)

ISCFVS. 1';(b) I~~~VS. J~~~; (c) l? vs. F; and (d) ISCFVS. J~.Given the large differences we found between SCF and Gordy bond orders, plots of type (d) would be expected to and do give a very poor correlation. Type (a) plots illustrate the vast difference between the SCF and Gordy bond orders, because they look very scattered. The only index that gives a sornewhat close to linear type (a) graph is RCI. Types (b) and (c) are the relations which we expect should give a true illustration of the connection between the indexes.

Type (b) plots do show a definite correlation, as we will discuss shortly. But first, it is important to note that of the six possible type (c) plots, only POZPC~vs. I,~looks 179 similar to the corresponding SCF plot. The other five type (c) plots Vary from somewhat scattered to very scattered, in cornparison to the corresponding type (b) plots. A typical example is shown in Figure 5.4. A general trend can be observed, but overall there is much scatter. Looking at the molecule subgroups (e.g., azoles) leads to even lower correlation.

m azine 3E azole O oxazole x thiazole

O l I I I I I I 30 40 50 60 70 80 90 100 HOMAp with Gordy bond orders

Figure 5.4 Correlation between POZPC and HOM%, both calculated with Gordy bond orders, at the MP2 geometries.

Consider the correlation of type (b), that is, with SCF bond orders only. Examples are shown in Figures 5.5 and 5.6. Figure 5.5 shows a tight line being followed, but it is not a straight line. In fact, that Figure depicts a quadratic relationship between POZPC 100--- 100---

90- azine

i! i! 80- I azole Q) 70 rn oxazole m Y 3 MW thiazole # = w,,"" 5B 40- '"in*

30 MO 2 20- Mo O O O0 10- Cl 0

O 1 I I I I I 1 I 1 I 50 60 70 80 90 1O0 HOMAp with SCF bond orders

Figure 5.5 Correlation between POZPC and HOM%, both calculated with SCF bond orders, at the MP2 geometries. and HOM%, due to their respective dependence on the differences between pairs of bond orders. This message is lost in Figure 5.4. The three plots of type (b) which are vs.

HOM%, al1 show that quadratic hook. The three of RCI vs. another al1 have a fair degree of scatter, but a pattern is still discernible showing that RCI gives results similar to

POZPC. There is a special trend in al1 the type (b) plots that include I,, and that can be seen in Figure 5.6. It is obvious that there are two sections of the curve: the six- membered ring molecules are on a different straight line than the 5-rings. The equation

POZPC = -50 + 145.5 x 1, fits al1 the data of Figure 5.6 with a correlation coefficient O ! I I I I 1 1 30 40 50 60 70 80 90 11 Bird IA with SCF bond orders

Figure 5.6 Correlation between POZPC and I,, both calculated with SCF bond orders, at the MP2 geornetries.

R~ = 0.92, but does not do justice to the data. A better fit is obtained using two lines, one

for each ring size or by using a two-parameter equation POZPC = -29.6 + 103.5 x 1, +

24.6 x L6, where L6 is 1 if the molecule is a six-membered ring but O for a five-

membered ring. For the last equation, R~ = 0.995. The explanation of al1 this is that the

Bird index was ~nified~~(using = 1.235 x 1,) in such a manner as to cause such a large distinction between itself and the other bond-order-based indexes. Whether or not this distinction is good is debatable; but the point is that the difference has not previously been mentioned and thus it deserves to be stressed here. 182

Another gross feature is clear from looking at the difference between the Gordy

and SCF 1,. The Gordy IA ranks 5 azoles as the most aromatic of al1 the 42 molecules,

even above benzene. However, the SCF IA gives a more sensible ranking, with benzene

and pyridine higher than pyrrole and the azoles. This order generally holds true for the

other SCF indexes as well.

5.4.4 Bond-Length-Based HOMA, Index. Our HOMA, bond-length-based

aromaticity scales agree reasonably well with those of Krygowski and ~~raiiski.~'For

HOMAd, only for 3 of 20 molecules are Krygowski and Cyraiiski's values different from ours by more than about 10%: 1,2,4-triazine, thiophene, and furan, for which we find 96,

87, and 25 versus their 79, 65, and 3. Larger percent differences were found for EN and

GEO, in part because the numbers themselves are small. Al1 differences should be attributed to the different geometries used. For the molecules in this study, al1 EN values were positive (whereas, for example, Krygowski and Cyrahki's EN for s-triazine was negative). This is because Our MP2 geometries accurately emulate the gas-phase, for which r,, = 138.8 pm-their28 suggested value-is too short. Benzene is the most obvious example: its HOMA value ends up lower than that of pyridine.

We found that HOMA, and EN both correlate well with GEO. Correlation R~'S are 0.99 for HOMA vs. GEO, 0.92 for HOMA vs. EN, and 0.85 for EN vs. GEO.

Although this contradicts what Krygowski and Cyrafiski ~tated,~~it should be mentioned that this trend can also be observed in their data for monoheterocycles.2g The confusion arises because for most of the molecules, EN is very small, and indeed for Our molecuIes 183

a plot of O < EN c 0.3 vs. GE0 does look very scattered. The oxazoles in particular have

higher values of EN which correlate better with GEO.

Correlation plots of HOMAd, EN, and GE0 vs. bond-order-based indexes gave

several interesting results: from our data, the (inverse) correlation of either EN or of GE0

to 1, looks nearly the same (unlike what JSrygowski and Cyrafiski suggested2'). A

quadratic dependence can be observed, due to the power dependence of the HOMA-based

indexes. The bond-order-based indexes correlate a little better with the HOMAd scales if

Gordy bond orders rather than SCF ones are used. This is presumably because both

Pauling bond numbers and Gordy bond orders are empirical relations that depend on the

use of standard bond lengths.

HOMA, has the advantage over HOMAd that we cm use SCF bond orders instead

of being dependent on empirical relations to include the different types of heterobonds.

It might prove worthwhile in future work to divide the HOMA, index into two parts, EN

and GEO, as Krygowski and Cyrdski did for HOMAd.

5.4.5 Angle-Based Indexes. As expected from the similar bond-order-based

indexes, plots of either APOZPC or AVGNEXT vs. ADIFFPC show the same quadratic

dependence on ADIFFPC as was found between POZPC and HOM% (cf. Figure 5.5).

APOZPC and AVGNEXT give qualitatively sirnilar results, except that the actual numbers

for APOZPC are lower, such that it covers.evenly the range from 100 down to below

zero.

The angle-based indexes stil1 have a problem with syrnmetric molecules: Dsh moIecules such as hexazine al1 receive precisely the sarne index value as benzene, even 184 though their aromaticity should be different. Oxatetrazole ends up third from the top. This problem suggests that an angle-based index cannot be used as a stand-alone scale of aromaticity.

The angle-based indexes give the thiazoles a relatively low ranking, because the angle at second-row S is rather small compared to the average five-mernbered-ring angle.

ADIFFPC and APOZPC treat the thiazoles sirnilarly, whereas AVGNEXT ranks them relatively higher. That is because the small angle at S appears two and four times, respectively, in the other scales but only once in AVGNEXT.

By contrast, the bond-order-based indexes rank the oxazoles the lowest, primarily due to the low bond orders in the bonds with O. A graph of POZPC versus APOZPC divides the molecules into their four groups without overlapping: azines, azoles, and oxazoles are al1 ranked between 40 and 100 by APOZPC, but thiazoles are between 40 and 0; meanwhile, POZPC ranks azines the highest, oxazoles the lowest, and mixes the azoles and thiazoles in the middIe.

It is possible to combine angle-based aromaticity indexes with bond-order-based ones: the two obvious ways are by averaging or multiplying. We attempted this in various ways. Perhaps the best one to recommend is (POZPC + APOZPC)/2, since both indexes are of the same form. It is difficult to judge whether the resulting scale predicts the aromaticities more accurately or not, except for the obvious improvement for Djh molecules. Of course, a major flaw in any such scheme is that the angle-alternation and bond-order-alternation indexes treat al1 Doh molecules the same. 185

5.4.6 Polarizability-Based Indexes. Consider the predictions for aromaticity given by the indexes derived from polarizability. The MP2 polarizability anisotropy follows a step-wise aza-substitution-causes-decrease pattern. In fact, so do the in-plane and the out-of-plane polarizabilities. This pattern is so regular (with only minor differences between geometric isomers), that the only further description needed is to specify the relative polarizability of the four non-aza-substituted parents: benzene, thiophene, pyrrole, furan. We have discussed this previously for a~oles,'~and oxa~oles.'~For the n-polarizability, the same trend is followed for the out-of-plane and in-plane polarizability, except that for anl,s-triazine is lower than two tetrazines.

But the ordering is markedly different for the x-polarizability anisotropy. It has in every five-rnembered-ring group some molecufes higher and sorne Iower than the family-head molecule. For each set of geometric isomers, the AaR index predicts differences. One of the clearest is for 5-rings with three locant numbers: 1,2,5- isomers will be more arornatic than 1,3,4- isomers. In fact, for al1 the five-membered rings the

1,2-, 1,2,3-, 1,2,5- and 1,2,3,S- isomers have a Aan greater than that of the corresponding parent compound. That 1,S- is ranked higher than 1,3- says that putting the N adjacent to the parent heteroatorn is the better option. That 1,2,3,5-is higher than 1,2,3,4- says that bonding the heteroatom to two N's is better than to only one. In other words, the favorable situation is to have C's adjacent and away from the parent heteroatom. For the thiazoles, enhanced Aa'l for an isomer is correlated to enhanced in-plane x-polarizability.

For azines, the order is exactly that of aza-substitution with two exceptions: pyrazine

(D2h)is predicted to be more aromatic than pyridine, and s-triazine (IlJh)is predicted to 186 be less arornatic than the tetrazines or pentazine. This Aa'E index echoes the large

distinction found between isomers in the bond-order-based aromaticity indexes. See, e.g.,

Figure 5.7, which plots POZPC against AaK.

90- m I m X m 5 m .-E 80- a 70- V: I m m 9 Ma 6 60- M irc E ! 9 50- Azine r M I( z nY LI .-2 40- M%" Thiazole X l! l! 30- m 0 x C O 8 20- CI n ~3 O op 10 04- Oxazole l9

0- 1 1 1 1 I I 4 6 8 10 12 14 16 1 Pi-polarizability anisotropy (a.u.)

Figure 5.7 Correlation between POZPC, calculated with SCF bond orders, and Aa", calculated according to Eq 11, both at the MP2 geometries.

The other notable feature for the n-polarizability anisotropy is that the values for the 5-rings are much lower than those for the 6-rings. Compare, Say, pentazine and pentazole, 12.6 vs. 7.2: the difference is two N's replaced by one N-H. In this case, a'; is almost the same (13.6 vs. 12.9), but anlis very different (26.2 vs. 20.1). Compare benzene and pyrrole, 17.6 vs. 6.9, with two C-H groups replaced by one N-H: anLis just 187 a little different (28 vs. 26.4), but anl is very different (45.6 vs. 33.3). Thus the ir- electrons associated with pyrrolic N are less polarizable in the plane compared to other

.n-electrons associated with N or C in these molecules. The average difference between the types of five-membered rings (comparing pairs of molecules with identical locant

numbers, e.g., " 1,2,5-") is quite regular: 1.2 a.u. between thiazoles and azoles, and 1.5 a.u. between azoles and oxazoles.

It is profitable to note the differences between the average (6 = (2ai + (w,)/3) and the anisotropic polarizability. In general, the former should not be taken as an index of aromaticity because it is a property that is not limited to aromatic molecules. However, for the molecules in Our study the two scales agree in the following ways: in al1 cases, aza-substitution causes a decrease; and, there are only srnall differences among isomers.

There are several important disagreements: the relative ranking of the four groups of molecules differs (e.g., for a, thiophene is close to pyridine, whereas for Aa, thiophene is close to the tetrazines); the plot of à vs. Aa has a slightly different average slope for each molecule group; the plot for each group is somewhat curved, with each additional aza-substitution decreasing Aa by a larger arnount than it decreases a.

What about the physical meaning of a polarizability-based index? Aromaticity is related to the concepts of electron delocalization and the preference of a planar ring over a nonplanar ring or chain conformation, but the exact connections have not been fully understood. The common factor for al1 these molecules is the number of n-electrons. This in itself suggests that some measure of the n-polarizability may give the proper relative scaling. Bulgarevich et a2." had suggested the in-plane x-polarizability, an,(normalized 188 with respect to benzene) as an aromaticity index (to get a measure of the ring current).

However, the n-electrons will be polarizable out-of-plane, to some extent, too. The polarizability out-of-plane will not be a result of any ring current. Thus the part of the in- plane polarizability that is different from (an enhancement, over and above) the out-of- plane polarizability is the part to attribute to the electron delocalization and aromaticity.

Hence, Aan should be a good aromaticity index. Fowler and ~teiner~~considered this as a possible aromaticity index, but thought it was not good because of the strange result that for cyclopentadienyl (C,H<), Aan is very close to O.

It is interesting to note that some of the recent work on ring currents done in the context of rnagnetic properties has also tried to measure the separate roles of s- and n- electrons. An example is the work of Fleischer et a1.,6* who compared isomers of benzene to see which had the greatest n-contribution to the out-of-plane magnetic susceptibility.

Though the patterns in the n-polarizability anisotropy seem quite clear and interesting, it is important to remember that the estimate of oh separation cornes from the UCHF polarizabilities, which are not as accurate nor as diable as the MP2 polarizabilities.1s-17 The total anisotropic polarizability, Au, does not suffer from this weakness: it can be found from experiment or from calculations done at the most accurate levels.

5.4.7 Hardness. In previous work, Zhou and pad4 suggested, on the bais of

Hückel method HOMO-LUMO gaps, that hardness was correlated with aromaticity.

~ird~~also used hardness as an index of aromaticity. He criticized Zhou and ~arr's'~ theory-based hardnesses. Instead, he wanted an "experimentaI" source of hardnesses, so 189

he chose a formula from ~ornorowski~~relating hardness to molar refraction,

q = 19.6 x (RD)-'. However, in Komorowski's article63 it seems that the relation to

molar refraction is only supposed to be for ions and atoms (because they are

approxirnately spherical). Komorowski even says it works well for ions, but not very well

for atoms. No mention is made of rnolec~les.~'

Our SCF/C and Cl hardnesses are expected to be more accurate than those

previously published. The hardnesses of five- and of six-membered rings are intermingled,

with pyrrole the least hard. In general, ma-substitution increases the hardness. Plots of

our calculated hardnesses versus either Zhou and ~arr's~~or ~ird's~' show no overall

correlation. However, if we distinguish the individual sub-groups, such as azines, then

within each group there is rough correlation with Bird's values."

~ird~lquestioned whether Zhou and Parr's finding34 of a correlation between thei?4 hardnesses and aromaticity was accurate. But, Bird did find a good correlation between his own hardnesses and other indicators of ar~maticity.'~We could find'no good correlation between any HOMO-LUMO gap property and a bond-order-based index. The good correlations Bird found when comparing hardness with IA or resonance energy benefitted from the fact that Bird used a large selection of varied rnolec~les.~~If one looks only at his data for monocycles,35the correlations worsen considerably.

We plotted Aa and Aan against l/q, 1/~,and EHOMO(ELUMO is uncorrelated to either). The plots of Aa to the three are close to linear, whereas those of A$ are not.

Because Aa and Aa'F are quite different indexes, as discussed above, they should be expected not to behave in the sarne way. For al1 three plots, not one correlation line is 190

found, but four: each group of molecules has its own line, because of the effect of the

lead heteroatom. The graph of Aa vs. 1/~is sirnilar to that of Aa vs. llq for azoles and

oxazoles, but for the (lower) azines, the latter plot has a more randorn distribution; also,

the former overlaps the thiazoles and azines on almost the same line. The best linear

relationship is between Aa and 1/~,shown in Figure 5.8.

40 azines R I axoles m rn - oxazoles

thiazoles

B *O "'O O

IO! 1 1 I I 1 I I 4 4.5 5 5.5 6 6.5 7 7.5 8 l/absolute value of chemical potential

Figure 5.8 Conelation between MP2 Aa and the reciprocal of the absolute value of the chemical potential, both with the C (Cl for thiazoles) basis at the MP2 geornetries.

Roy et al." found the cube of the average polarizability to be proportional to Ilq, but that was for how the properties varied in a molecule as a bond was stretched. We found that cubing the polarizabilities (a,Aa) did improve the correlations with llq, again 191

treating each group of molecules separately. However, the correlations of a3 with I/x

were still better, and just marginally better than those of a with 1/~.

For the correlations in Our data of the average polarizability to the HOMO-LUMO

gap properties, the effects cm already be predicted by what was said above about the corrdation of 5 to Aa. Corresponding plots are rather similar, given the differences between a and Aa. Particularly for the mines, plots using Aa give the better correlations.

5.5 Guideposts for Accuracy of Aromaticity Indexes

There is much that is already known about the relative aromaticities of the molecules in this study. However, the main points of confusion are nevertheless fundamental. Any index claiming to be quantitative for the molecules we consider here must be able to predict the effect of aza-substitution, heteroatom type, and ring size on aromatici ty .

5.5.1 Aza-substitution. The effect of aza-substitution on arornaticity is tricky to determine. It is tempting to think that any aza-substitution should have the same effect, no matter whether it is the first, second, or other nitrogen being added to the ring, and regardless of the neighboring atom types. This is the major prediction of Aa. However, the bond-order- and angle-based indexes predict that the aromaticity of isomers varies greatly. POZPC ranks the three triazines (in order of increasing locant number) at 84, 65, and 100; and the four triazoles at 62,38,50, and 27. APOZPC gives 76, 54, 40; and 64,

38, 36, and 77. What differences could occur? Aza-substitution in 6-rings could be different from in 5-rings because the angle requirements are different (6 vs. 5 angles 192 needed to round out the ring). Maybe the -N= fits more comfortably into one rather than the other, considering that in 6-rings the angle at the N will be considerably more than in 5-rings. The other major source of variance is the neighbor effect. The 1,3,4- isomers of the five-membered rings aiways end up at the Iow end of the bond-order-based aromaticity scales. This is the arrangement which has single C's interspersed between heteroatorns.

5.5.2 Heteroatom Type. The order of decreasing aromaticity (cal1 it [O*]): pyrrole, thiophene, furan is pretty well established, and also in most cases, furan is quite a bit below, while the other two are placed dose together. This is supported by al1 four of our SCF-bond-order-based indexes. This is different from Pozharski 's values for his own index,7 which showed the most common variant in the series, [oB]:thiophene more aromatic than pyrrole, with furan last. [O*] is the result for Schieyer et aL's published nucleus-independent chernical shifts (NICS)?~magnetic susceptibility exaltations (A)," aromatic stabilization energies (ASE)'~ and Julg (bond length) indexlg values,12

Krygowski and Cyraiiski's HO MA:^ and the Hess-Schaad resonance energy per n- electron (REPE) (cf. ref 2., p.20). Others in support of [oB]are experimental A quoted by ~ozharskii' and B ird, l and Bird's hardne~s.~~

Trying to obtain [O*]puts a fairly tight constraint on the polarizability-based indexes. Thiophene's polarizability anisotropy is quite a bit greater than that of pyrrole, and this inequality holds for the other polarizability-based indexes, too. The thiazoles contain more electronsnnthan the other molecules in Our study. Dividing by the total 193 number of electrons puts the thiazoles more "properly" in line with the azoles and oxazoles.

5.5.3 Ring Ske. Bird has touted his I, index because it predicts that sorne 5- rings are more aromatic than ben~ene.~~However, with Our SCF bond orders, it Nrns out that they are substantially lower than benzene again (less than 0.9 to benzene's 1.0). Other evidence: NICS, suggested by Schleyer et rates the 5-rings, even furan, higher than ben~ene.'~Surely that is wrong. What meaning is there to a definition of aromaticity that rates furan more aromatic than benzene? Fowler and ~teiner's~~three magnetic criteria have benzene higher than C5H5-or pyrrole. Al1 of our calculated bond-order-based indexes have the 6-rings ranked higher than the corresponding (according to level of aza- substitution) 5-rings. In fact the 6-rings are all rated more aromatic than the 5-rings for the three indexes other than the Bird index, which, by contrast, predicts that pyridine and the first 5-rings will be similarly aromatic.

For polarizability-based scales, there can be a danger of having the five-membered rings rated more aromatic than the 6-rings only if we divide by too many scaling factors: number of electrons, number of ring atoms, ring area (A); because, of course, the 6-rings are larger and have an extra ring atom. Dividing by area has an interesting effect on the x-polarizability; for 6,thiophene is shifted to be lower than pyrrole, but also benzene is shifted to be Iower than those two and even Iower than furan. However for AaXIA,the

6-rings are al1 above the 5-rings and 1,4-diazine is at the top of the scale. Dividing by number of electrons brings the thiazoles lower so that they partially overlap the other 5- rings; but thiophene is still higher than pyrrole. 194

The only other question of importance is whether there is any problem with a

particular index regarding the molecule's syrnrnetry group. High symmetry (e.g., D,,)

skews some indexes. The only syrnmetry constraint on our polarizability-based indexes

of aromaticity is that they are applicable to planar molecules. An extension would have

to be added, in order for them to apply to nonplanar aromatics.

5.6 Conclusion

Our consistent MP2 geometries calculated specifically for the monocycles concerned help to prevent al1 the aromaticity scales from being skewed by geometric effects.

We expect the SCF calculated bond orders to be more accurate than those based on simple empirical relations such as the Gordy bond orders. The SCF bond orders can mode1 the particular environment in the individual molecule, as opposed to assuming that a bond length combined with the two atom types completely determines the bond order.

The bond-order-based indexes we determine show rnany differences frorn the values previously published for the same indexes. Ours are expected to represent the geometric aromaticity more accurately.

Angle-based aromaticity indexes were defined, implemented and combined with bond-order-based indexes. However, they provide only limited help, improving the bond- order-based indexes' poor treatment of DJhmolecules, but not helping for D6h molecules.

The reciprocal of the hardness as an aromaticity scale was shown to be similar to the polarizability anisotropy, but the reciprocal of the electronegativity correlated even 195 better to Aa. Neither was correlated to bond-order-based indexes, within the set of molecules we studied.

AI1 polarizability-based indexes were similar except for the ones based on the 7c- polarizability anisotropy, which, like the geometric scales, predicted notably different aromaticity for isomers. The best total anisotropic polarizability is more accurately calculated and hence more reliable than the sl; measure. According to da,the aromaticity in these molecules will be reduced by each aza-substitution, so that isomers have very similar aromaticity; six-membered rings wilI be more aromatic than corresponding five- membered ones; and the aromaticity of the five-membered rings will decrease as the lead heteroatom is changed frorn S to N-H to O.

Our main emphasis in this project has not been to prove or disprove that the criteria of aromaticity are saying the same thing. Rather, Our approach is that if a given aromaticity scale is to be used, then it must be implemented properly. It is fascinating to see what uses people have made of the bond-order-based indexes, considering the flawed ways in which they have been implemented. For polarizability, Our concern has been to make the data availabIe for future studies of aromaticity, plus to focus on how polarizability can be related to aromaticity. To be thorough, we checked whether the geometric and polarizability criteria predict the same results. But it is important to remember that they really are rneasures of different things, electron delocalization and electron delocalizability, neither of which is precisely aromaticity. 5.7 References and Notes

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Schomaker, V.; Stevenson, D. P. J. Am. Chem. Soc. 1941, 63, 37-40.

Fleischer, U.; Kutzelnigg, W.; Lazzeretti, P.; Mühlenkamp, V. J. Am. Chern. Soc.

1994, 116, 5298-5306.

Komorowski, L. Chem. Phys. 1987, 114, 55-71. Chapter 6. Conclusion 201 6.1 Main Results of Thesis

In this thesis, a systematic study of heteroaromatic azaboracycles has been described. The approach has been to use the same reasonably accurate methods to calculate the various properties for each molecule. Thus, on average the results should be reliable, although fhture work with more accurate methods may show significant differences in particular cases.

Cornparison with experiment suggests that the geometries and relative stabilities are well represented by MP2/6-3 lG(d) and that the vibrational frequencies are reasonably accurate. Distinct differences were noted between the three levels of calculation, MNDO,

HF, and MP2.

The B-N and other bond lengths and the polarizabilities of azaboracycle isomers

Vary greatly depending on the ordering of atoms around the ring. EIectron correlation sometimes increases, sornetimes decreases the polarizability. Thus, the properties would be hard to predict with simple tramferrable models.

Including the results in this thesis and previous work, we have now looked at al1

64 possible 36-electron five-membered rings containing one N-H or O, zero to four CH's or N's, and one or two NHBH sets; and 56 six-membered rings consisting of benzene, al1 azines, and al1 BN-substituted benzenes and pyridines. In the future, it will be worthwhile to calculate stabilities and other properties of the remaining azaborinines.

Recent interest in quantitative aromaticity indexes has focussed on structural, magnetic, or energetic criteria. The analysis in Chapter 5 showed that polarizability indexes are also usefur indicators of aromaticity and that geometric indexes must be 202

detemined using accurate geometries and bond orders in order to be able to describe the

arornaticity with sufficient accuracy. The various scales of aromaticity do not agree.

Perhaps "theory" of aromaticity is really just "model" and hence each model will have

some areas of applicability and some of failure. Thus the criteria of aromaticity must be applied with great care in each molecular case in order to obtain useful insight.

6.2 Possible Follow-up Work

The list of the most stable azaboracycles suggest particular isomers which may be relatively easy to synthesize (with appropriate substituents) but have not yet been made.

Future work on polarizabilities of heteroaromatic rings should include further gas-phase experiments, since the available data are relatively scarce.

The data obtained in this thesis will continue to be used for additional study of the trends in polarizability and aromaticity for heteroaromatic rings. One possible project is to determine the relative aromaticity of the BN-rings. The aromaticity of borazine and other inorganic rings has already received considerable attention (Archibong and Thakkar

1994; Fowler and Steiner 1997; Schleyer et al. 1997; Jemmis and Kiran 1998).

The additive models of polarizability could be developed in several ways. One would be to estimate the components of each bond polarizability. The additive models could be used to predict the polarizabilities of other molecules. It would be possible to use the polarizability data to test others' additive models of polarizability.

There are many molecules that differ from those in this thesis by only one factor.

It would be worthwhile to study further the polarizability of phosphabenzene, boroxine 203 (Archibong and Thaldcar 1994) and related molecules, and the properties and aromaticity

of phospha- and thiazaboracycles (some of which have been made [Katritzky et al.

1996]), three- (ternu~aket al. 1997) and four-membered (Batandjieva et al. 1997) rings

containing B and N, fluorobenzenes (for which experimental polarizabilities are available

[Aroney et al. 1974; Gentle et al. 1990]), fluorinated azaboracycles, including borazines

(Parker and Davis 1997), and aluminum-containing rings such as Al,N3H, (Nyulhi 1997).

6.3 Comparison with Relevant Recent Results

In this thesis, 1 have not used the largest possible basis sets or highest levels of

correlation that could be applied to molecules the size of borazine. For the properties of

most of the molecules there are no experimental or theoretical reference data with which

to compare. One can be tricked by certain apparently regular trends into believing that a

smal1 basis set is al1 that is required. Increasingly-accurate calculations and experiments

are desirable to veriQ results. Also, it is worthwhile to make careful comparison in any case that such other data are available. There have been several recent accurate studies on the properties of heteroaromatic rings.

6.3.1 Geometry of s-triazine. Morrison et al. (1997) reported a detailed experimental and theoretical analysis of the geometry of s-triazine. They carefully combined electron diffraction, high resolution infrared, liquid crystal NMR, and ab initio data. Their calculations included a basis set as large as 6-31 lG(2df,2pd) and some coupled cluster calculations with the smalIer 6-31 1G(d,p). Results for the C-N bond length are shown in Figure 6.1. 1 C-N in s-triazine

-

L -2 ' ). 1 t 1 1 1 1 HF MP2 MP2(all) MF3 MP4SDQ CCSD CCSD(T) Calculation method

Figure 6.1 Basis set and electron correlation effects on the C-N bond Iength of s-triazine, relative to the recomrnended r, bond length, 133.6 pm (Morrison et al. 1997). The basis sets are [l] 6-31G(d), 121 6-31 lG(d,p), [3] 6-31 lG(df,p), and [4] 6-31 lG(2df,Zpd). Fully correlated MP2(all) from Doerksen and Thakkar (1996); other results frozen core, from Momson et al. Also shown is the estimated r, of Momson et al.

Morrison et al. were disappointed that their r, prediction was shorter than the largest basis MP2 bond length by 0.9 pm. It may be necessary to use even larger bais sets to obtain such a short MP2 bond length. The adjustment of their r, lengths relied on an MP2(frozen core)/6-3 lG(d) force field and the assumption of a Morse potential for an isolated C-N bond, Earlier HFl4-21G work (Pyckhout et al. 1986) had predicted the difference, r, - re, to be 0.6 pm. Perhaps it is better to suppose that the true re will be close to Morrison et aL's best estimated geornetry, since that also is close to the r, 205 geometry they detennined. Interestingly, the agreement between MP2/6-3 lG(d) and their

IR r, geometry is good, 0.5 pm for C-N, 0.1 pm for C-H, and 0.2" for LCNC, similar to the agreement found with microwave r, geometries for non-BN-containing rings.

6.3.2 Resuits for polarizabilities. In considering the usefulness of the polarizabilities of this thesis, it is worthwhile to examine recent calculated polarizabilities for benzene and for furan, and to take an overview of the difference between MP2K and experimental polarizabilities for borazine and any azoles, azines, and oxazoles for which such results are available.

Among al1 rings, benzene is the one whose polarizability has been studied the most. Some workers have been distracted by the fact that the effect of electron correlation on benzene's average polarizability is relatively small (only 2% in the C basis [Appendix

11). The group of Agen have supported using a relatively small basis set for polarizability

(and hyperpolarizability) calculations, together with the random-phase approximation

(RPA) approach, which should give polarizabilities very simiIar if not identical to coupled

Hartree-Fock (CHF). Previously, they used the 4-31G basis supplemented with a diffuse p and d set on carbon and nitrogen to deterrnine the polarizabilities of the azines (Knuts et ah 1993). Most recently, Luo et al. (1998) used the Sadlej basis (similar to the C basis) with RPA, which had already been used to determine the CHF polarizability of benzene

(Lazzeretti et al. 1990), and noted that the 4-32G+(pd) results really were too low (Luo et al. 1998).

The vibrationaI polarizability of benzene has received careful attention (Alrns et al. 1975; Bishop and Cheung 1982; GentIe and Ritchie 1989; Luo et al. 1998). Luo et al. 206

found that, as expected, the frequency-dependent vibrational polarizability is much smaller

than the static value. The average static vibrational polarizability is about 2 a.u. for

benzene, but the anisotropic (a,- aL)vibrational polarizability is -5 am. The minus sign

indicates that the out-of-plane oscillations are more significant for benzene. Thus the

electronic anisotropic polarizability should be larger than the total by 5 a.u. This led Luo

et al. to a total anisotropic polarizability of 28.6 a.u. (with RPA), compared to the best

experimental result, 33.2 I 1.2 a.u. (Gentle and Ritchie 1989). However, the MP2lC

electronic polarizability, 36.1 a.u. (with zero-point vibrational corrections not included)

(Appendix l), is rather close to experiment, 35.0 a.u. of Alms et al. (1975) and 38.5

1.2 a.u. of Gentle and Ritchie (1989). In any case, more work needs to be done on the

anisotropic polarizability of benzene.

Figure 6.2 shows the difference between MPZC and other average and anisotropic polarizabilities for furan. Recently, Millefiori and Alparone (1998) studied the static dipole polarizability of furan by density functional theory (DFT) rnethods, using the

Sadlej basis. Several functionals were tested, using a DFT geometry. Perhaps the best results were with B3LYP, shown in Figure 6.2.

Also, Christiansen et al. (1997) have performed coupled cluster (CC) calculations on furan at its experimental geometry. These should be more accurate than DIFTlSadlej or MP2K (El-Bakaii Kassimi et al. 1996). The MP2 and CC results shown in Figure 6.2 are static polarizabilities obtained with an augmented double-zeta basis set just slightly smaller than the C basis. Christiansen et al. dso considered a triple-zeta basis. Their

CCSD program can include frequency dependence. In Figure 6.2, the extra lines at the Polarizability of furan

7y Anisotropy

Average f

i I 1 I I 1 I 1 B3LYP MP2 CCSD CCSD(T) est(0) est(f) expt

Figure 6.2 Polarizability of furan, relative to MP2/C. MP2/C from El-Bakali Kassimi et al. (1996), B3LYP from Millefiori and Alparone (1998), other calculated results from Christiansen et al. (1997), including their best estimated polarizabilities. Cf. text. Experimental liquid-phase average polarizability (Dennis et al. 1983) and gas-phase anisotropy (Coonan et al. 1992), both at 632.8 nm. Error bars are shown for the experimental polarizabilities (with dotted lines for the anisotropy).

CCSD position are for the average (lower line) and anisotropy at 633 nm, the wavelength used for the best available experimental polarizability (Coonan et al. 1992). Dispersion increases both the average and anisotropic polarizability. The label est refers to the best estimate of Christiansen et al. for the static (O) or frequency-dependent (f) polarizability, allowing for remaining deficiency in basis set and in correlation. The MP2K results are satisfactory, albeit that for furan (as for benzene) correlation effects are rather small. Average polarizability '5 A T

liquid

I -7 ' 1 1 I 1 1 1 1 1 1 1 1 benzene pyrazine pyrimidine pyrrole irnidazole borazine pyridine pyridazine s-triazine pyrazole furan

Figure 6.3 Average polarizabilities for heteroaromatic rings, in liquid or in solution as noted. For experimental details, see the following: Iiquid benzene (Alms et al. 1975) and borazine (Hough et al. 1955); those in dioxane (Battaglia and Ritchie 1977b), except pyrroie, pyrazole, and irnidazole (Calderbank et al. 1981); in carbon tetrachloride (Le Fèvre et al. 1959); in cyclohexane (with error bars) (Dennis et al. 1983).

Figures 6.3 and 6.4 show percent differences between experiment and MPUC for the average and anisotropic polarizability (A2@, respectively. Figure 6.3 shows that the biggest difference for the average is 62.Figure 6.4 shows that the anisotropy in solution

is much lower than in the gas phase. Future work including more gas-phase experiments and/or more calculations of the frequency dependent polarizability will be helpful. In cyclohekne -40' 1 I 4 i I i I a i I i benzene pyrazine pyrimidine pyrrole imidazole borazine pyridihë pyridGine s-trlaziÏne pyrazole furan

Figure 6.4 Anisotropic polarizabilities for heteroaromatic rings, in solution or gas-phase as noted. For experimental details, see the following: gas-phase benzene (Gentle and Ritchie 1989, with enor bars; Alms et al. 1975) and furan (Coonan et al. 1992); in dioxane and in cyclohexane for benzene and pyridine (Battaglia and Ritchie 1977a) and the diazines (Battaglia and Ritchie 1977b); s-triazine in dioxane (Battaglia and Ritchie 1976) and cyclohexane (Blanch et al. 199 1); pyrrole, pyrazole, and imidazole in dioxane (Calderbank et al. 1981); in carbon tetrachloride (Le Fèvre et al. 1959), except for benzene (Battaglia and Ritchie 1977b); in cyclohexane, furan (Dennis et al. 1983) and borazine (Dennis and Ritchie 1993). Error bars shown for al1 in cyclohexane, except for furan (for which, cf. El-bakaii Kassimi et al. 1996). Not shown is the Stark effect Aafor s-tetrazine (Heitz et al. 1991) which is 80% lower than MP2/C. 6.4 References

Alms, G. R.; Burnham, A. K.; Hygare, W. H. 1975 Measurement of the dispersion in polarizability anisotropies. J. Chern. Phys. 63, 3321.

Archibong, E. F.; Thakkar, A. J. 1994 Polarizabilities of aromatic six-membered rings: Azines and "inorganic benzenes." Mol. Phys. 81, 557.

Aroney, M. J.; Cleaver, G.; Pierens, R. K.; Le Fèvre, R. J. W. 1974 Molecular polarisability. The anisotropy of the Cs-F group. J. Chem. Soc. Perkin 11, 3.

Batandjieva, B.; Miadokova, 1.; çemu~ak,1. 1997 An ab initio study of four-membered rings. Boranes HBXYBH (X, Y = C, N, O). Adv. Quantum Chem. 28, 220.

Battaglia, M. R.; Ritchie, G. L. D. 1976 Magnetic anisotropies of s-triazine and 2,4,6- trichloro-s-triazine. Mol. Phys. 32, 148 1.

1977a Kerr constants, Cotton-Mouton constants, and magnetic anisotropies of pyridazine, pyrimidine, and pyrazine. J. Chem. Soc. Perkin 11, 897.

197713 Molecular magnetic anisotropies frorn the Cotton-Mouton effect. J. Chem. Soc. Faraday 11, 209.

Bishop, D. M.; Cheung, L. M. 1982 Vibrational contribution to molecular dipole polarizabilities. J. Phys. Chem. Rej Data 11, 1 19.

Blanch, E. W.; Dennis, G. R.; Ritchie, G. L. D.; Wonnell, P. 1991 Cotton-Mouton effect, magnetic anisotropy and charge delocalization of 2,4,6-tris(dimethy1amino)- 1,3,5-triazine. Cornparison with 1-3,s-triazine. J. Mol. Struct. 248, 201.

Calderbank, K. E.; Calvert, R. L.; Lulcins, P. B.; Ritchie, G. L. D. 1981 Magnetic anisotropies and relative aromaticities of pyrrole, pyrazole, irnidazole and their N-methyl derivatives. Aust. J. Chem. 34, 1835.

~ernukilc,1.; Fowler, P. W.; Steiner, E. 1997 Ring currents and magnetic properties of the cyclopropenyI cation and isoelectronic triangular 2n: electron systems. Mol. Phys. 91, 401.

Christiansen, O.; Halkier, A.; Jorgensen, A. 1997 Coupled cluster calculations of the polarizability of furan. Chem. Phys. Le#. 281, 438.

Coonan, M. H.; Craven, 1. E.; Hesling, M. R.; Ritchie, G. L. D.; Spackman, M. A. 1992 Anisotropic molecular polarizabilities, dipole moments, and quadrupole moments of (CH,)&, (CH,)$, and C4H$ (X = O, S, Se). Comparison of experimental results and ab initio calculations. J. Phys. Chem. 96, 7301.

Dennis, G. R; Ritchie, G. L. D. 1993 Molecular quadrupole moments, magnetic anisotropies, and charge distributions of borazine, B-trichloroborazine, N- trimethylborazine, and B-trichloro-N-trirnethyiborazine. Comparison with benzene and its derivaîives. J. Phys. Chem. 97, 8403.

Dennis, G. R.; Gentle, 1. R.; Ritchie, G. L. D.; Andrieu, C. G. 1983 Field-gradient- induced birefringence in dilute solutions of furan, thiophen, and selenophen in cyclohexane. J. Chem. Soc., Faraday Trans. 2 79, 539.

Doerksen, R. J.; Thakkar, A. J. 1996 Polarizabilities of heteroaromatic molecules: Azines revisited. Int. J. Quantum Chem. 60, 42 1.

El-Bakali Kassimi, N.; Doerksen, R. J.; Thakkar, A. J. 1996 Polarizabilities of oxazoles: Ab initio calculations and simple modelii. J. Phys. Chem. 100, 8752.

Fowler, P. W.; Steiner, E. 1997 Ring currents and aromaticity of monocyclic n-electron systems C6H6, B3N3H6,B303H3, C3N3H3, C5H& C7H7+, C3N3F3, C6H3F3, and C,F,. J. Phys Chem A 101, 1409.

Gentle, 1. R.; Hesling, M. R.; Fütchie, G.L.D.1990 Kerr effects, Rayleigh depolarization ratios, polarizabilities, and hyperpolarizabilities of fluorobenzene and pentafluorobenzene. J. Phys. Chern. 94, 1844.

Gentle, 1. R.; Ritchie, G.L.D. 1989 Second hyperpolarizabilities and static and optical- frequency polarizability anisotropies of benzene, 1,3,5-trifluorobenzene, and hexafluorobenzene. J. Phys. Chem. 93, 7740.

Heitz, S.; Weidauer, D.; Hese A. 1991 Measurement of static polarizabilities on s- tetrazine. J. Chem. Phys. 95, 7952.

Hough, W. V.; Schaeffer, G. W.; Dzurus, M.; Stewart, A. C. 1955 The preparation, identification and characterization of N-Trialkylborazoles. J. Am. Chem. Soc. 77, 864.

Jemmis, E. D.;Kiran, B. 1998 Aromaticity in X3Y3H6(X = B. Al, Ga; Y = N, P, As), X3Z3H, (Z = O, S, Se), and phosphazenes. Theoretical study of the structures, energetics, and magnetic properties. Inorg. Chem. 37, 21 10.

Katritzky, A. R.; Rees, C. W.; Sctiven, E. F. V., eds. 1996 Comprehensive Heterocyclic Chernistry II: Revîew of the Literature 1982-1995: The Structure, Reactions, Synthesis, and Uses of Heterocyclic Compounds; Pergarnon: Oxford. Knuts. S.; Vahtras, O.; Agren, H. 1993 Multiconfiguration response theory calculations of singlet and triplet spectra of the azabenzenes. THEOCHEM 279, 249.

Lazzeretti, P.; Tossell, J. A. 1991Coupled Hartree-Fock calculations of the electric dipole polarizability and first hyperpolarizability of some "inorganic benzenes." THEOCHEM 236, 403.

Le Fèvre, C. G.; Le Fèvre, R. J. W.; Rao, B. P.; Smith, M. R. 1959 Molecular polarizability. Ellipsoids of polarizability for certain fundamental heterocycles. J. Chem. Soc., 1188.

Luo, Y.; Norman, P.; Agren, H.; Sylvester-Hvid, K. O.; Mikkelsen, K. V. 1998 Dielectric and optical properties of pure liquids by means of ab initio reaction field theory. Phys. Rev. E 57, 4778.

Millefiori, S.; Alparone, A. 1998 (Hyper)polarizability of chalcogenophenes C4H4X (X = O, S, Se, Te). Conventional ab initio and density functiond theory study. THEOCHEM 431, 59.

Morrison, C. A.; Smart, B. A.; Rankin, D. W. H.; Robertson, H. E.; Pfeffer, M.; Bodenmüller, W.; Ruber, R.; Macht, B.; Ruoff, A.; Typke, V. 1997 Molecular structure of 1,3,5-triazine in gas, solution, and crystal phases and by ab initio calculations. J. Phys. Chem. A 101, 10029.

Nyulki, L. 1997 About the aromaticity of A12N3H5. J. Chern. Soc., Dalton Trans. 2373.

Parker, J. K.; Davis, S. R. 1997 Ab initio study of the relative energies and properties of fluoroborazines. J. Phys. Chern. A 101, 9410.

Pyckhout, W., Callaerts, 1.; Van Alsenoy, C.; Geise, H. J.; Almenningen A.; Seip, R. 1986 The molecular structure of s-triazine in the gas phase determined from electron diffraction, infrared/Rarnan data and ab initio force field calculations. J. Mol. Struct. 147, 321.

Schleyer, P. v. R.; Jiao, H.; van Eikema Hommes, N. J. R.; Malkin, V. G.; Malkina, O. L. 1997 An evaluation of the aromaticity of inorganic rings: Refined evidence from magnetic properties. J. Am. Chern. Soc. 119, 12669. Appendixes Appendix 1. Polarizabilities of Heteroaromatic Molecules: Azines ~evisited*

Robert J. Doerksen and Ajit J. Thakkar

* Published in Znt. J. Quantum Chem. 1996,60,421. The Abstract and Acknowledgements have been removed for this thesis; also, in Table A1.2, some of the entries for 1,2-diazine have been corrected. Al.1 Introduction

Polarizabilities determine long range intermolecular induction and dispersion forces, low energy electron-molecule scattering cross sections, Langevin capture cross sections in ion-neutral collisions, and various phenornena such as collision-induced spectral line shifts [1,2]. We have begun a systematic study of the polarizabilities of heteroaromatic molec.ules [3-61 to build a sizeable body of accurate polarizability data for such molecules, examine the utility and limits of simple models for these polarizabilities, and to study the connection between polarizability and aromaticity. We have previously studied four families of heteroaromatic molecules: azines [3], azoles 143, oxopurines [5] and oxazoles [6].However, Our study of the azines (aza-benzenes) [3] differs from Our subsequent ones [4-61 in that i) a rnix of experirnental and calculated geometries was used rather than a consistent set of calculated geometries, ii) a poorer basis set was used for some of the polarizability calculations, iii) four of the 12 possible aza-substitutions were not considered, iv) additive models were not reported, and v) no attempt was made to separate the O- and z-electron contributions to the polarizability.

The purpose of this paper is to revisit the azines and remove the above shortcomings. We report systematic, electron correlated, ab initio calculations of the equilibrium geometries, dipole moments and static dipole polarizabilities of benzene and al1 twelve aza-benzenes. Uncoupled Hartree-Fock calculations are used to determine the fraction of the mean polarizability that arises from the n-electrons. Additive and other simple models for the polarizabilities of the 13 azines, 10 azoles and 10 oxazoles are 216 discussed, and empirical formulas are given that fit the polarizabilities of these 33 moiecules quite well.

A1.2 Methods and Definitions

Only a concise surnmary of Our methods is given here because full details were given in previous work [4] which used the same computational methods. Equilibriurn geometries for al1 13 azines were computed using al1 electron, second-order Mgller-Plesset

(MP2) perturbation theory [7,8] and the 6-31G* basis set [9]. The dipole moments and polarizabilities were calculated by the finite-field method [10,11] using self-consistent- field (SCF) and valence MP2 field-dependent energies computed with GAUSSIAN-90

1121. The finite-field SCF polarizabilities are equivalent to coupled Hartree-Fock (Cm) ones [Il, 133. We used a [Ss3p2&3s2p] basis set, denoted C in ref [4], of contracted

Gaussian-type functions (GTF)for the finite-field calculations. It consists of a [4s2p/2s] double zeta substrate Cl41 augmented by [lslplls] diffuse GTF, and [2d/2p] polarization

GTF - one optirnized for electron correlation [15] and the other for polarizabilities [4].

We compute uncoupled Hartree-Fock (UCHF) polarizabilities [16]to estimate the relative contribution of the n-electrons.

The molecules were placed on the yz-plane to block diagonalize the polarizability tensor whose eigenvalues al=a, I 5 (-, determine the rnean:

- 1 I a = -(a, + a, + a,) = -(a, + 02 + ~j) 3 3 and the difference between the mean in-plane and out-of-plane components: 1 1 al - a, = -(a, + a,) - a, = 2(a, + CS) - a, 2

The Kerr anisotropy and the in-plane anisotropy respectively are:

Atomic units are used in this work for polarizabilities.

A1.3 Equilibrium Geometries

Figure Al. 1 is a scde drawing of our MP2/6-3 1G* equilibrium geometries for the

thirteen azines. The MP2/6-3 lG* bond lengths were remarkably sirnilar in al1 the azines;

CC bonds were between 138.5 and 139.6 pm, CN and NN bonds between 133.5 and

134.7 pm, and CH bonds between 108.5 and 108.8 pm. A cornparison of our structures

with experimental 117-251 and other ab initio calculations [26-351 follows.

Seven of our geometries are not tabulated because they are very similar to

previously reported caicu1ations. MP2/6-31G* geometries for benzene, pentazine and

hexazine were reported previously [26,27,29]. Raman [17] and infrared (IR) [25]

spectroscopy leads to a similar geometry for benzene. There are no published experimental geometries for pentazine and hexazine because they have not yet been isolated [27,29]. Our 1,3,5-triazine geometry coincides exactly with a MP2/6-31G** calculation of Creuzet and Langlet [31]. They discussed the good agreement with an Figure Al.l The azines at their MF'2/6-3 lG* optirnized geometry: benzene (l), pyridine (2), 1,2-diazine or pyridazine (3), 1,4-diazine or pyrazine (4), 1,3-diazine or pyrimidine (S), 1,2,3-triazine (6), 1,2,4-triazine (7), 1,3,5-triazine or s-tnazine (8), 1,2,3,4-tetrazine (9), 1,2,4,5-tetrazine or s-tetrazine (IO), 1,2,3,5-tetrazine (Il), pentazine (12), and hexazine (13). MP2IC dipole moments are shown to scale, with the arrow head pointing to the negative end. The solid line is the a-axis of inertia and the dotted line is the a,-axis of pofarizability (not specified for syrnmetric tops). 219 experimental Raman structure [la]. There is even better agreement with the combined electron diffraction (ED), Raman and calculated r, geometry of 1,3,5-triazine 1221 which differs from ours by less than 0.1 pm and 0.1" except that their CH bond is 1.8 pm longer. Our unlisted MP216-3 1G* geometries for 1,2,3-triazine and 1,4-diazine are within

0.2 pm and 0.1" of their frozen core counterparts [32,33]. Fischer et al. [32] found fair agreement between their MP2 calculation and an X-ray structure [20] for 1,2,3-triazine.

Our geometry for 1,4-diazine differs by about 0.5 pm and < 1.1 O from the r, structure obtained by Cradock et al. [23] using electron diffraction and liquid-crystal (LC)NMR data. Our unreported geometry for 1,2,4,5-tetrazine is very similar to an MP2/6-3 11G** calculation [35] except that Our NN bond is 0.4 pm longer. The same work [35] included a CASPT2lANO geometry in closer agreement to the microwave (MW) r, structure [20].

However, the experimental parameters have rather large uncertainties; for example, the

NN bond length of 132.56 + 1.6 pm 1201 does not rule out Our MP216-3IG* value of

133.9 pm.

Table Al.l lists our calculated geometries for the other six azines. We are unaware of previous electron correlated geornetries for pyridine and 1>-dimine, although

SCF geometries have been published in 6-31G* and 6-31G** basis sets [36,37] arnong others. Our MP2/6-3 lG* geometry for 1,2-diazine has bond lengths that differ from their

MP216-3 1 1G** counterparts [30] by up to 1.7 pm. Table Al. 1 shows that our pyridine geometry agrees with the microwave r, substitution structure [t 93 within 0.6 pm and 0.3".

Our geornetries for 1,2- and 1,3-diazine are in reasonable agreement with the r, structures obtained by Cradock et al. [23,24] by combining ED, MW, and LC-NMR data. Table TABLE Al.1: Comgarison of MP2/6-31G* Geometries with Others. Bond lengths in picometers and bond angles in degreee. Pyr idine 1.2-diazine 1.3-diazine l,2.4-triazine 1.2,3..4- 1.2,3.,5- tetrazlne tetrazine MP2 Expea MP2 EX^.^ MP2 ~xp.~MP2 MP2' MP2 CISD~ MP~ CISD~

' MW r, structure. ref [19]. ED. MW. LC-NMR structure. refs I23.241. TZVP basis. ref [341. DZP basis, h, ref [28]. 8 22 1

Al.1 shows that the average absolute discrepancies are 0.7 pm and OS0, but the largest

bond length differences are 1.3 pm for CN in 1,3-diazine and 1.5 pm for C4H in 1,2-

diazine.

Table Al.l compares our geornetries for the other three azines with previous

calculations because we are unaware of any experimental geometries. The recent

MP2/TZVP geometry 1341 for 1,2,4-triazine has virtually identical bond angles, but longer

ring bonds and shorter CH bonds with an average difference of 0.5 pm. Table A 1.1 shows that Our bond lengths differ from CISDIDZP ones [28] by an average and maximum of

1.6 and 3.8 pm for 1,2,3,4-tetrazine, and 1.1 and 2.7 pm for 1,2,3,5-tetrazine. Al1 but the

CC bond are shorter. Angles differ by as much as 0.8".

A1.4 Dipole Moments

Five of the molecufes have a zero dipole moment; Figure Al.l shows, to scale, the magnitudes, orientations and polarities of the dipole moments of the other eight azines. Figure Al. 1 also shows the inertial a-axis corresponding to the srnallest principal moment of inertia. The orientation of the dipole moment is deterrnined by syrnrnetry except in I ,2,4-triazine whose MP2K dipole moment makes an angle of 45.19" with the a-axis. In each azine, the negative end of the dipole moment points to the region with the rnost nitrogens.

Our MP2IC dipole moments are listed in Table A1.2. Half are higher and half lower than the unlisted SCFIC dipole moments by an average of 1.5% and a maximum of 3%. The dipole moments obtained by microwave Stark spectroscopy, 2.215D for TABLE Al.2: MPS/C Dipole Moments in Debyes aad Polarizabilities in Atomic Units at the MP2/6-31G* mornetriea. ~hen-fractions f of the UCHF polarizabilitiee are given as percentages.

Benzene Pyridine 1,2-diazine 1,4-diazine 1,3-diazine 1,2,3-triazine 1,2,4-triazine 1,3,5-triazine 1,2,3,4-tetrazine 1,2,4,5-tetrazine 1,2,3,5-tetrazine Pentazine Hexaz ine 223

pyridine [38], 4.221) for 1,2-diazine [39] and 2.334D for 1,3-diazine [40], are 0.13-0.16D

(or 4-6%) lower than Our MP2K values. Somewhat better agreement with experiment was

obtained by Palmer et al. who reported MRCI/TZVP dipole moments of 2.275D, 4.343D

and 2.383D for pyridine [411, 1,2-diazine [42] and 1,3-diazine [43] respectively. For the

other azines, there are no experimental dipole moments available, and Our calculations are

more accurate than previous ones. Palmer et al. 1441 reported minimal basis SCF dipole

moments (at ad hoc geometries) for al1 eight azines which are on average 16% lower than

our SCFK values. Mo et al. reported [45] SCF/6-3 1Gs//SCF/6-3 1G dipole moments for

seven azines; they differ from our SCFK values by an average of 1.6% and a maximum

of 4.4%.

A1 "5 Polarizabilities

Figure Al. 1 shows the MP2IC principal axis of highest polarizability - the g-axis.

It lies on a symmetry axis except in 1,2,4-triazine where it makes a 86.3 1O angle with the

inertial a-mis. The %-axis is generally more distant from the less polarizable nitrogen atoms than the g-ais. The three symmetric tops have an isotropie in-plane polarizability, and a non-unique ct+xis which is therefore not shown in Figure Al. 1.

Our MP2IC polarizabilities are listed in Table A1.2. Figures Al .2 and A1.3 show that MP2IC polarizabilities are higher than the CHF/C values by an average of 4.2% and

7% for E and Ala, respectively, with maximum differences of 7.2% and 12% for s-triazine. The UCHFIC E and Ala, respectively, are 3% and 10% different from the

CHFIC results on average, but are 12% and 26% higher for the worst-case hexazine. 12

'O Mean polarizability UCHFE 8-

6- T 4- -E c 2- .-O iCI .Io O -m tr5 >"Y--' m- CHFIS

4O*; ?-- X xA CHF/G -6-

-8 1 1 t 1 1 1 1 1 1 1 1 1 1 12345 678910111213 Azine

Figure A1.2 Percent differences between the mean polarizability computed by a given method and its counterpart calculated at the CHF/C level. The numbering systern for the azines is defined in Figure Al. 1.

A cornparison with previous CHF calculations is also made in Figures A1 -2 and

A 1.3. Hinchliffe et al.'s [37,46] CHF calculations using a 6-3 1 1G+3d,3p bais set, denoted CHF/G, are consistently 3% and 5% below Our CHWC results for and A,a respectively because their basis set is less appropriate for polarizabilities. By contrast,

CHF results 13,471 obtained using Sadlej's basis [48] designed specifically for polarizabilities, denoted CHF/S, are very close to our CW/C values; the differences average only 0.6% for a and 1.4% for A,a. Even these small differences are chiefly due to small differences in geometry. To confirm this, we calculated UCHFK polarizabilities Figure A13 Percent differences between the polarizability anisotropy A,a computed by a given method and its counterpart calculated at the CHF/C level. The numbering system for the azines is defined in Figure AI. 1. at the geometries (denoted //exp) used in refs [3,47]. The (UCHF/C//MP2/6-31G* -

UCHF/C//exp) differences always agreed in sign and rough magnitude with the

(CHF/C//MP2/6-3 1G* - CHF/S//exp) differences.

A cornparison with previous correlated calculations is also made in Figures A1.2 and A1.3. Some older calculations discussed previously [3] are not mentioned here. The

MCLR a of Knuts et al. [49] are significantly lower (4.5% on average) than even our

CHF/C values. Moreover, their MCLR results are lower than their own RPA values whereas dl other calculations predict that correlation raises à for the azines. Their 226

MCLR values, obtained at the geometry of benzene for al1 the azines, are probably

unreliable. By contrast, the MPZlS(ad1ej) polarizabilities [3] for three azines differ from

the MP2IC values by only 1% and 2% for and A,a, respectively. Again the small

differences are due to small geometrical differences. We found the MP2IC values for s-

tetrazine using the same geometry [2O] as in ref [3] to be 49.44, 25.66, and 26.07 for a,

Ap, and ALa,respectively which is within 0.1 of the MP2lS result [3]. Schütz et al. [35]

reported very similar CASPT2/Sl/CASPT2/ANO values of 49.2, 25, and 25.3 a.u. for a,

Alor, and A2a, respectively. Archibong and Thakkar 133 reported hybrid polarizabilities

calculated by combining CHF/S or MP2/S values with higher order correlation corrections

computed with Spackman's smaller basis set 1501. Figures A1.2 and A1.3 show that there

is no uniformity to the differences, averaging 1.2% and 3.6% for and A,a,respectively,

between their hybrid results and Our MP2IC values. Our uniforrn MP2/C results are likely

to be more reliable.

There have been rnany experimental measurements of the polarizability of benzene but only two in the gas-phase. Alms et al. [SI] extrapolated refractivity data and Rayleigh scattering depolarization ratios to zero frequency to find E = 67.48 and A,a = 35.0.

Gentle and Ritchie [52] used the temperature dependence of electrooptical Kerr effect measurements, and Rayleigh scattering depolarization ratios at optical frequencies to find a zero frequency value of Ala= 38.5 I1.2 after correcting for vibronic effects. These measured values are in good agreement with our MPUC values of a = 69.3 and A,a =

36.2 given that we have omitted the effects of zero-point vibrational motion. 227

There has been much less experimental work on the polarizabilities of the other azines. The only gas-phase result we are aware of is a value of A,a = 5.4 for s-tetrazine obtained by laser Stark spectroscopy 1531; however, this value is almost five times as small as theoretical calculations [3,35,53] which are al1 close to Our MP2K value of 26.2.

This huge discrepancy has not yet been resolved. Battaglia and Ritchie [54] extracted polarizabilities from dipole moments, molar refractions, and molar Kerr and Cotton-

Mouton constants measured at 633 nm for benzene and five azines dissolved in dioxane.

They had to assume 4a=O to carry out the analysis; Table A1.2 shows that this assumption is most nearly valid in 1,2-diazine and least valid in 1,4-diazine. They deterrnined oi to be 70.4, 64.3, 56.7 and 50.4 for benzene, pyridine, the diazines and s- triazine respectively. These values differ from our MP2K results by an average of 3.4% and a maximum of 6.2%. Battaglia and Ritchie [54] found Ala = A2a = 25.4, 25.5, 26.4,

27.3, 29.2 and 29.1 for benzene, pyridine, 1,2-diazine, 14-diazine, 1,3-diazine and s- triazine respectively. These anisotropies are lower than Our MP2lC values by amounts ranging from 30% to 9% except for s-triazine where their value is only 1.5% higher than ours. Given that Battaglia and Ritchie's polarizabilities are for a wavelength of 633 nm in dioxane solution and based on a questionable assumption, the agreement with Our infinite wavelength calculations for non-vibrating isolated moIecules is as good as cm be hoped for.

Finally, note from Table A1.2 that the UCHF mfraction of the transverse polarizability decreases with aza-substitution from 62% in benzene to 47% in hexazine.

The UCHF n-fraction of the mean in-plane polarizability varies more erratically between 228

50% and 56%. These fractions are very sirnilar to those obtained from a non-empirical

Ünsold approximation by Mulder et al. [SS].

A1.6 Polarizability Models

Molecular polarizabilities have long been modelled as a sum of contributions from the constituent atoms [56,57]. Such modeIs work rather well for ab initio polarizabilities of many organic [58] and heteroaromatic molecules [4,6]. Here we examine models that can describe simu~taneouslythe polarizabilities of 10 azoles [4], 10 oxazoles 161, and the

13 azines. The simplest is the additive atom model:

in which ni is the number of atoms of type i. The parameters, obtained by linear regression of Our MP2lC polarizabilities for 33 molecules, are listed in Table A1.3 with the model's average absolute error 6, and maximum error 6,. Mode1 5 predicts the average polarizability of 33 heteroaromatic molecules with 6, = 1.1 % and 6, = 3%. The coefficient of the ni term can be interpreted as the polarizability of an atom of type i in a planar heteroaromatic molecule; Table A1.3 shows that the parameters are smaller than free atom polarizabilities [2]. When model 5 is applied to the anisotropy Ala, the errors are tripled. Addition of a rnolecular "volume" term, taken to be as in ref [6],where

A is the area enclosed by the ring yields the model a - B,n, + Bznc + B3nN + hnO + B~A3'2 TABLE A1.3: Parametere (in ~todcunit81 and Errors of various Polarizability Modela. Mode1 numbers refer to equations in the main text and an asterisk indicates tbat a nonuniform data base W~Bused. a model b,ût model 230 Table A1.3 shows that this model improves 6, to 0.66% and 3% for or and A,a,

respectively. Unfortunately, the coefficients of the ni terms no longer have a simple

interpretation.

A more elaborate model uses additive contributions from bond connections [4]

where n, is the number of connections (or bonds making no distinction between single

and multiple bonds) between atoms of types i and j. For Our 33 molecules, thcre are two constraints upon the n,: 2ncH = 2ncc + ncN + nco and the number of intra-ring connections = 6 - (nM + n,, + nco). Thus Eq. (7)reduces to

Table A1.3 shows that model 8 is a slight improvement over model 6. Adding a terni to model 8 does not improve it. Numerical experiments, and previous correlations between polarizability and electronegativity 1591, led us to a term involving the energies of the frontier orbitals:

in which x = (E,,,, - e,,,)/2 can be thought of as an approximation to the hardness.

Table A1.3 shows that model 9 gives an excellent prediction of a (6, = 0.3696, 6, =

1.2%), and a reasonable one of A,a (6, = 2%. 6, = 6%).

The accuracy of Our simple models is somewhat higher than might have been expected. We think this is so partly because the 33 heteroaromatic molecules are closely related and partly because Our polarizability data base is of uniform quality. To 23 1 dernonstrate this latter point, we include in Table A1.3 models 8* and 9* obtained by fitting Eqs. (8) and (9) to Our polarizability data base with eight azine polarizabilities of this work replaced by the hybrïd results of ref 131. Table A1.3 shows clearly that the error measures for models 8* and 9* are noticeably larger than those for models 8 and 9.

These models utilize only isotropic information and therefore work better for the mean polarizability E than for the anisotropy Ala. The Kerr anisotropy 4acan be described by these models with an accuracy slightly worse than that obtained for A, a.

However, an accurate mode1 for the in-plane anisotropy A3a requires anisotropic information. We are trying to develop such models.

A1.7 References

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[59] J. K. Nagle, J. Am. Chem. Soc. 112, 4741 (1990). Appendix 2. The C Basis set*

* From El-Bakali Kassimi, N.; Doerksen, R. J.; Thakkar, A. J. J. Phys. Chem. 1995, 99, 12790; J. Phys. Chern. 1996, 100, 8752; and this work. Table ~2.1 The C Basis Set: Double-zeta, Diffuse, and ~olarizationFunctione. No. H B C N O

a c a c a c a c a C s 1 13.011 0.033484 4710.8 0.001133 6779.9 0.001222 9155.6 0.001327 11852 O. O01445 1.9623 0.234719 706.74 0.008740 1017.2 0.009430 1373.6 0.010240 1778.2 0.011147 0.44454 0.813774 160.88 0.044488 231.57 0.048023 312.73 0.052146 404.86 0.056764 45.518 0.168530 65.547 0.182197 88.547 0.197926 114.66 0.215473 14.724 0.458573 21.253 0.496064 28.756 0.538041 37.279 0.584655 5.1823 0.440665 7.5339 0.385105 10.246 0.320789 13.334 0.246860 2 0.12195 1. S. 1823 0.409712 7.5339 0.471311 10.246 0.524498 13.334 0.569590 1.9068 0.632099 2.8031 0.571296 3.8443 0.517846 5.0385 0.471791 3 0.033 1. 0.33302 1. 0.52151 1. 0.7465 1. 1.0136 1. 4 0.10434 1. 0.15957 1. 0.22475 1. 0.3025 1. 5 O. 033 1. O. 049 1. 0.068 1. 0.09 1. P 1 O -727 1. 12.053 0.017293 18.734 0.018170 26.667 0.018796 34.493 0.020236 2.612 0.105484 4.1362 0.112687 5.9557 0.117689 7.7562 0.126799 0.74689 0.365660 1.2004 0.376170 1.744 0.383034 2.282 O. 394680 0.23873 0.663256 0.38346 0.648667 0.55629 0.638961 0.71691 0.624014 2 O .O5 1. 0.077218 1. 0.12129 1. 0.17315 1. 0.21461 1. 3 O. 025 1. 0.038 1. O. 054 1. 0.064 1. d 1 0.343 1. O. 55 1. O. 817 1. 1.185 1. 2 O. 15 1. 0.1 1. 0.15 1. 0.25 1. Appendix 3. Specifics on Making Scale Structural Diagrams 239

A3.1 Getting started. A normal scale drawing of a molecule displays the

coordinates of the atoms to scale. In my work 1 wanted (a) to have a scale drawing of a

whole set of molecules, with the key element that the relative size of the different

mo1ecules be accurately displayed; and (b) to represent the dipole moment to scale and

to display the orientation of the dipole moment, the inertial axes, and the polarizability

axes. This was surprisingly difficult. No software could automatically satisfy those two

conditions and then yield a satisfactory final picture. A supplementary goal was to be able

to specify the orientation of al1 the molecules in a standard way, so that it would be easy

to see similarities and differences when comparing the final pictures. Also, since this

thesis includes 100 molecules, it was important to have a system that was as automatic

and as quick as possible. Two types of drawings were required, planar and nonplanar, the

latter requiring consideration of perspective.

In the following the base filenarne is assumed to be "a"; some filenarne extensions

refer to standard formats; .FOR names refer to programs 1 wrote for this thesis; and

external software is referenced, only once each.

The present procedure starts with PCOORD.FOR, which requires the following

input: a.xyz (coordinates in the standard form, including atomic numbers); a.ctl,

containing the number of ring atoms, number of atoms, and a specification of first and

second atorn in proper locant order. The output, a.cor, contains the coordinates, now reordered to go around the ring. PCOORD.FOR also produces files containing the ring area, maximum distances, and al1 bond lengths and angles in neat forrn. 240 A3.2 Scale drawings of planar molecules. abe el' can take a.cor (or any xyz-

coordinate file) as input and generate a.sdf, which is easy to edit (e.g., to change whether

a bond will be shown as single or double). This can then be imported into ISISID~~W~as

a-mol, if there is nothing else to be put into the picture.

The next challenge is to prepare for a picture that includes both scale geometry

and other add-ons such as the dipole moment, inertial axes, and polarizability axes. That

information is available as output a.grf from POL201,3 the finite-field polarizability fitting

program. The output contains the coordinates of the following: the atoms; the center of

mass; the dipole moment vector tip, to be drawn with origin at the center of mass; and

unit vectors for the inertial and polarizability axes, to be drawn crossing at the center of

mass.

The final stage is to use ~~~~~~~FOR to generate a different a.mol that includes

the atoms plus other items. GRFMODFOR takes as input both a.grf and a.sdf. It makes

sure the two are properly matched and then adds the appropriate information from a.grf

into the a.sdf format, and outputs it al1 as a.mol. Axes are al1 represented as C-C bonds, and can be displayed as such in ISIS/Draw (with the "C" suppressed), or overlaid with a dotted line. ISIS/Draw has a lirnited selection of mows available which can be used to show the dipole moment direction. GRFMOD.FOR also connects the first (in Iocant order) atom to the center of mass; in ISIS/Draw the whole picture can be rotated to orient this

"bond" in a standard way (cf. Figures 3.1-3.5), to make a good picture.

A3.3 Scale drawings of nonplanar moleeules. GTORT4.FOR operates on a.cor to prepare a.ins input for ORTEP-3 for ~indows.~The only other input 24 1

GTORT4.FOR requires is the number of atoms. ORTEP-3 is useful for preparing

ball-and-stick pictures that show perspective (cf. Figure 3.7). Also, 1 used it to help

prepare the azaborinines picture, Figure 2.1. ORTEP-3 can picture only one molecule at

a tirne. While it can print such a picture directly, it can also generate a.hpg output which

can be read into ~raw~erfec?or Corel wordperfect6 (WP) graphics programs for further

modification. Also, in WP several of the pictures cm be collected and arranged to forrn

a group portrait of many molecules (such as Figure 3.7).

A3.4 Concluding remarks. In my early work, the .grf file was designed to be

input to MOLWIN? in which the molecule could be rotated to give a good view, and

which could then generate input for DrawPerfect. But in DrawPerfect a whole new picture

had to be drawn piece-by-piece, overlaid on top of the MOLWIN-generated one, because

the individual elements of the MOLWIN image could not be further modified. The

individual molecule pictures in Figure Al.l were made in this way. MOLWIN and

ORTEP-3 do not preserve the relative scale for pictures of different molecules. In fact, they are designed to zoom in and give the "best" picture based on the maximum dimensions to be displayed. Thus oxatetrazole (with no hydrogens) initially appears the sarne size as benzene. Once in DrawPerfect, a set of molecule pictures had to be individually scaled, for instance, by checking that al1 C-H bonds were displayed as exactly the same length. This route required a trernendous amount of work and included some inevitable loss of accuracy.

Most chernisis use software such as ISISIDraw to draw molecules using standard bond lengths and angles. However, that software is also very useful for handfing scale 242 drawings. The key is to load in a structure in the forrn of a .mol file. Initially, certain settings must be selected. Then, ISIS/Draw can maintain the relative scale of a set of molecules. In other words, if given coordinates as input, it will display them appropriately. Many .mol files can be loaded into one ISISIDraw .skc file. They can be easily manipulated: moved, sized, rotated. ISIS/Draw portrays the molecule pictures in 2D without perspective, though it keeps the full 3D information in memory.

A3.5 References

(1) Babel Version 1.3, Copyright 1992-1996, P. Walters and M. Stahl.

(2) lslsTM/Draw2.1.1, Copyright 1990-1997, MDL Information Systems, Inc.

(3) POL201, A. J. Thakkar, R. J. Doerksen, and N. El-Bakali Kassimi.

(4) ORTEP-3 for Windows 1.04P, Copyright U. of Glasgow, by L. J. Farrugia, J.

Appl. Cryst. 1997, 30, 565. A GUI for ORTEP-III Version 1.0.2, C. K. Johnson

and M. N. Burnett.

(5) DrawPerfect 1.1, Copyright 1991, WordPerfect Corporation.

(6) Corel WordPerfect 7, Copyright 1996, Corel Corporation Ltd.

(7) Molwin Version 2.3, Copyright 1993-1994, P. V. Ganelin, Dept. of Chem.,

Catholic U. of America. Appendix 4. Relevant Hantzsch-Widman Names for Molecules 244

According to the International Union of Pure and Applied Chemistry (Powell

1983), heteromonocycles should be named according to specific niles based on a

substitution scheme which originated in the work of Hantzsch and Weber (1887) and of

Widman (1887). Here, 1 note the relevant basic rules of the extended Hantzsch-Widman

system, and its rules regarding saturation, indicated hydrogen, and locant numbering.

A4.1 Basic rules. Some of the rules included in the Hantzsch-Widman system

are standard ones that also apply to other molecules, such as the numbering scheme and

prefixes. The individual elements of the system have their origin in common names. Aza-

for N, bora- for B, oxa- for 0, thia- for S, are al1 cornmon ro other branches of

chemistry; if followed by a vowel, then the final "a" is dropped, as in oxaza- or borin ....

The Hantzsch-Widman system treats heterocycles as derivatives of carbocycles.

Substitution names make no direct reference to the C's in the molecule, except that the

total number of ring atoms decides the stem (suffix). Instead a prefix is included to refer

to each heteroatom. For example, "diazaborinine": diaza- means 2 N's; bor-, short for bora- means 1 B; -inine means a six-membered ring; thus, there rnust be 4 C's.

Problems have arisen because of conflict with common names. Thus the endinp

-borine was rejected in favor of -borinine (though -borine is still used as a comrnon name ending). There is a special ending for saturated five-membered rings containing nitrogen,

-01idine. The Hantzsch-Widman system permits use of common narnes for a particular list of heterocycles (Powell 1983).

A4.2 Unsaturation. The definition of a saturated compound is one that has the standard valence for each atorn, B-3, C-4, N-3, 0-2. The reference structure for 245

unsaturated is a structure containing "the maximum number of non-cumulative double

bonds" (Powell 1983). In other words, an unsaturated six-membered ring would have

altemating single and double bonds, whereas for a five-mernbered ring there would be

two double and three single bonds.

Under the rules, a compound that is neither fully unsaturated nor fully saturated

can be named with reference to either extrerne, though it is prefenced to refer to the

unsaturated name (Powell 1983). For example, N3B2H,is completely saturated by the

Hantzsch-Widman definition (N and B each have a valence of 3), so that it is a

triazadiborolidine, or in unsaturated terminology a triazadiborole. Note that the molecules

of Chapter 2 could also be named with respect to the saturated reference as -inanes.

The prefix dehydro- is used to indicate where a saturation has been removed, i.e.,

that an H is missing. Similarly, hydro- indicates that an unsaturation has been removed.

For example, each NBHz pair substituted into a six-membered mine or benzene

necessitates adding two hydro- prefixes, in the unsaturated naming scheme.

The saturated name is occasionally shorter (and hence, more convenient), e.g.,

1,2,3,4,5-triazadiborolidine,which is 2,3,4,5-tetrahydro- 1H-1,2,3,4,5-triazadiborole with

respect to the unsaturated triazadiborole. Nevertheless, I prefer always to stick with unsaturated names, though sample alternative names are also specified in the captions to

Figures 3.1-3.5.

A4.3 lndicated hydrogen. When a five-membered ring is drawn with two double bonds, then there will be one ring atom joined into the ring by two single bonds. 246

If that atom has an H, then the H must be indicated with a special locant number (Powell

1983). Otherwise, the molecule's name would not uniquely specify that location.

For a BN-containing molecule, an indicated H is mandated ~henits reference

molecule (such as an mole) has an indicated H. The 16 azaboroles each must have a

single indicated H. Neither the O-containing five-membered-rings nor the six-membered

rings require an indicated H, since their reference unsaturated structures do not have one.

A4.4 Locant numbering. The priorities for numbering are: a. heteroatoms; b.

indicated H; c. "hydro" or "dehydro:"

a. i. The priority is O, S, N, B. Use that to choose onIy the cftrst atom. ii. Next, assign locants to al1 heteroatoms in al1 possible ways, starting from any heteroatoms of the priority type (e.g., if the first heteroatom is N, then starting from any N prepare any possible locant set). Compare the locant sets, without regard to type of heteroatom.

Compare first locants, then second locants, ... stopping at the first point of difference and choosing as the final set the one with the lowest locant in such a comparison. iii. If a final order is not yet resolved, then compare locant sets considering priority of heteroatom tYPe.

b. If there are several possible orderings based on heteroatom locants, then choose the numbering that rnakes the indicated H (if any) numbering the lowest.

c. Finally, if there is still choice available, choose the numbering that gives the lowest set of "hydro" or "dehydro" coefficients. A4.5 References.

Hantzsch, A.; Weber, J. H. 1887 Ber. Dtsch. Chem. Ges. 20, 31 18.

Powell, W.H. 1983 Pure Appl. Chem. 55, 409.

Widman, 0. 1887 J. Prakt. Chern. 38, 185. Appendix 5. Supplementary Material for Chapter 2, Azaborinines: Structures, Vibrational Frequencies, and Polarizabilities TABLE A5.1: Calculated MP2/6-31G(d) Geometries for C,, and CZh ~zaborinines*.

120.5 a A11 Czv except 11 (CZh). Bond lengths in picometers and bond angles in degrees. TABLE A5.2: Calculated MP2/6-31G(d) Geometries for C, Azaborininesa.

1 2 4 5 8 13 16 15 a Al1 ~P2/6-31G(d)except 4 which is ~P2/6-31G(d,p).Al1 planar except 15. Bond lengths in picometers and bond angles in degrees. TABLE A5.3: Harmonie Vibrational Frequencies for CZo and Cl,, Azaborininesa. 1122 1019 1075 1019 947 973 1007 995 1273 1211 1290 1215 1055 1156 1192 1215 1324 1307 1319 1320 1323 1258 1291 1366 1457 1365 1428 1425 1467 1489 1490 1523 1581 1449 1520 1500 1616 1652 1533 1538 1629 1585 1657 1719 2707 2645 2670 2720 3218 3192 2696 2679 3177 3191 3260 3218 3236 3593 3253 3617 3633 3598 3593 3554 a Al1 ~P2/6-31G(d)//MP2/6-31G(d).Al1 C2, except 11 ChIn ci1. TABLE A5.4: Harmonic Vibrational Frequencies for C, ~zaborinines'.

1 2 4 5 8 13 16 15 a Al1 ~~2/6-310(d)//~~2/6 -310 (d) except 4 which is MP2/6 -31~(d,p) //~~2/6-310 (d,p) . A11 planar except 15. In cm''. Appendix 6. Supplementary Material for Chapter 3, Heteroaromatic Azaboracycles: Structures and Vibrational Frequencies

TABLE A6.3: XE Bond ~engths. and XYEl Angleab for Al1 Oxazaboroles. 2-H 3 -H 4-H 5-H 12H 23H 3 4H 45H N. m. w w O CU FI rl

P. m. 0 w di rl rlrl

bi. (U. 00 m rl CU 4 4

cri. m . w di O N 4 rl cri U'OJ. . rl rl aD O O 4 rl rl rl

rl. rl. m rl 4 0 rl rl

4. rl . QI rl 40 rl rl

e. eo. rl DO O rl rl rl

w b m QD TABLE A6.4 : Dihedral Anglesa1', 8tabilitiesC with respect to Planar Conformation and Dipole Moment angle^^'^ for Ail Nanplanar Five-Membered Rings; MP2/6-310(d) and Cl Symmetry Excegt as Noted.

a In degrees. Parameter labels refer to canonical locants (as in Figures 3-5 and 7). In kcal/mol. MP2./C. 4 - 180°. ' Except as Noted. C2. MP2/6-31G(d,p). MP2/6-311G(dlp). Cs (nonplanar). KNCH. TABLE A6.5 : Harmonic vibrational ~requenciea' and Infrared lntensitiesb for Diazaborinines, Al1 C,. 18 3 9 20 21 2 2 23 24 25 26 27 a In cm-'. * Given qualitatively, with vw < 2 I w < 20 5 m < 60 I s < 125 5 vs, in km mol-l. ~11 MP216-31G(d)//MP2/6-31G(d). TABLE A6.6: Ilarmonic Vibrational Frequencies and Infxared Intensities for C, ~riazadiborinines.'

28 29 30 31 33 3 4 35 36 a Cf. notes on Table A6.5. TABLE A6.7: Barmonic Vibrational Frequencies and Infrared Intensities for Triazadiborinines; C, Except as ~oted..

974w 929w 9 3 Ovw 94lw 921w 997w 895w 87 9vw

1022w 1018vw 1004w 1003w 1017m 1014w Bl 285~ 292~ 580w 47 5m 732w 792m 9 1lvs 871vs 2690vs 27 08"s 2759s 2741w 2727s 2735s 1507s 1509vs 3274w 3255" 3198~ 3222w 3101s 3148111 2534s 1747~s 3543rn 3586m 3540m 3508m 3627m 3627m 2703"s 2657~s 3588m 359% 3575m 3557m 3642s 3636m 3586s 3608s a Cf. notes on Table A6.5. Cz,. TABLE A6.8: Hamonic Vibrational Frequencies and Infrared ~ntensitiesfor diazaborolee and Triazadiboroles; Synmietry as N~ted.~

44b 45= 5gc 4 6d 47' 58f

TABLE A6.9: Harmonie Vibrational Frequencies and Jnfrared Intensitiea for C, Triazabaroles, Oxazaboroles, and ~xadiazadiborole.~

49 52 53 60 61 63 6 4 65 85

a Cf. notes on Table A6.5. TABLE A6.10: Harmonic Vibrational Frequenciee and Infrared Intenaitiee for Non-C, Triazaboroles, Oxazaborole, and Oxadiazadiboroles; Cl Except as N~ted.~

48 50 51 62 84 86b 87=

3615s 3654s 3674w

a Cf. notes on Table A6.5. C2,. C2. TABLE A6.11: Harmonic Vibrational Frequenciei and Infrared Intensities for the C, Tetrazaborole and ~xadiazaboroles.~

a Cf. notes on Table A6.5. TABLE A6.12: Harmonic Vibrational Frequenciee and Infrared Intensities for Tetrazaboroles and Oxadiazaboroles; Cl Excegt as ~oted.~

5 4 66 68 7 7 56b 57b

a Cf. notes on Table A6.5. C,,. TABLE A6.13: ~annonicVibrational Frequencies and Infrared Intensities for Oxatriazaboroles; C, Except as Noted.'

389~s A" 550~ 509~ 209s 5 4 6vw 506~ 521m 656vw 63Sm 577w 653vw 5849 638w 772s 713s 657m 749s 624m 812w 849m 864m 827w 830m 841" 87 3m A' 829~ 746m 710m 801w 372~s 910w 932w 851vw 820w 86 8m 809vw 9 6 Ow 988v 949w 931vw 894w 878w 1074w 1066w 997w 970rv 969vw 936vw 1117~ 1114m 1102~ 1089111 1094" 1014m 1134s 1149w 1157" 1163m 1197vw 1132w 1247m 1245s 1310s 1278m 1248w 1255m 1346s 1410m 1368s 1371~ 1374m 1422m 1441w 1550m 1470m 1447, 1480m 1498s

2841s 2840s 2849s 2 82 Os 2833s 2833s 3596~s 3583s 3697s 3598~s 3628vs 3683vç

a Cf. notes on Table A6.5. Cl.