Aromaticity and Antiaromaticity E-Book

Miquel Solà

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 508

Veröffentlichungsjahr: 2022

Beschreibung

Aromaticity and Antiaromaticity A comprehensive review of the science of aromaticity, as well as its evolution, from benzene to atomic clusters In Aromaticity and Antiaromaticity: Concepts and Applications, a team of accomplished chemists delivers a comprehensive exploration of the evolution and critical aspects of aromaticity. The book examines the new global criteria used to evaluate aromaticity, including the Nucleus Independent Chemical Shift (NICS) index and the electronic indices based on electronic properties. Additional discussions of inorganic aromatic compounds developed in this century, which give rise to new concepts like multifold aromaticity, are included. Three-dimensional aromaticity found in fullerenes and nanotubes, Möbius aromaticity present in some annulenes, and excited state aromaticity are explored as well. This volume explores the geometrical, electronic, magnetic, and thermodynamic characteristics of aromatic and antiaromatic compounds and their reactivity properties. It also provides: * A thorough historical overview of aromaticity, as well as simple electronic and structural models * Comprehensive explorations of organic and inorganic aromatic compounds, concepts of stability and reactivity, and geometric, energetic, magnetic, and electronic criteria of descriptors of aromaticity * Practical discussions of heteroaromaticity, as well as Möbius aromaticity and excited state aromaticity * In-depth examinations of sigma, pi, delta, and phi aromaticity Perfect for graduate students, researchers, and academics interested in aromaticity, organometallic chemistry, and computational chemistry, Aromaticity and Antiaromaticity: Concepts and Applications will also earn a place in the libraries of professionals and researchers working in organic, inorganic, and physical chemistry.

Leseprobe

Aromaticity and Antiaromaticity

Concepts and Applications

Miquel Solà

Universitat de Girona, Spain

Alexander I. Boldyrev

Utah State University, UT, USA

Michał K. Cyrański

University of Warsaw, Poland

Tadeusz M. Krygowski

University of Warsaw, Poland

Gabriel Merino

Cinvestav Mérida, Mexico

This edition first published 2023

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The right of Miquel Solà, Alexander I. Boldyrev, Michał K. Cyrański, Tadeusz M. Krygowski, and Gabriel Merino be identified as the authors of this work has been asserted in accordance with law.

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Foreword

One of the specific features of chemistry compared to physics is the importance of heuristic models, which describe, interpret, and classify the structures and reactivities of molecules. For the evaluation of chemical models, it is important to recognize that they are not right or wrong, but more or less useful for the interpretation and prediction of experiments. To put it in the words of a pioneer of theoretical organic chemistry: “The only criterion for a model is its usefulness, not its ‘truth’” (M. J. S. Dewar, J. Am. Chem. Soc. 1984, 106, 669). Many of the models still valid today were developed before the first physically correct description of the chemical bond was presented by Heitler and London in 1927. This is remarkable because all of the models proposed so far were based on ideas from classical physics, although the chemical bond in molecules is a quantum theoretical phenomenon that is subject to its own postulates, which cannot be derived from classical physics. It seems that the models obtained by experimental observations and filtered by discussions, which served as a neural network, contain quantum theoretical phenomena.

With the advent of quantum theory and its application in chemistry, the models were partially modified and they received a quantum chemical basis that allowed a theoretical understanding and an extension of the rules associated with the model. A prominent example is aromaticity, which is the subject of this book. The concept of aromaticity, which goes back to the introduction of a planar ring structure with three double bonds for benzene by Kekulé in 1865, was given a quantum theoretical basis by Erich Hückel in 1931 that was later translated in 1951 by William Doering into the famous 4n + 2 “Hückel's rule.” The emergence of quantum chemical methods combined with numerous theoretical models and the development of sophisticated experimental techniques in recent decades have led to an enormous expansion in the scope of aromaticity far beyond the original chemistry of benzene derived compounds. The concept of aromaticity was extended from unsaturated π‐bonded systems in organic chemistry to saturated σ‐bonded compounds in inorganic chemistry and the synthesis of metallaaromatic compounds following a theoretical prediction by Roald Hoffmann among many other discoveries and suggestions greatly expanded its scope. The restriction of aromaticity to cyclic compounds was removed by the proposal of Y‐conjugation in tri‐substituted molecules by Gund and the relevance for aromatic stabilization in electronically excited states by Baird extended the scope of aromaticity beyond ground state properties. A major impact to the concept of aromaticity is the finding that π‐delocalized aromatic compounds exhibit particular magnetic properties, which were suggested to identify aromatic systems and to estimate the degree of aromaticity.

The scope and limitation of aromaticity and the introduction of new classes of aromatic compounds have been at the center of numerous controversial discussions in recent years. It is appropriate and very timely to summarize the present standing of theoretical and experimental research on the topic of aromaticity. The authors of this book have provided an overview of the current state of the art in 14 chapters. The well‐chosen selection summarizes the progress and the most important facets of the current experimental and theoretical research on aromatic compounds and the different points of view on aromaticity, which have been considerably expanded in the last two decades. The question of the occurrence and strength of aromaticity in molecules will continue to be the subject of controversial discussions in the future. It is questionable whether the almost 50 different types of aromaticity proposed in recent decades are a useful addition to the original arsenal. It remains to be seen whether the observation that aromatic compounds exhibit certain magnetic properties also allows the reverse conclusion that the occurrence of such magnetic quantities automatically indicates aromatic stabilization. These are just two of the exciting questions about aromaticity that will be the focus of ongoing research in the field. This book is a timely and important contribution to the current state of the debate, which will be continued.

Gernot Frenking                        

Philipps‐Universität Marburg

Preface

The concept of the chemical bond as the link that connects atoms in a molecule is at the core of chemistry. A chemical reaction is nothing but the breaking of some chemical bonds and the formation of others. Despite its importance, the chemical bond is not exactly defined and this is the origin of many past and ongoing controversial debates. Aromatic compounds bear a particular type of chemical bond characterized by extensive electron delocalization in closed circuits. Aromaticity as an idea is more than 150 years old and is still not only alive, but even briskly developing. Like for the chemical bond, aromaticity is a not well‐defined property that cannot be measured experimentally. However, the chemical consequences derived from this property are evident and can be quantified. Moreover, this concept is extremely useful to rationalize the structure, stability, and reactivity of many species that otherwise would not be understood.

The initial Kekulé and Erlenmeyer concept of aromaticity was based on structural and reactivity criteria, respectively (cyclic systems resistant to some kind of reactivity). Later it was found that, classically, aromaticity is associated with electron delocalization (at the beginning only by means of π‐electrons), which in many cases was associated with reactivity (and some other properties as magnetic ones). Some root understanding of the aromatic character has in the last decades expanded wider, introducing many aromaticities with various prefixes to underline slightly different understanding of the original meaning. New criteria introduced by means of new “indices” of aromaticity allowed significant broadening of the original term aromaticity that now covers the entire Periodic Table and different electronic states. This has been exemplified with the electron localization functions of the ground state of benzene and the first excited state of octahedral 5A1g Be6 species depicted in the cover graphics of the book (pictures courtesy of Dr. Ouissam El Bakouri).

Over the last 30 years, there has been a significant expansion in the number and types of aromatic compounds and in our understanding of the concept of aromaticity. This book was written to provide students at the advanced undergraduate level and graduate level with a deep discussion of the recent advances in the field of aromaticity. A good knowledge of standard first‐year courses in organic and theoretical chemistry is recommended.

Chapter 1 starts with a brief historical account of the concept of aromaticity. Next, in Chapter 2, we focus on simple electron counting rules that are followed by a large number of organic and inorganic aromatic compounds, such as Hückel's, Baird's, Wade‐Mingos', and Hirsch's rules together with other less known aromaticity rules.

In 2001, Boldyrev, Wang, and coworkers detected a series of bimetallic compounds containing Al42−, the first all‐metal aromatic cluster identified, which is σ‐ and π‐aromatic and, therefore, it shows multifold aromaticity. It is now widely accepted that the π‐aromaticity of classical organic compounds is not unique and that there are chemical compounds having σ‐, δ‐, and ϕ‐aromaticity, together with combinations of these different types of aromaticity. Aromaticity and antiaromaticity in metal systems and the new types of aromaticity found in these molecular entities are discussed in Chapters 3 and 11.

The connection among aromaticity, stability, and reactivity is well established. However, among a series of isomers or different electronic states, the most stable and less reactive is not always the most aromatic. The role of aromaticity on stability and reactivity is analyzed in Chapter 4.

The number and variety of molecules that have aromatic character has grown exponentially in the last years. It is important to have methods at hand that allow careful classification of any system as aromatic/nonaromatic/antiaromatic. The last two decades brought a number of powerful tools to diagnose aromaticity. Readers interested in a detailed account of the existing descriptors of aromaticity, are referred to Chapters 5–8. In these chapters, we present the definition of the most used indices of aromaticity based on geometric, energetic, magnetic, and electronic properties of molecules.

The list of the different types of aromaticity is very large. Among them, one of the most important is heteroaromaticity, a particular case of aromaticity found in organic classical molecules in which a CH group or C atom of the aromatic ring is replaced by a heteroatom like O, N, or S. Chapter 9 refers to this type of aromaticity.

A conjugated monocyclic molecule of 4n π‐electrons with a set of p‐orbitals in which there is a single (or an odd number) out‐of‐phase overlap constitutes a Möbius aromatic species. Chapter 10 provides a detailed description of Möbius aromaticity.

Properties derived from the aromatic character of classical organic molecules are due to the π‐electrons. However, it has been shown that the π‐electrons of benzene possess a distortive tendency that transforms the D6h molecular symmetry into a D3h symmetry structure. Therefore, D6h symmetry in benzene is not due to the π‐electrons, but to the σ‐system. Chapter 12 gathers a series of works discussing this paradox.

The concept of aromaticity was initially developed to account for the unusual stability of planar benzene derivatives. However, certain molecules as closo boranes and charged fullerenes are examples of species with aromaticity in three dimensions. Chapter 13 analyzes three‐dimensional aromaticity with a special emphasis on spherical aromaticity.

In 1972, Baird showed that the lowest lying triplet excited states of conjugated monocyclic compounds with 4n π‐electrons are aromatic, introducing the so‐called excited state aromaticity (ESA). Nowadays, it is widely accepted that ESA is important to understand many photochemical reactions. Chapter 14 focuses on the recent developments of ESA, which is presently one of the hot topics in the field of aromaticity.

For all chapters, the references are placed at the end of each to allow the reader to quickly get acquainted with the field.

A number of people have contributed to this book throughout the past years. The authors wish to express their gratitude to all of them, especially, to all PhD and postdoc students that have been working in our groups performing studies that improved our knowledge on aromaticity. Finally, yet importantly, we thank our families for their patience and continuous support.

Finally, the authors wants to dedicate this book to the late Prof. Dr. Paul von Ragué Schleyer for his ground breaking contributions to aromaticity.

Miquel Solà                      

Girona, Catalonia, Spain

Alexander I. Boldyrev    

Kauai, Hawaii, USA        

Michał K. Cyrański        

Warsaw, Poland              

Tadeusz  M. Krygowski

Warsaw, Poland              

Gabriel Merino              

Mérida, México              

List of Abbreviations

2c‐2e

two‐center two‐electron

2cBO

Orthogonalized two‐center Bond Orbitals

2c‐ESI

Two‐center Electron Sharing Indices

5‐MR

Five ‐Membered‐ring

6‐MR

Six‐Membered ring

ACID

Anisotropy of the Induced Current Density

AdNDP

Adaptive Natural Density Partitioning

AIM

Atoms‐in‐Molecules

AO

Atomic Orbital

AOM

Atomic Overlap Matrix

ARCS

Aromatic Ring Current Shieldings

ARE

Adiabatic Resonance Energy

ASE

Aromatic Stabilization Energy

ATI

Average Two‐Center Index

BAC

Bond Alternation Coefficient

BE

Bonding Energy

BH

Bingel‐Hirsch addition

BLA

Bond Length Alternation

BLE

Bond Length Equalization

BLW

Block‐Localized Wave function

BOIA

Bond Order Index of Aromaticity

BV

Bifurcation Value

CASSCF

Complete Active Space Self‐Consistent Field

CCRE

Conjugated Circuits Resonance Energy

CMO

Canonical Molecular Orbital

COT

Cyclooctatetraene

CTED

Corrected Total Electron Density

DA

Diels‐Alder cycloaddition

DBU

1,8‐diazabicycloundec‐7‐ene

DDQ

2,3‐dichloro‐5,6‐dicyano‐1,4‐benzoquinone

DFT

Density Functional Theory

DI

Delocalization Index

DRE

Dewar Resonance Energy

EC

Energy Content

ECRE

Extra Cyclic Resonance Energy

ED

Electron Density

EDA

Energy Decomposition Analysis

EDLA

Electron Density Localized on Atoms

EDDB

Electron Density of Delocalized Bonds

EDLB

Electron Density of Localized Bonds

ELF

Electron Localization Function

EMF

Endohedral Metallofullerene

ESI

Electron Sharing Indices

ESIPT

Excited State Intramolecular Proton Transfer

ETS

Extended Transition State

FLU

Aromatic Fluctuation Index

HCM

Hollow Cylinder Model

HF

Hartree‐Fock

HMO

Hückel Molecular Orbital

HOMA

Harmonic Oscillator Model of Aromaticity

HOMED

Harmonic Oscillator Model of Electron Delocalization

HOMHED

Harmonic Oscillator Model for Heterocycles of Electron Delocalization

HOMO

Highest Occupied Molecular Orbital

HOSE

Harmonic Oscillator Stabilization Energy

HRE

Hückel Resonance Energy

HS

High Spin

HSRE

Hess‐Schaad Resonance Energy

HtFfA

Heat of Formation of a Molecule from Atoms

IPR

Isolated Pentagon Rule

ISE

Isomerization Stabilization Energy

LCAO

Linear Combination of Atomic Orbitals

MCI

Multicenter Electron Delocalization Index

MHP

Maximum Hardness Principle

MO

Molecular Orbital

MPP

Minimum Polarizability Principle

MSO

Molecular Spin‐Orbital

NAO

Natural Atomic Orbital

NBO

Natural Bond Orbital

NCI

Non‐Covalent Index

NDDO

Neglect of Diatomic Differential Overlap

NICS

Nucleus‐Independent Chemical Shift

NIR

Near‐Infrared

NMR

Nuclear Magnetic Resonance

LP

Lone Pair

LUMO

Lowest Unoccupied Molecular Orbital

ON

Occupation Number

PAC

Predictive Aromaticity Criteria

PAH

Polycyclic Aromatic Hydrocarbon

PCH

Polycyclic Conjugated Hydrocarbon

PDI

Para

‐Delocalization Index

pEDA

π Electron Donor‐Acceptor

PJT

Pseudo‐Jahn‐Teller

PLR

Para

Linear Response

QC

QuasiClassical

QTAIM

Quantum Theory of Atoms in Molecules

RAHB

Resonance‐Assisted Hydrogen Bond

RCBV

Ring‐Closure Bifurcation Value

RCM

Ring Current Model

RE

Resonance Energy

REC

Ring Energy Content

REPE

Resonance Energy Per π‐Electron

SA

Salicyldenaniline

SCF

Self‐Consistent Field

sEDA

σ Electron Donor‐Acceptor

SOJT

Second‐Order Jahn‐Teller

SSAdNP

Solid State Adaptive Natural Density Partitioning

TDDA

Dehydrodianthracene

TFVC

Topological Fuzzy Voronoi Cell

TPA

Two‐Photon Absorption

TRE

Topological Resonance Energy

TS

Transition State

VB

Valence Bond

VDE

Vertical Detachment Energy

1Historical Overview

Source: T. M. Krygowski, H. Szatyłowicz, ChemTexts 2015, 1, 12.

“Classification and theory are not ends in themselves. If they generate new experimental work, new compounds, new processes, new methods – they are good; if they are sterile – they are bad.” (E. D. Bergman, in Aromaticity, Pseudo‐aromaticity, Anti‐aromaticity, Proceedings of an International Symposium held in Jerusalem 31 March – 3 April 1970 [Eds. E. D. Bergmann, B. Pullman], Israel Academy of Science and Humanities, Jerusalem, 1971, pp. 392–392)

Aromaticity belongs to one of the four most commonly used terms in organic chemistry and related fields, along with conformation, H‐bonding, and polymerization [1]. As it can be seen from the histogram in Figure 1.1, the topic of aromaticity appears on average more than 40 times per day in 2020. So great interest in research related to aromaticity is associated with a large number of works on the definition of the term, as well as on experimental and theoretical studies on the physicochemical characteristics attributed to the aromatic character of chemical species in their ground, excited, and transition states [2].

Figure 1.1 The number of scientific papers, appearing per day, in which the indicated term appears in title, abstract or keywords (asterisk denotes any adequate ending, i.e., s, ed, ing, ity, or nothing) [1, 2].

The term aromatic applied for the classification of organic species was used for the first time in 1856 by Hoffman in his work on monocarboxylic derivatives of benzene [3]. However, discovery of the archetypic aromatic compound benzene and estimation of its empirical formula as well as its physical properties had been done already 30 years earlier in 1825 by Faraday [4]. Then for almost half of century, most efforts were devoted to the investigation of the structure of benzene and chemical properties of its derivatives. Two aspects were taken into account at that time. Kekulé regarded as aromatic compounds those that are structurally similar to benzene [5, 6], whereas Erlenmeyer [7] considered as aromatic compounds those having alike chemical properties (reactivity) as benzene. Later Kekulé also accepted the importance of a similarity to benzene in chemical properties, and in his Lehrbuch der organischen Chemie[8], classified alcohols, aldehydes, and acids into either aromatic or aliphatic ones. From the stoichiometry, it was known that benzene is built up of single and double bonds, but troubles appeared in trying to understand its structure. The first cyclic structure of benzene as a hexagon was suggested by Laurent in 1855 and presented by Loschmidt [9], but much closer to the actual viewpoint is a structural formula by Kekulé [8], as shown in Figure 1.2.

In the Kekulé formula, the carbons are arranged in a hexagon with alternating double and single bonds between them. In 1869, Ladenburg [10] questioned Kekulé's structure, indicating that there should be four isomers for disubstituted benzene derivatives (Figure 1.3), whereas only three of them were known. Moreover, he suggested another view of its structure, i.e. by presenting it as – actually named – prismane. Figure 1.4 presents relations between isomers of benzene and relates them to isomers of prismane.

Figure 1.2 The Loschmidt (a) and the Kekule (b) cyclic formulas of benzene.

Figure 1.3 Assumed isomers by Ladenburg for disubstituted benzene derivatives.

In defense of his cyclic structure, Kekulé accepted some assumption that the double and single bonds are in a permanent exchange – again an ingenious intuition? The concept of prismane‐like structure of benzene was questioned and rejected by Baeyer [13] since 1,2‐disubstituted benzene derivatives did not exhibit optical activity.

At the same time as benzene, another classically aromatic hydrocarbon, naphthalene, was found in 1821 as a result of distillation of tar coal [14]. Its structure as two benzenes fused was proposed in 1866 by Erlenmeyer [7]. Other important aromatic compounds were those containing heteroatoms. They were discovered at the end of the 19th century. Anderson in 1849 [15] obtained the first heteroaromatic molecule named pyridine through his studies on the distillation of bone‐oil and other animal matter. In 1870, Limpricht obtained furan [16], whereas in 1890, Hantzsch presented a method of synthesis of pyrrole derivatives [17]. Thiophene accompanying benzene in tar coal was discovered by Meyer in 1883 [18]. It exhibited chemical properties similar to benzene. In 1891, Bamberger [19] proposed a cyclic structure for all these compounds.

Figure 1.4 Relationships between substituted benzene and prismane (as a structure of benzene) derivatives [11, 12].

At the end of 19th century, quite a number of aromatic compounds were known, and it was known that these compounds are less reactive than their olefinic analogues. The question that was posed at that time was: Is the resistance of aromatics, thus formally unsaturated compounds, due to their cyclic structure? The question was answered by the work of Willstätter [20], who obtained a derivative of cyclooctatetraene and found that its chemical behavior is similar to a typical olefinic molecule.

A new impetus came from physics. Discovery of the electron and then its role in atoms, and later in molecules, gave a new view on molecular structure. At that time, structural formulae were presented basing on an octet‐rule introduced by Lewis [21] and then developed by Langmuir [22]. A modern understanding of the chemical bond was established suggesting that a chemical bond is a pair of electrons shared by two atoms. In 1922, Crocker [23] presented a new version of the benzene structure built up in agreement with the Lewis–Langmuir theory. It is shown in Figure 1.5.

Figure 1.5 Crocker's electronic structure of benzene.

Crocker's structure took into account six equivalent carbon atoms with six electrons in between. This idea was, for a few years, not noticed, until about 1925 when Armit and Robinson [24] made use of it and introduced the concept of electron sextet, and noted that “the group of six electrons…resists disruption,” describing for the first time the concept of aromatic sextet. They also introduced an assignment of the aromatic sextet by a circle inside the six membered rings. All these works inspired Clar [25] to formulate the π‐sextet rule as an extension of the Hückel's 4n + 2π‐electron rule from monocyclic species to benzenoid systems. Moreover, in 1938, Evans and Warhurst [26] already noted the analogy between the π‐electrons of benzene and the six delocalized electrons in the cyclic transition state of the Diels–Alder reaction between butadiene and ethylene, and suggested for the first time the existence of aromatic transition states.

The next important step in description and understanding aromaticity has come as a result of development of quantum mechanics [27] and its applications to chemical problems [28–30]. Then an important contribution by Hückel has to be stressed. He introduced the concept of σ and π orbitals, describing electrons differing in their properties (symmetry), and then worked out a simplified molecular orbital theory, nowadays known as Hückel Molecular Orbital (HMO) theory [28]. Application of the HMO theory allows interpretation of the nonaromatic properties of cyclobutadiene and cyclooctatetraene, giving a solution for the older question posed in the time of Willstätter. A famous “magic” rule from these works resulted: 4n + 2 saying that cyclic π‐electron systems containing 4n + 2 electrons are stable, whereas those containing 4n electrons are less stable. This rule was formulated by Doering et al. [31], who noted that heptatrienyl cation is more stable than cyclopentadienyl cation since the first one possesses 4n + 2π‐electrons whereas the latter has 4n. A wide and a strong documentation of this rule has been given by Roberts, Streitwieser, and Regan [32], and the rule may be more generally stated as follows: “those monocyclic coplanar systems of trigonally hybridized atoms which contain 4n + 2π electrons will possess relative electronic stability” [33].

The determination in 1959 of the structure of the closo borane B10H102− ion by Lipscomb [34], and the synthesis three years later of the first derivatives of closo‐dodecaborate and closo‐decaborate by Muetterties's group [35], released the concept of aromaticity from the two dimensions and moved it to three dimensions. In 1972, Baird [36] predicted the existence of the lowest‐lying triplet state aromaticity for monocyclic conjugated rings of 4n π‐electrons. This prediction was confirmed by the identification of planar triplet ground states of C5H5+ and C5Cl5+ [37, 38]. The concept of σ‐aromaticity was proposed first by Dewar in 1979 [39] to account for the unexpected small strain in cyclopropane. In 1982, Roper et al. [40] synthesized the first metallabenzene, an osmabenzene, and initiated a new group of aromatic species, the so‐called metalla‐aromatic compounds [41]. Boldyrev, Wang, and coworkers detected in 2001 a series of bimetallic clusters containing Al42−, the first type of all‐metal aromatic clusters known, face capped by an M+ cation (M = Li, Na, Cu) [42].

The development of quantum chemical methods as well as computational techniques allowed an effective application of various theoretically based models related to energetic, magnetic properties, and geometry of the π‐electron structure of molecules. There have appeared in various definitions of resonance energy (RE), recalling the most popular and important, the Dewar resonance energy (DRE) [43], and the Hess and Schaad RE's [44, 45] estimated for hundreds of π‐electron hydrocarbons. At that time, appeared also an idea of isodesmic and homodesmotic reactions [46–50], which have given rise to numerous estimations of variously defined aromatic stabilization energy (ASE) values [51]. Parallel to these achievements, theoretical estimations of magnetic properties appeared, as the anisotropy of magnetic susceptibility Δχ [52] and the magnetic susceptibility exaltation Λ [53] supplementing the data obtained by experimental measurements. Introduced in 1996 by Schleyer, Nucleus Independent Chemical Shift (NICS) [54] has been recognized as an extremely important tool in estimations of aromaticity [55, 56]. Extremely appealing are the maps of induced ring currents, which illustrate not only the ways of cyclic π‐electron (de)localization, but also quantify its extent [57–64]. At that time, molecular geometry became relatively easily accessible as well as available by theoretical models. Then, Julg et al. [65] defined the variance of the perimeter bond lengths in a molecule as a numerical characteristic of aromaticity. This work was the beginning of a plethora of papers that applied the analysis of molecular geometry for describing the aromaticity of molecules [2, 66]. Development of Quantum Theory of Atoms in Molecules [67] as well as other theories that propose new definitions of atomic charges [68] has resulted in a number of approaches applying atomic charges, electronic delocalization as well as physical properties in bond or ring critical points for the quantification of aromaticity [69, 70]. Presentation of these results as well as the appropriate discussion are, among others, the main aims of this book.

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2Simple Electron Counting Rules

“Rules are rules but no laws. They can be broken.”

G. Frenking (the 11th Congress on Electronic Structure: Principles and Applications 2018 held in Toledo)

2.1 Introduction

Archetypal aromatic compounds such as C6H6, C5H5+ in its lowest lying triplet state, Al42−, B12H122−, or C6010+ present high symmetry. Symmetry is a common characteristic of quintessential aromatic compounds. In many cases, these symmetric molecular structures bring about shells of degenerate highest‐occupied molecular orbitals (MOs) that can be fully occupied, resulting in a closed‐shell structure, or can can have the highest‐energy occupied shell only half‐filled with same‐spin electrons. These two types of electronic structure provide an extra energetic stabilization to aromatic molecules that resembles the stabilization gained by the main‐group elements having eight valence electrons in its valence shell (octet rule), transition metals following the eighteen valence electron rule or having d5 parallel spin occupation, and f‐elements that obey the 32‐electron principle. This closed‐shell or half‐filled shell electronic structure is the source of several rules of aromaticity, the most relevant being Hückel's 4n + 2 rule. Given the imprecise character of the aromaticity concept, the existence of a series of very simple mathematical rules based solely on the counting of valence or π‐electrons that can account for the aromaticity of a large number of organic and inorganic molecules is remarkable. Substituents, counterions, external fields, and so on may reduce the symmetry of these representative aromatic compounds and remove the degeneracy of highest‐occupied MOs. Still, in many cases, the electronic structure of these less symmetric species resembles that of typical aromatic compounds and, consequently, they retain a significant aromatic character. This means that the rules that are discussed in the next sections of this chapter can be applied not only to compounds with high symmetry but also to these compounds with less or nonsymmetry at all.

2.2 Hückel's 4n + 2 Rule

In 1931, Hückel [1–4] applied his simplified MO theory to conjugated molecules. The results obtained were used by Doering and Detert to derive the 4n + 2 π‐electron rule [5]. This was the first and most important of a series of rules for aromaticity. The Hückel's 4n + 2 π‐electron rule [1–4] states that monocyclic conjugated hydrocarbons (annulenes, CnHn) of Dnh symmetry with 4n + 2 π‐electrons are aromatic, whereas those with 4n π‐electrons are antiaromatic. The origin of this rule is the MO distribution of π‐orbitals in Dnh annulenes that can be determined with the Hückel molecular orbital (HMO) method or with more sophisticated methodologies. In the HMO method, the following assumptions are made: (i) only π valence electrons are considered, σ electrons are ignored; (ii) the diagonal matrix elements are set equal to α (Coulomb integral), the energy of an electron in a carbon 2p orbital; and (iii) the off‐diagonal elements are set equal to β (resonance integral) if the two atoms are next to each other and zero otherwise. The secular equation to be solved for a cyclic conjugated π‐system of Dnh symmetry with N atoms is the following N × N secular determinant:

(2.1)

The solution of this Hückel determinant is:

(2.2)

The most stable MO (j = 0) is formed by the in‐phase interactions of all p‐atomic orbitals. The rest of the orbitals, except the last one for even N values that has all p‐orbitals out of phase, come by pairs (j = i and j = N − i have the same energy). As can be seen in Figure 2.1, closed‐shell electronic structures are then obtained for the following number of π‐electrons: 2, 6, 10, 14,…, that is, 4n + 2 π‐electrons (n = 0, 1, 2,…). This closed‐shell electronic structure provides aromatic stabilization. On the contrary, the singlet ground state of 4n π‐electrons annulenes does not lead to a closed‐shell electronic structure. As a result, these systems (e.g., butadiene, cyclooctatetraene,…) suffer Jahn–Teller distortions that break the Dnh symmetry and localize the π‐electrons. These annulenes with 4n π‐electrons are antiaromatic. The preparation of the tropylium cation, C7H7+, by Doering and Knox [6] in 1954 is considered the first experimental verification of Hückel's 4n + 2 π‐electron rule. From a theoretical point of view, the first well‐documented proof of the 4n + 2 π‐electron rule has to be attributed to Roberts, Streitwieser, and Regan who computed the resonance energy for a series of cyclic conjugated hydrocarbons [7]. The Hückel's rule can also be applied to species different from classical aromatic annulenes. For instance, the inorganic cluster Li3+ is considered a σ‐aromatic species [8] with two delocalized σ‐electrons that follows Hückel's rule.

Figure 2.1 The distribution of π‐molecular orbitals in Dnh annulenes and the origin of the (a) Hückel's 4n + 2 π‐electron rule and (b) Baird's 4n π‐electron rule.

2.3 Baird's 4n π‐Electron Rule for the Lowest‐Lying Triplet Excited State

The Baird's 4n π‐electron rule [9] for the lowest‐lying triplet excited state, T1, asserts that annulenes of Dnh symmetry that are antiaromatic in their singlet ground state are aromatic in their lowest‐lying triplet state (T1