Chem. Rev. 2005, 105, 3911−3947 3911 Theoretical Evaluation of Electron Delocalization in Aromatic Molecules by Means of Atoms in Molecules (AIM) and Electron Localization Function (ELF) Topological Approaches Jordi Poater,‡ Miquel Duran,‡ Miquel Sola`,*,‡ and Bernard Silvi*,§ Institut de Quı´mica Computacional and Departament de Quı´mica, Universitat de Girona, 17071 Girona, Catalonia, Spain, and Laboratoire de Chimie The´orique, Universite´ Pierre et Marie Curie, UMR-CNRS 7616, 4 place Jussieu, 75252 Paris Cedex 05, France Received December 28, 2004 Contents 1. Introduction 1. Introduction 3911 Benzene is the emblematic example of an aromatic 2. Electron Delocalization 3915 molecule, and the problem of its structure has given 2.1. General Aspects 3915 rise to a chemical serial story running over several 2.1.1. Partition of Molecular Space 3915 decades. The epistemological digest of this story writ- 2.1.2. Multivariate Analysis of Electron Densities 3915 ten by Stephen G. Brush1,2 shows how this problem 2.1.3. Spin Pair Composition 3917 has been at the root of important concepts such as 2.2. Electron Delocalization in the Framework of 3918 those of mesomery and resonance. Before the advent Atoms in Molecules Theory of quantum mechanics, chemists had thought of the 2.2.1. Technical Aspects 3918 benzene structures in terms of two center bonds 2.2.2. Examples and Applications in Chemistry 3922 attempting to preserve the valence of the carbon atom and to explain its chemical properties. Kekule´’s 2.3. Electron Delocalization from the Electron 3925 3 Localization Function theory of the structure of the benzene molecule in- 3. Applications to the Study of Aromaticity 3927 vokes the oscillatory hypothesis in which “the fourth 3.1. View of Aromaticity from the Atoms in 3927 valence of each carbon oscillates between its neigh- Molecules Theory bors, synchronously with all the other fourth va- lences, so that the structure switches rapidly between 3.1.1. Electron Delocalization in Aromatic 3927 1 Systems: Preliminary Studies the two structures”, whereas Claus proposed a 4 3.1.2. New Electronic Aromaticity Criterion: 3929 diagonal hypothesis in which the fourth valence of Delocalization Index each carbon is directed toward the carbon in the para 3.1.3. Homoaromaticity, a Particular Case 3931 position. The latter hypothesis has been rejected because it enables only two derivatives, and it has 3.1.4. Aromaticity of Planar and Bowl-Shaped 3932 Polycyclic Aromatic Hydrocarbons and been revised to remove this inconsistency: instead Fullerenes of forming a bond, the fourth valence stops near the - 5 3.1.5. Substituent Effect and Aromaticity 3934 center of the ring in the Armstrong Baeyer formula, or there is only one bridging bond as in the Dewar’s 3.1.6. Aromaticity Divergences in Pyracylene 3936 bridged benzene formula.6 In Thiele’s partial valence 3.1.7. Fluctuation of Electronic Charge as 3938 model,7 the adjacent carbon-carbon bonds are con- Aromaticity Descriptor sidered as intermediate between single and double 3.1.8. Summary of the Application of AIM to 3939 Aromaticity bonds. These formulas were later considered by K. C. Ingold to set up his intra-annular tautomerism,8 3.2. Electron Localization Function Contributions 3939 to Aromaticity which appears to be the generalization of Kekule´’s oscillatory hypothesis. Ingold’s tautomerism hypoth- 3.2.1. Electrophilic Substitution Indicators 3939 esis was later called mesomerism.9 The mesomery is 3.2.2. Correlation with Other Indicators 3940 an important concept in chemistry, which implicitly 3.2.3. ELFπ Scale 3941 introduces the electron delocalization in the context 3.2.4. ELF Covariance Analysis 3941 of the prequantum electronic theory. The first ap- 4. Concluding Remarks 3942 plications of quantum chemistry to the benzene 5. Acknowledgments 3943 problem led on the molecular orbital (MO) side Erich 6. Abbreviations 3943 Hu¨ ckel to propose his famous 4n + 2 rule10 and on 7. References and Notes 3943 the valence bond (VB) side Pauling and Wheland to identify resonance with Ingold’s mesomerism.11,12 An enlighting contribution of modern VB theory on the 13 * Authors to whom correspondence should be addressed (tele- benzene structure has been brought by Shaik et al., phone +34-972-418912; fax +34-972-418356; e-mail miquel.sola@ who have shown that the hexagonal symmetry of udg.es or [email protected]). ‡ Universitat de Girona. benzene is due to the σ-system because the π com- § Universite´ Pierre et Marie Curie. ponent is distortive along a Kekulean distortion. 10.1021/cr030085x CCC: $53.50 © 2005 American Chemical Society Published on Web 07/13/2005 3912 Chemical Reviews, 2005, Vol. 105, No. 10 Poater et al. Jordi Poater was born in Banyoles, Catalonia, Spain, in 1977. He received Miquel Sola` was born in Fonteta, Catalonia, Spain, in 1964. He received his bachelor’s degree in chemistry in 1999 at the University of Girona. At his diploma of chemistry at the University Autonomous of Barcelona in the same university, in 2003 he obtained his Ph.D. in computational and 1986 and his Ph.D. from the same university in 1991, both with academic theoretical chemistry, under the supervision of Profs. Miquel Sola` and honors. His doctoral research on the carbonic anhydrase enzymatic Miquel Duran. His thesis was based on the application of the delocalization catalysis under the direction of Joan Bertran and Agustı´ Lledo´s was index (DI), derived from the bielectronic density, to the analysis of and awarded the Catalan Saint Albert Prize. After several months as a applications in molecular structure, chemical reactivity, and aromaticity. consultant with a private company, in 1993 he moved to the University of In particular, his most relevant works are based on the introduction of a Girona (UdG) as assistant researcher. In 1994 he did postdoctoral research new aromaticity criterion based on electron delocalization (PDI). At this in Amsterdam with Evert Jan Baerends and in 1995 in Calgary with Tom time he is carrying out postdoctoral research with the group of Dr. F. Ziegler. He was appointed assistant professor at UdG in 1997. In 2001, Matthias Bickelhaupt (Vrije Universiteit Amsterdam, The Netherlands), in he was awarded the Distinguished University Research Promotion of the the quantum biological field. Generalitat de Catalunya. Since 2003, he has held a permanent position as full professor with UdG. He has published about 120 papers on theoretical studies of organic and organometallic reaction mechanisms, the use of quantum molecular similarity measures, the development of conceptual density functional theory, and the analysis of electron delocalization and aromaticity. Miquel Duran was born in Catalonia, Spain, in 1957 and was educated at the Universitat Auto`noma de Barcelona (UAB), where he received his Ph.D. in 1984. After pursuing postdoctoral work with Prof. H. F. Schaefer at the University of California, Berkeley, he became first assistant professor at UAB and later in 1992 full professor at the University of Girona. His Bernard Silvi (1946) is full professor at the University Pierre et Marie primary research interests lie in the development of methodology for the Curie in Paris. He is the director of the Laboratoire de Chimie The´orique study of nonlinear optical effects, intermolecular interactions, and molecular (UMR7616), a joint University−CNRS research group involved in both electron density. methodological and applied quantum chemistry. He is author or coauthor The concepts of electron localization and delocal- of about 150 publications in scientific journals. His research activity has been successively devoted to vibrational spectroscopy, intermolecular ization emerging from the analysis of the electronic interactions, modeling of minerals, and silicated materials and in the past structure of benzene are relevant not only for under- 10 years to the topological approaches of chemical bonding. standing the molecular structure in aromatic mol- ecules14 but also for general modern chemistry. Thus, explain the nature of the chemical bond17-19 and with delocalization of π-electrons has been long invoked the aim of establishing a link between the rigorous to rationalize the molecular structure, stability, but abstract wave function and the classical chemical magnetic properties, and chemical reactivity of π-con- concepts based on the Lewis theory20,21 and the jugated molecules. Electron localization is also es- valence shell electron pair repulsion (VSEPR) model sential for descriptive chemistry because in this field of molecular geometry.22,23 Electronic localization/ one needs to know where local groups of electrons delocalization plays also a key role in the analysis of such as core or valence electrons, electron pairs, electronic fluctuation and electron correlation bonding pairs, unpaired electrons, or π-electron effects16,24-27 and, therefore, this concept is relevant subsystems are placed. In particular, appropriate in the development of new density functionals.28-30 auxiliary tools allowing electron pair localization15,16 A simple search on the Science Citation Index of the have been long pursued in quantum chemistry to ISI Web of Knowlegde31 gives about 2500 entries for Electron Delocalization in Aromatic Molecules Chemical Reviews, 2005, Vol. 105, No. 10 3913 the use of the “electron* localization” or “electron* energy unchanged. Two principal criteria have been delocalization” expressions in titles, keywords, or used to compute localized MOs from the delocalized abstracts