ARTICLE pubs.acs.org/JPCA Understanding Conjugation and Hyperconjugation from Electronic Delocalization Measures Ferran Feixas,*,† Eduard Matito,‡ Jordi Poater,† and Miquel Sola*,† † Institut de Química Computacional and Departament de Química, Universitat de Girona, Campus de Montilivi, 17071 Girona, Catalonia, Spain. ‡ Kimika Fakultatea, Euskal Herriko Unibertsitatea, and Donostia International Physics Center (DIPC), P.K. 1072, 20080 Donostia, Euskadi, Spain bS Supporting Information ABSTRACT: The concepts of conjugation and hyperconjugation play an important role to provide an explanation for several fundamental phenomena observed in organic chemistry. Because these effects cannot be directly mea- sured experimentally, their assessment became a primary concern for chemists from the very beginning. In general, the stabilization produced by both phenomena has been studied by means of isodesmic reactions and energy based analysis such as the energy decomposition analysis. In recent years, electronic delocalization measures have been successfully applied to elucidate the nature of chemical bonding and the aromatic character of all kind of molecules. Because conjugation and hyperconjugation stabilizations are strongly linked to the concept of electron delocalization, this paper will give an account of both effects from the point of view of electronic delocalization measures calculated within the framework of the quantum theory of atoms in molecules. In particular, we focus our attention in the controversial case of the stabilization by conjugation in 1,3-butadiyne and 1,3-butadiene. Unexpectedly, theoretical calculations based on the scheme proposed by Kistiakowsky to quantify the extent of stabilization due to conjugation predicted that the conjugation of 1,3-butadiyne was zero. Subsequent energetic analyses contradicted this observation. These studies pointed out the presence of hyperconjugation stabilization in the hydrogenated product of 1,3-butadiyne and 1,3- butadiene that were used as reference systems in the Kistiakowsky’s scheme. Consequently, the extra stabilization of 1-butyne due to hyperconjugation hides the stabilization by conjugation of 1,3-butadiyne. Our results based on electron delocalization measures confirm both the presence of conjugation in 1,3-butadiene and 1,3-butadiyne and hyperconjugation stabilization in their respective hydrogenated products, 1-butene and 1-butyne. 1. INTRODUCTION Despite their importance and widespread use, neither con- The concepts of conjugation and hyperconjugation have been jugation nor hyperconjugation can be experimentally directly measured. Therefore, many theoretical works have been devoted extensively employed to rationalize the structure, stability, and fi reactivity of several organic compounds. Conjugation basically to nd methodologies that can help us to study the role played by these effects. Thus, several tools have been used to characterize involves interactions between π-orbitals, although its definition À them, including isodesmic reactions,5 9 energy decomposition has been extended to p-orbitals to cover lone pair interactions 10À16 17,18 with the π-system. The concept of hyperconjugation, which was analyses (EDA), valence bond theory, the Laplacian of the charge density and the ellipticiy as derived from the atoms-in- introduced by Mulliken in 1939, accounts for the interaction 19,20 21,22 π molecules theory, natural bond orbital (NBO) analyses, between two orbitals with -symmetry where one or both of 23 1,2 fi the concept of protobranching, and the block-localized wave them come from a saturated moiety. It can also be de ned as 23,24 the interaction between the orbitals involved in a σ-bond (usually function (BLW) method. The use of isodesmic reactions À À π may lead to contradicting results (see for instance refs 21 and 16) C HorC C) with those related to an adjacent -bond (usually ffi CdC) or another σ-bond. Both conjugation and hyperconjuga- because in isodesmic reactions it is di cult to isolate the conjugation tion effects lead to stabilizing interactions that influence the geometry, electron density, dissociation energies, or nuclear Special Issue: Richard F. W. Bader Festschrift magnetic resonance properties among many other physicochem- ical observables.3 For instance, it has been shown that hypercon- Received: June 1, 2011 jugation effects are decisive to explain CÀH bond dissociation Revised: August 29, 2011 energy differences in alkanes.4 Published: September 21, 2011 r 2011 American Chemical Society 13104 dx.doi.org/10.1021/jp205152n | J. Phys. Chem. A 2011, 115, 13104–13113 The Journal of Physical Chemistry A ARTICLE À and hyperconjugation stabilizations from other effects.8,13,14,23,25 27 pairs of atoms. The most popular measures of electron deloca- In addition, different isodesmic reactions may provide significantly lization use two-electron quantities such as the pair-density, the different results for the same species.25 These drawbacks have been conditional probability, or the exchangeÀcorrelation density partially solved by the EDA that allows a direct estimation of (XCD).33 The work of Wiberg,34 was the first to define an ESI conjugation and hyperconjugation effects by analyzing the interac- based on the XCD. In a landmark paper of 1975, Bader and tion energy between different fragments of the molecule13,14 and by Stephens35 described how the XCD can be used to quantify the NBO and BLW approaches. Although the electron conjugation is delocalization or sharing of electrons over different regions of the represented by the electronic delocalization along the π-system, molecular space; this measure was going to be called the there have been very few attempts to address this effect from delocalization index (DI) by Fradera, Austen, and Bader in electron delocalization measures. In this context, Bader and co- 1999.36 The DI is obtained by double integration of the XCD Γ workers proposed to analyze the electron density at a bond critical ( XC(Br1B,r2)) over the regions of atoms A and B: point to study conjugation and hyperconjugation effects of CÀC Z Z bonds in hydrocarbons.19 δð ; Þ¼ À Γ ð ; Þ ð Þ A B 2 XC Br1 Br2 d Br1 d Br2 1 One of the most striking examples of the conjugation and A B ff π hyperconjugation e ects shows in the analysis of the -conjuga- Other definitions of ESIs are available in the literature, related tion stabilization in polyenes and polyynes. In 1936, Kistiakowsky in one way or another to the original definition of the DI by Bader À proposed a method to evaluate the conjugation stabilization of and Stephens.37 43 The calculation of the ESI needs a scheme to 1,3-butadiene that consists on the calculation of the energy define the domains of the atoms in the molecule. In this study we ff 28 di erence between its two hydrogenation steps. In 2003, will use the quantum theory of atoms in molecules (QTAIM) Rogers et al. applied Kistiakowsky’s scheme to quantify the 44 fi 26 molecular space partition introduced by Bader, which de nes conjugation stabilization of 1,3-butadiyne. Unexpectedly, they the atomic regions from the condition of zero-flux gradient in the found that the stabilization produced by two conjugated triple density. fi bonds was zero. This nding triggered subsequent energetic For monodeterminantal closed-shell wave functions the DI studies that analyzed the connection between conjugation reads and hyperconjugation effects in the two hydrogenation steps 8,13 occ:MO proposed by Kistiakowsky. These works concluded that δð ; Þ¼ ð Þ ð ÞðÞ Kistiakowsky’s scheme is not adequate to determine conjugation A B 4 ∑ Sij A Sij B 2 i;j energy in butadiynes because it does not take into account the role of hyperconjugation in 1-butyne, the first hydrogenated The summations in eq 2 run over all occupied molecular product. orbitals. Sij(A) is the overlap between molecular orbitals i and j In this work, we put forward a new method to quantify con- within the basin of atom A. δ(A,B) provides a quantitative idea of jugation and hyperconjugation effects from electron delocaliza- the number of electrons delocalized or shared between atoms A tion measures. Unlike isodesmic reactions, the method proposed and B, and they are strictly positive for single-determinant wave in the present work does not rely on reference systems. The functions. To study the effect of π-conjugation on the electron method is applied to compare the strengths of π-conjugation delocalization, we split the value of the DI into σ and π stabilization in 1,3-butadiene and 1,3-butadiyne, to refute Rogers’ components. This separation is only strictly possible for planar 8 observation, and to corroborate the results of Houk and systems with only σ- and π-occupied orbitals where Sσπ(A) = 0 Frenking13 from the electron delocalization viewpoint. Subse- and, thus, δ(A,B) can be exactly split into σ and π contributions. quently, the role played by hyperconjugation in 1-butene and δð ; Þ¼δ ð ; Þþδ ð ; ÞðÞ 1-butyne is discussed. We anticipate here that our results show A B σ A B π A B 3 that there is a good correlation between the results from EDA The definition of DI can also be extended to analyze the and those derived from electron delocalization measures. amount of electrons shared between more than two regions. These are called multicenter indices and they were defined 45 2. METHODOLOGY by Giambiagi and co-workers and extended to QTAIM by Bochicchio et al.46 For the conjugation and hyperconjugation The analysis of the electronic distribution in molecules has analysis we are interested in four-center delocalization indices occupied a central position in chemistry since the seminal work (4c-DIs), which at the monodeterminantal closed-shell level read of Lewis.29 After the advent of quantum mechanics, the quanti- : fication of electron delocalization became one of the challenges occ MO δðA;B;C;DÞ¼16 ∑ SijðAÞ SjkðBÞ SklðCÞ SliðDÞð4Þ for theoretical chemists to characterize the electronic structure of i;j;k;l molecules.