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Revealing Electron Delocalization through the Source Function Emanuele Monza,†,|| Carlo Gatti,*,‡,§ Leonardo Lo Presti,*,†,§ and Emanuele Ortoleva†,|| †
Dipartimento di Chimica Fisica ed Elettrochimica, Universita degli Studi di Milano, via Golgi 19, I-20133 Milano, Italy Istituto di Scienze e Tecnologie Molecolari del CNR (CNR-ISTM) and Dipartimento di Chimica Fisica ed Elettrochimica, Universita di Milano, via Golgi 19, I-20133 Milano, Italy § Center for Materials Crystallography, Aarhus University, Langelandsgade 140, DK-8000 Aarhus C., Denmark ‡
bS Supporting Information ABSTRACT: The source function (SF) introduced in late 90s by Bader and Gatti quantifies the influence of each atom in a system in determining the amount of electron density at a given point, regardless of the atom’s remote or close location with respect to the point. The SF may thus be attractive for studying directly in the real space somewhat elusive molecular properties, such as “electron conjugation” and “aromaticity”, that lack rigorous definitions as they are not directly associated to quantum-mechanical observables. In this work, the results of a preliminary test aimed at understanding whether the SF descriptor is capable to reveal electron delocalization effects are corroborated by further examination of the previously investigated benzene, 1,3-cyclohexadiene, and cyclohexene series and by extending the analysis to some benchmark organic systems with different unsaturated bond patterns. The SF can actually reveal, order, and quantify π-electron delocalization effects for formal double, single conjugated, and allylic bonds, in terms of the influence of distant atoms on the electron density at given bond critical points. In polycyclic aromatic hydrocarbons, the SF neatly reveals the mutual influence of the benzenoid subunits. In naphthalene it provides a rationale for the changes observed in the local aromatic character of one ring when the other is partially hydrogenated. The SF analysis describes instead biphenyl as made up by two weakly interacting benzene rings, only slightly perturbed by the combination of mutual steric and electronic effects. Eventually, a new SF-based indicator of local aromaticity is introduced, which shows excellent correlation with the aromatic index developed by Matta and Hernandez-Trujillo, based on the delocalization indices. At variance with this latter and other commonly employed quantum-mechanical (local) aromaticity descriptors, the SF-based indicator does not require the knowledge of the pair density, nor the system wave function, being therefore promising for applications to experimentally derived charge density distributions.
I. INTRODUCTION Electron localization and delocalization are undoubtedly among the most important and frequently exploited cornerstones of chemistry.1,2 In their topical review on the theoretical evaluation of electron localization/delocalization in terms of topological approaches, Poater, Duran, Sola, and Silvi (PDSS),1 report about 2500 entries in the Web of Science for the use of the “electron localization” or “electron delocalization” expressions in titles, keywords, or abstracts of articles published between 1990 and 2004. An analogous search for the 20052011 period yields more than 8200 entries. This indicates that this topic has, in the last five years, even increased rather than diminished in its attractiveness in spite of, or perhaps because of, electron localization/delocalization not being an observable and directly measurable quantity,1 though its effects are.1,2 Similar arguments and features may be applied to the concept of aromaticity,14 another cornerstone of chemistry, generally ascribed to the effects of electron delocalization in cyclic systems, but whose univocal definition is likely to be impossible, certainly controversia,l3,4 and potentially full of ambiguities.4 Nevertheless, more than 61 000 entries for the use of “aromatic” or “aromaticity” expressions in titles, keywords, or r 2011 American Chemical Society
abstracts of articles published between 2005 and 2011 are obtained when such a query is posed to the Science Citation Index of the ISI Web of Knowledge. I.1. Conjugation through Electron Localization/Delocalization. Many years ago, Richard Bader and co-workers5 and Dieter Cremer, Richard Bader and co-workers6 published two landmark papers on the description of conjugation, hyperconjugation, and homoaromaticity in terms of electron distributions and of characteristic properties of such distributions at the bond critical points (bcps).7,8 These seminal papers paved the way toward an understanding of the various phenomena related to electron delocalization using electron-based descriptors. In those pioneering papers, dating back to 1983, properties now very frequently employed, like the electron-density-based bond orders, the bond ellipticities, and the extent of the alignment of axes Special Issue: Richard F. W. Bader Festschrift Received: April 29, 2011 Revised: June 23, 2011 Published: July 15, 2011 12864
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The Journal of Physical Chemistry A uniquely defining the plane of the π-electron distribution for each carboncarbon bond, were introduced and explored for the first time. The great merit of those papers was the demonstrated possibility to translate the electronic effects predicted by orbital models into observable properties of the electron density (ED) distribution, F(r).5 A great, practical advantage of the ED description, besides and because of being based on an observable, was the possibility to be equally applied to nonplanar systems, where the σπ separation of the molecular orbital models is no longer feasible. Examples of successful applications were the deciphering of whether or not homoaromatic conjugation was present in a series of nonplanar cations featuring a cyclopropyl ring variously conjugated with an unsaturated fragment (including the highly debated case of the homotropylium cation)6 or the analysis of the competing electron conjugation pathways in some 11,11-disubstituted 1,6-methane[10]annulenes.9 Later, it became clear that the very mechanism of electron delocalization/localization should be only related to and revealed through the two-electron density or pair density, which is the simplest quantity that describes the pair behavior, i.e., the correlated motion of a pair of electrons.1,2 This belief led to the introduction of several descriptors, defined in terms of the exchangecorrelation density (XCD), which measures the deviation, due to Coulomb and Fermi correlation of electron motions, between the true pair density of a system and that given by the purely classical description of a product of independent EDs. Such descriptors, collectively indicated as electron sharing indexes (ESI)1012 according to Fulton’s terminology, measure the extent to which electrons are shared between two or more atoms (or other chemical “entities”). They differ, among each other, as to the way atoms or chemical entities are defined12,13 (e.g., fuzzy atoms12 or quantum theory of atoms in molecules,7 QTAIM, atomic basins) and as to how the XCD is expressed and approximated.12 Probably the nowadays most frequently used ESI formulation is due to Fradera, Austen, and Bader,14 where one of the various possible forms for the XCD is doubly integrated over the same QTAIM atomic domain, A, or over two different QTAIM atomic domains A and B to yield the so-called localization λ(A,A) or delocalization δ(A,B) indices, respectively. It is worth stressing that such descriptors deeply rest upon the concepts developed by Bader and Stephens in their 1974 seminal paper on the “Fluctuation and correlation of electrons in molecular systems”15 and provide a very insightful statistical interpretation of electron localization/delocalization effects,1 because the variance and covariances of the electron populations of the domains are shown to be intimately related to their associated λ and δ values, respectively. Delocalization indices (DI’s) provide an estimate of the number of electron pairs delocalized (shared) between different topological atoms, no matter whether they are directly linked or not by a bond path. DI’s have been largely employed to highlight delocalization effects in nontrivial chemical bonding situations, like those typically occurring in organometallic systems,16,17 and also as ingredients of cumulative indices to quantify aromaticity (vide infra).3 On top of their clear physical meaning, the DI’s have a great advantage over local electron-based descriptors of electron delocalization such as the bond ellipticity, the bond order, the alignment of axes denoting bond π-electron distributions, etc. Indeed, rather than reflecting in an indirect way the influence of more or less remote atomic or bond distributions, the DI’s provide a direct measure of how two bonded or not bonded
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atoms are “talking” to each other in a given system. Such an important difference also holds with respect to those integral indicators which yield a measure of π-density accumulation and delocalization over a conjugated systems, like the out-of-molecular plane principal component of the traceless atomic quadrupole moment tensors.7,18 However, at variance with ED-based descriptors, the DIs necessitate the pair density (or at least the firstorder density matrix in single determinant theoretical approaches) for their evaluation—a fact that generally precludes their use when dealing with EDs derived from experiment or with ab initio computations on periodic systems (e.g., crystals).17 Though lacking the direct physical connection to electron pairing associated to the DI’s, the source function (SF)19 offers an interesting way to directly relate the ED at a point to the influence that more or less distant atomic domains have on its value. Furthermore, the SF, being defined in terms of the Laplacian of the ED, retains the important advantage of not requiring the pair density for its evaluation. It is so applicable on the same grounds to both experimentally derived and theoretical electron densities (and, in general, to all cases where a pair density is not easily available). In the next subsection, the SF is briefly introduced, whereas subsection I.3 discusses whether such descriptor may or may not be able to reveal electron delocalization. I.2. The SF. Bader and Gatti19 have shown that the ED at any point r within a system may be regarded as determined by a local source LS(r,r0 ) operating at all other points of the space: Z ð1Þ FðrÞ ¼ LSðr;r0 Þ 3 dr0 The local source (LS) is expressed as LS(r,r0 ) = (4π 3 |r r | ) 3 r2F(r0 ), where the Green’s or influence function,20 (4π 3 |r r|)1, represents the effectiveness of the cause, the Laplacian of the density at r0 , r2F(r0 ), to produce the effect, the electron density at r, F(r). Integrating LS over the atomic basins of a system, may give F(r) as a sum of atomic contributions 0 1
FðrÞ ¼ Sðr;ΩÞ þ
∑ 0
Sðr;Ω0 Þ
r∈Ω
ð2Þ
Ω 6¼ Ω
each of which named as the source function of atom Ω to F(r). Assuming Ω as the basins of QTAIM ensures a rigorous association of S(r;Ω) to the atoms or group of atoms of “chemistry”.7,19 The electron density at point r, hereinafter referred to as the reference point rp, may be seen (eq 2) as determined by an internal SF self-contribution, S(r,Ω), and by a sum of SF contributions, ∑Ω0 6¼ΩS(r,Ω0 ), from the remaining atoms or groups of atoms within a molecule. Decomposition afforded by eq 2 foresees the SF as a tool able to provide chemical insight.21,22 When chemical bonding is discussed, the bcps are generally taken as the least biased choices for rps19,21 and the relative contribution of an atomic domain Ω to the ED at the rp is called the source function percentage contribution of Ω to rp, SF%(rp,Ω). It is obviously given by SF%(rp,Ω) = [S(rp,Ω)/ F(rp)] 3 100. In general, the larger is the bond order and the more covalently bonded are two atoms, the higher is their SF% contribution to the ED value at their bcp.21,22 On the contrary, for less localized bonding interactions, the SF contributions become much more delocalized throughout the molecule.21,22 The SF analysis has been applied to several classes of bonding types, including hydrogen-bonds, multicenter bonds, and metal metal and metalligand bonds in organometallic systems, and it has also been exploited for assessing typical chemical features, 12865
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The Journal of Physical Chemistry A like chemical transferability, the effect of substituents, etc.22 A very recent paper reviews SF theory and applications and illustrates many of the ongoing SF developments.22 I.3. SF and Electron Delocalization. The potential uses of the SF are yet not fully explored. Among them, the capability or not of the SF to reflect in some way electron localization/delocalization, is certainly one worth of investigation. In a preliminary work,22 one of us (C.G.) addressed such a question challenged by a recent claim, published on this same journal, according to which “π-electron delocalization in the benzene ring is not manifest in the SF when the rp is taken at the CC bcp”.23 Reasoning behind this assertion is the null contribution from π-molecular orbitals to the ED in their nodal plane. However, as shown by Bader et al.5 long ago in the seminal paper referred to earlier, and as clearly demonstrated in several following studies,18,24 σ- and π-distributions are self-consistently interrelated, rather than being independent one from the another. Thus, it was conjectured that some, albeit small, effect of electron conjugation could also be visible when the rp lies in the π-nodal plane, even though π-orbitals do not give direct contributions to F in that plane. Data on a series of increasingly π-conjugated systems (cyclohexene, cyclohexadine, benzene) demonstrated that this is actually the case and the obtained results, along with new considerations and yet unpublished data are summarized in section III. As expected, it was also shown22 that the effects of enhanced or decreased electron delocalization can be greatly magnified if rps, for which the effect of π-electron conjugation takes place directly through π-electron distribution, rather than indirectly through σπ electron interdependency, are chosen. Such an ability of the SF to reflect π-electron conjugation is thoroughly independent from σ/π-separation of the ED because the SF tool was applied to the total density, though the adoption of a standard LCAO-MO wave function also allowed us to dissect the separate σ and π-contributions to the SF values.22 Were this separation not realizable, the same SF results would be obtained by analyzing an equivalent ED, though expressed in a completely different form, e.g., by a numerical representation or in terms of an experimentally derived pseudoatom multipolar expansion.25 This observation highlights the possibility, using the SF tool, to recover electron conjugation effects using both F’s derived experimentally (hence without σ- and π-separation being allowed) and F’s where the departure from planar symmetry inhibits a proper separation of σ- and π-contributions. I.4. Local Aromaticity and Its Measures. A detailed description of electron delocalization/localization is of utmost importance to improve our understanding of the effect it has on the structural, magnetic, energetic, and reactivity-based properties of a system. However, despite the merits of such descriptions, chemists have also been always looking for criteria able to rank, in a simple way, classes of systems in terms of their supposedly higher or smaller electron delocalization/localization features.13 In particular, a kind of Holy Grail search has indeed been the hunt for suitable criteria to quantify aromaticity,4 with several indices having been proposed in the last decades to define global and local26 aromaticity in polycyclic aromatic hydrocarbons (PAH). Each of them exhibits its own strengths and weaknesses: for a comprehensive and critical review on aromaticity indices in PAH, the reader is referred to Bultinck.3 To start with, here we briefly recall the most popular and widely employed descriptors, such as the harmonic oscillator model of aromaticity (HOMA),27,28 the nucleus independent chemical shift (NICS),29,30 DI-based1,31,12,24,3234 and multicenter
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indices35 (even within the QTAIM framework36), and molecular quantum similarity-based criteria.37 Each of these descriptors relies on one or more different physical properties that are commonly associated with the concept of “aromaticity”. It should be noted that the latter may be correlated to a number of quantities, such as bond length equalization (as for HOMA), diamagnetic susceptibility and shielding (that are in turn related to ring currents),4,38,39 electron delocalization (accessible through DI’s), and polarizability.4,40,41 It is not truly surprising that the above-mentioned methods disagree in ranking classical aromatic compounds on an absolute (overall and local26) aromaticity scale. This divergence, as claimed by Bultinck,3 is not a consequence of a real multidimensional character of aromaticity but it is rather due to “confusion and vagueness of the term (local) aromaticity”. Whatever definition of (local) aromaticity is used, however, it should be taken in mind that a completely different π-network with respect to benzene is obtained upon fusing together two or more aromatic rings. This is the consequence of the mutual influence of each delocalized (aromatic) system on the other(s). Overall, the final result is not simply the sum of two or more benzene rings, but the setting up of a truly different aromatic system in terms of electronic properties, bond lengths and orders, and chemical reactivity.42 In other words, PAH compounds show emerging properties that are peculiar to the overall molecule and that are lost if their subunits are considered separately. To avoid confusion on this topic, two possible solutions have been proposed in the literature. (i) Following Lazzeretti,4 only observables should be used to quantify aromaticity, as “any quantitative theory of aromaticity merely based on a priori benchmarks that cannot be experimentally checked would be very unsatisfactory, or even useless”. (ii) According to Bultinck,3 such a first position, although epistemologically very strong, “would reject nearly every current aromaticity index”. Therefore, from this point of view, a good compromise could be that “all studies dealing with molecular or local aromaticity should specify exactly what the term aromaticity is used for”. Being aware that a full rationalization of the concept of (local) aromaticity (and other unicorns43) is far from trivial, according to Lazzeretti4 we try to tackle it on observable grounds. Namely, we define our new index of local aromaticity (source function local aromaticity index, SFLAI, section V.4) by using the ED quantummechanical observable and its analysis in terms of the “local” and “distant” influence effects on such an observable as dissected by the SF tool. I.5. Outline of the Paper. Section II briefly gives the computational details pertinent to the evaluation of the in vacuo electron densities (ED’s) discussed in this work, together with some details related to their QTAIM and SF analyses. Section III reviews and extends the main findings recovered by Gatti22 on the three six-membered ring (6MR) test cases he considered (see Scheme 1 below). Section IV, on the other hand, analyzes whether Scheme 1. Unsaturated 6MR Systems with Increasing Extent of Electron Conjugation
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Table 1. Bond Lengths, SF and SF% Contributions for the Shortest Bond in Benzene, C6H6, 1,3-Cyclohexadiene, C6H8, and Cyclohexene, C6H10a system
d/Å
Fb
SFba
SFnn
SFothers
SFba%
SFnn%
C6H6-D6h
1.402
0.301
0.254
0.016
0.005
84.3 (72.2)
5.3 (9.1)
1.5 (2.6)
C6H8-C2v C6H10-Cs
1.345 1.339
0.331 0.334
0.289 0.291
0.010 0.008
0.002 0.000
87.1 (78.2) 87.2 (78.7)
3.1 (4.7) 2.5 (3.6)
0.7 (1.3) 0.0 (0.1)
SFothers%
a
SFba(%), SFnn(%), and SFothers(%) are the SF(percentage) contributions of directly bonded C atoms, nearest neighbor C atoms, and remaining (other) carbon atoms, respectively. As regards the SF% values, the first entries refer to atomic SF% contributions to the electron density at bcp (in-plane, ip, contributions), while those in parentheses to contributions at 1 au above the molecular plane and along the bond major axis direction (out-of-plane, oup1.0, contributions).
the SF description is stable or not against changes in the adopted computational level on the same systems. Eventually, section V extends to polycyclic (aromatic) hydrocarbons (P(A)H) our analysis of the capability of the SF to reveal electron delocalization and introduces a new descriptor of local aromaticity based on the SF. The main conclusion are drawn in section VI. Application of the SF to the description of hyperconjugation and to electron delocalization effects in nonplanar systems and heterocycles is deferred to a forthcoming paper. Extension to the case of experimentally derived EDs is also in progress in our lab.
II. COMPUTATIONAL DETAILS The gas-phase optimized geometries of the investigated systems have been obtained at the DFT/B3LYP level of theory, with the DZVP2 basis set,44 exploiting in each case the maximum point symmetry possible. The GAUSSIAN09 program was used throughout.45 All the topological properties, DI’s and the atomic SF contributions were calculated with a modified version46 of the AIMPAC suite of programs.47 The accuracy of the numerical integration has been checked by ensuring that atomic Lagrangian integrals7 and percentage errors (ER%)48 in the reconstruction of the ED at the rp through the SF contributions are reasonably low (typically, atomic Lagrangian integrals were lower than 103 au for C atoms and ER% lower than 1% in all examined cases). The difference between the sum of the atomic charges and the molecular charge is another useful criterion. Such a difference never exceeds 0.005 e for any of the investigated molecules. Atomic SF contributions were evaluated using the CC bcp’s as suitable and unbiased rp choices for the study of such bonding interactions. To emphasize π-electron delocalization effects on a given bond, SF contributions were also evaluated at rp’s chosen by moving away from the bcp, along the major axis—the direction associated with the L2 eigenvector of the ED Hessian matrix evaluated at the bcp along which the magnitude of the negative curvature of the ED is a minimum.5,7 Bader et al.5 showed how the major axis points in the direction of the maximum in the π-electron distribution of the molecular orbital theory. These are, in our view, the least biased rp choices one may do to inspect whether the SF tool is able or not to reveal electron delocalization effects. III. SIX-MEMBERED RINGS The selected 6MR compounds are characterized by different unsaturated patterns and increasing π-electron delocalization (see Scheme 1, from the left to the right). Table 1 reports the values of atomic SF for the C1C6 bond, which has the shortest internuclear distance and highest π-bond character in all
systems (clearly such a bond is by symmetry equivalent to C2C3 in 1,3-cyclohexadiene and to all remaining CC bonds in benzene). SF contributions at bcp and 1 au above/below the molecular plane and along the major axis direction (see section II) are hereinafter referred to as in-plane (ip) and out-of-plane (oup-1.0) contributions.49 From Table 1, it should be noted that both the SF and the SF% ip contributions from the C atoms other than those directly involved in the analyzed bond increase with decreasing double bond character and bond electron localization for such a bond (Figure 1, top), with the contribution of the next-neighbor atoms [C2 and C5, SFnn(%)] being several times larger than that of the farthest ones [C3 and C4, SFothers(%)]. Conversely, the contribution of the directly bonded C1 and C6 atoms [SFba(%)] decreases both in value and in percentage with increasing π-delocalization through the series (though only a hardly detectable reduction of 0.003 au and 0.1 percentage points occurs on going from cyclohexane to 1,3-cyclohexadiene). Interestingly, at variance with the SFba% descriptor, the trends of the SFnn% and SFothers% values are always clearly evident throughout the series. This is pictorially demonstrated in Figure 1 (top). All this evidence agrees with an increasing π-electron delocalization and decreasing localized nature of the shortest CC bond along the series and with the standard organic chemistry wisdom that would depict the π-bond(s) in cyclohexene as a purely vinylic one, those in 1,3-cyclohexadiene as those of a conjugated 1,3-diene, and those in benzene as part of a fully aromatic system. Inspection of individual atomic SF contributions discloses even more subtle details of π-delocalization in the studied series (Figure 1, top and middle). In 1,3-cyclohexadiene, the two nearest neighbor atoms C2 and C5 yield 60% and 40% of the SFnn% contribution to the density at the C1C6 bcp, as due to their being part of the conjugated and of the saturated moieties of the molecule, respectively. Analogously, atom C3 accounts for 97% of the overall contribution of the two farthest atoms to the electron density at C1C6 bcp, i.e., more than 30 times the contribution from the C4 atom. The C1 and C6 contributions to the density at their intervening bcp are, instead, hardly differentiated, though, as expected, the contribution of C6 exceeds that from C1 (43.7% vs 43.4%). As shown in Figure 1 (top and middle), the asymmetries in the 1,3-cyclohexadiene C1C6 bcp SF contributions of the nearest neighbors or of the farthest atoms are neat, revealing the capability of the SF to clearly distinguish electronic from geometrical effects. Such asymmetries signal how the local different π-bond pattern of two topologically equivalently connected atoms impacts on their ability to determine a smaller or larger electron density contribution at a remote bcp. 12867
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Figure 1. SF% contributions for the electron density of the shortest bond in benzene, 1,3-cyclohexadiene, and cyclohexene. Reference point: top, the bcp [(red) dot]; middle, at 1 au above the molecular plane and along the bond major axis direction; bottom, at 2 au, along this same axis. Only SF% contributions for the next-neighbors and the farthest C atoms are shown and displayed as spheres whose volume is proportional to the contribution magnitude (blue, positive; yellow, negative).
Table 2. Bond Lengths, Bonded Atoms Delocalization Indices, δba, and SF% Contributions for Some of the Formally Single CC Bonds in 1,3-Cyclohexadiene, C6H8, and Cyclohexene, C6H10a
a
system
bond
bond type
d/Å
δba
SFba%
SFnn%
SFothers%
C6H8-C2v
C1C2
conjugated
1.473
1.10
81.4 (65.8)
7.4 (13.2)
0.3 (0.5)
C6H8-C2v C6H10-Cs
C3C4 C1C2
allylic allylic
1.514 1.505
1.02 1.03
79.0 (59.6) 79.1 (59.7)
4.9 (8.7) 5.0 (8.9)
1.2 (2.2) 0.1 (0.1)
For the SF% values, the first entry refers to the CC bcp, the one in parentheses to oup-1.0, contributions (see caption of Table 1).
In agreement to what previously found,22 π-delocalization effects are already made evident by the in-plane density contributions, where the π-density has only an indirect influence. When the rp is moved above (or below) the molecular plane, πelectrons play a direct role, enhancing therefore the ability of the SF to reveal π-delocalization effects, as shown numerically in Table 1 and pictorially in the middle and bottom rows of Figure 1.50 Indeed, the SF% contribution from the non directly bonded atoms increases in all the systems at the expense of the contribution of C1 and C6 atoms, and the more so the larger is the extent of π-delocalization in the system (benzene > 13
cyclohexadiene > cyclohexene) and the higher is the distance of the rp from the molecular plane. The only exception is for the farthest atoms in cyclohexene whose contribution continues to be negligible and becomes even slightly negative. However, this is only an apparent exception, because in cyclohexene there is just one isolated double bond and no electron delocalization through a conjugated π-system may take place. As shown in Figure 1, the SF translates, in an easy-to-catch representation, the enhancement of the capability of atomic regions distant from a given bond to determine the density distribution along such bond, which is caused by electron delocalization. 12868
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Figure 2. Variation of SF%, F, ε, and delocalization indices δ with Hamiltonian and basis set for benzene (rhombuses), 1,3-cyclohexadiene (squares), and cyclohexene (triangles). In each panel, the first four data, from left to right, refer to B3LYP calculations with the indicated basis set. The TZVP basis set was used in RHF and CISD calculations. The bcp of the shortest bond, C1C6 (Scheme 1), in the three systems is taken as reference point for SF calculation and topological local properties. The delocalization indices refer to the C1, C6 pair of atoms.
The analysis of the formally single CC bonds provides further physical insights (Table 2). The C1C2 bond in 1,3-cyclohexadiene (C2v) lies within two double bonds, and it is thus part of a conjugated system, whereas the other two bonds considered in Table 2 are formally allylic bonds. The conjugated C1C2 bond is characterized by the largest (7.4%) contribution to its bcp from the nearest neighbors C6 and C3 atoms, consistently with a widespread charge delocalization involving the C6dC1—C2dC3 sequence in 1,3-cyclohexadiene. The other two allylic single bonds in Table 2 are characterized by similar nearest neighbors contributions and a more localized nature (SFnn ≈ 5%). Interestingly, the contributions from the farthest atoms, SFothers%, permit one to differentiate the allylic bonds in cyclohexene and 1,3-cyclohexadiene. It is worth noting that the contribution from nearest neighbors atoms to the C1C2 bcp density in 1,3-cyclohexadiene is even larger, in percentage, than that provided from the same kind of atoms in benzene. Indeed, in 1,3-cyclohexadiene, due to the lack of conjugation, these atoms are unable to significantly determine the electron density at the single C4C5 bond and thus
concentrate their nearest neighbor contribution mostly on the conjugated C1C2 bond. Furthermore, the lower bcp density of C1C2 with respect to the CC bcp density in benzene, comparatively enhances the SFnn% contribution to C1C2 in 1,3-cyclohexadiene, despite a closer similarity in the corresponding SFnn values (0.020 au in 1,3-cyclohexadiene and 0.016 au in benzene). As found for formally double bonds, the π-electron conjugation effects become even more evident (see Table 2) when the rp is moved away from the molecular plane (SFnn% oup-1.0 values being as large as ≈13% and ≈9% for the conjugated and allylic bonds, respectively).
IV. EFFECTS OF BASIS SET AND CORRELATION To test how robust is the SF picture of electron delocalization effects, we carried out calculations on the three 6MR test cases of section III, using four different basis sets and three Hamiltonians in various combinations (B3LYP/DZ, B3LYP/DZVP2, B3LYP/ TZVP, B3LYP/cc-pTZV; RHF/TZVP, B3LYP/TZVP, CISD/ TZVP).45 12869
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Table 3. Difference, Δ, in SF% Values between Benzene and 1,3-Cyclohexadiene/Cyclohexenea method
Δ(SFba %)
Δ(SFnn%)
Scheme 2. Connectivity of Two-Ring Hydrocarbons, with the Atom Numbering Scheme
Δ(SFothers%)
Basis Set Effect B3LYP/DZ
2.8/3.1
2.4/2.9
0.9/1.5
B3LYP/DZVP2 B3LYP/TZVP
2.8/2.9 3.1/2.9
2.2/2.8 2.2/2.9
0.8/1.5 0.8/1.5
B3LYP/cc-pTZV
2.7/2.8
2.2/2.7
0.8/1.5
Correlation Effects B3LYP/TZVP
3.1/2.9
2.2/2.9
0.8/1.5
RHF/TZVP
3.0/3.2
2.4/3.0
0.9/1.5
CISD/TZVP
2.9/3.2
2.5/2.9
0.8/1.5
a
The bcp of the formal double bond (C1C6 in Scheme 1) is taken as reference point in atomic SF calculations.
In Figure 2 we first analyze how the various SF% contributions considered in section III for the electron density Fb at the shortest bond in the three systems (C1C6, Scheme 1) vary with method/basis set. Corresponding changes for the Fb itself, the C1C6 bond ellipticity ε, and the delocalization indices δ(C1, C6) are also displayed. It may be easily seen that the SF% values are indeed stable, remaining quantitatively invariant with respect to both the choice of the computational method and basis set quality. Percentage changes are invariably less than 1 percentage point for the dominant contribution from C1 and C6 atoms and less than 0.1 of such points for the remaining atoms. On the contrary, the Fb values may change by about 1015%, the Laplacian of the density at the bcp by 4045%, and the bond ellipticity values by even 6080%. This outcome is to be stressed because it neatly distinguishes51 the SF% descriptor from the customarily used local QTAIM descriptors, whose values are known to significantly depend on the basis set52 and, though to a lesser extent, upon inclusion of the electron correlation.53 Except when the ED at rp is very small, the SF percentage contribution has been shown to represent a more robust chemical descriptor than the SF contribution itself whose changes with basis set and computational method necessarily (almost) parallel those for the Fb values.51 Interestingly, although not explored in the present theoretical study, the SF% descriptor was proved to be generally much more stable than the usual bond topological descriptors also against the multipole-model bias in the evaluation of X-ray derived EDs. This was revealed by comparing SF% data and local bond properties of multipole-derived and theoretical EDs, the former being obtained from the ab initio structure factors.51 The DI values, which result from an integration as do the SF values (although in the case of the SF on just one atomic basin instead of two) are rather stable against basis set change but, at variance with SF%, undergo a noticeable decrease (Figure 2) when they are evaluated through a multideterminant wave function (CISD).54 Actually, it is well-known that DI values as computed at the RHF or DFT level are generally rather overestimated for shared interactions.12,55,56 On the other hand, the SF is obtained through a mathematical identity from the ED, the latter feeling only indirectly, as it is a monoelectronic property, the effects of Coulomb correlation. Having demonstrated the stability of the individual SF% contributions for the various set of atoms (ba, nn, and others), we test now whether also the SF picture of electron
delocalization effects remain unaffected by basis set/methods changes. Table 3 shows the differences Δ(SF%), with respect to benzene, for the atomic SF percentage contributions to the ED at the bcp of the formally double C1C6 bond in 1,3-cyclohexadiene and cyclohexene. It may be easily seen that the Δ(SF%) values are indeed very stable, remaining quantitatively invariant with respect to the choice of both the computational method and basis set quality. This result is remarkable for a couple of important reasons. First, it shows that even the small and indirect effects due to π-electron localization in the molecular plane are stable and neatly recovered by all adopted computational levels. Second, it enables one to confidently use any one of such levels (hereinafter DFT/B3LYP/DZVP2 in our case) for the study of more complex systems.
V. POLYCYCLIC HYDROCARBONS Polycyclic hydrocarbons (PH) and polycyclic aromatic hydrocarbons (PAH) have been largely investigated because they are potent atmospheric pollutants57 with carcinogenic properties.58 More recently, they have been also recognized as basic building blocks of C-based nanostructures of potentially high technological interest.59 PAH of increasing size and complexity have also served as suitable test sets for the several indices that have been proposed in the last decades to study overall and local aromaticity.13,26 In this section, the ability of the SF to reveal and dissect electron delocalization when it ideally results from the mutual influence of more than one delocalized (aromatic) system is explored. Some benzenoid (i.e., “benzene-like”) rings in a number of simple planar PAH’s will be taken as suitable test cases (Scheme 2): (i) naphthalene (C10H8-D2h, subsection V.1); (ii) three partially hydrogenated PAH’s (subsection V.2), i.e., 1,2-dihydronaphthalene (C10H10-Cs), 1,2,3,4-tetrahydronaphthalene (also called tetraline, C10H12-C2v), and 1,4dihydronaphthalene (C10H10-C2v); (iii) three different conformational isomers of biphenyl (subsection V.3). Eventually, the new index of local aromaticity, based on the SF descriptor, is introduced in subsection V.4. The systems shown in Scheme 2 have different degrees of partial hydrogenation, exhibiting therefore different π-networks. In naphthalene, the two fused rings constitute a single aromatic system and, due to symmetry, experience the same kind of πelectron delocalization; in 1,2-dihydronaphthalene, the π-delocalized network only partially involves ring II, which is characterized by a vinylic bond conjugated with the ring I benzenoid system; in 1,4-dihydronaphthalene and 1,2,3,4-tetrahydronaphthalene (tetraline), the π-electron delocalization between the two rings is switched off, at least in terms of neutral resonance 12870
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Table 4. Bond Lengths, d, Bonded Atoms Delocalization Indices δba, and SF% Contributions for the Unique CC Bonds in Naphthalene, As Compared to Those for the CC in Benzene (in Italic) bond
d/Å
δba
SFnn%a
SFba%
SFothers%b
z = 0.0 au C7C8
1.380 1.49
85.3
4.4 (2.4, 2.0, 0.0)
2.3 (0.7, 0.4, 1.2)
C6C7
1.424 1.29
83.7
5.9 (5.9, 0.0, 0.0)
1.9 (0, 0.8, 1.1)
C8C9
1.427 1.25
82.5
7.3 (3.0, 2.0, 2.3)
3.0 (1.4, 0.0, 1.6)
C9C10
1.434 1.22
81.2
9.8 (4.9, 0, 4.9)
3.2 (1.6, 0.0, 1.6)
CC (C6H6) 1.402 1.39
84.3
5.3
1.5
C7C8 C6C7
74.2 70.4
7.2 (3.9, 3.3, 0.0) 4.3 (1.3, 0.6, 2.4) 10.1 (10.1, 0.0, 0.0) 3.7 (0, 1.5, 2.2)
C8C9
68.3
12.2 (5.1, 3.3, 3.8)
C9C10
65.4
16.6 (8.3, 0.0, 8.3)
CC (C6H6)
72.2
z = 1.0 au
9.1
5.9 (2.6, 0.0, 3.3) 6.0 (3.0, 0.0, 3.0) 2.6
a
Values within parentheses are (from left to right): the percentage SF contributions of the nearest neighbor C atoms belonging to the same ring (I) of the bond being analyzed and not being shared with the other 6MR; the same contributions from the C atoms common to the two 6MRs; the same contributions from atoms belonging only to the other 6MR (II). b As in footnote a, but referred to the “other” C atoms.
structures. Finally, though biphenyl is in principle a fully conjugated system, any resonance structure implying a different arrangement of the π-bonding network would be appreciably less stable due to charge separation and loss of local aromaticity in both rings. V.1. Naphthalene. V.1.1. Mutual Influence of the Two Benzenoid Rings As Revealed by the SF Descriptor: General Considerations and SF Percentage Contribution Trends. Table 4 shows that the SF percentage contributions from the two bonded atoms, SFba%, follow the same trend as of that of bond distances and of electron sharing (DI’s) between these same pairs of atoms. In particular, C9C10 has the lowest SFba% value, in agreement with its longer distance and lower CC electron sharing in the series of bonds. Apart from C7C8, all bonds exhibit lower SFba% and δba values and higher bond distances than in benzene, whereas the SF% contributions from distant atoms show an opposite behavior and trend. By analyzing them, one gets the following interesting insights. Due to the fusion of two benzenoid rings, four unique CC bonds are formed in naphthalene. These bonds differ in the number of nearest neighbor C atoms, as well as in the ring the latter C atoms belong. More in detail, the nearest neighbor atoms (i) may be part of the same 6MR of the bond being analyzed, (ii) may be instead (at least for part of them) common to both 6MRs, or (iii) they may even belong to the other 6MR. In particular, C7C8 and C6C7 type bonds have two nearest neighbor C atoms (both belonging to their own 6MR (category i) or, in the case of C7C8, one being shared with the other 6MR (category ii)), whereas the C8C9 bond type has three nearest neighbor C atoms and C9C10 four such atoms. For C9C10, two of them belong to one 6MR and the other two to the other ring, while for the C8C9 bond type one (category i) belongs to the same ring as the C8C9 bond, one to the other 6MR (category iii) and one nearest neighbor C is shared between the two 6MRs (category ii). Analogously, the “other” atoms increase in number with respect
to benzene and belong, in all cases, to at least two of the three categories (iiii) sketched above. If one cuts up a molecule in cycles (graph theory) rather than in domains in 3D space, naphthalene may be seen as composed of two aromatic 6MR and one aromatic 10MR system, with C atoms taking part of either two or three (C9, C10) of such rings.26 Clearly, rather then being simply the sum of its composing moieties, the real system is a new aromatic molecule that results from their mutual influence.42 The SF enables one to analyze such an influence in terms of contributions to CC bcp densities from distant C atoms. The SFnn% value increases with the increasing number of nearest neighbor C atoms, each of them bringing a contribution of about 2.5% to the Fb value. Therefore, C9C10, with its four nearest neighbor C atoms, has a SFnn% value of 9.8, which is almost twice that in benzene or that of the C7C8 and C6C7 bonds in naphthalene. An intermediate value of 7.3% is found for C8C9 bond, which has three nearest neighbor C atoms. On the other hand, the contributions from SFothers% increase, with respect to benzene, for all CC bonds, as due to the enhanced number of such atoms from 2 to 8 in naphthalene, but the increment is, as expected, larger the less peripheral is the location of the bond being analyzed. Therefore, SFother% is only slightly larger (1.9%) than in benzene for C6C7, the most peripheral type of bond, and more than twice that big for the central C9C10 bond. The separate contributions to SFnn% and SFothers% from the three categories of distant carbon atoms listed earlier are also reported in Table 4. These values neatly show that the enhancements of SFnn% and SFothers% with respect to benzene are the result of additional contributions coming from atoms which do not belong to the 6MR including the bond being analyzed. In the case of the central C9C10 bond, each 6MR contributes to the bcp density roughly as in benzene. In other words, none of the bonds can be considered as exclusively pertaining to an isolated 6MR, but all of them, in different degrees and various forms, are influenced by their involvement in at least two of the three composing resonance systems. Analysis of data at 1 au above/ below the molecular plane and in the direction of the major axes of the various bonds, confirm, also quantitatively, the enhancement of the π-electron delocalization effects found for benzene. For instance, the increase of SFnn% and SFothers for the C9C10 bond, on passing from the molecular plane to 1 au above/below such plane is 170% and 187% to be compared with the corresponding and quite comparable values of 172% and 173% in benzene. Similar enhancements are found for the other bonds, showing that π-electron delocalization takes place through both 6MRs and through the 10MR, and either indirectly through σπ interdependency (data for rps in the molecular plane) or in a direct way through the π-electron distribution (data for rps above/below such plane). V.1.2. Mutual Influence of the Two Benzenoid Rings As Revealed by the SF Descriptor: A Closer Look. Having rationalized the general trends in the different SF contributions from nearest neighbor and more distant C atoms to the bcp density of the four unique CC bonds in naphthalene, a closer look to such contributions and to the strictly related SF contributions from bonded atoms is briefly taken in the following. The bond common to both rings, C9C10, has the lowest SFba% value (81.2%) because of its highest SFnn% and SFothers% values with respect to all other CC bonds in naphthalene. Basically, C9C10 comes out to be the most delocalized bond as due to its largest number (4) of nearest neighbor C atoms 12871
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Table 5. As in Table 4, Complementing the Data for the Ring I in Naphthalene with Those in Its Partially Hydrogenated Derivatives (Scheme 2) bond C7C8
C8C9
C9C10
C5C10
C5C6
C6C7
hydrogenated positions
d/Å
δba
SFba%
SFnn%a
SFothers%b
none
1.380
1.49
85.3
4.4 (2.4, 2.0, 0.0)
2.3 (0.7, 0.4, 1.2)
1, 2 1, 4
1.401 1.396
1.38 1.41
84.2 84.7
5.2 (2.7, 2.5, 0.0) 5.0 (2.7, 2.3, 0.0)
1.8 (0.7, 0.5, 0.6) 1.9 (0.7, 0.6, 0.6)
1, 2, 3, 4
1.394
1.42
84.7
4.9 (2.6, 2.3, 0.0)
1.7 (0.7, 0.6, 0.4)
none
1.427
1.25
82.5
7.3 (3.0, 2.0, 2.3)
3.0 (1.4, 0.0, 1.6)
1, 2
1.399
1.38
83.8
6.1 (2.6, 2.2, 1.3)
2.2 (1.4, 0.0, 0.8)
1, 4
1.406
1.35
83.6
6.4 (2.7, 2.4, 1.3)
2.3 (1.4, 0.0, 0.9)
1, 2, 3, 4
1.410
1.33
83.4
6.6 (2.7, 2.5, 1.4)
1.8 (1.4, 0.0, 0.4)
none
1.434
1.22
81.2
9.8 (4.9, 0.0, 4.9)
3.2 (1.6, 0.0, 1.6)
1, 2 1, 4
1.411 1.405
1.31 1.35
82.1 82.8
8.4 (5.3, 0.0, 3.1) 7.8 (5.2, 0.0, 2.6)
2.2 (1.5, 0.0, 0.7) 2.3 (1.5, 0.0, 0.8)
1, 2, 3, 4
1.400
1.37
83.0
7.6 (5.1, 0.0, 2.5)
1.5 (1.5, 0.0, 0.0)
none
1.427
1.25
82.5
7.3 (3.0, 2.0, 2.3)
3.0 (1.4, 0.0, 1.6)
1, 2
1.407
1.34
82.7
6.9 (2.7, 2.4, 1.8)
2.4 (1.4, 0.0, 1.0)
1, 4
1.406
1.35
83.6
6.4 (2.7, 2.4, 1.3)
2.3 (1.4, 0.0, 0.9)
1, 2, 3, 4
1.410
1.33
83.4
6.6 (2.7, 2.5, 1.4)
1.8 (1.4, 0.0, 0.4)
none
1.380
1.49
85.3
4.4 (2.4, 2.0, 0.0)
2.3 (0.7, 0.4, 1.2)
1, 2 1, 4
1.398 1.396
1.40 1.41
84.5 84.7
4.9 (2.7, 2.2, 0.0) 5.0 (2.7, 2.3, 0.0)
2.0 (0.7, 0.6, 0.7) 1.9 (0.7, 0.6, 0.6)
1, 2, 3, 4
1.394
1.42
84.7
4.9 (2.6, 2.3, 0.0)
1.7 (0.7, 0.6, 0.4)
none
1.424
1.29
83.7
5.9 (5.9, 0.0, 0.0)
1.9 (0.0, 0.8, 1.1)
1, 2
1.400
1.39
84.3
5.3 (5.3, 0.0, 0.0)
1.6 (0.0, 1.1, 0.5)
1, 4
1.403
1.38
84.2
5.4 (5.4, 0.0, 0.0)
1.7 (0.0, 1.2, 0.5)
1, 2, 3, 4
1.405
1.36
84.0
5.5 (5.5, 0.0, 0.0)
1.5 (0.0, 1.2, 0.3)
a
Values within parentheses are (from left to right): the percentage SF contributions of the nearest neighbor C atoms belonging to the same ring (I) of the bond being analyzed and not being shared with the other 6MR; the same contributions from the C atoms common to the two 6MRs; the same contributions from atoms belonging only to the other 6MR (II). b As in footnote a, but referred to the “other” C atoms.
(section V.1.1). The C8C9 bond has one less of such atoms, two belonging to its own ring (yielding a total contribution of 5.0%, which is less than in benzene, because C10 is common to both rings and contributes for only 2%, Table 4) and the other belonging to ring II (accounting for the remaining 2.3% contribution). This implies that SFba% for this bond (82.5%) is higher than for C9C10, at the same time being lower than in benzene (84.3%). In other words, C8C9 exhibits a remarkable degree of electron delocalization in terms of influence from distant carbons, even greater than in benzene. The C7C8 bond follows the trend just outlined, having only two contiguous carbon atoms and with one of them, C9, being shared with ring II, able to contribute by just 2%. Therefore, such a bond shows not only the greatest double-bond character (SFba% = 85.3%) with respect to all other CC bond types in naphthalene but also a larger SFba% value than in benzene.60 In particular, the nearestneighbors SF contribution amounts to an overall 4.4% (to be compared with 5.3% in benzene), resulting in higher localization (and strength) of this bond. Also the C6C7 bond has just two nearest-neighbor carbon atoms (C5 and C8), but both atoms not being shared with ring II, they can contribute by 5.9% to the C6C7 bcp density, a value significantly larger than found for the C7C8 bond and even slightly larger than is observed in benzene. Such a SFnn% increase sums up to the concomitant increase of SFother% with respect to benzene (1.9% vs 1.5%) yielding a slightly weaker and less localized CC bond (SFba% = 83.7%) than in benzene (Table 4).
Overall, each benzenoid ring in naphthalene is less aromatic than benzene, in the sense that in C10H8 the π-density is not equally distributed among all the CC bonds, as evidenced by the trends in the SF% values. In turn, this is due to the mutual influence of each ring to the other, which is particularly evident when SF% contributions from nearest neighbor and even more distant C atoms are considered and dissected in their composing terms. V.2. Partially Hydrogenated PAH. Upon hydrogenation, πelectron delocalization between the two fused rings is partially (1,2-dihydronaphthalene) or totally (1,4-dihydronaphthalene and 1,2,3,4-tetrahydronaphthalene) broken and rings I and II (Scheme 2) undergo remarkable changes (Table 5). The following discussion will be devoted to the variations in the former ring, while the analogue data on the CC bonds of ring II are collected within the Supporting Information. As a general observation, we first note that the SF contributions from nearest neighbor atoms, SFnn%, are ordered, as in naphthalene, according to the number of such atoms for the various bonds in the variously hydrogenated PAHs. Therefore, SFnn% is largest for C9C10, intermediate for C8C9 and C5C10 bonds, and smaller for those bonds with only two nearestneighbor C atoms. However, for all systems and bonds (C8C9, C5C10, C9C10) closest to ring II, SFnn% values are lower than the corresponding ones in naphthalene as a result of a significant decrease of the contribution from the atoms belonging exclusively to ring II (Table 5). In turn, this is the consequence of the 12872
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Table 6. Delocalization Indices (δ) vs Source Function (SF) (at z = 1 au) Picture of Delocalisation Effects on C9 and C10 of Naphthalene and Its Derivatives C10H8-D2h δ(C9,I)
a
SF%(C9,I)b δ(C9,II)a
3.18 73.5 3.18
C10H10-Cs 3.50 79.1 2.92
C10H10-C2v 3.52 79.3 2.94
C10H12-C2v 3.51 79.2 2.96
SF%(C9,II)b
73.5
69.1
70.4
70.4
δ(C10,I)a SF%(C10,I)b
3.18 73.5
3.45 76.6
3.52 79.3
3.51 79.2
δ(C10,II)a SF%(C10,II)b
3.18 73.5
3.02 72.7
2.94 70.4
2.96 70.4
δ(Cx,I) and (δ(Cx,II) are the summation of all DI’s among Cx and all the other C atoms belonging to ring I (II). C9 and C10 belong simultaneously to both rings. b SF%(Cx,I) and SF%(Cx,II) are the sum of the SF% contributions of Cx to the electron density of the chosen rp (1 au above/below the bcp along the bond major axis) for all CC bonds belonging to ring I and II, respectively. The C9C10 bond belongs simultaneously to both rings. a
(partial) interruption of conjugation upon hydrogenation in such ring. Because the observed decrease is not sufficiently compensated for by the concomitant increase of the contributions from the nearest neighbor atoms belonging to ring I, a general lowering for SFnn% occurs with respect to naphthalene. For the remaining bonds, upon hydrogenation, both SFnn% and SFothers% become closer to the corresponding values in benzene (Table 2). It should be also noted that, upon hydrogenation, the contribution to SFothers% due to the atoms belonging exclusively to ring II diminishes, with respect to naphthalene, in all compounds and for all ring-I bonds, while that due to the carbon atoms shared by the two 6MRs slightly increases. This expected result agrees with the enhanced benzenoid character of ring I and the (partial) interruption of conjugation between the two rings. As shown in Table 5, SFnn% is largest in naphthalene (9.8%) and the ratio of separate contributions from the two rings is fixed to one by symmetry in this molecule, while for the last member of the hydrogenated series, SFnn% is diminished to 7.6% and the contribution from ring I becomes more than twice as big as that from ring II. Similar reasoning applies to SFothers%. Data in Table 5 are now discussed in more detail. Compared to naphthalene, the central C9C10 bond is strengthened on going from 1,2-dihydronaphthalene to 1,2,3,4-tetrahydronaphthalene as shown by the trend in bond lengths, DI’s, and SFba%. Moreover, contributions to the C9C10 bcp from nearest neighbors and other distant C atoms belonging to ring I comparatively increase with respect to those of carbons of ring II. More in detail (Table 5), SFnn% for this bond is largest in naphthalene (9.8%) and the ratio of separate contributions from the two rings is fixed to one (4.9%:4.9%) by symmetry. On the contrary, in tetraline SFnn% is diminished to 7.6% and the contribution from ring I becomes more than twice as big as that from ring II (5.1%:2.5%). Similar reasoning applies to SFothers%. This result indicates that C9 and C10, despite their central geometric position, are more effective in determining the electron density in ring I, through their SF contributions, than they can in ring II. In turn, this implies that their involvement into the π-electron circulation in ring I is comparatively increasing while that in ring II progressively reduces along the series. Once more, and in spite
of the presence of the 1/|r r0 | term in the formula for the local source contribution at the reference point r, it is found that the SF contributions neatly distinguish the electronic effects from the purely geometric ones. The C6C7 bond undergoes a significant strengthening on passing from naphthalene to 1,2-dihydronaphthalene, while its strength remains almost constant throughout the rest of the series (Table 5). On the other hand, upon hydrogenation C7C8 and C5C6 become weaker than in naphthalene and have similar features in all the partially hydrogenated derivatives. The latter evidence implies that the effects on ring I due to the double C3C4 bond in 1,2-dihydronaphthalene are indeed small. It can be therefore deduced that a significant delocalized π-electron system no longer exists in ring II to influence the ring I, even for the most favorable case of 1,2dihydronaphthalene. Eventually, C8C9 and C5C10 bonds are also invariably reinforced with respect to naphthalene. It is interesting to note that the major changes affecting ring I occur as soon as the aromatic π-electron circulation in ring II is disrupted, whereas significantly smaller variations in both geometrical and electronic features of the ring I CC bonds are evident throughout the rest of the series. Nevertheless, the surviving double bonds in ring II may or may not still interact with the aromatic system in ring I, and the SF is able to detect these subtler effects. For example, considering the bridging C9C10 bond, it is worth noting that the decrease, from 8.4% to 7.8%, of the SFnn% value on going from the 1,2 to the 1,4 dihydrogenated compound is almost entirely due to the significantly diminished contribution from ring II atoms. In the first compound, the double bond C3C4 is conjugated with the aromatic ring I, therefore being able to provide some influence at the C9C10 bcp. On the contrary, the C2C3 bond in 1,4dihydronaphthalene is distant and necessarily more localized. Table 6 compares SF and DI values for the two central C9 and C10 atoms while “interacting” with both the fused rings. To emphasize the specific effects of π-electrons, SF contributions are calculated at rp’s above/below the main molecular plane, located at z = (1.0 au from the bcp and along the bond major axes. The overall capability of C9 or C10 atoms to exchange electrons with any of the other C atoms of ring I or II is represented by the sum of the corresponding DI’s, listed as δ(Cx,I) and δ(Cx,II) in Table 6, with X = C9 or C10. The parallel aptitude of C9 or C10 atomic distribution to determine the density at representative points for the π-electrons distributions of CC bonds in ring I or II, is given by the sum of the SF% contributions of such atoms to the mentioned bond rp’s. These sums are listed in Table 6 as SF %(Cx,I) and SF%(Cx,II), respectively, with X = C9 or C10. It is indeed gratifying that both kinds of cumulative descriptors of electron information transmission provide a similar picture. Upon hydrogenation, the number of electrons exchanged by C9 (or C10) with ring I atoms increase by about 0.3 e, whereas that with ring II decreases by a similar amount, with respect to naphthalene. Analogously, SF%(Cx,I) increases by about 6% and SF%(Cx,II) decreases in general by about 3%, which clearly indicates an overall enhanced ability of central atoms in determining the π-electron distribution in ring I and a diminished capability of these atoms to do the same in ring II. Integral quantities in Table 6 reveal also the consequences of the asymmetrical electron conjugation in 1,2-dihydronaphthalene. As a matter of fact, the indicators for C9 in this molecule are closer to those computed for 1,4-dihydronaphthalene and 1,2,3,4-tetrahydronaphthalene, while those for C10 are like to those of this C atom in naphthalene. These asymmetries may also 12873
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Table 7. Bond Lengths, Delocalization Indices δ, and Source Function Percentage (SF%) Contributions for Relevant CC Bonds in Biphenyl Isomers bond C1C6
C5C6
C4C5
C1C10
ϕ/deg 37
d/Å 1.409
Fb, au 0.299
δba 1.34
SFba% 83.4
SFnn%
SFothers% a
3.1 (1.5)b
a
6.7 (5.3)
0 90
1.411 1.405
0.298 0.300
1.33 1.36
83.4 83.5
6.7 (5.3) 6.6 (5.3)a
3.1 (1.5)b 3.0 (1.5)b
37
1.399
0.303
1.40
84.3
5.0
2.2 (1.4)b
0
1.398
0.304
1.40
84.6
5.0
2.2 (1.4)b
90
1.401
0.302
1.39
84.3
5.0
2.1 (1.4)b
37
1.401
0.302
1.39
84.6
5.3
1.8 (1.2)b
0
1.399
0.303
1.38
84.5
5.4
1.8 (1.3)b
90
1.401
0.302
1.39
84.5
5.3
1.7 (1.2)b
37 0
1.488 1.495
0.261 0.257
1.05 1.06
79.1 79.4 (61.0)c
10.6 10.6 (18.3)c
3.8 3.9 (7.9)c
90
1.497
0.259
1.01
78.7
10.9
3.8
Percentage contribution from nearest neighbors C that belong to ring I. b Percentage contribution from all the “other-type” C atoms that belong to ring I. c Percentage contribution at z = 1 au. a
be revealed by inspecting the inter-ring SF contributions or delocalization indices (see Tables S2 and S3 in the Supporting Information). V.3. Biphenyl. The equilibrium geometry of biphenyl is staggered with a dihedral angle ϕ between the two rings as large as ≈37° for the adopted basis set and computational model. The other two optimized structures here considered correspond to transition states with ϕ = 0° and ϕ = 90°, respectively. Table 7 reports geometric and electronic parameters for CC bonds in biphenyl as a function of ϕ. In all the isomers considered, the ring CC bond lengths are close to those in benzene, with the exception of C1C6 type of bonds, which are slightly elongated because of the H 3 3 3 H close contacts arising from the proximity of the two rings.61 Not unexpectedly, C1C6 is longest in the planar and shortest in the 90° transition states, because H 3 3 3 H steric interactions are maximized in the former and minimized in the latter state. The central C1C10 bond, which connects the two rings, has essentially a σ nature, as testified by 1.01 e δba e 1.06 and lower SFba% contributions with respect to the ring CC bonds. Moreover, δba stays roughly the same upon moving from the twisted minimum to the planar geometry, but the bond is lengthened due to the effect of the steric repulsions between the hydrogen atoms that dominate over the bond shortening provoked by the π-overlap maximization occurring at ϕ = 0°. When, at ϕ = 90°, the π-systems within the two rings become orthogonal to each other, δba decreases, and the C1C10 bond length increases because of the dominance of lack of πoverlap, despite the H 3 3 3 H steric repulsion being at a minimum. Ellipticity values are consistent with the changes in the πcharacter of the C1C10 bond as a function of the dihedral ϕ angle discussed above. More in detail, ε is 0.103 when ϕ = 0° and then it reduces to 0.084 for the equilibrium geometry (ϕ = 37°), and eventually becomes exactly 0.000 when ϕ = 90°. It is worth noting that the trend of δba values against the dihedral angle ϕ, is nicely paralleled by the trend in the SFba% contributions, at variance with both the trend in C1C10 bond lengths and Fb values. The δba values and SFba% contributions for this bond (Table 6) similarly reflect the delicate balance between the bond order increase due to the π-character enhancement on passing
from ϕ = 90° down to ϕ = 0° and the concomitant bond lengthening due to the closer H 3 3 3 H approach. Although surprising at a first sight, the very large contribution to the C1C10 bcp density from next neighbor C atoms (SFnn% = 10.6 at equilibrium geometry) may be easily explained. The formally single bonds C1C2 in cyclohexene and in 1,3-cyclohexadiene, which have both two next-neighbor C atoms, exhibit SFnn% values of 5.0 and 7.4, respectively (Table 2). This difference is due to the allylic nature of the former and the conjugated nature of the latter (see also section III above). In biphenyl, the formal C1C10 single bond has four nearest neighbor C atoms and it is linked to four partially double bonds. As a consequence, SFnn% is largely increased with respect to the mentioned formal single bonds in cyclohexene and 1,3-cyclohexadiene, and to a similar extent to what is observed for the central C9C10 bond in naphthalene (SFnn% = 9.8, Table 4), which has a comparable chemical environment. Moreover, the following interesting observation can be made. In 1,3-cyclohexadiene, the overall SF% contribution to the C1C2 bcp coming from the nearest neighbor C atoms (SFnn%) plus that coming from the H atoms linked to C1 and C2 amounts to 13.8%. When the same procedure is applied to the allylic C1C2 bond in cyclohexene, one gets an even higher value of 15.5%. In practice, the large SFnn% for C1C10 as compared to the corresponding values for the C1C2 bonds in 1,3-cyclohexadiene and in cyclohexene, may be interpreted in terms of the (partial) replacement of the SF contributions of two (or three) nearest neighbor hydrogen atoms in the latter two bonds with those of two nearest neighbor carbon atoms in the former bond.62 Note, also, that the SFnn% value for C1C10 is almost insensitive to the dihedral angle, pointing out that it is not affected by the four nearest neighbor C atoms being or not being in the same plane. In conclusion, the effects due to the relative orientation of the two phenyl rings are small and mostly absorbed by the changes observed in the δba and SFba% values for the central bond. However, the presence of a second ring, regardless of the value of ϕ, has some impact on each benzenoid ring, in particular for C1C6 type of bonds. Relative to the CC bond in benzene (Table 2), C1C6 has a decreased δba and SFba% value and an enhanced SF contribution to its bcp density from nearest neighbor C atoms (6.7% vs 5.3% in benzene) and from the 12874
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comparison of the value of the index as obtained from experiment and theory; on the other hand, it enables one to evaluate such an index for systems experiencing important environmental effects, like a crystal field, or for systems difficult to handle theoretically (e.g., a defective crystal or a crystal with a very large unit cell and with many atoms within it). But, how can an aromatic index be defined in terms of SF contributions? We start from the functional form of the Fermi hole delocalization density (FHDD) index developed by Matta and Hernandez-Trujillo,24 and analogously define a source function local aromaticity index (SFLAI) as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 6 c ð3Þ ðk SFΩb %Þ2 SFLAI ¼ 1 6 Ω¼1 b¼1
∑
farthest C atoms (3.1% vs 1.5% in benzene). As shown in Table 7, the increase in the SFnn% and SFother% contributions is entirely due to the extra contributions coming from ring II atoms. The sum of such extra contributions (3.0% and a total of 4.1% including the H sources) is, however, quite close to the SF contribution of the H in benzene (2.9%) being replaced by a phenyl group in biphenyl. The situation is reminiscent of what is found in a saturated hydrocarbon chain where the sum of the SF contributions to the CH bcp density in methyl group for all atoms external to the CH group remain constant, regardless of the chain length. As shown by Bader and Gatti19 such a result for the SF is a physical necessity if one is to observe characteristic (i. e., “transferable”) atomic and bond properties in linear hydrocarbons. The fact that the linked phenyl group closely mimics the contributions of the replaced H in benzene corroborates the substantial electronic decoupling of the two 6MRs in biphenyl. At variance with naphthalene, biphenyl may be considered as made up of two rather independent benzene rings, which are weakly interacting between each other and which are slightly perturbed by the combination of steric and electronic effects, whose relative weight depends on the value of ϕ. V.4. New Electron-Density-Based Descriptor of Local Aromaticity. Ranking the amount of local aromaticity in real systems on a relative scale may be useful to highlight what chemical regions in a complex PAH have the largest similarity/ dissimilarity with respect to a suitable reference system, therefore being subject to comparable or unlike chemical reactivity. As mentioned in section I, this is one of the main reasons for introducing proper aromatic indices. In formal agreement with Bultinck’s et al. suggestion,26 and referring to 6MR, “local aromaticity” will be used hereinafter as a synonym of similarity with respect to benzene in terms of the various introduced SF% contributions. Nevertheless, this should not necessarily imply that an extended aromatic network within a PAH can be rigorously partitioned into individual 6MR’s with different degrees of “aromaticity”. In general, fused rings that have small similarity with benzene will likely exhibit also a quite different chemical behavior. At variance with aromatic indices based on DI’s or on other nonobservable quantum-mechanical objects, the peculiar feature of an aromatic index defined in terms of the SF contributions is its potential applicability to both theoretical and experimental EDs, like the X-ray derived multipole EDs.22,51 This characteristic offers several advantages: on the one hand, it allows a direct
∑
In (3), the SF% contributions replace DI values in the original formulation of FHDD. In more detail, ∑SFΩb% is the sum of all SF% contribution of the carbon atom Ω to each bth CC bond critical point in the benzenoid ring, with k being the analogue quantity in benzene and SFLAI thereby being equal to 1 for benzene. c is a normalization constant such that SFLAI = 0 for cyclohexane.63 SFLAI was tested on the systems discussed earlier, including 6MR’s and PAH’s, supplemented by anthracene and phenanthrene (Scheme 3). The corresponding outcomes are compared to those obtained from other well established structural and electron delocalization descriptors such as HOMA,27,28 bond order index of aromaticity (BOIA),3 para-delocalization index (PDI),31,33 and FHDD:24 HOMA ¼ 1 BOIA ¼ 1 PDI ¼
a 6 ðdi do Þ2 6 i¼1
∑
1 6 ðBi Bref Þ2 6 i¼1
∑
1 3 δi 3 i¼1
∑
ð4Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 6 c FHDD ¼ 1 ðk δij Þ2 6 i¼1 j¼1
∑
∑
In HOMA, aromaticity is measured by evaluating the deviation of all bond lengths di from an optimal value do (here taken as 1.388 Å), and a is a constant. In BOIA, Bi and Bref are bond index measures of the CC bond i of the analyzed and of the reference systems, respectively; here we take the δba as bond indices and Bref δref is the value of δba in benzene, calculated with the DFT/B3LYP level of theory and DZVP2 basis. For what concerns PDI, δi is the DI between one of the three pairs of para related carbon atoms in a 6MR, whereas FHDD accounts for all CC delocalization indices δij in the benzenoid ring, with k being their sum in benzene and c a constant yielding FHDD equal to zero in cyclohexane. The above cited formulas are reported only for the sake of clarity; the interested reader is referred to the literature for more detailed explanations. Table 8 collects the outcomes for the investigated systems. Correlation coefficients among SFLAI and other descriptors are quite satisfactory (0.93/HOMA, 0.95/BOIA, 0.94/PDI, 0.99/ FHDD). It is easy to see that SFLAI is also able to correctly detect the progressive increase of π-delocalization on passing from cyclohexane, through cyclohexene and 1,3-cyclohexadiene, to benzene (see section III). Moreover, it shows that ring I is less aromatic in naphthalene than in its hydrogenated derivatives and 12875
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Table 8. Source Function Local Aromaticity Index, SFLAI, and Other Aromaticity Indices (See Text) As Evaluated for Isolated and Fused Six-Membered Hydrocarbon Rings systema
HOMA BOIA
PDI
FHDD SFLAI
cyclohexane
4.899 0.828 0.0082 0.000
0.000
cyclohexene
3.950 0.841 0.0145 0.204
0.131
1,3-cyclohexadiene
2.120 0.869 0.0418 0.452
0.400
benzene
0.951 1.000 0.1029 1.000
1.000
naphthalene 1,2-dihydronaphthalene (I)
0.718 0.983 0.0746 0.771 0.937 0.998 0.0927 0.910
0.757 0.831
1,4-dihydronaphthalene (I)
0.944 0.999 0.0977 0.938
0.887
1,2,3,4-tetrahydronaphthalene (I)
0.937 0.998 0.0969 0.934
0.885
biphenyl (C1)
0.940 0.999 0.0969 0.938
0.901
biphenyl (D2h)
0.937 0.999 0.0956 0.932
0.902
biphenyl (D2d)
0.947 1.000 0.1004 0.957
0.906
anthracene (I)
0.572 0.971 0.0651 0.711
0.712
anthracene (II) phenanthrene (I)
0.654 0.982 0.0647 0.687 0.827 0.990 0.0806 0.817
0.678 0.794
phenanthrene (II)
0.400 0.963 0.0458 0.608
0.635
a
The entries in parentheses refer, when appropriate, to the symmetry label of the compound or to the specific 6-membered ring that is considered within a PAH system.
it correctly highlights the greater similarity with respect to benzene of the peripheral rings (I) in phenanthrene. As regards anthracene, it should be noted that SFLAI, FHDD, and PDI determine the peripheral ring (I) as that most like to benzene, whereas BOIA and HOMA agree in drawing just the opposite conclusion. Anyhow, it should be noted that the SFLAI outcome on anthracene can be quite easily rationalized if the arguments used above to explain the loss of local aromaticity in naphthalene are extended to this three-fused rings system. In naphthalene, the sharing of two C atoms (C9, C10) between the two fused rings weakens the local electron circulation, as C9C10 π-electrons are involved in both the interacting aromatic systems. In anthracene, the central ring shares four of its six C atoms with the two peripheral rings, therefore being somewhat impoverished of π-density. Obviously, the same holds true also for phenanthrene. Such a picture agrees well with the outcomes by Bultinck and coworkers, who demonstrated3,3537 a decreasing benzene-likeness of the central ring in linear polyacenes. The failure of HOMA and BOIA is probably due to the fact that these indices consider only bond length and bond order equalization, without taking explicitly into account how other atoms within the structure contribute to determine the bond length or the bond order value of a given bond.64 In spite of the present rationalization of the interchanged relative aromaticity of the two different rings in anthracene found by HOMA and BOIA, one cannot, however, overlook that this is still a controversial issue.65 It should be remarked that SFLAI gives very similar values to FHDD, even on an absolute scale. This result is likely to be the consequence of two concurring facts. On the one hand, as we observed for these PAH systems and in other cases,22,48 DI’s and SF contributions correlate generally very well (see for instance the trends for these two quantities in Table 6). On the other hand, SFLAI and FHDD share, by definition, a formally identical functional form, although one has also to remember
that the corresponding ingredients are conceptually different. Indeed, in FHDD, the electron sharing among all pair of atoms is considered, while in SFLAI, properties relating a given atom and the ED at a suitable rp for a given CC bond are taken into account for all atom-bond pairs. The impressive agreement between FHDD and SFLAI trends is a remarkable result. Being also applicable to experimentally derived ED’s, SFLAI appears to be an attractive index for the investigation of local aromaticity and electron delocalization in condensed phase.
VI. CONCLUSIONS In this work, the results of a preliminary test aimed at understanding whether the source function (SF) descriptor is capable of revealing to some extent electron delocalization effects directly in real space and independently from any MO scheme or decomposition are corroborated by further examination of the previously considered six-membered ring systems and by extending the analysis to a series of benchmark polycyclic aromatic hydrocarbons (PAH) and biphenyl. Moreover, using the same SF tool, we investigated electronic effects ascribable to the changes in the amount of electron delocalization in cyclic hydrocarbon systems with different unsaturation degrees. Eventually, we propose our own local aromaticity index based on the SF descriptor. Overall, the following conclusions can be drawn. (i) Generally speaking, the SF translates in an easy-to-catch representation (see Figure 1 for an example), the enhancement of the capability of atomic regions distant from a given bond to determine the density distribution along such bond, which is caused by electron delocalization. With respect to an indirect picture of the electron delocalization through the influence it has, for instance, on the internuclear distances or on the local electron properties at a given CC bcp, the SF has the advantage of offering a nonlocal and more direct representation of such a (not measurable) property. The possibility to magnify or dampen delocalization effects through the choice of various and suitable rps adds further potentiality and insights. (ii) The SF was found to be a useful tool to study and break down into a number of insightful figures the peculiar features of electron delocalization in six-membered ring systems, being able to disclose even subtle differences among fully aromatic and partially hydrogenated systems with various π-electron patterns. Overall, the SF descriptor seems capable of revealing, ordering, and quantifying π-electron delocalization effects for both formal double bonds and single conjugated or allylic bonds, through the consideration of the influence of distant (“nearest neighbor”, “others”) atoms to the electron density at a given bcp. (iii) As regards polycyclic aromatic hydrocarbons, the SF descriptor is able to detect and dissect in a number of very insightful contributions the mutual influence of each benzenoid ring to all the others. The mutual influence of rings is particularly evident when the SF contributions from nearest neighbor and even more distant C atoms are considered. For instance, according to the SF analysis, none of the bonds in naphthalene can be considered as exclusively pertaining to an 12876
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’ ASSOCIATED CONTENT
bS
Supporting Information. Local and nonlocal topological descriptors for the symmetry-independent chemical bonds in 6MR systems. Bond lengths, DI’s and SF % contributions for ring II in naphthalene. Overall DI’s and SF % values for ring-crossing interactions in naphthalene and its derivatives, with some comments. Bond lengths, DI’s and SF % contributions for the symmetry-independent CC bonds in anthracene and phenanthrene. This information is available free of charge via the Internet at http://pubs.acs.org
’ AUTHOR INFORMATION Corresponding Author
*C.G.: e-mail,
[email protected]; fax number, +39 (02) 50314300. L.L.P.: e-mail,
[email protected]; fax number, +39 (02) 50314300 Notes
)
isolated 6MR, but all of them, in different degrees and various forms, are influenced by their involvement in at least two of the three composing resonance systems. When the aromatic system in ring II of naphthalene is destroyed upon (partial or total) hydrogenation, the formally single CC bonds in ring I become stronger, whereas the formally double CC bonds weaken, with the only exception of the central C9C10 one. The observed changes in the SF% contributions indicate that ring I gradually increases its similarity to benzene at the expense of ring II, or put in another way, it increases its amount of local aromaticity. Interestingly, and despite their central geometric position, C9 and C10 atoms become progressively more able in determining the electron density in ring I, through their SF contributions, than they do in ring II, showing that the SF contributions neatly distinguish the electronic effects from the purely geometric ones. (iv) The relative invariance of the various SF contributions to the density at the ring CC bcp’s in the three biphenyl isomers indicates that in this case the aromatic systems of the two six-membered rings are quite independent of each other or, more correctly, that they influence each other in a way that is substantially unaffected by their relative orientation. At variance with naphthalene, biphenyl may be considered as made up of two independent, weakly interacting benzene rings that are only slightly perturbed by the combination of mutual steric and electronic effects. (v) A new descriptor of local aromaticity, SFLAI, has been proposed that shows excellent correlation with the formally similar, DI-based FHDD index. As SFLAI is based on SF, it is applicable in all cases where a pair density is unavailable, including the case of experimentally derived charge densities. This implies that it can provide a direct comparison, on the same real-space grounds, of theoretical and experimental electron delocalization estimates, allowing at the same time to single out important environmental solid-state effects on the molecular properties. A final remark appears to be worthy of note. We showed that SF remains quantitatively invariant with respect to the choice of both the computational method and basis set quality, demonstrating that the small and indirect effects due to π-electron localization are stable and neatly recoverable even at a quite low theory level. Last, but not least, such a stable behavior is promising in the perspective of applying the SF descriptor of aromaticity even to multipole or maximum entropy (experimentally derived) charge densities.
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E-mail: E.M.,
[email protected]; E.O., emanuele.
[email protected].
’ ACKNOWLEDGMENT C.G. warmly thanks Richard Bader for the longstanding, invaluable friendship and his precious scientific advice throughout the years. C.G. is also deeply indebted to Richard Bader for his fundamental contribution to the seminal work on the Source Function. All other authors join C.G. in congratulating Richard Bader for the gift he gave to chemistry through the development of Quantum Theory of Atoms in Molecules. C.G. and L.L.P. thank the Danish National Research Foundation for partial funding of this work through the Center for Materials Crystallography (CMC). Financial support by the Italian MIUR is also gratefully acknowledged by L.L.P. (Fondi PUR 10%, 2010). ’ REFERENCES (1) Poater, J.; Duran, M.; Sola, M.; Silvi, B. Chem. Rev. 2005, 105, 3911–3947. (2) Merino, G.; Vela, A.; Heine, T. Chem. Rev. 2005, 105, 3812–3841. (3) Bultinck, P. Faraday Discuss. 2007, 135, 347–365 and references therein. (4) Lazzeretti, P. Phys. Chem. Chem. Phys. 2004, 6, 217–223. (5) Bader, R. F.W.; Slee, S.; Cremer, D.; Kraka, E. J. Am. Chem. Soc. 1983, 105, 5061–5068. (6) Cremer, D.; Kraka, E.; Slee, S.; Bader, R. F. W.; Lau, C. D. H.; Nguyen-Dang, T. T.; MacDougall, P. J. J. Am. Chem. Soc. 1983, 105, 5069–5075. (7) Bader, R. F. W. Atoms in molecules. A quantum theory; Oxford University Press, Oxford, U.K., 1990. (8) The bond critical points (bcps) are locations where the gradient of the electron density, rF, vanishes and where F is at a minimum along a line of maximum electron density, with respect to any lateral displacement, linking two nuclei. These lines are called bond paths when the system is at an equilibrium geometry. Bond paths may be put in one to one correspondence with chemical bonds and the density properties evaluated at their associated bcps are known to be highly representative of the nature of the bonds. (9) Gatti, C.; Barzaghi, M.; Simonetta, M. J. Am. Chem. Soc. 1985, 107, 878–887. (10) Fulton, R. L.; Mixon, S. T. J. Phys. Chem. 1993, 97, 7530–7534. (11) Fulton, R. L. J. Phys. Chem. 1993, 97, 7516–7529. (12) Matito, E.; Sola, M.; Salvador, P.; Duran, M. Faraday Discuss 2007, 135, 325–345 and references therein.. (13) (a) Bultinck, P.; Cooper, D. L.; Ponec, R. J. Phys. Chem. A 2010, 114, 8754–8763. (b) Heyndrickx, W.; Salvador, P.; Bultinck, P.; Sola, M.; Matito, E. J. Comput. Chem. 2011, 32, 386–395. (14) Fradera, X.; Austen, M. A.; Bader, R. F. W. J. Phys. Chem. A 1999, 103, 304–314. (15) Bader, R. F. W.; Stephens, M. E. Chem. Phys. Lett. 1974, 26, 445–449. (16) Macchi, P.; Sironi, A. Coord. Chem. Rev. 2003, 238239, 383–412. (17) Gatti, C. Z. Kristallogr. 2005, 220, 399–457. (18) Gatti, C.; Ponti, A.; Gamba, A.; Pagani, G. J. Am. Chem. Soc. 1992, 114, 8634–8644. 12877
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The Journal of Physical Chemistry A (19) Bader, R. F. W.; Gatti, C. Chem. Phys. Lett. 1998, 287, 233–238. (20) Arfken, G. Mathematical Methods for Physicists; Academic Press: Orlando, FL, 1985. (21) Gatti, C.; Cargnoni, F.; Bertini, L. J. Comput. Chem. 2003, 24, 422–436. (22) Gatti, C., Struct. Bonding (Berlin) 2011, 1 $DOI: 10.1007/ 430_2010_31 2. (23) Farrugia, L. J.; Macchi, P. J. Phys. Chem. A 2009, 113, 10058–10067. (24) Matta, C. F.; Hernandez-Trujillo J. Phys. Chem. A 2003, 107, 7496–7504. (25) Koritsanszky, T.; Coppens, P. Chem. Rev. 2001, 101, 1583–1627. (26) Bultinck, P.; Fias, S.; Ponec, R. Chem.—Eur. J. 2006, 12, 8813–8818. According to these authors a possible definition of “local” aromaticity in PAH systems is “the retention of aromatic character in a molecular fragment compared to the same fragment in an archetypical molecule”. (27) Kruszewski, J.; Krygowski, T. M. Tetrahedron Lett. 1972, 13, 3839–3842. (28) Krygowski, T. M. J. Chem. Inf. Comput. Sci. 1993, 33, 70–78. (29) Elser, V.; Haddon, R. C. Nature (London) 1987, 325, 792–794. (30) Schleyer, P. v. R.; Maerker, C.; Dransfeld, A.; Jiao, H.; van Eikema Hommes, N. J. R. J. Am. Chem. Soc. 1996, 118, 6317–6318. (31) Bader, R. F. W.; Streitweiser, A.; Neuhaus, A.; Laidig, K. A.; Speers, P. J. Am. Chem. Soc. 1996, 118, 4959–4965. (32) Matito, E.; Duran, M.; Sola, M. J. Chem. Phys. 2005, 122, 014109. (33) Poater, J.; Fradera, X.; Duran M. Sola, M. Chem.Eur. J. 2003, 9, 400–406. (34) Matta, C. F.; Boyd, R. J. The quantum theory of atoms in molecules. From Solid State to DNA and Drug Design; Wiley: New York, 2007. (35) Bultinck, P.; Ponec, R.; Van Damme, S. J. Phys. Org. Chem. 2005, 18, 706–718. (36) Bultinck, P.; Rafat, M.; Ponec, R.; Van Gheluwe, B.; CarboDorca, R.; Popelier, P. J. Phys. Chem. A 2006, 110, 7642–7648. (37) Bultinck, P.; Ponec, R.; Gallegos, A.; Fias, S.; Van Damme, S.; Carbo-Dorca, R. Croat. Chem. Acta 2006, 79, 363–371. (38) Gomes, J. A. N. F.; Malion, R. B. Chem. Rev. 2001, 101, 1349–1384. (39) Keith, T. A.; Bader, R. F. W. Chem. Phys. Lett. 1993, 210, 223–231. (40) Millefiori, S.; Alparone, A. J. Mol. Struct. (THEOCHEM) 1998, 431, 59–78. (41) De Proft, F.; Geerlings, P. Chem. Rev. 2001, 101, 1451–1464. (42) For example, the resonance energy of naphthalene is greater than benzene (61 kcal 3 mol1 vs 36 kcal 3 mol1, i.e., 6.1 kcal 3 mol1 vs 6.0 kcal 3 mol1 per carbon atom, see: Brown, W. H. Organic Chemistry; Saunders College Publishing: Philadelphia, 1995). On the contrary (see section V.1), both rings of naphthalene are less aromatic than benzene in terms of geometrical and electronic similarity. Which then is considered the “more” aromatic system between the two? Or, posed in other words, can aromaticity in naphthalene really be expressed in terms of a sum of the aromatic degree—whatever such an expression means—of the two composing six-membered rings such as to give rise to an overall system (slightly) more aromatic than benzene? These are likely to be ill-posed questions because the π-system of naphthalene should be in fact considered as an emerging property due to the mutual influence of each benzenoid ring to the other. (43) Frenking, G.; Krapp, A. J. Comput. Chem. 2007, 28, 15–24. (44) Godbout, N.; Salahub, D. R.; Andzelm, J.; Wimmer, E. Can. J. Chem. 1992, 70, 560–571. (45) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A. et al. Gaussian 09, Revision A.1; Gaussian, Inc.: Wallingford, CT, 2009. (46) Gatti, C. Unpublished result (available upon request).
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(47) Available from Prof. R. F. W. Bader’s Laboratory, McMaster University, Hamilton, ON, Canada L8S 4M1; http://www.chemistry. mcmaster.ca/aimpac/. (48) Gatti, C.; Lasi, D. Faraday Discuss. 2007, 135, 55–78. (49) The specific distance of 1 au was chosen as close to the one (1.05 au) at which the π-MO density reaches its maximum along the major axis of a CC bond in benzene, at the adopted theory level. (50) The reader is addressed to Table 4 of ref 22 for values of the separate σ and π-contributions to the SF when rps lie out of the molecular plane. (51) Lo Presti, L.; Gatti, C. Chem. Phys. Lett. 2009, 476, 308–316. (52) Engels, B.; Schmidt, T. C.; Gatti, C.; Schirmeister, T.; Fink, R. F. Struct. Bonding (Berlin), 2011, 151, DOI:10.1007/430_2010_36 and references therein (53) Gatti, C.; MacDougall, P. J.; Bader, R. F. W. J. Chem. Phys. 1988, 88, 3792–3804. (54) Wang, Y.; Matta, C.; Werstiuk, N. H. J. Comput. Chem. 2003, 24, 1720–1729. (55) Poater, J.; Sola, M.; Duran, M.; Fradera, X. Theor. Chem. Acc. 2002, 107, 362–371. (56) This is due to the fact that (i) the Coulomb correlation is lacking by definition within the RHF method, and (ii) in the DFT method density matrices and the pair density are not defined. For the sake of evaluating DI’s in DFT, the pair density is tout court constructed employing the HartreeFock formalism on KohnSham wave functions.55 In other words, RHF provides just an upper limit of the correct DI value between two atoms, as the Coulomb correlation has generally the effect of reducing the number of shared electron pairs. The DFT-derived DI estimates, on the other hand, are usually overestimated and may be even larger than the corresponding RHF ones, as the Coulomb correlation is only in part taken into account by currently employed exchangecorrelation functionals. (57) Samanta, S. K.; Singh, Om. V.; Jain, R. K. Trends Biotechnol. 2002, 20, 243–248. (58) Goldman, R.; Enewold, L.; Pellizzari, E.; Beach, J. B.; Bowman, E. D.; Krishnan, S. S.; Shields, P. G. Cancer Res. 2001, 61, 6367–6371. (59) Wu, D.; Liu, R.; Pisula, W.; Feng, X.; M€ullen, K. Angew. Chem., Int. Ed. 2011, 50, 2791–2794. (60) Not unexpectedly, the bond C1C2 in naphthalene, equivalent by symmetry to C8C7, is particularly subject to electrophilic attack. (61) Matta, C. F.; Hernandez-Trujillo, J.; Tang, T.-H; Bader, R. F. W. Chem.—Eur. J. 2003, 9, 1940–1951and references therein. (62) If a similar analysis is made for the 6MR series in section II, one finds that the electron delocalization effects revealed by the SF analysis are real and not simply the result of the replacement of H atoms linked to C by an increased number of available π-electrons on these same C atoms. (63) Different suitable expressions should clearly be formulated for N-membered rings with N 6¼ 6 and/or for rings containing heteroatoms (64) SFLAI takes into account correlations among a bond and those atoms not directly involved in such bond in a quite distinct manner, i.e., by measuring directly the influence of various atoms on specific bonds and then summing together the various contributions. Because SF% contributions have to sum up to 100%, the SFLAI index is implicitly affected also by the cumulative SF% contribution from atoms of other rings. A similar situation occurs also for FHDD. (65) Feixas, F.; Matito, E.; Poater, J.; Sola, M. J. Comput. Chem. 2008, 29, 1543–1554.
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