Article pubs.acs.org/JPCA
Topological Hückel−London−Pople−McWeeny Ring Currents and Bond Currents in altan-Corannulene and altan-Coronene Timothy K. Dickens*,† and Roger B. Mallion‡ †
University Chemical Laboratory, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom School of Physical Sciences, University of Kent, Canterbury CT2 7NH, United Kingdom
‡
ABSTRACT: The Hückel−London−Pople−McWeeny (topological) approach for calculating π-electron ring currents and bond currents in conjugated systems is applied to the structures altan-corannulene and altan-coronene, recently designed by Monaco and Zanasi. These authors applied the ab initio ipso-centric formalism to calculate π-electron current-density maps and individual bond current intensities in these structures. In a parallel investigation the key findings of Monaco and Zanasi’s ab initio study are here qualitatively confirmed by this much more rudimentary topological assessment: namely, that, in their three layers, the two structures altan-corannulene and altancoronene reveal patterns of contra-rotating paramagnetic/diamagnetic/paramagnetic π-electron circulations in their innermost, middle, and outer cycles, respectively.
■
INTRODUCTION Much attention has been devoted over many decades to the calculation of π-electron ring currents and bond currents in conjugated hydrocarbons, both extant and hypothetical; please see refs 1−4 for reviews. In recent work, some groups have been adopting optimized geometries and have then applied ab initio approaches, the most frequent appeal being made to what has become known as the ipso-centric formalism; refs 5−12 are representative of many, and a detailed description of the method and its history is available in ref 9. An entirely different philosophy attempts to establish how much insight may be gleaned by adopting an idealized geometry and applying the classic Hückel−London−Pople−McWeeny (HLPM) method,13−17 fully documented in ref 18. Application of this latter formalism invokes the minimum number of assumptions and parameters; in its simplest form,19−21 this approach depends only on a knowledge of the carbon−carbon connectivity immediately evident from the structure’s molecular graph22 and the areas that are assumed for its constituent rings.18 (A third way23−29 of considering these π-electron circulations involves the concept of circuits of conjugation/conjugated circuits,30,31 though these will not be our concern here.) These studies have frequently involved consideration of specially designed8,10,11,32−34 structures, largely with a view to investigating the so-called annulene-within-an-annulene (AWA) rule5−8,10−12,32−40 for super-ring41−43 systems; (please see Figure 1 of ref 18 for a diagram illustrating a generalized super-ring structure). Such were, for example, [10,5]-coronene (1) (Figure 1), discussed in refs 8, 32−34, 40 and first © 2014 American Chemical Society
Figure 1. Molecular graphs22 of [10,5]-coronene (1) and 7-coronene (2).
considered more than 40 years ago44, and the p-coronene series, introduced and defined in ref 34; the latter includes, again, [10,5]-coronene (1) as well as, for example, 7-coronene (2), the molecular graphs22 of which are both illustrated in Figure 1. It is appropriate also to make passing mention here of the other three regular [r,s]-coronenes.44,40,34 Sometimes, qualitative agreement is observed between the predictions of the ab initio5−12 and HLPM13−21 approaches, and sometimes it is not. For example, there is agreeReceived: November 23, 2013 Revised: January 15, 2014 Published: January 15, 2014 933
dx.doi.org/10.1021/jp411524k | J. Phys. Chem. A 2014, 118, 933−939
The Journal of Physical Chemistry A
Article
ment8,32−34,40 that [10,5]-coronene (1) appears to obey the AWA rule,35−37 whereas there are conflicting qualitative predictions between the two methods in the case of, for instance, the recently discussed34,12 7-coronene (2) (Figure 1): HLPM calculations34 predict that 2 does conform to the AWA model,35−37 while computations12 arising from an application of the ipso-centric ab initio approach based on optimized geometries predict12 that it does not. Monaco and Zanasi10,11 have recently taken this designer policy8,32,33 (literally) to another level and have considered what might be thought of as three-layered super-ring systems, in the forms of (a) corannulene (structure 3 of Figure 2) further surrounded by a peripheral ensemble of alternating five-membered
[24]-annulene, respectively) in such a way that an outgoing C− H bond from the inner structure and an internal C−H bond from the annulene are substituted by a C−C bond that joins the carbon-atom skeleton of the inner structure to alternating carbon-atoms of the annulene.) As a result of their ab initio ipso-centric approach5−12 (as well as of their calculations effected by invoking the concept of circuits of conjugation23−31) Monaco and Zanasi report10 that these structures reveal “...unprecedented patterns of three contra-rotating paramagnetic/diamagnetic/paramagnetic circulations...” in the innermost, middle, and outer cycles, respectively, of 5 and 6. The purpose of this article is to draw attention to the fact that, on this occasion, the HLPM method,13−18 even in its most rudimentary, so-called topological, incarnation,19−21 comes to identical qualitative (and even, as will be seen, semiquantitative) conclusions.
■
CALCULATIONS The standard HLPM formalism,13−17 as described in exhaustive detail in ref 18, was used to calculate ring-current intensities in 5 and 6. Those for 3 and 4 were already available from refs 45−48, 38, 39, and 18. The simplest (topological) version19−21 of this method13−18 was adopted; please see eqs 14 and 16 of ref 18. This approach involves the assumption of regular polygons of uniform side-length forming an idealized planar structure.49 In the course of this, the area of a five-membered ring was taken to be smaller than that of a standard hexagonal (benzene) ring by the factor16 (5 cot(π/5)/(6 cot(π/6)). Once the relative ring-current intensities had been computed and expressed as a (dimensionless) ratio to the ring current intensity calculated, by the same method, for benzene, a consistent set of bond current intensities51 was obtained by repeated application of the microscopic analogy of Kirchhoff’s Law of Conservation of Current at a Junction1−4,23−29 in macroscopic electrical networks. Maps for ring-current intensities and for the bond currents that are consistent with them are depicted in Figures 4 and 5. Comparisons of individual bond-current intensities51 by the ipso-centric5−12 and HLPM13−21 approaches are presented for altan-corannulene (5) in Table 1 and for altan-coronene (6) in Table 2; later (in
Figure 2. Molecular graphs22 of corannulene (3) and coronene (4).
and six-membered rings to form what Monaco and Zanasi call10 altan-corannulene (or altan-1, structure 5 in Figure 3) and (b) coronene (structure 4 of Figure 2) likewise further surrounded in the way just described to form what Monaco and Zanasi10 name altan-coronene (or altan-2, structure 6 of Figure 3). (The altan terminology arises from the fact that Monaco and Zanasi10 think of the formation of 5 and 6 as arising by notionally placing an unsaturated hydrocarbon (corannulene (3) in the one case, coronene (4) in the other) inside a [4n]annulene (in the present examples, it is a [20]-annulene and a
Figure 3. Molecular graphs22 of the specially designed8,10,11,32,33 structures10 altan-corannulene (altan-110) (5) and altan-coronene (altan-210) (6). In the depictions of structures 5 and 6, the bonds of the corannulene (3) system situated at the core of 5 and the bonds of the coronene (4) system at the core of 6 are both emphasized by being drawn with thicker lines (red). Bond labelings indicated on 5 and 6 for the symmetrically distinct bonds of 3−6, which will be referred to later, are identical to those adopted by Monaco and Zanasi in ref 10. 934
dx.doi.org/10.1021/jp411524k | J. Phys. Chem. A 2014, 118, 933−939
The Journal of Physical Chemistry A
Article
Figure 6), the two types of calculations are also compared by means of comparitor diagrams.
■
DISCUSSION OF RESULTS It can immediately be seen from the ring current and bond current maps for both altan-corannulene (5) and altancoronene (6) shown in Figures 4 and 5 that (a) the net current-flow around the central ring (that is, the innermost cycle, involving bonds symmetrically equivalent to bond a in Figure 3) is in the clockwise (that is, paramagnetic) direction; (b) the net flow around the middle cycle (involving the symmetrically distinct bonds c, d, and e of Figure 3) is counterclockwise (that is, in the diamagnetic sense), and (c) the circulation around the outer, peripheral cycle (involving the symmetrically distinct bonds g, h, j, and k of Figure 3) is, as with the innermost cycle, again clockwise (that is, paramagnetic). There is, therefore, qualitative agreement between the predictions of these topological HLPM computations19−21 and those arising from the ipso-centric, ab initio calculations5−12 of Monaco and Zanasi,10 namely, that, in their three layers, the two structures altan-corannulene (5) and altan-coronene (6) each reveal patterns of contra-rotating paramagnetic/diamagnetic/paramagnetic π-electron circulations (in their innermost, middle, and outer cycles, respectively). Monaco and Zanasi also pointed out in the context of their study10 by the ab initio ipso-centric method5−12 that (a) patterns in the innermost and middle cycles of altancorannulene (5) and altan-coronene (6) are qualitatively unchanged from what they are in the corresponding cycles of their respective parent-systems (corannulene (3) and coronene (4)) and (b) that the innermost cycle has a smaller current in the corresponding altan systems, 5 and 6, than in 3 and 4, respectively, while (c) the intensity on the middle cycle of 5 and 6 (which corresponds to the outer cycle in 3 and 4, respectively) is (semi)-quantitatively similar in the two sets of structures. In other words, these authors claim that the behavior of the circulating π-electron currents in the core of 5 and 6, which consists of corannulene (3) and coronene (4), respectively, is qualitatively preserved when it is encased in the extra layer of five- and six-membered rings. Precisely the same conclusion could be drawn from the calculations presented here, which, as already noted under Calculations, have been effected by applying the HLPM13−18 approach in its simplest (topological) form.19−21 The paramagnetic circulations around the central rings (what we have been calling the innermost cycles) of 3 and 5 are about 1.0 and 0.92, respectively; the same figures for the central rings of 4 and 6 are (approximately) 0.42 and 0.22. The diamagnetic current around the peripheral cycle of 3 and the average diamagnetic current around the middle cycle of 5 are ca. 1.22 and 0.94, respectively. The corresponding data for the analogous cycles in 4 and 6 are about 1.46 and 1.04. Monaco and Zanasi also observe10 that, on the basis of their ab initio ipso-centric calculations,5−12 the outermost circuits host strong paramagnetic circulations, of roughly half the benzene reference value. In the case of our topological HLPM calculations,19−21 the average paramagnetic current around the periphery of 5 is ca. 0.41, and around the outer cycle of 6, it is
Figure 4. Map of the topological ring currents (in black) and the associated topological bond currents (in red) for altan-corannulene (altan-110) (5). The topological ring currents and bond currents are dimensionless quantities, all appropriately depicted here as pure numbers, without units. The ring current in benzene on this same scale is 1.000. Positive (diamagnetic) ring currents are considered to circulate counterclockwise around their respective rings, while negative (paramagnetic) ring currents flow in the clockwise sense around those rings. The numerous bond currents flow in the direction indicated by the arrow pointing along each bond.
Figure 5. Map of the topological ring currents (in black) and the associated topological bond currents (in red) for altan-coronene (altan-210) (6). For an explanation of the conventions on the depiction of ring currents and bond currents, please see the caption to Figure 4.
935
dx.doi.org/10.1021/jp411524k | J. Phys. Chem. A 2014, 118, 933−939
The Journal of Physical Chemistry A
Article
Table 1. Comparison of Individual ab Initio (ipso-Centric) and HLPM (Topological) Bond Currents in Corannulene (3) and altan-Corannulene (5) symmetrically distinct bonda a c d e g h j k
bond current in corannulene (3) (by the ipso-centric5−12 ab initio method)b,c
bond current in corannulene (3) (by the HLPM13−21 topological method)b,d
bond current in altan-corannulene (5) (by the ipso-centric5−12 ab initio method)b,c
bond current in altan-corannulene (5) (by the HLPM13−21 topological method)b,e
−0.76 0.79 0.80 0.79
−1.00 1.22 1.22 1.22
−0.39 0.80 0.73 0.78 −0.69 −0.69 −0.64 −0.64
−0.92 0.88 1.05 0.88 −0.33 −0.33 −0.50 −0.50
a
Symmetrically distinct bonds in corannulene (3) (columns 2 and 3) and altan-corannulene (5) (columns 4 and 5) are labeled (in column 1) as in Scheme 1 of ref 10 (illustrated here in Figure 3). Only bonds from the central and peripheral cycles in corannulene (3) and from the central, middle, and peripheral cycles in altan-corannulene (5) are included here; the spokes bonds55 are omitted from consideration. bA positive sign indicates a diamagnetic (counterclockwise) circulation and a negative sign indicates a paramagnetic (clockwise) circulation around the cycle containing the bond under consideration. Reference to Figure 3 shows that this means (i) for corannulene (3), the central cycle (the bonds symmetrically equivalent to bond a) and the outer, peripheral cycle (involving bonds c, d, and e); and (ii) for altan-corannulene (5), the central cycle (the bonds equivalent to bond a), the middle cycle (pertaining to bonds c, d, and e), and the outer, peripheral cycle (involving bonds g, h, j, and k). cipso-Centric bond currents for 3 and 5 are transcribed from ref 10. dHLPM topological bond currents for corannulene (3) are from refs 45 and 38. eHLPM topological bond currents for altan-corannulene (5) are transcribed from Figure 4.
Table 2. Comparison of Individual ab Initio (ipso-Centric) and HLPM (Topological) Bond Currents in Coronene (4) and altanCoronene (6) symmetrically distinct bonda a c d e g h j k
bond current in coronene (4) (by the ipso-centric5−12 ab initio method)b,c
bond current in coronene (4) (by the HLPM13−21 topological method)b,d
bond current in altan-coronene (6) (by the ipso-centric5−12 ab initio method)b,c
bond current in altan-coronene (6) (by the HLPM13−21 topological method)b,e
−0.46 1.27 1.24 1.27
−0.42 1.46 1.46 1.46
−0.28 1.06 1.03 1.07 −0.40 −0.42 −0.42 −0.42
−0.22 0.97 1.20 0.97 −0.04 −0.04 −0.27 −0.27
a
Symmetrically distinct bonds in coronene (4) (columns 2 and 3) and altan-coronene (6) (columns 4 and 5) are labeled (in column 1) as in Scheme 1 of ref 10 (illustrated here in Figure 3). Only bonds from the central and peripheral cycles in coronene (4) and from the central, middle, and peripheral cycles in altan-coronene (6) are included here; the spokes’ bonds55 are omitted from consideration. bA positive sign indicates a diamagnetic (counterclockwise) circulation and a negative sign indicates a paramagnetic (clockwise) circulation around the cycle containing the bond under consideration. Reference to Figure 3 shows that this means (i) for Coronene (4): the central cycle (the bonds symmetrically equivalent to bond a) and the outer, peripheral cycle (involving bonds c, d and e), and (ii) for Altan-Coronene (6): the central cycle (the bonds equivalent to bond a), the middle cycle (pertaining to bonds c, d, and e), and the outer, peripheral cycle (involving bonds g, h, j, and k). cipso-Centric bond currents for 4 and 6 are transcribed from ref 10. dHLPM topological bond currents for coronene (4) are from refs 46−48, 38, 39, and 18. eHLPM topological bond currents for altan-coronene (6) are transcribed from Figure 5.
approximately 0.15. The ab initio ipso-centric 5−12 and HLPM13−21 calculations agree that the peripheral paramagnetic circulation in 5 is of greater magnitude than that in 6. Both the ab initio ipso-centric5−12 and the HLPM13−21 approaches are in qualitative accord that, when 3 and 4 are converted to 5 and 6, respectively, by the addition of an outer layer of alternating five- and six-membered rings (as in Figures 2 and 3), (a) the patterns of current circulation around the core of 5 and 6, that is, their innermost and middle cycles, are effectively preserved as what they are in the inner and outer cycles of the parent structures, 3 and 4, respectively, and that (b) the peripheral bonds of the newly added layer of rings support paramagnetic circulations, that in 5 being larger than that in 6.
Now, we note that altan-corannulene (5) is a system that may potentially be regarded as a [5]-annulene within a [15]annulene, which is itself surrounded by an outer [20]-annulene, while altan-coronene (6) would appear potentially to be a candidate for being considered as a [6]-annulene within an [18]-annulene within a [24]-annulene. If the latter behaved according to an intuitive extension of the AWA rule35−37 to three layers, then the behavior of the π-electron circulations displayed would have to be diamagnetic/diamagnetic/paramagnetic (in the innermost, middle, and outer cycles, respectively). However, paramagnetic/diamagnetic/paramagnetic is what is in fact actually predicted by both the ipsocentric5−12 and the HLPM approaches.13−21 The situation with altan-corannulene (5) is more complicated: (a) If transfer of one electron f rom the inner [5]-cycle to the middle [15]-cycle were postulated, the pattern would be 936
dx.doi.org/10.1021/jp411524k | J. Phys. Chem. A 2014, 118, 933−939
The Journal of Physical Chemistry A
Article
Figure 6. Comparitor diagrams for (a) corannulene (3), (b) coronene (4), (c) altan-corannulene (5), and (d) altan-coronene (6). The comparison in each case is between the predictions of ab initio ipso-centric calculations5−12 (blue lines) and those of topological HLPM calculations13−21 (brown lines). These diagrams are based on the data presented in Tables 1 and 2. (Dimensionless) Bond currents are displayed along the vertical axes, and letters denoting individual bonds in each of the structures 3−6, labeled as in Figure 3, are ranged along the horizontal axes.
ones; that is, diamagnetic bond currents are more strongly diamagnetic and paramagnetic ones are more strongly paramagnetic; please see columns 4 and 5 of Tables 1 and 2 and Figure 6c,d for support of this assertion. As was pointed out by one of us (and others),56−59 many years ago, this is to be expected in an HLPM calculation on structures in which some rings are predicted to bear paramagnetic ring currents and no allowance is made for resonance-integrals to be made iteratively self-consistent with respect to calculated bond orders.56−59,40 Such an extension has, however, not been entertained here because doing so would have deprived the calculations of being what we have called purely topological18−21 in nature, and our aim has always been to see to what extent the simplest, parameter-free, form of the HLPM approach, which takes account only of carbon-atom connectivity and treats rings as regular polygons in an assumed planar network, can mimic the trends reflected by more-sophisticated calculations based on more-realistic geometries.
expected to be paramagnetic/paramagnetic/paramagnetic (in the innermost, middle, and outer cycles, respectively); instead, paramagnetic/diamagnetic/paramagnetic is predicted. The circulation in the middle cycle would, therefore, violate an intuitive extension of the AWA rule to this three-layered system in this way. (b) If the postulated transfer of one electron were in the opposite direction, f rom the [15]-membered cycle to the [5]-membered cycle, the expected pattern for the three cycles would be diamagnetic/diamagnetic/paramagnetic. The middle cycle (and, as before, the peripheral cycle) would now comply with the AWA rule, but the direction of circulation in the innermost cycle would not conform to the AWA rule. Overall, therefore, concerning altan-corannulene (5) and altan-coronene (6), it is seen that the qualitative predictions of both the ab initio ipso-centric method5−12 and the HLPM formalism,13−21 though consistent and in agreement, between themselves, are not in accord with the provisions of the AWA rule.35−37 That is the situation qualitatively. However, in addition to providing pictorial current-density maps, as has become traditional when results of the ipso-centric ab initio calculations are presented,5−12 Monaco and Zanasi, in their recent paper,10 have also displayed actual, quantitative bond current intensities (expressed relative to the similarly calculated benzene value) for the symmetrically nonequivalent bonds in structures 3−6 (which are all explicitly depicted and labeled in Figure 3). This therefore affords the opportunity for a direct, quantitative comparison between the predictions of the ipso-centric and HLPM formalisms for these four structures.51 Such a comparison is made in Tables 1 and 2. The same data are also presented as comparitor diagrams in Figure 6. In the case of corannulene (3) (Table 1, columns 2 and 3, and Figure 6a) and coronene (4) (Table 2, columns 2 and 3, and Figure 6b), where all HLPM ring currents are diamagnetic, trends between the two approaches compare well. In the case of 5 and 6, however, there is a tendency for the HLPM bond currents to be more exaggerated than the corresponding ab initio ipso-centric
■
CONCLUSIONS Calculated π-electron currents based on the simplest HLPM approach,13−21 which requires1,2,4,18−21 only (a) knowledge of the molecular graph22 of the structure under study (taken to be geometrically planar) and (b) the assumption that the areas of its constituent rings are those of regular polygons of standard side-length, are in semiquantitative agreement with those effected by appeal to the more-sophisticated ipso-centric ab initio method based on more-realistic geometries.5−12 The predictions of both formalisms agree that, in their three layers, the two structures altan-corannulene (5) and altan-coronene (6) each reveal patterns of contra-rotating paramagnetic/ diamagnetic/paramagnetic π-electron circulations, in their innermost, middle, and outer cycles, respectively. Nevertheless, with regard to 5 and 6, the predictions of both the ab initio ipso-centric method5−12 and the HLPM formalism,13−21 though consistent and in (semiquantitative) agreement between themselves, are not in accord with the provisions of any intuitive extension of the AWA rule35−37 to such three-layered systems. 937
dx.doi.org/10.1021/jp411524k | J. Phys. Chem. A 2014, 118, 933−939
The Journal of Physical Chemistry A
■
Article
(21) Dickens, T. K.; Mallion, R. B. Topological Ring-Currents in the “Empty” Ring of Benzo-Annelated Perylenes. J. Phys. Chem. A 2011, 115, 351−356. (22) Trinajstić, N. Chemical Graph Theory; 1st ed.; CRC Press: Boca Raton, FL, 1983; Vol. I, (a) Chapter 3, pp. 21−30; (b) Chapter 2, pp. 12−14. (23) Gomes, J. A. N. F.; Mallion, R. B. A Quasi-Topological Method for the Calculation of Relative “Ring-Current” Intensities in Polycyclic, ̀ Conjugated Hydrocarbons. Revista Portuguesa de Quimica 1979, 21, 82−89. (24) Gayoso, J.; Boucekkine, A. Application de la Théorie des Graphes à la Détermination des Exaltations de Susceptibilité des Hydrocarbures Conjugés. C. R. Hebd. Séances Acad. Sci. (Paris) 1979, C288, 327−330. (25) Mandado, M. Determination of London Susceptibilities and Ring Current Intensities Using Conjugated Circuits. J. Chem. Theory Comput. 2009, 5, 2694−2701. (26) Ciesielski, A.; Krygowski, T. M.; Cyrański, M. K.; Dobrowolski, M. A.; Aihara, J. Graph-Topological Approach to Magnetic Properties of Benzenoid Hydrocarbons. Phys. Chem. Chem. Phys. 2009, 11, 11447−11455. (27) Randić, M. Graph Theoretical Approach to π-Electron Currents in Polycyclic Conjugated Hydrocarbons. Chem. Phys. Lett. 2010, 500, 123−127. (28) Dickens, T. K.; Gomes, J. A. N. F.; Mallion, R. B. Some Comments on Topological Approaches to the π-Electron Currents in Conjugated systems. J. Chem. Theory Comput. 2011, 7, 3661−3674. (29) Fowler, P. W.; Myrvold, W. The “Anthracene Problem”: ClosedForm Conjugated-Circuit Models of Ring Currents in Linear Polyacenes. J. Phys. Chem. A 2011, 115, 13191−13200. (30) Gomes, J. A. N. F. Some Magnetic Effects in Molecules. D. Phil. thesis, University of Oxford, 1976, pp. 69−73. (31) Randić, M. Conjugated Circuits and Resonance Energies of Benzenoid Hydrocarbons. Chem. Phys. Lett. 1976, 38, 68−70. (32) Monaco, G.; Fowler, P. W.; Lillington, M.; Zanasi, R. Designing Paramagnetic Circulenes. Angew. Chem. Int. Ed. 2007, 46, 1889−1892. (33) Monaco, G.; Zanasi, R. On the Additivity of Current Density in Polycyclic Aromatic Hydrocarbons. J. Chem. Phys. 2009, 131, 044126. (34) Dickens, T. K.; Mallion, R. B. π-Electron Ring-Currents and Bond-Currents in [10,5]-Coronene and Related Structures Conforming to the ‘Annulene-Within-an-Annulene’ Model. Phys. Chem. Chem. Phys. 2013, 15, 8245−8253. (35) Barth, W. E.; Lawton, R. G. Dibenzo[ghi,mno]fluoranthene. J. Am. Chem. Soc. 1966, 88, 380−381. (36) Barth, W. E.; Lawton, R. G. The Synthesis of Corannulene. J. Am. Chem. Soc. 1971, 93, 1730−1745. (37) Benshafrut, R.; Shabtai, E.; Rabinowitz, M.; Scott, L. T. πConjugated Anions: from Carbon-Rich Anions to Charged Carbon Allotropes. Eur. J. Org. Chem. 2000, 1091−1106. (38) Dickens, T. K.; Mallion, R. B. Topological Ring-Current Assessment of the ‘Annulene-Within-an-Annulene’ Model in [N]Circulenes and some Structures Related to Kekulene. Chem. Phys. Lett. 2011, 517, 98−102. (39) Dickens, T. K.; Mallion, R. B. Topological Ring-Currents and Bond-Currents in Some Nonalternant Isomers of Coronene. J. Phys. Chem. A 2011, 115, 13877−13884. (40) Dickens, T. K.; Mallion, R. B. The Regular [r,s]-Coronenes and the ‘Annulene-Within-an-Annulene’ Rule. RSC Adv. 2013, 3, 15585− 15588. (41) Aihara, J. Is Superaromaticity a Fact or an Artifact? The Kekulene Problem. J. Am. Chem. Soc. 1992, 114, 865−868. (42) Aihara, J. A Simple Method for Estimating the Superaromatic Stabilization Energy of a Super-Ring Molecule. J. Phys. Chem. A 2008, 112, 4382−4385. (43) Aihara, J.; Makino, M.; Ishida, T.; Dias, J. R. Analytical Study of Superaromaticity in Cycloarenes and Related Coronoid Hydrocarbons. J. Phys. Chem. A 2013, 117, 4688−4697. (44) Bochvar, D. A.; Gal’pern, E. G.; Gambaryan, N. P. Calculation of the Molecule of Coronene-10,5 (C30H10). Izv. Akad. Nauk SSSR, Ser.
AUTHOR INFORMATION
Corresponding Author
*(T.K.D.) E-mail: [email protected]. Phone: +44 1223 763 811. Notes
The authors declare no competing financial interest.
■
REFERENCES
(1) Haigh, C. W.; Mallion, R. B. Ring Current Theories in Nuclear Magnetic Resonance. In Progress in Nuclear Magnetic Resonance Spectroscopy; Emsley, J. W., Feeney, J., Sutcliffe, L. H., Eds; Pergamon Press: Oxford, U.K., 1979/1980; Vol. 13, pp 303−344. (2) Gomes, J. A. N. F.; Mallion, R. B. The Concept of Ring Currents. In Concepts in Chemistry; Rouvray, D. H., Ed.; Research Studies Press Limited: Taunton, U.K., 1997; and John Wiley & Sons, Inc.: New York, 1997; Chapter 7, pp 205−253. (3) Lazzeretti, P. Ring Currents. In Progress in Nuclear Magnetic Resonance Spectroscopy; Emsley, J. W., Feeney, J., Sutcliffe, L. H., Eds; Elsevier: Amsterdam, The Netherlands, 2000; Vol. 39, pp 1−88. (4) Gomes, J. A. N. F.; Mallion, R. B. Aromaticity and Ring Currents. Chem. Rev 2001, 101, 1349−1383. (5) Steiner, E.; Fowler, P. W.; Jenneskens, L. W. Counter-Rotating Ring-Currents in Coronene and Corannulene. Angew. Chem. Int. Ed. 2001, 40, 362−366. (6) Steiner, E.; Fowler, P. W. Patterns of Ring Currents in Conjugated Molecules: a Few-Electron Model Based on Orbital Contributions. J. Phys. Chem. A 2001, 105, 9553−9562. (7) Fowler, P. W.; Steiner, E. Pseudo π-Currents: Rapid and Accurate Visualisation of Ring Currents in Conjugated Hydrocarbons. Chem. Phys. Lett. 2002, 364, 259−266. (8) Monaco, G.; Viglione, R. G.; Zanasi, R.; Fowler, P. W. Designing Ring-Current Patterns: [10,5]-Coronene, a Circulene with Inverted Rim and Hub Currents. J. Phys. Chem. A 2006, 110, 7447−7452. (9) Steiner, E.; Soncini, A.; Fowler, P. W. Full Spectral Decomposition of Ring Currents. J. Phys. Chem. A 2006, 110, 12882−12886. (10) Monaco, G.; Zanasi, R. Three Contra-Rotating Currents from a Rational Design of Polycyclic Aromatic Hydrocarbons: altanCorannulene and altan-Coronene. J. Phys. Chem. A 2012, 116, 9020−9026. (11) Monaco, G.; Memoli, M.; Zanasi, R. Additivity of Current Density Patterns in Altan-Molecules. J. Phys. Org. Chem. 2013, 26, 109−114. (12) Monaco, G.; Zanasi, R. Investigation of the p-Coronene Series in the Context of the ‘Annulene-Within-an-Annulene’ Model by means of Ipso-Centric Ab-Initio Calculations of π-Electron Currents. Phys. Chem. Chem. Phys. 2013, 15, 17654−17657. (13) Coulson, C. A.; O’Leary, B.; Mallion, R. B. Hückel Theory for Organic Chemists; Academic Press: London, 1978. (14) Yates, K. Hückel Molecular Orbital Theory; Academic Press: New York, 1978. (15) London, F. Théorie Quantique des Courants Interatomiques dans les Combinaisons Aromatiques. J. Phys. Radium (7e Série) 1937, 8, 397−409. (16) Pople, J. A. Molecular Orbital Theory of Aromatic Ring Currents. Mol. Phys. 1958, 1, 175−180. (17) McWeeny, R. Ring Currents and Proton Magnetic Resonance in Aromatic Molecules. Mol. Phys. 1958, 1, 311−321. (18) Dickens, T. K.; Mallion, R. B. An Analysis of Topological RingCurrents and their Use in Assessing the Annulene-Within-anAnnulene Model for Super-Ring Conjugated Systems. Croat. Chem. Acta 2013, 86, 387−406. (19) Mallion, R. B. Topological Ring-Currents in Condensed Benzenoid Hydrocarbons. Croat. Chem. Acta 2008, 81, 227−246. (20) Balaban, A. T.; Dickens, T. K.; Gutman, I.; Mallion, R. B. Ring Currents and the PCP Rule. Croat. Chem. Acta 2010, 83, 209−215. 938
dx.doi.org/10.1021/jp411524k | J. Phys. Chem. A 2014, 118, 933−939
The Journal of Physical Chemistry A
Article
Khim. 1970, 3, 435−437 An English translation is available from: Consultants Bureau, A Division of Plenum Publishing Corporation, 227 West 17th Street, New York, NY 10011, via the website http:// www.springerlink.com/content/l5np5207n8q10240/. (45) Mallion, R. B. Ring Currents in Corannulene, a Prototype Pattern-Molecule for Buckminsterfullerene Math/Chem/Comp 1988. Proceedings of an International Course and Conference on the Interfaces between Mathematics and Computer Science, Dubrovnik, Yugoslavia, 20th−25th June, 1988; Graovac, A., Ed., Studies in Physical and Theoretical Chemistry; Elsevier Science Publishers, B. V.: Amsterdam, The Netherlands, 1989; Vol. 63, pp 505−510. (46) Maddox, I. J.; McWeeny, R. Ring Currents in Aromatic Molecules. J. Chem. Phys. 1962, 36, 2353−2354. (47) Jonathan, N.; Gordon, S.; Dailey, B. P. Chemical Shifts and Ring Currents in Condensed Ring Hydrocarbons. J. Chem. Phys. 1962, 36, 2443−2448. (48) Aihara, J. The Origin of Counter-Rotating Rim and Hub Currents in Coronene. Chem. Phys. Lett. 2004, 393, 7−11. (49) The calculations of Monaco and Zanasi10 have indicated that the two Altan structures considered here (5 and 6, depicted in Figure 3) are bowl-shaped. Now, the ring-current concept is strictly defined in the HLPM approach13−17 only for geometrically planar structures.1,2,4 Accordingly, in the calculations being reported here, we take the symmetries of altan-corannulene (5) and altan-coronene (6) to be D5h (i.e., noncrystallographic) and D6h (crystallographic), respectively. If these structures had been considered to be nonplanar, they would have been ascribed symmetries of C5v and C6v, respectively. Furthermore, when they were adapting the McWeeny method,17 in order to make it approximately applicable to the nonplanar helicenes, Haigh and one of the present authors50 stated that the assumptions required to do this were valid only if molecular overcrowding is such that “...the skeletal distortion about any bond is comparatively mild.” They further pointed out that, “...because of the way in which the strains from overcrowding are spread over many degrees of freedom, this condition does appear to be satisfied in the case of the helicenes, even though the overall nonplanarity between well-separated parts of such molecules may in fact be very large.” This requirement does appear to be fulfilled in the case of the HLPM calculations being presented here, on structures 5 and 6. (50) Haigh, C. W.; Mallion, R. B. Proton Magnetic Resonance of Non-Planar Condensed Benzenoid Hydrocarbons. II. Theory of Chemical Shifts. Mol. Phys. 1971, 22, 955−970. (51) It is pertinent to observe that the quantitative bond currents obtained from the ab initio ipso-centric method5−12 do not in general strictly obey Kirchhoff’s Law of Current Conservation at Junctions28,29 unless an infinite basis-set is adopted.12 However, bond-currents calculated by the HLPM approach13−21 do, by contrast, rigorously obey the Kirchhoff Conservation Law, thereby drawing attention to an agreeable analogy,52−54 in the context of the microscopic, molecular systems being considered here, with classical macroscopic electricalnetworks, of the type addressed by Kirchhoff in his pioneering work of more than 150 years ago.52−54,28,29 (52) Mallion, R. B. Some Graph-Theoretical Aspects of Simple Ring Current Calculations on Conjugated Systems. Proc. Royal Soc. (London) 1974/1975, A341, 429−449. (53) Mallion, R. B. On the Magnetic Properties of Conjugated Molecules. Mol. Phys. 1973, 25, 1415−1432. (54) Mallion, R. B. On the Number of Spanning Trees in a Molecular Graph. Chem. Phys. Lett. 1975, 36, 170−174. (55) Balaban, A. T.; Bean, D. E.; Fowler, P. W. Patterns of Ring Current in Coronene Isomers. Acta Chim. Slov. 2010, 57, 507−512. (56) Coulson, C. A.; Mallion, R. B. On the Question of Paramagnetic “Ring Currents” in Pyracylene and Related Molecules. J. Am. Chem. Soc. 1976, 98, 592−598. (57) Mallion, R. B. Some Comments on the Use of the “RingCurrent” Concept in Diagnosing and Defining “Aromaticity. Pure Appl. Chem. 1980, 52, 1541−1548.
(58) Gomes, J. A. N. F.; Mallion, R. B. Calculated Magnetic Properties of Some Isomers of Pyracylene. J. Org. Chem. 1981, 46, 719−727. (59) Mallion, R. B. Ring-Current Effects in C60. Nature 1987, 325, 760−761.
939
dx.doi.org/10.1021/jp411524k | J. Phys. Chem. A 2014, 118, 933−939
Topological Hückel−London−Pople−McWeeny Ring Currents and Bond Currents in altan-Corannulene and altan-Coronene Timothy K. Dickens*,† and Roger B. Mallion‡ †
University Chemical Laboratory, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom School of Physical Sciences, University of Kent, Canterbury CT2 7NH, United Kingdom
‡
ABSTRACT: The Hückel−London−Pople−McWeeny (topological) approach for calculating π-electron ring currents and bond currents in conjugated systems is applied to the structures altan-corannulene and altan-coronene, recently designed by Monaco and Zanasi. These authors applied the ab initio ipso-centric formalism to calculate π-electron current-density maps and individual bond current intensities in these structures. In a parallel investigation the key findings of Monaco and Zanasi’s ab initio study are here qualitatively confirmed by this much more rudimentary topological assessment: namely, that, in their three layers, the two structures altan-corannulene and altancoronene reveal patterns of contra-rotating paramagnetic/diamagnetic/paramagnetic π-electron circulations in their innermost, middle, and outer cycles, respectively.
■
INTRODUCTION Much attention has been devoted over many decades to the calculation of π-electron ring currents and bond currents in conjugated hydrocarbons, both extant and hypothetical; please see refs 1−4 for reviews. In recent work, some groups have been adopting optimized geometries and have then applied ab initio approaches, the most frequent appeal being made to what has become known as the ipso-centric formalism; refs 5−12 are representative of many, and a detailed description of the method and its history is available in ref 9. An entirely different philosophy attempts to establish how much insight may be gleaned by adopting an idealized geometry and applying the classic Hückel−London−Pople−McWeeny (HLPM) method,13−17 fully documented in ref 18. Application of this latter formalism invokes the minimum number of assumptions and parameters; in its simplest form,19−21 this approach depends only on a knowledge of the carbon−carbon connectivity immediately evident from the structure’s molecular graph22 and the areas that are assumed for its constituent rings.18 (A third way23−29 of considering these π-electron circulations involves the concept of circuits of conjugation/conjugated circuits,30,31 though these will not be our concern here.) These studies have frequently involved consideration of specially designed8,10,11,32−34 structures, largely with a view to investigating the so-called annulene-within-an-annulene (AWA) rule5−8,10−12,32−40 for super-ring41−43 systems; (please see Figure 1 of ref 18 for a diagram illustrating a generalized super-ring structure). Such were, for example, [10,5]-coronene (1) (Figure 1), discussed in refs 8, 32−34, 40 and first © 2014 American Chemical Society
Figure 1. Molecular graphs22 of [10,5]-coronene (1) and 7-coronene (2).
considered more than 40 years ago44, and the p-coronene series, introduced and defined in ref 34; the latter includes, again, [10,5]-coronene (1) as well as, for example, 7-coronene (2), the molecular graphs22 of which are both illustrated in Figure 1. It is appropriate also to make passing mention here of the other three regular [r,s]-coronenes.44,40,34 Sometimes, qualitative agreement is observed between the predictions of the ab initio5−12 and HLPM13−21 approaches, and sometimes it is not. For example, there is agreeReceived: November 23, 2013 Revised: January 15, 2014 Published: January 15, 2014 933
dx.doi.org/10.1021/jp411524k | J. Phys. Chem. A 2014, 118, 933−939
The Journal of Physical Chemistry A
Article
ment8,32−34,40 that [10,5]-coronene (1) appears to obey the AWA rule,35−37 whereas there are conflicting qualitative predictions between the two methods in the case of, for instance, the recently discussed34,12 7-coronene (2) (Figure 1): HLPM calculations34 predict that 2 does conform to the AWA model,35−37 while computations12 arising from an application of the ipso-centric ab initio approach based on optimized geometries predict12 that it does not. Monaco and Zanasi10,11 have recently taken this designer policy8,32,33 (literally) to another level and have considered what might be thought of as three-layered super-ring systems, in the forms of (a) corannulene (structure 3 of Figure 2) further surrounded by a peripheral ensemble of alternating five-membered
[24]-annulene, respectively) in such a way that an outgoing C− H bond from the inner structure and an internal C−H bond from the annulene are substituted by a C−C bond that joins the carbon-atom skeleton of the inner structure to alternating carbon-atoms of the annulene.) As a result of their ab initio ipso-centric approach5−12 (as well as of their calculations effected by invoking the concept of circuits of conjugation23−31) Monaco and Zanasi report10 that these structures reveal “...unprecedented patterns of three contra-rotating paramagnetic/diamagnetic/paramagnetic circulations...” in the innermost, middle, and outer cycles, respectively, of 5 and 6. The purpose of this article is to draw attention to the fact that, on this occasion, the HLPM method,13−18 even in its most rudimentary, so-called topological, incarnation,19−21 comes to identical qualitative (and even, as will be seen, semiquantitative) conclusions.
■
CALCULATIONS The standard HLPM formalism,13−17 as described in exhaustive detail in ref 18, was used to calculate ring-current intensities in 5 and 6. Those for 3 and 4 were already available from refs 45−48, 38, 39, and 18. The simplest (topological) version19−21 of this method13−18 was adopted; please see eqs 14 and 16 of ref 18. This approach involves the assumption of regular polygons of uniform side-length forming an idealized planar structure.49 In the course of this, the area of a five-membered ring was taken to be smaller than that of a standard hexagonal (benzene) ring by the factor16 (5 cot(π/5)/(6 cot(π/6)). Once the relative ring-current intensities had been computed and expressed as a (dimensionless) ratio to the ring current intensity calculated, by the same method, for benzene, a consistent set of bond current intensities51 was obtained by repeated application of the microscopic analogy of Kirchhoff’s Law of Conservation of Current at a Junction1−4,23−29 in macroscopic electrical networks. Maps for ring-current intensities and for the bond currents that are consistent with them are depicted in Figures 4 and 5. Comparisons of individual bond-current intensities51 by the ipso-centric5−12 and HLPM13−21 approaches are presented for altan-corannulene (5) in Table 1 and for altan-coronene (6) in Table 2; later (in
Figure 2. Molecular graphs22 of corannulene (3) and coronene (4).
and six-membered rings to form what Monaco and Zanasi call10 altan-corannulene (or altan-1, structure 5 in Figure 3) and (b) coronene (structure 4 of Figure 2) likewise further surrounded in the way just described to form what Monaco and Zanasi10 name altan-coronene (or altan-2, structure 6 of Figure 3). (The altan terminology arises from the fact that Monaco and Zanasi10 think of the formation of 5 and 6 as arising by notionally placing an unsaturated hydrocarbon (corannulene (3) in the one case, coronene (4) in the other) inside a [4n]annulene (in the present examples, it is a [20]-annulene and a
Figure 3. Molecular graphs22 of the specially designed8,10,11,32,33 structures10 altan-corannulene (altan-110) (5) and altan-coronene (altan-210) (6). In the depictions of structures 5 and 6, the bonds of the corannulene (3) system situated at the core of 5 and the bonds of the coronene (4) system at the core of 6 are both emphasized by being drawn with thicker lines (red). Bond labelings indicated on 5 and 6 for the symmetrically distinct bonds of 3−6, which will be referred to later, are identical to those adopted by Monaco and Zanasi in ref 10. 934
dx.doi.org/10.1021/jp411524k | J. Phys. Chem. A 2014, 118, 933−939
The Journal of Physical Chemistry A
Article
Figure 6), the two types of calculations are also compared by means of comparitor diagrams.
■
DISCUSSION OF RESULTS It can immediately be seen from the ring current and bond current maps for both altan-corannulene (5) and altancoronene (6) shown in Figures 4 and 5 that (a) the net current-flow around the central ring (that is, the innermost cycle, involving bonds symmetrically equivalent to bond a in Figure 3) is in the clockwise (that is, paramagnetic) direction; (b) the net flow around the middle cycle (involving the symmetrically distinct bonds c, d, and e of Figure 3) is counterclockwise (that is, in the diamagnetic sense), and (c) the circulation around the outer, peripheral cycle (involving the symmetrically distinct bonds g, h, j, and k of Figure 3) is, as with the innermost cycle, again clockwise (that is, paramagnetic). There is, therefore, qualitative agreement between the predictions of these topological HLPM computations19−21 and those arising from the ipso-centric, ab initio calculations5−12 of Monaco and Zanasi,10 namely, that, in their three layers, the two structures altan-corannulene (5) and altan-coronene (6) each reveal patterns of contra-rotating paramagnetic/diamagnetic/paramagnetic π-electron circulations (in their innermost, middle, and outer cycles, respectively). Monaco and Zanasi also pointed out in the context of their study10 by the ab initio ipso-centric method5−12 that (a) patterns in the innermost and middle cycles of altancorannulene (5) and altan-coronene (6) are qualitatively unchanged from what they are in the corresponding cycles of their respective parent-systems (corannulene (3) and coronene (4)) and (b) that the innermost cycle has a smaller current in the corresponding altan systems, 5 and 6, than in 3 and 4, respectively, while (c) the intensity on the middle cycle of 5 and 6 (which corresponds to the outer cycle in 3 and 4, respectively) is (semi)-quantitatively similar in the two sets of structures. In other words, these authors claim that the behavior of the circulating π-electron currents in the core of 5 and 6, which consists of corannulene (3) and coronene (4), respectively, is qualitatively preserved when it is encased in the extra layer of five- and six-membered rings. Precisely the same conclusion could be drawn from the calculations presented here, which, as already noted under Calculations, have been effected by applying the HLPM13−18 approach in its simplest (topological) form.19−21 The paramagnetic circulations around the central rings (what we have been calling the innermost cycles) of 3 and 5 are about 1.0 and 0.92, respectively; the same figures for the central rings of 4 and 6 are (approximately) 0.42 and 0.22. The diamagnetic current around the peripheral cycle of 3 and the average diamagnetic current around the middle cycle of 5 are ca. 1.22 and 0.94, respectively. The corresponding data for the analogous cycles in 4 and 6 are about 1.46 and 1.04. Monaco and Zanasi also observe10 that, on the basis of their ab initio ipso-centric calculations,5−12 the outermost circuits host strong paramagnetic circulations, of roughly half the benzene reference value. In the case of our topological HLPM calculations,19−21 the average paramagnetic current around the periphery of 5 is ca. 0.41, and around the outer cycle of 6, it is
Figure 4. Map of the topological ring currents (in black) and the associated topological bond currents (in red) for altan-corannulene (altan-110) (5). The topological ring currents and bond currents are dimensionless quantities, all appropriately depicted here as pure numbers, without units. The ring current in benzene on this same scale is 1.000. Positive (diamagnetic) ring currents are considered to circulate counterclockwise around their respective rings, while negative (paramagnetic) ring currents flow in the clockwise sense around those rings. The numerous bond currents flow in the direction indicated by the arrow pointing along each bond.
Figure 5. Map of the topological ring currents (in black) and the associated topological bond currents (in red) for altan-coronene (altan-210) (6). For an explanation of the conventions on the depiction of ring currents and bond currents, please see the caption to Figure 4.
935
dx.doi.org/10.1021/jp411524k | J. Phys. Chem. A 2014, 118, 933−939
The Journal of Physical Chemistry A
Article
Table 1. Comparison of Individual ab Initio (ipso-Centric) and HLPM (Topological) Bond Currents in Corannulene (3) and altan-Corannulene (5) symmetrically distinct bonda a c d e g h j k
bond current in corannulene (3) (by the ipso-centric5−12 ab initio method)b,c
bond current in corannulene (3) (by the HLPM13−21 topological method)b,d
bond current in altan-corannulene (5) (by the ipso-centric5−12 ab initio method)b,c
bond current in altan-corannulene (5) (by the HLPM13−21 topological method)b,e
−0.76 0.79 0.80 0.79
−1.00 1.22 1.22 1.22
−0.39 0.80 0.73 0.78 −0.69 −0.69 −0.64 −0.64
−0.92 0.88 1.05 0.88 −0.33 −0.33 −0.50 −0.50
a
Symmetrically distinct bonds in corannulene (3) (columns 2 and 3) and altan-corannulene (5) (columns 4 and 5) are labeled (in column 1) as in Scheme 1 of ref 10 (illustrated here in Figure 3). Only bonds from the central and peripheral cycles in corannulene (3) and from the central, middle, and peripheral cycles in altan-corannulene (5) are included here; the spokes bonds55 are omitted from consideration. bA positive sign indicates a diamagnetic (counterclockwise) circulation and a negative sign indicates a paramagnetic (clockwise) circulation around the cycle containing the bond under consideration. Reference to Figure 3 shows that this means (i) for corannulene (3), the central cycle (the bonds symmetrically equivalent to bond a) and the outer, peripheral cycle (involving bonds c, d, and e); and (ii) for altan-corannulene (5), the central cycle (the bonds equivalent to bond a), the middle cycle (pertaining to bonds c, d, and e), and the outer, peripheral cycle (involving bonds g, h, j, and k). cipso-Centric bond currents for 3 and 5 are transcribed from ref 10. dHLPM topological bond currents for corannulene (3) are from refs 45 and 38. eHLPM topological bond currents for altan-corannulene (5) are transcribed from Figure 4.
Table 2. Comparison of Individual ab Initio (ipso-Centric) and HLPM (Topological) Bond Currents in Coronene (4) and altanCoronene (6) symmetrically distinct bonda a c d e g h j k
bond current in coronene (4) (by the ipso-centric5−12 ab initio method)b,c
bond current in coronene (4) (by the HLPM13−21 topological method)b,d
bond current in altan-coronene (6) (by the ipso-centric5−12 ab initio method)b,c
bond current in altan-coronene (6) (by the HLPM13−21 topological method)b,e
−0.46 1.27 1.24 1.27
−0.42 1.46 1.46 1.46
−0.28 1.06 1.03 1.07 −0.40 −0.42 −0.42 −0.42
−0.22 0.97 1.20 0.97 −0.04 −0.04 −0.27 −0.27
a
Symmetrically distinct bonds in coronene (4) (columns 2 and 3) and altan-coronene (6) (columns 4 and 5) are labeled (in column 1) as in Scheme 1 of ref 10 (illustrated here in Figure 3). Only bonds from the central and peripheral cycles in coronene (4) and from the central, middle, and peripheral cycles in altan-coronene (6) are included here; the spokes’ bonds55 are omitted from consideration. bA positive sign indicates a diamagnetic (counterclockwise) circulation and a negative sign indicates a paramagnetic (clockwise) circulation around the cycle containing the bond under consideration. Reference to Figure 3 shows that this means (i) for Coronene (4): the central cycle (the bonds symmetrically equivalent to bond a) and the outer, peripheral cycle (involving bonds c, d and e), and (ii) for Altan-Coronene (6): the central cycle (the bonds equivalent to bond a), the middle cycle (pertaining to bonds c, d, and e), and the outer, peripheral cycle (involving bonds g, h, j, and k). cipso-Centric bond currents for 4 and 6 are transcribed from ref 10. dHLPM topological bond currents for coronene (4) are from refs 46−48, 38, 39, and 18. eHLPM topological bond currents for altan-coronene (6) are transcribed from Figure 5.
approximately 0.15. The ab initio ipso-centric 5−12 and HLPM13−21 calculations agree that the peripheral paramagnetic circulation in 5 is of greater magnitude than that in 6. Both the ab initio ipso-centric5−12 and the HLPM13−21 approaches are in qualitative accord that, when 3 and 4 are converted to 5 and 6, respectively, by the addition of an outer layer of alternating five- and six-membered rings (as in Figures 2 and 3), (a) the patterns of current circulation around the core of 5 and 6, that is, their innermost and middle cycles, are effectively preserved as what they are in the inner and outer cycles of the parent structures, 3 and 4, respectively, and that (b) the peripheral bonds of the newly added layer of rings support paramagnetic circulations, that in 5 being larger than that in 6.
Now, we note that altan-corannulene (5) is a system that may potentially be regarded as a [5]-annulene within a [15]annulene, which is itself surrounded by an outer [20]-annulene, while altan-coronene (6) would appear potentially to be a candidate for being considered as a [6]-annulene within an [18]-annulene within a [24]-annulene. If the latter behaved according to an intuitive extension of the AWA rule35−37 to three layers, then the behavior of the π-electron circulations displayed would have to be diamagnetic/diamagnetic/paramagnetic (in the innermost, middle, and outer cycles, respectively). However, paramagnetic/diamagnetic/paramagnetic is what is in fact actually predicted by both the ipsocentric5−12 and the HLPM approaches.13−21 The situation with altan-corannulene (5) is more complicated: (a) If transfer of one electron f rom the inner [5]-cycle to the middle [15]-cycle were postulated, the pattern would be 936
dx.doi.org/10.1021/jp411524k | J. Phys. Chem. A 2014, 118, 933−939
The Journal of Physical Chemistry A
Article
Figure 6. Comparitor diagrams for (a) corannulene (3), (b) coronene (4), (c) altan-corannulene (5), and (d) altan-coronene (6). The comparison in each case is between the predictions of ab initio ipso-centric calculations5−12 (blue lines) and those of topological HLPM calculations13−21 (brown lines). These diagrams are based on the data presented in Tables 1 and 2. (Dimensionless) Bond currents are displayed along the vertical axes, and letters denoting individual bonds in each of the structures 3−6, labeled as in Figure 3, are ranged along the horizontal axes.
ones; that is, diamagnetic bond currents are more strongly diamagnetic and paramagnetic ones are more strongly paramagnetic; please see columns 4 and 5 of Tables 1 and 2 and Figure 6c,d for support of this assertion. As was pointed out by one of us (and others),56−59 many years ago, this is to be expected in an HLPM calculation on structures in which some rings are predicted to bear paramagnetic ring currents and no allowance is made for resonance-integrals to be made iteratively self-consistent with respect to calculated bond orders.56−59,40 Such an extension has, however, not been entertained here because doing so would have deprived the calculations of being what we have called purely topological18−21 in nature, and our aim has always been to see to what extent the simplest, parameter-free, form of the HLPM approach, which takes account only of carbon-atom connectivity and treats rings as regular polygons in an assumed planar network, can mimic the trends reflected by more-sophisticated calculations based on more-realistic geometries.
expected to be paramagnetic/paramagnetic/paramagnetic (in the innermost, middle, and outer cycles, respectively); instead, paramagnetic/diamagnetic/paramagnetic is predicted. The circulation in the middle cycle would, therefore, violate an intuitive extension of the AWA rule to this three-layered system in this way. (b) If the postulated transfer of one electron were in the opposite direction, f rom the [15]-membered cycle to the [5]-membered cycle, the expected pattern for the three cycles would be diamagnetic/diamagnetic/paramagnetic. The middle cycle (and, as before, the peripheral cycle) would now comply with the AWA rule, but the direction of circulation in the innermost cycle would not conform to the AWA rule. Overall, therefore, concerning altan-corannulene (5) and altan-coronene (6), it is seen that the qualitative predictions of both the ab initio ipso-centric method5−12 and the HLPM formalism,13−21 though consistent and in agreement, between themselves, are not in accord with the provisions of the AWA rule.35−37 That is the situation qualitatively. However, in addition to providing pictorial current-density maps, as has become traditional when results of the ipso-centric ab initio calculations are presented,5−12 Monaco and Zanasi, in their recent paper,10 have also displayed actual, quantitative bond current intensities (expressed relative to the similarly calculated benzene value) for the symmetrically nonequivalent bonds in structures 3−6 (which are all explicitly depicted and labeled in Figure 3). This therefore affords the opportunity for a direct, quantitative comparison between the predictions of the ipso-centric and HLPM formalisms for these four structures.51 Such a comparison is made in Tables 1 and 2. The same data are also presented as comparitor diagrams in Figure 6. In the case of corannulene (3) (Table 1, columns 2 and 3, and Figure 6a) and coronene (4) (Table 2, columns 2 and 3, and Figure 6b), where all HLPM ring currents are diamagnetic, trends between the two approaches compare well. In the case of 5 and 6, however, there is a tendency for the HLPM bond currents to be more exaggerated than the corresponding ab initio ipso-centric
■
CONCLUSIONS Calculated π-electron currents based on the simplest HLPM approach,13−21 which requires1,2,4,18−21 only (a) knowledge of the molecular graph22 of the structure under study (taken to be geometrically planar) and (b) the assumption that the areas of its constituent rings are those of regular polygons of standard side-length, are in semiquantitative agreement with those effected by appeal to the more-sophisticated ipso-centric ab initio method based on more-realistic geometries.5−12 The predictions of both formalisms agree that, in their three layers, the two structures altan-corannulene (5) and altan-coronene (6) each reveal patterns of contra-rotating paramagnetic/ diamagnetic/paramagnetic π-electron circulations, in their innermost, middle, and outer cycles, respectively. Nevertheless, with regard to 5 and 6, the predictions of both the ab initio ipso-centric method5−12 and the HLPM formalism,13−21 though consistent and in (semiquantitative) agreement between themselves, are not in accord with the provisions of any intuitive extension of the AWA rule35−37 to such three-layered systems. 937
dx.doi.org/10.1021/jp411524k | J. Phys. Chem. A 2014, 118, 933−939
The Journal of Physical Chemistry A
■
Article
(21) Dickens, T. K.; Mallion, R. B. Topological Ring-Currents in the “Empty” Ring of Benzo-Annelated Perylenes. J. Phys. Chem. A 2011, 115, 351−356. (22) Trinajstić, N. Chemical Graph Theory; 1st ed.; CRC Press: Boca Raton, FL, 1983; Vol. I, (a) Chapter 3, pp. 21−30; (b) Chapter 2, pp. 12−14. (23) Gomes, J. A. N. F.; Mallion, R. B. A Quasi-Topological Method for the Calculation of Relative “Ring-Current” Intensities in Polycyclic, ̀ Conjugated Hydrocarbons. Revista Portuguesa de Quimica 1979, 21, 82−89. (24) Gayoso, J.; Boucekkine, A. Application de la Théorie des Graphes à la Détermination des Exaltations de Susceptibilité des Hydrocarbures Conjugés. C. R. Hebd. Séances Acad. Sci. (Paris) 1979, C288, 327−330. (25) Mandado, M. Determination of London Susceptibilities and Ring Current Intensities Using Conjugated Circuits. J. Chem. Theory Comput. 2009, 5, 2694−2701. (26) Ciesielski, A.; Krygowski, T. M.; Cyrański, M. K.; Dobrowolski, M. A.; Aihara, J. Graph-Topological Approach to Magnetic Properties of Benzenoid Hydrocarbons. Phys. Chem. Chem. Phys. 2009, 11, 11447−11455. (27) Randić, M. Graph Theoretical Approach to π-Electron Currents in Polycyclic Conjugated Hydrocarbons. Chem. Phys. Lett. 2010, 500, 123−127. (28) Dickens, T. K.; Gomes, J. A. N. F.; Mallion, R. B. Some Comments on Topological Approaches to the π-Electron Currents in Conjugated systems. J. Chem. Theory Comput. 2011, 7, 3661−3674. (29) Fowler, P. W.; Myrvold, W. The “Anthracene Problem”: ClosedForm Conjugated-Circuit Models of Ring Currents in Linear Polyacenes. J. Phys. Chem. A 2011, 115, 13191−13200. (30) Gomes, J. A. N. F. Some Magnetic Effects in Molecules. D. Phil. thesis, University of Oxford, 1976, pp. 69−73. (31) Randić, M. Conjugated Circuits and Resonance Energies of Benzenoid Hydrocarbons. Chem. Phys. Lett. 1976, 38, 68−70. (32) Monaco, G.; Fowler, P. W.; Lillington, M.; Zanasi, R. Designing Paramagnetic Circulenes. Angew. Chem. Int. Ed. 2007, 46, 1889−1892. (33) Monaco, G.; Zanasi, R. On the Additivity of Current Density in Polycyclic Aromatic Hydrocarbons. J. Chem. Phys. 2009, 131, 044126. (34) Dickens, T. K.; Mallion, R. B. π-Electron Ring-Currents and Bond-Currents in [10,5]-Coronene and Related Structures Conforming to the ‘Annulene-Within-an-Annulene’ Model. Phys. Chem. Chem. Phys. 2013, 15, 8245−8253. (35) Barth, W. E.; Lawton, R. G. Dibenzo[ghi,mno]fluoranthene. J. Am. Chem. Soc. 1966, 88, 380−381. (36) Barth, W. E.; Lawton, R. G. The Synthesis of Corannulene. J. Am. Chem. Soc. 1971, 93, 1730−1745. (37) Benshafrut, R.; Shabtai, E.; Rabinowitz, M.; Scott, L. T. πConjugated Anions: from Carbon-Rich Anions to Charged Carbon Allotropes. Eur. J. Org. Chem. 2000, 1091−1106. (38) Dickens, T. K.; Mallion, R. B. Topological Ring-Current Assessment of the ‘Annulene-Within-an-Annulene’ Model in [N]Circulenes and some Structures Related to Kekulene. Chem. Phys. Lett. 2011, 517, 98−102. (39) Dickens, T. K.; Mallion, R. B. Topological Ring-Currents and Bond-Currents in Some Nonalternant Isomers of Coronene. J. Phys. Chem. A 2011, 115, 13877−13884. (40) Dickens, T. K.; Mallion, R. B. The Regular [r,s]-Coronenes and the ‘Annulene-Within-an-Annulene’ Rule. RSC Adv. 2013, 3, 15585− 15588. (41) Aihara, J. Is Superaromaticity a Fact or an Artifact? The Kekulene Problem. J. Am. Chem. Soc. 1992, 114, 865−868. (42) Aihara, J. A Simple Method for Estimating the Superaromatic Stabilization Energy of a Super-Ring Molecule. J. Phys. Chem. A 2008, 112, 4382−4385. (43) Aihara, J.; Makino, M.; Ishida, T.; Dias, J. R. Analytical Study of Superaromaticity in Cycloarenes and Related Coronoid Hydrocarbons. J. Phys. Chem. A 2013, 117, 4688−4697. (44) Bochvar, D. A.; Gal’pern, E. G.; Gambaryan, N. P. Calculation of the Molecule of Coronene-10,5 (C30H10). Izv. Akad. Nauk SSSR, Ser.
AUTHOR INFORMATION
Corresponding Author
*(T.K.D.) E-mail: [email protected]. Phone: +44 1223 763 811. Notes
The authors declare no competing financial interest.
■
REFERENCES
(1) Haigh, C. W.; Mallion, R. B. Ring Current Theories in Nuclear Magnetic Resonance. In Progress in Nuclear Magnetic Resonance Spectroscopy; Emsley, J. W., Feeney, J., Sutcliffe, L. H., Eds; Pergamon Press: Oxford, U.K., 1979/1980; Vol. 13, pp 303−344. (2) Gomes, J. A. N. F.; Mallion, R. B. The Concept of Ring Currents. In Concepts in Chemistry; Rouvray, D. H., Ed.; Research Studies Press Limited: Taunton, U.K., 1997; and John Wiley & Sons, Inc.: New York, 1997; Chapter 7, pp 205−253. (3) Lazzeretti, P. Ring Currents. In Progress in Nuclear Magnetic Resonance Spectroscopy; Emsley, J. W., Feeney, J., Sutcliffe, L. H., Eds; Elsevier: Amsterdam, The Netherlands, 2000; Vol. 39, pp 1−88. (4) Gomes, J. A. N. F.; Mallion, R. B. Aromaticity and Ring Currents. Chem. Rev 2001, 101, 1349−1383. (5) Steiner, E.; Fowler, P. W.; Jenneskens, L. W. Counter-Rotating Ring-Currents in Coronene and Corannulene. Angew. Chem. Int. Ed. 2001, 40, 362−366. (6) Steiner, E.; Fowler, P. W. Patterns of Ring Currents in Conjugated Molecules: a Few-Electron Model Based on Orbital Contributions. J. Phys. Chem. A 2001, 105, 9553−9562. (7) Fowler, P. W.; Steiner, E. Pseudo π-Currents: Rapid and Accurate Visualisation of Ring Currents in Conjugated Hydrocarbons. Chem. Phys. Lett. 2002, 364, 259−266. (8) Monaco, G.; Viglione, R. G.; Zanasi, R.; Fowler, P. W. Designing Ring-Current Patterns: [10,5]-Coronene, a Circulene with Inverted Rim and Hub Currents. J. Phys. Chem. A 2006, 110, 7447−7452. (9) Steiner, E.; Soncini, A.; Fowler, P. W. Full Spectral Decomposition of Ring Currents. J. Phys. Chem. A 2006, 110, 12882−12886. (10) Monaco, G.; Zanasi, R. Three Contra-Rotating Currents from a Rational Design of Polycyclic Aromatic Hydrocarbons: altanCorannulene and altan-Coronene. J. Phys. Chem. A 2012, 116, 9020−9026. (11) Monaco, G.; Memoli, M.; Zanasi, R. Additivity of Current Density Patterns in Altan-Molecules. J. Phys. Org. Chem. 2013, 26, 109−114. (12) Monaco, G.; Zanasi, R. Investigation of the p-Coronene Series in the Context of the ‘Annulene-Within-an-Annulene’ Model by means of Ipso-Centric Ab-Initio Calculations of π-Electron Currents. Phys. Chem. Chem. Phys. 2013, 15, 17654−17657. (13) Coulson, C. A.; O’Leary, B.; Mallion, R. B. Hückel Theory for Organic Chemists; Academic Press: London, 1978. (14) Yates, K. Hückel Molecular Orbital Theory; Academic Press: New York, 1978. (15) London, F. Théorie Quantique des Courants Interatomiques dans les Combinaisons Aromatiques. J. Phys. Radium (7e Série) 1937, 8, 397−409. (16) Pople, J. A. Molecular Orbital Theory of Aromatic Ring Currents. Mol. Phys. 1958, 1, 175−180. (17) McWeeny, R. Ring Currents and Proton Magnetic Resonance in Aromatic Molecules. Mol. Phys. 1958, 1, 311−321. (18) Dickens, T. K.; Mallion, R. B. An Analysis of Topological RingCurrents and their Use in Assessing the Annulene-Within-anAnnulene Model for Super-Ring Conjugated Systems. Croat. Chem. Acta 2013, 86, 387−406. (19) Mallion, R. B. Topological Ring-Currents in Condensed Benzenoid Hydrocarbons. Croat. Chem. Acta 2008, 81, 227−246. (20) Balaban, A. T.; Dickens, T. K.; Gutman, I.; Mallion, R. B. Ring Currents and the PCP Rule. Croat. Chem. Acta 2010, 83, 209−215. 938
dx.doi.org/10.1021/jp411524k | J. Phys. Chem. A 2014, 118, 933−939
The Journal of Physical Chemistry A
Article
Khim. 1970, 3, 435−437 An English translation is available from: Consultants Bureau, A Division of Plenum Publishing Corporation, 227 West 17th Street, New York, NY 10011, via the website http:// www.springerlink.com/content/l5np5207n8q10240/. (45) Mallion, R. B. Ring Currents in Corannulene, a Prototype Pattern-Molecule for Buckminsterfullerene Math/Chem/Comp 1988. Proceedings of an International Course and Conference on the Interfaces between Mathematics and Computer Science, Dubrovnik, Yugoslavia, 20th−25th June, 1988; Graovac, A., Ed., Studies in Physical and Theoretical Chemistry; Elsevier Science Publishers, B. V.: Amsterdam, The Netherlands, 1989; Vol. 63, pp 505−510. (46) Maddox, I. J.; McWeeny, R. Ring Currents in Aromatic Molecules. J. Chem. Phys. 1962, 36, 2353−2354. (47) Jonathan, N.; Gordon, S.; Dailey, B. P. Chemical Shifts and Ring Currents in Condensed Ring Hydrocarbons. J. Chem. Phys. 1962, 36, 2443−2448. (48) Aihara, J. The Origin of Counter-Rotating Rim and Hub Currents in Coronene. Chem. Phys. Lett. 2004, 393, 7−11. (49) The calculations of Monaco and Zanasi10 have indicated that the two Altan structures considered here (5 and 6, depicted in Figure 3) are bowl-shaped. Now, the ring-current concept is strictly defined in the HLPM approach13−17 only for geometrically planar structures.1,2,4 Accordingly, in the calculations being reported here, we take the symmetries of altan-corannulene (5) and altan-coronene (6) to be D5h (i.e., noncrystallographic) and D6h (crystallographic), respectively. If these structures had been considered to be nonplanar, they would have been ascribed symmetries of C5v and C6v, respectively. Furthermore, when they were adapting the McWeeny method,17 in order to make it approximately applicable to the nonplanar helicenes, Haigh and one of the present authors50 stated that the assumptions required to do this were valid only if molecular overcrowding is such that “...the skeletal distortion about any bond is comparatively mild.” They further pointed out that, “...because of the way in which the strains from overcrowding are spread over many degrees of freedom, this condition does appear to be satisfied in the case of the helicenes, even though the overall nonplanarity between well-separated parts of such molecules may in fact be very large.” This requirement does appear to be fulfilled in the case of the HLPM calculations being presented here, on structures 5 and 6. (50) Haigh, C. W.; Mallion, R. B. Proton Magnetic Resonance of Non-Planar Condensed Benzenoid Hydrocarbons. II. Theory of Chemical Shifts. Mol. Phys. 1971, 22, 955−970. (51) It is pertinent to observe that the quantitative bond currents obtained from the ab initio ipso-centric method5−12 do not in general strictly obey Kirchhoff’s Law of Current Conservation at Junctions28,29 unless an infinite basis-set is adopted.12 However, bond-currents calculated by the HLPM approach13−21 do, by contrast, rigorously obey the Kirchhoff Conservation Law, thereby drawing attention to an agreeable analogy,52−54 in the context of the microscopic, molecular systems being considered here, with classical macroscopic electricalnetworks, of the type addressed by Kirchhoff in his pioneering work of more than 150 years ago.52−54,28,29 (52) Mallion, R. B. Some Graph-Theoretical Aspects of Simple Ring Current Calculations on Conjugated Systems. Proc. Royal Soc. (London) 1974/1975, A341, 429−449. (53) Mallion, R. B. On the Magnetic Properties of Conjugated Molecules. Mol. Phys. 1973, 25, 1415−1432. (54) Mallion, R. B. On the Number of Spanning Trees in a Molecular Graph. Chem. Phys. Lett. 1975, 36, 170−174. (55) Balaban, A. T.; Bean, D. E.; Fowler, P. W. Patterns of Ring Current in Coronene Isomers. Acta Chim. Slov. 2010, 57, 507−512. (56) Coulson, C. A.; Mallion, R. B. On the Question of Paramagnetic “Ring Currents” in Pyracylene and Related Molecules. J. Am. Chem. Soc. 1976, 98, 592−598. (57) Mallion, R. B. Some Comments on the Use of the “RingCurrent” Concept in Diagnosing and Defining “Aromaticity. Pure Appl. Chem. 1980, 52, 1541−1548.
(58) Gomes, J. A. N. F.; Mallion, R. B. Calculated Magnetic Properties of Some Isomers of Pyracylene. J. Org. Chem. 1981, 46, 719−727. (59) Mallion, R. B. Ring-Current Effects in C60. Nature 1987, 325, 760−761.
939
dx.doi.org/10.1021/jp411524k | J. Phys. Chem. A 2014, 118, 933−939
![Topological ring-currents and bond-currents in the altan-[r,s]-coronenes.](/assets/img/hold.png)
