{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,1,21]],"date-time":"2024-01-21T01:37:26Z","timestamp":1705801046553},"reference-count":72,"publisher":"American Chemical Society (ACS)","issue":"11","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Chem. Theory Comput."],"published-print":{"date-parts":[[2011,11,8]]},"DOI":"10.1021\/ct2002539","type":"journal-article","created":{"date-parts":[[2011,10,5]],"date-time":"2011-10-05T15:24:51Z","timestamp":1317828291000},"page":"3661-3674","source":"Crossref","is-referenced-by-count":24,"title":["Some Comments on Topological Approaches to the \u03c0-Electron Currents in Conjugated Systems"],"prefix":"10.1021","volume":"7","author":[{"given":"Timothy K.","family":"Dickens","sequence":"first","affiliation":[{"name":"University Chemical Laboratory, University of Cambridge, Lensfield Road, Cambridge, CB2 1EW, England, United Kingdom"}]},{"given":"Jos\u00e9 A. N. F.","family":"Gomes","sequence":"additional","affiliation":[{"name":"REQUIMTE, Departamento de Qu\u00edmica, Faculdade de Ci\u00eancias, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal"}]},{"given":"Roger B.","family":"Mallion","sequence":"additional","affiliation":[{"name":"School of Physical Sciences, University of Kent, Canterbury, CT2 7NH, England, United Kingdom"}]}],"member":"316","published-online":{"date-parts":[[2011,10,5]]},"reference":[{"key":"ref1\/cit1a","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1016\/0009-2614(76)80257-6","volume":"38","author":"Randi\u0107 M.","year":"1976","journal-title":"Chem. Phys. Lett."},{"key":"ref1\/cit1b","doi-asserted-by":"crossref","first-page":"1905","DOI":"10.1016\/0040-4020(77)80375-X","volume":"33","author":"Randi\u0107 M.","year":"1977","journal-title":"Tetrahedron"},{"key":"ref1\/cit1c","doi-asserted-by":"crossref","first-page":"444","DOI":"10.1021\/ja00444a022","volume":"99","author":"Randi\u0107 M.","year":"1977","journal-title":"J. Am. Chem. Soc."},{"key":"ref1\/cit1d","doi-asserted-by":"crossref","first-page":"1587","DOI":"10.1351\/pac198052061587","volume":"52","author":"Randi\u0107 M.","year":"1980","journal-title":"Pure Appl. Chem."},{"key":"ref2\/cit2","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/j.cplett.2010.09.064","volume":"500","author":"Randi\u0107 M.","year":"2010","journal-title":"Chem. Phys. Lett."},{"key":"ref3\/cit3","doi-asserted-by":"crossref","first-page":"2694","DOI":"10.1021\/ct9002866","volume":"5","author":"Mandado M.","year":"2009","journal-title":"J. Chem. Theory Comput."},{"key":"ref4\/cit4","doi-asserted-by":"crossref","first-page":"11447","DOI":"10.1039\/b913895a","volume":"11","author":"Ciesielski A.","year":"2009","journal-title":"Phys. Chem. Chem. Phys."},{"key":"ref5\/cit5","first-page":"303","volume-title":"Progress in Nuclear Magnetic Resonance Spectroscopy","volume":"13","author":"Haigh C. W.","year":"1979"},{"key":"ref6\/cit6","first-page":"205","volume-title":"Concepts in Chemistry","volume":"7","author":"Gomes J. A. N. F.","year":"1997"},{"key":"ref7\/cit7a","first-page":"1","volume-title":"Progress in Nuclear Magnetic Resonance Spectroscopy","volume":"39","author":"Lazzeretti P.","year":"2000"},{"key":"ref7\/cit7b","doi-asserted-by":"crossref","first-page":"1349","DOI":"10.1021\/cr990323h","volume":"101","author":"Gomes J. A. N. F.","year":"2001","journal-title":"Chem. Rev."},{"key":"ref8\/cit8","doi-asserted-by":"crossref","first-page":"3449","DOI":"10.1021\/cr9903656","volume":"103","author":"Randi\u0107 M.","year":"2003","journal-title":"Chem. Rev."},{"key":"ref9\/cit9","first-page":"507","volume":"57","author":"Balaban A. T.","year":"2010","journal-title":"Acta Chim. Slov."},{"key":"ref11\/cit11","doi-asserted-by":"publisher","DOI":"10.1002\/qua.23081"},{"key":"ref12\/cit12","first-page":"327","volume":"288","author":"Gayoso J.","year":"1979","journal-title":"C. R. Hebd. S\u00e9ances Acad. Sci."},{"key":"ref14\/cit14a","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1080\/00268975800100211","volume":"1","author":"Pople J. A.","year":"1958","journal-title":"Mol. Phys."},{"key":"ref14\/cit14b","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1080\/00268975800100381","volume":"1","author":"McWeeny R.","year":"1958","journal-title":"Mol. Phys."},{"key":"ref14\/cit14c","doi-asserted-by":"crossref","first-page":"1056","DOI":"10.1051\/jcp\/1962591056","volume":"59","author":"Veillard A.","year":"1962","journal-title":"J. Chim. Phys. Phys. Chim.\u2013Biol."},{"key":"ref14\/cit14d","first-page":"184","volume":"272","author":"Gayoso J.","year":"1971","journal-title":"C. R. Hebd. S\u00e9ances Acad. Sci."},{"key":"ref14\/cit14e","doi-asserted-by":"crossref","first-page":"558","DOI":"10.1021\/ja00497a011","volume":"101","author":"Aihara. J.-I.","year":"1979","journal-title":"J. Am. Chem. Soc."},{"key":"ref14\/cit14f","doi-asserted-by":"crossref","first-page":"2005","DOI":"10.1246\/bcsj.60.2005","volume":"60","author":"Mizoguchi N.","year":"1987","journal-title":"Bull. Chem. Soc. Jpn."},{"key":"ref14\/cit14g","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1007\/BF01169015","volume":"3","author":"O\u2019Leary B.","year":"1989","journal-title":"J. Math. Chem."},{"key":"ref14\/cit14h","unstructured":"It should be noted that, despite its authorship, ref14gis actually a historical paper reporting previously unpublished work (dating from 1953) by the late Professor C. A. Coulson, F.R.S., in which the London theory(14i)of diamagnetic susceptibility in conjugated systems is couched in terms of Coulson\u2019s own contour-integral formalism.(14j)"},{"key":"ref14\/cit14i","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1051\/jphysrad:01937008010039700","volume":"8","author":"London F.","year":"1937","journal-title":"J. Phys. Radium (7e S\u00e9r.)"},{"key":"ref14\/cit14j","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1017\/S0305004100017175","volume":"36","author":"Coulson C. A.","year":"1940","journal-title":"Proc. Cambridge Philos. Soc."},{"key":"ref15\/cit15","unstructured":"Gomes, J. A. N. F.Some Magnetic Effects in Molecules. D. Phil. Thesis,University of Oxford (Linacre College):England, United Kingdom, 1976; pp69\u221273."},{"key":"ref16\/cit16","first-page":"82","volume":"21","author":"Gomes J. A. N. F.","year":"1979","journal-title":"Rev. Port. Qu\u00ecm."},{"key":"ref17\/cit17","doi-asserted-by":"crossref","first-page":"765","DOI":"10.1080\/00268978000101861","volume":"40","author":"Gomes J. A. N. F.","year":"1980","journal-title":"Mol. Phys."},{"key":"ref18\/cit18","first-page":"561","volume":"53","author":"Gomes J. A. N. F.","year":"1980","journal-title":"Croat. Chem. Acta"},{"key":"ref19\/cit19","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1007\/BF02402398","volume":"59","author":"Gomes J. A. N. F.","year":"1981","journal-title":"Theoret. Chim. Acta"},{"key":"ref21\/cit21","first-page":"13","volume":"88","author":"Randi\u0107 M.","year":"2011","journal-title":"J. Indian Chem. Soc."},{"key":"ref22\/cit22","doi-asserted-by":"crossref","first-page":"592","DOI":"10.1021\/ja00418a045","volume":"98","author":"Coulson C. A.","year":"1976","journal-title":"J. Am. Chem. Soc."},{"key":"ref23\/cit23","first-page":"227","volume":"81","author":"Mallion R. B.","year":"2008","journal-title":"Croat. Chem. Acta"},{"key":"ref24\/cit24","first-page":"209","volume":"83","author":"Balaban A. T.","year":"2010","journal-title":"Croatica Chem. Acta"},{"key":"ref25\/cit25","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1021\/jp1096103","volume":"115","author":"Dickens T. K.","year":"2011","journal-title":"J. Phys. Chem. A"},{"key":"ref26\/cit26","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1002\/andp.18451400402","volume":"64","author":"Kirchhoff G.","year":"1845","journal-title":"Anal. Phys. Chem."},{"key":"ref27\/cit27","unstructured":"Cundy, H. M.S.M.P. Advanced Mathematics Book 3 [Metric];Cambridge University Press:London, 1970; p911."},{"key":"ref28\/cit28","unstructured":"Smart, D.Linear Algebra & Geometry;Cambridge University Press:Cambridge, England, United Kingdom, 1988; S. M. P. Further Mathematics Series, pp307\u2013308."},{"key":"ref29\/cit29","first-page":"429","volume":"341","author":"Mallion R. B.","year":"1974","journal-title":"Proc. R. Soc. London, Ser. A"},{"key":"ref32\/cit32","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1515\/zna-1967-0117","volume":"22","author":"Baer F.","year":"1967","journal-title":"Z. Naturforsch. A"},{"key":"ref36\/cit36a","unstructured":"In view of the theoretical formulation of the method proposed by Gomes,(17-19)this rule\u2014printed in the text initalics\u2014is here given a wording that is more general than that adopted for the corresponding rule (iii) in ref16. (The original wording in ref16was \u201c(iii) If at least one bond of a particular ring forms a part of a given \u201cconjugation circuit\u201d, this ring shall be said to \u201cparticipate\u201d in that \u201cconjugation circuit.\u201d\u2019). Professor P. W. Fowler(36b, 36c)has recently proposed an equivalent version of rule (iii) that is couched entirely in terms ofbondcurrents, rather thanringcurrents:"},{"key":"ref36\/cit36b","unstructured":"Fowler, P. W.Personal Communication, June 19th, 2011."},{"key":"ref36\/cit36c","author":"Fowler P. W.","journal-title":"J. Phys. Chem."},{"key":"ref37\/cit37","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1016\/0301-0104(79)80033-6","volume":"41","author":"Gutman I.","year":"1979","journal-title":"Chem. Phys."},{"key":"ref38\/cit38a","unstructured":"An anonymous reviewer has pointed out that a coefficient,cL, is defined in Mandado\u2019s approach(3)as the ratio of the number of conjugated circuits of a certain type,L, and the number of Kekul\u00e9 structures (nKin Mandado\u2019s paper)\u2014please see ref3for a definition of these quantities. This ratio emerges from the valence-bond treatment of the resonance energy proposed in ref3, which starts with the valence-bond wave function proposed by Herndon,(38b)in which the root of the number of Kekul\u00e9 structures is introduced in order to normalize the wavefunction (equation1in Mandado\u2019s paper\u2014see also ref19). It seems that Mandado effectively \u201cnormalizes\u201d by dividing byK\/2 (2 being the number of Kekul\u00e9 structures in benzene). Regarding the approach of Ciesielski et al.(4)we are unclear why the so-called \u201clocal\u201d and \u201cglobal\u201d quantities calculated from the bond currents are normalized (by division by (1\/2)K(K\u2013 1), the number of sets of distinct conjugation circuits) butnotthe actual bond currents themselves\u2014which, of course, are what we ourselves are primarily interested in in this paper:"},{"key":"ref38\/cit38b","doi-asserted-by":"crossref","first-page":"2404","DOI":"10.1021\/ja00788a073","volume":"95","author":"Herndon W. C.","year":"1973","journal-title":"J. Am. Chem. Soc."},{"key":"ref40\/cit40a","unstructured":"Even more elaborate \u201cnormalizations\u201d have been discussed and argued for by Randi\u0107(40b)in the context, specifically, of structures related to perylene:"},{"key":"ref40\/cit40b","unstructured":"Randi\u0107, M.Personal communication to R.B.M., May 3, 2011."},{"key":"ref42\/cit42a","unstructured":"The method of Gomes and Mallion(16)takes into account the effect of the number of centers on the ring-current intensity by incorporating the numerical values of the annulenic ring-current intensities reported by Baer et al.(32)An anonymous reviewer has pointed out that this is also taken into consideration, in an equivalent way, in Mandado\u2019s method.(3)In fact, Mandado introduces a variablefLdefined as the number of benzene rings enclosed by the circuit, in order to represent the size of the circuit. In the method of Gomes and Mallion,(15-19)the size of the circuits is measured in idealized molecular geometries with equal bond lengths; the energy and the current associated with a annulene are assumed to depend on the number of centers of that annulene. These energy terms depend mainly on the number of alternating double and single bonds, which, in turn, depend on the number of centers. Gomes and Mallion(15, 16)adopted the annulenic ring currents of Baer et al.(32)as an alternative to a possibly arbitrary parametrization. Mandado(3)has instead opted for a parametrization. The following is a summary(42b)of how all four \u201cconjugation circuits\u201d approaches(15, 16, 2-4)count the contributions of conjugation circuits as they increase in size. In both the Gomes\u2013Mallion(15, 16)and Mandado(3)models, contributions taper off(42b)with ring size\u2014they start off large for small conjugation circuits, settle down, and then virtually disappear altogether at [N] = 22. In the Randi\u0107(2)model, all conjugation circuits, of whatever size, give an equal contribution per occurrence. In the approach of Ciesielski et al.(4)larger conjugation circuits contribute more per occurrence, as they have larger area. In Mandado\u2019s model,(3)the larger conjugation circuits contribute less per occurrence, because, in this formulation, the area is in the denominator. In the approach of Gomes and Mallion,(15, 16)although circuit area is in the numerator, contributions from the larger conjugation circuits fall off rapidly with the size of the conjugation circuit because of the fact that the Baer et al.(32)\u201cannulenic\u201d currents decrease virtually to zero at annulene size [22]). This phenomenon of the decrease in importance of the contributions from the larger conjugation circuits is thus essentially the same in both the Mandado(3)and Gomes\u2013Mallion(15, 16)formalisms:"},{"key":"ref42\/cit42b","unstructured":"Fowler, P. W.Personal communication, July 3, 2011."},{"key":"ref43\/cit43","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/0166-1280(90)80032-J","volume":"210","author":"Gomes J. A. N. F.","year":"1990","journal-title":"THEOCHEM"},{"key":"ref44\/cit44a","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1098\/rspa.1960.0164","volume":"257","author":"Longuet-Higgins H. C.","year":"1960","journal-title":"Proc. R. Soc. London, Ser. A"},{"key":"ref44\/cit44b","volume-title":"The Molecular Orbital Theory of Conjugated Systems","author":"Salem L.","year":"1966"},{"key":"ref45\/cit45a","doi-asserted-by":"crossref","first-page":"3613","DOI":"10.1016\/S0040-4020(01)93810-4","volume":"28","author":"Haddon R. C.","year":"1972","journal-title":"Tetrahedron"},{"key":"ref45\/cit45b","unstructured":"idem ibid.3635\u20133655."},{"key":"ref46\/cit46","doi-asserted-by":"crossref","first-page":"1415","DOI":"10.1080\/00268977300101211","volume":"25","author":"Mallion R. B.","year":"1973","journal-title":"Mol. Phys."},{"key":"ref47\/cit47a","unstructured":"Mallion, R. B.Empirical Appraisal and Graph Theoretical Aspects of Simple Theories of the \u201cRing-Current\u201d Effect in Conjugated Systems. D. Phil. Thesis:University of Oxford (Christ Church):England, United Kingdom, 1979; pp124\u2013131."},{"key":"ref47\/cit47b","first-page":"1","volume":"62","author":"Haigh C. W.","year":"1989","journal-title":"Croat. Chim. Acta"},{"key":"ref48\/cit48","doi-asserted-by":"crossref","first-page":"1073","DOI":"10.1080\/00268977100101051","volume":"20","author":"Amos A. T.","year":"1971","journal-title":"Mol. Phys."},{"key":"ref49\/cit49a","unstructured":"Kirchhoff\u2019s Law of current conservation(26-29)is, by contrast,notviolated in, for example, the \u201ccoupledHartree\u2013Fock\u201d procedure described in:"},{"key":"ref49\/cit49b","doi-asserted-by":"crossref","first-page":"713","DOI":"10.1080\/00268977500102281","volume":"30","author":"Coulson C. A.","year":"1975","journal-title":"Mol. Phys."},{"key":"ref50\/cit50a","volume-title":"H\u00fcckel Theory for Organic Chemists","author":"Coulson C. A.","year":"1978"},{"key":"ref50\/cit50b","doi-asserted-by":"crossref","DOI":"10.1016\/B978-0-12-768850-3.50005-5","volume-title":"H\u00fcckel Molecular Orbital Theory","author":"Yates K.","year":"1978"},{"key":"ref52\/cit52","unstructured":"Fowler, P. W.Personal communication to R. B. M., June 8, 2011."},{"key":"ref54\/cit54","unstructured":"Fowler, P. W.; Bean, D. E.Personal Communication to R. B. M. at the Fifth Conference on Computers in Scientific Discovery, Sheffield, England, United Kingdom, July 2010."},{"key":"ref55\/cit55","doi-asserted-by":"crossref","first-page":"1063","DOI":"10.1080\/00268977600102461","volume":"32","author":"Atkins P. W.","year":"1976","journal-title":"Mol. Phys."},{"key":"ref59\/cit59a","doi-asserted-by":"crossref","first-page":"4811","DOI":"10.1021\/ja00973a009","volume":"88","author":"Pople J. A.","year":"1966","journal-title":"J. Am. Chem. Soc."},{"key":"ref59\/cit59b","first-page":"103","volume":"16","author":"Haddon R. C.","year":"1971","journal-title":"Top. Curr. Chem."},{"key":"ref59\/cit59c","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1021\/ar50051a001","volume":"5","author":"Sondheimer F.","year":"1972","journal-title":"Acc. Chem. Res."},{"key":"ref60\/cit60","doi-asserted-by":"crossref","first-page":"737","DOI":"10.1021\/ja00732a031","volume":"93","author":"Trost B. M.","year":"1971","journal-title":"J. Am. Chem. Soc."},{"key":"ref61\/cit61a","unstructured":"This observation is re-enforced by Figure 3 of ref61b, in which the bond currents in the 60 bonds of kekulene are expressed in terms of just three independent parameters,A,B, and \u03b4. These are not actually the ring currents in the three symmetrically non-equivalent rings of kekulene, but they are related to them in the following way: the ring current in the six-membered ring that Steiner et al.(61b)label \u201cI\u201d is equal to (A+ \u03b4); the ring current in ring II is (A\u2013 \u03b4), and that in the internal, 18-membered, ring III is (A\u2013B). Once again, therefore, in this example, a bond-current description is entirely equivalent to a ring-current one, with as many independent variables being needed to specify the several bond currents as there are symmetrically non-equivalent rings (and, hence, distinct ring currents) in the structure under consideration:"},{"key":"ref61\/cit61b","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1039\/b009763m","author":"Steiner E.","year":"2001","journal-title":"Chem. Commun."}],"container-title":["Journal of Chemical Theory and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubs.acs.org\/doi\/pdf\/10.1021\/ct2002539","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,5]],"date-time":"2023-04-05T12:54:23Z","timestamp":1680699263000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubs.acs.org\/doi\/10.1021\/ct2002539"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,10,5]]},"references-count":72,"journal-issue":{"issue":"11","published-print":{"date-parts":[[2011,11,8]]}},"alternative-id":["10.1021\/ct2002539"],"URL":"https:\/\/doi.org\/10.1021\/ct2002539","relation":{},"ISSN":["1549-9618","1549-9626"],"issn-type":[{"value":"1549-9618","type":"print"},{"value":"1549-9626","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,10,5]]}}}