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The Three-Center, /- E Le C Tro N Chemical Bond The Three-Center, /-Electron Chemical Bond Myriam S. de Giambiagi, Mario Giambiagi Centro Brasileiro de Pesquisas Fisicas, 22290-180 Rio de Janeiro—RJ, Brasil Jorge Herrera Centro Latino Americano de Fisica, Av. Wenceslau Braz 71, fundos, 22290 R io de Janeiro—RJ, Brasil Z. Naturforsch. 49a, 754-758 (1994); received March 23, 1994 A definition is proposed for the number of electrons x involved in a three-center bond, based upon the corresponding two-center bond indices. The number of electrons is fractionary, ranging from about 1 to somewhat more than 4 in the systems considered here. It is seen that the sign of the three-center index does not depend on %■ Key words: Three-center bond; Multicenter bond indices; Active charge. 1. Introduction sical bond with population close to 2 with a small fluctuation from this value. The overwhelming major­ The concept of three-center bond is almost contem­ ity of population analysis, of which Mulliken’s [13] is porary of the Lewis two-center electron pair bond [1], by far the most popular, assigns a bond value of 1 , 2 , A distinction is made between two-electron (2e) and and 3 for single, double and triple bonds, respectively. four-electron (4e) three-center (3 c) bonds. or di- When all-valence-electrons calculations were intro­ borane are taken as paradigms of 3c-2e bonds and duced, Hoffmann’s EHT (extended Hückel theory) (FHF)“ of 3c-4e bonds [1]. [14] used M ulliken’s population analysis, which has We have introduced a three-center bond index [2] in the drawback of becoming zero for orthogonal bases. satisfactory agreement with certain experimental data. The other approximation dealing with all-valence- The values obtained have negative sign. Sannigrahi electrons at that time was C N D O (complete neglect of and Kar [3] affirm that 3 c bond indices have no chem­ differential overlap) with its alternative IN D O (in­ ical significance unless they are positive. In [4], it is complete neglect of differential overlap) and, in par­ asserted that negative values could be associated to ticular, the CNDO/2 variation [15]. Mulliken’s or­ (3c-2e) bonds, while positive values should indicate bital-orbital population lends itself to be contracted (3c-4e) bonds. We show elsewhere that the sign has a to an atom-atom population; CNDO orbital-orbital straightforward meaning [5] and here that in M O the­ population does not. In order to compare EH and ory the number of electrons taking part in such a bond CN D O /2 results, Wiberg [16] was obliged to devise a may be seen from a different viewpoint. new bond index for C N D O , by the way in a very The original Coulson bond order [6 ] has been de­ humble footnote. Trindle [17] proposed for it a sub­ vised for the ti electrons of alternant hydrocarbons; n division into self-charge and active charge analogous systems including heteroatoms led to bond orders to Mulliken’s atomic and overlap populations. which are not comparable with C - C bond orders [7]. MINDO/3 [18] and M NDO [19] came later. Almost universal consent assigns for example a bond A few years after the work of Wiberg, which did not order of 2 for the C = C bond in ethylene [8 ]. Very few receive the full deserved attention, we proposed a examples may be found in early literature which bond index which generalized Wiberg’s when non- throw ti population wholly on the bonds [9,10]. More orthogonal bases were used. Although we used it in recently, a theory considering fluctuations in orbital all-valence electrons calculations [7,20], nothing pre­ domains [1 1 , 1 2 ] describes these domains along a clas- vented taking into account the core electrons or en­ larging the basis. Reprint requests to Prof. D r. M ario Giambiagi, Centro Coming back to our three-center index, we are led Brasileiro de Pesquisas Fisicas, Rua Dr. Xavier Sigaud, 150, to seek an answer to the question; how many electrons 22290-180 Rio de Janeiro-RJ, Brasilien. are there in a three-center bond? 0932-0784/94 /0700-0754 $ 06.00 © - Verlag der Zeitschrift für Naturforschung, D-72027 Tübingen Dieses Werk wurde im Jahr 2013 vom Verlag Zeitschrift für Naturforschung This work has been digitalized and published in 2013 by Verlag Zeitschrift in Zusammenarbeit mit der Max-Planck-Gesellschaft zur Förderung der für Naturforschung in cooperation with the Max Planck Society for the Wissenschaften e.V. digitalisiert und unter folgender Lizenz veröffentlicht: Advancement of Science under a Creative Commons Attribution Creative Commons Namensnennung 4.0 Lizenz. 4.0 International License. M. S. de G iam biagi et al. • The Three-Center, /-Electron Chemical Bond 755 2. The Three-Center Bond Index Sannigrahi and Kar [3] have made an interesting ob­ servation, namely that a higher-order bond index can­ We restrict ourselves to the ground state of closed- not generally be determined from a knowledge of shell systems, which can be described through a single- lower-order bonds. The summation in (7) includes the determinant wave function. The first-order density possibility of equal subindices {A, B o r C); i.e,, terms matrix is a mixed tensor [2 1 ] in one, two and three centers appear in N. IABC is invariant for any permutation of A, B ,C , cyclic [3] or 2 n ba= 2 Z x iax ‘b, (1 ) i not. This is easily seen when writing the tensor equiv­ alent to (2 ): where x ia(xib) are the covariant (contravariant) coeffi­ rbca_rabc___jbac /q\ cients of the i-th occupied M O . This tensor is repre­ *abc *cab *acb * sented by a matrix. We may build an orbital-orbital For the contraction to be carried out, the order of the tensor [2 1 ] indices is immaterial, as long as it involves a covariant i?a = 4 n bang, (2) with a contravariant index; however, it cannot involve the covariant and contravariant index of the same II which may also be represented by a matrix; this one matrix, for in this case the three-center ABC index is symmetric (otherwise, the tensor could not be repre­ splits into one-center and two-center indices. sented by a matrix), while the matrix in (1 ) is asymmet­ If we write the expression for IABC equivalent to (4) ric for non-orthogonal bases. In these cases, there ex­ for IAB, we obtain [5] ists a convention stating that the covariant index indicates rows and the contravariant index columns of <(4a ~< 4a >) (4b -< 4 b >)(4c - < 4 c » y = Iabc /2- (9) the corresponding matrix. Contracting the tensor in uses to be positive (although it is not positive (2 ) we obtain the bond index I AB for the bond (formal definite in non-orthogonal bases). Thus, (4) means that or not) linking atoms A and B [7,21] if qA fluctuates in one sense from its average value, qB iAB = 4 z n bari£. (3) fluctuates in the opposite sense. As to IABC, the three aeA fluctuations in (9) are not likely to be all in the same beB sense; if two are in one sense and one in the opposite This may be related with the correlation between the sense, either positive or negative values are to be fluctuations of the charges qA and qB from their aver­ found, with no evident connection at all with two or four age values [22,23]: electrons in the three-center bond. Recently, covalent bond orders have been proposed < (4 a - < 4 a » { 4b -Q b » > = - Ia b / 2, (4) [24] within the framework of Bader’s topological the­ ory of atoms in molecules [25]; they are compared 4 a being the charge operator and [22, 23] with Wiberg indices, stating (incorrectly) that both < 4 a > - 4 a = 2 Z n aa; iV = 2T ri7. (5) give zero for so-called noninteracting pairs of atoms aeA [24], On the contrary, Wiberg indices and their gener­ The idempotency of the 17 matrix allows us to write alization emphasize the role of “long bonds” or “sec­ [7,21] ondary bonds” [7,21,26,27]; they are most important in three-center bonds [2, 3, 4, 28, 29]. A sharing index has also been developed [8 ] and has been compared w - *-(i)(<“ +.S. 4 (6) with covalent bond orders [30], sharing indices being half of covalent bond orders in some cases and quite The charge qA is M ulliken’s gross population with a close in value in most of therm. We think that some of quite different partition into self-charge and active the values reported [8 , 24, 30] are too low when com­ charge [17]. pared with those one could expect from chemical intu­ Similarly [2] ition. We find, for example, that the indices for LiH should be closer to 1 [29]. N = ( ~ ) X l ABc\ W = (? ) Mayer [31] has proposed a model for three-center \ 4J A,B ,C bonds, where a single atomic orbital is assigned to beB ceC each of the three centers. This model is used in [4, 32, 756 M . S. de Giam biagi et al. ■ The Three-Center, ^-Electron Chemical Bond 33]. Reference [32] deals withIABC as being a (3c-2e) (4), the fluctuations of the qH's around their average index for the examples chosen. In [4], K ar states that values are either zero or not correlated to each other. qA + qB + qc = 2 for a (3c-2e) bond. It is curious that We have found this feature only for this type of com­ neither of them takes into account the partition into pounds, being thus, it seems, specific of the H H ' corre­ active and self-charge.
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