J. Serb. Chem. Soc. 7 (10) 967 973 (007) UDC 54 74+537.87:53.74+539.194 JSCS 369 Origial scietific paper The McClellad approximatio ad the distributio of -electro molecular orbital eergy levels IVAN GUTMAN* # Faculty of Sciece, Uiversity of Kragujevac, P.O. Box 60, 34000 Kragujevac, Serbia (Received 9 March 007) Abstract: The total -electro eergy E of a cojugated hydrocarbo with carbo atoms ad m carbo carbo bods ca be approximately calculated by meas of the McClellad formula E g m, where g is a empirical ttig costat, g 0.9. It was claimed that the good quality of the McClellad approximatio is a cosequece of the fact that the -electro molecular orbital eergy levels are distributed i a early uiform maer. It will ow be show that the McClellad approximatio does ot deped o the ature of the distributio of eergy levels, i.e., that it is compatible with a large variety of such distributios. Keywords: total -electro eergy, McClellad formula, Hückel molecular orbital theory. The total -electro eergy E is oe of the most thoroughly studied theoretical characteristics of cojugated molecules that ca be calculated withi the Hückel molecular orbital (HMO) approximatio. 1, Research o E is curretly very active. 3 8 Log time ago McClellad proposed the simple approximate formula: 9 E g m (1) where is the umber of carbo atoms ad m the umber of carbo carbo bods, ad where g is a empirically determied ttig parameter, g 0.9. I the meatime a large umber of other (,m)-type approximate expressios for E have bee proposed, but, as demostrated by detailed comparative studies, 1013 oe of these could exceed the accuracy of Eq. (1). I 1983 the preset author discovered 14 that a result closely similar to Eq. (1) ca be obtaied by assumig that the HMO eergy levels are uiformly distributed. Evetually such a distributio-based approach to E was elaborated i more detail. 15,16 The coclusio of the works 1416 was that the McClellad approximatio (Eq. (1)) is coected with the assumptio that the HMO -electro eergy levels of cojugated hydrocarbos are distributed i a (early) uiform maer. * E-mail: gutma@kg.ac.yu # Serbia Chemical Society member. doi: 10.98/JSC0710967G 967
968 GUTMAN The reasoig by meas of which this coclusio was obtaied will be briefly repeated. If 1,,..., are the Eige values of the molecular graph represetig the respective cojugated molecule, the: 13 i1 E () As is well kow, 1, the graph Eige values satisfy the relatio: i E ( ) m (3) Without loss of geerality, Eqs. () ad (3) may be rewritte as: ad i1 E - - i x ( x) dx E x ( x) dx m where (x) is the probability desity of the distributio of the graph Eige values. It should be metioed i passig that the exact expressio for (x) is: 1 ( x) ( x i ) i1 with deotig the Dirac delta-fuctio. I situatios whe the actual form of the probability desity (x) is ot kow (i.e., whe the spectrum of the molecular graph is ot kow), oe tries to guess a approximate expressio for it, deoted by *(x), which must satisfy the coditios: *( x )dx 1 (4) m x *( x)dx ad, of course, *(x) 0 for all values of x. The the quatity E*, (5) E * x *( x)dx (6) - is expected to provide a reasoably good approximatio for the total -electro eergy E. I the works, 14,16 the simplest possible choice for *(x) was tested, amely, *(x) = b for a x +a ad otherwise *(x) = 0 (7)
McCLELLAND APPROXIMATION 969 The form of the fuctio (7) is show i Fig. 1. The parameters a ad b ca easily be determied from the coditios Eq. (4) ad (5), resultig i 6 m a ad b (8) 4 m By isertig the coditios give by Eq. (7) back ito Eq. (6), oe obtais: E* = a b which combied with Eq. (8) yields: E * g * m (9) with g* beig a costat equal to 3 /. Not oly is the algebraic form of the expressio (9) idetical to the McClellad approximatio (Eq. (1)), but also the value of the multiplier g* = 0.8660 is remarkably close to the (earlier) empirically determied value for g. Fig. 1. The form of the probability desity (Eq. 7) for a = 3 ad b = 1/6. The Eige values of the molecular graph are assumed to be uiformly distributed withi the iterval ( a,+a), i.e., withi the iterval (a, a), the probability desity is assumed to be costat (equal to b). Outside this iterval, the probability desity is set to be equal to zero. Thus, it ca be see that by assumig a uiform distributio of the Eige values of a molecular graph, the McClellad formula (Eq. (1)) ca be reproduced. What has hitherto bee overlooked is that formula (1) ca also be deduced by usig may other probability desities. OBTAINING FORMULA (1) FROM A VARIETY OF MODEL FUNCTIONS *(x) Suppose that the model based o Eq. (7) is required to be upgraded by icludig the iformatio that the MO eergies aroud the o-bodig level (correspodig to x = 0) are more umerous tha those far from the o-bodig level, see diagram 1 i Fig.. This ca be achieved by meas of the fuctio: x *( x) b[1 ] for a x +a ad otherwise *(x) = 0 (10) The, by direct calculatio i a fully aalogous maer as described i the precedig sectio, formula (9) is obtaied with g* = 5 / 4.
970 GUTMAN If, however, the opposite is assumed, amely that the MO eergies aroud the o-bodig level are less umerous tha those far from the o-bodig level (see diagram i Fig. ), ad therefore set x * ( x ) b for a x +a ad otherwise *(x) = 0 (11) the Eq. (9) is agai obtaied, this time for g* = 15 / 4. The model fuctio * may be made still more complicated, with two miima or two maxima (diagrams 3 ad 4 i Fig. ), i.e., x x *( x ) b [1 ] for a x +a ad otherwise *(x) = 0 (1) or x 1 *( x ) b[ ] for a x +a ad otherwise *(x) = 0 (13) but Eq. (9) is still obtaied with g* = 5 1 /(1 ) ad g* = 5 7 /( 187), respectively. Hitherto, it was required that the model fuctio be symmetric with regard to x = 0, i.e., that *( x) = *(x), i.e., that the pairig theorem be obeyed. 1, However, eve this plausible restrictio is ot ecessary, as show by the examples: ad x *( x ) b[ 1] for a x +a ad otherwise *(x) = 0 (14) x *( x ) b[ 1] for a x +a ad otherwise *(x) = 0 (15) Also the fuctios (14) ad (15) imply the validity of Eq. (9), with g* = 3 3 / 16 ad g* = 9 5 /(16 ), respectively. The forms of the fuctios (14) ad (15) are show i diagrams 5 ad 6 i Fig.. I order to further demostrate the arbitrariess of the form of the model fuctio that leads to the McClellad approximatio, a example with a sigularity at x = 0 was costructed (see diagram 7 i Fig. ): 1/ x *( x ) b for a x +a ad otherwise *(x) = 0 (16) I spite of the (physically impossible) property of the model fuctio (16) that (x) for x 0, Eq. (9) is also obtaied with g* = 5 / 3.
McCLELLAND APPROXIMATION 971 1 3 4 5 6 Fig.. Several probability desities resultig i approximate expressios for the total -electro eergy of the McClellad type. Note that whereas i the models 1 (Eq. (10)), (Eq. (11)), 3 (Eq. 1)), 4 (Eq. (13)) ad 7 (Eq. (16)), the probability desity is symmetric with respect to x = 0. I the models 5 (Eq. (14)) ad 6 (Eq. (15)), it is chose to be highly asymmetric. I the models 1 ad, the probability desity is chose so as to have, respectively, a maximum ad a miimum at x = 0. I the models 3 ad 4, there are two maxima ad two miima, respectively. I the model 7, the probability desity has a sigularity at x = 0. The parameters a ad b are chose to be the same as i Fig. 7 1.
97 GUTMAN CONCLUDING REMARKS I the seve examples for *(x) give i the precedig sectio, Eq. (9) is always arrived at, but the multiplier g* assumes differet umerical values. I our opiio this detail is of lesser importace. Namely, it is possible to costruct model fuctios *(x), such that g* i Eq. (9) has ay desired value. For istace, if for some t 1, the x *( x ) b[1 t ] for a x +a ad otherwise *(x) = 0 (17) 3 5 t g* = 4 t 33t 5 By varyig the parameter t, the multiplier i Eq. (9) assumes values betwee 3 5 / 8 = 0.8385 ad 15 / 4 = 0.968, see Fig. 3. Therefore, the model fuctio (17) ca always be chose so as to exactly reproduce the empirically determied value of g i the McClellad formula (Eq. (1)). This, of course, would be fully articial ad without ay scietic justicatio. Fig. 3. Depedece of the multiplier g* i Eq. (9) o the parameter t of the probability desity (17). I order that *(x) be positive valued, it must be t > 1. The mai coclusio of the preset work is that the McClellad approximatio (Eq.(1)) has othig to do with the distributio of the HMO -electro eergy levels ad that o iferece o this distributio ca be made based o the fact that Eq. (1) i a surprisigly accurate maer reproduces the actual E-values. Ackowledgemet: This work was supported by the Serbia Miistry of Sciece, through Grat No. 144015G, Graph Theory ad Mathematical Programmig with Applicatios to Chemistry ad Egieerig.
McCLELLAND APPROXIMATION 973 - Prirodo matemati~ki fakultet Uiverziteta u Kragujevcu - m E g m, g, g 0,9. - -., -,. ( 9. 007) REFERENCES 1. A. Graovac, I. Gutma, N. Triajsti, Topological Approach to the Chemistry of Cojugated Molecules, Spriger-Verlag, Berli, 1977. I. Gutma, O. E. Polasky, Mathematical Cocepts i Orgaic Chemistry, Spriger Verlag, Berli, 1986 3. I. Gutma, J. Serb. Chem. Soc. 70 (005) 441 4. M. Peri, I. Gutma, J.Radi Peri, J. Serb. Chem. Soc. 71 (006) 771 5. G. Idulal, A. Vijayakumar, MATCH Commu. Math. Comput. Chem. 55 (006) 83 6. B. Zhou, MATCH Commu. Math. Comput. Chem. 55 (006) 91 7. A. Che, A. Chag, W. C. Shiu, MATCH Commu. Math. Comput. Chem. 55 (006) 95 8. J. A. de la Peña, L. Medoza, MATCH Commu. Math. Comput. Chem. 56 (006) 113 9. B. J. McClellad, J. Chem. Phys. 54 (1971) 640 10. I. Gutma, L. Nedeljkovi, A. V. Teodorovi, J. Serb. Chem. Soc. 48 (1983) 495 11. I. Gutma, S. Markovi, A. V. Teodorovi, Z. Bugari, J. Serb. Chem. Soc. 51 (1986) 145 1. I. Gutma, Topics Curr. Chem. 16 (199) 9 13. I. Gutma, T. Soldatovi, MATCH Commu. Math. Comput. Chem. 44 (001) 169 14. M. Fischerma, I. Gutma, A. Hoffma, D. Rautebach, D. Vidovi, L. Volkma, Z. Naturforsch. 57a (00) 49 15. I. Gutma, MATCH Commu. Math. Comput. Chem. 14 (1983) 71 16. I. Gutma, M. Raškovi, Z. Naturforsch. 40a (1985) 1059 17. A. Graovac, I. Gutma, P. E. Joh, D. Vidovi, I. Vlah, Z. Naturforsch. 56a (001) 307.