Annulenes were studied using the concept of aromatic stabilization.
A simple expression is presented for the calculation of
aromatic stabilization of annulenes. The index of aromatic stabilization
differentiates correctly between the [4n + 2] - and [4n]
-annulenes. The analysis of the approach of Kruszewski and
Krygowski shows that their KK index differs only by a constant
from the value of aromatic stabilization of a particular annulene.
Similarly, the relation between REPE values of Hess and Schaad
and indices of aromatic stabilization is discussed

I. Gutman

M. Milun

N. Trinajstić

English

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'.::CA-696 CR 0 AT I CA CHEMIC A ACT A 44 (1972) 207 539.19:547.5 Original Scientific Paper Hiickel Molecular Orbital Calculations of Aromatic Stabilization of Annulenes I. Gutman, M. Milun, and N. Trinajstic I nstitute »Ruder Boiikovic« P. 0. B. 1016, 41001 Zagreb, Croatia, Yugoslavia Received September 20, 1971 Annulenes were studied using the concept of aromatic stabili-zation. A simple expression is presented for the calculation of aromatic stabilization of annulenes. The index of aromatic sta-bilization differentiates correctly between the [4n + 2] - and [4n] -annulenes. The analysis of the approach of Kruszewski and Krygowski shows that their KK index differs only by a constant from the value of aromatic stabilization of a particular annulene. Similarly, the relation between REPE values of Hess and Schaad and indices of aromatic stabilization is discussed. Recently we1-3 developed the concept of aromatic stabilization within the framework of the ffuckel molecular orbital (HMO) theory. The aromatic stabilization (As) has been defined as the difference between the HMO energy of a given cyclic conjugated molecule and the HMO energy of the correspond-ing reference structure. The HMO energy of the reference structure is equal to the sum of the effective energy of the polyene carbon-carbon »double« bonds and the effective energy of the polyene carbon-carbon »single« bonds. This approach has been applied to a number of polycyclic conjugated molecules giving in all cases an excellent correlation between calculated As values and chemical behaviour1-3. We thought that the annulenes would be an interesting group of mono-cyclic conjugated molecules to test our approach. The reference structure of the particular annulene (cyclic polyene) is best approximated by the corresponding linear polyene. Therefore, the index of aromatic stabilization of annulenes is equal to the difference between the HMO energy of a given annulene and that of a corresponding linear polyene. Thus, the index of aromatic stabilization of annulenes really corresponds to the energy change resulting from the ring closure. In the past a number of authors studied these compounds from various aspects4-12• Especially interesting are studies of Dewar and Gleicher13, and Figeys14, who predicted resonance energies of annulenes, the former using a variant13·15 of the SCF n:-molecular orbital method16•17 and the latter using the LCAO-BETA method18. CALCULATION AND RESULTS The total n:-electron energy (E,, ) of the annulene is evaluated using the following simple expression9 : 208 I. GUTMAN, M. MILUN, AND N. TRINA.JSTIC I 4 cosec En (annulene) = 4 cotan Jt N , for [ 4 n + 2]-annulenes Jt - . for [4 n]-annulenes N' (1) Similarly, the total n:-electron energy of the corresponding linear polyene is calculated from the expression: Jt En (polyene) = 2 cosec -2 2N + 2 (2) In above expressions N is the number of carbon atoms in annulenes (1) or polyenes (2). The energies obtained from the eXipressions (1) and (2) are given in units of B· Aromatic stabilization is then given as follows: Jt 4 cosec --N Jt (3) As = or -2 cosec +2 2N + 2 Jt 4 cotan N The HMO energies of annulenes from [4] to [30] and that of the corres-ponding linear polyenes with the values of aromatic stabilization are sum-marized in the Table. Similarly, the REPE values of Hess and Schaad19, the KK values of Krygowski12, the resonance energies (ER) calculated by Figeys14 using the LCAO-BETA method18, and by Dewar and Gleicher13 using the SCF MO method113•15 are also given in the Table. The Table is divided in two parts: the first part contains results for [4n]-annulenes, while the second part contains results for [4n + 2]-annulenes. The geometries of the investigated annulenes are shown in Fig. 1. All annulenes are assumed to be planar. This assumption is not completely correct20,21, because in some cases the repulsion between the annulene inner hydrogen atoms might be considerable. Therefore, the molecule, in order to avoid the effect of hydrogen interference, might distort from planar con-formation. In fact, there is some experimental evidence22- 25 that up to [18]--annulene the nonplanarity of the molecules is significant, but not exceptio-nally great. The plot of As as a function of the annulene ring size is shown in Fig. 2. We note that As as a function of the ring size divides the annulenes in two groups: aromatic [4n + 2] - and non-aromatic [4n]-annulenes. Our result is in agreement with earlier theoretical studies (REPE, KK, ER)12•14•19 and with the predictions obtained by considering NMR spectra of annulenes. The proton chemical shifts are sensitive to the magnetic properties of the n:-electrons and therefore, NMR also differentiates between the [ 4n + 2]-and [4n]-annulenes26. Because our approach is very simple we cannot predict when the change of the aromaticity of the [4n+2]-annulenes should occur. Such a change should be the result of the bond alternation which appears in higher annulenes.8 Dewar and Gleicher13 predicted that the critical size of the annulene, at which TABLE I HMO Energies (E" ) of Annulenes and Corresponding Linear Polyenes Indices of aromatic stabilization (As), RE!'E and KK values, and resonance energies (ER) calculated by Figeys (LCAO-BETA method), and by Dewar and Gleicher (SCF MO method). [N]-annulene En (annulene) I E" (polyene) As [13] REPE [13] I KK[13] I ER (Figeys)13 I EH (Dewar [~] (13] [kcal/mole] and Gleicher)12 [eV] [4] 4.000 4.472 -0.472 -0.2680 2.000 -24.34 -0.534 [8] 9.657 9.518 0.139 -0.0609 2.669 -10.78 -0.108 [12] 14.928 14.593 0.335 -0.0239 2.875 -6.58 0.131 [4n] [16] 20.109 19.676 0.433 -0.0111 2.975 -4.47 0.043 [20] 25.255 24.762 0.493 -0.0520 3.035 -3.11 -0.120 [24] 30.383 29.852 0.531 -0.0020 3.075 -2.03 -0.305 [28] 35.501 34.944 0.557 -0.00001 3.105 -1.33 -0.339 [6] 8.000 6.988 1.012 0.0653 3.528 10.01 1.318 [10] 12.944 12.053 0.891 0.0264 3.426 6.30 1.061 [14] 17.976 17.134 0.842 0.0161 3.383 4.89 0.689 [4n + 2] [18] 23.035 22.220 0.815 0.0118 3.359 4.26 0.382 [22] 28.106 27.308 0.798 0.0096 3.344 3.86 0.229 [26] 33.187 32.396 0.791 0.0090 3.335 3.76 -0.054 [30] 38.267 37.486 0.781 0.0080 3.323 3.57 -0.263 :i:. ::0 0 s;: :i:. >-'l H (') r.n .-,] :i:. b:J ...... t"' ...... N :i:. >-'l ..... 0 z 0 ":l :i:. z z d t"' trj z trj r.n t-:> 0 '° 210 I. GUTMAN, M. MILUN, AND N. TRINAJSTIC the bond alternation occurs, is between [22]- and [26]-annulene (see Table I, last column.) They predicted [26]-annulene to be a non-aromatic molecule. There are some indications in the literature26 that they may be correct in their prediction. For example, the NMR spectrum of tridehydro-[26]-annulene27 indicated that tli.is compound is non-aromatic. Apparently, the bond alter-D 0 0 w [4] (6] [8] (10] l) 0 0 [12] [16] . [14) QOQ [20]. 0 (24] [26) Fig. 1. Geometries of annulenes [4]-[30]. nation appearing in [26]-annulene quenches the ring current, and therefore, the NMR spectrum of tridehydro-[26]-annulene does not show discrete inner and outer proton resonances; its NMR spectrum consists of broad multiplets at -r 2.0 to 4.5. A typical aromatic annulene NMR spectrum would be that of [18]-annulene28 , taken at - 60° C with the band of the six inner protons at -r 12.99 and the band of the twelve outer protons at -r 0.72. The NMR results26 support the theoretical results of Dewar and Gleicher 13, according AROMATIC STABILIZATION OF ANNULENES 1.0 0.0 0.5 o,----"-0----(4 o] - ANNUL EN ES w u • • • w n H ~ m m N - -Fig. 2. Index of aromatic stabilization (A,) vs . the annulene ring size. 211 to which the [4n + 2]-annulenes up to n = 5 are aromatic, while the [4n + 2]-annulenes with n > 5 are non-aromatic. Finally, we would l~ke to discuss an additional interesting point. Our results agree also well with the approach of Kruszewski and Krygowski29 • They introduced the KK index as a measure of aromaticity of a particular conjugated molecule, and it was defined as the resistance of a conjugated molecule to addition reactions. The KK index can be calculated from the expression: KK = E ( conjugated ) - E (product) " molecule " (4) which is describing the re-energy change during the course of the addition reaction: ~)QJ ff' r + XY .... H (5) .... conjugated product molecule as a result of the formation of new bonds C-X and C-Y in the positions r ands. E" (product) is the n:-energy of the most favourable product of addition. Proof that 1,2 addition product is the most favourable one in the annulene addition reactions will be reported elsewhere3°. It is not surprising that our results are very close to those by Krygowski12 (see Table I, column 5) obtained using the KK index29, because the expres-sion (4) written in a different form is very similar to our expression (3) : 11: 4 cosec N 11: KK= -2 cosec ---- + 2 (6) 2N-2 11: 4 cotan N 212 I. GUTMAN, M. MILUN, AND N. TRINAJSTIC Consequently, the difference between KK and As values of annulenes is nearly constant ( ~ ~ 2.55). To conclude, we would like to discuss the relation between As indices and REPE values (see Table I, column 4) of Hess and Schaad19• The REPE index is defined as follows: REPE= ( reference ) En (annulene) - En structure N (7) The :rt-energy of the annulene reference structure19 differs from the :n:- energy of the linear polyene by a nearly constant value: En ( :~;~~~~~=) - En (polyene) = 0.0052N - 0.73 = Constant (8) 2~~bining eqs. (7) and (8) and utilizing the definition of As one obtains the following relation between As and REPE values: As - n. REPE = Constant This expression can be simplified by using: n. REPE =RE Thus, eq. (9) becomes: As - RE = Constant (9) (10) (11) It can be easily shown that the constant in the above expressions is inde-pendent of the choise of bond energies in the reference structure. Acknowledgements. We thank Dr. L. Klasinc (Zagreb) for helpful comments, Dr. T. Cvitas (Zagreb) for helping us in the presentation of the manuscript, and Professor M. Randie (Salt Lake City) for critical reading of the manuscript. REFERENCES 1. M. Mi 1 u n and N. Trina j st i c, Proceedings of the Meeting of Croatian Chemists, Zagreb, Croatia, February 1971, p. 72. 2. M. Mi 1 u n and Z. Sobotka, B. Sc. Theses, Croatian University, Zagreb, 1971. 3. M. Mi 1 u n, Z. Sobotka, and N. Trina j st i c, J . Org. Chem 37 (1972) 139. 4. J. E. Le n n a rd - Jone s, Proc. Roy. Soc. (London) A 158 (1937) 280. 5. C. A. C o u 1 so n, Proc. Roy. Soc. (London) A 164 (1938) 383. 6. H. Kuhn, J. Chem. Phys. 17 (1949) 1198. 7. M. J. S. Dewar, J . Chem. Soc. (1952) 3546. 8. H. C. Lo n g u et - Higgins and L. S a 1 em, Proc. Roy. Soc. (London) A 251 (1959) 172; ibid. A 257 (1960) 445. 9. 0. E. Po 1 ans k y, Monat. Chem. 91 (1960) 916. 10. J. N. Murr e 11 and A. Hinch 1 if fe, Trans. Faraday Soc. 62 (1966) 2011. 11. N. Trina j st i c and R. J. W rat ten, J. Mol. Structure 3 (1969) 395. 12. T. M. Kry go w ski, Tetrahedron Lett. (1970) 1311. 13. M. J. S. Dewar and G. J . G 1 e i ch er, J. Am. Chem. Soc. 87 (1965) 685 . 14. H. P. Fig e y s, Tetrahedron 26 (1970) 5225. 15. A. L. H. Chung and M. J. S. Dewar, J. Chem. Phys. 42 (1965) 756. 16. C. C. J . Rooth a an, Rev. Mod. Phys. 23 (1951) 69. 17. J. A. Pop 1 e, Trans. Faraday Soc. 49 (1953) 1375. 18. H. P. Fig e y s, T etrahedron 26 (1970) 4615. AROMATIC STABILIZATION OF ANNULENES 213 19. B . A. Hess, Jr. and L. J. Schaad, J . Am. Chem. Soc. 93 (1971) 305, 2413. J. Org. Chem. 36 (1971) 3418. 20. K. Mis 1 ow, J. Chem. Phys. 20 (1952) 1489. 21. 0. E. Po 1 ans k y, Monat. Chem. 91 (1960) 898. 22. O. B as ti a n sen, L. He db erg, and K. He db erg, J . Chem. Phys. 27 (1957) 1311. 23. J. Bergman, Nature (London) 194 (1962) 679. 24. J. B erg ma n, F. L. Hi rs c hf e 1 d, D. Rab in o vi ch, and G. M. J . Schmidt, Acta Cryst. 19 (1965) 227. 25. S. M. Johnson and I. C. Paul, J. Am. Chem. Soc. 90 (1968) 6555. 26. R. C. Ha d do n, V. R. Had do n, and L. M. J a ck man, The NMR Spectro-scopy of Annulenes, Fortschritte Chem. Forschung, Band 16, Heft 2, pp. 103-220, Springer-Verlag, Berlin-Heidelberg-New York 1971. 27. C. C. Le z no ff and F. Sondheimer, J . Am. Chem. Soc. 89 (1967) 4247. 28. L. M. Jackman, F. Sondheimer, Y. Amie 1, D. A. Ben - E fr aim, Y. Gao n i, R. W o 1 o vs k y, and A. A. Bothner - By, J. Am. Chem. Soc. 84 (1962) 4307. 29. J . Kruszewski and T. M. Kry go w ski, Tetrahedron Lett. (1970) 319. 30. I. Gutman and N. T r in a j s ti c, to be published. IZVOD Aromaticka stabilizacij a anulena izracunata Hiickelovom metodom molekularnih orbitala I. Gutman, M. Milun i N . Trinajstic Anuleni su studirani pomocu indeksa aromaticke stabilizacije (A8). Izveden je jednostavan izraz za numericko racunanje A3-indeksa. A8-indeks tocno razlikuje [4n + 2]-anulene od [4n]-anulena. Analizirana je takoder metoda Kruszewskog i Krygowskog, i pokazano je da se njihov KK-indeks razlikuj e od A8-indeksa odgo-varajuceg anulena samo za konstantnu vrijednost. Takoder je pokazano da se REPE-vrijednosti H essa i Schaada razlikuju od As-indeksa promatranog anulena za gotovo konstantnu vrijednost. INSTITUT »RUDER BOSKOVIC« 41000 ZAGREB, HRVATSKA Primljeno 20. rujna 1971. 

Huckel Molecular Orbital Calculations of Aromatic Stabilization of Annulenes

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