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CROATICA CHEMICA CCACAA 59 (3) 533-538 ACTA (1986) YU ISSN 0011-1643 UDe 541 Original Scientific Ptiper CCA-1666 Proof of the Formulae for the Molecular Orbitals and Energy Levels of Mobius Annulenes, Based on the Theory of Skew-Circulant Matrices A. C. Day Dyson Perrins Laboratory, University of Oxford, United Kingdom R. B. MaHion The King's School, Canterbury, United Kingdom and M. J. Rigby St. Patrick's House 47, Park Hill Road, WaHington, Received September Surreu, United Kingdom 4, 1985 The formulae for the molecular orbitals and corresponding energy-levels that arise in a Hiickel molecular-orbital (HMO) treatment of Mi:ibius systems are derived by appeal to the theory of skew-circulant matrices. The approach adopted is analogous to that previously used to obtain the orbital energiea and HMO's of 'Hiickel' annulenes from the theory of circulant matrices. INTRODUCTION In his classic 1964 paper, Heilbronner! stated (without proof) the expressions for the molecular orbitals, and corresponding energy-levels, that arise in a Hiickel-molecular-orbital (HMO) treatment of Mčbius systems - results which very quickly inspired others=? to an intensive theoretical study of M6bius molecules.? These same formulae were subsequently proved in Zimmerman's book"; this proof is, however, not a short one, making appeal, as it does, to the ideas of group theory and the use of trial solutions. In this note, therefore, we show how these resuits may be elegantly obtained, outside the immediate realms of HMO theory, by judicious exploitation of the theory of skew-circulant matrices.P'"! The arguments called upon are entirely analogous to those previously invoked by one of US13,14 to derive the orbital energies and HMO's of 'Hiickel' annulenes from the theory of circulant matrices (e. g., ref. 13). A. C. DAY ET AL. 534 THE The circulant PROPERTIES OF CIRCULANT MATRICES matrix 1 / °2 an ••••••• ( . \ 02 ~ 0t.. • • • • •• °1 k = 1,2, ... , n, and {Ak'} has been shown10-15 to have eigenvalues, vectors, {Vk'} k 1, 2, ... , n, where = eigen(1), and (2), v'k and (Uk is an nth root of unity - i. e., one of the n roots of the scalar equation = wn THE We POW PROPERTIES (3). 1 OF SKEW-CIRCULANT show that the skew-circulant B, ···.. . ···... ·. ·... .... ... ..... ...... ···...- 01 -an B matrix MATRICES O2 03 an 01 02 ~-1 01 °r>-2 -on_1 -an -02 -03, -04 01 has eigenvalues and eigenvectors formally given by equations provided that (Uk in those equations is everywhere replaced by is an nth root of minus one. (1) and (2), where ek @k, First consider the quantity (4) FORMULAE where @k is one particular of the scalar equation Equation MtlBIUS 535 ANNULENES root of minus one - nth i. e., one of the n roots 19n=-1 (5) has solutions 19k = cos [(2k FOR {19k} k (5), = 1,2, ... , n, which are given by + 1) 1r/nJ + i sin [(2k + 1) 1rlnJ = exp [(2k + 1)i 1rlnJ (6). It will be clear from equations (4) and (5) that the following equations true: lk = al + a2 19k + a319k2 + ... + an 19kn-l 19k1k = =- 19k2lk This set of equations al -an + al an_l - 19k + a2 19k2 + an 19k + al 19k! + implies the following matrix O 0 3 an al O 2 an 1 Qk -an al °n_2 Q2 k 2 .................. ..... -03 which, on the left-hand the vector* -04 + an_l + an_2 19kU-l 19kn-l (7) equation: /'\ Qk = Ak al siđe, contains are \ Q2 k (8) :~) Q k the skew-circulant matrix B. Renee, (9) is an eigenvector of B, with eigenvalue Ak given by equation (4). Since @k is just one of the n roots (equation (6)) of equation (5), it follows that there will be a Vk (equation (9)) and a Ak (equation (4)) corresponding to every distinct @k, k = 1,2, ... , n. {vd k = 1,2, ... , n and {Ad k = 1,2, .. , n are thus the complete set of eigenvectors and corresponding eigenvalues of B. * Or, (ef. Heilbronner's comment-), the appropriate real linear-combination of Vk and Vk*, to give linear-combination-of-atomic-orbital (LCAO) molecular-orbitals (MO's) with real coefficients. A. C. DAY ET AL. 536 It should be noted that the set of eigenvectors, {vd, is the same for all skew-circulant matrices, but that the eigenvalues depend explicitly upon the elements that comprise any row of the particular skew-circulant matrix under consideration. APPLICATION TO MOBIUS ANNULENES As may be seen from earlier work (e. g. refs. 1, 8, 16-19), a suitable adjacency-matrix for the cyclic graph CnM representing the carbon-atom eonnectivity of a Mobius [n]-annulene would be a skew-circulant matrix with first row (O, 1, O,... , O, - 1) - that is, a matrix of the form** o o o A(C~) o o -1 o o o o o 1 o o o o -1 o o •••• In the terminology of the matrix B, therefore, are non-zero, and so, from equation (4), lk It Iollows from equation equation (11) as only a2 (= 1) and an (= -1) = ek - ek (11). =- Bk-i, which allows us to write n 1 - (5) that Bkn-l lk (lO). = ek + ek- 1 whence, by equation (6), we obtain }'k = 2 cos [(2k + 1) n/n] (12). The eigenvalue list for the matrix A (CnM) is thus {lk} k = 1,2, ... , n. It should be observed that the eigenvalues are, of course, entirely real, since A (CnM) is real-symmetric, even though this is not the case for the general skew-circulant matrix B. Finally, since, in the HMO interpretation, orbital energies (e kM) are measured in units of a resonance integral {J' and with reference to a Coulomb integral a, namely, we may write equation (12) as €kM = a + 2 j3' cos [(2k + 1) n/n] The solution for the corresponding H €k =a (13). Hiickel system takes the form + 2 j3 cos (2k n/n) (14). ** In its quantum-mechanical interpretation, equation (10) implies the choice of a basic set of atomic orbitals that has onlyasingle < I}) , IH Il})j > - term negative. This choice is always possible for a Mobius ring.l.8.16-1D FORMULAE FOR MdBIUS ANNULENES 537 Writing P' in equation (13) and P in equation (14) allows for the faet that the dihedral angles (eu) between adjacent p-orbitals are different for a regular Mčbius and a regular Hiickel system - namely, n/n (= eu) and zero, respeetively, the resonanee integrals being related by ~' = .~eos (n/n). Comparison of the eosine term in equation (13), which may be expressed as eos (2krc/n + n/n), with that in equation (14) shows that the Mčbius and. the Hiickel solutions for a given n are n/n »out of phase« with eaeh other. In Frost and Musulin's mnemonie device'" for representing the RMO energy-lev el spectrum of Huckel annulenes, the regular n-gon is oriented with one vertex »down«. The phase angle of n/n by which equations (13) and (14) differ eorresponds in this eontext to a rotation of Frost and Musulin's polygon20 through an angle of n/n - i. e., one half of the angle subtended by an edge of the polygon at its centre. That is why, in the eorresponding mnemonie deviee introdueed by Zimrnerrnanš-š'" for Mčbius systems, the regular polygon is oriented, in a circle of radius 2 P', with one edge »down«. To eomplete the RMO interpretation, we recall-! that the eomponents of the eigenveetor Vk of A (enA!) (or appropriate linear-eombinations of Vk and Vk* if real eoefficients are required-A'") are the weighting coefficients of the individual basis atomie-orbitals, {c1\JIl = 1,2, ... , n, of the kth linear-eombination-of-atomie-orbitals Hiickel MO, Pk, for the M6bius [n]-annulene i. e. (ef. refs. 1, 8): pk = n-1j2 n ~ (19k)rr1 <Pil p=l. = the last step being via equation n n~lj2 . ~ p=l {exp [(2k + 1) (tA- - 1) i n/n] } <Pil (15),. (6). In the eourse of diseussion of this paper at the Symposium in Dubrovnik, Professor D. J. Klein kindly pointed out21 to the author who read the paper at the Symposium (R. B. M.) that the solution to the Mčbius/Huckel matrix-problem may also be effeeted (a) by use of double-group representation? and (b) by applieation of a suitable (diagonal) unitary-transformation to yielđ an Herrnitian (though not real) circulant-matrix." Professor Klein also made the intriguing observation'" that the eigenvalue problem pertaining to the Mčbius/Huckel type of matrix may be generalised in other ways: one generalisation is to a matrix in which the two elements that, in the conversion of a 'Htiekel' matrix into a 'Mčbius' one, are ehanged from + 1 to -1 are replaced by ei<l> and e-i<l> (eP being real). (A 'Huckel' matrix therefore corresponds to eP = 2krc, k an integer, while the 'Mčbius' matrix is reprodueed when eP = (2k + 1) rc). The eigenvalue solution to this matrix may then be obtained'" via method (b), above, and constitutes a simple generalisation of the Frost-Musulin'" 'shadow diagram'š": their polygon is simply rotated by eP/n. A seeond generalisation is in terms of a Pariser-Parr-Pople type of model that includes eleetron-eleetron interaction=; in this case, simple, elegant solutions do not seem to be known=, though some general properties of the ground-state eigen-solution have recently been established." 538 A. C. DAY ET AL. REFERENCE S 1. 2. 3. 4. 5. 6. 7. 8. E. He i 1b r o n ner, Tetrahedron Letters (1964) 1923. S. F. Ma s o n, Nature 205 (1965) 495. H. E. Zi m m e r man, J. Am. Chem. Soc. 88 (1966) 1564. M. J. S. De war, Tetrahedron Supplement 8 Part I (1966) 75. H. E. Zi m m e r man, Angew. Chem. Int. Ed. Eng!. 8 (1969) 1. H. E. Zi m m e r man, Accounts Chem. Res. 4 (1971) 272. H. E. Zi m m e r man, Tetrahedron 38 (1982) 753. H. E. Zi m m e r man, Quantum Mechanics For Organic Chemists, Academic Press, London, United Kingdom, 1975, pp. 113-116. 9. A. G rao vac and N. Tri naj s t i ć, Croat. Chem. Acta 47 (1975) 95. 10. G. Ko w ale w s k i, Determinantentheorie, Third Edition, Chelsea Publishing Co., New York, United States of America, 1943, p. 26. 11. W. Ka u z man n, Quantum Chemistry: An Introduction, Academic Press, New York, United States of America, 1957, pp. 48-50. 12. P. J. Da v i s, Circulant Matrices, John Wiley & Sons, New York, United States of America, 1979. 13. R. B. Mall i o n, BuU. Soc. Chim. France (1974) 2799. 14. C. A. C o u 1s o n, B. O' Lea r s, and R. B. Mall i o n, Hilckel Theory For Organic Chemists, Academic Press, London, United Kingdom, 1978, Appendix C, pp, 156-158. 15. W. L. Fer r a r, Algebra, Second Edition, Oxford University Press, Oxford, United Kirigdom, 1957, pp. 23-24. 16. A. G rao vac and N. Tri naj s t i ć, J. Molec. Structure 30 (1976) 416. 17. E. C. Hass and P. J. Plath, Z. Chem. 22 (1982) 14. 18. A. C. Day, R. B. Mall i o n, and M. J. R i g b y, On The Use of Riemannian Surfaces In The Graph-Theoretical Representation of Mobius Systems, in Proceedings of the Symposium on Chemical AppLications of Topology and Graph Theory, held at the University of Georgia, Athens, United States of America, April 18th April 22n", 1983 (Editor: R. B r u ceK i n g), Elsevier Science Publishing Co., New York, United States of America, 1983, pp. 272-284. 19. N. Tri naj s t i ć, Chemical Graph Theory, Vol. I, C. R. C. Press, Inc., Boca Raton, Florida, United States of America, 1983, pp. 84 & 85. 20. A. A. F r o s t and B. J. M u s u 1i n, J. Chem. Phys. 21 (1953) 572. 21. D. J. K 1e i n, Personal Communication to R. B. M., at Dubrovnik, September, 1985. 22. R. A. Ha r r i s, Chem. Phys. Letters 95 (1983) 256. 23. D. J. K 1e i n and N. Tri naj s t i ć, J. Am. Chem. Soc. 106 (1984) 8050. SAŽETAK Dokaz formula za molekularne orbitale i energijske razine Mobiusovih anulena zasnovan na teoriji specijalnih cikličkih matrica A. C. Day, R. B. MaHion i M. J. Rigby Izvedene su formule za molekularne orbitale i energijske razine, koje se javljaju u okvirima Hiickelove MO teorije, i to za Mčbiusove anulene, s pomoću teorije specijalnih cikličkih matrica.